High Temperature Structures and Materials. Proceedings of the Third Symposium on Naval Structural Mechanics Held at Columbia University, New York, N.Y., January 23â€“25, 1963

O F F I C E O F NAVAL R E S E A R C H

S T R U C T U R A L M E C H A N I C S S E R I E S

HIGH TEMPERATURE

STRUCTURES AND MATERIALS

PROCEEDINGS OF THE THIRD SYMPOSIUM

ON NAVAL STRUCTURAL MECHANICS

H e l d a t C o l u m b i a Univers i ty , N e w York, N Y . J a n u a r y 2 3 - 2 5 , 1963

Sponsored by the Office of Naval Research and Columbia University

Edited by

A. M. FREUDENTHAL B. A. BOLEY

H. LIEBOWITZ

S Y M P O S I U M P U B L I C A T I O N S D I V I S I O N

PERGAMON PRESS

O X F O R D • L O N D O N • N E W Y O R K • PARIS

1964

P E R G A M O N PRESS L T D . Headington Hill Hall, Oxford

4 & 5 Fitzroy Square, London, W. 1

P E R G A M O N PRESS INC. 122 East 55th Street, New York 22, JV.T.

G A U T H I E R - V I L L A R S ED. 55 Quai des Grands-Augustins, Paris 6

P E R G A M O N PRESS G.m.b.H.

Kaiserstrasse 75, Frankfurt am Main

Distributed in the Western Hemisphere by

T H E M A C M I L L A N C O M P A N Y • N E W Y O R K pursuant to a special arrangement with

P E R G A M O N PRESS I N C .

New York 22, N.Y.

Copyright © 1964

P E R G A M O N PRESS INC. Reproduction in whole or part is permitted for any purpose of the

United States Government

Library of Congress Card No . 63-22053

Printed in Great Britain by

Adlard & Son Ltd., Bartholomew Press, Dorking

PREFACE

T H E Third Symposium on Naval Structural Mechanics was held from January 23 to January 25, 1963, at Columbia University. It was sponsored jointly by the Structural Mechanics Branch of the Office of Naval Research of the U.S. Navy and the Department of Civil Engineering and Engineering Mechanics of Columbia University.

The symposium was devoted to structural mechanics under conditions of elevated temperatures. It was intended to cover the various aspects of structural design for elevated temperature service. For this purpose papers were invited covering the following areas:

Material aspects of elevated temperature design. Effects of high speed environment. Thermal stress analysis. Design criteria and reliability.

The sessions were planned so that each of them would be devoted to one specific aspect of the field. By requesting the speakers to present reviews of the present status of the field with particular emphasis on their own contributions to it, it was hoped to provide an integrated picture of the various aspects of elevated temperature design. It was also hoped that the collaboration in one conference of material physicists, mechanicians, and structural designers concerned with the problem of elevated temperature effects on the performance and safety of modern structures would at least in part bridge the gaps existing between the three groups.

The first day of the conference was devoted entirely to material aspects, the second day to the effects of high speed environment and thermal stress analysis and the third day to design criteria and reliability. The present volume contains the full text of all papers presented at the conference, including written con-tributions to the discussion, as well as the panel discussion following the session on design criteria and reliability in edited form.

The complete program of the conference follows:

Wednesday, January 23, 1963

Session I : 9.15 a.m.

Chairman: Harold Liebowitz, Head, Structural Mechanics Branch, and Engineering Advisor, Office of Naval Research.

Opening of the Symposium by Dr. Grayson Kirk, President, Columbia University, and Rear Admiral L. D . Coates, Chief of Naval Research, Department of the Navy.

v

vi P R E F A C E

Chairman: Clarence Zener (Westinghouse Central Laboratory)

Material Aspects of Elevated Temperature Design

Warren P. Mason (Bell Telephone Laboratories): Temperature Depend-

ence of Elastic and Anelastic Properties in Solids.

Joseph A. Pask (University of California): Thermo-Mechanical Be-havior of Ceramics.

Session I I : 2 p.m.

Chairman: S. Stanford Manson (Lewis Research Center, NASA, Cleveland, Ohio)

Material Aspects of Elevated Temperature Design

John E. Dorn and Jim D . Mote (University of California): Physical

Aspects of Creep.

Nicholas J. Gran t and Arthur W. Mullendore (Massachusetts Institute of Technology): Creep Fracture and the Third Stage of Creep.

Shuji Taira (Kyoto University): Thermal Fatigue and Its Relation to Creep Rupture and Mechanical Fatigue.

Thursday, January 24, 1963

Session I I I : 9 a.m.

Chairman: Maurice A. Biot (New York, New York)

Effects of High Speed Environment

F . Philip Bowden and John H. Brunton (Cavendish Laboratory, Cam-

bridge, England): The Behavior of Materials at Supersonic Speeds.

E. Scala (Cornell University): Material Aspects of the Re-entry Problem.

Bruno A. Boley (Columbia University): Analysis of Problems of Heat Conduction and Melting.

Session IV. 2 p.m.

Chairman: Raymond D. Mindlin (Columbia University)

Thermal Stress Analysis

Heinz Parkus (Technische Hochschule, Vienna, Austria): Solution of

Thermoelastic Boundary Value Problems.

Eli Sternberg (Brown University): Analysis of Thermal Stresses in Viscoelastic Solids.

L. M. Kachanov (Leningrad State University, U.S.S.R.): Analytical Methods of Creep Design, Especially within the Nonlinear Range, (presented by title)

Banquet: 7 p.m.

P R E F A C E vii

Friday, January 25, 1963

Session V: 9 a.m.

Chairman: Walter J. Trapp (Materials Laboratory, ASD, Dayton, Ohio)

Design Criteria and Reliability

Alfred M. Freudenthal (Columbia University): Reliability under Ele-vated Temperature Conditions.

M. O. W. Wolfe (Royal Aircraft Establishment, Farnborough, England): Elevated Temperature Design and Design Criteria

Panel Discussion on Design Criteria

W. Boccius (Lockheed Missiles and Space Company, Sunnyvale, Cali-fornia).

A. M. Freudenthal (Columbia University, New York, New York).

J. M. Hedgepeth (Martin Company, Baltimore, Maryland).

H. Lowndes (ASD, Wright-Patterson Air Force Base, Dayton, Ohio).

M. O. W. Wolfe (Royal Aircraft Establishment, Farnborough, Hants, England).

The editors would like to express their appreciation of the cooperation received from authors, sessions chairmen, discussors, and panel members. Columbia University gratefully acknowledges the support of the Structural Mechanics Branch of the Office of Naval Research of the U.S. Navy in the organization of the Conference and the valuable assistance given by Mr. J. M. Crowley of that Office and Mr. Irving S. Rudin of the Naval Research Labor-tory, and also expresses thanks to those members of the staff of the Department of Civil Engineering and Engineering Mechanics of Columbia University who contributed to the success of the Conference; in particular to Professor J. M. Garrelts, Executive Officer of the Department, to Professor R. A. Heller of the Committee on Arrangements, and to Miss Joan Breslin for her assistance in the organization and editorial work.

WELCOMING REMARKS

GRAYSON K I R K

President, Columbia University

MORE than a dozen years ago I recall being with President Eisenhower, then here at Columbia, when, thanks to the cooperation which had been given to us by the Navy, we dedicated our new cyclotron. At that time it was one of the largest, I believe, anywhere. Now, after only a dozen years, it is small in com-parison with similar research tools elsewhere.

But it was then a most impressive facility and the crowd that had gathered for the dedication was greatly interested in it. Quite clearly, General Eisenhower had to say something about a machine whose actual workings he did not understand in any detail. So he said, " I wish I had the time to tell you all about this new cyclotron and how it works; but I have time only to push the button and let some of the other people talk to you about what will go on here."

Everyone laughed because he had said what was felt by most of the non-scientists in the audience.

I must confess that I have a similar feeling this morning when I attempt to say a few words of greeting to the members of a Conference assembled to study high temperature structures and materials.

As a political scientist and as an administrator of the University for a decade or more, I am somewhat more familiar with the high temperatures of faculty members than of structures and materials. But this is not your topic, and I lose, therefore, my only possible opportunity to make a substantial contri-bution to your work.

It is one of the very pleasant duties of a university to arrange meetings like this one; to bring together from all parts of this country and from other coun-tries experts with a major interest in a special subject of great importance. Thus, the university serves as a medium by which the latest ideas can be pooled and examined in that spirit of give-and-take which is characteristic of modern science and which has brought about such rapid development.

The university, as handmaiden and host, of course is a beneficiary because of its opportunity to participate and the opportunity for its colleagues to listen. I trust that during today, tomorrow, and Friday there will be a series of extremely interesting discussions. I know the program has been planned carefully over a long period of time, and so, not to waste any more of your time, I will now end my remarks and permit you to begin the work which you have come together to undertake. I wish you a good and fruitful meeting and I hope the University may have the pleasure of welcoming you all again, at some future time, to Morningside.

x i

OPENING OF THE SYMPOSIUM

REAR ADMIRAL L. D . COATES, U.S.N.

Chief of Naval Research

I T is indeed a pleasure to join with President Kirk in welcoming you to the Third Symposium on Naval Structural Mechanics. Every other year since 1958 the Office of Naval Research has sponsored a symposium in this field, and we are especially pleased to have Columbia University serve as the host for this year's meeting.

ONR's association with Columbia in the field of structural mechanics, parti-cularly in the field of thermal stress, extends back for nearly 15 years. Begin-ning first with low temperature work and then moving on to elevated tempera-tures, Columbia's Department of Civil Engineering with O N R support has consistently pioneered in laying a foundation of fundamental theory needed to solve the problems of structural and material stress.

In this connection I am happy to announce that the Navy in a joint program with the Air Force is completing plans for setting up at Columbia this country's, if not the world's, first educational research institute specifically designed for the study of fatigue. As all of you well know, the increasing severity of operating conditions for both military and civilian vehicles and equipment has created a demand for longer operational life and increasing use of both high temperature and high strength structural materials. At the same time, although there has been a great deal of experimentation and testing in exploring the fatigue pheno-menon in metals, what has really been needed is emphasis on the interdiscip-linary study of fatigue. Only through a broad, coordinated program that cor-relates and evaluates all the numerous and various factors that contribute to fatigue failures can we design structures efficiently where fatigue is a known rather than an unknown quantity and thereby greatly increase reliability. We at O N R feel that the Fatigue Institute at Columbia, which is an expansion of the work at the Fatigue Laboratory here, will be a major step toward filling the gap.

In another unusual program here, O N R is encouraging advanced study and research in the aero-structural sciences by providing funds to permit Columbia each year to give three promising structural mechanics researchers an oppor-tunity to perform studies and research here on a post-doctoral level. N o t only will this encourage urgently needed structural engineers to engage in theoretical work but the Navy will benefit by establishing contact with some of the most brilliant of the coming generation of engineers and scientists.

This type of program will also help in correcting a serious deficiency that has resulted from obsolescence in engineering training in this country. Too many

xiii

xiv O P E N I N G O F T H E S Y M P O S I U M

engineering schools and engineering students have been inclined to accept handbook knowledge as sufficient for advanced engineering design. The fact is that there is a lack of sound theoretical understanding of engineering principles now essential to accomplish significant technological advances. As a result of this lack, the Navy and other consumers of engineering technology are finding themselves shortchanged in acquiring the talent and information they need to solve their problems. More than ever, academic institutions must foster educational and research programs aimed at attaining this critical objective.

A distinctive feature of this symposium series is that we have been able to attract the leading experts in the field from abroad who have accepted our invitation to submit papers. This year we have an excellent representation on the program, including two papers by investigators from Great Britain, as well as papers by prominent scientists from Japan, Austria and Russia. I relish the international flavor of this symposium since it clearly demonstrates that science cannot be contained within national boundaries. It also underlines that there are distinct advantages to be gained from cooperative efforts, a philosophy in which O N R ardently believes and which it practices. By exchanging information from all parts of the world in a symposium such as this one, we can all move more rapidly and effectively toward the goals that we all desire to reach.

These O N R symposia are somewhat different from the annual technical society meetings in that our purpose is to survey and evaluate particular fields rather than to report solely on new research. The organizers of this symposium stated their intention of selecting speakers for their eminence in significant phases of the several disciplines involved in elevated temperature mechanics, and I am sure that a perusal of the program will convince you that they have admirably succeeded.

The high temperature field has been of major interest to the Navy for some time. The unusual and extreme environments in which our equipment has to function explains that interest. For more than a decade, for example, we have been working with nuclear reactors and the unique high temperature problems they create. We are constructing aircraft and missiles that are steadily pro-gressing in speeds from the supersonic to the hypersonic range. These structures are being increasingly exposed to the mechanical, physical, and chemical effects of hypersonic airflow. The ablation of missiles during the re-entry phase is of particular concern. If we had not been able to achieve some solution to this problem, the Navy's Polaris missile, which is now one of this nation's major deterrents against threatened aggression, would not be able to function as effectively as it does.

Just after the war, our objective was to design materials and structures which could withstand temperatures of a few hundred degrees; today we are con-quering temperatures of several thousand degrees. Although we have come a long way, the challenges ahead are still as great and even greater than those we have faced in the past. For example, we certainly need to know much more about the ablation phenomenon. This could permit us to increase the angle of re-entry and thereby significantly increase a missile's striking ability. We are

O P E N I N G O F T H E S Y M P O S I U M XV

now concerned with velocities so great that thermal effects in certain cases actually can cause catastrophic deterioration of the surface. This phenomenon has yet to be thoroughly investigated either by experimental or by theoretical methods, and our research is aiming toward that end.

There is no question that theoretical analyses are of major importance in determining the stresses in the behavior of structures. A fundamental under-standing of the mechanics not only of ablation but also of such phenomena as creep and stability is needed to help us solve the problems in continuum mech-anics.

The gathering of such data requires the intensive, coordinated efforts of applied mechanics people, chemical engineers, solid state physicists, and applied mathematicians as well as metallurgists. In addition to original basic research, there is also a constant demand for improved instrumentation to measure high temperature phenomena.

Although the U.S. Navy has its own special interests in gathering and applying this information, it should be obvious that application of this knowledge goes well beyond the military area. Human progress over the past centuries has been largely geared to the pace at which man learned how to escape from the re-strictions of his natural environment. One of his chief weapons in this step-by-step mastery of his environment has been his increasing ability to create more and more sophisticated materials and structures both for his comfort and his protection against natural and human hostile forces. This will continue to be the key to our progress, especially now as we raise our eyes more boldly to the challenge of outer space.

All of the work covered in the papers of this symposium is an important part of this critical effort. I congratulate you on your past achievements, and I look forward to the many triumphs that are ahead of you.

TEMPERATURE DEPENDENCE OF ELASTIC

AND ANELASTIC PROPERTIES IN SOLIDS

WARREN P . MASON

Bell Telephone Laboratories Incorporated, Murray Hill, N e w Jersey

1 . I N T R O D U C T I O N

T H E temperature and frequency dependence of the elastic and anelastic proper-ties of solids has been used to investigate a large number of solid state pheno-mena. The list of processes so investigated is now very long and includes such phenomena as thermal relaxation, phonon-phonon scattering, phonon-electron-magnetic field effects, point and line imperfections, magnetic and electrostatic domain motion, phonon-magnon interactions, interaction of acoustic waves with electron and nuclear spins, phonon masers and ultrasonic amplification in piezoelectric semiconductors. Obviously, all of these cannot be discussed. The present paper selects from these subjects those which appear to be more closely related to the subject matter of this conference, namely thermal and phonon-phonon interactions, effect of imperfections on thermal conductivity, the stress induced motion of electrons and holes in semiconductors and the internal friction due to point and line imperfections (i.e. dislocations). The motion of dislocations at high stress values is probably related to creep and fatigue occurring in metals.

1.1. Thermoelastic Effects

The type of data obtained by acoustic means can be illustrated by two well-known examples which date back to 1940. In the first example, many metals vibrating in flexure were shown by Bennewitz 1 and Roetger to have internal friction peaks of the type shown by Fig. 1, at low frequencies. This curve is for Geiman silver. This effect was explained by Zener 2 to be due to the fact that there is a temperature difference between the expanded and the compressed parts of the bar which diffuses across the width w with a time constant equal to

where D is the thermal diffusion constant K/pCp, K is the thermal conductivity and pCp the specific heat per unit volume and w the width of the vibrating bar. The height of the peak is determined by the relaxation strength A which in this

(1)

2 1

2 W A R R E N P . M A S O N

case is determined by the difference between the adiabatic and isothermal value of Young's moduli or

(2)

where a is the temperature expansion coefficient, Y® the isothermal Young's modulus, YQ the adiabatic value and T the absolute temperature. For German silver, a = 18.36 X 10" 6 , Y% = 1.075 x 10 1 2 dynes/cm 2 , P = 8-4 g/cm 3 ,

x i<r 3

60 4 0 2 0 10 8 6 4 2

FREQUENCY IN CYCLES PER SECOND

1.0 0 . 6

F I G . 1. Measurements of internal friction of a bar in flexure, as a function of the frequency (after Bennewitz and Rotger).

with this value of J . It is seen that there is a good agreement with the measured results.

It can be seen that the data are in good agreement with a single thermal relaxation time for the effect. Thermal conductivity in German silver is deter-mined 3 about equally by phonon and electron propagation. Figure 2 shows the energy distribution as a function of the lattice frequencies and with a Debye temperature of about 315°K (that of the principal component copper) most of

(3)

Cp = 0.393 x 10 7 ergs/g and hence A .= 0.00325 at 25°C. The solid line of Fig. 1 shows a plot of the usual relaxation equation

E L A S T I C A N D A N E L A S T I C P R O P E R T I E S I N S O L I D S 3

the energy will be concentrated at frequencies near the cutoff frequency vg. Hence a single relaxation time is to be expected. At lower temperatures, the energy is spread over a wide frequency range—with each frequency having a different relaxation time—and later data indicates a spread in relaxation times.

At low frequencies, the velocity and Young's modulus are determined by isothermal conditions while at frequencies above the relaxation frequency, adiabatic conditions prevail. Since the relaxation time varies only very slowly with temperature, no activation energy is involved. For a longitudinal mode of

f .6

F I G . 2. Relative thermal energy per frequency interval plotted against a factor propor-tional to the frequency. Dashed lines show limiting frequencies for 315°K and 70°K for

German silver.

where the numerical value is for copper. This corresponds to a frequency of 3 .5 X 1 0 1 0 cycles for which the velocity goes from its adiabatic velocity to its isothermal velocity.

Although this frequency is so high that it has not been observed experi-mentally, the thermoelastic effect does contribute to the attenuation of a sound wave, particularly in metals. From equation (3) and the relation between the

radians/sec (5)

where V is the sound velocity. The relaxation frequency /o has two solutions of which one is zero and the other is the high frequency

(4)

motion the relaxation time is determined by equation (1) with a half wavelength taking the place of the width w; since


attenuation A in nepers per cm, the value of the internal friction Q'1 and the phase shift B in radians per cm, one finds

(6)

Since no change of volume is involved for a shearing motion, there is no thermo-elastic effect for shear waves and ^ = yfi.

The thermoelastic effect is important in metals and when no other source of dissipation such as dislocation motion is present, it accounts for the attenuation quite closely. Figure 3 shows measurements given by Liicke 4 of the attenuation

thermo-elastic

0 50 100 150 Frequency in megacycles per second

F I G . 3. Ultrasonic attenuation in zinc single crystals along the hexagonal axis as a function of the frequency (after K. Liicke).

of a single zinc crystal, measured along the hexagonal axis, for longitudinal and shear waves. In hexagonal crystals dislocations move in a direction perpendicular to the hexagonal axis so that longitudinal waves along the axis produce no shearing stresses capable of moving them. The measured loss is nearly accounted for by the thermoelastic effect, as shown by the comparison curve. Shear waves can move dislocations and the high loss for shear waves is attributed to the motion of dislocations.

Thermoelastic effects in semiconducting and insulating crystals are quite small and do not account for much of the loss which can be shown to be due to purely thermal processes. For example, the loss for longitudinal waves in germanium measured at 306 Mc as a function of the temperature is shown by Fig. 4, which shows also the attenuation in db per cm for shear waves. Since investigations 5 have shown that dislocations are few in number and moreover the attenuation is independent of the number of dislocations, the cause for the attenuation is the direct conversion of acoustic energy into heat which proceeds

where A for a plane wave at high frequencies, when expressed in terms of the Lame A and fi constants, is

Bulk Modulus (7) where


by phonon-phonon interactions. The thermoelastic effect for a cubic crystal with longitudinal waves propagated along a cube axis has the same form as equation (6) with A equal to

(8)

where en and c i 2 are two of the elastic moduli for a cubic crystal. For ger-manium at 300°K, a = 5.5 x 10~6; en - 12.92 X 10 1 1 and cvt = 4.79 X 10 1 1

dynes/cm 2 ; P = 5.32 g/cm 3 ; Cv = 0.317 X 107 ergs/g; # = 5 . 8 5 x l 0 6

ergs/cm x °K. Hence the calculated attenuation is 1.15 X 10~2 nepers/cm or 0.1 db per cm at 300 °K. This accounts for only a small part of the total attenua-

2 . 8

2 . 4

EN

UA

TIO

N

IN

DB

/C

M

0 . 4

TLONG. ( 3 0 6 M C )

S H E A R ( 3 0 0 M C )

5 0 1 0 0 1 5 0 2 0 0 2 5 0 T E M P E R A T U R E I N D E G R E E S K E L V I N

3 0 0

F I G . 4 . Longitudinal and shear-wave attenuations in germanium along < 1 0 0 > axis.

tion measured. Shear waves are also attenuated by a thermal process and here thermoelastic effects vanish. Another mechanism is proposed 6 in Section II, which appears to account for the added loss of both types of waves.

1.2. Relaxations Due to the Motion of Interstitial and Substitutional Atoms

The other example considered is the Snoek 7 effect which is caused by the presence of interstitial carbon or nitrogen impurity atoms in iron or steel samples. By using a wire of steel in the form of a fiber in a torsional pendulum, the decrement 8 which is related to the internal friction parameter g _ 1 by the equation

8 = ^ e - 1 (9)

can be measured by determining the rate for which the vibration dies down.


With a 4 .9 second period, Fig. 5 shows the decrement of a wire free from nitrogen and carbon over the temperature range — 50°C to + 5 0 ° C . As can be seen from the curve 0, the decrement is nearly independent of the temperature. The curve labeled 3 is the result obtained by annealing the wire in a hydrogen atmosphere, with 4 per cent ammonia, at a temperature of 600 °C. A very large increase in the internal friction occurs with a maximum at about 0°C. An analysis of the steel shows about 0.06 per cent nitrogen, which is the saturation value. Smaller amounts of ammonia in the hydrogen produce smaller peaks.

o.ioo

0 . 0 7 5

0 . 0 5 0 h

0 . 0 2 5

- 8 0 - 6 0 - 4 0 - 2 0 0 2 0 4 0 T E M P E R A T U R E I N DEG C

F I G . 5. Attenuation relaxation in annealed wire due to 4 per cent ammonia in hydrogen (after Snoek).

If the frequency of vibration is changed, the temperature of the peak changes according to the relation

T o e +i6 ,40o /*r w h e r e r 0 = 2 .3 X 10" 1 3 sec (9a)

This is a typical Arrhenius equation with an activation energy of 16.4 kilo-calories per mole. The relaxation satisfies the usual formula

(10)

with the relaxation strength AEjE having a value of 0.052 for the curve of Fig. 5. The solid line shows a plot of equation (10) with this value of AEjE and with T given by equation (9). It will be seen that a single relaxation energy agrees very well with the measurements.

The explanation of this effect proposed by Snoek 7 is that nitrogen—and also

T


carbon which was shown to produce a similar effect with an activation energy of 18 kilocalories per mole—can settle interstitially half way between the iron atoms in the positions marked x, y and z in Fig. 6. The effect of a stress along the x direction is to separate the iron atoms in this direction and to com-press the atoms along y and z directions by a Poisson's ratio effect. As a result there is a tendency foi atoms at y and z to move to x positions To do so requires a thermal energy imparted to them that is equal to the activation energy for this jump, i.e. 16.4 kilocalories for nitrogen and 18 kilocalories per mole for carbon. It is the delay caused by this jump time which produces the lag in the strain with respect to the applied stress and hence the internal friction. This interpretation was confirmed by Dykstra 8 , who applied stresses along [111] axes and found no relaxation since such stresses affect the x, y and z positions equally.

F I G . 6. Proposed mechanism for stress induced diffusion of nitrogen in iron (after Snoek).

Many other interstitial solute atoms have been investigated 9 by similar tech-niques. The requirement 9 that a point defect shall produce an internal friction peak is that the point defect shall introduce distortions which have a lower symmetry than the lattice. For body centered crystals, interstitial atoms intro-duce internal friction when the octahedral sites (i.e. the x, y, z positions of Fig. 6) are occupied. Many examples are known. 9 Hydrogen has recently been shown 1 0 to produce a peak at 29 °K at a frequency of one cycle per second and it is inferred that a quantum mechanical tunneling process must be involved. A much smaller activation energy peak has recently been found 1 1 due to water vapor in sodium chloride which also involves a tunneling process and this will be discussed in Section III. This process occurs with such rapidity that it affects thermal conduction processes in sodium chloride.

Relaxation due to substitutional atoms was first observed by Zener 1 2 for the case of a brass crystals. Figure 7 shows measurements for a 620 cycle vibrating bar and the internal friction reaches a maximum at 417°C. This peak had an activation energy of 35 kilocalories. The original source for this relaxation was assumed to be that the zinc atoms were larger than the copper atoms and the


preferential distribution was affected by the applied stress. Zener type relaxa-tions have been observed in body centered, face centered and hexagonal close packed systems. Some of the materials are Li-Mg, Ag-Zn, Mg-Cd, Au-Ag-Zn , Cd-Ag. Although the original assumptions were that this was a nearest neighbor interaction, measurements 9 of the relaxation strength as a function of crystal orientation have suggested that next nearest neighbors may be responsible. The breadth of the peak is usually wider than that for a single relaxation and Nowick 1 3 and Berry have shown that the major part of the distribution of relaxation times is due to a distribution of the activation energies. They interpret this distribution in terms of local concentration fluxuations in the solid solutions.

TEMPERATURE IN DEGREES CENTIGRADE

550 525 500 475 4 5 0 425 4 0 0 375 350 325

V~!—n—n—n-n—n—n—n—H~

1.20 125 1.30 135 1.40 L45 1.50 1.55 160 1.65 1.70

1 0 0 0 / T

F I G . 7. Relaxation in attenuation in a-brass crystals (after Zener).

2 . P H O N O N - P H O N O N I N T E R A C T I O N S I N S E M I C O N D U C T O R S

A N D I N S U L A T I N G C R Y S T A L S

Most of the recent measurements have been carried out at frequencies up to the kilomegacycle range and for these measurements special techniques are required. In the range from 10 Mc up to 1000 Mc, a common experimental arrangement is that shown by Fig. 8. Here a bar in the form of a single crystal or a polycrystal is driven by a quartz crystal attached to the bar by a thin layer


of liquid or cement. Longitudinal waves can be generated by X-cut quartz crystals while shear waves of any polarization can be generated by AT, AC, BT or BC shear vibrating crystals with the particle displacement axis (the x axis of the crystal) mounted at the desired orientation with respect to the crystal. The transducer acts not only to launch the wave but also to pick up reflections from the far end. A series of pulses is received and the attenuation can be determined by the rate at which the pulses are reduced. If the faces of the rod are parallel within close limits and the axis coincides with the direction of energy propaga-tion, an exponential series of pulses can usually be obtained. Correct ions 1 4 for the spreading losses and the seal losses have to be made in order to obtain the true losses. Velocities can be obtained from the timing of the pulses or by a pulse superposition method 1 4 for which pulses are elongated in time so that they overlap. By changing the frequency, individual cycles can be made to add or subtract and a very accurate velocity determination can be made.

0 0 1

F I G . 8. Typical arrangement for measuring attenuation in solids. Dislocation mechanism is that for an alkali halide crystal.

For a wide temperature range, considerable difficulty is encountered with the seals on account of the difference in the temperature expansion coefficient of the transducer and the bar. If this difference is not too large a light (20 centipoise) silicon liquid is useful. The seal is made as thin as possible. The longitudinal measurements for germanium of Fig. 4 were made this way using a tourmaline transducer since this has a uniform expansion coefficient of 3.9 parts per million along the main surface of the transducer, which is perpendicular to the hexagonal axis. For silicon such a seal fails at about 60 °K. For lower temperature measure-ments in silicon and in sodium chloride, a new bond mater ia l 1 1 4 methyl, 1-pen-tene has been found useful. This freezes at 120°K and boils at 50 °C. On account of the high vapor pressure at room temperature, the usual procedure is to cool down the transducer, the sample and the liquid down to dry ice temperature (—78 °C). The seal is formed by putting a drop on and squeezing out all excess after which the sample is put in the Dewar and cooled down to a low temperature. The seal becomes hard at 80 °K and since the relative thermal expansion between the sample and the transducer is small down to helium temperature, good low temperature measurements have been made for these materials for the first time.

Figure 4 shows longitudinal measurements for germanium at 306 Mc over a wide temperature range and similar shear wave measurements at 300 M c

2§


Figure 9 shows similar measurements for silicon. The longitudinal wave measure-ments were made at 286 Mc and 480 Mc, while the shear wave measurements were made at 495 Mc. As discussed below, dislocations in silicon and germanium do not contribute to the acoustic loss and the only known source of attenuation at temperatures below room temperature 1 5 is direct conversion of acoustic waves into heat which procedes by means of phonon-phonon interactions.

To show that other sources of attenuation are not appreciable, and to indicate the elastic properties of silicon as a function of the temperature, results are shown from two invest igat ions. 5 ' 1 5 Figure 10(a) and (b) show data relating to the stress-strain curves of silicon bars and whiskers. 5 U p to 600 °C the stress-

2 . 4

2 . 0

1.6

. 2

).8

0 . 4

4 8 0 MC : ( L O N G . }

< y 2 8 6 MC ( L O N G . ) ,

( S H E A R ) ^ 4 9 5 M C

( L O N G . ) ,

( S H E A R )

5 0 1 0 0 150 2 0 0 2 5 0 T E M P E R A T U R E I N D E G R E E S K E L V I N

3 0 0

F I G . 9. Longitudinal and shear-wave measurements in silicon along a <100> axis.

strain curve is linear and fracture results at the highest stresses without any flow taking place. The final stress depends somewhat on the size of the specimen, being about 2 X 1 0 1 0 dynes/cm 2 (strain of about 1 per cent) for bars having a cross-section less than 2 X 10~ 5 cm 2 , but dropping to 3 X 10 9 dynes/cm 2 (strain of about 0 .2 per cent) for bars having a cross-section greater than 2 x 10~ 4 cm 2 .

Figure 10(b) shows the resolved shear stress 1 6 (which is about one-third of the longitudinal stress) for germanium at 500°C and some of the quantities that enter into the curves. Similar results are obtained for silicon at a temperature of 1000°C. U p to the shearing stress r c , no new dislocations are created and the stress-strain curve is linear. Above this stress, dislocations are being created in increasing numbers, but the velocity is so low that only a small change from linearity is observed. When the stress reaches r w , the rate of creation times the velocity—which determines the plastic strain rate—is large enough so that the shear stress drops to a low value. The original measurements 5 indicated that the

AT

TE

NU

AT

ION

IN

D

B/C

M


F I G . 10. (a) Load deflection curves for silicon bars and whiskers at several temperatures, (b) Stress-deflection curves for a silicon rod showing yielding and aging (after Pearson,

Read and Feldmann).

been established at 2 .55 eV. Figure 11 shows measurements of the internal friction of silicon at 100 kc and 300 kc as a function of the temperature. An oxygen peak appears at 1300°K for 100 kc and this moves up in temperature as the frequency increases according to an Arrhenius type equation (9a). The activation energy found for this motion is 2 .55 eV which is close to the value of 3.1 eV measured by diffusion measurements. The frequency fo = coo/2rr found is 7 x 10 1 3 c/s.

A lower peak is shown also which has been ascribed to an effect connected with the carrier life time. The activation energy for this process is 1.45 eV. Both of these processes produce less internal friction than other sources below about 600°K. Below 500°K all the internal friction can be ascribed to other sources. Since dislocations are completely immobilized, the remainder is ascribed to direct conversion of acoustic energy to heat (phonon-phonon interactions).

(a) (b)

dislocations were repinned by oxygen atoms, but the later measurements 1 6 do not show any recovery of the elastic modulus. At very low shearing stresses used in the measurements of the internal friction, the independence of the result 5 on the number of dislocations present indicates either that the dislocations are pinned by impurity atoms or that the Peierl's barrier is so high that no appreciable motion is possible for the dislocation.

Internal friction measurements have been used 1 5 to show independently that oxygen can move in silicon and an activation energy for such movements has


This is confirmed by the disappearance of this loss below the condition wr > 1 where r is the thermal relaxation time.

It has already been pointed out that the thermoelastic loss is inadequate to account for the measured attenuation in both silicon and germanium. Another source of attenuation involving only the direct conversion of acoustic energy to thermal energy was first pointed out by A. Akheizer 1 7 . In this effect a suddenly applied strain causes an effective temperature change for each of the phonon

branches. The temperature change—which is a function of direction—results from changes in the dimensions and in the elastic moduli which are functions of the applied strain. These changes will in general be different for each branch and each direction. This temperature difference is equalized in the very short relaxa-tion time T j which ordinarily will be different for each phonon frequency range. At relatively high temperatures, i.e. above about 150°K for germanium, the energy is concentrated in the high frequency phonon range and a single time

F I G . 1 1 . Internal friction of a silicon single crystal measured at 1 0 0 k c and 3 0 0 kc as a function of the temperature (after Southgate).


constant T—which agrees with the thermal time constant—represents the results. Below this temperature a distribution of time constants is required.

As discussed in detail in this section, the changes in temperature of each branch depends on the change in the acoustic wave velocity and the dimensions caused by the strain associated with the acoustic wave. Hence, to evaluate the effect requires a measurement of the third order elastic moduli, i.e. the deviations from linearity of the second order or normal elastic moduli. Not having such measurements available, Akheiser assumed an isotropic substance and evaluated the effect from the thermal expansion coefficient which also depends on an average value of nonlineaiity through the Griineisen 1 8 constant y which is related to a, the linear temperature expansion coefficient by

d o

where pCv is the specific heat per unit volume and B the bulk modulus. The average Griineisen constant depends on the limiting frequency vg of the Debye spectrum according to the eauation—V eauals the volume—

(12)

As shown below, vg is a function of the third order moduli. Using this relation, Akheiser concluded that the attenuation of acoustic waves due to this source was just equal to the thermoelastic effect or

(13)

As discussed previously, this loss is too small to agree with experiment. Further-more, this derivation does not take into account the finite relaxation time con-nected with thermal equilibrium processes.

Bommel and Dransfe ld 1 9 a have generalized the treatment of Akheiser to take account of the finite relaxation time. They assume two different branches with different Griineisen constants and obtain an average value of y which is deter-mined from the relation

(14)

where (AT/T)it 2 are the two increases in temperature for the two branches and Ap/p the corresponding changes in density due to the sound wave. With this concept, the attenuation expressed in the form of decibels per inch b e c o m e s 1 9 a > 6

(15)

where r is taken as the thermal relaxation time given by

(16)


where C is the specific heat per unit volume and V2 is the square of the Debye average velocity. For an isotropic material

(17)

where Vi and Vt are respectively the longitudinal and transverse sound velocities of the solid. For more complicated crystals other methods are avai lable 2 0 for determining V. In particular

V2 = 1.257 x 10 1 1 (Ge); V2 = 3.45 x 10 1 1 (Si) (18)

Although this expression agrees reasonably well with the measurements in quartz, the agreement is not good for silicon and germanium. The dashed curves of Fig. 12 show the value of equation (15) for arbitrary value of y A V = 4 . 8

3 . 5 ,

0 5 0 1 0 0 150 2 0 0 2 5 0 3 0 0 T E M P E R A T U R E IN D E G R E E S K E L V I N

F I G . 12. Measured and calculated ultrasonic attenuations in silicon and germanium as a function of the temperature. Dashed lines are a plot o f Eq. (15). Small-dash lines are a

plot of Eq. (19), with r being taken as the thermal relaxation time.

for Ge and y A V = 4 .1 for silicon. Both values are larger than seem reasonable and the trend of the curves with temperature is not in good agreement with the measurements.

A new derivation from fundamentals has been made which results in an expression for the attenuation caused by the Akheiser effect in the form

where Eo is the thermal energy per unit volume, r is the relaxation time for the

(19)


interchange of thermal energy between the various phonon modes and D is a constant whose value can be derived when the values of the third order moduli are known.

If an average value of T equal to the thermal value of (16) is used, the attenua-tion becomes

The small-dashed lines of Fig. 12 show a calculation from this equation using the measured thermal values of (20) and values of D = 10.8 for germanium and 9.0 for silicon. The 10.8 value is close to that calculated from the measured third order moduli of germanium. Below about 150°K the measured value de-

1.0

0.8

flQ Q

0.6

z o

z LU

t 0 .2

G e ( 3 0 6 MC :) (ATHEORETICAL POINTS) <>

^ Si ( 4 9 5 MC) (O THEORETICAL POINTS)

50 100 150 2 0 0 2 5 0 TEMPERATURE IN DEGREES KELVIN

3 0 0

F I G . 1 3 . Measured and calculated shear-wave ultrasonic attentuation in silicon and germanium.

creases faster than the calculated value using a thermal relaxation time for both silicon and germanium. This agrees with the fact that low frequency longitudinal waves interact with phonons of the same frequency r a n g e 2 1 and hence the relaxation r is greater than the thermal relaxation time which involves frequen-cies ~kT/h. Figure 13 shows similar curves for the shear waves in germanium and silicon. The best empirical value for D is unity for both curves. Since the low frequency shear mode can interact with high frequency phonons of thermal frequency (~kT/h)21, the relaxation time should be the thermal relaxation time in agreement with Fig. 13.

The method for deriving equation (19) is as follows. In contrast to the thermo-elastic effect, the Akheiser effect is an energy dissipation taking place under a uniform (but time dependent) strain. Its origin is that stretching at constant phonon occupation number Ni for each mode is not the same as stretching under adiabatic reversible conditions. This nonequilibrium distribution results

2

(20) (nepe rs / cm


in different temperatures in different phonon branches which are returned to thermal equilibrium by means of phonon-phonon collisions. This process occurs with a relaxation time r. For an alternating stress, the temperature difference lags behind the applied stress causing a relaxation effect governed by the usual relaxation equation

(21)

where Eo is the elastic modulus pertaining to the wave under consideration and AE the instantaneous increase in modulus caused by the temperature separation of the phonon modes. Using the well-known formula 2 2 for the relation between the attenuation and the internal friction, we have

(22)

after using the relation that the velocity, V = <\/(E/p.) B is the phase constant co/V. Hence to determine the attenuation caused by this effect, one has to calculate how the nonequilibrium temperature separation of the phonon modes by the applied stress affects the elastic modulus.

The standard state of the solid is taken as that for the unstressed solid at the temperature T. The total internal enerev of such a solid is

(23-

where v% is the frequency of the vibration mode corresponding to the subscript i. Vibration modes are usually described by a wave vector q and a discrete variable p which labels the polarization direction and the acoustical or optical character. The index i is a short hand designation for all these factors. The last term of (23) represents the zero point energy which can be lumped with Uo since it does not vary with stress.

The quantity Nt equals the number of phonons present in state /, which in a thermal equilibrium state is given by

(24)

where k is Boltzmann's constant and Tthe absolute temperature. It is important to recognize that equation (23) holds whether or not a state of thermal equili-brium exists since it merely adds up the kinetic and potential energies asso-ciated with all the lattice motions. Uo represents the added energy present when all the Ni go to zero and hence it will be the potential energy contribution.

When a strain Sj is applied to the crystal, it causes a change in the mode frequency given by

(25)

where v ?: 0 is the frequency of the mode i in the standard state.


In order to make use of the ultrasonically measured deviations of the elastic moduli from linearity, we have to make use of the Debye approximation which assumes that there is no dispersion in the velocity. To take account of the variations of yi

i with direction and with the measured shear and longitudinal waves, we have to assume an anisotropic Debye approximation with three limiting frequencies. Since variations along the <100>, <110> and <111> axes have so far been used to evaluate the constants, the form of the thermal energy term will be

(28)

We wish to show that the integral remains invariant to the strain. It follows directly from the fact that a dimensional change does not alter the entropy (i.e. does not affect the occupations numbers) that the integral is invariant to a dimensional change. For the more restricted Debye approximation this result is shown by the following considerations. When the temperature is small so that the energy above vg. is small, it is obvious from Fig. 2 that a change in vg will not affect the integral. In the Debye theory there is no dispersion so that all the intermediate eigenfrequencies must vary in the same way as the limiting frequency which from (25) is

(29)

(27)

where Nt is the number of modes in a solid angle containing one of the directions evaluated, vg. is the limiting frequency for each mode and direction, G is a constant determined by the number of modes in the sector under consideration, Vi the wave velocity of the mode under consideration, and / the path length.

If we suddenly change the dimension along the x axis by a strain Si, the stress will be obtained by differentiating U by the strain. Since neither isothermal or adiabatic conditions prevail, the elastic constant will be determined by a constant occupation number Nt of all the modes. Since, however, it does not have any relaxation component, the attenuation can be obtained by considering only the thermal part. Then the stress is given by

In agreement with measurements of third order moduli made over a wide temperature r ange 2 3 it is assumed that the constants y\ do not vary appreciably with temperature. By direct differentiation we see that

(26)


Hence for the integral to remain constant, the ordinate must vary in the inverse ratio. It follows then that for any value of the temperature that the integral is invariant to the strain.

Hence, equation (28) can be written in the form

where E% is the thermal energy associated with each direction and each mode. The values of y{ can be calculated directly from the measured third order

moduli. A numerical calculation has been made for germanium by using the three waves propagated along the [100], [110] and [111] axes. While this is only an approximation for a complete integral for all directions—which would re-quire the use of a computer and a complicated program—it should give an indication of the degree of agreement between the theory and the experiment. For a longitudinal phonon wave along a [100] axis with a single strain Si (which corresponds to an acoustic wave along [100]), the value of v'g, the value for the strained sample, i s 2 4

The last term represents the stress required to keep the volume constant as the temperature varies. The other term adds to the elastic modulus which will instantaneously increase by the amount inside the bracket summed for all directions. From the values given in Table 1 and assuming that all modes have the same thermal energy (high temperature approximation) the instantaneous stiffness added is

3 x 1.29 x Eo = 1.44 x 10 1 0 dynes/cm 2 (33)

This value will not relax down to the unstrained value but rather down to the value obtained for an adiabatic vibration since the time constant for going from a suddenly strained crystal is very small compared to the adiabatic-iso-thermal time constant. It is well k n o w n 2 5 that the effect of an increase in tem-perature AT is to cause a decrease in the modulus of a longitudinal wave along x equal to

Acn = - a (cn + 2c X 2 ) AT/Sx (34)

(30)

(31)

Using the values en = 1.2885 dynes/cm 2 , C m = — 1.22 x 10 1 2 dynes/cm 2 , y\ — + 1.34. Similar values for the other directions for both shear and longitu-dinal waves are shown by Table 1. The value of the equation (30) then becomes

(32)

EL

AS

TIC

A

ND

A

NE

LA

ST

IC

PR

OP

ER

TIE

S I

N

SO

LID

S 19

+ 1.34

1 . 3

0.627

0.9

- 0.297

- 0.009

2.54

0.9

Equation for y\Si

CO

—̂1

CO C

o

co

^L

c

o+

^l

5

' „

. «o

.

^

r'T

<o

"\L ^

^

^

^

<o

^

-"

"^

<

M

sj

to

<0

rj

Cj

. 2

^

'

C

S 3

- 4-

2

U

H i

3

^

4iL

^

ri

g

+

<N

+

1

Number of

waves Polarization

direction

100

010, 001

010, 001

100, 100

001, 010

110, 101

OTl, 011

100, 100

Direction of

propagation

100

100

010, 001

010, 001

010, 001

110, 101

011, OTl

011, OTl

Type of

wave

Long

Shear

Long

Shear

Shear

Long

Shear

Shear

TABLE 1

TABLE 1—continued

Type of

wave

Direction of

propagation Polarization

direction

Number of

waves Equation for y\Si

Value of

y)

Long 110,1T0 101,101

110,1T0 101, 101 4 p ( c i i + ci2 + c 4 4 ) + 3 C m + 3Cn2 + Cieel 0

L 2 ( c n + ci2 + 2 c 4 4 ) J 1 1.47

Shear 110, 1T0 101,101

1T0,110 101, 101 4 \2cn + 3 C m - C m ! 0

L 2 ( c n - C 1 2 ) J b l - 0 . 2 2 7

Shear 110, 1T0 101, 101

001, 001 010, 010 4 \Cll + C 1 2 - C 4 4 + ( C l 4 4 + C l 6 6 ) / 2 1 a

L 2cm J 6 1 + 0 . 5 0

Long 111, 111 111, ITT

111, 111 11T, ITT 4 H e n + 4ci2 + 6 c 4 4 + 3 C m + 2 C n 2 + Cieel «

L C l l + 2C12 + 4C44 J 1 1.56

Shear 111, 1T1 111, 111

TT2, T12 112,112 4 ["4C11 + 5C12 - C 4 4 + 3 C m + 2Cll2 + ( C l 4 4 + 5 C l 6 6 ) / 4 - 2Cl23l 0

L 6 ( C 1 1 - C 1 2 + C 4 4 ) J S l 1.35

Shear 111, 111 11T, ITT

T10,110 110,110 4 p C l l + C 1 2 + C 4 4 + 3 C l l l - Cll2 + ( C l 4 4 + C l 6 6 ) / 2 ]

L 2 ( C 1 1 - C 1 2 + C 4 4 ) J ^ 0 . 5 5

20 W

AR

RE

N

P.

MA

SO

N

Since Eo = 3 .7 X 10 9 ergs/cm 3 we find for D a value of 9.75 in relatively good agreement with the experimentally measured value of 10.8.

A similar calculation was made for shear waves along [100] with the result shown by Table 2. Eighteen out of 39 modes are actuated. They occur in pairs of opposite sign so that there is no average change in temperature. The average value of (37) becomes

0.525 E0; hence D = 1.58 (40)

compared to a measured value of 1.0. It is probable that better agreement could be obtained by taking a larger average since most of the divergence is caused by two large values. It was shown previously 2 6 that dimensional changes alone resulted in a value of D = 1. Hence it appears that the theory presented results in values close to the experimental results.

This source of loss is much smaller in metals since only a small part of the thermal conductivity is associated with phonon propagat ion. 2 7 Furthermore the relaxation time is only about 1/100th of that for insulators and hence this effect is usually small compared to the thermoelastic effect.

(38)

(39)

3 . E F F E C T O F I N T E R S T I T I A L S D U E T O W A T E R V A P O R O N T H E

A T T E N U A T I O N A N D V E L O C I T Y O F U L T R A S O N I C W A V E S ,

A N D O N T H E T H E R M A L R E S I S T A N C E O F

S O D I U M C H L O R I D E C R Y S T A L S

Recently measurements 1 1 of the attenuation and velocity of shear and longi-tudinal ultrasonic waves in sodium chloride crystals have shown a spectrum of relaxation frequencies at temperatures below 10°K. These effects occur for

From (19) and (2:

AE = (14.4 - 2.4) x 10 9 = 12.0 X 10 9 dynes/cnv

where y is the Gruneisen constant denned by equation (11). For the case con-sidered here

(37)

using (11) and the value of 3B = e n + 2 c i 2 for a cubic crystal. Hence the AE of equation (22) becomes

(36) Acn = a ( c n + 2ci2) yTo = y2pCvTo

Hence the increase in modulus for an adiabatic variation is

(35)

for a cubic crystal. F rom equation (14), the relation between the increase in temperature and the strain Si = — Aolo is

21 E L A S T I C A N D A N E L A S T I C P R O P E R T I E S I N S O L I D S

Type of

wave

Direction of

propagation Polarization

direction

Number V a l u e

o f of waves Equation for y{Siz y\

Long 101 101 t \Cll + C12 + 4C44 + 2Cl66l ^ . _

L 2 ( c u + c 1 2 + 2 c 4 4 ) J * 1 3 + 1 2 3

Shear 101 010 1 [4C44 + C 4 5 6 l c

1 " L 8C44 J 5 1 3 - 0 1 9 2

Shear 101 T01 t \~ C l l + Cl2 + 2C44] o ~~ L 2 ( c n - c 1 2 ) J 5 1 3 - ° ' 3 3 5

Long T01 T01 1 _ L T C l l + C 1 2 + 4C44 + 2Cl66l . _ + I 2 (c 1 1 + c 1 2 + 2c44) J 5 1 3 - 1 ' 2 3

Shear T01 010 i f 4 C 4 4 + C 4 5 6 l c

1 + [ 8C44 J 5 1 3 + 0 1 9 2

Shear T01 101 1 _L_ \~ C H + C 1 2 + 2C441 0

+ L 2 ( c n - c 1 2 ) J * 3 + 0 3 3 5

Long 111, 1T1 111, 1T1 j \2C11 + 4 C l 2 + 15C44 + 3Cl44 + 5Cl66 + 2C4561 „ L 6 (cn + 2 c i 2 + 4 c 4 4 ) ~~J S l z + 0 ' 6 5 5

TABLE 2

22 W

AR

RE

N

P.

MA

SO

N

EL

AS

TIC

A

ND

AN

EL

AS

TIC

P

RO

PE

RT

IES

IN

S

OL

IDS

13 o

^

>

+0.46

-0.482

-0.655

-0.46

+0.482

CO

CO

.

a c

1 cr W

w eo

w

<0 C

o

^

to

1 1

<D

+ ^

+

d +

a ^

a

^

1

G1

| G

1 ^

G1

| G

1 |

G1

1 5

1

+ <?

1

1

5 1

« '

' +

„

1

'

^

1 2

«

§

+

t *

+

,1 ft

, |

1 +

+

Number of

waves

<N

<N

<N

<N

Polarization direction

<N

O

|<-t

O

1-< T

-l 1—

1

1-H

II—•

1—1

i-H

H

-• I-H

C

C

£

P

C

Direction of

propagation

^ |R

-T

h-H

Type of

wave

Shear

Shear

Long

Shear

Shear

23

pBtiuiiuoo—z aiavx


crystals grown in the atmosphere (Harshaw) but are absent for vacuum grown crystals (Isomet). The impurity producing the effect has been shown to be water vapor, for by annealing the Isomet crystals in water atmosphere at 750 °C for 11 hr, the same set of relaxations can be made to occur as were measured for the Harshaw crystals.

The alkali halide crystals have a very simple dislocation structure, as shown by Fig. 8. The dislocations lie in <100> directions and in (110) planes and are actuated by shear stresses in the (110) plane with polarizations in the <110> directions. It has been shown 2 8 that a shearing wave along a <110> direction with its particle motion along a <100> direction will not actuate dislocations whereas the shear wave propagated along <110> with its particle motion along <110> will strongly actuate the dislocations. This relation was first used by Merkulov 2 8 to study the effect of dislocations on wave propagation. One polarization is damped only by the phonon-phonon interaction while the perpendicular orientation shows in addition the effect of dislocation motion. The work discussed in this section was originally undertaken to extend the measurements down to very low temperatures.

In the course of this work the very low temperature relaxation mentioned above was found and it is the purpose of this section to describe it in detail. The first measurements were made to investigate the sensitivity of the effect to stress direction and it was found to have the same stress sensitivity as the dislocation effect, namely the relaxation occurs if the ultrasonic wave produces a shearing stress in the (110) planes with the stress direction along <lT0> axes. Since the relaxations do not occur in vacuum grown crystals whereas the dis-location effects occur for both vacuum and atmosphere grown crystals, it is found not to be connected with the presence of dislocations.

For the most sensitive orientation—the <110> axis with <U0> stress direc-tions—Fig. 14, solid line, shows the measured attenuation for a 57 Mc shear wave. The relaxation effect is the difference between the solid line and the short dashed line, which is caused mostly by dislocation damping. This result is con-firmed by measurements of the same orientation in Isomet crystals which show a curve very similar to the dashed line. Measurements were made from 18 Mc to 200 Mc, and if one plots the natural logarithm of the measuring frequency against the inverse temperature of the attenuation peak, as shown by Fig. 15, a straight line results. Hence, the relaxation satisfies an Arrhenius type equation

ft ^foe-H^RT w h e r e f0 = 5 X 1 0 9 ± 0 - 2 ; ft = 21 ± 2 .3 cal/mole (41)

Figure 14, for the highest dashed line, shows a plot of a relaxation equation of the type of equation (10) with on = 1 / T following the form of equation (41). The agreement is good above the peak value but below the peak value it is seen that a distribution of relaxation frequencies is needed to match the measured value. For a distribution of relaxation frequencies, the attenuation can be written in the form

(42)


0.28

0.24

J0.20

0.0, 1

16

,0 .12

o 08

0.04

<* — *

' A it /

/

\ \ \ \\

• /

0 > 1

I T

1

w

0 /

/ ( \ '

1. 5 a s (X2 0 4 O6 0£M> 2 4 6 8 10

TEMPERATURE IN DEGREES KELVIN 2 0 4 0

F I G . 1 4 . Measured attenuation for a shear wave along the < 1 1 0 > axis with a particle velocity along < 1 1 0 > at 5 7 M c (solid line). D a s h e d lines show three relaxation curves as

discussed.

F I G . 1 5 . Plot of attenuation peak temperatures against the logarithm of the measuring frequencies.

2 O

g o

QT

S o 1 Z

Q

< i o UJ

o


where the B% are a series of constants which determine the heights of the relaxa-tion peaks. Taking a ratio of about 1.23 between the activation energies, it is found that a series of constants Bt can be chosen which can match a smooth internal friction curve within ± 5 per cent. The top three relaxations are shown by the dashed lines. Below 1.4°K, which was the lowest temperature attained with the available cryostat, the attenuation was assumed to be between the dashed and the dot-dash lines. This requires a wide distribution of relaxation frequencies which is in agreement with the model proposed. It appears that an attenuation close to the dot-dash line is the most probable.

TEMPERATURE IN °K

F I G . 16. Shear-wave velocities along <110>, p.V. along < l l 0 > as a function of the temperature for Harshaw and Isomet crystals.

The velocities of the shear waves for both Isomet and Harshaw crystals are shown by Fig. 16. From 20°K to 1.4°K, the vacuum grown crystal has a velocity independent of the temperature. The Harshaw crystal, however, has an unusual type of velocity change in that the velocity decreases as the temperature decreases, contrary to most relaxations.

On account of the small activation energy and the high frequency fo% one would expect that these relaxations would exert an effect at very high frequencies and it occurred to the writer that this relaxation might explain the loss of thermal conductivity previously found in Harshaw sodium chloride crystals by Klein 2 9 . Figure 17(a) shows the thermal conductivity in watts/cm x °K plotted


against the absolute temperature. The top curve is for a vacuum grown crystal while the bot tom curve is for a Harshaw crystal of the type used for measuring the ultrasonic relaxation. It is seen that there is a thermal conductivity drop of 100 to 1 caused by the impurities present in the sodium chloride.

It was shown by Kle in 2 9 that there was a one to one correspondence between the drop in thermal conductivity and the height of an ultraviolet absorption band as shown by Fig. 17(b). F rom the height of this peak and Smakula's rule,

2 0 | 1

F I G . 1 7 . (a) Thermal conductivity of sodium chloride for various crystals, (b) Relat ion between the thermal conductivity and the height of the "hydroxyl" ultraviolet absorption

band (after M. V. Klein).

as well as by direct chemical analysis, it was concluded that the number of impurity atoms was between 2 X 10 1 6 to 2 X 10 1 7 per cm 3 giving concentra-tions between 10~ 6 and 10~ 5 with the higher figure being more probable. If this were a direct scattering process it was pointed out that the scattering cross-section would have to be 3000 times the Rayleigh scattering cross-section and also the temperature dependence was wrong for a scattering process.

If the relaxation spectrum found ultrasonically is to produce thermal effects, it must introduce attenuations in the frequency range from 2 x 1 0 1 0 cycles to

Temperature, °K

(a)

Photon wavelength, Ufj.

(b )


2 X 10 1 1 cycles which are the dominant frequencies—determined from the equation

hf=kT (43)

for the temperatures 1°K to 10°K. The attenuation produced at higher fre-quencies can be determined from equation (42) if we know the activation energies Hi and the attempt frequencies fot for the various relaxations of the spectra, fo and Hi are known from the measurements for the highest component. This attempt frequency of 5 X 10 9 cycles per second indicates a rather low proba-bility of occurrence for the effect. A relation of the form

ft = 10 1 2 e^Sil1® e-(HilRT>> (44)

is assumed for the relaxation frequency of the process. The value of 10 1 2 cycles per sec is a commonly observed value for a single relaxation process. For the highest energy value — Hi = 21 cal/mole — AS has to be negative and equal to 10.6 cal/mole. The lower the activation energy, the fewer successive pro-cesses have to occur and hence the lower the disorder so that A St is assumed proportional to Hi. Then equation (44) becomes

ft = 10 1 2

e-°-252Hie-(Hi/RT) (45)

With these values of fi9 and the values of Bi to correspond to the measured attenuation values of Fig. 14, one can calculate from equation (42) the attenua-tion to be expected at any other frequency. These constants were tested on the available data up to 200 Mc and reproduced the measured data with good precision.

To show that the ultrasonic relaxation spectrum accounts for the thermal loss observed, the only quantity in common between the two sets of data is the mean free path. The acoustic wave can be regarded as being composed of a number of phonons which are scattered out of the path by the impurities. When the attenuation equals one neper, the number of phonons left will be e~x times the original number and this length is by the usual definition, the mean free path. One has to take account of the fact that heat waves consist of longitudinal and shear waves which come from all directions. The most efficient direction of scattering is along the <110> direction with the displacement along the <ll0> directions but most of the phonons will have a smaller component of stress in the (110) planes. A calculation indicates a factor of 1/9 for shear phonons which carry most of the energy. Hence the mean free path is calculated by finding the distance for which the extrapolated attenuation from equation (42) equals 9 nepers. The result for the two extremes of Fig. 14 is shown by Fig. 18. The lower curve corresponds to the dashed line of Fig. 14 while the upper one corresponds to the dot-dash line.

The mean free path for thermal phonons is calculated from the heat con-ductivity equation

K = \CVX (46)

where C is the specific heat per unit volume, V the Debye average velocity,


K the thermal conductivity and A the mean free path. Since C varies as the third power of the temperature and V is practically independent of the tem-perature, then A depends on the temperature variation of K—assumed to be Tn—according to

XaT^) (47)

F I G . 18. Comparison of the ultrasonic and thermally determined mean-free paths for phonons in Harshaw sodium chloride crystals.

Using the measured values of C, V and assuming K is proportional to the square of the temperature, Klein derived the dashed curve labeled equation (1). The agreement in fair with the solid line labeled 2. The slope of K from Fig. 17(a) is more nearly n = 1.8, which would give a better agreement with the calculated value. Hence it appears that a relaxation process causes the thermal resistance. It is a very efficient process on account of the large effective cross-section and one which is not described by any of the ordinary mechanisms. 3 0

The facts that the elastic constant decreases as the temperature decreases and that the relaxations obey Boltzmann's statistics indicate that we are dealing


with a number of energy levels. The model proposed for the effect is the one shown by Fig. 19. Water is assumed to take the place of the chlorine atoms. Whether water is dehydrated at the high growing temperature and enters the crystal as an O - H radical or whether the water molecule itself enters the lattice is not certain. The best evidence obtained by cleavage 3 1 and paramagnetic resonance indicates that the water molecule itself enters the lattice and this is the form shown by Fig. 19. However, the qualitative results discussed here could be obtained by either the water molecule or the hydroxyl molecule. To agree with the stress sensitivity observed, the hydrogens have to make bonds to the sodiums. For a (110) shear stress, directed as shown, the bond distance of one set of sodiums is increased while that of another set is decreased. Two other possibilities, i.e. bonding to chlorines, or positions in <111> directions, do not react to this stress.

F I G . 19. Proposed model for ultrasonic and thermal relaxation process in sodium chloride crystals.

The bond energy between the hydrogens and the sodiums is large in sodium hydride and must be much larger than 21 cal/mole even when the hydrogens are connected to the oxygen. Hence the small activation energy observed re-quires a quantum mechanical tunneling process in which the hydrogen atoms successively occupy the six equivalent positions adjacent to the sodium. Al-though the model with six equivalent positions and either one or two tunneling atoms is too complicated to calculate in detail, a comparison with the familiar tunneling process in ammonia shows that the energy values are of the right order of magnitude. In ammonia as shown by Fig. 20, the nitrogen atom tunnels between the triangle of hydrogens. The two energy levels are determined by the antisymmetric and symmetric combinations of the wave functions of the two positions. Their difference is determined by the equation

E2 — Ei = hf(ergs) (48)


where / is the tunneling frequency 23,870 Mc. This corresponds to an energy difference of 2 .3 cal/mole or (1/9) that of equation (41). Figure 20 shows that the energy levels can be separated by applying an electric field which lowers one energy level with respect to the other. The energy separation increases by the amount 2AE, where AE is the amount the original potential wells are raised or lowered.

I »

Inversion levels Electric field (Stork effect)

F I G . 20. Model for a tunneling process in the ammonia molecule and the separation of the two energy states by an applied electric field.

€ 2 - € , = h f 6 2 -€ , = (%-€, ) 0

+ 2 A E

F I G . 21. Separation of energy levels in water molecule impurity caused by a shearing strain.

For the model of Fig. 19 there will be many combinations of wave functions of the six positions and hence a spectrum of energy levels in agreement with the measurements. These energy levels can be separated by a shearing strain in the (110) planes with its polarization direction along <110>, and the amount of separation is proportional to the square of the strain as shown by Fig. 21. The lines marked c x and e 2 are the highest and lowest energy levels, while the dashed


lines indicate the presence of many other energy levels which are also separated by a strain. This separation of energy levels adds a term to the internal energy of the form

where No and Eo are the numbers and energy of the lowest ground state. Hence at absolute zero all of the impurity molecules are in their lowest energy state and the function F(T) = 0, i.e. no energy proportional to the square of the strain exists. At a high temperature all the states will be equally populated and an impurity energy (CNAK/2V) Sf2 is added to the internal energy. This increases the effective shearing constant JJL = (en — c\<i)\2 and hence the elastic modulus increases with temperature. From Fig. 14—dot-dash line—and Fig. 16, this increase adds about 0 .6 per cent to the 0°K shear modulus of 2.37 x 10 1 1

dynes/cm 2 . Hence CNAK/V= 14 X 10 9 dynes/cm 2 and K= 6.25 x 10" 9 . To double the separation of energy levels requires a strain of about 1 0 - 3 .

For an alternating stress the rate for which the population goes from the ground state to any one of the other energy levels is controlled by an Arrhenius type equation such as (41) or (44). A direct measurement of the constants is possible only for the highest energy state, but the effect on the thermal con-ductivity is indicative of a relation such as equation (45).

4 . U S E O F E L A S T I C A N D A N E L A S T I C M E A S U R E M E N T S I N D E -

T E R M I N I N G T H E P R O P E R T I E S O F E L E C T R O N I C A N D H O L E E N E R G Y

S U R F A C E S — D E T E R M I N A T I O N O F T H E T I M E C O N S T A N T O F T H E

P I E Z O R E S I S T I V E E F F E C T

Attenuation and elastic constant measurements have been used extensively in determining the energy surfaces (Fermi surfaces) along which electrons move. For a free electron model, the energy surface—which is portrayed as a sphere in momentum space—represents at 0°K a division between the filled energy states and the states having an energy larger than the Fermi energy

(49)

where F(T) is a function of the temperature, C the impurity concentration (10~ 5), V the molal volumes is 26.8 cm 3 , NA is Avogadro's number and E% are the successive energy barriers which produce the total relaxation spectrum. These energies sum up to 1630 ergs/cm 3.

The relative numbers in the various states are determined by the Boltzmann equation

(50)

(51)


where ft is Planck's constant h divided by 2TT, mo the mass of an electron (9.1 X 1 0 - 2 8 g) and N is the number of electrons per cm 3 .

As the temperature is increased, some electrons acquire a thermal energy kT, and the distribution is no longer sharp. For metals kT is much smaller than WF up to temperatures in the order of 8 x 10 4°K. The temperature 7z>, for which the thermal energy equals the Fermi energy WF, is known as the de-generacy temperature. For semiconductors for which N is much smaller than for metals, the degeneracy temperature is often much less than room tempera-ture. Above Tn, Boltzmann's statistics are valid, while below TD, Fermi-Dirac statistics must be employed.

In actual crystals, the Fermi surface is not spherical but takes up a shape determined by band theory. Electrons still move on the surface but they have an effective mass determined by the curvature of the energy surface according to the equation

where k — co/V is the wave vector. Attenuation measurements at low temperatures and in magnetic fields have

been used considerably to delineate the shape of Fermi surfaces in metals. An acoustic wave can distort the shape of the surface slightly causing the electrons to flow in the direction of the lowest energy points. If the mean free path between collisions with phonons or impurity atoms is large enough, considerable energy can be abstracted from the acoustic wave by this process and a high attenuation can result. The effect of a magnetic field is to cause the electrons to revolve in a plane perpendicular to the direction of the magnetic field. If the mean free path is long enough, the electrons can be made to flow in re-entrant orbits which alternately cause peaks and valleys in the acoustic attenuation as a function of the magnetic field. Such variat ions 3 2 can be related to the shape of the Fermi surface.

Since a number of review articles 3 2 have been written on this subject, the present section is limited to a description of recent work on the motion of electrons on semiconductor energy surfaces. This work is related to the piezo-resistance effect in semiconductors which is now being used 3 3 to provide very sensitive strain gages and pressure transducers. The attenuation and velocity changes discussed here provide an answer to how fast a pressure change can be measured by such devices as far as the piezoresistance mechanism itself is concerned. The results indicate that pressure changes occurring up to intervals as small as 10~ 1 2 seconds should produce an effect.

Silicon and germanium crystallize in the diamond-type structure shown by Fig. 22. There are four adjacent silicons which form covalent bonds (exchange of electrons) with each silicon. This is a strong bond which takes about 1.1 eV (25.4 kilocalories per mole) to break. When such a bond is broken, the electron can move off leaving a partial bond behind. Since electrons from adjacent bonds can fill the broken bond, this can move also and acts like a positive

(52)

3


electron or "hole". When all the electrons or holes are produced by thermal agitation, the semiconductor is called intrinsic and has a resistivity which has a temperature slope determined by half the energy gap value since the Fermi surface can be shown to be midway between the valence and conduction band. The resistivity of an intrinsic silicon is shown by Fig. 23. For very pure silicon (less than 10 1 4 impurity atoms per cm 3 ) the resistivity can increase to 1000 ohm cm at 200°C before impurity conduction occurs.

When atoms of valence + 5 are substituted for silicon, four of the bonds can be satisfied, but one is left over and the electron is contributed to the conduction band. At low temperatures, the electron will have a lower energy if it is captured

by an impurity which has a residual charge of + 1 . The electron will revolve in a hydrogen-like orbit with a radius given by the equation

where e is the dielectric constant (16 for Ge) and m* = mo. The ionization energy for such an impurity atom is in the order of 0.05 eV as shown on Fig. 23. Hence, at low temperatures for a lightly doped material, most of the electrons will be frozen out and the resistivity becomes very high.

For higher doping levels the wave functions for the various impurity atoms tend to overlap and the lowering of energy becomes less. For a doping in the order of 10 1 8 impurity atoms per cm 3 the trapping energy disappears, the electron density is not a function of the temperature, and the behavior is similar to that for a metal.

When atoms of valence + 3 are substituted for silicon or germanium there is one unsatisfied bond which acts as a hole. This also is more stable if it performs

F I G . 22. Types of bonds found in silicon and germanium.

a = eh2/m*e2 = 8.5 x 10~ 8 cm (53)


CONDUCTION . B A N D

I

ENERGY GAP

i . i i e v I I

V A L E N C E BAND

FERMI LEVEL

O + H O L E

D O N O R ' IMPURITY

L E V E L

1 0 3

2 o

o z

> 1

t

!«>"' w K O

0 . 0 8 ev

A C C E P T O R I M P U R I T Y , L E V E L

/

J f O + . H O L E

T 1 '/ 1 1 1

/ / PURE / S I L I C O N

t (•— (•— —— —— — — — _ — — m m m

/

BORON

6.7 X 1 0 1 7 PER CC

6 .7 X I 0 1 8

1.3X I 0 1 9

0 0 . 0 0 2 0 . 0 0 4 0 .006 1 / T ° K

F I G . 2 3 . Intrinsic and p- and /T-doped silicon semiconductors and their corresponding energy surfaces. Bot tom figure shows resistivity of boron-doped polycrystalline silicon as

a function of the temperature.

The velocity will increase until the electron suffers a collision with either a phonon or an impurity atom. If the average time between collisions is r, then the electron or hole attains an average velocity of Vo determined from (54) by replacing t with r. The mobility ii is the average velocity divided by the field. In M K S units this will be in units of m 2 /V sec = 10~ 4 cm 2 /V sec. The electronic charge is 1.6 X 1 0 - 1 9 coulombs while m* is expressed in kilograms. For example for lightly doped ^-germanium, the mobility is about 3700 cm 2 /V sec = 0.37

o » - e

600 400 200 100 50 0 -100°C

(54)

an orbit around the negatively charged impurity atom. For boron this energy is in the order of 0.08 eV as shown by the line on Fig. 23.

When an electron or hole is free in the conduction or valence bands it is accelerated in an electric field and acquires a velocity equal to


m 2 /V sec. The average mass is 0 . 1 2 times the electron mass or 1 .1 X 1 0 ~ 3 1 kg. Hence r the average collision time is r ^ 2 . 5 x 1 0 ~ 1 3 sec. For a heavily doped sample of 3 x 1 0 1 9 arsenic atoms per cm 3 , the mobility is reduced to 0 . 0 3 0 0 m 2 /V sec, and r = 2 x 1 0 ~ 1 4 sec. /?-type silicon, for which measurements are presented later, has relaxation times of 7 X 1 0 ~ 1 4 and 1 X 1 0 ~ 1 4 sec for lightly and heavily doped samples.

The conductivity for a semiconductor with one type of carrier is given by the equation

where p is the resistivity in ohm meters ( 1 0 ~ 2 ohm cm) if N is expressed as the number of carriers per cubic meter, e — 1 .6 x 1 0 - 1 9 coulombs and p, is the mobility in m 2 /V sec. Figure 23 shows typical resistivities plotted against doping for polycrystalline silicon doped with boron atoms. Lightly doped samples first decrease in resistance as the temperature is increased and then increase again. The decrease is due principally to the increase in number of carriers as the temperature is increased due to release from bound states. The increase at higher temperatures is due principally to the shorter time constant r caused by increased collisions with phonons. For a doping of about 6 . 7 x 1 0 1 8 boron atoms per cm 3 , the resistivity is nearly constant up to 300 °K but increases above this temperature. This behavior reflects the fact that all the holes are free at all temperatures and r is determined below 300 °K by collisions of electrons with impurity atoms, which produce a constant value of r. Above 300 °K, r is shortened by increased collisions with phonons causing an increase in resistivity.

The energy surfaces of silicon are shown on Fig. 24 for two directions in momentum space for both p- and «-type doping. There are three energy surfaces but in the case of w-type silicon the energy is much less for one surface along the <100> direction and electrons tend to congregate in these valleys. There are six such valleys, four of which are shown by Fig. 25. Choosing one of these minima as an origin, the surfaces of constant energy can be represented by

where mt and mi are the transverse and longitudinal masses. The energy surfaces above the band edges are then ellipses in momentum space as shown by Fig. 25. As shown by cyclotron resonance experiments in which the electrons are constrained to move in a path perpendicular to the magnetic field, for silicon and germanium, the values of the transverse and longitudinal masses are

(55)

(56)

mt = 0 . 1 9 m 0 ; raz = 0 . 9 8 m 0 (Si) (57)

mt = 0 . 0 8 2 m0; mi =1.57 m0 (Ge)

Nep, = Ne2r/m' P


i t ! 1 1.56V 1 1 1 1 1 1 1 1

2 5 ev | 1 ± ! J ~ i i

"^CONDUCTION' j ELECTRONS

i i i

1.08 ev

' LIQHT AND 1HEAVY HOLES

? k= ( 0 , 0 , 0 ) ^

[III]-* DIRECTION IN k SPACE • [100]

F I G . 24. Energy surfaces of /?- and «-type silicon.

F I G . 25. Multivalley energy surfaces for n silicon. Solid lines are equal energy contours for unstressed crystal while dashed lines show effect of a strain Si a long the x-axis.

ENER

GY-


One of the first effects to confirm the multivalley energy surfaces for ger-manium and silicon was the piezoresistive effect. This phenomenon expresses the fact that the electrical resistivity of a semiconductor changes when a stress is applied in certain crystallographic directions. The amount of change is nearly two orders of magnitude larger than resistance changes in metal strain gages and such semiconductor gages are now being widely used in industry. 3 3 The origin of the effect for n-type silicon, whose energy surfaces are shown by Fig. 25, is that a strain Si along the X crystallographic axis raises the energy surfaces along the direction of the strain and lowers those at right angles as shown by the dashed lines which represent equal energy surfaces. This change causes the population of electrons in the <100> valleys to decrease and those in the <001> and <010> valleys to increase by a process known as intervalley scattering.

If the time of the application of the stress is longer than the intervalley scat-tering time T , equilibrium conditions can be assumed to hold and for non-degenerate material the ratio of the number of electrons in the two types of valleys is given by the Boltzmann ratio

(58)

where a±Si is the amount the <100> energy surfaces are increased and a 2 the amount the <010> surfaces are decreased. The deformation potential Su is defined as the energy difference between the two types of surfaces for a linear stress Si equal to unity. Another relation is that the sum of the electrons remains constant or

4iVoio + 2Nioo = No (59)

Solving for the number in each type of surface, we find to a first approximation

(60)

If we introduce these terms in the expression for the conductivity for a field applied in the same direction as the strain,

(61)

(62)

where po is the resistivity and /x 0 the unstrained mobility which is equal to

(63)

Since /x„ is inversely proportional to mi and n± to mt, the value of the expression in brackets is — 1.1 for silicon and — 1.39 for germanium. The effect of a stress

and separate the part out due to the strain, we find

a = e [2iVioo j&n + 47Voio / * J


at right angles to the current in ^-silicon is to produce an effect mi2 one-half

the value of mu and of opposite sign or

The effect of a shearing strain S12 in the crystallographic plane is to produce the same change in the valleys in the (001) plane and hence to produce no effect. Experimental results 3 4 confirm this concept except that small deviations occur which are ascribed to other minor effects.

For germanium with <111> valleys it can be shown that mu = mi2 = 0 while

Both effects in w-type material produce a decrease in resistance. F rom the values of mu and mu measured at room temperature it is found that the value of the deformation potential is about 16 eV = 2 .5 X 1 0 - 1 1 ergs per electron for germanium and 8.6 eV = 1.37 X 1 0 - 1 1 ergs per electron for ^-silicon. Other types of data give values of the same order for these materials at room temperature.

The multivalley surfaces do not hold for p-type silicon or germanium, which have the three energy surfaces shown by Fig. 24. Two of the energy surfaces are degenerate in that they make contact at the central zone point. The third surface is spherical with an energy value that is 0.035 eV above the other two surfaces for silicon. Even at room temperature, less than 25 per cent of the holes can reside in this surface and it is usually neglected. In any case the effect of a stress on this surface should be negligible compared to its effect on the other two. Cyclotron resonance effects 3 5 have been used to investigate the curvature of the two degenerate surfaces and they have been found to satisfy an equation of the type

where the negative sign refers to the Vi band and the positive to the V2 band. Since the higher surface Vi has the smallest curvature, holes have the highest mass. This has been evaluated as 0.49 mo. For the V2 surface the average mass is 0.16 mo. At thermal equilibrium the relative number of holes in the two bands is proportional to (mi /m2) 3 / 2 = 5.4. Hence about 85 per cent of the holes are in the Vi band and 15 per cent in the V2 band.

It was first pointed out by A d a m s 3 6 that a piezoresistance effect would result due to the separation of the energy surfaces by the two types of shearing stresses that actuate the <111> and <100> valleys of «-type materials. The most complete calculations are those of Pikus and Bir 3 7 , who derive a formula for the piezo-resistance effects in the form

mi2 = — m n / 2 (64)

(65)

(66)

(67)


where B = ^B2 + C 2 / 5 ; D = s/3B2 + C 2 , b and d are deformation potentials and (on — ai)/ao = (\/mi/m2 — ^)l{y/m^m+ 1) = 0.212 is the difference in the conductivities produced by the two types of holes divided by the sum. With the values 3 5 of B = IA, C = 3.7 for silicon the measured value of mu = + 110 for /7-silicon at 300 °K, the value of the deformation potential d becomes about 34 eV.

These considerations account well for the equilibrium piezoresistance effects. If the doping level is high enough to cause the semiconductor to become de-generate in the temperature range of interest, Fermi-Dirac statistics have to be used. 3 8 In general the effect is to lower the value of the piezoresistance effect but to make it more nearly independent of the temperature. However, none of these measurements answers the question of how fast the piezoresistance effect will respond to a stress since this depends on the intervalley or intersurface time constant T which determines how fast an electron or hole can change from one valley or one surface to another.

These questions have been answered 3 9 recently for ^-germanium and /7-silicon by measurements of the ultrasonic attenuation and modulus changes of heavily doped specimens and by measurements 4 0 of the acoustoelectric effect for more lightly doped samples. The acoustoelectric effect arises from the radiation pressure of an acoustic wave on electrons in a multivalley system for which bunching can occur. It results in a d.c. potential across the specimen as long as the acoustic wave is traversing the semiconductor. The ratio of the attenuation to the acoustoelectric field Eae is given by the equation

(68)

where e is the electric charge and N the density of the carriers, V the sound velocity and S the acoustic power density. It turns out that the acoustoelectric voltage can be measured at lower doping levels than can the difference of attenuation caused by the presence of electrons or holes in the semiconductor. However, for dopings above 10 1 8 atoms per cm 3 attenuations have been measured for both ^-germanium and ^-silicon.

The attenuation caused by the intervalley relaxation time in an intervalley semiconductor has been investigated by Blount 4 1 and as long as COT <^ 1 and coX/Y < 1, the attenuation follows a relaxation equation of the form of (22) with r being the intervalley relaxation time. The second condition, where A is the mean free electronic path, results from the large difference between electron velocities on the Fermi surface and the velocities of sound waves. In equation (22), AE is the elastic modulus change caused by the shift in electron popula-tion from one set of valleys to another and T is the time taken for the electron to go from one valley to another. That there is a change in the elastic modulus for /i-silicon is readily seen from equation (60) and the fact that the energy levels for <100> valleys are raised by (2/3) SuSi while those for perpendicular


in the degenerate range where m* is now the density of states mass (m 2 ra j ) 1 / 3

which has a value 0.22 for germanium. In between the variation as a function of temperature is shown by Fig. 26. Measurements have been m a d e 4 3 for a doping of 3.5 X 10 1 9 arsenic atoms per cm 3 . The difference in modulus Ac±± between the pure and doped sample is shown by Fig. 27. This is in fair agreement with the calculated value.

Using a value of Su of 20 eV which is usually observed at low temperatures, m* = 0*22 mo which corresponds to the average mass, one obtains a value of Acu = 5.4 x 10 1 0 compared to a measured value of 3.76 x 10 1 0 dynes/cm 2 . The dashed line shows a calculation for Acu for a doping of 1 X 10 1 8 arsenic atoms per cm 3 obtained by reducing the measured value in the ratio (Ni/N2)1/3, and the degeneracy temperature by (NiINz)2!*.

To obtain a value for r, the intervalley relaxation time, one has to measure the attenuation or the proportional acoustoelectric voltage. It turns out that the attenuation difference between the doped and undoped sample is too small to measure for doping less than 1 0 1 8 atoms per cm 3 . However, since there is no other source producing an acoustoelectric voltage except the attenuation due to electrons, this technique can be used at lower doping levels. Figure 28 shows the intervalley scattering rate—inverse of relaxation time r—as a function of temperature for doping levels from 1 0 1 4 to 1 0 1 6 arsenic atoms per cm 3 . At low

in the normal range and

(72)

(71)

In a similar manner by employing a shear wave along <110> with polarization along <1T0>, the second relation holds and since a shear in the x-y plane (£12) does not affect the energy valleys, the last relation is evident.

For w-germanium a similar calculation 4 2 shows that

(70)

Hence there is no change to first powers of Si but a negative contribution occurs proportional to the square of the strain. If we add this to the elastic energy (1/2) c n S 2 , it is seen that a lowering of the elastic modulus occurs of a value

(69)

valleys are lowered by —(1/3) EuSi. Hence the energy change AW for the de-formed surface is given by the equation

3§


0 0 . 5 1.0 1.5 2 . 0 2 .5 3 .0 3 . 5

R A T I O O F T / T D

F I G . 2 6 . Relative variation of the shearing modulus Ac^ in going from a degenerate to a nondegenerate region (after Keyes).

temperatures the scattering is due to impurities while at high temperatures most of the scattering is due to thermal phonons. Figure 29 shows measurements of the attenuation of longitudinal waves in the <110> directions for low doping (less than 10 1 4 ) , dopings of 1 0 1 8 and 3 X 1 0 1 9 arsenic atoms per cm 3 . The values are not given above 80 °K since the difference is in the order of the experimental error above this temperature. The longitudinal wave along <110> is controlled by the elastic constant ( en + C12 + 2c44)/2. Since only c 4 4 has a component which changes with doping, the attenuation for this case is

O L I I I I I 1

0 5 0 1 0 0 150 2 0 0 2 5 0 3 0 0 T E M P E R A T U R E IN D E G R E E S K E L V I N

F I G . 2 7 . Elastic shear modulus difference A C 4 4 between pure and doped germanium having 3 . 5 X 1 0 1 9 arsenic atoms per c m 3 (after Bruner and Keyes).

F I G . 2 8 . Intervalley scattering rate as a function of the temperature for arsenic doped germanium having dopings from 10 1 4 to 10 1 6 (after Weinreich, Sanders and White).


where

(74)

Using the measured attenuation difference and the values of dcu from Fig. 25, the time constants are

(75) r = 2 .3 X 1 0 - 1 3 ; T - 4 . 0 x 1 0 - 1 3

N= 3 .0 X 1 0 1 9 ; N= 1 x 10 1 8


F I G . 3 0 . Intervalley relaxation time for various dopings and conditions.


tering at room temperature. The time of 2 x 10~ 1 3 sec as compared to the overall scattering time of 2 X 10~ 1 4 sec calculated previously, indicates that about 1 in 10 collisions results in a change of valley at a doping of 3 x 10 1 9

arsenic atoms per cm 3 . As a practical device for all doping levels the piezo-resistance effect should respond in a time in the order of 10~ 1 2 sec at room temperature for «-type material.

The response for p-type silicon is somewhat different than «-type material. Figure 31 shows attenuation measurements for longitudinal waves along <100> and <110> as a function of the temperature and the frequency. At very low temperatures the attenuation increases very rapidly. An even larger effect occurs for shear waves along <110> as shown by Fig. 32. From the measurements of Figs. 31 and 32, value of AEr can be calculated from equations of the form of (22) or (73). At 5°K we have the values of Table 3.

F I G . 3 1 . Attenuation for longitudinal waves in pure and p-type silicon with a doping of 2 . 5 x 1 0 1 8 boron atoms per c m 3 .

with r being independent of the temperature up to 300 °K. Figure 30 shows the intervalley relaxation time for various conditions. The top curve shows the relaxation time as limited by impurity scattering as determined from Fig. 28 and Fig. 29. For temperatures of 150°K, phonon scattering is dominant up to dopings of 10 1 6 arsenic atoms per cm 3 but for higher dopings impurity scattering predominates. For 300°K the crossing point is about 10 1 7 atoms per cm 3 . F rom the data of Fig. 30 it appears that the relaxation time of impurity scattering is determined by the ratio of the number of collisions which cause a change of valley to the total number of collisions. For low doping levels the ratio may be rather small for impurity scattering but is in the order of 0.025 for phonon scat-


2

£ 6

Z

I I D 2 H 5c

o <110>; PV<001> (475 MC)

• <110>; PV<1T0> (475 MC)

• PURE SAMPLE

o <110>; PV<001> (475 MC)

• <110>; PV<1T0> (475 MC)

• PURE SAMPLE

10 20 30 4 0 50 60 TEMPERATURE IN DEGREES KELVIN

70 80

F I G . 3 2 . Measured attenuation of shear waves in pure and /?-type silicon with a doping of 2 . 5 x 1 0 1 8 boron atoms per c m 3 .

Since doping does not change the crystal structure, there can be only three changes in modulus and from Table 3, these are

The last measurement gives a check value

(77)

compared to the measured value of 0 . 1 . By using a pulse superposition method , 4 1 velocity measurements of a sufficient

accuracy have been made to determine the difference Acu between the pure and the doped sample. Figure 33 shows the velocity measurements for a shear wave along the <100> direction. From these measurements the values of Acu are directly determined. Table 4 shows values measured from 5°K to 100 °K. The second row shows the measured attenuation difference between the pure and doped samples, while the last row shows the corresponding relaxation time.

TABLE 3

Propagation Particle Elastic Velocity Aa — Ap

direction direction modulus cm/sec nepers/cm

1 0 0 1 0 0 Cll 8 . 4 9 x 1 0 5 0 . 0 6 8 J C I I T = 0 . 0 2 1 9

1 1 0 0 0 1 C44 5 . 8 6 x 1 0 5 0 . 8 6 ACAA^ = 0 . 0 8 9 5

1 1 0 1 1 0 (cn — Cl2)/2 4 . 6 9 x 1 0 5 0 . 3 1 7 (Acu- ACI2)T = 0 . 0 3 3 7

1 1 0 1 1 0 Cll + C12 + 2C44

2 9 . 1 7 5 x 1 0 5 0 . 2 5

= 0 . 1

(76)


F I G . 33. Shear wave velocity along <100> for pure and/?-type silicon doped with 2 . 5 X 1 0 1 8

boron atoms per c m 3 .

r varies approximately according to an equation of the form

r = 1 .6 x K H 2 ^ / * 2 1 (78)

where # the activation energy is about 0 . 0 0 0 7 5 eV = 17 cal/mole. This is a different type relation from that found for ^-germanium.

Recently measurements have been made for doping levels from 5 X 1 0 1 7

boron atoms per c m 3 to 1 0 2 0 boron atoms per cm 3 which clarify the mechanisms affecting the intervalley relaxation times. Since the measurements given by Table 3 show that Aci2 = — Acn/2, use is made of longitudinal wave measure-ments along the <111> directions which can be made more easily than shear wave measurements at the low temperatures. For this direction the elastic modulus is ( cu + 2ci2 + 4c44)/3 and hence the change in velocity, due to doping, determines the value (4/3)J<?44.

Figure 34 shows measurements for the longitudinal velocities for samples having resistivities of 200 ohm cm, 0 . 0 1 ohm cm, 0 . 0 0 4 2 ohm cm and 0 . 0 0 1

TABLE 4

Temp. ° K 5° 10° 20° 50° 80° 100°

ACAA d / cm 2 1 . 0 5 X 1 0 1 0 6 . 4 7 X l O 9 4 54 x l O 9 2 . 5 x 1 0 s 2 . 0 2 x l O 9 1.8 x l O 9

Ad -Ap 0 . 8 4 0 . 2 3 0 12 0 . 0 4 6 0 . 0 3 5 0 . 0 2 9 T in sec 8 . 5 x l O - 1 2 3 . 7 X l O - 1 2 2 . 8 x 1 0 - 1 2 1.8 X l O - 1 2 1.8 x l O - 1 2 1.7 x l O - 1 2


ft-aal I I I I 1 1 1 1 1 2 5 10 20 50 100 200 500

TEMPERATURE IN DEGREES KELVIN

F I G . 3 4 . Measured velocities for longitudinal waves along the < 1 1 1 > direction in silicon for boron dopings of 1 0 1 4 , 1 0 1 9 , 3 X 1 0 1 9 and 1 0 2 0 a toms /cm 3 .

0.61 . 1 1 1 1 1 1 I

F I G . 3 5 . Measured attenuations for longitudinal waves along the < 1 1 1 > direction in silicon for boron dopings of 1 0 1 9 and 3 x 1 0 1 9 a toms /cm 3 .


where Ac is the change in elastic modulus deduced from the difference in velocities of the doped and undoped samples. The attenuations are given by Fig. 35.

Measurements 4 5 of resistivity as a function of the temperature for various dopings of boron in silicon show that 5 X 1 0 1 7 is the lowest doping level for which the resistivity is independent of the temperature. Hence the measurements of Fig. 32 and 33 are nearly for the lowest doping level for which the activation energy 0 can be obtained. For lower doping levels the electrons are captured by traps and do not contribute to the conductivity at low temperatures.

The intervalley scattering time for a doping of 1 0 1 9 boron atoms per cm 3

satisfies the equation r = 2 . 1 X K H 3 ^ / * * 7 (79)

where 0 = 8 . 7 5 X 1 0 ~ 1 6 ergs per hole or 1 2 . 5 cal/mole = 5 . 4 X 10" 4 eV. This is less than the value of 17 cal/mole found for a doping of 2 . 5 X 1 0 1 8 . The doping of 3 X 1 0 1 9 boron atoms per cm 3 produces a constant value of 1 0 ~ 1 3 sec without an activation energy 0. The same is true for the doping of 1.3 x 1 0 2 0 boron atoms per c m 3 and the scattering time is less than 1 0 ~ 1 3 sec.

If we plot the multiplying factor of the exponential term—i.e. the high temperature relaxation time—the result is shown by the solid line of Fig. 30 marked /7-type silicon. For a doping of 3 x 1 0 1 9 boron atoms per cm 3 , there is no activation energy and it is thought that the intervalley time is determined by the number of collisions of the holes with the impurity centers that result in a transfer from one surface to another. Since the relaxation time associated with the total number of collisions is 1 0 ~ 1 4 sec, it appears that one in ten collisions results in a transfer of energy surfaces. If this were the only mechanism opera-tive, one would expect that this time would parallel that found for fl-germa-nium for which a collision process is definitely dominant, and hence should follow the dashed line.

The difference between the solid and dashed lines indicates that in the doping region below 2 . 5 x 1 0 1 8 , the hole is captured by the impurity atom and spends some time there before it becomes free again. If we subtract 2 X 1 0 - 1 3 sec— which corresponds to the time between two successive captures as derived from the dashed line—this time is 1 .4 x 1 0 ~ 1 2 sec for the doping of 2 . 5 X 1 0 1 8 . Below the 2 . 5 X 1 0 1 8 doping level, it is assumed that the energy of trapping

(73) n e p e r s / c m

ohm cm. These resistivities correspond to doping levels of 1 0 1 4 , 1 0 1 9 , 3 X 1 0 1 9

and 1 .3 X 1 0 2 0 boron atoms per cm 3 . The corresponding attenuation differences are shown by Fig. 35. For the highest doping level, scattering of the wave occurs due probably to segregation of the impurities and attenuation measure-ments can be made only to 100 Mc. The indicated relaxation time is less than 1 0 ~ 1 3 sec independent of the temperature.

The intersurface relaxation times can be calculated from the equation


remains constant. It decreases for a doping level of 10 1 9 and disappears entirely above a doping level of 3 X 10 1 9 boron atoms per cm 3 . This is consistent with the ac t ion 4 5 found for deeper traps where the overlapping of the wave functions from adjacent boron atoms eliminates the trapping energy. In accordance with the interpretation given below the intervalley relaxation time is extended along the dashed line until it approaches the collision line. An attempt was made to verify this concept by measuring the velocity and attenuation for a doping of 5 X 10 1 7 boron atoms per cm 3 . The relaxation time follows the same equation as (78) with 0 = 18.4 cal/mole, and with T 0 shown by Fig. 30.

Since the degeneracy temperature is 30 °K or larger for all of the doping levels used, the valence bands of Fig. 30 are filled up to levels at least 0.0025 eV above the band minimums. Hence, it is believed that the relaxation measured is between different points on the same energy surfaces rather than an interchange between surfaces. This is confirmed by substituting gallium for boron. This material gives two activation energies 75 cal/mole and 0 .8 cal/mole for the heavy and light hole surfaces. Such a distortion of the energy surfaces can be caused by their warping by the applied stress or by a fixed warping caused by the shearing strains pro-duced by the boron or gallium atoms in the lattice. As suggested previously, 3 9

the time of 1.4 x 1 0 - 1 2 sec is consistent with the interpretation that the trapping energy is stored as a warping of the energy surfaces. This warping results in a quantum mechanical tunneling between points on the surface having a frequency.

5 . D I S L O C A T I O N M E C H A N I S M S A N D T H E I R E F F E C T S O N

E L A S T I C A N D A N E L A S T I C P R O P E R T I E S

Dislocations can move due to a resolved shearing stress applied along the glide plane by the ultrasonic wave and these motions are a source for ultrasonic

(80)

The time of a cycle is 5.6 X 10~ 1 2 sec and since the measured time spent in the neighborhood of the boron atom is 1.4 x 10~ 1 2 sec at high temperatures, this indicates that at high temperatures the time between capture and release is in the order of a quarter of a cycle.

So far no calculations for the change in elastic modulus Acu have been made for the condition that the energy surfaces are separated. However for the highest doping of 1.3 x 1 0 2 0 boron atoms per cm 3 the calculation of Keyes 4 2 should be valid. This gives a change in shearing modulus of

(81)

Using a density of states mass of 0.28 mo, d=5A5xl0~11 and N=l. 3 X l O 2 0

atoms per cm 3 , the calculated value becomes 7.25 X 10 1 0 dynes per cm 2 com-pared to a measured value of 5.15 X 1 0 1 0 which is 3/4th the indicated change in the longitudinal constant Ac.


LOW STRESS FIELD HIGH STRESS FIELD

( A )

INCREASING STRESS •

F I G . 3 6 . Possible dislocation motions as a function of the applied stress (after Granato and Luecke).

Figure 36 shows some of the dislocation motions proposed by Granato and Luecke 4 8 to explain various types of internal friction and modulus changes. At low stress amplitudes dislocations are considered to be pinned by impurity atoms and to be bowed out by the stress in the form of loops as shown by Figs. 36(b) and (c). During this regime a modulus decrease occurs since a plastic strain is added to an elastic strain and for very pure crystals the elastic modulus may be reduced by as much as 10 per cent over that for a metal with a large number of pinning points. During this regime the internal friction is assumed to be contributed by damping of dislocations by such phonon pro-cesses as the thermoelastic effect, 4 9 the Akheizer effect 5 0 and phonon scattering by dislocations. 5 1 At higher strain amplitudes, dislocations break away from their pinning impurities out to nodal points determined by the intersection of the dislocation networks, which do not move under the effect of stress. For each half cycle the dislocations become repinned by the impurities. This process causes 4 8 a modulus change and an internal friction change which are amplitude dependent. Both changes rise to a maximum and then decrease with further

( B ) (C) (D) (E) (F) (G)

attenuation and velocity changes. An example has already been given for shear waves in zinc transmitted along the hexagonal axis which has the attenuation shown by Fig. 3. Dislocations can affect the elastic modulus of a solid and as a consequence the corresponding internal friction, as shown by Fig. 10(b) for silicon, when the applied stress is very large. The action of such dislocations is very complex and as a consequence it is often not possible to determine uniquely what dislocation mechanism is causing the modulus or internal friction change. This is particularly true of high strain dislocation motions and as a result it is often more profitable to consider practical problems from a continuum mechanics point of view even though, without much doubt, dislocation mechanisms are responsible for such phenomena as creep, fracture and fatigue in metals. It is the purpose of this section to mention briefly some of the models that have been proposed to explain some of the observations. Complete reviews and discussions are given in two recent re fe rences . 4 6 ' 4 7


increase in strain amplitude. When the strain becomes large enough, disloca-tions can be generated by the Frank-Read mill process shown by Fig. 36(f) and (g) and this causes a very rapidly rising increase in the modulus and in the internal friction.

There is considerable experimental verification for all three regimes. Figure 37 shows measurements 5 2 of the decrement and the modulus change, in per cent, of a single lead crystal as a function of strain amplitude. The lowest regime extends from low strains up to strains of 5 x 10~ 6, the next regime up to strains of 5 X 1 0 - 5 while the third region exists for strains above this. Similar measurements have been made by the wri ter 5 3 and by Lazan 5 4 . In the last regime fatigue occurs in metals. This process is usually ascribed to the generation of vacancies caused by the dragging of dislocation jogs—produced by dislocations

to

8

6

4

2

0

o X

P B - N O . 1 3 - 5

AY j -

j -

-

3 . 0

2 , 5

2 . 0

o 1,5 x

SI* 1.0

0 .5

10 1 0 0 1 0 0 0

€ X 1 0 '

F I G . 37. Internal friction and change in Young's modulus observed in a single lead crystal as a function of the strain amplitude (after G. S. Baker).

cutting other dislocations—through the metal. While all these mechanisms agree with experiments in a number of details, they are not without their difficulties. For example, it has been questioned 5 5 whether dislocation loops move out as shown by Fig. 36(b) and (c) or whether the motion consists in the generation and diffusion of jogs along the dislocation lines. In the highest stress region, dislocations can be generated by other processes than the Frank-Read multi-plication process. Such a process is the multiple cross-glide mechanism. 5 6

Another internal friction mechanism of considerable interest for the study of the fundamentals of dislocation theory is the Bordoni 5 7 peak which occurs in plastically strained metals and single crystals. Figure 38 shows measurements 5 8

of the internal friction of a copper polycrystalline sample strained by 8.4 per cent. A large peak occurs at about 68 °K for a frequency of 1100 c/s and a small peak at 29 °K. Both peaks move to higher temperatures for higher frequencies


and satisfy an Arrhenius type equation such as equation (9a) with the activa-tion energy equal to 0.122 eV (2800 cal per mole) and the frequency fo=6 X 10 1 0 , for the high temperature peak and 0.041 eV for the lower peak. Some of the features established for these peaks are (as summarized by Truell and E l b a u m 4 6 ) :

1. The peak occurs for both polycrystal and single crystals after slight plastic deformation. It is, however, more evident in polycrystalline material.

2. The height of the peak increases and the peak maximum shifts slightly to higher temperatures with increasing amounts of plastic deformation; these effects saturate at about 2 to 3 per cent plastic strain.

3. The peak disappears upon annealing at elevated temperatures.

20

INT

ER

NA

L

FR

ICT

ION

, C

T1

4v

a>

ro

o>

A MEASU RED

/ /

/

CALCULATED "BACKGROUND

ic -— SCREW

F MIXED 1

i c -— SCREW

F MIXED 1

4 0 60 120 160 200 240 TEMPERATURE IN DEGREES KELVIN

260

F I G . 38. Bordoni type internal friction peak observed in polycrystal copper bar (after D . Niblett and J. Wilks).

4. The height of the peak and the temperature of the peak maximum are independent of the wave amplitude.

5. Impurities reduce the height of the peak and shift the peak maximum slightly towards lower temperatures; similar effects are observed as a result of neutron irradiation.

The most widely accepted explanation of these peaks 5 9 is that they are con-nected with the thermal generation of a loop, of the type shown by Fig. 39, which causes a dislocation lying along a minimum energy line to be thrown across the Peierl's energy barrier to the next minimum energy position. The energy associated with the generation of the two oppositely-directed jogs and that required to take the dislocation across the Peierl's barrier is the activation energy of the process. When the measuring frequency approaches the rate of formation of such loops, the attenuation is a maximum. Details of the calculation are given in reference 59. The two peaks are due to two types of dislocations, which can lie along minimum energy positions. These have Burger's vectors


which are perpendicular to and at an angle of 60° to the direction of the length. The model in general agrees with a number of features measured. Most measure-ments indicate that the width of the peak is about twice as large as would be expected from a single relaxation.

In conclusion elastic and anelastic measurements have produced a large amount of data which show that there are attenuation and velocity changes associated with the presence and motion of dislocations in metals and other crystals. The complexity of the dislocation motion is such that it is difficult to isolate a particular type of motion and ascribe any one effect to this motion.

>MAXIMA . MINIMA OF ^POTENTIAL

/ ENERGY

DISLOCATION LINE

F I G . 3 9 . Mode l for thermally activated dislocation loop (after A. Seeger).

6 . C O N C L U S I O N S

The examples cited and many others not discussed, show that physical acoustic methods, which depend primarily on the measurement of elastic and anelastic properties of acoustic wave propagation, are powerful means for investigating the mechanisms contributing to the mechanical properties of the materials. When the theory of such mechanisms is straightforward, the measure-ments can contribute materially to evaluating the constants entering the theory. When the processes are more complex, physical acoustic methods can be used to provide data which can lead to a more exact evaluation of the solid state motions occurring.

R E F E R E N C E S

1. B E N N E W I T Z , H . and ROTGER, H . Phys. Zeitsch. 37, 5 7 8 ( 1 9 3 6 ) ; Zeitsch. f. tech. Physik, 19, 5 2 1 ( 1 9 3 8 ) .

2 . ZENER, C. Phys. Rev. 5 2 , 2 3 0 ( 1 9 3 7 ) .

3 . B E R M A N , R. Phil. Mag. 42 , 6 4 2 ( 1 9 5 1 ) .

4 . L U C K E , K . / . Appl. Phys. 27 , 1 4 3 3 ( 1 9 5 6 ) .

5 . PEARSON, R E A D and F E L D M A N . Acta Met. 5, 1 8 1 ( 1 9 5 7 ) ; L A M B , R E D W O O D and STEIN-

SCHLIFER. Phys. Rev. Letters, 3 , 2 8 ( 1 9 5 9 ) .

6. M A S O N , W. P. , M C S K I M I N , H . J. and BATEMAN, T. B . Paper K 2 6 , Fourth Int. Congress

on Acoustics, Copenhagen (August 2 1 - 2 8 , 1 9 6 2 ) .

7. SNOEK, J . L . Physica, 8, 7 1 1 ( 1 9 4 1 ) .

8. D Y K S T R A , L . J . Phillips Research Reports, 2 , 3 5 7 ( 1 9 4 7 ) .


9. A review article on ,'Internal Friction D u e to Point Defects" has recently been given by B. S. B E R R Y , Acta Met. 10, 271 (1962).

10. HELLER, W. Acta Met. 9, 600 (1961). 11. M A S O N , W. P. and B R U G G E R , K. Phys. Rev. Letters, 7, 270 (1961); Paper J42, Fourth Int.

Congress on Acoustics, Copenhagen (August 21 -28 , 1962). 12. ZENER, C. Elasticity and Anelasticity of Metals, p. I l l , Chicago: University of Chicago

Press (1948). 13. N O W I C K , A . S. and BERRY, B. S. Acta Met. 10, 312 (1962).

14. Calculations of diffraction effects are reviewed in Chapter IV by H. J . M C S K I M I N in Physical Acoustics, Vol. I , edited by W. P. Mason (1963).

15. A high temperature relaxation due to oxygen has been found in silicon by P. D . SOUTHGATE, Proc. Phys. Soc. London, 76, 385 (1960), but the effect vanishes at r o o m temperature and lower.

16. PATEL, J. R. and C H A U D H U R I , A . R. "Macroscopic Plastic Properties of Dislocation-Free Germanium and Other Semi-conducting Crystals"; sent to / . Applied Physics.

17. AKHEIZER, A . J. Phys. U.S.S.R. 1, 277 (1939). 18. See KITTEL, Introduction to Solid-State Physics, Chapter 5, Wiley (1953), for a discussion

of these relations. 19a.B6MMEL, H. E . and D R A N S F E L D , K. Phys. Rev. I l l , 1244 (1960). 19b.A recent calculation by W O O D R U F F and EHRENREICH, Phys. Rev. 123, 1553 (1961) gives a

more general evaluation of the terms considered by Bommel and Dransfeld. 20. D E L A U N A Y , J. "The Theory of Specific Heats and Lattice Vibrations", Solid-State Physics,

Vol. II, N e w York: Academic Press (1956). 21. H E R R I N G , C. Phys. Rev. 95, 954 (1954). 22. See M A S O N , W. P. Physical Acoustics and the Properties of Solids, p. 91 , N e w York: Van

Nostrand (1958). 23. M C S K I M I N , H. J. and A N D R E A T C H , P. "Elastic Modul i of Germanium versus Hydrostatic

Pressure at 25°C and - 1 9 5 . 8 ° C " , / . Appl. Phys. 34 , 651 (1963). 24. BATEMAN, T., M A S O N , W. P. and M C S K I M I N , H. J. / . Appl. Phys. 32 , 928 (1961).

25. See M A S O N , W. P. Piezoelectric Crystals and Their Applications to Ultrasonics, p. 32, N e w York: Van Nostrand (1950).

26. M A S O N , W. P. J. Acoustical Soc. Amer. 32 , 458 (1960). 27. See KITTEL, C. Introduction to Solid-State Physics, p. 243, N e w York: Wiley (1953). 28. M E R K U L O V , L. G. Akusticheskie Zhurnal, 5 , 4 3 9 (1959); M E R K U L O V , L. G. and Y A K O V L E V ,

L. A. Akusticheskie Zurnal, 6, 244 (I960). 29. K L E I N , M. V. Phys. Rev. 122, 1393 (1961). 30. KLEMENS, P. G. in Solid-State Physics, Vol . 7, pp. 1-98, F. Seitz and D . Turnbull (Editors),

N e w York: Academic Press (1958). 31. W H E T T E N , N . R. and VANDERSLICE, T. A. Bull. Am. Phys. Soc. 6, 360 (1961); / . Chem. Phys.

37, 535 (1962). 32. The Fermi Surface, W. A . Harrison and M. B. W e b b (Editors), N e w York: John Wiley

(1960); Encyclopedia of Physics, Vol. X I - 2 , pp. 208-218 , R. Truell and C. Elbaum (Editors), Berlin: Springer-Verlag (1962).

33. Semiconductor and Conventional Strain Gages, Mills D e a n III (Editor), N e w York: Academic Press (1962).

34. SMITH, C. S. Phys. Rev. 94 , 42 (1954). 35. DRESSELHAUS, G., K I P , A . F . and KITTEL, C. Phys. Rev. 98 , 368 (1955).

36. A D A M S , E. N . Phys. Rev. 96 , 803 (1954). 37. P I K U S , G. and B I R , G. Soviet Physics—Solid State, 1, 1502, 1675 (1959). 38. POLLACK, M . Phys. Rev. I l l , 798 (1958). 39. M A S O N , W. P. and B A T E M A N , T. Phys. Rev. Letters, 10, 151 (1963). 40. WEINREICH, G., S A N D E R S , T. M. , Jr. and W H I T E , H. G. Phys. Rev. 114, 33 (1959).

41. B L O U N T , E. I. Phys. Rev. 114, 418 (1959). 42. KEYES, R. W. IBM Journal, 5, 266 (1961). 43. B R U N E R , L. J. and K E Y E S , R. W. Phys. Rev. Letters, 7, 55 (1961). 44. M C S K I M I N , H. J. / . Acous. Soc. Amer. 34 , 609 (1962). 45. M O R I N , F . J. and M A I T A , J. P. Phys. Rev. 96 , 28 (1954).

46. "Internal Friction Conference", Acta Met. 10, 271 (1962). 47. T R U E L L , R. and E L B A U M , C. "High Frequency Ultrasonic Stress Waves in Solids", Ency-

clopedia of Physics, Vol . X I - 2 , S. Flugge (Editor), Berlin: Springer-Verlag (1962).


4 8 . G R A N A T O , A . and L U E C K E , K . / . Appl. Phys. 27, 5 8 3 and 7 8 9 , ( 1 9 5 6 ) .

4 9 . ESHELBY, J . D . Proc. Roy. Soc. {London) A193 , 3 9 6 ( 1 9 4 9 ) .

5 0 . M A S O N , W . P. / . Acous. Soc. Amer. 32 , 4 5 8 ( 1 9 6 0 ) .

5 1 . LEIBFRIED, G . Z. Physik, 127, 3 4 4 ( 1 9 5 4 ) .

5 2 . B A K E R , G . S. / . Appl. Phys. 28 , 7 3 4 ( 1 9 5 6 ) .

5 3 . M A S O N , W . P. Seminar Amer. Soc. of Metals, Chapter VIII ( 1 9 6 2 ) . 5 4 . L A Z A N , B . J . Fatigue, Amer. Soc. for Metals, Chapter 2 ( 1 9 5 4 ) ; Int. Conference on Fatigue

in Metals, London ( 1 9 5 6 ) , Paper 2 : Section 2 ( 1 9 5 6 ) ; Murray Lecture, Soc. Exp. Stress Analysis, 1 ( 1 9 5 7 ) .

5 5 . SEEGER, A . and SCHILLER, P. Acta Met. 10, 8 2 ( 1 9 6 2 ) ; BRAILSFORD, A. D . Phys. Rev. 128,

1 0 3 3 ( 1 9 6 2 ) .

5 6 . KOEHLER, J . S. Phys. Rev. 86, 5 2 ( 1 9 5 2 ) ; O R O W A N , E. Dislocations in Metals, 69 , A I M E , N e w York ( 1 9 5 4 ) .

5 7 . B O R D O N I , P. / . Acous. Soc. Amer. 26, 4 9 5 ( 1 9 5 4 ) .

5 8 . NIBLETT, D . H . and W I L K S , J . H . Phil. Mag. 1, Eighth Series, 4 1 5 ( 1 9 5 6 ) .

5 9 . SEEGER, A. , D O N T H , H . and PFAFF, F. Disc. Faraday Soc. 23 , 1 9 ( 1 9 5 7 ) .

D I S C U S S I O N

GEORGE H E R M A N N (The Technological Institute, Northwestern University, Evanston, Illinois): Mechanicians concerned with the behavior o f structural elements at elevated temperatures should be most thankful to Dr . Mason , who has brought within the body o f a single paper valuable recent information on temperature dependence of elastic and anelastic properties in solids. A s seen from the bibliography to this paper, this information, in its original form, is scattered widely in the periodical literature and is of difficult access to someone whose main interest lies in incorporating the temperature dependence of certain material properties in studies of structural response, such as stress distribution, stability of equilibrium, etc.

In connection with thermoelastic effects, mention might be made of some recent and as yet unpublished work, completed a short time ago, by Dr . James Tasi (presently with Martin-Marietta Corporation, Denver, Colorado), and in part by the present writer, on thermoelastic dissipation in high frequency vibrations of crystal plates.

In the course of this work an integral formula for dissipation was derived and applied first to an infinite isotropic plate, focusing attention on the three lowest, real, symmetric branches of the spectrum. It was shown that low frequency extensional vibrations of plates can be adiabatic or isothermal, depending upon whether adiabatic or isothermal boundary conditions, respectively, are specified on the faces of the plate.

Using these exact solutions of thermoelastic equations as a guide, Mindlin's two-dimensional equations of coupled thickness-shear, flexural and extensional mot ion o f crystal plates were extended to include thermoelastic effects.

Numerical calculations have been carried out for a finite AT-cut quartz plate with traction-free edges and 0 . 1 cm thick, vibrating freely in vacuo at a temperature of 2 9 8 . 2 ° K , with a non-conducting film coating the plate faces. Thus, heat transfer across the plate faces by conduction or convection was prevented, while heat transfer by radiation turned out to be so small in the range considered, such as to justify the assumption of adiabatic boundary conditions.

The results indicate that the dissipation measure Qr1 is of the order of 1 0 ~ 8 near the thickness-shear frequency and that the order of mode , mode-coupling at a free edge and dimensional ratios have important effects on dissipation near the thickness-shear frequency. In the funda-mental thickness-shear m o d e the dissipation diminishes with increase in the order o f the flexural mode , which is coupled to the thickness-shear m o d e through the edge condition. The dissipation also diminishes as the length-thickness ratio of the plate approaches that for which the ampli-tude ratio, of thickness-shear to flexure, is a maximum.

A s indicated by Dr . Mason , the thermoelastic effect is more important in metals, such as zinc. It might be worthwhile to explore this effect for a wider range of frequencies than has been done so far.

THERMO-MECHANICAL BEHAVIOR OF CERAMICS

JOSEPH A . PASK and STEPHEN M . COPLEY

Ceramic Laboratories, Department of Mineral Technology, and Inorganic Materials Research Divis ion, Lawrence Radiation Laboratory, University of California,

Berkeley, California


CERAMIC, or inorganic nonmetallic, materials are of potential interest for appli-cations at elevated temperatures and under corrosive conditions. The usually available ceramic materials, however, exhibit low strength and brittleness. Theoretically, they should be strong; there are also indications that certain

F I G . 1. Electron photomicrograph of insulator porce la in . 3 5

crystal structures may possess capabilities of plastic deformation. Most of the difficulties associated with normally available ceramic bodies are due to their extremely complex microstructures as shown in Fig. 1. This complexity results from the presence of a number of crystalline phases and generally a glass or

57

58 J O S E P H A . P A S K A N D S T E P H E N M. C O P L E Y

vitreous phase, and often results in an internally stressed condition. Most bodies also contain some pores. Recent efforts toward the improvement of capabilities of ceramic materials have led to the development of "oxide bodies" approaching theoretical density with considerably simpler microstructures, as shown in Fig. 2, and correspondingly with improved properties.

Generally, studies on strength and ductility are related in the sense that dis-locations are involved in both cases. As a first requirement, high strengths are achieved when dislocations, if present, are immobile, and ductility is obtained when dislocations can move without causing fracture. In addition, high strength is dependent upon the absence of other defects, such as pores and microcracks,

F I G . 2. Microstructure of a transparent magnesia body.

which can act as stress raisers resulting in the nucleation and propagation of fracture. This paper will be concerned with the problems of developing ductility in polycrystalline single-phase ceramics as presently understood and thus con-stitutes a progress report or a state-of-the-art analysis of the subject.

In a normal polycrystalline piece the grains or crystals are randomly oriented. If such a piece is to be ductile under an applied stress, each grain must be able to deform to any arbitrary shape in order to maintain its grain boundary con-tacts without excessive elastic strain. If the deformation of each grain occurred in an isotropic or uniform manner, the problems would be considerably simpli-fied or essentially not exist. Plastic deformation, however, occurs along certain planes in certain directions, referred to as slip systems, and a grain thus exhibits slip or plastic anisotropy. Taylor 1 , based on a previous analysis by von Mises2, postulated that the grains in a polycrystalline piece must have at least five inde-

T H E R M O - M E C H A N I C A L B E H A V I O R OF C E R A M I C S 59

pendent slip systems in order to exhibit ductility. Even with the fulfillment of this requirement additional problems exist because of slip nonuniformity and, in many cases, the occurrence of cleavage fracture after some initial dislocation movement.

Therefore, specific problems of interest in understanding the behavior of polycrystalline ceramics are the nature of slip anisotropy and nonuniformity, and cleavage fracture. In this report it is thus necessary to discuss first single-

F I G . 3 . A model illustrating the "rock-salt" crystal structure and showing displacement along a { 1 1 0 } plane.

crystal behavior both at room and elevated temperatures. Most studies of this nature have been done on compounds with the "rock-salt" crystal structure, particularly LiF and MgO. This paper is therefore restricted to these materials and covers: (1) the crystal structure and dislocations, (2) plastic behavior of single crystals, and (3) plastic behavior of polycrystals.

2 . C R Y S T A L G E O M E T R Y

2 . 1 . "Rock-Salt" Crystal Structure

On the assumption that the ions behave as spheres, the space-filling ions in this structure are the anions which are close-packed according to a face-centered cubic arrangement similar to that for many metals, as seen in Fig. 3. Such a


packing of spheres has two types of holes: octahedral (six neighbors) and tetra-hedral (four neighbors). There are as many octahedral holes as spheres, and twice as many tetrahedral holes. The cations in this structure are of a size that fit into the octahedral holes. This size must be such that the cation-anion ratio falls between 0.732 and 0.414. Because of equal valences these compounds have equal numbers of cations and anions, and thus all the octahedral holes are filled. The tetrahedral holes remain unfilled. The resulting structure consists of two

{l I 0 } < ! I 0 > GLIDE

{lOO} < I I 0 > GLIDE

F I G . 4. Translation gliding in the same direction, but on two different planes in crystals with the "rock-salt" structure. 2 8

interpenetrating f.c.c. lattices, one of cations and the other anions, as schematically represented in Fig. 4.

In a highly ionic compound, the ions can be treated as if they were rigid spheres. The lattice energy for the compound is then the resultant of all the attractive and repulsive forces between the ions. Distortion or displacement of ions, particularly in certain directions, would be resisted because of the resulting strong repulsive forces. However, if the ions are polarizable or the electron clouds can be easily distorted, these repulsive forces are modified. The nature of the bonding mechanisms may also vary between compounds. Therefore, a number of compounds with the same crystal structure, based on distribution of atoms in space, will exhibit variations in behavior and properties.


2 .2 . Slip Systems

In a given crystal structure glide or slip occurs on certain planes in well-defined directions. For f.c.c. and "rock-salt" structures the direction of glide, as determined by the direction of densest rows of like atoms or ions, is always a <110> direction. For the latter, however, a <100> direction is specifically ruled out because such a movement results in strong electrostatic repulsion between cations. The f.c.c. metals have the {111}<110> family of slip systems on the basis of the requirement that a glide plane should be the one of densest packing and widest spacing. In the "rock-salt" structure the same geometry applies; however, the {111} planes are inoperable because glide on them would result in offsets at the faces of the crystal either with an excess or a deficiency of positive charges, and thus produce an electrostatic unbalance or faulting.

Experimentally it has been found that at room temperature the easy slip planes for NaCl, LiF, and MgO are {110}, although movement on {100} planes can also be realized, particularly at elevated temperatures. The preferred slip systems in PbS and PbTe, on the other hand, are {100}<110>; whereas AgCl, KI, and KC1 show approximately equal tendency for glide on both slip systems. These differences in behavior have been attributed to the varying polarizability of the ions. 3

Geometrically, both are possible, but the {100}<110> slip systems are more favorable than the {110}<110> slip systems on the basis that the former requires the least distortion of an anion in moving to its next position. However, for the more ionic crystals like NaCl and MgO, movement on {100} results in stronger electrostatic repulsive forces because in the process the cations come closer and are less screened from each other, as seen in Fig. 4. In the case of the less ionic compounds like PbS and PbTe, the polarizability of the ions results in a reduction of this repulsive force.

2 . 3 . Resolved Shear Stresses

If a uniaxial stress <J is applied to a single crystal, the resolved shear stress or acting on a specified slip plane in a specified slip direction is given by

( i )

where a$, and y% are direction cosines of the stress axis, slip plane normal and slip direction, respectively. For convenience, the direction cosines can be referred to basis vectors pointing in the [100], [010], and [001] crystallographic directions.

In LiF and MgO easy glide occurs on the {110}<110> family of slip systems. In this family there are three orthogonal sets of slip planes, as shown in Fig. 5, in each of which the Burgers vectors are also at right angles. The resolved shear stress on each plane of an orthogonal pair can be shown to be equal for any uniaxial stress. Because of the random orientation of grains in a polycrystalline piece, it is desirable to know the shear stresses developed on all potential glide planes for any orientation of a grain. A uniaxial stress diagram for a given slip


system is constructed by setting the resolved shear stress in equation (1) equal to the critical resolved shear stress required to operate the slip system and then plotting the absolute value of the applied uniaxial stress as a function of stress axis orientation. 4 Stress axes may be represented by their points of emergence through the (111) plane, the other reference point for the axes being the origin. Stress axes corresponding to the same applied uniaxial stress may then be represented as a curve lying in the (111) plane.

Each of the three orthogonal sets of {110}<110> slip systems will in general be operated by a different uniaxial stress for a given stress axis orientation. For this orientation, these sets can be designated as I, II, and III in the order of increasing applied uniaxial stress to operate them. This procedure can be repeated for all stress axis orientations. Uniaxial stress diagrams for sets I, II , and III slip systems, respectively, can then be constructed, as shown in Figs. 6, 7,

[001] [100]

F I G . 6 . A uniaxial stress diagram for set I , { 1 1 0 } < 1 1 0 > slip systems. 4

F I G . 5 . A diagram of the six slip planes of the { 1 1 0 } < 1 1 0 > slip systems.


[ 0 1 0 ]

F I G . 8. A uniaxial stress diagram for set I I I , {110}<110> slip systems. 4

TABLE 1. T H E A P P L I E D U N I A X I A L STRESSES CORRESPONDING TO THE

LETTERED C U R V E S IN F I G S . 6, 7 A N D 8

Curve a (25°C) x 1 0 - 7 k g / m 2 a (600°C) X l O 7 k g / m 2

or (1200°C) x 1 0 - 7 k g / m 2

A 1.20 0 . 5 0 0 . 2 8 B 1 .40 0 . 5 8 0 . 3 3 C 2 . 8 0 1 .17 0 . 6 6 D 4 . 2 0 1.75 0 . 9 8 E 5 . 6 0 2 . 3 4 1.31 F 00 00 00


and 8. Table 1 lists the applied uniaxial stresses corresponding to the lettered curves in these figures at 25°, 600°, and 1200°C for MgO.

In Figs. 6, 7 and 8, three orientations of particular interest. The first one is the <100> orientation and is the most extensively used in single crystal studies. It is easily obtained because of the strong cleavage along {100} planes which is taken advantage of in the preparation of specimens. A <100> stress axis produces the maximum resolved shear stress on two orthogonal sets of slip systems, each set being symmetrical with the stress axis. Movement on either or both sets of these slip systems results in deformation without any lateral translation or rota-tion of the ends of the crystal. A <110> stress axis again produces a shear stress on two orthogonal sets of slip systems, but its magnitude is only half of that obtained with a <100> orientation. Complete symmetry of the resultant shear strain is present around the stress axis when both sets of orthogonal slip systems move equally, and thus deformation of the crystal can again occur without lateral translation or rotation of the ends. A < 111 > stress axis orientation results in no plastic deformation due to glide on the {110}<110> slip systems because the resolved shear stress on them is zero.

The <111> direction, however, is of particular interest because it is favorably oriented for developing a shear stress, which is equal to 0.472 a, on three of the six {100}<110> slip systems; the resolved shear stress acting on the remaining three slip systems vanishes. Also, in some of the other random orientations of single crystals a higher resolved shear stress may result on one of these slip systems instead of a {110}<110> slip system, particularly at elevated temperatures. Furthermore, because this family is normally favored in some of the "rock-salt" type of crystals, it is of interest to determine a uniaxial stress diagram for the most easily operated of the {100}<110> slip systems, as shown in Fig. 9. Table 2 lists the applied uniaxial stresses corresponding to the lettered curves at 600° and 1200°C for MgO. The critical resolved shear stress required to operate these slip systems was taken to be the resolved shear stress required to yield a single crystal of MgO in compression with a < 111 > loading axis.

All other orientations of a single crystal relative to a uniaxial stress axis result in some degree of nonsymmetry relative to both families of slip systems. The deformation becomes complex under these conditions.

These discussions of shear stresses have considered the deformation of single crystals with the only requirement that no translation or rotation of their ends should occur. In polycrystalline specimens the additional requirement exists that a grain should be able to assume any arbitrary shape imposed upon it by interaction with its neighbors. As mentioned earlier, Taylor has indicated that the crystals must have at least five independent slip systems to fulfill this re-quirement. Kelly 5 has shown that the {110}<110> family of slip systems has only two independent slip systems and that the {100}<110> family has three indepen-dent slip systems. Thus, it is evident that, under any circumstances, a knowledge of the plastic behavior of both families of slip systems in single crystals under various conditions becomes a prerequisite to the subsequent understanding and analysis of the mechanical behavior of a single-phase polycrystalline specimen.


TABLE 2 . STRESSES CORRESPONDING TO LETTERED C U R V E S I N F I G . 9

AT 6 0 0 ° A N D 1 2 0 0 ° C

Curve OR ( 6 0 0 ° Q x 1 0 - 7 k g / m 2 <r(1200°C) x 1 0 - 7 k g / m 2

A 00 00 B 1 2 . 3 0 2 . 8 0

C 7 . 0 0 1 . 5 9

D 5 . 8 5 1 . 3 3

E 4 . 7 2 1 . 0 8 F 4 . 4 0 1 . 0 0

G 4 . 1 6 0 . 9 4

H 4 . 4 0 1 . 0 0

3 . T H E P L A S T I C D E F O R M A T I O N O F " R O C K - S A L T " T Y P E

S I N G L E C R Y S T A L S

As mentioned, an understanding of the plastic deformation of a single-phase polycrystalline specimen is dependent on an understanding of the plastic be-havior of single crystals. This understanding will be developed in terms of the observed behavior of individual dislocations.

Several experimental techniques have been developed for observing the behavior of individual dislocations in crystalline materials. One is chemical etching. If a crystal is placed in a suitable etchant, etch pits are formed which have been demonstrated to correspond to the sites where dislocations meet the crystal surface. 6 An etched crystal surface thus gives a two-dimensional picture of the distribution of dislocations in the crystal. Another technique is that of transmission electron microscopy. Dislocations in thin foils can be observed using the electron microscope because the strained lattice around them reduces

4


the local transmittance of the foil. Dislocations in MgO have been successfully studied using this technique. 7 ' 8 A third technique for observing dislocations is to form a visible precipitate on them. 9 Dislocations revealed in this manner are said to be decorated. By application of these techniques, considerable infor-mation has been obtained about the origin of slip, dislocation velocities, and dislocation multiplication in "rock-salt" type crystals.

3 . 1 . The Origin of Slip

It has been d e m o n s t r a t e d 6 ' 1 0 that slip in a single crystal is usually initiated by the expansion of dislocation half loops which are present at its surface.

F I G . 1 0 . Etched rosette of dislocations at the surface of an M g O crystal ( 2 8 0 x ) .

These half loops are formed during the cleaving of the sample and also during its handling. Figure 10, for example, shows an etched rosette of dislocations which was produced by sprinkling the surface of an MgO crystal with 120-grit SiC and then pressing on it lightly. This rosette consists of dislocation half loops lying on the six {110}<110> slip systems (see Fig. 5).

In addition to the dislocations produced by cleaving and handling, "rock-salt" type single crystals have a grown-in dislocation substructure. This sub-structure consists of networks of dislocations forming subgrain boundaries* and randomly spaced individual dislocations. In MgO, for example, the subgrains

* Boundaries between volumes of the crystal which are slightly misoriented with respect to one another.


are 1-10 mm in diameter. Within these subgrains grown-in dislocation densities of about 5 X 10 4 /cm 2 are observed. 1 1

Figure 11, a transmission electron photomicrograph of MgO, illustrates two important characteristics of grown-in dislocations; they contain impurity pre-

F I G . 11. Grown-in dislocation which does not lie on a slip plane. Dot ted lines show approximately the projection of a { 1 1 0 } . 1 1

cipitates and they often do not lie on slip planes. As a result of either or both of these factors, grown-in dislocations are not observed to move during de-formation. There is evidence, however, that they may play an important part in dislocation multiplication and in the formation of elongated edge dislocation pairs which are the dominant feature observed in transmission electron photo-micrographs of slip bands in M g O . 8


3.2. Dislocation Velocities

A dislocation half loop will expand when the shear stress acting on it exceeds some critical value. The velocity at which it expands depends strongly on the resolved shear stress. It can be measured by etching the initial position of the half loop, applying a shear stress of known magnitude for a known length of time and then etching the final position of the half loop. From the distance between the final and the initial positions of the half loop the average velocity can be calculated. Gilman and Johns ton 1 2 have measured the stress dependence

10 V

IO*

I ! i 1 XT -"T— t—r— 1 i l l I 1 ~ r -

Sound velocity

Edge co mponents-^

7 /"^Screw comf >onents

h V V

astic limit

i l i

l!i 1 i i i i M M

Applied sheor stress, kg/mm 2

F I G . 12. Dis location velocities of a typical LiF crystal . 1 4

of dislocation velocities in LiF crystals. Their data for one such crystal are shown in Fig. 12. Johnston has also measured dislocation velocities in MgO and has observed a similar behavior at a higher stress level. 1 3

The stress required to move dislocations at a given velocity in LiF varies from one crystal to another as is shown in Fig. 13. This variation is attributed to differences in impurity content and differences in distribution of the impurities in the crystal as determined by its thermal history. Further discussion of this effect will be given later.

3 .3 . Dislocation Multiplication

If a dislocation half loop at the surface of a "rock-salt" type crystal is ex-panded by a sufficiently large shear stress, a profuse multiplication of dislocations

From Mechanical Behavior of Materials at Elevated Temperatures, edited by John E. D o r n . Copyright 1961. McGraw-Hil l Book Company. U s e d by permission.

10

10

10

10

o

S! 10'

* 1 0

G

| 10'

I 10_

o o o 10":

o

10 ":

10"

10"!

io-< 10"'


takes place resulting in the formation of a slip band. Figure 14, for example, shows a slip band in MgO initiated by half loops at a dislocation rosette.

The manner in which this multiplication takes place is not completely under-stood. There is general agreement, however, that the sources of multiplication are elongated edge dislocation pairs such as those observed in transmission electron photomicrographs of slip bands in MgO (Fig. 15). Pairs having a sep-aration h such that

where G is the shear modulus, b is the Burger's vector, y is Poisson's ratio, and o> is the resolved shear stress, can move past each other as shown in Fig. 16

io~7l 1 '—i—i i i i i I 1 i i i i i i i i I 0 2 I 0 3 | 0 4

RESOLVED SHEAR STRESS, <j/mm 2

F I G . 13. Dis location velocities of different LiF crystals . 2 5

to form spiral dislocation sources. Pairs whose separation is less than the critical value are stable and lie immobile in the slip band in the absence of climb. A number of mechanisms have been suggested for the formation of these edge dislocation p a i r s . 8 ' 1 5 - 1 8 Details of these mechanisms are discussed in these references.

3.4. The Kinetics of Single Crystal Yielding

In the case of a compression or tension test, the difference between the true stress and that calculated by dividing the applied force by the undeformed

(2)


cross-section of the sample is small for the first few percent strain. In this range a constant force rate test may be safely regarded as a constant stress rate test. A similar approximation cannot be made, however, for strain rate and dis-placement rate in a constant displacement rate test. In this case the elastic strain of the testing machine must be considered.

100/x F I G . 1 4 . A slip band in M g O initiated by half loops at the dislocation roset te . 1 1

Johns ton 1 9 has developed a simple dislocation model which correctly predicts the yielding behavior of LiF single crystals under a wide variety of experimental conditions. Following his analysis the interaction of a single crystal with a testing machine is shown schematically in Fig. 17. In a constant displacement rate test, the crosshead motion Rt deforms the machine as well as the sample so that

Rt = AS + ALE + ALP (3)

where AS is that part of the crosshead motion going to the elastic strain of the

F I G . 1 6 . A n edge dislocation pair acting as a spiral dislocation source.



apparatus and ALe and ALp are the changes due to the sample's elastic and plastic strains, respectively. The difference between the apparent strain rate e and the plastic strain rate of the sample iP is

(4)

provided that

Crosshead -

Sample

R - displacement rate

R/LQ= e

Ks force constant

LQ

S sample height A - cross section E - Young's modulus

F I G . 1 7 . A schematic representation of a deforming apparatus. The crosshead moves at a constant speed R tending to compress the crystal. Part of the crosshead mot ion goes to the elastic strain of the machine and is represented by an imaginary spring with spring

constant K.19

These symbols are identified in Fig. 17. Using the formula

y — pbv (5)

where y is the shear rate, p is the dislocation density, b is the Burger's vector and v is the average dislocation velocity, the following expression is obtained

(6)

where m is a geometrical factor relating the axial strain rate e to the shear rate y . In the case of LiF, the velocity of dislocations as a function of shear stress has

been measured. The dislocation density as a function of strain can be estimated by counting etch p i t s . 1 2 Thus equation (6) can be graphically integrated.

Johnston's analysis predicts the yield drop observed at room temperature in a constant displacement rate test of LiF and also predicts correctly how the variation of such parameters as displacement rate and machine hardness will


affect its magnitude. The analysis can be used to predict the yield behavior of other crystalline materials if sufficient information about dislocation velocities and densities are available. It can also be easily extended to predict yielding behavior in a constant force tes t . 2 0 In this case

(7)

for small strains.

4 0 0 0

' 3500 !

£ 3 0 0 0

5 0 0

iU-AS ( : U T ;UT, POL SHED

S CLEAN/ ED

k ^CUT

V POLISHE D, ANNEALED

! ^ / * ~

T, ANNEALED

\ x E A V ED, ANNE :ALED i

x INC MCATES F

1

RACTURE

1 0 0 . 5 1.0 1.5 2 .0 2 .5

CALCULATED OUTER FIBER S T R A I N , P E R C E N T

F I G . 1 8 . Stress-strain curves in bending showing the effects of specimen treatment on single-crystal LiF.

3.5 . Stress-Strain Behavior for Slip on the {110}<110> Family of Slip Systems

At room temperature most crystals having the "rock-salt" type of structure cleave readily on {100} planes. Thus samples with {100} faces are easily formed. As already mentioned, cleaving introduces dislocation half loops on the {110}<110> slip systems at the surface. The density and depth of penetration of these half loops are dependent on the hardness of the crystal. When such samples are stressed in a <100> direction or bent around a <100> axis with a {100} neutral plane, slip occurs on these slip systems.

Figure 18 shows stress-strain curves obtained by bending single crystals of LiF, which had been treated in several ways, at room temperature in a constant displacement rate machine . 2 1 The curves indicate that the plastic behavior of otherwise similar single crystals is strongly influenced by their t reatment:

CA

LC

UL

AT

ED

O

UT

ER

F

IBE

R

ST

RE

SS

, P

SI

4§


" 0 0 0 0 0 2 0 0 4 0.06 0 .08 0.10 0.12 0.14 016 018 TRUE STRAIN - € (IN./INJ

F I G . 1 9 . Stress-strain curves at temperatures ranging from — 1 9 6 ° C to 8 2 0 ° C for M g O single crystals loaded in compression with a < 1 0 0 > loading axis at a stress rate of 2 0 lb

i n " 2 s e c - 1 . 2 4

of yield stress with temperature is shown in Fig. 22. The yield stress of single crystals is very sensitive to dislocation mobility. Consequently, the shape of the yield stress vs. temperature curve depends upon the impurity content and the thermal history of the samples tested. Gorum, Luhman, and P a s k 2 3 have shown that impurities increase the yield stress of MgO at room temperature. The amount of this increase was dependent upon thermal history and was minimized by quenching. Hulse and P a s k 2 4 further showed that the effect of quenching on yield stress was decreased with increase of test temperature. Recently, Johns ton 2 5 has obtained yield stress vs. temperature curves for slow-cooled and quenched samples of LiF containing 75 ppm of Mg and also for "pu re" LiF samples containing 3 ppm Mg. His data are shown in Fig. 23 and clearly indicate a hardening at moderate temperatures by the Mg. This hardening is believed to depend on the distribution of the Mg in the crystal lattice. Quench-ing presumably keeps most of the Mg in solution. The exact mechanism of

whether they were cut, cut and polished, or cleaved. It was suggested that the cutting of the specimen resulted in a severely work-hardened surface which acted as a barrier to the motion of dislocations. All of these specimens after annealing at 810°C showed similar stress-strain curves with a pronounced yield drop. Stokes and L i 2 2 by careful preparation and annealing of MgO single crystals apparently removed all dislocation sources at the surface as indicated by a tensile stress at fracture of about 138,000 psi without the indication of any plastic flow.

Figures 19, 20, and 21 show stress-strain curves for slow-cooled MgO single crystals compressed in a <100> direction at an effective constant stress rate of 20 lb i n - 2 s e c - 1 for temperatures ranging from —196 to 1500°C. The variation

3 5 , 0 0 0

3 0 , 0 0 0

2 5 , 0 0 0

£ 2 0 , 0 0 0

to t/) £ 1 5 , 0 0 0 h-to

1 0 , 0 0 0

5 , 0 0 0


impurity hardening in these types of crystals is not understood at this t ime; however, considerable research is being done in this area.

Figure 24 shows the dependence of the initial work-hardening rate on tem-perature, which is defined as the slope of the first linear part of the stress-strain curve following the yield. At about 1200°C the initial work-hardening begins to decrease rapidly. Vacancies would be expected to become quite mobile in this

I 6 h

2h

0.02 0 .04

True

0 .06

strain

0 . 0 8 0.10

F I G . 2 1 . Stress-strain curves at temperatures ranging from 1 3 0 0 ° C to 1 5 0 0 ° C for M g O single crystals loaded in compression with a < 1 0 0 > loading axis at a stress rate of 2 0 lb

F I G . 2 0 . Stress-strain curves at temperatures ranging from 1 0 0 0 ° C to 1 2 0 0 ° C for M g O single crystals loaded in compression with a < 1 0 0 > loading axis at a stress rate o f 2 0 lb

i n - 2 s e c - 1 . 2 0


6 0 0

Temperature (°C)

F I G . 22. Temperature dependence of the yield stress for M g O single crystals loaded in a <100> direction in compress ion . 2 0

6 r

TEMPERATURE, °K

F I G . 23. Temperature dependence of critical resolved shear stress for LiF. The solid curve for the "pure" crystal corresponds to slowly cooled specimens, and the dashed line to air-

cooled specimens. The two cooling rates were ^ 0 . 0 0 2 ° C / m i n and 50°C/min . 2 5


5 1 ^ 0 0 'o

5 1 2 0 0

£ 1 0 0 0 o> c. S 8 0 0 -o o

f 6 0 0

"1 1 1 r A Hulse and Pask

o Copley and Pask

- 2 0 0 4 0 0 8 0 0

Temperature (°C)

1200 1600

F I G . 2 4 . The temperature dependence of initial work-hardening for M g O single crystals loaded in a < 1 0 0 > direction in compress ion . 2 0

F I G . 2 5 . The total strain at fracture in compression for single crystals of M g O loaded in a < 1 0 0 > direction and polycrystalline M g O . 4


temperature range, because half the melting point for MgO is about 1250°C. Gi lman 2 6 has recently suggested that the elongated edge dislocation pairs ob-served in slip bands of MgO contribute considerably to its work-hardening. Washburn et al.8 have observed that these pairs break up to form a series of small prismatic loops when MgO is heated in the electron microscope by in-creasing the beam current. In view of the possibility of diffusion above 1200°C, it seems likely that the rapid decrease in initial work-hardening observed is due to the breakup or complete annealing out of this kind of defect.

F I G . 26. M g O single crystal deformed in <100> direction in compression to 1 4 . 8 per cent at 1 4 0 0 ° C 2 0

Figure 25 shows the total strain at fracture of MgO single crystals tested at temperatures up to 1200°C at a loading rate of 20 lb i n - 2 s e c - 1 . Two reasons can be given for the large increase in ductility at about 1200°C. First, the prob-ability of nucleating a crack decreases at temperatures approaching half the melting point because of the possibility of stress-induced climb at dislocation pileups. Second, at high temperatures dislocations move rapidly at low stresses. Stress concentrations at crack tips can then relax plastically, and thus the cracks will propagate less easily. The amount of ductility observed at high temperatures, however, depends on the loading rate. Increasing the rate of loading decreases the ductility.


20,000

o o

o o

o o

o o

o_ to"

O

in" dsd)

sS3dlS

0 0.01 0 02 0.03 0.04 0.05 006 0.07 0.08 0 TRUE STRAIN - £ (IN/IN)

F I G . 2 7 . The effect of loading rate on the stress^-strain behavior of M g O compressed in the < 1 0 0 > direction at r o o m temperature. 2 4

3.6. Stress-Strain Behavior for Slip on the {100}<110> Family of Slip Systems

When a uniaxial stress is applied to a "rock-salt" type single crystal oriented in a < 111 > direction, the resolved shear stress on the {110}<110> slip systems is zero, and thus if slip occurs, it must take place on other slip systems. Figure 28 shows etched slip bands on two adjoining {100} faces of an MgO single crystal deformed at 650 °C in compression in this orientation. The vertical line in the middle of the photograph marks the edge between the adjoining faces; the bands, therefore, have traveled along {100} planes. Because the shortest slip vector in this plane is a/2 <110> and because electrostatic faulting would result from slip in a <100> direction, it is concluded that these bands must be caused by dislocations having an a/2 <110> Burger's vector lying on a {100} plane.

Figure 29 shows typical stress-strain curves for MgO single crystals tested in compression with a < 111 > loading axis, and Fig. 30 shows the yield stress vs. temperature data. The latter figure also shows data for yield stress vs. tem-perature for specimens with <100> and <110> loading axes. The high shear

At temperatures greater than 1200°C, samples loaded at 20 lb i n - 2 s e c - 1 can often be strained 10 per cent without the occurrence of cleavage. Figure 26 shows a sample deformed to 14.8 per cent strain at 1400°C. The markings which appear on the surface of the sample are the result of the separation of slip alternating on planes at 45 degrees to the surface and at 90 degrees to the surface, thus forming distinct interpenetrating regions.

Figure 27 shows the effect of loading rate on the stress-strain behavior of MgO at room temperature. The variation of the yield stress is qualitatively consistent with that predicted by equation (6). It can be seen that higher loading rates cause greater initial work-hardening and less ductility.


stress required for slip on the {100}<110> slip systems in comparison with flow on {110} glide planes was discussed earlier in the text. It can be seen that the magnitude of the yield stress for flow on the {100} glide planes also depends more strongly on temperature. The ratio of the resolved shear stresses for yielding on the {100} and {110} slip planes was 10: 1 at 350°C and 3 : 1 at 1200°C.

Of still greater interest is the fact that crystals compressed in the <111) direction showed no plastic deformation below about 350°C; they failed below this temperature by brittle shattering. Another point of interest is that the yield stress for this orientation was still decreasing up to 1200°C, whereas those with the <100> orientation showed an essentially constant value above about 950 °C (Fig. 22). Further studies on the temperature effects are being extended in the writers' laboratories.

The high initial rates of work-hardening for the < 111 > orientation, as shown in Fig. 29, may be due to the intersection of dislocations whose Burger's vectors form an angle of 60 degrees with each other. There is evidence that slip band


intersections of this type lead to extensive tangling of dislocations. 2 7 This factor may also be responsible for the smaller amount of strain at fracture for this orientation in comparison with the <100> orientation at a given temperature.

Similar data for the resolved yield stress vs. temperature for the two slip systems for LiF have also been obtained and are shown in Fig. 3 1 . 2 8 The data for the {110}<110> slip system was obtained in bending and for {100}<110> in

I4i 1 1 1 j . 1 1 1 ' 1 1

qI l I I 1 I I t 1 I I I 0 0.01 0.02 0.03 0.04 0.05

TRUE STRAIN, 6

F I G . 2 9 . Stress-strain curves for single crystals of M g O with a < 1 1 1 > loading axis . 4

torsion. Although no plastic flow was observed on {100} planes in torsion below about 225 °C, Johns ton 2 9 has obtained flow at room temperature in compression after reaching a stress level about 15 times that necessary for flow on {110} planes.

4 . P O L Y C R Y S T A L L I N E S P E C I M E N S

4 . 1 . Stress-Strain Data

Stress-strain data have been obtained with temperature as a parameter for polycrystalline MgO and LiF in the writers' laboratory. 4 ' 21> 3 0 The following discussion will be primarily concerned with these materials.

J O S E P H A . P A S K A N D S T E P H E N M. C O P L E Y

400 600 TEMPERATURE , 'C

F I G . 30. Yield stress vs. temperature for single-crystal and polycrystalline M g O . 4

3 0 0 4 0 0 5 0 0 6 0 0 TEMPERATURE. (°K)

F I G . 31. Comparison of stress for {100}<110> slip with that for {110}<110> slip in LiF crystals . 2 8

82


a. Polycrystalline MgO. Polycrystalline samples were sawed from cylinders prepared by cold pressing Nor ton M212 MgO and firing to about 1785°C at a rate of approximately 100°C per hour. The typical impurity content of this batch was reported to be : A I 2 O 3 — 2 . 2 per cent, S 1 O 2 — 1 . 8 per cent, Fe203—0.05 per cent, CaO—0.45 per cent, and loss on ignition—0.14 per cent. The visual grain size was about 40 /x. The average porosity was 9.3 per cent with about 95 per cent of the observed pores being in the grain boundaries.

The samples were loaded at 20 psi i n - 2 s e c - 1 in compression at a number of temperatures. Typical stress-strain curves are shown in Fig. 32. Bulk yield

stresses were taken as the stress at the intersection of the stress-strain curve and a straight line drawn parallel to its elastic portion with a strain off-set of 0.00025. The dependence of these values on temperature is indicated by a curve in Fig. 30, and the total strains at fracture are shown in Fig. 25.

At room temperature, it can be seen that the stress -strain behavior of these polycrystalline specimens was essentially brittle. Fracture usually occurred at a few tenths of a percent plastic strain. Etching and microscopic examination of the deformed specimens showed, however, that slip had occurred in both surface and internal grains. Figure 33 shows the etched surface grains of one such specimen. Because the etch only revealed dislocation intersections with {100} faces, many of the grains do not show any slip bands. In those grains showing slip bands, sometimes three or perhaps four slip systems appear to be acting. On the basis of the amount of slip observed, it seems likely that slip

84 J O S E P H A . P A S K A N D S T E P H E N M . C O P L E Y

F I G . 3 3 . Etched slip bands in polycrystalline M g O deformed at room temperature ( 1 6 0 x ) . 4

occurred in most grains to some extent. Thus, it is concluded that the brittle-ness of polycrystalline MgO at room temperature cannot be attributed just to a lack of dislocations in the individual grains or to a failure of those present to move and multiply.

Figure 34 shows the etched slip band structure of a polycrystalline specimen deformed at 1240°C. The etching procedure used here revealed dislocation

F I G . 3 4 . Etched slip bands in polycrystalline M g O deformed at 1 2 4 0 ° C ( 3 2 0 x ) . 4


intersections with all faces. At this temperature it can be seen that slip bands are curved and forked. Similar slip behavior has been observed in polycrystal-line AgCl at room temperature and in NaCl at elevated temperature and has been called "wavy s l ip" . 3 1 Figure 35 shows "wavy slip" in AgCl. Carnahan et al?1 have attributed "wavy slip" to additional dislocation movement on the {100}<110> slip systems and have pointed out a correlation between the appear-ance of "wavy slip" and the ductile brittle transitions in these materials. N o similar correlation can yet be made in the case of MgO, however, because at 1250°C only about 2 per cent plastic strain has been observed.

F I G . 35. "Wavy slip" in polycrystalline AgCl deformed at — 72°C. Average grain size is 0 . 1 2 m m . 3 1

It is of further interest to note, as seen in Fig. 30, the similarity between the temperature dependence of the yield stress for the polycrystalline specimens and for single crystals with a <111> loading axis above 600 °C. This similarity strongly suggests that in this temperature range the yielding of fine-grained specimens such as these is determined by slip beginning on the {100}<110> slip systems.

This laboratory is now in the process of extending the MgO data to tempera-tures of about 1600°C. Efforts are also being made to obtain MgO polycrystal-line specimens of theoretical density.

b . Polycrystalline LiF. Theoretically dense polycrystalline samples were pre-pared by crystallization from a melt of purified LiF, which contained 0.002 per cent Mg, less than 0.001 per cent Al, and less than 0.0005 per cent Ca. The minimum grain size obtained by this method was about 3 mm. Specimens,


approximately 2 in. by 0.350 in. by 0 .1 in., were cut from these ingots and chemically polished.

Stress-strain curves were obtained in four-point bending using a rig with a support span of f in. and a load span of £ in. and a strain rate of about 0.016 in. /min. 2 1 ' 3 0 The data obtained at temperatures up to 500°C are shown in Fig. 36. It is of interest to note that for these large-grain specimens the yield stress at room temperature was essentially the same as that for flow on the {110}<110> slip systems.

The strain in the outer fiber ranged from about 0 .2 per cent at room tem-perature to about 3 per cent at 350 °C, and the curves exhibited a high rate of

I 1 1 1 1 n

Crosshead movement x specimen thickness ( i n 2 )

F I G . 36. Stress-strain curves for large-grained polycrystalline LiF deformed in bending at r o o m temperature. 3 0

work-hardening. At 400° and 500°C, the outer fiber strain was about 7 per cent, and the stress-strain curves showed a sharp yield point and less work-hardening. The change in the character of the yield, together with the increased ductility which occur between 350°C and 400°C, must almost certainly be attributed to the greater activity of the {100}<110> family of slip systems. Gi lman ' s 2 8 measure-ments, obtained in torsion tests, show that the resolved shear stress on this system in this temperature range approximates that for flow on the {110}<110> slip systems. Wavy slip is also observed at this temperature.

The fractures observed at temperatures up to 250 °C were principally of the cleavage type but initiated by a region of grain boundary fracture associated with a three-grain junction. At 500 °C failure appeared to be completely of grain boundary type. At the intermediate temperatures the fracture lengths were divided between grain boundary and cleavage. This behavior is associated with the fact that cleavage becomes less evident at the elevated temperatures. With the tendency to cleave thus reduced, failure occurs at the grain boundaries which are regions of high stress concentration.


4 .2 . Interaction Between Grains in Polycrystalline Specimens

The LiF specimens mentioned earlier were not considered to be the best for mechanical purposes because of their large grain size. However, they were suitable for studying the interactions between grains because of the develop-ment of techniques for determining the orientations of the grains in the surface of a specimen and for revealing slip plane t races . 2 1 The orientations were deter-

F I G . 37. Etched slip bands in tension surface of large-grained LiF deformed at room temperature. Tensile axis vertical . 2 1

mined by stereographic projections. An etch (4 per cent HBF4 in 200 proof ethyl alcohol) was developed that revealed dislocations on all faces of an LiF crystal.

A polycrystal specimen which had been etched after deformation in bending is shown in Fig. 37; the position cf the tension axis is indicated by T's. The major (or the most numerous and heavy long-range) slip bands have considerable


edge component. At the fracture stress of 2280 psi for this specimen, the resolved shear stress for all the major slip systems except two ranged from 590 to 1090 psi. The slip system, in the upper right grain, which slants steeply down to the right, had a resolved shear stress of only 230 psi resulting from the applied load; and in the upper left grain the slip system sloping to the right had a resolved shear stress of 430 psi. The resolved-yield stress for single crystals was found to be about 600 psi. These figures indicate that the additional shear stress for flow on the latter two systems must have been provided by inter-action between the grains.

j l .O mm

F I G . 38. A magnified and extended view of grains in the lower right-hand corner of Fig. 3 7 . 2 1

An expanded and extended view of the lower right portion of Fig. 37 is shown in Fig. 38. The slip systems operating in grain H-3 at the boundary between H-3 and H-4 and the nearly vertical and horizontal slip in H-5 which originated at the boundary between H-4 and H-5 were accommodating the deformation in H-4 which appears to be the dominant grain in the specimen. The resolved shear stress from the applied load on these two systems was only 370 psi. The fine horizontal slip at the top and bottom and the light vertical slip in H-4 had resolved shear stresses from the applied load of only 210 psi. These are considerably below the critical yield stress. The additional stress for opera-


tion of these systems must again have developed from the interaction of de-forming grains in the aggregate.

A photomicrograph of a three-grain junction in the tension surface of another specimen is shown in Fig. 39. The degree of grain boundary migration during annealing at 800 °C for 2 hr in this type of specimen is indicated by the position of the three-grain junction before annealing in grain J -3 ; the original junction is visible because of the slight selectivity of the polishing solution for different crystal orientations, which was applied prior to annealing. Local deformation

F I G . 39. A three-grain junction in the tension surface of a polycrystalline LiF specimen deformed at room temperature. Tension axis horizontal . 2 1

in the form of fine slip can be seen spreading from the junction into grains J-l and J-3. Of particular interest is the local deformation in grain J-2 at its boundary with J-3. A magnified view of a part of this boundary is shown in Fig. 40. On the stereographic projection for this crystal the fine boundary slip corresponds to the (010) slip plane. As shown in the figure the nearest {110} type plane that might correspond to the observed trace is the (T01) plane. However, this plane is 15 degrees away from the measured position of the slip trace and was outside the range of error in these measurements. The slip direction was probably [101] on the basis that this direction would result in the traces being nearly edge-type


which would be expected to accommodate the deformation in the adjoining grain J-3 whose diagonal slip traces were edge bands.

The resolved shear stress on the fine slip system in grain J-2 due to the applied load was calculated to be 880 psi at fracture. Johns ton 2 9 found that the stress necessary for {100}<110> type slip was about 9000 psi. It is thought that this concentrated stress at the boundary resulted from large-scale deformation of the adjacent grain J-3. Although undoubtedly unusual because of its uniqueness, it is of great interest that such a high stress was realized without the initiation of fracture or separation at the grain boundary. In some less favorable orienta-

F I G . 4 0 . A magnified view of the local deformation in grain J-2 at its boundary with J-3 (Fig. 3 9 ) . 2 1

tion of the two grains, the high stress concentration undoubtedly would have resulted in the nucleation and propagation of a crack at the grain boundary.

4 . 3 . Problems in Polycrystalline Behavior

As previously mentioned, the three factors of (1) cleavage fracture, (2) slip nonuniformity, and (3) slip anisotropy appear to be responsible for the observed brittle behavior of polycrystalline MgO and LiF. These factors will now be discussed in greater detail.

When dislocations moving on a slip plane encounter a barrier, a dislocation


pileup results. Zener 3 2 was the first to suggest that the concentrated stresses which develop at the head of such a pileup might cause the leading dislocations to coalesce forming a crack nucleus. There is much evidence that the fracture of crystalline materials in which dislocation motion occurs can be initiated in this m a n n e r . 3 3 ' 3 4

Grain boundaries are effective barriers to dislocation motion. In polycrystal-line materials, the formation of crack nuclei where dislocations encounter grain boundaries would seem to be an inherent feature associated with plastic de-formation. In order to cause fracture, however, crack nuclei must grow to a sufficient size to be able to propagate in the stress field resulting from the applied load. Thus, it is important to understand the factors which influence the growth of crack nuclei.

In the case of polycrystalline MgO and LiF at lower temperatures, conditions are extremely favorable for the growth of crack nuclei. These materials exhibit a strong cleavage because there are certain crystallographic planes of low co-hesive energy (in this case {100} planes) along which cracks can propagate easily. Also, relatively large stresses are required to move dislocations at high velocities in LiF and MgO at these temperatures. Thus, under these conditions crack propagation is not excessively hindered by plastic stress relaxation.

Because the growth of a crack nucleus requires a local tension stress, the normal stresses present in a deforming body are very important. In a tension or bend specimen conditions are obviously favorable for the growth of crack nuclei while in a compression or a hydrostatically loaded specimen, such growth is suppressed. The stress distribution in a grain of a polycrystalline specimen depends upon the load applied to the specimen. In addition, however, it depends on the capacity of the grain to change its shape to a shape defined by the de-formed shapes of its neighboring grains.

Slip in MgO and LiF tends to be nonuniform. Etch s t u d i e s 6 ' 1 0 of {110}<110> slip in these materials at room temperature show that prior to yielding slip bands are usually few and randomly spaced. Further straining corresponds not to the appearance of new slip bands as much as to the widening of already existing ones.

The localization of slip into wide bands causes tensile stress concentrations to develop in regions where slip bands intersect the grain boundary. In these regions, conditions are thus extremely favorable for the growth of crack nuclei.

Also, a grain cannot change shape as required if the slip in it is nonuniform. For such a change to be possible slip must occur on closely spaced planes, the amount of shear varying continuously from one plane to the next. Conse-quently, with nonuniform slip, large elastic strains must necessarily occur in the grains of a deformed polycrystalline specimen. Such strains favor the growth of crack nuclei and help propagate cleavage fracture.

In order for a grain in a polycrystalline specimen to be able to deform to an arbitrary shape by slip alone, Taylor 1 has pointed out that it must have five independent slip systems. Kelly 5 has shown that of the six {110}<110> slip systems, only two are independent. Thus, at the lower temperatures where slip


is restricted to the {110}<110> slip systems, the plastic deformation of poly-crystalline MgO and LiF is not possible because of slip anisotropy. Any plastic strain must be accompanied by a large elastic strain thus enhancing the prob-ability of cleavage fracture.

As the temperature is increased, conditions become less favorable for cleavage fracture to occur. At temperatures approaching half the melting point, self-diffusion becomes rapid and as a result stress-induced climb can occur. Thus, it becomes more difficult to nucleate cracks. Also, dislocations become increas-ingly mobile at higher temperatures and thus plastic stress relaxation becomes more important.

Kelly 5 has shown that the {110}<110> and the {100}<110> slip systems when taken together provide five independent slip systems thus satisfying Taylor's criterion. There is evidence that {100}<110> slip can and does contribute to deformation of polycrystalline Mgo above 600°C. A comparison of Figs. 9 and 30 shows that for MgO at 600° and 1200°C slip on the {100}<110> slip systems is likely over a range of loading axis orientations but that the required uniaxial stresses for slip are higher than those for {110}<110>. Also, the curved and forked slip bands in MgO shown in Fig. 34 are evidence that dislocation movement is no longer confined to slip on the {110}<110> systems. Finally, it was noted earlier that above 600 °C the yield stress of polycrystalline MgO is similar to that required to yield a single crystal with a <111> loading axis. These observations suggest that the increase in strain observed above 600°C was a result of {100}<110> slip and that such slip is essential for yielding of polycrystalline samples.

5 . S U M M A R Y A N D C O N C L U S I O N S

Slip anisotropy, slip nonuniformity, and ease of crack propagation by cleavage presently appear to be the principal difficulties in the development of ductile polycrystalline ceramic materials. In general, ceramic materials can be divided into three groups on the basis of the nature of their behavior:

1. Completely brittle fracture. In this case either the crystals do not have any dislocations, or, if they are present, the dislocations are completely immo-bile. The objective, then, is to attain maximum strength which can generally be achieved by eliminating all defects or stress raisers, particularly if the crystals possess a cleavage.

2. Slightly ductile behavior but brittle fracture. In this case some ductility is realized, but initial dislocation movement leads to the nucleation and growth of crack nuclei and results in cleavage fracture. This type of behavior is exhibited by most of the "rock-salt" type crystalline materials as discussed in this chapter.

3. Ductile behavior and fracture. In this case the crystals are ductile and do not have any cleavage. Failure thus occurs by grain boundary separation or "tear-ing". AgCl polycrystalline specimens and, in general, "rock-salt" structures above approximately half the melting temperature exhibit this behavior.

Considerable work has been done toward understanding the mechanical behavior of single crystals. As part of this activity procedures have been de-


veloped for predicting the shapes of stress-strain curves which apply fairly well at low temperatures. An intriguing question still unanswered is the possible effect of impurities on reducing the spread in the critical shear stresses on the two slip systems {110}<110> and {100}<110> and on reducing the tendency for existing cleavage fracture. It is known, for instance, that the cleavage of LiF is sensitive to the degree of its purity.

Although a complete understanding of single-crystal behavior has not yet been achieved, it has been adequate to start some preliminary studies toward the understanding of polycrystalline specimens. Problems associated with slip anisotropy have been complicated by slip nonuniformity both in single-crystal and polycrystalline specimens. The resulting stress system does not lend itself easily to mathematical analysis. Besides the indicated effect of impurities on the behavior of single crystals, there is the additional question of their effect on the grain boundaries. Another problem of importance on which little work has been done is that of impact strength. This factor may be critical because preliminary studies have indicated more brittle behavior under such c o n d i t i o n s . 2 0 ' 2 4

A final point of interest is the apparent relationship between the grain size of the polycrystalline piece and the dependence of the critical shear stress on one of the families of slip systems. It appears that flow of the normally easy glide systems is blocked in the small-grain specimens, and measurable macro-flow does not occur until some flow can be realized on the {100}<110> slip systems. In turn, it appears that flow of the easy glide system can occur sufficiently in the large-grain specimens to realize some macro-flow. The checking of this relation-ship requires the capabilities of preparation of specimens with controlled grain sizes and controlled densities. As an example, no one has as yet succeeded in making a specimen of extremely fine-grained, theoretically dense and trans-parent LiF. Similar problems exist for polycrystalline MgO and other materials. One of the deficiencies thus in the development of our understanding of the whole problem of mechanical behavior of polycrystalline ceramic materials is the inability to produce specimens of controlled microstructure.

This chapter thus constitutes a progress report on the understanding of the mechanical behavior of polycrystalline ceramic materials at room and elevated temperatures. Certain major and critical problem areas have become apparent : the effect of "doping" or impurities on slip anisotropy and cleavage fracture, the slip requirements for realization of plastic deformation in polycrystalline specimens, and the capability of processing and producing specimens with controlled and characterized microstructures.

A C K N O W L E D G M E N T S

This paper is based primarily on a background of knowledge and a number of papers developed through research work in the writers' laboratory. The initial studies were made under the financial support of the National Aero-nautics and Space Administration; the later studies, under the auspices of the U.S. Atomic Energy Commission.


R E F E R E N C E S

1. TAYLOR, G. / . Inst. Metals, 62 , 307 (1938). 2. V O N MISES, R. Z. angew. Math. u. Mech. 8, 161 (1928). 3. M U L L E R , H. Am. Mineralogist, 16, 237 (1931). 4. H U L S E , C. O., COPLEY, S. M. and PASK, J. A. / . Am. Ceram. Soc. 46, 317 (1963).

5. GROVES, G.W. and K E L L Y , A. Phil. Mag. 8, 877 (1963).

6. G I L M A N , J. J. and JOHNSTON, W. G. Dislocations and Mechanical Properties of Crystals, p. 116. N e w York: Wiley (1957).

7. W A S H B U R N , J., K E L L Y , A . and WILLIAMSON, G. K . Phil. Mag. 5, 192 (1960).

8. W A S H B U R N , J., GROVES, G. W., K E L L Y , A. and WILLIAMSON, G. K . Phil Mag. 5, 991

(1960). 9. PULLIAM, G. R. / . Am. Ceram. Soc. 46, 202 (1963).

10. STOKES, R., JOHNSTON, T. and Li, C. Third Tech Rept. under Office of Naval Research Project Nonr-2456(00) N R - 0 3 9 - 0 4 .

11. E L K I N G T O N , W. E. , THOMAS, G. and W A S H B U R N , J. / . Am. Ceram. Soc. 46, 307 (1963).

12. JOHNSTON, W. G. and G I L M A N , J. J. / . Appl Phys. 30, 129 (1959). 13. JOHNSTON, W. G. Private communication. 14. G I L M A N , J. J. Mechanical Behavior of Materials at Elevated Temperatures, p. 41 . N e w

York: McGraw-Hil l (1961). 15. JOHNSTON, W. G. and G I L M A N , J. J. / . Appl Phys. 31 , 632 (1960). 16. FOURIE, J. T. and WILSDORF, H. G. F. / . Appl Phys. 3 1 , 2219 (1960). 17. L o w , J. R., Jr. and T U R K A L O , A. M. Acta Met. 10, 215 (1962). 18. TETELMAN, A. S. Acta Met. 10, 813 (1962). 19. JOHNSTON, W. G. / . Appl. Phys. 33 , 2716 (1962). 20. COPLEY, S. M. and P A S K , J. A . T o be published.

21. SCOTT, W. D . and PASK, J. A . / . Am. Ceram. Soc. 46, 284 (1963).

22. STOKES, R. J. and Li, C. H. , Sixteenth Tech. Rept. under Office of Naval Research Project Nonr-2456( 0 0 ) N R - 0 3 9 - 0 4 .

23. G O R U M , A. £ . , L U H M A N , W. J. and PASK J. A. / . Am. Ceram. Soc. 43 , 241 (1960).

24. H U L S E , C. O. and P A S K , J. A . J. Am. Ceram. Soc. 43 , 373 (1960).

25. JOHNSTON, W. G. J. Appl. Phys. 33 , 2050 (1962). 26. G I L M A N , J. J. / . Appl. Phys. 33 , 2703 (1962). 27. HOOVER, D . B. and W A S H B U R N , J. / . Appl. Phys. 33 , 11 (1962). 28. G I L M A N , J. J. Acta Met. 7, 608 (1959). 29. JOHNSTON, W. G. Private communication. 30. B U D W O R T H , D . W. and P A S K , J. A. Trans. Brit. Ceram. Soc. (in press). 31. C A R N A H A N , R. D . , JOHNSTON, T. L. , STOKES, R. J. and Li, C. H. Seventh Tech. Rept.

under Office of Naval Research Project N o n r - 2 4 5 6 ( 0 0 ) N R - 0 3 2 - 4 5 1 . 32. ZENER, C. In Fracturing of Metals, (American Society for Metals, Cleveland, 1948), p. 3. 33. STOKES, R. J., JOHNSTON, T. L. and Li, C. H. Phil. Mag. 3 , 718 (1958).

34. W A S H B U R N , J., G O R U M , A. E. and PARKER, E. R. Trans. AIME 215, 230 (1959).

35. L U N D I N , S. T. "Electron Microscopy of Whiteware Bodies", Transactions of IVth Inter-national Ceramic Congress. Florence, Italy (1954).

PHYSICAL ASPECTS OF CREEP

JOHN E. D O R N * and JIM D . M O T E J

1. I N T R O D U C T I O N

OVER the past 1 0 0 years, engineers, metallurgists and ceramists have expressed ever-increasing technical interest in the high temperature plastic behavior of materials. Most of this period was characterized by extensive creep testing of numerous alloys and some ceramic materials leading to empirical descriptions of the dependency of the creep rate on stress and temperature, as well as qualita-tive concepts concerning the various metallurgical and structural factors that provide creep resistance. In spite of the substantial effort expended on this problem, no simple unique rationale developed over the first 9 0 per cent of this period, either to permit predictions of creep behavior, or to provide the basis for the scientific design of creep-resistant alloys.

Slightly more than ten years ago, however, based principally on the marriage of dislocation theory with the theory for kinetics of reactions, a new atomistic approach to unraveling the complicated nature of creep was initiated. Although much yet remains to be done before a satisfactorily complete picture of creep matures, and despite the continued need for semi-empiricism in the development of creep-resistant materials, it is, nevertheless, now rather clearly established that the continued pursuit of this new method of attack on the problem will eventually lead to the desired knowledge. It is the major purpose of this report to indicate those areas where some progress has been made in the theory of creep and to identify those issues that need additional clarification.

Since creep is the result of thermally activated flow mechanisms, the shear-strain rate, yt, arising exclusively as a result of the operation of the ith mechanism, can be formally established to be

* Miller Professor (1962-1963) of Materials Science, Department of Mineral Technology, and Research Metallurgist of the Lawrence Radiation Laboratory, University of California, Berkeley, California.

t Research Metallurgist of the Lawrence Radiation Laboratory, University of California, Berkeley, California.

95

Whereas T , T, and k have the usual meanings of the applied shear stress, the absolute temperature, and Boltzmann's constant, the remaining terms deserve at least a brief description. The quantity h\ is the enthalpy of activation for the reaction in the forward direction. With rare exceptions, the pressure under con-sideration is so low that the enthalpy of activation very nearly equals the

r i . i )

96 J O H N E . D O R N A N D JIM D . MOTE

activation energy. It does depend, however, on the applied shear stress, which assists the forward reaction ( + ) and hinders the reverse reaction (—), thus giving magnitude and directionality to the strain rate. Since the dependence of the enthalpy of activation on the stress arises as a result of the force acting on the dislocations and thus assists the reaction, it also depends on the sub-structure, st, that is present. Although in certain cases, e.g. those mechanisms that are diffusion controlled, the enthalpy of activation is practically insensitive to the temperature, in other cases it might depend on the temperature in harmony with the effect of temperature on the shear modulus of elasticity. The frequency term, f\, into which we have incorporated the activation entropy, can also depend on r, T, and st. The influence of temperature on this term, however, is always small in contrast to its effect in the exponential Boltzmann term. As in all kinetics of reaction problems, the net forward rate is determined by the total contribution to the forward rate minus the total contribution to the reversed rate, which gives rise to the two terms of equation (1.1) that contribute to the net creep rate due to the zth process.

When processes are sequential, such that / follows k which follows /, etc., the creep rate for the sequence is dictated by the slowest process, which we designate here as the zth process. But when the total creep rate arises from a series of parallel, and independently operative mechanisms, the total creep rate, y — ^yu is given by the sum of the contributions over all such independent processes. When more than one process determines the creep rate, the difficulties of identifying these mechanisms become almost insurmountable. Consequently, most of the existing information about creep has been deduced for those condi-tions of experimentation at which one mechanism predominates. Only those mechanisms for which hi < 50 kT can be stimulated by thermal fluctuations. Consequently only the easier (lower enthalpy) mechanism can take place at low temperatures. As the temperature is increased, such easier mechanisms occur so rapidly that they permit almost instantaneous straining. Continued creep at these intermediate temperatures thus occurs as a result of the operation of more difficult processes having higher values of h\. At the highest tempera-tures only the most difficult processes determine the creep rate, and the easier processes account for the almost instantaneous initial strain. Dislocation theory, however, is not yet sufficiently sophisticated to be able to predict unambiguously which of the various individual rate-controlling mechanisms might be the operative one. Nevertheless it is occasionally possible to identify, from experi-mental evidence in specific examples, several temperature regions over each of which different dislocation mechanisms of creep predominate.

It is the objective of creep theory to determine the functional dependency of ft and hi on the pertinent mechanical and structural variables for each of the series of permissible thermally activatable dislocation reactions. Our analyses here will be limited to discussion of such high temperature creep phenomena that are controlled by diffusion mechanisms. Such diffusion-controlled processes assume significance at temperatures above about one-half of the melting tem-perature. In this range of temperatures the creep rate is almost always determined

P H Y S I C A L A S P E C T S OF C R E E P 97

by one or more of several diffusion-controlled mechanisms. The only known exceptions to this generalization concern the high temperature creep of single crystals of Al , 1 M g 2 and Zn . 3 No light has yet been shed on why these materials gave exceptions to the general rule. Perhaps in the single crystal tests of Al and Mg the types of barriers that are essential for blocking dislocations, and thereby necessitating the operation of the climb processes to provide additional creep, were not formed, whereas they are formed in polycrystals which are subjected to polyslip. The possible validity of this explanation is supported by the fact that the high temperature creep of polycrystalline Al is diffusion con-trolled. Prismatic slip in Zn may be controlled by the Peierls mechanism.

The various diffusion-controlled mechanisms for high temperature creep include:

1. Stress-directed diffusion of vacancies in polycrystals 2. Thermally activated motion of jogged screw dislocations 3. Climb of edge dislocations 4. Solute atom diffusional processes and viscous drag 5. Structural instability of dispersed phases

The importance of high temperature creep is well documented in a number of outstanding reviews, symposia, and monographs on this s u b j e c t . 4 - 1 0 It would indeed be impossible, in terms of the limitations of time and space, to review everything that has been done and said about creep here. Rather we plan to concentrate our attention on the mechanisms for creep; we will briefly review some of the significant evidence and illustrate known mechanisms with some of the more recent data ; we will also point out where creep theory is weak and, in some instances, how it might be improved.

2. N A B A R R O C R E E P

Creep by the stress-directed diffusion of vacancies, formulated originally by N a b a r r o 1 1 , is one of the best understood, theoretically, of the various mechanisms for high temperature creep. Although Herr ing 1 2 subsequently presented a more sophisticated and detailed analysis of this process, we shall review the original Nabarro approach here, because it provides a more vivid physical picture of the mechanism.

This creep mechanism applies only to polycrystalline aggregates at very high temperatures and differs from the other high temperature creep processes that crystalline materials might undertake insofar as it does not depend on the motion of dislocations. Consequently it can occur in polycrystals when the grains contain no dislocations or where all dislocation motion is blocked either by high Peierls stresses or by some other very effective locking mechanism. If a polycrystalline aggregate cannot creep by any other mechanism, it will, under appropriate conditions of stress and temperature, creep by means of stress-directed diffusion of vacancies.

In reviewing Nabarro creep, we will first consider a polycrystalline aggregate of one phase and one component. The grain boundaries of the aggregate are

5

98 J O H N E. D O R N A N D JIM D . MOTE

regions of discontinuity in the structure and are therefore excellent regions which serve as sources and sinks for vacancies. When a tensile stress, p, is applied normal to a boundary, it can assist in the formation of a vacancy since it will do work pQv per vacancy produced, where Qv, the volume of a vacancy, is about equal to the atomic volume. In the absence of a stress, the probability of finding a vacancy at a given site is

B

I ^ J

, rr c D

p F I G . 2 . 1 . Vacancy flow in a grain.

Consider now a cubic grain of dimensions d, as shown in Fig. 2 . 1 , so stressed to give a shear stress, /?, at 45° to the specimen axis. Whereas the probability of finding a vacancy along AB and CD is proportional to exp(pQf/kT), that of finding a vacancy along AD and BD is proportional to exp — (pQfjkT). Due to the concentration gradient, vacancies are presumed to flow as shown by the arrows in the figure. This flow of vacancies will be matched by an equal and opposite flow of atoms resulting in a longitudinal extension and lateral contraction of the grain with time under stress.

It is not immediately apparent, however, that the vacancies should enter the volume of the grain and undertake volume diffusion in the grain as was postulat-ed by both Nabarro and Herring. This question is emphasized by the fact that both the free energy of formation and the free energy of activation of motion

(2.1)

where nv is the number of vacancies among ns lattice sites and gf is the free energy of formation of a vacancy. Consequently, assuming equilibrium can be established, the probability of finding a vacancy in the lattice just below a grain boundary subjected to a tensile stress, p9 is

(2.2)

a quantity which is somewhat greater than the equilibrium value in the absence of a stress.


of a vacancy in the grain boundary is appreciably less than that in the grain itself. Obviously the total creep rate will depend on the sums of the contribu-tions from the volume and boundary diffusion processes. For the present, however, we will assume that the volume diffusion mechanism predominates and we will describe in some detail later the thus far neglected mechanism involving grain boundary diffusion.

Three steps are involved in the original stress-directed diffusion model for creep, namely, the formation of a vacancy in the grain volume near the grain boundary under a positive stress, its migration to the boundary that is under a negative stress and its annihilation at that boundary. Both Nabarro and Herring assumed that the generation and annihilation steps were so rapid in contrast to the diffusional step that the rate of creep was dictated by the diffusional flux. In view of the disorder in the grain boundary, this assumption appears to be quite good and it will be accepted here, particularly because it enjoys some experimental verification.

Under these assumptions, the difference in concentration of vacancies between the grain sides AB and AC becomes about

(2.4)

If h is taken as the height of an atom, the area of an atom is Q/h. The flux of atoms across an atomic area at d/4 from the corner along AC and at d/4 from the corner along AB is

(2.3)

where Q is the atomic volume and a is to be taken as slightly less than unity. The value a thus corrects somewhat for the fact that under diffusional conditions the concentration of the vacancies at the grain boundaries differs slightly from the equilibrium value. Equation (2.3), however, can only represent an ap-proximation to the facts since vacancies will have a shorter diffusion path near the corners of the grain than in the centers of the edges. Thus, under steady state conditions, the stress is somewhat relaxed near the corners and increases as the center of the boundary is approached. Consequently, p varies over the surface of the grain and has its maximum value at the center of the boundary. At d/4 from the corner of a grain, we assume that p = pr where T is the macro-scopic shear stress and j8 has a value of near unity. For this point the diffusion path is approximately n/2(d/4) and therefore the concentration gradient for this path is about

or, since firQ/kT < 1 for the usual values of the applied stress,

(2.5)


where Dv is the diffusivity of a vacancy. Each such atomic transfer results in increasing AC by h and decreasing AB by h. The shear strain is therefore y = 2h/d per atom transferred. Consequently the creep rate approximates

(2.6)

where D, the self-diffusivity, is equal to Dve~^kT. The more sophisticated analysis of Herring gives approximately the same answer. Herring also discussed the effect of various grain shapes on the creep rate.

>-

_i m < 2 0 . 5 o oc CL

QT

- VOLUME -DIFFUSION

- GRAIN BOUNDARY DIFFUSION -

— I d = I O O O b i g m b / k T = 2 0

H d = I O O O b - , g m b / k T = 4 0

HI d = 100 b-, g m b / k T = 2 0

0 I i i l i I I i i 1 1 1 1 1

0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

<9f + 9 m - < W / k T

F I G . 2 . 2 . Probability that a vacancy will remain in the grain boundary during diffusion.

The significant characteristics of creep by means of the stress-directed diffusion of vacancies are as follows:

1. Absence of a primary stage or any pronounced transients 2. An activation energy that equals that for volume self-diffusion 3. A creep rate that increases linearly with the stress 4. A creep rate that varies inversely with the square of the grain diameter 5. A creep rate that can be approximated by equation (2.6) allowing for

small variations of (a/3) from unity because of variations in grain size and irregularities in grain shape.

The Nabarro approach assumes that a vacancy has a high probability of moving out of the boundary so that diffusion occurs primarily through the volume of the grain. We will now explore this assumption. If diffusion takes place along the boundary, say over an average distance d/2, the time, t, for this migration will be given by approximately

(2.7)


where DVb is the vacancy diffusivity in the boundary. We wish now to determine what the probability might be for a vacancy to remain in the boundary for a time as long as t. The probability that a boundary vacancy will leave the boundary in any one vibration is

/?+ = 1/6 e-(9/-9fbVkT e-ffm^T (2.8)

The one-sixth term arises from the fact that the vacancy can leave the boundary only in the + z direction, the x and y directions being selected to be the boundary. In order to escape from the boundary, the vacancy must acquire the additional energy gf — g/& where gf is the free energy of formation of a vacancy in the grain and gfb, which is smaller than gf, is the free energy of formation of a vacancy in the grain boundary. Furthermore, to move the vacancy it must also acquire the free energy for activation of motion of a vacancy across the boundary, namely gm. Vacancies can also enter the boundary from the grain. For the purpose of the present calculation, we assume that there are ample sources and sinks in the form of jogs on the dislocations to maintain the equilibrium number of vacancies in the grain. Consequently the probability that a vacancy will enter the boundary is

To estimate this probability we let

(2.12)

(2.13)

and write equation (2.12) as

(2.14)

We note that the larger d is, and the larger gmb is, the less probable is grain boundary diffusion. But the larger gf + gm — g/& is, the more probable grain

(2.9)

and the net probability that a vacancy will leave the boundary is then given

by

(2.10)

The probability that a vacancy will not move out of the boundary in one vibration is (1 — p), and therefore the probability it will remain in the boundary for n shakes is

p = (l -P)n = (i -pybt (2.11)

where v& is the frequency of vibration of an atom in the boundary. Therefore the probability that grain boundary diffusion will predominate is about


and arrive at the values of P given in Fig. 2 .2 . These data reveal that the prin-cipal variable that discriminates between volume and grain boundary creep is gf + gm — gfb/kT. For the lower melting temperature metals, where this is less

I O 7 I 0 8 I 0 9

F I B E R S T R E S S , D Y N E S / C M 2

F I G . 2 . 3 . Effect of stress on the strain rate of AI2O3. (After Folweiler).

than about 30, volume diffusion will predominate. But where gf + gm — g/b/kT is more than about 30, which might apply to the refractory metals and some ceramic materials, the process should occur predominately by grain boundary diffusion.

The derivation of the equation for stress-directed diffusion of vacancies exclusively by means of grain boundary diffusion follows the same principles already established by Nabarro for volume diffusion. The result is easily shown to be approximately

(2.15)

boundary diffusion becomes. In order to illustrate this problem, we take the following sets of values

I II III

d = 10006 d = 10006 d = 100b

P H Y S I C A L A S P E C T S OF CREEP 103

The only significant difference between the creep rate by grain boundary diffu-sion, equation (2.15), as contrasted to that for volume diffusion, equation (2.6), concerns the diffusivity: Therefore the activation energy for the grain boundary diffusion mechanism is somewhat less than that for the volume diffu-sion mechanism.

Whereas equations (2.6) and (2.15) represent the two extremes that might be encountered, in many cases both mechanisms might contribute simultaneously. The creep rate in such cases is not given by a linear average of the two alternate mechanisms. The theory for this general case must yet be developed.

At high temperatures and low values of the stress, polycrystalline C u , 1 3

A g 1 4 and A u 1 5 have been shown to creep at rates that increase linearly with the applied stress. The following table permits a comparison of the activation energies obtained from the creep tests with those for self-diffusion.

TABLE 2 . 1 . ACTIVATION ENERGIES (CAL/MOLE)

For volume Metal For creep self-diffusion

Cu 56,800 46,500

A g (not determined) 46,000 A u 51,000 53,000

The agreement between the activation energies for creep and volume self-diffusion in Au are excellent. It is not evident, however, why the activation energy for creep of Cu is higher than that for self-diffusion. In either event, however, volume diffusion and not grain boundary diffusion appears to be the preferred mechanism.

For providing a basis of comparing theory with experiment, we write equation (2.6) as

TABLE 2 . 2 . N A B A R R O CREEP

T D d A A Metal °K cm 2 / s ec c m calc observed

Cu 1324 4 0 . 4 x l 0 ~ 1 0 1 9 x l 0 - 3 0 . 0 0 3 4 0 . 0 0 5 2 A u 1315 2 . 3 x l O " 1 0 6 . 6 x l 0 - 3 0 . 0 4 0 . 4

These data reveal that the Nabarro equation gives the correct order of magni-tude for the creep rate.

It is necessary to emphasize, however, that other mechanisms of creep can also give a linear dependence of creep rate of stress and an activation energy for

(2.16)

Using the data recorded in Table 2 .2 , we arrive at the recorded values of A.


volume self-diffusion. Harper and D o r n 1 6 have clearly demonstrated that al-though these conditions are fulfilled for the high temperature low stress creep rate of polycrystalline aluminum, the Nabarro mechanism cannot apply. Markers on the specimen revealed that creep was not due to increase of atoms in the horizontal grain boundaries but that the interior of the grains strained. Single crystals gave about the same creep rate of polycrystals. And the creep rate was about 1300 times greater than that suggested by the Nabarro equation.

10" 4

O UJ CO

LLF

Si

<

(-CO

10"

O GRAIN S I Z E ^ 7 M I C R O N S

A GRAIN S I Z E ^ 1 3 M I C R O N S

T = I 8 7 3 °K

I O " 5

J I I I I I I

JO" 4

S T R E S S / ( G R A I N S I Z E ) 2 , D Y N E S / C M 4

F I G . 2 . 4 . Effect of grain size on the strain rate. (After Folweiler).

It will be shown later in this report that the creep observed by Harper and Dorn can be attributed to the motion of jogged screw dislocations.

Considerable interest has developed recently in ascertaining the basic mechan-isms for high temperature creep of ceramic materials. We will review here those data that appear to be pertinent to the Nabarro mechanism. A number of ceramic materials give creep rates at high temperatures that increase linearly with the applied stress. The activation energies for creep of these materials are compared with those for sintering and volume diffusion in Table 2 . 3 .


TABLE 2.3. ACTIVATION ENERGIES F O R CERAMIC MATERIALS

Ref. Material Temperature

range °K

Grain size

A*

Ac tivation ene rgies in cal /mole


range °K

Grain size

A*

Creep Sintering

Vol. diffusion


range °K

Grain size

A*

Creep Sintering Anion Cai ion

17 AI2O3 Single X I 152,000 18 AI2O3 1773-2073 7-34 130,000

152,000

19 AI2O3 130,000

150,000 20 AI2O3 1873-2073 3-13 130,000

150,000

20 AI2O3 1873-2073 50-100 200,000 21 AI2O3 1773 coarse grain 200,000 21 BeO 1923-2018 25 120,000 22 BeO 1873-2273 80-100

120,000 116,000

±10,000 23 BeO 1644-1811 7.4-10.5 95,800 24 UO2 1073-1273 — 95,000 24 UO2.06 1073-1273 — 72,000 24 UO2.16 1073-1273 — 65,000 25 UO2 29,700

±2300 26 uo2(?) 68,000

Although these data exhibit considerable scatter and must be deemed some-what preliminary, several interesting correlations nevertheless emerge. The best data for creep of AI2O3 give activation energies for creep that are slightly below those for sintering and anion volume diffusion. It is therefore possible that the high temperature creep of AI2O3 might be controlled principally by means of volume anion diffusion although some grain boundary diffusion may also take place. It is expected that vacancy diffusion in AI2O3 is principally controlled by anion diffusion since the activation energy for cation diffusion is expected to be much lower than that for anion diffusion. The agreement between the activation energy for creep of BeO and the cation volume diffusion seems to be somewhat fortuitous since anion diffusion is expected to have a higher activation energy and therefore O ion is the slower diffusing species that would control the rate of vacancy diffusion. It is possible here that the high tempera-ture creep of BeO occurs primarily as a result of grain boundary diffusion in which event the observed activation energy of 95,800 cal/mole for creep might refer to the activation energy for grain boundary diffusion of the anion. The activation energies for creep in defect lattices of U O 2 are sensitive to com-position as would be expected.

The data of Folweiler 1 8 on the high temperature creep of polycrystalline AI2O3, shown in Figs. 2 .3 and 2 .4 , reveal that both the effects of stress and grain size on the creep rate agree with the expectations of the Nabarro mech-anism. Similar results have been given by Warshaw and N o r t o n 2 0 .

5§

106 J O H N E. D O R N A N D J I M D . M O T E

Assuming

DAho3 = 1.5 X E -ÔOOART

d = 1 x 10~ 4 cm

= 14 x 1 0 - 2 4 cm 3

we calculate, for volume diffusion, that A of equation (2.6) is 0.825 x 10~ 1 2

cm 2 /dyne sec. The experimentally determined value is 2 .4 x 10~ 1 2 cm 2 /dyne sec which constitutes good verification of the Nabarro mechanism.

3 0 KEY

T, = 1644 °K T 2 = 1700 °K T 3 = l 7 5 5 ° K i

= 18 1 I

T,

d, = 8 . 8 x l 0 _ 4 c m d 2 = 9.4* I0" 4 cm d 3 = 9 .7x |0" 4 cm d 4 =10.1 x I 0 " 4 c m

10 15 2 0

7 7 d 2 , I 0 1 3 D Y N E S / C M 4

30

F I G . 2 . 5 . ykT/D versus T/^/ 2 (Data of Vandervoort and Barmore).

The most complete data in support of the Nabarro mechanism of creep was reported recently by Vandervoort and Barmore 2 3 . Their data for high tempera-ture creep of BeO covered ranges of temperature, stress and grain size. If we assume that for grain boundary diffusion in BeO that

£) = I S e-92,000/RT

we can plot their data as shown in Fig. 2 .5 , as

(2.17)


The experimental slope of this curve is Kexp = 9 .0 X 10~ 2 3 cm 3 whereas the theoretical value deduced from equation (2.15) is 2 x 10~ 2 3 cm 3 . Thus, good confirmation of the theory has been achieved with experiment.

Additional critical experiments, however, are needed to provide information on the activation energies for creep and for diffusion of the anions and cations in the volume and along the grain boundary. Such experiments should explore the effects of all of the significant variables. Further theoretical work is needed to establish the creep equation under conditions where grain boundary and volume diffusion can occur simultaneously.

3 . H I G H T E M P E R A T U R E C R E E P O F P U R E M E T A L S

Predictions based on theoretical considerations of the climb of edge disloca-tions agree more closely with the known experimental facts on the high tempera-

I 2 3 4 5 6 T I M E , HOURS

F I G . 3 . 1 . Typical creep curve at elevated temperature.

ture steady-state creep of pure polycrystalline metals, some dilute alpha solid solutions, and some rather stable dispersion strengthened alloys, than predic-tions based on any other single mechanism. Unfortunately, however, none of the several different versions of the climb theory formulated so far provide predictions that agree precisely with all of the known facts. Therefore a critical re-analysis will be made of this theory, leading to a better appreciation of primary creep and consequently, to a more realistic version of steady state creep that is in better harmony with current knowledge. Preliminary to such theoretical discussion, however, we will restate here some of the major experi-mental observations for which any good theory of high temperature creep must account:

1. During constant elevated-temperature creep at high stress levels, creep


curves of polycrystalline pure metals exhibit, as shown by the example in Fig. 3 . 1 , 2 7 an initial plastic strain on loading, a primary stage of decreasing creep rates, a secondary stage of constant creep rate, and a tertiary stage of accelerating creep rates leading to rupture. We shall not be concerned here with this terminal stage of creep in spite of its importance, because it involves the auxiliary factor of microfracturing, and we wish to confine our attention to the simple case of creep per se.

2. The initial straining is due to the motion of dislocations. Because of the high temperatures, such low activation energy mechanisms as the Peierls process, intersection of dislocations and cross-slip, especially in high stacking fault metals, occur almost without delay. But as the dislocations move they interact with each other and produce barriers to further motion.

0.12 r

T I M E , HOURS

F I G . 3 . 2 . Plot illustrating increasing strain over primary stage of creep with increasing stress (After Sherby and Dorn) .

3. Once a pattern of barriers is established, further motion of dislocation can only occur as a result of thermal activation over these barriers. Only those pro-cesses that have activation energies of less than about 50 kT can be operative.

4. During primary creep, dislocations are being thermally activated principally over the lower energy barriers. But since the creep rate is continuously de-creasing over this range, more barriers are being introduced as a result of such creep. Since more dislocations are being held up at barriers, this means that the reduction in the creep rate over the primary stage is due to the fact that the density of moving dislocations is decreasing.

5. If the barriers are removable by recovery, a balance is eventually achieved between the rate of recovery of the barriers and the rate at which they are reintroduced as a result of creep straining. This happens at the secondary stage of creep.


6. The strain over the primary stage of creep increases with increasing stress as shown in Fig. 3 . 2 . 2 8 This arises because the higher strain rate due to the higher stress causes a more rapid introduction of barriers. Therefore higher strains must be reached, in higher stress tests, before a balance can be achieved between the rate of recovery and the rate of reintroduction of barriers.

7. During high temperature creep of high stacking fault energy metals, sub-grains are produced as a result of the formation of both twist and tilt boundaries. Low energy twist boundaries are produced by patterns of screw dislocations. The formation of the tilt boundaries is due to polygonization arising from the introduction of edge dislocations into the tilt wall and the subsequent climb of such edge dislocations.

8. McLean 2 9 has shown, in a case where grain boundary shearing was small, that about one-half of the moving dislocations that account for the creep rate enter the subgrain boundaries. Evidently the remainder disappear as a result of mutual annihilation.

9. The angle between adjacent subgrains continues to increase uniformly throughout both the primary and secondary stages of creep. Consequently the disorientation of the subgrains does not affect the secondary creep rate. The subgrain boundaries therefore appear to act merely as good sinks for excess dislocations. This concept is in harmony with the knowledge that low angle boundaries exhibit only very local stress fields.

10. The subgrain sizes during the primary and secondary stages of creep are independent of the temperature of test and the strain. They decrease, however, with increasing stresses, as shown in Fig. 3 . 3 . 3 0

11. These observations suggest that the subgrain sizes are due to the non-homogeneous straining in any one grain as a result of the dissimilarly oriented neighboring grains. More severe bending and twisting and finer subgrain sizes are therefore expected for the higher stress tests.

12. Assuming that the degree of disorientation of the subgrains does not affect the creep rate, and that the subgrains merely act as suitable sinks for excess dislocations, the barriers that build up during the initial straining and primary creep must be found within the subgrains.

13. Transmission electron micrography clearly reveals that most of the dis-locations in crept metals that have high stacking fault energies lie in the sub-grain boundaries. Garafalo, Zwell, Keh and Weissman 3 4 have shown that throughout primary creep of Fe, the subgrain size remains substantially constant but that the disorientation of the subgrains increases and the density of the dislocations within the subgrains decreases throughout primary creep.

14. N o piled-up arrays of dislocations have been seen in crept specimens of high stacking fault metals. The dislocation arrangements within the subgrains of crept metals are characterized by (a) isolated dislocations, (b) dipoles and (c) entanglements. Electron transmission micrographs, shown in Figs. 3 .4 to 3.8, that illustrate typical arrangements of dislocations in crept metals, were kindly furnished by Dr. Gareth Thomas.

15. More barriers are produced following creep at a higher stress to the


same strain. This fact is illustrated in Fig. 3 . 9 . 3 5 When following creep to point A under a high stress at 4000 psi, the stress is changed to 2000 psi, the resulting creep rate immediately following the change in stress is lower than that obtained at the same strain when crept exclusively at the lower stress level.

16. All of the barriers, however, are recoverable. Regardless of differences in the original creep stress history, as shown in Fig. 3 . 9 , 3 5 the same final secondary creep rate is reached as is obtained by testing exclusively at the final stress level. Incidently, the subgrain size is also recoverable and it is dictated only by the final stress level.

O.OI I I 4 ? * 1 2 4 ? ? i 2 4 6 6 1

10 100 1 0 0 0 10 ,000

STRESS, a , PSI

F I G . 3 . 3 . The variation of subgrain diameter with creep stress (After Shepard and Dorn) .

17. The creep curves for a given stress agree well with the empirical relation-ship

y = f(te-^kT) (T = const)

An example of this is given in Fig. 3 .10 . 2 7 The value of the apparent activation energy, q, so obtained, at least within somewhat limited ranges of the creep rate for high stacking fault metals, appears to be independent of the stress and strain.


and consequently the substructural changes during primary creep are a function of te~^kT and have about the same apparent activation energy as creep itself.

19. When we define the apparent activation energy for creep by q\ where

F I G . 3 . 4 . Regions showing entangle-ments and dipoles (Al -1 % M g crept under 2200 psi to e = 0 .078 at 330°C).

F I G . 3 . 5 . Twist boundary and isolated dislocations ( M o strained 0 . 0 5 and

recovered at 1200°C for 3 hr).

F I G . 3 . 6 . Region of severe entanglement F I G . 3 . 7 . Region showing more or less leading to large density of loops and free dislocations (Al -1 % M g crept to edge dipoles (Al -1 % M g crept to e = 0 . 0 2 6 at 303°C under 2200 psi).

e = 0 .037 at 300°C under 3745 psi).

18. The creep rate therefore can be approximated by

(r = const]


F I G . 3 . 8 . More or less free dislocations and some dipoles (Al -1 % M g crept to e = 0 . 0 2 6 at 303°C under 2200 psi).

^ 4 0 0 0 P

\ 51

v

t D

\ 2 0 0 0 I

r

2 0 0 0

\*

• F

B

q = 3 4 , 0

T = 47

0 0 CALOI

r 8 ° K

R\ES PER M O L E

0 O. IO 0 . 2 0 0 . 3 0 0 . 4 0 0 . 5 0

£ , T R U E C R E E P S T R A I N

F I G . 3 . 9 . Effect o f change in stress on creep rate o f aluminum (After Sherby, Trozera and Dorn) .

I 0 1 8

I 0 1 7

I 0 1 6

1 0 "

I 0 1 4

1 0 "

I 0 1 2

z=£

-e^

H/

RT

) H

0U

RS-i


we again find that q' is a constant that is insensitive to the stress level and the strain.

20. The values of q and q' obtained by the techniques described above agree very well with each other.

21. Additional confirmation of the significance of diffusion to the creep rate was illustrated by the interesting correlation shown by Sherby 3 6 . The self-diffusivity of Fe in austenite is known to increase with carbon content. As shown in Fig. 3 . I I , 3 6 an analogous increase in the secondary creep rate also is obtained.

22. Furthermore, as shown in Fig. 3 . 1 2 , 4 0 the values of q and q\ for many pure metals, agree quite well with the activation energies for self-diffusion.

23. For low and moderate stresses, as shown in Fig. 3 . 1 3 , 2 7 ' 4 1 the secondary creep rate is related to the stress by

°s = Arn T = const

where 2 .5 < n < 5 .5, dependent on the metal under test. 24. Introducing an empirical activation energy suggests that

The nominal validity of this equation for a number of metals was recently re-viewed by McLean and Hale, as shown in Fig. 3 .14 . 4 2 The agreement is good at the lower stress levels but the creep rate appears to increase slightly more rapidly than with a fixed power of the stress at the higher stress levels.

25. Decrease in stress tests, for creep at the higher stress levels, as shown in Fig. 3 .15 , 2 7 > 5 2 reveal that

y oc ePT T = const

where p appears to be insensitive to the temperature and substructure. 26. When, in the course of a high temperature creep test, the stress is re-

moved, the elastic strain recovers immediately, following which some time-dependent negative creep takes place as shown by Bayce, Ludemann, Shepard, and D o r n 5 3 . We necessarily conclude that some dislocations are restrained in moving forward by barriers that provide back stresses.

27. Butcher 5 4 has shown that the creep rate for a constant shear stress test decreases when the hvdrostatic pressure, z?, is increased according to

where Qo is the atomic volume. In the cross-slip region K ~ 1.4, but at higher temperatures where the activation energy approximates that for self-diffusion K ~ 0.80. This is in good agreement with the value expected when the activation enthalpy for creep is equal to that for self-diffusion and hc = ha = Ua + p&a-

The above-mentioned experimental observations constitute the major factors for which any reasonable theory of high temperature creep must account. Although other issues of lesser importance might be added to this summary,


2 < cr to Q-UJ

O UJ tr h-

0.1

0.01

0 . 0 0 !

•8 -6

-4

-2

8 -6

-4—C

-2

8 ° K -6

5 3 1 0 8 A 4 •

-4 — 4 7 4 2

1 0 8 A 4 •

-2 —

2 J

4 > 3 > T 1 ?

B 1 1

2 4 4 3 1 J

3 > 4 \ 1 ) ( 1 ! *• ( » i 1

10" i o - | 0 - I 7 10-16 I 0 " 1 5

B = t e _ 3 4 0 0 0 / R T ( t IN HRS., T IN °K)

K T 1 4

F I G . 3 . 1 0 . Creep strain as a function of the temperature compensated time for a constant true stress of 3000 psi.

i o - r

o UJ cj) , 0 - n

eg o

10"'

o UJ ^ | 0 " 4

O GRUZIN, K0RNEV AND KURDIM0V (37)

MEAD AND BIRCHEN ALL (38)

1 0 0 I 2 3 4 5

ATOMIC P E R C E N T C A R B O N

F I G . 3 . 1 1 . Effect of carbon on the diffusion rate and creep rate o f gamma iron at 1000°C (After Sherby).

a.

a

b.

P H Y S I C A L A S P E C T S OF CREEP

F I G . 3 . 1 3 . Effect of small stresses on the creep rate of aluminum containing 3 . 1 at. % magnesium (After Laks).

115

J O H N E . D O R N A N D JIM D . MOTE

6.01 7 8 9 10 II 12 13 14 15 16 17 18 19 2 0 21

Symbol O • + V O • A Metal ond A 1 9 n f t o A l 9 n A » A l , 7 I . 7 0 - 3 0 b r a s s r d 2 5 - i 4 9 ° C u , 4 0 0 ° , 5 0 0 * aFe, 550°, temp. °C A l , 2 0 0 A l , 2 0 4 A l , 371 4 0 Q . CO, 6 3 0 ° . 700° 7 0 0 °

Ref. 4 3 4 4

Symbol • • •

Metal and X F e , 9 5 0 , ° N j 4 0 0 » N i M O o # P b , 2 5 ° Sn, l30° -230* a T l , 6 2 - 2 1 5 ° /8Tt,234-280* temp, c 1000 ' , \ v r _ i

Ref. 4 8 4 9 5 0 4 6 51

4 0 0 *

4 5 4 6

X

6 3 0 ° , 700°

4 7

7 0 0 °

4 2

F I G . 3 . 1 4 . Correlation of creep data according to equation of paragraph 2 4 (After McLean and Hale).

1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 CT, T R U E S T R E S S , PSI

6 0 0 0

F I G . 3 . 1 5 . Effect o f stress on the creep rate for a constant structure (After Sherby, Frenkel, Nadeau and Dorn) .

116


they do not appear to be as critical for the formulation of a theory as those

recorded above.

4. Q U A L I T A T I V E D E S C R I P T I O N O F A T H E O R Y F O R C R E E P

O F P U R E P O L Y C R Y S T A L L I N E M E T A L S

A T H I G H T E M P E R A T U R E S

We will now formulate the theory for creep of pure polycrystalline metals at high temperatures based on the motion of jogged screw dislocations and the climb of edge dislocations. In doing so we will borrow extensively from previous theoretical deductions. But we will so adjust the assumptions to agree more accurately with the known facts, which were described in Section 3 of this report, than was done in previous formulations of this problem. Although the end result will provide a better appreciation of creep than was available hereto-fore, particularly the primary stage, a complete solution of the problem in all of its generalities cannot yet be given. This arises because several significant issues are not yet understood in sufficient detail to provide the desired analytical formulation, and also because high Peierls energy materials and very low stacking fault energy metals, might creep by different mechanisms than those formulated here.

We will confine our attention to pure polycrystalline high stacking fault energy metals that are creeping at temperatures above one-half of their melting temperature. Under these conditions we will assume that the activation energies for the Peierls mechanism, intersection and cross-slip are so much less than 50kT, that these processes take place almost instantaneously.

As creep takes place dislocations move to the subgrain boundaries. But since the creep rate is not affected by the disorientation of adjacent subgrains, the subgrain boundaries merely serve as inks for the dislocations. As shown by L i , 5 5 stress fields due to subboundaries are short range and, therefore, they do not introduce any appreciable long-range back stresses on the moving disloca-tions. Subgrains are formed in high stacking fault energy metals by the non-homogeneous stressing at the grain boundary due to the different orientations of the adjacent grains. The local bending moments and torques to which each grain is subjected also increase with the applied stress. Consequently, finer sub-grain sizes are produced by the higher stresses. On this basis we assume that the significant creep-rate controlling mechanism is determined by the motion of dislocations within each subgrain and we here dissociate our analysis from the subgrain size considerations that played a role in some of the previous theories of high temperature creep.

Although most of the dislocations in creep specimen reside in the subgrain boundary, transmission electron microscopy reveals that there are also disloca-tions in the subgrain volume. Some of these dislocations are responsible for continued creeping; but the question arises as to how free dislocations are formed. They might arise as a result of escape from the subgrain boundaries and grain boundaries; they may also be produced at grain boundaries from


permanent Frank-Read sources in the subgrain, from generation of dislocations at points of high stress concentration in the volume or on the boundary of the subgrains, by multiplication due to cross-slip, or by climb over barriers and from entanglements. Since the subgrain boundaries are presumed to be sinks, it is unlikely that they permit escape of dislocations when the stress is held constant. There is no evidence of multiplication of dislocations in high tem-perature creep by either primary Frank-Read sources or at points of stress concentration. Cross-slip and climb, however, can provide the necessary genera-tion of free dislocations. As a result of cross-slip and climb, each subgrain continues to contain in its volume a number of moving dislocations. Subgrain volume dislocations are never completely exhausted as a result of migration to the subgrain boundaries because of facile cross-slip and because of climb from entanglements. Therefore there is no need to postulate the existence of special Frank-Read sources, as was done in some of the previous theories on high temperature creep, which neglected the role of screw dislocations, facile cross-slip, and climb from entanglements in their formulation.

Dislocations produced by cross-slip migrate under the applied stress. As the cross-slipped segment bows out, edge components are formed. Such edge com-ponents, although jogged, can move very rapidly until they are arrested at some barrier, because the jogs on edge dislocations can move conservatively. But the forward motion of the jogs on screw segments of dislocations restrain the motion of these segments because they move nonconservatively and therefore require the formation, diffusion and annihilation of either vacancies or inter-stitials. Sifrce the energy of formation of an interstitial is extremely high, how-ever, the motion of jogged screw dislocations must be almost exclusively attri-buted to the vacancy mechanism.

Screw dislocations can undertake several different processes, dependent on the local geometry: (1) Screw dislocations of opposite signs on the same slip plane will annihilate each other; (2) screw dislocations of the opposite sign on nearby slip plafies can cross-slip and annihilate each other; (3) some can leave the subgrain and enter twist boundaries; (4) and others will cross-slip to provide additional dislocations.

As shown by Friedel 5 6 , the length, //, between unit high jogs on a straight screw dislocation Will be given by

= e~^kT (4.1)

under equilibrium conditions where b is the Burger's vector, and gj is the free energy of formation of a jog. When, however, a screw dislocation is moving, additional jogs having heights of p atomic planes, where p can be more than unity, are formed. Therefore moving screw dislocations can have many more jogs than the static equilibrium number dependent on the frequency of double cross-slip. There are several consequences of cross-slip: (1) When the superjog length on the cross-slip plane is longer than ir/b(r — r*), the jog itself will act like a Frank-Read source and so can provide a supply of dislocations on the cross-slip plane. Here r is the line energy per unit length, r is the applied

P H Y S I C A L A S P E C T S OF C R E E P 1 1 9

stress and T * is the back stress due to other dislocations. Other possibilities are shown in Fig. 4 . 1 , (2) when L ' is longer than 2F{T — r*) and the jog HI is sufficiently high so that the positive and negative screw edge segments of the dislocations can pass each other, the section IJ permits continued multiplication of dislocations like a Frank-Read source; (3) when L ' is longer than 2r/b(r - r*), but FG is sufficiently short so that F G < [ G £ / 8 T T ( 1 - ^ ) ( T - T * ) ]

F I G . 4 . 1 . Classes of jogs .

(G = the shear modulus of elasticity and ^ is Poisson's ratio), a dipole FF' and GG' of positive and negative edge dislocations is formed; ( 4 ) but when L is shorter than 2r/b(r — r*), the situation over ABCD prevails. Here the con-tinued motion of the dislocation results in either the formation or adsorption of vacancies at the jog, dependent upon whether the jog at BC is a vacancy forming or an interstitial forming jog respectively.

Dipoles also form when positive and negative edge segments of dislocations on two nearby slip planes cannot pass each other. Because piled-up arrays of

120 J O H N E.' D O R N A N D JIM D . MOTE

dislocations are not seen in transmission electron micrographs of crept metals, long-range back stresses cannot arise from this source. But there are other sources of back stress fields that provide barriers that cannot be surmounted by thermal fluctuations. We have already mentioned the formation of positive and negative edge dislocation dipoles, which constitutes one example of stress field interactions. Such dipoles, although they have no long-range stress fields, can nevertheless interact with other dislocations in their immediate vicinity. Other stress field barriers that cannot be surmounted by thermal fluctuations can arise from screw dislocations under certain conditions. When screw dislocations are severely jogged, they move slowly and introduce back stresses on other nearby screw dislocations. Entanglements are undoubtedly excellent sources of interacting stress fields. The recovery of some of the plastic strain where the stress is removed following high temperature creep attests to the presence of such back stresses.

The preceding deductions then suggest the following summary of our model for high temperature creep: (1) Dislocations in high stacking fault energy crystals cross-slip with ease at high temperatures; (2) Edge components move rapidly but become arrested near dislocations of the opposite sign on nearby slip planes forming dipoles; (3) Other dipoles are formed by cross-slip of the screw segments of dislocations; (4) Entanglements are produced in the subgrain volumes perhaps due to interactions between dipoles and glissile dislocations and as a result of facile cross-slip; (5) Continued creep, however, depends on the motion of jogged screw dislocations; (6) As we shall see, both vacancy forming and interstitial forming jogs can move with about the same velocity at high temperatures; (7) But the presence of the dipoles and entanglements limits the length of the glissile screw segments of dislocations; (8) And back stresses restrain their motion; (9) If the dipoles did not recover, the subgrains would become cluttered up with dipoles and the creep rate would become vanishingly small; (10) The positive and negative dislocations of the dipoles annihilate each other by the climb mechanism; (11) The creep rate, however, as seen by this model, depends on the motion of jogged screw dislocations; (12) The substructure in the volume of the subgrain provides more substantial barriers and the length of the glissile screw dislocations decreases over the primary stage of creep, thus accounting for the decreasing creep rate; (13) But the secondary creep rate is reached when the rate of annihilation of the dipoles and entanglements by climb equals their rate of formation; (14) The subgrain boundaries play only a rather passive role in this process so long as the stress is constant.

5. Q U A N T I T A T I V E F O R M U L A T I O N O F A T H E O R Y F O R

C R E E P O F P U R E P O L Y C R Y S T A L L I N E M E T A L S

A T H I G H T E M P E R A T U R E

The quantitative theory for high temperature creep that will be presented here is formulated in terms of the concepts discussed qualitatively in Section 4. Although some of the details of each issue that is involved have been stated

P H Y S I C A L A S P E C T S OF C R E E P

previously in the literature, they will be briefly reviewed here again for the sake of completeness. The only new issue concerns the viewpoint that will be taken and how the individual items will be specialized and introduced into this re-formulation to provide a more complete and more realistic theory of high temperature creep than those that were presented previously. We shall particu-larly emphasize the primary stage of creep since this has been largely neglected in the past.

High temperature creep will be considered to be controlled by the thermally activated motion of jogged screw dislocations. This subject has already been discussed by M o t t 5 7 , Seeger 5 8 and Friedel 5 9 . The approach we will use here is based essentially on the discussion by Hirsch and Warr ington 6 0 . The strain rate due to the motion of jogged screw dislocations is simply

y = spbsv

where the superscripts refer only to the screw components, sp being the total length per unit volume and sv is the mean velocity of the active screw disloca-tions. We emphasize the point that sp is the total length of the moving screw dislocations. It does not include screw dislocations held up at barriers or incorporated in the entanglements. The reason for this definition of sp will become clear later.

The mean velocity of a screw dislocation depends on the details of the mechanism that takes place at the jog. As shown in Fig. 5 . 1 , the motion of the dislocation is restrained at the jogs P2, P i , and P3, etc. But as a result of the local stress r — T * , the screw dislocation will bow out and acquire a radius of curvature given by

(5.1)

Thus a force

acts to move the jog forward. But since

(5.3)

and

(5.4)

the angle 6/2 depends only on // and (T — r*). If the most difficult process occurs at P i , the point P i will lag behind P 2 and P3 causing a to decrease and therefore the force on P i to increase. Thus local adjustments of this type give a rather uniform forward motion of all of the jogs. As a statistical average, there-fore, the mean force per atom plane height is

(5.2)

(5.5)

121


But since long-range back stresses vary periodically over the volume of a lattice such that jr*dv = 0, the mean force acting on a unit height of a jog on a moving dislocation is simply about

(5.6)

As described by Friedel 5 6 thermal fluctuations assist the mean force acting on each atomic plane height of the jog to either produce or absorb vacancies respectively, dependent on whether the jog is a vacancy forming or an interstitial

F I G . 5 . 1 . Force on a jog due to a stress.

forming jog. For example, the net frequency of the absorption of a vacancy at an interstitial forming jog is given by

(5.7)

where v = the Debye frequency

ns

= the probability of a vacancy being adjacent to the jog

ns = the number of lattice sites per unit volume n = the number of vacancies per unit volume of the jog


Z = the coordination number

e-gjkT — t h e probability of a thermal fluctuation needed to move the vacancy to the jog

gm = the free energy of activation for the motion of a vacancy rljb2lp = the average work done by the applied stress per vacancy

absorbed gf — the free energy for the formation of a vacancy.

Whereas the first term following the equality gives the frequency of absorption of a vacancy, the second term refers to the frequency with which a vacancy can be produced at this jog. Diffusion of vacancies is known to occur so rapidly at temperatures above one-half of the melting temperature that n~ ~ no, the equilibrium concentration of vacancies. Therefore n~jns ^ no/ns. Since the super jog of the screw dislocation moves forward one Burger's vector per p vacancies that are produced

(5.8)

where ga = gf + gm is the free energy for activation of diffusion. A similar expression is readily obtained for the forward motion of a vacancy forming super jog. N o special consideration need be given the interstitial forming jogs, since they can move with about the same velocity as vacancy forming jogs. Very likely, however, some of the interstitial forming jogs will slide conserva-tively along the screw dislocation and thus be annihilated as suggested by Hirsch 6 1 .

Although the qualitative bases for the formulation of equation (5.8) for high temperature creep are rather sound, it is yet necessary to show that it is in good quantitative agreement with the experimental facts. The major issues con-cern whether it predicts reasonably accurately both the effect of temperature and the effect of stress on the creep rate. Unfortunately a simple direct com-parison is not obtainable inasmuch as the substructural changes taking place over the primary stages of creep are dependent on the stress. Thus the sub-structural parameters spjp and lj/p appearing in equation (5.8) might depend implicitly on the stress as well as the substructure.

The apparent activation energy, obtained by the effect of an abrupt change in temperature on the creep rate is given by

(5 .9 )

where ha is the enthalpy for diffusion, and is obtained from the well-known thermodynamic relationship

(5.10)

The presence of the term involving the change in the density of screw disloca-tions with temperature arises from the fact that the interaction stress fields of


dislocations decrease as the temperature increases. Consequently some dis-locations that are just at rest near a peak interaction stress will be released when the temperature is increased. It is expected, however, that this contributes only a small positive term to q. Since the observed activation energy for high tem-perature creep is known to be about equal to that for self-diffusion, it necessarily follows also that rljb2 < pkT for the cases studied. Under these conditions, the apparent activation energy for creep is independent of the stress and differs only slightly from that for self-diffusion.

A further justification of the smallness of lj/p is obtained when we consider the known effects of abrupt decreases in stress on the creep rate. We suggest here that

(5.11)

The major difference, therefore, between previous formulations of the theory for the motion of jogged screw dislocations and the present one concerns reten-tion of the first term following the second equality sign. It arises from the fact that dislocations within the subgrain interact with each other. As shown schematically in Fig. 5.2, the interaction stresses T * vary over the slip plane.

r*

F I G . 5 . 2 . Schematic of variation of interaction stresses T * over the slip plane.

When a stress T 2 is applied, all the dislocations excepting / will be moving through the crystal. But when the stress is changed to T± dislocations c and d will no longer contribute to the total length of moving dislocations. Thus sp decreases for a given substructure as r is decreased. It has been shown in Section 3 of this report that ft is insensitive to the temperature, an experimental fact which requires that

This suggests that most of the effect of changes in stress on the creep rate must arise from the change in the density of moving dislocations with stress. Accord-


ing to this interpretation, the creep data for Al require that lj cannot be much greater than about 40b. Therefore the distance between the jogs of moving dislocations is less than the equilibrium distance

I} = be~ffi^T (5.12)

where g% is the free energy of formation of a jog. But since we have confined our attention to high stacking fault metals in which cross-slip can occur easily, the small value suggested for lj is quite reasonable.

0 . 0 0 4 0 i — I — | — I — | — I — I — I — I — I — I — I — I — i — I — I — I — I — I — |

-cvooogl 1 1 1 . 1 1 1 1 1 1 1 0 2 4 6 8 D 12 14 16 18 2 0 22 2 4 26 2 8 3 0 3 2 3 4 3 6 3 8

APPLIED S T R E S S , <rA, PS I

F I G . 5 . 3 . Effect of low stresses on the secondary creep rate of high purity aluminum at 920° K (After Harper and Dorn) .

Partial confirmation of the validity of equation (5.8) was obtained by the low stress tests on creep of Al which were reported some time ago by Harper and D o r n 1 6 . For low stresses equation (5.8) reduces to the linear relationship between strain rate and stress

(5.13)

The creep curves at the lower stress levels gave practically zero initial straining and gave the secondary creep rate almost immediately, indicative of the concept that only negligible changes in substructure took place. The apparent activation energy for creep was found to be 35,500 cal/mole, and the steady state creep


Z = 12

b = 2 .86 x 10" 8 cm

we deduce that splj/p2 has the value of about 1.6 x 10~ 5 c m - 1 . In view of the low stresses employed here, the density of moving dislocations as low as is suggested by the small value of splj/p2 is quite possible.

When stresses above about 1.2 x 10 6 dynes/cm 2 are applied, the usual initial creep strain followed by primary creep is obtained, the creep rate no longer being linear with the stress. Under these circumstances equation (5.8) should apply. Since l3- is not expected to vary much with stress or temperature, we must assume that sp is the major pertinent structural variable. The initial straining upon loading increases the number of dislocations present. During the course of primary creep, however, sp must decrease as dislocations enter either the subgrain boundaries or entanglements and as more effective barriers to the motion of dislocations are established. The decreasing creep rate over the primary stage of creep is therefore directly related to the decreasing density of the mobile dislocations.

Inasmuch as no direct measure of lj/p has yet been established, it is not possible to determine accurately the value of sp. The trends, however, can be illustrated by arbitrarily selecting lj/p to equal, for example, the not unreason-able value of 20b. The values of *p obtained for this rather arbitrary illustration are shown in Fig. 5 . 4 . 6 2 These values of sp are not unreasonable, particularly in view of the fact that selection of somewhat higher values of lj would result in lower estimates of sp. It therefore appears that this theory is consistent with all of the experimental facts.

The concept that sp is the major substructural variable in the high temperature creep of many metals is further confirmed by the study of the effect of cold work on the creep resistance. During cold work many dipoles and extensive entanglements are produced. We therefore expect that the creep rate of a cold-worked metal would therefore be much less than that for an annealed specimen and that recovery during creep, which would release extra dislocations for flow, would result in an increasing creep rate until the secondary stage is reached. The validity of this suggestion is shown in Fig. 5.5 for severely cold-worked Al. The activation energy for creep was determined to be 36,000 cal/mole. Using the same assumed values of lj/p = 20b, as in the example of the high temperature creep of annealed Al, the calculated value of sp increases with with creep of cold-worked Al as shown in Fig. 5.6. Again lower values of sp are possible by selecting higher values of lj.

The secondary stage of creep occurs when the rate of generation of sub-structure due to creep itself is balanced by the rate of disappearance of the

rate shown in Fig. 5.3 as a function of stress was obtained. Using the values of

P H Y S I C A L A S P E C T S OF CREEP 1 2 7

substructure by recovery. Therefore additional information concerning creep can be obtained from recovery investigations. As we have seen, however, the substructure is quite complex, consisting of liberated dislocations, dipoles and entanglements in addition to subgrains. It is for this reason that no wholly satisfactory theory has yet been developed for the secondary creep rate.

1 0 '

10'

i o 6

10*

T = 573°K

r = 5 0 0 P S I

r = 2 5 0 PSI

0 O 2 0 . 0 4 0 . 0 6 0 . 0 8 S T R A I N

0 . 1 0 0.12 0.14

F I G . 5 . 4 . Variation of the density of moving screw dislocations with strain in pure annealed aluminum (Unpublished data of Raymond) .

The entanglements that are observed, appear to be the most significant substructural details seen in crept specimens of high stacking fault energy materials. And therefore a complete understanding of high temperature creep must depend on more detailed knowledge of how and why they form and how they might recover. Lacking sufficient information on this issue, we might nevertheless attempt some crude alternate analyses: For example, the existence of a steady state during the secondary stage of creep demands complete micro-scopic balance between the rate for formation and the rate of annihilation of any one of the numerous significant substructural details. Thus, although we


cannot yet treat entanglements satisfactorily, we might first approach the easier problem of studying the formation and disappearance of dipoles. The analysis to be given here differs only slightly in content, but nevertheless significantly in concept from that suggested by Chang 4 8 .

TIME, SECONDS

F I G . 5 . 5 . Plot illustrating the increasing creep rate of pure aluminum.

We will let n{y) be the number of dipoles per unit volume where the positive and negative edge dislocations are a distance y apart. We will assume that such dipoles are eliminated by climb induced by the stress axx as shown in Fig. 5.7. Therefore the net rate of formation of dipoles of height y is given by

(5.14)

where vc is the frequency for a unit climb. Therefore (nvc)y+a is the number


6

(a) (b)

F I G . 5 .7 . Elimination of dipoles by stress-induced climb.


of dipoles that are gained as a result of climb from a separation (y + a) to a separation y, (nvc)y are those of height y that are lost due to climb, vc is the frequency of a unit climb and k{y}, the number of dipoles of height y that are formed as a result of slip of screw dislocations per unit area, can (to a good first approximation) be taken as a small constant. If L is the mean free path of cross-slip, elementary statistics demands that k{y} = koe~ylL and therefore at the secondary stage of creep, where dn/dt = 0, we have

[d (nsvc) a/dy] + Ue~y^L xs

Psv = 0.

This equation is readily integrated to give ns = ns{y). The maximum value of y above which dipoles cannot form i s 6 3

For steady state conditions of n at ym we have, from equation (5.14) that

or (5.15)

Thus the secondary creep rate can be expressed either by

where p% is the probability of finding a jog on the climbing edge dislocations, Q is about the atomic volume and

Consequently, expanding the term in the brackets of equation (5.16) into a Taylor's series

(5.16)

(5.17)

in terms of the motion of jogged screw dislocations or by

in terms of climb. Using Friedel 's 5 6 expression, for climb we have

(5.18)

(5.19)

Although this expression gives the experimentally observed activation energy for secondary creep, it is yet incomplete since ns at ym depends on the stress. The same criticism can be leveled at all previous theories purporting to deter-mine the stress laws for the secondary creep rate. All previous theories for


creep by climb developed the stress laws by introducing questionable assump-tions. The eventual solution of the problem of steady state creep will depend on a more intimate understanding of the formation, nature, and recovery of the substructure, particularly entanglements, and the introduction of these con-cepts into a simple theoretical structure.

In order to obtain the empirical relationship that the secondary creep rate increases with the stress to the fifth power, Chris ty 6 4 and Weer tman 6 5 had assumed that climb was induced by piled-up arrays of dislocations. But such arrays are not seen and therefore the details of their approach are highly suspect. On the other hand it is possible that the stress fields induced by entanglements lead to stress concentrations that are found necessary to give a creep rate that varies with about the fifth power of the stress.

From the viewpoint of understanding the variation of substructure during creep on the creep rate, the most important observation is contained in the empirical expression discussed in Section 3, that

y = f(8) when r = const (5.20)

where 6 = te~^kT. Therefore

(5.21)

Consequently, if the theory for creep based on the motion of jogged screw dislocations is to be acceptable, we must associate f'{0} with sp such that

(5.22)

for a given stress, assuming that q is to be associated with approximately Ud-Although a completely satisfactory proof of the nominal validity of equation

(5.22) is not yet available, it is possible to set forth crudely the fact that the theory for the motion of jogged screw dislocations, supplemented with recovery by climb, gives, at least very closely, the relationship demanded by experiment.

The rate of change of the density of screw dislocations depends on the rate at which they are born as a result of climb from entanglements, etc., minus the rate at which they enter the subgrain boundary or are otherwise arrested at entanglements or other points in the subgrain volume. We crudely let this be

(5.23)

where N is the number of regions for climb per unit volume, p* is the length of screw produced per released dislocation and lc is the mean time of release of a dislocation due to climb. The first term beyond the equality therefore refers to the rate of formation of dislocations by climb whereas the second term states that their rate of disappearance depends on pv. When the previously discussed values for fc and pv are introduced, we immediately see that dpjdt depends on the temperature principally through e~ualkT. The major virtue of the proposed


theory, therefore, resides in the fact that, at least to a first approximation, it does account for the observed relationship of equation (5.20).

It is not proposed that all materials creep at high temperatures as a result of the motion of jogged screw dislocations. The Peierl's mechanism, cross-slip, or some other process might be strain-rate controlling for prismatic slip in hexa-gonal metals and in some intermediate phases. Even when the motion of jogged screw dislocations determines the creep rate, some minor variants of the pro-posed theory might apply. For example, at temperatures near or slightly below one-half of the melting temperature, two variations of the proposed theory become possible: (1) Vacancies produced at jogs might not be able to diffuse away sufficiently rapidly and in this case ha must be replaced by the smaller quantity hf, where hf is the enthalpy for the formation of a vacancy in a super-saturation of vacancies. Here also the reverse reaction must be formulated to take such excess vacancies into account ; (2) Alternately, diffusion may take place along the cores of the dislocations. Here ha in the theory must be replaced by the smaller value of hd which equals the sums of the enthalpies for formation and motion of vacancies along the dislocations.

Metals having low stacking fault energies might form piled-up dislocation arrays which would necessitate retaining the long-range back stress term, T * , in the formulation of the theory and perhaps modify the role of climb of disloca-tions in the recovery mechanism. Low stacking fault energy metals are expected to cross-slip less frequently and thus should exhibit higher values of From the practical viewpoint this suggests that the jogged screw dislocations in such metals will have higher velocities, but they will not climb as rapidly. The latter suggests that such metals should reach the secondary stage of creep earlier and that their secondary creep rate should be lower than that for the higher stacking fault metals. F rom this basic viewpoint, lower stacking fault metals should give apparent activation energies, q, that decrease linearly with the applied stress. Furthermore, the strain rate sensitivity, /?, should be slightly temperature-sensitive. These two effects should also be noted in the higher stacking fault metals that are deformed at higher stress levels.

In the previous discussion it was assumed that pj, the probability of finding a jog, was always high, there being an ample supply of jogs to permit rapid climb. If this is not the case, pj might be given by the equilibrium number of jogs, e~9ilkT, where gi is the free energy of formation of a jog. In this event the enthalpy term, ha in the equation for climb must be replaced by ha + hj where hj is the enthalpy for the formation of a jog. It is also possible under forced climb that the jogs are being used up more rapidly than new ones are created. In this event the term ha in the climb equation must be replaced with ha + hj where h!i approaches the activation enthalpy for the formation of a jog and is therefore greater than hj.

Although these variations in the theory need to be considered, the current evidence suggests that equation (5.22) is nevertheless in good agreement with the known facts. The missing information concerns the theoretical determina-tion of how sp changes during the course of creep and how, in detail, it depends


on stress, structure and time. This formulation, of course, must be based on more complete evidence from electron microscopy relative to the types of sub-structure that are significant to creep.

6. P O L Y P H A S E A L L O Y S

All highly creep-resistant alloys contain at least two phases and at least one of these is hard and strong. Three separate types of creep-resistant polyphase

A. D ISPERSION ALLOY.

B. A G G R E G A T E D ALLOY.

C. C O N T I N U O U S HARD P H A S E A L L O Y .

F I G . 6 . 1 . Schematic diagram of creep-resistant polyphase structures.

structures, shown schematically in Fig. 6 . 1 , exist, and all three types of struc-tures can coexist at one time in a single polyphase alloy. Each type of structure is significantly different from the others, and each provides its own unique mechanism of creep strengthening. Such structures can change during the course of creep by more precipitation, overaging, and spheroidization; further-more, dislocations in the matrix can interact with the solute atoms that are


present. We will postpone consideration of such complications for the present and also confine our attention to cases where the polyphase structure remains reasonably stable during the course of creep.

Each structure has its own unique effect on the creep resistance of a polyphase alloy. The alloy that has a continuous hard phase about the ductile matrix can only creep as the hard phase itself either creeps or fractures. When the con-tinuous phase is tenacious, strong and stable, highly creep-resistant alloys are obtained, but they will be brittle at low temperatures.

Creep in each soft matrix grain of the aggregated alloy must take place in a fashion analogous to that of an alpha solid-solution alloy. But dislocations will pile-up at the boundary of the hard phase resulting in back stresses on other dislocations in the soft grains and thereby arresting their motion. Such back stresses can be relieved by recovery in the soft matrix grain, deformation of the hard phase as a result of high stress concentrations, and also fracturing of the hard brittle phase. The aggregated structure has lower creep resistance than the two remaining types.

The most commonly practised method of improving the creep resistance consists in forming a rather stable dispersion strengthened alloy by precipitation hardening. Dispersed particles in the boundaries of the matrix grains serve to prevent grain boundary shearing. This not only reduces the creep rate by the amount of grain boundary shearing, but it also results in the presence of higher back stresses on dislocations by the amount the back stresses would have been relaxed by grain boundary shearing. Moving dislocations in the grain of a dis-persion strengthened alloy can interact in various ways with the dispersed particles; they might cross-slip, climb, cause deformation in the particles, or fracture of the particles. It is not surprising that, in view of the complexity of this problem, very little quantitative experimental work or theoretical analyses have been made to foster a better understanding of the creep of polyphase alloys.

As discussed by Grant and Pres ton 6 6 in their 1956 review of the subject, and as presented more recently by G u a r d 6 7 in the 1960 A.S.M. Symposium on Strengthening Mechanisms in Solids, current theories of dispersion strengthening are based on Fisher, Hart and P r y ' s 6 8 extension of Orowan ' s 6 9 original concept. Orowan suggested that as slip takes place, dislocation loops can extrude between adjacent particles on the slip plane, as shown in Fig. 6.2. Consequently the yield strength T depends on the particle spacing A according to the Frank-Read equation

(6.1)

where r* is the back stress acting on the moving dislocations. Fisher, Hart and Pry further suggest that the loops of dislocations left about the particles result in a back stress

(6.2)


where

N = the average number of loops about the particle and depends on the

strain

r = the mean particle radius and

/ = the volume fraction of dispersed phases

which causes very rapid strain hardening. Therefore the flow stress for plastic deformation depends on the particle spacing A and the mean particle radius, r. The finer the dipersion, the greater is the strengthening. At low temperatures they assumed that the hardening increased with strain as a result of the increas-ing number of loops about the particles. The maximum hardness that can be achieved is either that which causes plastic deformation or fracture of the particle under the stress concentrations due to piled-up dislocations. Under a

(a ) D I S L O C A T I O N (b) R INGS L E F T

EXTRUDING B E T W E E N ABOUT P A R T I C L E S .

P A R T I C L E S .

F I G . 6 . 2 . Schematic o f dislocations extruding between particles as suggested by Orowan.

constant stress, a fixed strain results when the back-stress fields due to arrested dislocations and loops equals the applied stress.

Weer tman 7 0 and Ansel and Weer tman 7 1 have shown that deformation of a dispersion hardened alloy under constant stress can continue to take place at high temperatures as a result of climb of dislocations. They assumed that dislocations originated at fixed Frank-Read sources, of which there are M dis-tributed at random per unit volume. Under the action of an applied stress, dislocations move out from these sources producing a series of concentric rings about the sources to a maximum radius L9 until they interact with each other and thus stop the sources. When the dislocations in the outermost rings from the series of sources climb and annihilate each other, an area nL2 is swept out per source. Consequently, the steady state creep rate should be given by

y 9 = Mirm/lc (6.3)


where the height of climb, h, is now associated with the mean height of the particle.

Weer tman 7 0 suggested that two types of creep laws were operative dependent on the stress. When 2r/bL < r < 2T\b\ dislocations cannot form loops about the particles, but Weertman suggests that they nevertheless climb. F rom the frequency of a unit climb given by equation (5.18), we obtain

for the secondary creep rate. Unfortunately there is practically no good data available to check Weertman's

theory. The most pertinent appears to be that obtained by Giedt, Sherby and D o r n 7 2 a number of years ago on the creep of dispersions of CuAl2 in an Al solid solution matrix. An activation energy for creep of 37,000 cal/mole was

where fc is the average time for complete climb of a dislocation. If, however, the near sources reside on slip planes about a distance h apart

Mnm ~ 1

and, therefore

(6.4)

(6.5)

suggesting that, since <JXX is small here,

(6.6)

This coincides with Weertman's equation when vxx is taken to equal r. It is, however, difficult to justify this equality since the major stress field that is acting in this instance arises from the image field of the blocked dislocation which gives axx = 0 at the core of the dislocation.

At higher stress levels Weer tman 7 0 suggests that piled-up dislocations give

(6.7)

and loops do form about the particles. The distance a loop must climb before another can form is

(6.8)

But it is difficult to visualize the details of climb if all loops start out on the same slip plane because here again axx = 0. For the higher stresses Weertman gets, the equivalent of

(6.9)


obtained, a value only 1500 cal/mole above that estimated for self-diffusion of Al and therefore in crude agreement with the theory in this respect. The second-ary creep rate could be approximated in most cases by

ys = Arê~^^°/kT (6.10)

where the average constants shown in Table 6.1 were obtained.

TABLE 6.1.

Alloy Temperature A h %Cu °K cm c m A n

3.05 422 9 7 x l 0 - 3 1.57X10- 4 1.5 3.05 422 9 6 x l 0 4 0 .15X10- 4 8.85 x l O 1 0 0.88 3.05 477 9 7 x 1 0 - 3 1.57X10- 4 20X10 1 0 0.81 3.05 477 9 6 x 1 0 - 4 0 .15X10 ' 4 64 X 1 0 1 0 0.81 3.05 530 9 7 x 1 0 - 3 1.57X10" 4 13 XlO 1 1 0.63 3.05 530 9 6 x 1 0 - 4 0 . 1 5 x l 0 ~ 4 9 . 6 4 X 1 0 2 1.56 3.05 578 9 7 x 1 0 - 3 1.57X10- 4 20.2 x l O 1 1 0.49 3.05 578 9 6 x l 0 - 4 0 .15X10- 4 30 .6x10 1.6

As noted in Table 6 . 1 , the value of n was always less than that suggested by theory (namely, 4 as expansion of equation (6.9) into a Taylor's series reveals). Furthermore A was not proportional to X2/h. Consequently, the theory does not agree well with the limited, somewhat scattered, data currently available.

In their creep studies on a SAP alloy, Ansel and Weertman obtained an activation energy of about 150,000 cal/mole. Since climb in a dispersion hardened Al alloy should give a value of about 35,500 cal/mole, that estimated for diffusion, the creep of SAP, as we will describe more fully later, must under-take some other mechanism than that postulated for the usual dispersion strengthened alloys.

Our understanding of high creep resistance of dispersion strengthened alloys could be advanced appreciably by more critical experimental investigations, covering determination of the activation energy and the stress law as well as the nature of the primary stage of creep in terms of particle size and distribution. Additional work is required on the nature of the substructures and dislocation patterns that are obtained during creep of dispersion strengthened alloys. And finally, perhaps, better theories for the high temperature creep of dispersion strengthened alloys can be developed in extending Weertman's critical and most significant concept that the secondary creep rate at high temperatures is con-trolled by the dislocation climb mechanism.

A more realistic approach to the problem of dispersion strengthening is based on the concept that the Orowan model gives a gross idealization of the true state of affairs. Recent electron microscopic investigations by T h o m a s 7 3 have shown that in the initial dislocation pattern, dislocations go from one to another particle. Frequently upon stressing, new dislocations are formed not at F r a n k -Read sources, but at the boundary of the dispersed particles and they loop out

6§


into the matrix. Thomas, Nutting and Hirsch 7 4 have observed that the presence of hard dispersed particles induces cross-slip. And Mitchell, Mitra and D o r n 7 5

have shown that most of the low temperature strain hardening of dispersion strengthened alloys does not arise from the Fisher-Hart-Pry mechanism of back stresses from concentric planar loops but rather, is due to the extremely dense dislocation population in the entanglements. Although these observations were based on the low temperature behavior of dispersion strengthened alloys, it is nevertheless reasonable to believe that most of these features with only minor modifications also apply to the creep behavior.

A tentative concept of the issues involved in the high temperature creep of a dispersion strengthened alloy might now be suggested. In this discussion we will consider only the case of a stable dispersion embedded in a matrix wherein no interactions take place between the dislocations and solute atoms. Under these conditions, we believe that the primary stage of creep can be described by equation (5.16) for the motion of jogged screw dislocations. This creep should differ from that in the dispersion free alpha solid solution due to the extensive cross-slip induced by the dispersed particles. Poly-slip and cross-slip in the neighborhood of the dispersed particles should result in a greater number of centers for entanglements and should therefore reduce the number of freely moving screw dislocations more rapidly than in a similar alpha solid solution. Due to the particles and their associated entanglements, the density of moving screw dislocations will be less at the same stress level and temperature than in an alpha solid solution. When the dislocations are highly jogged, the activation energy for creep at the lower stress levels should approximate that for self-diffusion. But at higher stress levels and particularly for low stacking fault metals the activation energy for creep should decrease linearly with the applied stress. Because s p , as well as the activation energy, depends on the stress, it is not possible to make any reasonable predictions of the stress law for high tempera-ture creep of dispersion strengthened alloys at this time. The secondary creep rate should be given either by equation (5.16) for the motion of jogged screw dislocations or by the rate of release of arrested and entangled dislocations as a result of recovery by climb. In this model the value of o x x to be used in cal-culating the frequency for climb will be given by the steady state conditions of the nature and density of the dislocations in the entanglement. Such steady state conditions of course will depend on the applied shear stress. It is suggested therefore that the high creep resistance of dispersion strengthened alloys is due to the low rate at which dislocations issue by climb from the entanglements and particularly the short distances they move before they again become entrapped.

A qualitative discussion on the creep behavior of SAP in terms of dispersion strengthening was presented by Grant and Pres ton 6 6 . However, the data ob-tained by Ansel and Weer tman 7 1 that the activation energy for creep of SAP is about 150,000 cal/mole, disqualifies the thought that its creep resistance follows the dispersion strengthened model. More recently, Meyers and Sherby 7 6 , and Meyers, Shyne and Sherby 7 7 , investigated the creep of SAP and obtained very


high activation energies, much above those for the activation enthalpy for diffusion in Al and in much better agreement with that for volume diffusion of the anions in AI2O3. Specimens also remained rigid and were crept at tempera-tures above the melting temperature of pure Al. The creep behavior of SAP was therefore attributed to follow what is expected from the presence of a thin film of AI2O3 adhering more or less continuously to the fine grained Al.

7. A V E R A G I N G A N D S P H E R O I D I Z A T I O N D U R I N G

H I G H T E M P E R A T U R E C R E E P

It is well known that the creep resistance of two-phase alloy systems decreases during the course of high temperature creep as a result of spheroidization and particle growth. The driving force for this reaction is the decrease in free surface energy; and the rate at which it takes place depends on diffusion. The continuous hard film type of two-phase alloy spheroidizes, as shown schematically in Fig. 6 . 1 , and assumes the low creep-resistant characteristics of a two-phase agglo-meration. The dispersion strengthened alloy develops a more widely spaced configuration of larger dispersed particles. As a result of such coarsening there are fewer points per unit volume at which cross-slip is induced and therefore there is less jogging and fewer points where entanglements might be induced to form. This also results in a decrease in creep resistance.

Since the design of high temperature alloys for long-term high temperature creep resistance depends on the stability of the structure, we will analyze the factors that are involved, following an earlier discussion by Schoeck 7 8 . The problem will be simplified by considering the coarsening rate of a spherical dispersion of an incoherent precipitate. We will not concern ourselves here with the nucleation of the precipitate, coherency stresses, precipitate shape and other auxiliary problems such as changes in composition of the precipitate.

The free energy of a system consisting of a spherical particle of h b atoms embedded in a matrix solid solution of B atoms in A, assumed to be ideal, involves the free energies of the particle, the solution and the free surface energy. At equilibrium the change in free energy, when one B a tom is removed from the surface of the particle and deposited in solution, is zero. The free surface energy of the system is

7F=47rr2y (7.1)

where y is the free surface energy per unit area. And the number of B atoms in

where Q is the atomic volume. Therefore, the change in free surface energy upon adding one atom to the particle is

(7.2)

(7.3)


(7.4)

where *UB is the energy per atom in the particle phase, p is the pressure, vQB

is the atomic volume, and PSB is the entropy per B atom in the particle phase. Assuming an ideal solid solution with random mixing of nA atoms and nB atoms, the free energy of the solution is

(7.5)

where the last term arises as a result of the configurational entropy. Therefore the chemical potential of each B atom in solution is

(7.6)

The total change in free energy upon transfer of one atom of B to the solution is, at equilibrium

(7.7)

Letting nB\(nB + nA) = c, the composition of the solution, and noting that at SQB — V&B and the thermal entropy at high temperature, ssB ^ psB, we obtain

C ~ e-(9uB-*uB)lkT e2yQ/rkT ( 7 ̂ g )

where suB — i>uB is the heat of solution per B atom. Therefore, the concentra-tion CQ of B atoms in equilibrium with a flat surface (r = oc) is

(7.9)

(7.10)

and

where

(7.11)

Consequently, the equilibrium concentration of solute atoms about a small particle is somewhat greater than that around a large particle. Therefore, the smallest particles will dissolve and the solute will precipitate out on the largest particles until a uniform distribution of particle sizes is obtained. Final equi-librium will be achieved only when a single large spherical particle is formed in the matrix

We now wish to estimate how long it will take for a small particle of radius r to dissolve. An exact solution to this problem is very difficult because as the smallest particles dissolve, those of intermediate size first get larger. When all of the smallest sized particles have dissolved, those that originally had inter-

The free energy per atom of the particle is


mediate sizes and therefore grew at first, now being the smallest, begin to dis-solve. Fortunately an exact solution is not necessary to illustrate the principles and to identify the major factors involved.

For simplicity we consider the two particles shown in Fig. 7.1 having initial radii ro and r where ro > r. We assume that steady state diffusion is established and that the concentration gradient is linear between the two particles. Very little error is introduced if we assume p to be small relative to r. Therefore the

F I G . 7 . 1 . Schematic of precipitate dissolution.

In spite of the gross approximations that were made, equation (7.14) suggests how the various variables enter the problem. Because t oc r 4 , small particles dissolve in a short time. As the particles become larger the time for their dis-solution increases substantially, not only because r is increased, but also because A is increased.

Whereas y and Q usually change only mildly with the system in question, D plays a much more important role, since it can be quite different for different systems and temperatures. But the major factor in determining the time for dissolution of the smaller particles is co.

It is essential to point out, however, that the coarsening of a dispersion strengthened alloy during creep might proceed somewhat more rapidly than it would in the absence of creep. A small increase in the diffusivity is to be expected as a result of the excess vacancies generated at moving jogs on screw dislocations and as a result of climb of edge dislocation. More significant, however, is the

where the term to the left of the equality gives the number of atoms leaving the smaller particle per second, D is the diffusivity, and irr\ is the area of the larger particle through which the flux moves. When equation (7.13) is integrated for ro > r we obtain

(7.14)

(7.13)

The diffusing flux is

(7.12)

concentration gradient between particles a distance A apart is crudely estimated TN H P


fact that moving dislocations can considerably increase the average diffusivity because of rapid pipe diffusion along the dislocation core.

To illustrate the great importance of co, Schoeck 7 8 considered a Ni -Al alloy which might have either M 3 A I or AI2O3 precipitates and calculated the time t for the case ro = 10~ 6 cm, r = 10~ 7 cm, A = l/x and 700°C, as shown below,

Diffusing /

Precipitate element D Co sec

N i sAl A l 1 0 - 1 0 — 5 x l O - 2 ~ 1 0 2

AI2O3 O 1 0 - 7 ~ i o - 1 4 ~ 1 0 n

whereas the small NisAl particle would have dissolved in 100 seconds, the AI2O3 particle under similar conditions, and in spite of the higher diffusivity of O that was assumed, would take about 1000 years to dissolve.

This example illustrates clearly that the compounds having the greatest heats of formation should provide the greatest structural stability. F rom the view-point of stability, oxide dispersion strengthened alloys appear to be most promising although some slightly less stable systems might have other ad-vantages. Adherence between the matrix and the precipitate is also significant.

8. E F F E C T O F S O L U T E A T O M S O N

H I G H T E M P E R A T U R E C R E E P

Several excellent r e v i e w s 7 9 - 8 1 have been written on the influence of solute atoms on the mechanical behavior of alloys. Although most of what has been said pertains to the behavior of alpha solid solutions at low temperatures, some of the same principles also apply under conditions of high temperature creep. On the other hand, the relative importance of the various strengthening mechanisms changes as the temperature is increased and new factors enter into the analyses at high temperatures. Therefore, the factors that provide great solid solution strengthening at low temperatures are not necessarily the best for developing creep resistance at high temperatures. For this reason it is appro-priate to consider the special factors of solid solution effects that are particularly significant for high temperature creep.

The high creep resistance of some complex ceramic materials is due to a high Peierls activation energy, solute atom pinning of dislocations, and high activa-tion energies for diffusion. Among the more ductile materials, however, the highest creep resistance is achieved, as we have already discussed in Section 6, by the introduction of hard and stable phases in a ductile matrix of refractory body-centered cubic solid solutions. The creep resistance of such alloys is determined not only by the nature and distribution of the second phase but also on the creep resistance of the softer matrix in which the hard phase is embedded. Additional creep resistance can be achieved in a dispersion hardened alloy by further strengthening the ductile matrix. We will consider here, how-


ever, only single phase alpha solid solutions to illustrate what factors are involved.

There are a number of different effects of solid solution alloying on the creep resistance and these effects do not all vary linearly with the atomic percentage of the alloying element. Consequently the effects of two or more elements are not simply additive. Furthermore, the greatest benefits are achieved by using a series of solute elements (not just one) so selected as to provide optimum coverage of all factors involved. Although a quantitative approach to the design of alloys for creep resistance from this viewpoint has not yet been developed, qualitatively at least, these concepts are employed and the significance of this complexity effect has been recognized. Therefore, many of the highest strength high temperature alloys now in use have compositions that appear to represent a roll-call of almost all of the metallic and many of the non-metallic elements in the periodic table.

The following issues are significant in solid solution alloying for high tem-perature creep resistance:

A. Diffusion effects B. Stacking fault energy modification C. Peierls stress modification D . Mot t -Nabar ro hardening E. Guinier-Preston hardening F . Suzuki locking G. Cottrell locking

H. Fisher short-range order hardening.

We will discuss each in turn.

A. Diffusion Effects

Although we shall emphasize here the factors that lead to high temperature strengthening, it is necessary to point out that some alloying elements actually decrease the high temperature strength of alpha solid solutions. A typical example is the alloy of C in y-Fe. Whereas C in solution in a-Fe causes appre-ciable low temperature strengthening, when dissolved in sufficient quantity in y-Fe it increases significantly the creep rate under a given stress. Sherby 3 6 , as noted in Section 3, has shown that this arises from the fact that C increases the rate of self-diffusion of Fe in y-Fe. In general, therefore, an important effect of alloying elements on creep arises from their effect on the diffusivity. This factor, of course, has special significance to the high temperature plastic behavior of materials because diffusion is not involved in the low temperature plastic be-havior, and because the high temperature plastic behavior is usually diffusion controlled. The same trend is noted in defect lattices of intermediate phases; those compositions that have a greater equilibrium number of either vacancies or interstitials exhibit lower strengths at high temperatures because of their higher diffusivities. Vacancy-solute atom binding will increase the activation energy for diffusion and improve the creep resistance.


where a is the lattice constant. Since alloying elements influence y, they change the spacing between the partials. For example, as Co is added to Ni the com-position approaches that for the transition from the f.c.c. to the h.c.p. phase. Therefore hF — CF decreases as the Co content is increased, causing y to de-crease and thereby increasing the separation of the partial d. At high tempera-tures the separation of the partials becomes yet greater because solute atoms distributed themselves between the two phases in such a way as to further decrease the free energy of the system. Theoretical deductions by Seeger 8 3

suggest that increasing electron/atom ratio as a result of adding solute elements to Cu, Ag, or Au, causes a decrease in the stacking fault energy. This is con-sistent with the increased ease of twinning in the higher electron/atom ratio solutions of these metals. Howie and Swann 8 4 have also measured the change in stacking fault energy upon alloying in a number of cases. In general the stacking fault energy may either increase or decrease upon alloying, dependent on the system. The stacking fault energy can also vary slightly with temperature.

Relative to f.c.c. metals that creep by the motion of jogged screw dislocations and by the climb of edge dislocations, those alloying elements that lower the stacking fault energy should have greater values of /? because of their higher constriction energies and the lowered probability for cross-slip. Equally the rate of climb should be lower because pj is smaller.

C. Peierls Stress Modification

The Peierls stress for pure f.c c. metals is so small that it never plays a role in the plastic deformation of these metals even at low temperatures. Although the Peierls stress is greater for b.c.c. metals, the Peierls mechanism is so easily activated that it is never significant above room temperature. Alloying may modify the Peierls stress in b.c.c. metals, but it appears that such modification would not be large enough to play a role in high temperature creep. Recent

where h is the spacing in the [111] direction, V is the molar volume, and hF and CF are the molar free energies of the hexagonal and cubic phases. The separation of the partial dislocations is determined by y and is estimated to be a b o u t 8 2

(8.2)

(8.1)

B. Stacking Fault Energies

Dislocations lying in the (111) plane of f.c.c. metals dissociate with a decrease in energy into pairs of Shockley partials foiming between the particles two layers of atoms packed according to the h.c.p. arrangement. A twin boundary consists of a single layer of atoms packed according to the h.c.p. basis. Thus a stacking fault boundary has roughly about twice the twin boundary energy. The energy y of a unit area of a stacking fault is almost equal to


da t a , 8 5 however, suggest that the Peierls stress for prismatic slip in Mg is re-duced considerably by alloying with Li. The Peierls mechanism, however, may be important in the creep of intermediate phases and ceramic materials.

D . Mott-Nabarro Hardening

Periodic volumetric hydrostatic tension stresses are introduced in the lattice by substitutional solid solution alloying. The motion of the screw components of dislocations will not be resisted by such stress fields but such local stresses will restrain the motion of edge components. Although the dislocations are somewhat flexible their line tension tends to keep them more or less straight. Consequently, when the stress centers are close together, as they are in the usual case of solid solution alloying, the dislocations move more or less as rigid units through the average stress of the periodically varying field. Conse-quently these stress fields have almost zero effect on the flow stress.

The stress fields of interstitial solute elements interact with both edge and screw dislocations. Generally the concentration of interstitial elements is com-paratively low so that dislocations can bow out between the high stress centers. In this case the applied stress must force the dislocations over the peaks of stress. Consequently somewhat higher solid solution strengthening is obtained from interstitial alloying. So long as the alloying remains random, however, neither of these mechanisms aid much in improving high temperature creep resistance because they are easily thermally activated.

E. Guinier-Preston Zone Hardening

During the early stages of precipitation in some systems, like atoms cluster to form Guinier-Preston zones and finally stable phase precipitates. Such clusters of atoms and Guinier-Preston zones also restrain the motion of dis-locations but usually much less effectively than critical dispersions of stable phase incoherent precipitates. The presence of such clusters and Guinier-Preston zones have only very short lives at high temperatures and therefore are not usually significant to high temperature creep.

F. Suzuki Locking

Suzuki locking 8 6 constitutes a potential mechanism for improving the high temperature creep resistance of f.c.c. alloys and therefore deserves more detailed attention. The stacking fault between the Shockley partials of the f.c.c. system consists of two atomic layers of h.c.p. packing of atoms. At high temperatures, where diffusion can occur, solute atoms will distribute themselves between the fault and the f.c.c. crystal in a manner analogous to the distribution of solute atoms between two phases. Although a complete and accurate analytical description of such a distribution is available, 8 7 it will be convenient here, in emphasizing the physical aspects of the problem, to treat the stacking fault as a separate phase rather than a surface.

The geometry of the problem is illustrated in Fig. 8 . 1 . At equilibrium the


stacking fault between the partials A and B, shown as the cross-hatched region, has a composition c/ whereas the composition of the crystal remains about the average value of the alloy, namely c. If then, a unit length of the dislocation is moved a distance 8, assumed to be identical for both partials in this approxima-tion, the work done is (r — T * ) bo. This work must equal the increase in chemical free energy. In moving the partial dislocations an amount 6, the volumes of the phases changed 2h8; for the first dislocation this volume of the matrix phase was produced at a composition Cf resulting in the disappearance of the same stacking fault volume of the same composition; and at the second dis-location this volume of the stacking fault was produced resulting in the disap-pearance of the same volume of the matrix phase at the composition c. Conse-quently

F I G . 8 . 1 . Stacking fault in Suzuki locked alloy.

In general both {Ff — F)c and {Ff — F)Cf are positive as, for example in the case of Ni -Cu alloys where the stable phase is always f.c.c, the faulted region having the higher free energy regardless of composition. But {Ff — F)Cf < {Ff — F)c

since the composition c/ was obtained as a result of equilibrium. Consequently the total term in the braces of equation (8.4) is always positive.

An additional relationship between the variables of equation (8.4) is obtained by invoking quasi-equilibrium conditions between the matrix phase and the faulted region. Complete equilibrium, of course, is never achieved inasmuch as this would demand the disappearance of the faulted regions. We consider, therefore, open matrix and faulted phases for which, at equilibrium, the virtual change in free energy §F't for the total system is

where F' refers now to the free energy of the matrix containing "a" atoms

(8.4)

region, and the subscripts c/ and c refer to the compositions at which the free energies must be evaluated. Therefore

(8.3)

where V is the molar volume, assumed identical regardless of stacking, F is the free energy per mole of the matrix, Ff is the free energy per mole of the faulted


and rib "ft" atoms, etc. The partial derivatives are, therefore, the chemical potentials. At equilibrium, 8Ft for the total system is zero. To arrive at the equi-librium condition we let 8n = Sna = — Srib = 8n{ = 8nf

a. This is merely the expression for conservation of mass which yet permits atom species transfer of Sn atoms of type " a " from the fault to the matrix and 8n atoms of type "ft" from the matrix to the fault. For equilibrium, therefore,

(8.5)

In general, the thermodynamic data that are needed to achieve a simul-taneous solution of equations (8.4) and (8.5) are not available. Furthermore, theories on the thermodynamics of solid solutions have not yet been well enough developed to provide reliable analyses of the thermodynamics of un-stable solid solutions (e.g. hexagonal Cu-Ni alloys). But in order to obtain an initial concept of possible trends, the regular solution laws might be invoked. To simplify the analysis this will be done not only for the matrix but equally for the faulted volume, in spite of the fact that the stacking fault region should be treated as a surface.

The free energy of an open regular solution consisting of na "a" atoms and rib "ft" atoms can be written as

(8.6)

where N is Avogadro's number, Fa and are the free energies per mole of pure " # " and "ft" atoms and the last term refers to the contribution of the entropy for random mixing. A similar expression applies to the faulted region. When c is defined as

(8.7)

Equation (8.6) written for one mole of the alloy reduces to

with similar expressions for the remaining free energies in equation (8.4). Then

(8.8)

(8.9)

M = (1 - c) F{ + cFt + AHm(c) + RT{c In c + (1 - c) In (1 - c)}

(F* - F)c = (1 - c)(Ff - Fa) + c(Fl - Fb) + (AH*, - AHvl)e

where the entropy terms vanish. Also

(Ff - F)Cf = (1 - cf){Fl - Fa) + cf(F{ - Fb) + (AH^ - AHm)c, (8.10}

Introducing equations (8.9) and (8.10) into (8.4) gives

(8.11)


For equilibrium conditions, we apply equations (8.6), etc., to the condition given by equation (8.5). For example

and

recalling that

(8.12b)

(8.12a)

(8.13)

and similarly for the faulted region

(8.14)

Consequently the equilibrium condition obtained by introducing equations (8.12) and (8.13) into (8.5) is given by

(8.15)

equations (8.11) and (8.15) therefore constitute the solution to the problem. An explicit expression for c/ is not obtainable from equation (8.15) since (d/dcf) AHm also depends on Cf. But the quantities in the braces of the expo-nential term will be positive and therefore c/ will be somewhat smaller than c. This difference will be greatest at the absolute zero and as the temperature increases Cf will increase very slowly to approach c.

When the solution is ideal the AHm's are zero and the equations (8.11) and (8.15) reduce to the simple expressions, respectively, of

(8.16)

(8.17)

(8.18)

(8.19)

where

and AF=-{F{-Fb)-{Fl-F%)

yb and ya being the stacking fault energies in pure b and a. Therefore (r — r*) can vary mildly with temperature depending on the variation in stacking fault energies of pure a and pure b with temperature.


We now wish to illustrate that substantial high temperature strengthening can be achieved by the Suzuki mechanism. For this purpose we will consider reasonable values for the pertinent thermodynamic quantities. For example, let

!

50 ergs/cm 2

100 ergs/cm 2

150 ergs/cm 2

V = 10 cm 3

h = 2 . 5 x 1 0 - 8 cm

Then

A F x l O 1 4

c ERG/ATOM

O.I 1.66 O.I 3.32 O.I 4 .98 0 . 2 1.66 0 . 2 3 .32 0 . 2 4 . 9 8

( y d — Ya) =

10- 1 4 > j

1 0 - 1 4 ergs/atom i o - 1 4

Introducing this into equation ( 8 . 1 7 ) we find that the results given for c — c/ shown in Fig. 8 . 2 are obtained. Thus according to equation ( 8 . 1 6 ) , we obtain the r — r* vs. T d a t a shown in Fig. 8 . 3 .

F I G . 8 . 2 . Variation of c-c/ with temperature.


These estimates reveal that substantial strengthening can be obtained as a result of Suzuki locking. In general the strength of Suzuki locked alloys decrease only slightly with an increase in temperature. This trend, however, might be slightly modified by the additional effect of temperature on yb — ya which was neglected in this estimate.

5 0 0 6 0 0 7 0 0 8 0 0 T, °K

9 0 0 1 0 0 0

F I G . 8 . 3 . Theoretical variation of ( T - T * ) with temperature for several stacking fault energy differences and two composit ions.

When the stacking fault is only a few Burger's vectors wide, it may be possible to have thermal fluctuations aid in unlocking the dislocations. At high tem-peratures, however, such released dislocations would relock very rapidly as a result of diffusion of solute atoms. This would result in relatively few moving dislocations and thus give low creep rates. Such locking would also take place with all freshly generated dislocations.

When the stacking fault is more than a few Burger's vectors wide, the activa-


where A = 4 RdeGb, R being the atomic radius of the solvent and R(l + e) being the atomic radius of the solute atom.

Interstitial atoms interact with both edge and screw components of disloca-tions as described by Cochardt, Schoeck, and Wiedersich 9 0 . The interaction energy in this case differs in detail from that given for substitutional solid solutions but nevertheless, is reasonably well approximated in its essential features by equation ( 8 . 2 0 ) . We will therefore develop our analysis exclusively in terms of substitutional solutions since analogous descriptions apply to inter-stitial solid solutions.

In Fig. 8 . 4 are shown the circles of constant interaction energy. Taking e to be positive, an atom at r and a will be acted upon by a force equal to the potential energy gradient and directed along the conjugate circle shown broken in the graph. Consequently at sufficiently high temperatures where diffusion can take place the solute atoms will migrate along the orthogonal set of conjugate circles to concentrate just below the core of the dislocation. Simultaneously the region just above the core of the dislocation will become depleted in solute atoms.

We will now attempt to ascertain the situation that will prevail at equilibrium. For this purpose we will assume that we are always dealing with a solid solution. Such complications as arise from precipitation will be described later. As shown by Lucke 9 1 and also Thomson 9 2 , the interaction energy between a dislocation and a solute atom is independent of other sources of strain energy. This arises from the fact that the stresses in the elementary theory of elasticity are linear functions of the strains. Consequently the principle of superposition applies. Therefore, regardless of the presence of other sources of strain, the interaction energy between a dislocation and solute atom is always given by equation ( 8 . 2 0 ) . This statement is correct even when a precipitate forms at the dislocation

tion energy for unlocking the dislocations should be so great that the frequency of unlocking becomes negligibly small. Consequently, practically no creep will take place unless the stress itself is large enough to unlock the dislocations.

In spite of the rather good theory for Suzuki locking, no unique examples of high temperature strengthening by this mechanism have yet been announced.

G. Cottrell Interaction Effects

Cottre l l 8 8 interactions also play an important role in creep, particularly near the lower temperature limit of the high temperature creep range. Substitutional atoms that have atomic radii which differ from the solvent species introduce local strain centers in the lattice. Such strain centers can react elastically with the hydrostatic tension stress fields around edge dislocations. As shown by Cottrel l 8 8 , and Cottrell and Bilby 8 9 , the interaction energy between an edge dislocation lying along the z axis of a cylindrical coordinate system and a solute atom at r and a is given by

( 8 . 2 0 )


core. Therefore a dislocation does not saturate, as was suggested in some of the earlier literature on this subject, as a result of satisfying the strain at the dislocation core. Due to their strain fields solute atoms, of course, interact with each other. As shown by Crussard 9 3 , however, this interaction decreases with the sixth power of the distance between the atoms and therefore involves only local, almost only nearest neighbor, interactions. Solute atom migration occurs as a result of the gradient in the strain energy until the reversed diffusion due to the composition gradient balances the forward diffusion. Solute a tom inter-actions can be incorporated into the analyses by assuming the regular solution laws. In large measure this procedure accounts for the interaction stress fields between solute atoms.

Y

x

F I G . 8 . 4 . Circles of constant interaction energy.

We will first estimate the equilibrium distribution of substitutional solute atoms about an edge dislocation. For this purpose we consider solute atoms distributed between two potential energy levels, F a n d VQ as follows:

N o . N o . N o . Energies sites solute atoms solvent atoms

V n P n - p Vo no Po no — po

We will assume that the alloy forms a regular solution, so that the bond energy for random mixing of the atoms at a potential energy V is

(8.21)


where z is the coordination number, and €AA, *BB, and eAB are the energies of an A-A, B-B, and A-B bond, respectively. A similar result applies for the atoms at the potential energy VQ. Therefore, the free energy of the system at the potential energy V is

and the chemical potential of the solute atoms is

(8.22)

(8.23)

A similar chemical potential applies to the system at Vo. At equilibrium the chemical potentials for each system are identical and

where e = €AB — CAA/2 — €BB/2 is the increase in energy upon breaking an A-B bond and making one half an A-A and B-B bond. We now let the com-position be co = po/no where Vo is zero. Consequently

(8.24)

(8.25)

This constitutes a slightly more accurate and detailed description of the equi-librium concentration of solute atoms around an edge dislocation than that previously suggested by Loua t 9 4 .

The concentration of atoms c in the regions around a dislocation where V is negative is greater than average composition CQ of the alloy. The concentration of atoms, however, also depends on e , and since the interaction energy, i, can be positive or negative, the concentration of atoms about the dislocation can be smaller or greater than that which would be obtained if the solution were ideal. Since the product (c — Co) £ is usually small, however, the effect of non-ideality of the solution is small particularly at high temperatures.

To illustrate the effect of dislocations on the distribution of solute atoms, we will consider the special case of Mg in Al, assuming that e is negligibly small. We will calculate the concentration of Mg atoms at a point 2 . 5b below the core of the dislocation. There

(8.26


l/T x 1 0 3

F I G . 8 .5 . Concentration of M g 2.5b below an edge dislocation in aluminum versus l/T.

Neglecting the effect of local stress fields on the usual phase diagram as well as the problems involved in nucleation of new phases, a precipitate should form on the dislocation for all concentrations above the solubility limit. As seen in Fig. 8.5, precipitates might form near the cores of dislocations even in extremely pure metals. The rate at which such precipitates form will depend

for which we take the following reasonable values

R = 2.86 X 10- 8

6 = 0.12 G = 2.5 X 10 1 1 dynes/cm 2

The results are recorded in Fig. 8.5 on which the solubility limit of Mg in Al is also shown.

T,°K

8 0 0 7 0 0 6 0 0 5 0 0 4 0 0


on the diffusion rate of the impurity atom and the kinetics of nucleation and growth. The feed-in potential of the dislocation, however, will remain sub-stantially unmodified in spite of the formation of a precipitate. Nevertheless, the local strains resulting from the formation of the precipitate will interact with the solute atoms. In general, however, this effect is small. During this process the precipitate, originally strung out along the edge dislocations, will begin to spheroidize. Experience reveals that dislocations decorated in this way are strongly locked and will not ordinarily participate in slip. Such locking, since it takes place in a region where localized stressing has caused dislocation to move, is very effective in restraining the motion of other dislocations. Furthermore, the presence of the precipitate does not materially alter the stress field of the dislocation and therefore such locked dislocations yet react to restrain the motion of newly generated free dislocations. As shown in Fig. 8.5, at high temperatures at the left of the solubility curve, precipitation will not take place and an atmosphere of solute atoms will form about the dislocation. As the temperature is increased the atmosphere becomes less dense. The most dense atmospheres are obtained where the concentration of the solute atom approximately equals its solubility limit at the temperature in question.

A similar analysis applies for interstitial solid solutions. In this case, however, the free energy of the solution in the region where the potential energy is V is better approximated by

where e{pln} is the increase in energy of the solution per interstitial atom, a quantity, of course, that is some function of the occupancy of the sites, {p/n}, and is also dependent on the temperature. Consequently the chemical potential of the solute atom in an interstitial solid solution is

(8.28)

A similar expression is obtained for the region where the potential energy is Vo. Consequently, equating the chemical potentials of the two regions for equilibrium, we obtain

r (8-29)

We let the lattice be one which has q equivalent interstitial sites per solvent atom. Then

(8.30)

(8.27)


(8.31)

(8.32)

The locking trends of interstitial solid solutions are quite similar to those already described for substitutional solid solutions. Screw as well as edge dislocations, however, will attract interstitial atoms.

Dislocations surrounded by atmospheres are also pinned into position, but the pinning is much weaker than that which is obtained when the dislocations become decorated with a precipitate. Such solute-atom atmosphere pinning, however, loses its significance at elevated temperatures, since the unpinning can be thermally activated, as described by Cottrell. In contrast the binding of a dislocation that is completely decorated with a precipitate is very strong, and current evidence indicates that in general the release of such decorated dis-locations cannot be thermally activated.

In their classical paper on interactions between solute atoms and dislocations, Cottrell and Bilby 8 9 presented a beautiful analysis of the viscous drag of atmospheres on moving dislocations. At high temperatures, the atmosphere of solute atoms moves along with a very slowly moving dislocation by diffusion. Since the center of the atmosphere is located at the core of the dislocation in this case, it does not perturb the slow motion of the dislocation. Furthermore, an atmosphere cannot form about a very rapidly moving dislocation because the dislocation does not remain sufficiently long in any one place to permit diffusion of the solute atoms. When a dislocation is moving at some inter-mediate velocity, however, a partial atmosphere lags slightly behind the core of the moving dislocation. The back force due to the atmosphere, in this case, decreases the velocity of the dislocation. As shown by Cottrell and Bilby, the greatest drag on the dislocation will take place at the critical velocity, vc, as determined from Einstein's mobility relationship, when

and letting c = co at Vo = 0, we obtain

where

(8.34)

(8.33)

where Ds is the diffusivity of the solute atom. High temperature creep appears to be controlled by the motion of jogged

screw dislocations. Such dislocations, which only interact strongly with inter-stitial alloying elements, have been shown to have an average velocity

where ga now refers to the average free energy of activation for diffusion of all substitutional elements in the solid solution. Consequently the most effective


Consequently, that interstitial alloying element which is most effective for in-creasing the creep resistance of an alloy at one temperature and one stress, will not be effective at either higher or lower temperatures or stresses. It therefore follows that the highest creep-resistant alloys, from the viewpoint of solute atom drag on moving dislocations, should contain a series of judiciously selected interstitial alloying elements in order to provide solute atom atmosphere drag over the significant ranges of temperatures and stresses to be used. Since the moving dislocations are never in pure screw orientation, some effects due to viscous drag of substitutional alloying elements can also occur.

Solute atoms not only can affect the high temperature creep rate as a result of viscous drag on the moving dislocations but also as a result of their effect on the rate of climb of arrested edge dislocations. Those elements that reduce the rate of recovery by decreasing the rate of climb will result in a more rapid decrease in the creep rate over the primary stage and a lower secondary creep rate. The effect of alloying on recovery has been reviewed by Per ryman 9 5 . Whereas alloying elements frequently increase significantly the temperature for recrystallization, their effect on recovery is usually much less. Furthermore, as shown by Perryman, the recovery rate can either be increased or decreased as a function of alloying For example, Mg in solid solution increases the rate of recovery of Al. Perryman has shown that this is principally due to the fact that Mg increases the vacancy concentration in Al.

When the solute atom concentration is high enough to decorate the edge dislocations with a precipitate, climb is restrained. But it appears unlikely that a solute atom atmosphere per se can materially influence climb. Consequently the frequency of climb in an alloy might be approximated by equation (5.18), namely

V c = v{z - 1) Pje-ffd/*T {e°xxQl*T _ i}

where the two significant variables are the probability of finding a jog, pj, and ga, which here refers to the average activation energy for diffusion of the sub-stitutional alloying elements. Therefore those alloying elements that decrease Pi and increase ga should be most effective in decreasing the creep rate. The probability of finding a jog can be decreased in f.c.c. metals by decreasing the stacking fault energy. Other factors such as the rate of nucleation of additional jogs and the possible effect of solute atoms on the climb has not yet been ex-plored deeply enough to provide additional judgment of this subject.

Short-Range Order Strengthening

Several years ago Fisher 9 6 demonstrated that appreciable solid-solution strengthening is obtained as a result of short-range ordering. Our analysis will

alloying element for providing additional drag on the moving dislocation due to Cottrell interactions are interstitial solute elements for which

(8.35)


follow somewhat the more recent discussions by F l inn 9 7 on the details of this mechanism. Whereas each atom in a crystal is bound to all other atoms, the bond interaction is rather local and exists principally as interactions between nearest neighbors, the second nearest neighbor reactions being small, and third nearest neighbor reaction being almost zero. In our discussion we will con-sider only nearest neighbor interactions.

It has proven convenient to classify solid solution alloys into one of three idealized categories, namely ideal solid solutions, regular solid solutions, and short-range ordered solutions. This classification is easily described in terms of the bond energies of nearest neighbors. Consider, for example, the bond equation

i A-A + \B-B~> A-B (8.36)

for which the energy change is

€ = €A—B — i €AA — i €BB (8.37)

per \ A-A and \ B-B bond that is broken to form an A-B bond. When e is zero the alloy is ideal and the atoms are distributed among the lattice sites in a purely random manner. Although some solid solutions of metals begin to ap-proach ideality no truly ideal solutions have yet been found. Generally, the energy e differs from zero. When e does not differ too much from zero, the atoms may yet be distributed over the lattice in almost a random way and the thermodynamic behavior can be approximated by the regular solution laws. In most alloys, however, i is rather large, due either to electronic or strain energy effects, and short-range ordering is significant.

When i is negative, the energy of the system is lowered by making as many A-B bonds as possible. Consequently, under equilibrium conditions alloys for which e is negative will have a greater number of A-B bonds and fewer A-A and B-B bonds than a random solid solution. A atoms will prefer B atoms as neighbors, thus leading to local or short-range ordering. When e is positive, the opposite takes place and A atoms prefer A atoms as neighbors, and B atoms prefer B atoms as neighbors. This results in clustering. This clustering, however, is merely the negative of ordering and is quite distinct from the agglomeration of similar atoms over a wide region during the early stages of formation of Guinier-Preston zones; essentially only nearest neighbors participate in true clustering.

Since the grand partition function for a short-range ordered alloy has not yet been formulated, approximations to the exact statistical mechanical formula-tion of the problem must be employed. We will use Cowley 's 9 8 approach to short-range order here, and we will limit our discussion only to nearest neighbor interactions. Let a be the degree of order, and PAB the probability that an A atom is next to a B atom. By definition of a

PAB = mA(l — a) (8.38a)

where rriA is the mole fraction of A atoms. When the solution is random, PAB

must be equal to WIA. When a is positive there are fewer than the average number


of A atoms about a B and the alloy exhibits clustering; when a is negative there are more than the average number of A atoms about a B a tom and the alloy exhibits short-range ordering. Thus for an ordered alloy both e and a are negative whereas for a clustered alloy both are positive. The product ea is always positive or, in a random solid solution, zero. The range of a is limited. When PAB = 0, a has its maximum value of 1, and when PAB is 1, a has its minimum value of a = 1 — 1/rriA = (MA — 1 ImX) = — m B / m ^ . The probabili-ties for the remaining arrangements, shown below, follow directly from the definition given in equation (8.38a), namely,

PBB = 1 — PAB = 1 — mA + mAa = mB + mAa (8.38b)

PBA = mB(l - a) (8.38c)

PAA = (1 — PBA) = mA + mBa (8.38d)

We now consider an alloy containing TV atoms in a structure that has a coordination number Z . In this alloy there are a total of NZ/2 bonds and we plan to express the total energy of the alloy in terms of the degree of order a as

( 8 . 4 0 )

When equations ( 8 . 3 8 ) are introduced and equation ( 8 . 3 9 ) is solved,

e{a} = m A e A A + m2

BeBB + 2mAmB€AB — 2mAmBOL€ ( 8 . 4 1 )

where e is given by equation ( 8 . 3 7 ) . Since a and e always have the same sign, e{a} is always less than the average energy of a bond in the randomly arranged alloy where a = 0.

When a dislocation moves through the lattice, it displaces its nearest neighbors across the slip plane and replaces them by next nearest neighbors. But next nearest neighbors are almost random. Consequently if two or more dislocations pass, the alloy becomes almost completely disordered across the slip plane. The average increase in energy per bond that is broken is

€ = e{0} - e{a} = 2mAmBae (8 .42)

In order to illustrate the determination of the flow stress, we will consider as shown in Fig. 8 . 6 slip on the (110) plane in the [111] of a b.c.c. crystal. The dislocation line of length CD moves one b breaking the former bonds of A

( 8 . 3 9 )

where e{a} is the average energy per bond in an alloy that has order a. The total energy of the alloy is the sums of the energies of the A-A, the B-B. and the A-B bonds; which is


with B and C and making new bonds with B' and C . Therefore, the work done

Therefore,

(8.43)

(8.44)

This can indeed be quite a large quantity. It has been shown that the equilibrium degree of order a 0 is given b y "

(8.45)

© A T O M S BELOW

O A T O M S ABOVE

F I G . 8 . 6 . Change in neighbor arrangement during deformation of a short-range ordered alloy. (When atom A above slip plane moves to A', it is no longer coordinated with atoms C and B below the slip plane, but with its former next nearest neighbors B' and C . )

Therefore, at sufficiently high temperatures, where diffusion is possible, the absolute value of the degree of order a 0 decreases slowly with increasing tem-perature. Below somewhat less than half the melting temperature, however, the flow stress (r — T*) is dependent on the "frozen-in" degree of order a and therefore is insensitive to the temperature.

When a long segment of a dislocation moves forward in a short-range ordered alloy, the activation energy needed to overcome the short-range ordered bonding energy is extremely high. For this reason it has been suggested that the motion of a dislocation through a short-range ordered alloy cannot be thermally activated. Therefore at low temperatures where the degree of order a is frozen-in, the flow stress, as shown by equation (8.44), should be independent of the tem-perature. And correspondingly above about half the melting temperature wheie diffusion takes place to maintain the equilibrium degree of order, the flow


stress should decrease only mildly with an increase in temperature in accord with the rather mild decrease of the degree of order with increasing temperature. As we shall see, however, there now appears to be examples of high temperature creep of short-range ordered alloys which demand that dislocation motion in short-range ordered alloys becomes thermally activated and diffusion controlled at high temperatures. We therefore suggest that the high temperature creep of short-range ordered alloys is not yet fully understood. Perhaps the motion of small dislocation segments must become possible in ordered alloys at high temperatures.

The flow stress at a strain rate of y ^ 10+2 m i n - 1 for prismatic slip of Ag (67 at .%) Al (33 at .%) is shown in Fig. 8 . 7 1 0 0 as a function of temperature.

F I G . 8 .7 . Effect of temperature on the critical resolved shear stress for prismatic slip.

Current evidence suggests that the thermally activated Peierls mechanism is operative over Region I. Flow in Region II, over which the yield strength is insensitive to the temperature, must arise from some athermal mechanism. Since the dislocations on the prismatic plane are not dissociated, the athermal mechanism cannot be due to Suzuki locking. The only known mechanism that can account simultaneously for the athermal behavior and the high flow strength is the Fisher mechanism of short-range order strengthening. The equivalent of equation (8.44) for slip on the prismatic plane of the hexagonal system is

We might assume that diffusion is so slow below about 475 °K, which is about

7

(8.46)


one-half of the melting temperature, that the existing short-range order is that which is frozen-in at 475 °K, for which r = 15 x 10 8 dynes/cm 2 . Solving equa-tions (8.45) and (8.46) simultaneously at 475 °K suggests that e = — 760 cal/mole and a = 0.30, a reasonable value for short-range order. Using these values in equations (8.45) and (8.46), we determine that flow stress should be given by the broken curve of Fig. 8.8 in the absence of thermally aided flow mechanisms. Above 475 °K, the experimentally determined flow stress decreases precipitously with an increase in temperature and lies below the curve predicted in terms of short-range order flow stress. Obviously some thermally activated dislocation mechanism is operative above 475 °K.

Thorough investigations on the creep behavior of this alloy above 475 °K have shown that a thermally activated slip mechanism is operat ive . 1 0 1 Following a brief inverted primary stage of creep, over which the creep rate increases slightly, a steady state creep rate is obtained which obeys the relationship

y = (1 .4 ± 0.3) T 3 . 6 ± O . I e-Q/RT p e r s e c ( 8 4 7 )

The activation energy, Q = 3 .3 x 10 4 ± 0 . 1 x 10 4 cal/mole, is insensitive to the stress and is in reasonable agreement with the estimated activation energy for diffusion. Although equation (8.47) suggests that this creep might be con-trolled by the climb mechanism, the creep curve, which does not exhibit the usual decreasing creep rate with strain, reveals that edge dislocations are not being piled-up against barriers. Therefore, some other diffusion controlled


mechanism of creep must apply. Since the stress is much below the value that was calculated necessary to overcome short-range order as long lengths of dislocations move athermally through the lattice, it appears necessary to con-clude that the applied stress can be aided by thermal fluctuation in moving short segments of the dislocations in short-range ordered alloys.

Lowley, Coll, and C a h n 1 0 2 observed similar creep behavior in Fe alloys containing 22.0 to 25.5 a t .% Al over a range of temperatures where short-range order is known to prevail. These authors attributed the acceleration of creep over the initial stages to a Shoeck 1 0 3 reordering effect under stress. Such an explanation, however, cannot apply to the Ag (67 a t .%) Al (33 at .%) alloy since Al and Ag have almost identical atomic radii. They observed that the creep rate for their Fe-Al alloys was given by

£ = AerQt*rrn (8.48)

where 4 .6 < n < 6 .0 and where the activation energy closely approximates that for diffusion of Al in Fe. Although the power of the stress term in equation (8.48) was extremely high, these authors nevertheless correlated their results with Weer tman ' s 1 0 4 theory for viscous drag of a solute atmosphere on the dislocations. But again, such a viscous drag cannot be operative in the Ag-Al alloys just described because the dislocation to solute atom interaction is practically zero.

The data on the creep of the two above-mentioned short-range ordered alloys clearly reveal that the high strengthening that can be achieved by short-range ordering at low temperatures is not maintained at temperatures where diffusion is facile, an experimental fact that contradicts previous concepts on short-range order strengthening. Although the high temperature creep of short-range ordered alloys appears to be diffusion controlled, the dislocation mechanism for this process is not yet completely understood.

9. S U M M A R Y

Substantial strides have been made over the past 15 years in uncovering the role of many of the important factors that determine the physical mechanisms of high temperature creep. In certain special materials, high temperature creep might occasionally occur as a result of thermal activation of special non-diffusional dislocation processes. More frequently, however, creep at high temperatures is determined by diffusional processes, principally the stress directed diffusion of vacancies, motion of jogged screw dislocations and the climb of edge dislocations.

Whereas some of the proposed atomistic mechanisms for high temperature creep, such as the stress-directed diffusion of vacancies, are based on sound, well-understood, detailed theories, others, such as the motion of jogged screw dislocations and, especially, recovery as a result of climb of edge dislocations, yet require additional attention. The major problems in these areas concern not so much the formalities of the theories, which are now well-conceived, but


rather the effect of the details of dislocation patterns and arrangements on these mechanisms. In view of the current interest in this subject, it can be expected that considerable light will be shed on these issues as a result of observa-tion of dislocation patterns in crept metals by means of electron-transmission microscopy. On the other hand, the principle judgment of the validity of any theory will remain the actual experimentally determined creep behavior of materials. This is necessarily so because most of the significant details of creep mechanisms are on atomic scale so as to remain below the resolution of the electron microscope. Consequently the electron microscope will provide the essential preliminary data needed to cull the unrealistic theories from those that have some basis in fact. Theory must yet be extended to provide more detailed information on subgrain formation, all of the details on dipoles, en-tanglements, and all other significant structural features that might be revealed by the electron microscope. Additional theoretical advances coupled with ex-perimental verification are required on the problems of diffusion in multi-component systems including investigations on grain boundary diffusion and pipe diffusion along dislocations.

U p to the present, most of the effort to understand the physical basis for high temperature creep has been based on the results of carefully controlled experi-ments on rather pure metals. Furthermore, some progress has already been made in uncovering some of the basic issues related to the effects of dispersions and alpha solid solution alloying on the creep behavior of metals. Many of these issues, however, require more detailed study before a comprehensive coherent theory for high temperature creep can be formulated.

Today many of the concepts outlined above are being employed, at least qualitatively, in the design of creep-resistant alloys; and some fundamental research has already been undertaken to unravel the basic processes that con-tribute to the creep resistance of complex high temperature engineering alloys. It was not the purpose of this review to cover this most important and interesting area. A voluminous literature now exists on various aspects of this subject. At the risk of serious omission of other important investigations we should like, at least, to mention the outstanding contributions that are being made by G l e n 1 0 5 on this difficult subject.

R E F E R E N C E S

1. L Y T T O N , J. L . , S H E P A R D , L . A. and D O R N , J. E. "The Activation Energies for Creep of

Single Aluminum Crystals Favorably Oriented for ( 1 1 1 ) [ 1 0 1 ] Slip", Trans. AIME 212, 2 2 0 - 2 2 5 ( 1 9 5 8 ) .

2 . TROZERA, T. A. , M O T E , J. and D O R N , J. E. "Activation Energies for Basal Slip in Mag-nesium Single Crystals", Trans. ASM ( 1 9 6 0 ) .

3 . G I L M A N , J. J. "Plastic Anisotropy of Zinc Monocrystals", Trans. AIME 206, 1 3 2 6 ( 1 9 5 6 ) . 4 . "Creep and Fracture of Metals at Elevated Temperatures", Proceedings of Symposium

National Physical Laboratory ( 1 9 5 4 ) . H . M . Stationery Office ( 1 9 5 6 ) . 5 . SULLY, A. H. "Recent Advances in Knowledge Concerning the Process of Creep in Metals",

Prog, in Met. Phys. 6, 1 3 5 - 1 8 0 ( 1 9 5 6 ) .

6 . "Creep and Recovery", Seminar of Am. Soc. Metals, Cleveland ( 1 9 5 7 ) . 7 . M C L E A N , D . "Point Defects and the Mechanical Properties of Metals and Alloys at High

Temperature", Symposium on Vacancies and Other Point Defects in Metals and Alloys, pp. 1 5 9 - 1 9 8 . L o n d o n : The Institute of Metals ( 1 9 5 8 ) .

P H Y S I C A L A S P E C T S O F C R E E P 165

8. D O R N , J. E. (ed.). Mechanical Behavior of Materials at Elevated Temperature. N e w York: McGraw-Hil l (1961).

9. "Structural Processes in Creep", Report of a Symposium, Iron and Steel Institute and Institute of Metals, L o n d o n (1961).

10. D O R N , J. E. "Progress in Understanding High-Temperature Creep", Gillett Memorial Lecture, A S T M (1962); to be published.

11. N A B A R R O , F. R. N . "Deformation of Crystals by the Mot ion of Single Ions". Report of a Conference on the Strength of Solids, pp. 75-90 . L o n d o n : Phys. Soc. (1948).

12. H E R R I N G , C. "Diffusional Viscosity o f a Polycrystalline Solid", / . Appl. Phys. 2 1 , 437-445 (1950).

13. PRANTIS, A . L. and P O U N D , G. M . "Viscous F low of Copper at High Temperatures", Trans. AIME 203, 664-668 (1955).

14. G R E E N O U G H , A. P. "The Deformation of Silver at High Temperature", Phil. Mag. 4 3 , 1075-1082 (1952).

15. B U T T N E R , F. H. , F U N K , E. R. and U D I N , H. "Viscous Creep of Go ld Wires Near the Melting Point", Trans. AIME 194, 401-407 (1952).

16. H A R P E R , J. and D O R N , J. E. "Viscous Creep of Aluminum Near Its Melting Tempera-ture", Acta Met. 5, 654-655 (1957).

17. C H A N G , R. "Creep of AI2O3 Single Crystals", / . Appl. Phys. 31 , 484 (1960). 18. FOLWEILER, R. C. "Creep Behaviour of Pore-Free Polycrystalline Aluminum Oxide",

J. Appl Phys. 32 , 5, 773-778 (1961). 19. COBLE, R. L. and B U R K E , J. E. "Sintering in Crystalline Solids", in Reactivity of Solids,

Proc. 4th International Symposium on the Reactivity of Solids, pp. 38 -51 . J. H. DeBoer et al. (eds.). Amsterdam: Elsevier Publishing Company (1961).

20. W A R S H A W , S. I. and N O R T O N , F. H. "Deformation Behaviour o f Polycrystalline Alu-minum Oxide", / . Am. Ceram. Soc. 45 [10], 479-486 (1962).

21. C H A N G , R. "High Temperature Creep and Anelastic Phenomena in Polycrystalline Refractory Oxides", / . Nuclear Materials 2 , 174-181 (1959).

22. AUSTERMAN, S. B. Discussion in Kinetics of High-Temperature Processes, Endicott House Conference, pp. 67-68 . Cambridge: The Technology Press, M I T : N e w York: Wiley; L o n d o n : Chapman and Hall, Ltd. (1959).

23. VANDERVOORT, R. R. and BARMORE, W. L. "Compressive Creep of Polycrystalline Beryllium Oxide", UCRL-6748 (1962).

24. SCOTT, R., H A L L , A. R. and WILLIAMS, J. "The Plastic Deformation of Uranium Oxides A b o v e 800° C", / . Nuclear Materials 1, 39-48 (1959).

25. A U S K E R N , A . B. and BELLE, J. "Diffusion of Oxygen in Uranium Dioxide", / . Chem. Phys. 28 , 171 (1958).

26. K I N G E R Y , W. D . Introduction to Ceramics. N e w York: Wiley (1960). 27. D O R N , J. E. "Some Fundamental Experiments on High Temperature Creep", J. Mech.

Phys. Solids 3 , 81-116 (1954). 28. SHERBY, O. D . and D O R N , J. E. "The Creep Properties o f Cold Rol led Aluminum Al loy

Sheet Metals", Trans. ASM 43 , 611 (1951). 29. M C L E A N , D . "Grain Boundary Slip During Creep of Aluminum", / . Inst. Metals 81 ,

293-300 (1952-53) . 30. S H E P A R D , L. A . and D O R N , J. E. Role o f Subgrains in High Temperature Creep, 1st

Tech. Rept. , Ser. N o . 108, Issue 1, Contract AF33(616) -3860 , Material Res . Lab. , Inst, of Engr. Res . , Univ . of Calif., Berkeley.

31. SERvi , I. S., N O R T O N , J. T. and G R A N T N . J. "Some Observations of Subgrain Formation During Creep in High Purity Aluminum", Trans. AIME 194, 965 (1952).

32. M C L E A N , D . "Creep Processes in Coarse-Grained Aluminum", / . Inst. Metals 80, 507 (1951-52).

33. M C L E A N , D . and FARMER, M. H. "The Relation During Creep Between Grain Boundary Sliding, Subcrystal Size and Extension", / . Inst. Metals 85, 41 (1956-57) .

34. GAROFALO, F. , Z W E L L , L., K E H , A. S. and WEISSMAN, S. "Substructure Formation in

Iron During Creep at 600° C", Acta Met. 9, 721-729 (1961). 35. SHERBY, O. D . , TROZERA, T. A. and D O R N , J. E. "The Effect of Creep Stress History at

High Temperatures on the Creep of Aluminum Alloys", Proc. Am. Soc. Testing Materials 56, 789 (1956).

36. SHERBY, O. D . "Factors Affecting the High Temperature Strength of Polycrystalline Solids", Acta Met. 10, 135-147 (1962).


37. G R U Z I N , P. L. , K O R N E V , U . V. and K U R D I M O V , G. V. Dokl. Akad. Nauk S.S.S.R. 80 , 49 (1951).

38. M E A D , H. W. and BIRCHENALL, C. E. "Self-Diffusion of Iron in Austenite", Trans. AIME 206, 1336 (1956).

39. FELTHAM, P. "The Plastic F low of Iron and Plain Carbon Steels A b o v e the A 3 -Po int" , Proc. Phys. Soc. {London) B66, 865 (1953).

40. D O R N , J. E. "Progress in Understanding High Temperature Creep", The 1962 Horace W. Gillett Memorial Lecture, American Society for Testing and Materials; to be published.

41 . L A K S , H. "Effect of L o w Stresses on the Creep Rates of a Polycrystalline Aluminum Alloy under Constant Structure", M.S. Thesis, University of California, Berkeley, 1953.

42. M C L E A N , D . and H A L E , K . F. "The Stress Sensitivity of Creep", in Structural Processes in Creep, Report of a Symposium organized by the Iron and Steel Institute and the Institute of Metals, London, Special Report N o . 70 (1961).

43. M C L E A N , D . "Crystal Slip in Aluminum During Creep", / . Inst. Metals 81 , 133-144 (1952-53).

44. SERVI, I. S. and G R A N T , N . J. "Creep and Stress Rupture Behaviour of Aluminum as a Function of Purity", Trans. AIME 191, 909 (1951).

45. FELTHAM, P. and COPLEY, G. J. "Creep in Face-Centered Metals and Solid Solutions with Special Reference to a-Brasses", Phil. Mag. 5, 649 (1960).

46. FRENKEL, R. E., SHERBY, O. D . and D O R N , J. E. "Activation Energies for Creep of

Cadmium, Indium, and Lead", Acta Met. 3 , 472 (1955). 47. FELTHAM, P. and M E A K I N , J. D . "Creep in Face-Centered Cubic Metals with Special

Reference to Copper", Acta Met. 7, 614 (1959). 48. FELTHAM, P. "The Plastic F low of Iron and Plain Carbon Steels Above the A3-Point",

Proc. Phys. Soc. {London) B66, 865 (1953). 49. WEERTMAN, J. and S H A N I N I A N , P. "Creep of Polycrystalline Nickel", Trans. AIME

206, 1223 (1956). 50. FELTHAM, P. "On the Mechanism of High-Temperature Creep in Metals with Special

Reference to Polycrystalline Lead", Proc. Phys. Soc. {London) 69B, 1173 (1956). 51. SHERBY, O. D . "Creep of Polycrystalline Alpha and Beta Thallium", Trans. AIME

212, 708 (1958). 52. SHERBY, O. D . , FRENKEL, R., N A D E A U , J. and D O R N , J. E. "Effect of Stress on the Creep

Rates of Polycrystalline Aluminum Alloys Under Constant Structure", / . Metals, AIME, 275-279 (Feb. 1954).

53. B A Y C E , A . E. , L U D E M A N N , W. D . , S H E P A R D , L. and D O R N , J. E. "Effect of Stress on the

Creep o f Aluminum in the Dis locat ion Climb Range", Trans. ASM 52 (1960). 54. BUTCHER, Thesis, Cornell University, Ithaca, N e w York (1962). 55. Li, J. C. M. "Theory of Strengthening by Dis locat ion Groupings", in Electron Micro-

scopy and Strengthening in Crystals, J. Washburn and G. Thomas (eds.) p. 713. N e w York: Wiley (1963).

56. FRIEDEL, J. "Les Dislocations", Gauthier-Villars, Paris (1956).

57. M O T T , N . F. "A Theory of Work-Hardening of Metal Crystals", Phil. Mag. 43 , 1151 (1952).

58. SEEGER, A. "The Generation of Lattice Defects by Moving Dislocations and Its Appl ica-tion to the Temperature Dependence of the F low Stress of F C C Crystals", Phil. Mag. 46, 1194-1217 (1955).

59. FRIEDEL, J. "On the Linear Work-Hardening Rate of Face-Centered Cubic Single Crystals", Phil. Mag. 46, 1169-1193 (1956).

60. H I R S C H , P. B. and W A R R I N G T O N , D . H. "The F low Stress of Aluminum and Copper at High Temperature", Phil. Mag. 6, 735-768 (1961).

61. H I R S C H , P. B. See N . F. Mott , "The Work-Hardening of Metals", 1960 Institute of Metals Lecture, Trans. AIME 218, 962-968 (1960).

62. R A Y M O N D , L. Unpublished data. 63. COTTRELL, A. H. Dislocations and Plastic Flow in Crystals, p. 48. London: Oxford

University Press (1953). 64. CHRISTY, R. W. "Theory of Creep Limited by Self-Diffusion",/ . Appl. Phys. 30 , 60 (1959). 65. WEERTMAN, J. "Theory of Steady State Creep Based on Dislocation Climb", / . Appl

Phys. 26, 1213 (1955); "Steady State Creep Through Dislocation Climb", / . Appl. Phys. 28, 362 (1957).


66. G R A N T , N . J . and PRESTON, O. "Dispersed Hard Particle Strengthening of Metals", Trans. AIME 2 0 9 , 349 (1957).

67. G U A R D , R. "Mechanism of Fine-Particle Strengthening", in Strengthening Mechanisms in Solids, Seminar o f the A S M , pp. 253-278 (1960).

68. FISHER, J. C , H A R T , E. W. and P R Y , R. H. "The Hardening of Metal Crystals by Pre-cipitate Particles", Acta Met. 1 , 336-339 (1953).

69. O R O W A N , E. "Theory of Dispersion Strengthening", Symposium on Internal Stresses in Metals and Alloys, p. 451 (1948).

70. WEERTMAN, J. "Theory of Creep of Dispersion-Hardened Alloys", Nava l Research Laboratory Report N o . 5123 (1958).

71. A N S E L L , G. S. and WEERTMAN, J. "Creep of a Dispersion-Hardened A l Alloy", Trans. AIME 2\S, 838-843 (1959).

72. G I E D T , W. H. , SHERBY, O. D . and D O R N , J . E. "The Effect of Dispersions on Creep

Properties of Aluminum-Copper Alloys", pp. 57 -63 , Trans. Am. Soc. Mech. Engrs. (1955).

73. THOMAS, G. Private communication. 74. THOMAS, G., N U T T I N G , J. and H I R S C H , P. B. / . Inst. Metals 8 6 , 7 (1957).

75. MITCHELL, J . B. , M I T R A , S. K . and D O R N , J . E. "Dispersed Particle Strengthening at

Low Temperatures"; to be published Trans. AIME. 76. MEYERS, C. L., JR. and SHERBY, O. D . / . Inst. Metals, 9 0 , 380 (1961-62). 77. MEYERS, C. L. , J R . , S H Y N E , J . C. and SHERBY, O. D . "Relation of Properties to Structure

in Sintered Aluminum Powder", Dept . of Materials Science Report N o . 62-10 , Stanford University, Stanford, Calif.

78. SCHVECK, G. "Thermodynamic Principles in High-Temperature Materials", in Mech-anical Behaviour of Materials at Elevated Temperatures, pp. 70-74 . J . E. D o r n (ed.). N e w York: McGraw-Hil l (1961).

79. HONEYCOMBE, R. W. K . "The Effect of Temperature and Alloying Additions on Deforma-tion of Metal Crystals", Progr. in Materials Sci. 1 9 , 95 -130 (1961).

80. F L I N N , P. A. "Solid Solution Strengthening", in Strengthening Mechanisms in Solids, pp. 17-50, A S M (1960).

81. PARKER, E. R. and H A Z L E T T , T. H. "Principles of Solution Hardening", in Relation of Properties to Microstructure Seminar, p. 30, A S M (1953).

82. D O R N , J . E. "Energetics in Dislocation Mechanics", UCRL-10455 . 83. SEEGER, A. "Mechanism of Glide and Work-Hardening in Face-Centered Cubic and

Hexagonal Closed Packed Metals", in Dislocations and Mechanical Properties of Crystals, p. 243. N e w York: Wiley (1957).

84. H O W I E , A . and S W A N N , P. R. "Direct Measurement of Stacking-Fault Energies from Observations of Dislocation Nodes" , Phil. Mag. 6 , 1215 (1961).

85. A H M A D I E H , A. Lawrence Radiation Laboratory, Berkeley, unpublished data. 86. S U Z U K I , H. "The Yield Strength of Binary Alloys", in Dislocations and Mechanical

Properties of Crystals, pp. 361-390. N e w York: Wiley (1957). 87. D O R N , J . E. "Thermodynamics of Stacking Faults in Binary Al loys", to be published;

Letter to Editor, Acta Met.

88. COTTRELL, A. H. "Effect of Solute Atoms on the Behaviour of Dislocations", in Report of a Conference on Strength of Solids, p. 30. The Physical Society (1948).

89. COTTRELL, A. H. and BILBY, B. A. "Dislocation Theory of Yielding and Strain Ageing of Iron", Proc. Phys. Soc. {London) A 6 2 , 49 (1949).

90. C O C H A R D T , A. , SCHOECK, G. and WIEDERSICH, H. "Interaction Between Dis locat ions

and Interstitial A t o m s in Body-Centered Cubic Metals", Acta Met. 3 , 533 (1955). 91. L U C K E , K . See COTTRELL, A. H. "Interactions of Dislocations", in Relation of Properties

to Microstructure Seminar, Ref. 31, p. 144, A S M (1953).

92. THOMSON, R. "The Non-Saturability of the Strain Field of a Dis locat ion by Point Im-perfections", Acta Met. 6, 23 (1958).

93. C R U S S A R D , C Metaux et Corrosion 25, Ur. 301, 203. 94. L O U A T , N . "The Effect of Temperature on Cottrell Atmospheres", Proc. Phys. Soc.

(London) B 6 9 , 459 (1956).

95. PERRYMAN, E. C. W. "Recovery of Mechanical Properties", in Creep and Recovery Seminar, p. I l l , A S M (1956).

96. FISHER, J . C. "On the Strength of Solid Solution Alloys", Acta Met. 2, 9 (1954).


97. F L I N N , P. A. "Solute Hardening of Close-Packed Solid Solutions", Acta Met. 6 , 6 3 1 - 6 3 5 (1958).

98. C O W L E Y , J. M. "An Approximate Theory of Order in Alloys", Phys. Rev. 11, 350 (1950). 99. F L I N N , P. A. "Electronic Theory of Local Order", Phys. Rev. 104, 350 (1956).

100. M O T E , J., T A N A K A , K. and D O R N , J. E. "Effect of Temperature on Yielding in Single Crystals of the Hexagonal A g - A l Intermetallic Phase".

101. H O W A R D , E. M. , BARMORE, W. L., M O T E , J. D . and D O R N , J. E. "On the Thermally Acti-

vated Mechanism of Prismatic Slip in the A g - A l Hexagonal Intermediate Phase", U C R L -10588.

102. L A W L E Y , A. , C O L L , J. A . and C O H N , R. W. "Influence of Crystallographic Order on

Creep of Iron-Aluminum Solid Solutions", Trans. AIME 218, 166 (1960). 103. SNOEK, J. L. "Effect of Small Quantities of Carbon and Nitrogen on the Elastic and

Plastic Properties of Iron", Physica 8, 711 (1941). 104. WEERTMAN, J. "Steady State Creep of Crystals", / . Appl. Phys. 28 , 1185-1189 (1957). 105. See for example G L E E N , J. "A N e w Approach to the Problems of Creep", / . Iron and

Steel Inst. 189, 333 (1958); "The Shape of Creep Curves", Trans. AS ME, J. Basic Eng. Paper N o . 62 WA-133 , presented at Winter Annual Meeting, 1962.

CREEP RUPTURE AND THE TERTIARY STAGE

OF CREEP

A R T H U R W . MULLENDORE and NICHOLAS J . G R A N T

Massachusetts Institute of Technology, Cambridge, Massachusetts.


I N considering the nature of the third stage of creep, it is well to examine first our terms of reference. Tertiary creep is defined with reference to the creep strain versus time curve as the period of accelerating creep rate which precedes fracture. This definition is hardly adequate to indicate the nature of tertiary creep since we use it with reference to both the constant load test and the con-stant stress test. The conditions at the incidence of tertiary creep are quite different in the two tests. The beginning of third stage in the constant stress test represents a departure from steady state conditions of stress and creep rate. In the constant load test, however, the second stage of creep is a period of changing conditions. The stress is continuously increasing, structure is changing, and the first intercrystalline cracks are forming; "steady state" creep is really a slow transition in creep rate through an inflection point of the creep curve. Thus the transition to third stage is not particularly significant with respect to any change in the character of the deformation.

One can attempt to characterize third-stage creep in two different ways. First it can be examined with specific reference to the creep data and the criteria of stability of elongation, and secondly it can be treated in terms of structural changes and changes in the deformation and fracture modes. It is the purpose of this paper to examine both points of view and to see to what extent they can complement each other. The ultimate goal of such considerations is to judge the feasibility of designing, in particular applications, on the basis of third stage creep.

2 . T H E S T A B I L I T Y O F C R E E P E L O N G A T I O N

Three reasons for the existence of third-stage creep can be enumerated: (1) Uniform reduction of cross-sectional area in the constant load test resulting

in an increase of stress which exceeds the concurrent strengthening of the metal by work hardening.

(2) Non-uniform reduction of cross-sectional area by localized necking or intercrystalline cracking in either the constant load or the constant stress test.

7§ 169

170 A R T H U l T w . M U L L E N D O R E A N D N I C H O L A S J. G R A N T

(3) Work softening (which is contrasted to the inadequacy of work hardening in (1) by the characteristic that initial deformation actually makes subsequent deformation occur more easily).

In the low-temperature tension test, the condition for stable uniform elon-gation is clear-cut: it exists up to the point where the change of load, P, with strain 8, is zero, i.e. dP = d8 = 0. In the creep test where strain rate as well as strain becomes an important factor the conditions for the onset of unstable elongation are somewhat different. Just because of the shape of the creep curve and the connection of the tertiary creep with necking or intercrystalline cracking, one tends to regard the end of second stage as the beginning of unstable elonga-tion. This is not quite correct, since in the constant load test the strain rate increases with stress and unstable elongation results at the inflection point in the middle of what we loosely define as second-stage creep. In the constant stress test, the corresponding state of strain hardening would develop towards the end of the transient stage of creep. Thus, unstable elongation does exist during second-stage creep, and it is only when the instability increases to observ-able magnitude that we say we have entered the tertiary stage of creep. The amount of steady state elongation which occurs in the constant stress test beyond the critical condition for unstable elongation may be up to 40 per cent in binary solid solution aluminum alloys. 1

The magnitude of the instability of elongation occurring during the constant load creep test has been calculated by Hoff 2 assuming no transient creep, that the instantaneous creep rate in a test at a given temperature is a function of stress only, and that this function is given by the log stress-log minimum creep rate plot for the metal. He further assumes the starting cylindrical specimens to have a 1 per cent smaller radius at the center of the gauge length than at the ends. His results for 24S-T3 aluminum, using the data of Dorn and Tietz 3 , are shown in Fig. 1 as the fractional decrease in radius at various points along the gauge length as a function of time. It can be seen that although he assumes unstable elongation from the beginning of the test, it is only beyond about 0 .7 of the rupture time, tcr, that the difference in rate of decrease of radius of the neck (x/L = 9 / 1 8 ) and that at the end of the specimen gauge length (x/L = 0) is markedly different. The actual creep curves for this material enter third stage at about 0 .2 or 0 .3 of the rupture life.

If the assumptions in the Hoff calculations are valid, one has a basis for cor-relating third-stage elongation values with minimum creep rate data. One would assume that the degree of instability of elongation in a given test is pro-portional to the relative change of minimum creep rate dimiJimin with stress da/a at that stress. This factor.

should be inversely proportional to the third-stage elongation. From a typical creep data plot (Fig. 2) it is seen that one can represent the relationship of minimum creep rate to stress by an equation:

C R E E P R U P T U R E A N D T H E T E R T I A R Y S T A G E OF C R E E P 171

0 . 5 O.G 0 . 7 0 . 8 0.9 1.0 1.1 . 1.2

F I G . 1 . Radius- t ime relationship for an imperfect tension bar of circular cross-section (Hoff 2 ) .

Therefore, elongation should be constant over any straight line segment of the log (j — log e m i n plot at a given temperature and the same for any parallel segment at another temperature. The third stage elongation values above the 5.10 per cent Mg points at 700°F in Fig. 2 are sufficient to indicate that this correlation is far from correct.

The reason this criterion for stability fails is that in a given constant load creep test the change of creep rate for a given change of stress, due to the re-duction of cross-sectional area, is not the same as the change of e m i n in two separate tests at the corresponding differences of stress. Figure 3 shows the change of creep rate in a given constant load test vs. stress calculated on the basis of uniform elongation compared to the minimum creep rate data for the alloy. The creep rate after the inflection point in the creep curve is sub-stantially lower than the minimum creep rate in a separate test at the correspond-ing stress. Any error in this calculation of stress during the creep test due to non-uniform elongation would tend to reduce the departure from the minimum creep rate curve.

Figure 4 shows elongation data from both constant stress 1 and constant

log l m i n = Kx + m log or (1)

or € m i n = # 2 C 7 ™ (2)

where m is the slope of the line and Ki, the ordinate at I = 1, equals log Then the relative change in creep rate with stress is:

172 A R T H U R W . M U L L E N D O R E A N D N I C H O L A S J . G R A N T

MINIMUM CREEP RATE (HOUR"1)

F I G . 2. Log stress, log minimum creep rate plot for constant stress tests of A l - M g solid solution alloys (Gemmel l 9 ) . Figures above the 5.10 per cent M g points at 700°F show

percent elongation during third stage creep.

load 4 tests of three aluminum-magnesium alloys which are in the solid solution range at the temperatures of testing. The elongations at the end of second stage ( £ T r u e ) and the alongations at rupture (EToi) are given for each alloy with the superscripts L and cr indicating constant load and constant stress tests, respectively. Even though the elongations vary over wide limits with stress, temperature and alloy content, the trends of the values for both types of test show a consistency which is encouraging.

if) CL

? 2 X

(f) LJ cc h-(f)

1

A L - I . 9 2 % MG 7 0 0 ° F

I

- ~ * £MIN

(CONST. S I

I

"RESS)

^ 6 VS. CALC. S C O N S T A N T L(

<T0 = 1 4 0 0

1

T R E S S IN DAD T E S T

PSI

( C O N S T . LC

i

1 DAD)

•

o 0 2 0 3

I - I N / I N . / H R .

0 . 4 0 . 5

F I G . 3. Comparison of the change of creep rate with stress (calculated) in a constant load test and the change of minimum creep rate with stress in constant load and in constant

stress tests.

C R E E P R U P T U R E A N D THE T E R T I A R Y S T A G E OF C R E E P 173

With both kinds of creep test data available for a given material, it becomes possible to correlate third stage elongation data in another way. At the in-flection point of the constant load test (aiL, liL):

LI IL (4)

where 1{L is the length of the gauge portion of the specimen at that t ime: a is

8 0 f o

§ 6 0 o

i—i i M I "

A K 9 4 % M g 5 0 0 ° F "

i i i i i n AI- l .92%Mg " 5 0 0 ° F "

i i i i i

AI-5. IO%Mg 5 0 0 ° F

6 8 10

T R U E ' T R U E

3 4 6 8 10

o - P S I x l O " 3

z 2 0 0 o

£ 160 o

o 120 LLI

I- 8 0

r m 1 1—[

,o AI- .94%Mg / \ 7 0 0 ° F

UJ

o or UJ CL

4 0

0

' T O T

i i—i—i i i i i

AI -5 . l 0%Mg . 7 0 0 °F

6 8 1 0 2 0 4 0 10 2 0 4 0 10 2 0 4 0 6 0 100

c r - P S M O " 2

F I G . 4. Elongation data for constant stress 1 and constant l o a d 4 creep rupture tests of A l - M g solid solution alloys.

the real stress, L is the applied load, and V is the volume of the gauge portion. The change of stress with elongation is given by

(5)

and the fractional change of stress with length is then:

174 A R T H U R W . M U L L E N D O R E A N D N I C H O L A S J. G R A N T

Now in the constant test in the middle of the second stage (lio) the strain harden-ing, dvjdl, is zero. Thus from constant load and constant stress creep data for a given material, one can (for a pair of tests giving the same minimum creep rate) obtain two points on the strain hardening versus elongation curve. If one approximates the curve in Fig. 5 as a straight line, one can define a strain hardening coefficient, S, equal to the slope of the straight line.

If this is assumed to be an inverse function of the stability of elongation, then:

elongation - f[liL(li(T - liL)] (8)

F I G . 5. Hypothetical curve of 1/a x da/dl vs. strain at a constant strain rate.

in Fig. 6, the third-stage elongation values (ETot — ETme) for the constant load tests are plotted versus liL{li(T — liL) for the alloys of Fig. 4. It is seen that although there is a good deal of scatter a definite relationship exists. It is rather remarkable that third-stage elongation values obtained over this wide range of alloy composition, testing temperature, and creep rate should correlate so well. The correlation is no doubt aided by the fact that only a few of the creep specimens ruptured with less than 100 per cent reduction of area. Those which did give intercrystalline fractures with low reduction of area values were prim-arily the specimens of the 5.10 per cent Mg alloy tested at 500°F.

It is also found that the third-stage elongation values for the constant stress tests show approximately the same correlation with this parameter; however, the scatter of the data is somewhat greater.

Although the utility of the above relationship is limited by the necessity of having two sets of creep rupture data, it does indicate that wide variations in third-stage creep elongations may be rationalized in terms of a single parameter.

(7)

C R E E P R U P T U R E A N D T H E T E R T I A R Y S T A G E OF C R E E P 175

3 . D E F O R M A T I O N A N D F R A C T U R E M E C H A N I S M S

In Section 1, it has been indicated that the onset of third-stage creep does not correspond to the point at which unstable elongation begins. Recognizing that unstable elongation exists earlier, it is possible to obtain a measure of its severity from second stage elongation data and to relate this to the elongation occurring during third stage creep. These relationships evolve only from the creep data and make no reference to modes of deformation and fracture. In fact, the reason for the success of the elongation relationship over a wide range of test conditions is that the parameter is sensitive to changes in any deformation or fracture mechanism which produces either a real or an apparent difference in strain hardening response. Thus, it is necessary in order to have a more basic

I- 2 0 0 z UJ O

RR UJ Q_

i 150 z o < o z I O O

- J UJ

UJ

o

Q

cc I

0.94 1.92 510 % MG 500 eF 0 A • 700°F 0 A • y

m ' ° 900°F • • •

/ s s s

A &

' o /

/ A *

/ /

m A '

/ s s s

A &

' o

V

/

/ A *

0.05 0.1 0.15 0.2

F I G . 6. Variation of third stage creep elongation of A l - M g alloys with the parameter liL {ha — hi,).

understanding of tertiary creep that the nature of deformation and fracture processes be examined.

3 . 1 . Creep Fracture

Intercrystalline cracking is a major factor in third-stage creep and is to a large extent responsible for commonly observed scatter in elongation data.

Intercrystalline cracking is a general occurrence in high-temperature creep under conditions of stress and temperature where grain boundary sliding becomes an important mode of deformation. One commonly observes a break in the log stress-log rupture life (or log minimum creep rate) plot of creep rupture data which denotes this transition from transcrystalline to intercrystalline fracture. An example of this is shown in Fig. 7 for an 80 nickel-20 chromium


alloy from Widmer and Gran t 5 . To the right of breaks A, B, C, D, E, and F, intercrystalline cracking occurs with increasing severity as rupture times in-crease. This is indicated by the decreasing values of reduction of area figures above the data points. In some materials 1 ' 4 » 1 0 at the higher temperatures or still lower stresses, a decrease in intercrystalline cracking is again observed and there may even be a revision to transcrystalline fracture.

A variety of intercrystalline cracks occurs during creep, ranging from smooth wedge-shaped cracks, to those which form by the joining of a series of voids, to the isolated voids found on the grain boundaries of alloys which have reverted to transcrystalline failure at high temperatures and long rupture times. In some alloys, it is possible to see a gradual and uniform transition through the whole series of intercrystalline fracture types as temperature and stress are varied.

Q.

i io4

CO

U — A I L ^ J s I 0 0 0 ° F 64

) .69

3

48

- ^ _ 9 3

o ^"^^

r69

C

,80

"^^^ 83

1 8 0 0 ^

^ 4 ! ^ I 3 5 0 ° F

5 0 0 ° F

0 0 ° F

60___ . — "

64 ) .69

3

48

> ^ r 40-^-60?

/ " ^ \ ^ \ / , 9 A . / A * ^ ^ 2 3 / ^

Aoy> 'T/S J 2 0 -

<20%

— " 2 2

40% 3 7 39

16 0~-

2 3

36

0 . 0 0 1 0 .01 0.1 I 10 1 0 0 1 0 0 0

Rupture life, hours

F I G . 7. Log stress vs. log rupture life plot of 80 N i - 2 0 Cr alloy showing reduction of area values (Widmer and Grant 5 ) .

Wedge-type cracks are commonly observed originating at triple points under conditions of testing where grain boundary sliding occurs and where grain deformation processes are insufficient to relieve stress concentrations at the point of origin of the cracks or at the end of the propagating crack. Maintaining the continuity of the intensely concentrated strain associated with grain bound-ary sliding is particularly difficult at triple points where shear on a grain boundary plane must be continued in the opposite grain by the intense action of several slip planes to produce a fold. This process is shown in Fig. 8 . If the grain in front of the sliding boundary cannot deform at the necessary rate in order to accommodate the grain boundary shear, then stresses will build up at the triple

1 0 5


F I G . 8. Grain boundary offset without intercrystalline cracking showing complex fold formation in aluminium.

point and may exceed the strength of the grain boundary, resulting in the formation of an intercrystalline crack, as shown in Fig. 9.

Intercrystalline wedge-type cracks may also originate at points other than grain boundary junctures. They may occur at a twin-grain boundary juncture, at points of grain boundary curvatures, at a grain boundary precipitate— indeed, any point on a grain boundary where restrictions are placed on the ability of the metal to maintain a continuity of strain. At any such point, the stress concentrations which develop may augment the uniform applied stress to the extent that the cohesive strength of the boundary is locally exceeded.

Once an intercrystalline crack is formed, it can propagate by virtue of the

F I G . 9. Intercrystalline crack in n i cke l 1 1 showing little evidence of fold formation.


stress concentration at the tip of the crack and will continue to grow unless the stresses are relieved by deformation at the tip. This is likely to occur when there is a change of direction of the boundary or when it reaches another triple point. An example of crack propagation being prevented by concentrated grain de-formation at the tip is shown in Fig. 10. If adequate means are available to relieve stress concentrations or to strengthen the material at the ends of cracks, it is quite possible for them to stabilize and cease growing. Thus one may obtain very large numbers of small intercrystalline cracks before fraction occurs. In addition to relief of stress by grain deformation, stresses can be relieved by grain boundary sliding. For example, when a crack has progressed up to a triple point, sliding on the two other grain boundaries forming the triple point can relieve the stress concentration. Widmer and Gran t 6 have observed another form of crack stabilization which occurs in 80 nickel-20 chromium alloys. In

F I G . 10. Stopping of intercrystalline crack in high purity a lumin-2 per cent copper two-phase a l loy 8 at grain boundary curvature due to fold formation.

this material, one observes creep curves at temperatures in the range 1500 to 1800°F which, after entering tertiary creep, again revert to decelerating creep rate before ultimate failure (Fig. 11). This behavior is not observed in creep tests of the same material in argon and is attributed to a strengthening of the material surrounding an intercrystalline crack by a process of internal oxidation. As a result of this stabilization of cracks, one obtains a very large amount of third stage creep and very large numbers of discontinuous intercrystalline cracks as shown in Fig. 12. A similar observation has been made by Y i m 1 1 in creep rupture tests of nickel with small amounts of impurity. Shahinian and Achter 7 have also observed strengthening in nickel and a nickel-chromium-aluminum alloy by oxidation of intercrystalline cracks and attribute the effect to complete filling of cracks with oxide which essentially heals the fissure and strengthens the metal.


B

4 0

^ ° 3 0

c o

o

o 2 0

U J

1 1 A l l o y 2 V

i i

~ x T e s t e d in air

• T e s t e d in Argon

4 F r a c t u r e

Tes t in

A r g o n

V | T e s t in

f air S

v •

\ \

8 0

6 0

4 0

2 0

0 4 0 8 0 "0 2 0 0

T i m e in hours

F I G . 1 1 . Creep curves of 8 0 N i - 2 0 Cr 6 tested at 1 8 0 0 ° F and 2 5 0 0 psi. Plot B shows entire curve of specimen tested in air. x on both graphs represents same point.

Another form of continuous fracture which can occur during creep results from the linking of a whole series of small voids which form along the grain boundary. When grain boundary voids were first observed, it was thought that they might form by a process of vacancy condensation, that is, their formation could be described as a classical nucleation process where an excess concentra-

F I G . 1 2 . 8 0 N i - 2 0 Cr tested at 1 8 0 0 ° F , 2 7 5 0 psi. Rupture life, 3 4 7 hr; total elongation, 8 5 per cent. 1 0 x .

A


tion of vacancies, resulting from deformation, precipitate to form a void. It has been subsequently shown that the excess vacancy concentration necessary for this mechanism is much too high for the nucleation to occur. It is rather observed that voids occur at points on the grain boundary where intense stress concentrations exist which have developed in much the same way as those which cause triple point cracks.

An example of intercrystalline cracking which forms by this mechanism is shown in Fig. 13. These pictures are from a ruptured creep specimen of alu-minum-5.1 per cent magnesium which was sectioned, infiltrated with lucite, and then mechanically polished to reveal the true shape of the intercrystalline

F I G . 1 3 . Intercrystalline cracks and voids in A l - 5 . 1 per cent M g tested at 5 0 0 ° F 0 . 5 3 hr rupture life 1 3 .

(a) Impregnated and mechanically polished.

(b) Electrolytically polished.

cracks. The cracks have much the same appearance of the wedge-shaped cracks discussed previously. Following the mechanical polishing, a very light electro-polish reveals the small voids which had been smeared over and it can be seen in Fig. 13(b) that a row of voids exist ahead of the crack in the upper left-hand corner and the other grain boundaries in the picture which have not cracked also have numerous voids along them.

In Fig. 14, we have an indication of why the voids form as they do. The grain boundaries in this specimen have become roughened or serrated in the processes of creep and boundary migration and each of these serrations constitutes (in the presence of grain boundary sliding) a region of stress concentration where the cohesive strength of the metal may be exceeded locally. A grain boundary containing this array of voids can subsequently fail either under the intensified stress normal to the grain boundary or by shear stress on the grain boundary. It is seen in observing the grain boundary in Fig. 14 that a rather small amount

a b


F I G . 1 4 . Voids and cracks in A l - 5 . 1 per cent M g tested at 5 0 0 ° F ; 3 7 hr rupture life: 1 0 0 x .

(approximately 5 /x) of grain boundary sliding would be sufficient to shear off the remaining area of contact between the grains.

When applied stresses decrease, there is a decreasing tendency for inter-crystalline cracking. Figures 13 and 14 show two stages of this tendency. At the higher stresses, the voids are small and closely spaced (Fig. 13); at somewhat lower stresses they are more widely spaced (Fig. 14); and at still lower stresses or higher temperatures (Fig. 15) the voids are sufficiently separated so that there is no tendency for them to join up to form continuous cracks and trans-crystalline fractures are obtained.

F I G . 1 5 . Voids in A l - 1 . 9 2 per cent M g tested at 7 0 0 ° F ; 0 . 2 5 hr rupture life: 1 0 0 x .


It has been indicated that the number of voids on a grain boundary decreases with decreasing stress, and thus there is less tendency for intercrystalline failure and a greater ductility in creep. The reason for the decrease in the number of voids is related to the serrated structure of the boundary. The distance between the serrations on the grain boundary have been found to increase with decreasing stress, and since the distance between voids is related to this dimension, it also increases. Of course, other factors also play a role in this tendency—one import-ant consideration is that the decreased strain rate associated with the lower stress allows more time for recovery processes, such as grain boundary migration, and hence the stress concentrations are reduced.

3 .2 . Deformation Processes

When intercrystalline cracking is not the important factor in tertiary creep, one is concerned with necking down and ductile failure. While locally intensified deformation is an important feature of tertiary creep, a large amount of more or less uniform elongation can occur during third-stage creep. In creep rupture testing of aluminum-copper alloys 8 in the age-hardened condition, for example, virtually the entire creep curve can be tertiary creep and the total creep elonga-tion may be as much as 100 per cent. It is interesting to observe the variation in tertiary creep of this alloy in relation to structural changes produced by different aging treatments and changes in the nature of the deformation. Prior structures produced by two different treatments are shown in Fig. 16. The creep curves for these heat treatments are distinctly different. Structure, Q5, with a fine dispersion of platelet shaped particles, is much stronger than Q7, and its creep curve shows tertiary creep over its entire life. The Q7 structure yields a creep curve consisting of the usual three stages, an explanation of the differences in the creep curves is seen in Fig. 17 which shows the deformation structures of the two materials. While the Q7 structure displays very fine, uniformly spaced slip markings, and little grain boundary sliding, the Q5 structure shows slip concentrated in heavy bands, large amounts of grain boundary sliding and sharp folds at triple points in the grains opposite a sliding boundary. The fact that slip tends to concentrate in bands implies that work softening takes place in these areas. This may be due to slip breaking through the very thin precipitate platelets and creating an easier path for subsequent slip.

Creep curves similar to those for structure Q5 also are observed in other precipitation strengthened materials. Udimet 500, for example, a complex super-alloy hardened by a Nis(Al, Ti) precipitate, gives the creep curves presented in logarithmic form, shown in Fig. 18. Here the steady state creep stage is indi-cated by the portion of the curves with a slope of 1 and the tertiary creep stage gives a slope greater than 1. At stresses above 110,000 psi, second-stage creep exists nearly up to fracture, but below this stress the second stage is compressed to less than half of one per cent elongation, and third stage constitutes much of the larger portion of the creep curve. This trend is summarized in Fig. 19 by a plot of total elongations and the elongations at the end of second stage versus stress.


4 . S U M M A R Y A N D C O N C L U S I O N S

A number of aspects of tertiary creep in metals have been examined in an attempt to show that some of the general concepts of this stage of creep are faulty or have important exceptions.

F I G . 16a. Structure Q5 before t e s t 1 2 . Water quenched from 9 3 0 ° F and aged for 72 hr at 5 0 0 ° F . Large, black round spots are etch pits. 500 x .

F I G . 16b. Q7 structure before t e s t 1 2 . Water quenched from 9 3 0 ° F , then aged for 38 hr at 7 0 0 ° F . Stabilized for 72 hr at 5 0 0 ° F . Black, round points are etch pits. 500 x .

Although one tends to associate tertiary creep with the incidence of unstable elongation and incipient failure, one can more accurately associate these with the beginning of second-stage creep in the constant stress test. Elongation becomes unstable in second-stage and intercrystalline cracking begins during


second-stage creep. Furthermore, large amounts of more or less uniform elonga-tion can take place during the third stage both with and without intercrystalline cracking. From these considerations, it would appear that the distinction between second- and third-stage is rather arbitrary in reasonably ductile materials.

F I G . 17a. Structure Q5 after fracture 1 3 ; heavy folding occurred. Vertical lines are scratches. 150 x .

F I G . 17b. Q7 structure after 10.2 per cent elongation at 2000 p s i 1 3 . Straight lines at 45° are scratched lines. 1 5 0 x .

In the case of Al -Mg solid solution alloys, it has been possible to correlate third-stage elongation with elongation data taken much earlier in the creep test, indicating again that in reasonably ductile materials tertiary creep has a degree of consistency, and predictability which is encouraging.


Since intercrystalline cracking is perhaps the greatest problem associated with third-stage creep, its general characteristics have been reviewed with emphasis on the stress and temperature dependence of cracking. Intercrystalline cracking is most troublesome at the high stress, low temperature side of the range in

T I M E - H O U R S

F I G . 1 8 . L o g - l o g creep curves for Udimet 5 0 0 at 1 2 0 0 ° F .

F I G . 1 9 . Second stage and total elongations for Udimet 5 0 0 at 1 2 0 0 ° F as a function of stress.

which it occurs, since it is here that the rate of propagation of cracks is most severe and ductility is minimized. At higher temperatures or lower stresses, it is often found that stress relaxation mechanisms can operate to reduce the rate of crack growth and stabilize the intercrystalline crack size. The phenomenon of void formation and growth is shown to be a continuation of this sequence of crack initiation and growth.


R E F E R E N C E S

1. GEMMELL, G. D . and G R A N T , N . J . Trans. AIME, 9, (1957); / . Metals, All (April 1957). 2. H O F F , N . J . Trans. ASME, 7, 105 (1953). 3. D O R N , J . E. and TIETZ, T. E. Proc. ASTM, 49 , 815 (1949). 4. M U L L E N D O R E , A . W . and G R A N T , N . J . Trans. AIME, 200, 973 (1954). 5. W I D M E R , R. W . and G R A N T , N . J . Trans. ASME; J. Basic Eng., 829 (December 1960). 6. Ibid. 882. 7. S H A H I N I A N , P. and A C H T E R , M. R. Trans. AIME, 215, 37 (1959). 8. PELLOUX, R. N . M. , C H U D H U R I , A . R. and G R A N T , N . J . Trans. AIME, 215, 204 (1959).

9. GEMMELL, G. D . "Effects of Solid Solution Alloying on Creep Deformation of Aluminum" Sc .D. Thesis, M I T (1955).

10. H A P P , M. B. and G R A N T , N . J . "The Role of Recovery and Recrystallization o n the Creep Rupture Behavior of 2s Aluminum" Submitted for publication.

11. Y I M , W . M. and G R A N T , N . J . "The Effect of Prior Strain and Polygonization on the Creep Rupture Properties of Nickel". T o be published in Trans. AIME.

12. ISHIDA, Y . "Grain Boundary Contribution to Total Strain in Creep of A l - 3 % Cu". S.M. Thesis, M I T (1961).

13. M U L L E N D O R E , A . W . and G R A N T , N . J . Proc. of 8th Sagamore Ordnance Materials Research Conference (August 1961).

THERMAL FATIGUE AND ITS RELATION TO CREEP RUPTURE AND MECHANICAL FATIGUE

SHUJI TAIRA

K y o t o University, K y o t o , Japan

A B S T R A C T

The problem of thermal fatigue is the most complex problem in the field of strength of materials at high temperatures. The resistance of materials under conditions of thermal fatigue (completely reversed strain cycling) has been explored and a number of experi-mental results on various heat-resistant materials have been reported. These results are fairly well summarized by the simple equation due to C. F. Coffin.

However, many structural members are subjected under service conditions to cyclic thermal stress combined with mechanical stress and it is necessary to investigate the behavior of materials under the combined conditions of cyclic temperatures and loads. The experimental procedures for the study of such problems are, however, fairly complex and it would be desirable to be able to predict the strength characteristics o f materials subject to combined cyclic thermal and mechanical stress from the information obtained in simple tests, such as mechanical fatigue tests and creep or creep rupture tests.

In the present paper the manner of superposition of cyclic thermal and mechanical stress is discussed and classified. In this discussion the stress ratio (thermal stress ampli-tude divided by mechanical mean stress) is introduced as a parameter. The case of infinite stress ratio represents the problem of thermal fatigue or completely reversed strain cycling and the strength of the material is characterized by the stress or strain amplitude as a function of the cycles to fracture. The case of zero stress ratio corresponds to creep or creep rupture and the strength is characterized by life to fracture under the mechanical stress applied. Most of the practical problems lie in the intermediate range of stress ratios between zero and infinity. Under those conditions both creep and mechanical fatigue affect the strength significantly.

Experiments were performed using several types of heat-resistant metallic materials and taking stress ratios covering the whole range from zero to infinity. The results of those experiments are presented in the form of stress range diagrams for fracture, taking the thermal stress amplitude as ordinate and the mechanical mean stress as abscissa.

The experimental results are subsequently interpreted by analysis. A s a basis for this analysis, thermal fatigue under completely reversed strain cycling is discussed in relation to mechanical fatigue in the case of cyclic strain, applying the concept of "incremental fatigue damage". The case of zero stress ratio, being the problem of creep or creep rupture under steady load and temperature, is analyzed on the basis of the hypothesis of life consumption in creep rupture. T o obtain the strength of materials subject to the c o m -bined cyclic thermal and mechanical mean stress (intermediate stress ratio), both funda-mental analyses are combined and the analytical stress range diagram for fracture is computed. The analytically obtained diagrams are compared with the experimental results.


FRACTURE of materials caused by cyclic application of thermal strain after a finite number of cycles is called thermal fatigue. The investigation of thermal fatigue has recently attracted the interest of engineers and researchers in the field of strength of m a t e r i a l s 1 - 9 since practical design problems require know-ledge of the resistance of materials against thermal f a t i g u e . 1 0 - 1 3

187

188 S H U J I T A I R A

Thermal fatigue strength is influenced by various factors, as discussed in a series of excellent papers due to L. F . Coff in 1 - 6 , concerned with the resistance of materials in thermal fatigue under completely reversed strain cycling and reporting experimental results on various types of heat-resistant alloys; it is considered that these results are adequately summarized by the simple equation proposed by Coffin.

However, there are many structural members which under service conditions are subject to cyclic thermal stress combined with mechanical s t r e s s 1 0 ' 1 4 ' 1 5

and it is necessary to know the behavior of materials under the combined con-dition of temperature and stress. The experimental procedure for the investiga-tion of such problems is extremely complex, and it is therefore desirable to predict the strength characteristics of materials subject to such conditions, even though approximately, by using information from the usual simple tests of materials at elevated temperature, such as mechanical fatigue tests and creep or creep rupture tests.

Separate types of practical problems are to be considered. The one is the case when both thermal and mechanical stresses are cyclic and completely reversed; in this case, materials fail without significant deformation. There are other structural members which are subjected to a combination of cyclic thermal and sustained mechanical stress; in these cases the mechanical stress plays the role of the mean stress in a fatigue process.

It is well known that the thermal stress arises as a result of the constraint of free expansion or contraction of materials due to temperature change. Consider the case that the structural member is mechanically stressed and further de-formation prohibited by the constraint of neighbouring members; if this member is now subjected to cyclic change of temperature, the combined condition of cyclic thermal and static mechanical stress is realized. However, if the maximum strain level is held constant during the test, the mechanical mean stress tends to relax with the number of cycles. This is the problem of the cycle dependent stress relaxation, which has been discussed in detail for isothermal conditions by J. M o r r o w 1 6 ' 1 7 .

A problem that represents a different type of combination of cyclic thermal and static mechanical stress arises in the so-called incremental c o l l a p s e . 1 4 ' 1 5 Free elongation of the material is not restrained and the mechanical mean stress does not decrease with number of thermal stress cycles. The structural members that are subjected to such stress conditions, for instance, a thick-walled cylinder under internal or external pressure subjected to cyclic temperature change, fail due to deformation. Fracture of such members is accompanied by extensive deformation. 1 4

The primary purpose of the present study is to search for practical means of prediction of the life of structures subjected to combined thermal and mechanical stress cycling from the results of simple fatigue and creep rupture tests at elevated temperature. With regard to the mode of fracture it is tentatively assumed that frac-ture which occurs without noticeable deformation is of the fatigue type, while frac-ture that is accompanied by noticeable deformation is of the creep rupture type.

T H E R M A L F A T I G U E A N D C R E E P R U P T U R E 189

The question of direct correlation between thermal fatigue and mechanical fatigue has often been discussed. Coffin considers that failure in thermal fatigue is caused by accumulation of plastic strain absorbed by the material subject to cyclic temperature variation under constraining conditions, and that the plastic strain in each half cycle is an important variable that is related to the number of cycles to fracture. The mechanical fatigue that is to be correlated with thermal fatigue is of course the so-called cyclic strain fatigue. A number of experiments have been made to elucidate the r e l a t i o n s h i p . 3 ' 9 ' 1 8 However, experimental evidence of the relationship is as yet vague, for want of a rational basis for the comparison of experimental results in both tests.

The author has performed an analysis of this relation in order to obtain a basis for the choice of temperature conditions and strain amplitude in setting up the experiments to relate mechanical fatigue to thermal fatigue, and also to interpret the results of experiments published so far. The analysis is presented in this paper.

In the case of thermal fatigue combined with mechanical mean stress, it is convenient to take the stress ratio A, the ratio of amplitude AaT of cyclic thermal stress component to mechanical mean stress aM as parameter. Zero stress ratio is the case of simple creep test under steady oad and varying temperature; infinite stress ratio corresponds to the case of thermal fatigue under completely reversed strain cycling. Practical problems usually lie in the intermediate region of 0 < A < oo.

The case of zero stress ratio is analyzed as the problem of creep rupture under varying temperature and steady load, on the basis of the life consumption hypo-thesis. The case of infinite stress ratio is related with cyclic strain fatigue at elevated temperature. For the case of intermediate stress ratio, both analyses are combined and the result is presented by the stress range diagram for fracture. The analytical diagram is compared with the experimental results.

In the present paper emphasis is placed on the analytical consideration; therefore the description of the experiments and the presentation of experi-mental results are limited because of limitations of space.

2 . M E C H A N I C A L F A T I G U E A T E L E V A T E D T E M P E R A T U R E

It is known that the fatigue strength of metals at elevated temperature is different from that at room temperature, fatigue life being strongly affected by the frequency of stress cycling, especially in the range of high stress amplitudes. This is shown in Fig. 1, presenting the experimental results of P. G. Forrest and H. J. Tapsell 1 9 . Lower frequency stress cycling results in shorter lives to fracture at the same stress amplitude. The frequency dependence of fatigue life at ele-vated temperatures differs from the fatigue behavior at room temperature.

Forrest and Tapsell consider that the stress-strain relation of materials at elevated temperature is affected by the applied stress rate as shown in Fig. 2, which corresponds to the stress-strain diagrams taken at the stress rates cor-responding to two cycling speeds of 125 and 2000 c p m . 1 9 When the stress level


is low, the difference in strain is negligible; however, if the stress level is high, a considerable difference in strain appears. Taking this effect of strain amplitude into account, the cycling speed dependence of the fatigue life of materials at high temperature can be interpreted.

The experimental relations shown in Fig. 1, between stress amplitude A a versus logarithm of time, is converted into the diagram of plastic strain Aep

0.1

A

o

/ . 0 10 Endurance-hours to

8000 cpm + 2000 cpm X

125 cpm

100 fracture

30 cpm 10 cpm

IOOO

FIG. 1 . Fatigue strength of a low carbon steel (nominal stress vs. t ime to fracture). Tests were carried out under various cycling speeds (Forrest and Tapsell).

^ ± 2 4 o

8 ±16

c * ° 0

elastic : line ^2000 cp m

1 / 125 cpm

±4 ±8 ±12 ±16 ±20 ±24 Alternating strain

:28

F I G . 2 . Dynamic stress-strain curve of 0 . 1 7 per cent carbon steel at 5 0 0 ° C (Forrest and Tapsell).

versus log N as shown in Fig. 3 . 2 0 For the calculation of the plastic strain ampli-tude corresponding to the stress amplitude for each experimental point in Fig. 1, the stress-strain curves of Fig. 2 were used, and the fracture time was converted into the number of cycles to fracture by taking the frequency into account. It is interesting to note that the experimental data of high temperature fatigue of low carbon steel taken under various cycling speeds in terms of Aev versus log N lie approximately on one curve. This leads to the conclusion that the fatigue


I04 io9 /o« io7

Number of cycles to fracture N

Fig. 3. Plastic strain amplitude vs. number of cycles to fracture of 0 . 1 7 per cent carbon steel at 500°C. (Based on test results of Forrest and Tapsell.)

life of materials at high temperature is mainly dependent on the magnitude of the amplitude of the plastic component of strain.

In order to prove the above statement, a series of fatigue tests were carried out at elevated temperature on a few types of metallic m a t e r i a l s . 2 0 ' 2 1 Figures 4 and 5, for example, show the results of experiments on a low carbon steel at 450°C and a 13 chromium steel at 600°C, respectively. The experiments were carried out by employing a constant strain type rotary bending fatigue testing machine. The experimental points fall on one curve in the range of high strain amplitude, but not in the case of low strain amplitude. This feature of the experimental result is explained in the following argument.

In the range of high amplitudes the component of plastic strain in the ampli-tude of alternating strain is large and therefore in the range of low strain ampli-

x/ 0'4

20 R

CM

I ' 6

£ 14

12 IO4

x 1 , 1

O / 7 0 cpm -we? ^

\ ft

> 3000 ci >m — >m

X I

IO5 io6 to7

Number of cycles to fracture N 10*

F I G . 4. Strain amplitude vs. cycles to fracture for 0.10 per cent carbon steel at 450°C (Nishihara, Taira et al.)


tudes the proportion of plastic strain component in the total strain amplitude is small and it is dependent on cycling speed, as seen from the stress-strain curves shown in Fig. 2. It might therefore be expected that the results plotted in the form of the Aep — log TV relation would be similar to Fig. 3.

The character of fatigue at elevated temperature as described above offers a suggestion for expressing the life in high temperature fatigue. 2 1 Considering the fact that the experimental points of the log Aep — log N diagram for high tem-perature fatigue lie on a straight line, a formula for the criterion of fatigue life is derived by assuming that in the process of fatigue an increment of damage A<f)f is caused in the materials during one half cycle of plastic strain Aep and that this damage is a function of Aep of the form

A<f>f = X(Aepy (2.1)

2 0

18 CM \

-o 16

I o

12

i \ p

• o •

170 cpm 1500 cpm

10* to5 /0« I07


10*

F I G . 5 . Fatigue strength of 1 3 chromium steel at 6 0 0 ° C . Presented in strain amplitude vs. number of cycles to fracture (Taira and Koterazawa).

where A and n are constants. While, strictly speaking, the magnitude of the plastic strain component would not be constant during fatigue life, the change, if any, is slight, and is therefore taken as constant. It is also assumed that the material breaks when the sum of damage increments that are absorbed by the material in the case of the alternation of plastic strain reaches a critical value <£o. Thus

<j>o=2NA<f>f (2.2)

where N is the number of cycles to fracture. From equations (2.1) and (2.2)

or in another form AeP . N1^ = (fo/lX)1^ = const.

(2.3)

(2.4)


The value of \L can be determined as the reciprocal of the slope of the log A €V — log N diagram presented by a straight line. From the information of Forrest and Tapsell 1 9 , it was found that ^ is very close to 2, a value substantiated by experimental results of cyclic strain fatigue published so far. 3 ' 4 , 5 , 2 2 , 2 3

Hence the fundamental equation for fatigue at elevated temperature is

Aev . TV1/2 — ( c ^ A ) 1 / 2 = const. (2.5)

It is interesting to note that this equation is the same as Coffin's equation for thermal f a t igue . 1 - 6

3 . A N A L Y T I C A L R E L A T I O N B E T W E E N M E C H A N I C A L F A T I G U E

A N D T H E R M A L F A T I G U E

In the process of thermal fatigue, the strain amplitude Ae and the range of cyclic temperature variation between the lower temperature level Ti and the upper temperature level T2 (T = T2 — T±), are the variables that determine the fracture life, while in the case of mechanical fatigue the strain amplitude Ae and the test temperature T are the relevant variables. In the analysis of the relation between mechanical fatigue and thermal fatigue, the strain amplitude is considered as the common variable, and the idea of "equivalent temperature" for mechanical fatigue is introduced. 2 4

The aim of the present analysis is to find a practical means for prediction of the thermal fatigue life from information obtained in mechanical fatigue tests. In order to predict the life in thermal fatigue tests under the strain amplitude A e and the temperature cycling between T± and T2, it is convenient to assume a mechanical test at a temperature which under the same strain amplitude Ae gives the same fracture life as the thermal fatigue test. The temperature at which the equivalent mechanical fatigue test is to be performed is called the "equi-valent temperature".

The stress-strain relation in one temperature cycle during the process of thermal fatigue is shown in Fig. 6(a). Since the particular shape of the curve is inconvenient for analytical treatment, an approximate stress-strain relation as shown in Fig. 6(b) is adopted as the basis of analysis.

Considering a process of thermal fatigue of a bar both ends of which are constrained, the applied cycle of temperature between T\ and T2 induces an alternating thermal strain of

Ae= aAT= a(T2 - 7 i ) ( 3 . 1 )

where a is the coefficient of linear expansion. In this expression, the strain amplitude is composed of the elastic and the plastic component, denoted as Aee and Aev, respectively; the ratio of the plastic component Aep to the total strain A e is denoted by £.

In the schematic stress-strain curve (Fig. 6(b)), it is assumed that the material behaves elastically during the heating path from T± to T% and the cooling path of T2 to T$. The temperature at T$ should be lower than T2 and, likewise, that

8


at T4 has to be higher than 7 i . Plastic deformation takes place during the tem-perature path from T$ to T± in tension and Ts to T2 in compression. Therefore

T 3 = T 2 - (AT, r 4 = 7 i - (AT (3.2)

In order to consider the progress of damage in a material due to thermal fatigue, the concept of cumulative damage similar to that used in the case of mechanical fatigue at elevated temperature is applied, taking A as a function of the temperature. Thus

J * = A ( D ( J 6 „ ) 8 (3.3)

For further discussion it is convenient to express this equation in differential form, or

8 # = X(T)(Sepf (3.4)

Differentiating equation (3.4), we have therefore

d(8<f>) == 2X(T) 8ep . d(8cp) (3.5)

Although equation (3.5) is originally derived as an equation governing mech-anical fatigue at constant temperature at elevated temperature levels, it is tentatively assumed that this equation is applicable to thermal fatigue, where the variation of X(T) during the temperature cycling between T\ and T% has to be considered. Although the coefficient of linear expansion a is also dependent on temperature, its variation over the relevant temperature range is small and it is therefore introduced in the analysis as a constant.

F I G . 6. Stress-strain relation in thermal stress cycling.

(b) (a)


We could determine Te by using equation (3.11) if an analytical form of X(T) and the values of £ and f 1 were known.

To obtain an analytical form of X(T) is not easy, but the ratio of the value of A(JT) at a temperature T to that at a reference temperature A(7o) can be obtained by using the information of mechanical fatigue tests performed at various temperature levels. The ratio is obtained as the reciprocal of the number

a relation that is identical with the expression

2A</> = Afc + Acf>2 (3.10]

where f i is the ratio of the plastic strain component to the total strain amplitude in mechanical fatigue. Since

(3.9) 2Acf> = 2a?\(Te) £l(AT)*

We are aiming to find the equivalent temperature Te for mechanical fatigue which gives the same life as the thermal fatigue. If the mechanical fatigue test at temperature Te is equivalent to the thermal fatigue due to temperature cycling between T± and T2, the increment of damage during one cycle should be equal for both test conditions. The damage during one cycle of mechanical fatigue equivalent to that in thermal fatigue is obtained by substituting equation (3.1) in equation (3.2) and multiplying by 2, or

Thus the increment of damage in the course of one cycle is given by the sum of A<j>i and Afc.

(3.7)

The upper and lower limits of the integration with regard to temperature are taken as T2 and T% due to the basic assumption that plastic deformation in compression occurs in the course of the temperature change between T% and T2

in the heating path. In the same way, the increment of fatigue damage during the cooling path (T2Ti) is given by

(3.6)

Applying equation (3.5) to the case of thermal cycling as shown in Fig. 6(b), the fatigue damage A</> during the heating path (T1T2) is given by

(3.11)

(3.8)


/ .0

\ 0.5

AISI 34 7 type stainless steel

1 O authors (To s 600 C. A€*0-62%)

f ?

i i

i i

i i /

• CoffinW (Jo =600C, A€ =0.62%)

<§> johanssonW ( To s 500 C, A e = 0.40%)

f ?

i i

i i

i i / / /

/ y

L— o—\/ -®

200 400

Temperature T °C

600

F I G . 7. Temperature dependence of specific thermal coefficient in fatigue damage for austenitic steel.

1.0,

0.-5

200 400 Temperature T °C

O authors ( To "

# Johansson ,

— • 1

S50°C, A€ = 0.4%)

Cr, 0.39Mo, 0.40 Nil 500°C,A€*0.5%]

J

-• "

600 800

F I G . 8. Temperature dependence of specific thermal coefficient in fatigue damage for ferritic steel.

of cycles to fracture at temperature T to that at the reference temperature To or A(r)/A(7o) = NTO/NT for one strain amplitude.

Figures 7 and 8 show the A(JT)/A(7O) versus T relation determined from the experimental results of cyclic strain fatigue tests on an 18-8 Cb stainless steel and a 2\ Cr-1 Mo steel, respectively, performed at several levels at elevated


This means that the value of X(T) at the equivalent temperature is equal to the mean of X(T) in the temperature range from 7 \ to T2.

In the case that the magnitude of the strain amplitude is small, the plastic strain would be negligible in the total value of the strain amplitude. Hence i = iiÔ and therefore

In the integration it is assumed that, since f AT is very small, X(T) in the inte-grand is taken as a constant, being respectively A(T2) and A(7i) for the first and

(3.13)

(3.12)

Hence

temperature. In this relation it is worth mentioning that the temperature co-efficient of mechanical fatigue A(r) is an increasing function of test temperature, which is not linear. This means that the increment of fatigue damage due to a certain value of plastic strain during one half cycle increases with increasing temperature.

In cyclic strain fatigue tests are performed at constant temperature and the damage in one half cycle is obtained by multiplying A(Te) by the square of plastic strain. However, in the case of thermal fatigue the temperature is variable; therefore, the damage is the product of the momentary value of X(T) and the square of the infinitesimal plastic strain increment. Hence the equivalent tem-perature could not be the simple arithmetic mean of 7 i and T2 in thermal fatigue (see equation (3.11)) although the arithmetic mean temperature is often adopted as the test temperature for mechanical fatigue tests carried out for correlation with thermal fatigue tests.

For the correlation of temperature conditions in thermal fatigue and mech-anical fatigue, extreme cases of equivalent temperature should be considered.

In case the magnitude of strain amplitude Ae, taken as common value in both cases, is large enough for the component of plastic strain to make up most of the strain amplitude, it follows that f = f i = 1, and therefore from equation (3.11)


That is, the value of X(T) at the equivalent temperature is equal to the arith-metic mean of the value of X(T) at Ti and TV

The value of X(T) as a function of T in the form of X(T)/X(To) is determined from the results of mechanical fatigue tests made at various test temperatures, as described above. Using the X(T)/X(To) vs. T diagram, such as Figs. 7 and 8, we can determine the value of X(Te)/X(To) using equation (3.11). If we consider the problem either as one of large strain amplitude or of small strain ampli-tude, we can determine X(Te)/X(To) by equation (3.12) or equation (3.14). When the value of X(Te)/X(To) is known, the equivalent temperature is obtained from the X(T)/X(To) vs. T diagram.

In the comparison of test results of cyclic strain fatigue and thermal fatigue, the problem is frequently encountered that cyclic strain fatigue tests are per-formed by using indirect heating by an electric furnace, the temperature of the specimen being uniform along its length, while the thermal fatigue tests are performed by direct heating systems, the temperature distribution being in-evitably non-uniform along the specimen length. 3 ' 9 j 1 8 Since the plastic de-formation is concentrated in the part that is at higher temperature, the life would be different in the two cases of uniform and non-uniform distribution of temperature, even for mechanical fatigue under a specific strain amplitude. In order to overcome this difficulty, an analytical method to convert the result of tests under uniform distribution of temperature to the case of non-uniform distribution is developed in the Appendix.

Figure 9 shows 2 7* 2 8 the results of cyclic strain fatigue in two cases of uniform and non-uniform distribution of temperature in an 18-8 C b - M o stainless steel (AISI 318 type). We see from the figure that the life is different for the two cases. The full line in the figure connects the experimental results for uniform distribution of temperature, the dotted line represents the conversion of these results into those for non-uniform temperature distribution. It is found that the dotted line agrees well with the experimental points for the non-uniform tem-perature distribution.

Figures 10 and 1 1 2 5 show the test results of cyclic strain fatigue and thermal fatigue on 18-8 Cb stainless steel (AISI 347 type) and 2\ Cr-1 M o steel, which are the same materials to which Figs. 7 and 8, respectively, refer. The thermal fatigue tests were carried out by taking a fixed mean temperature of the tem-perature cycling, being 400°C and 300°C, for the respective materials. The cyclic strain fatigue tests were carried out by employing the indirect heating system. The full lines which present the experimental results for cyclic strain fatigue are converted into diagrams for non-uniform temperature distribution, presented by the dotted line. It is found from Fig. 10 that the results of thermal fatigue

(3.14)

the second term in the right-hand side of equation (3.11). If in this equation we put £iAT equal to £AT, that is, if we assume the ratio of the plastic strain component to the total strain amplitude to be equal in cases of mechanical fatigue and thermal fatigue, the relation is introduced


AISI 3 /8 type stainless steel 650mC 2.0

0.1

C

- test d d i s t r i

t e s t J d i s t r i

or but

'or b u t

1

u n i f o r m i o n a l o e

uneven t i o n a l o n

tern g 1

emp S i

I

p e r a t u r e e n g t h

e r a t u r e e n g t h

I

\ ^ _ 4 c pm

cor cas d i s

r —

_ 4 c pm

cor cas d i s

i v e r s i o n e of un t r i b u t i

o f ever on £

f u l l l i l t emper i l o n g lei

le t a t u i lgbr

o the *e L

—

/ 0 2 / 0 3 iO4 10* I06


F I G . 9 . Comparison of the results of strain cycling fatigue tests for the cases of uniform and non-uniform distribution of temperature along length of specimen (Taira and Ohnami).

tests carried out by taking the mean of Ti and T2 as 400°C are close to the converted line of the results of cyclic strain fatigue of the same material per-formed at 400°C. In the case of 2\ Cr-1 Mo steel, however, the test results of thermal fatigue for the mean temperature of 300°C are rather close to the converted line of the test results for cyclic strain fatigue at 600°C instead of the line for the test at 300°C (the mean temperature in the thermal fatigue test).

2.0

Ther

mal

st

rain

am

plitu

de

A€%

0.5

AISI 3 4 7 type stainless steel T"

O thermal fatigue test for uneven temperature distribution (T = 400°C)

^ m ' / cpm

• cyclic strain fatigue test for uniform temperature distribution ( TQ= 400°0 )

— conversion of experimental curve for cyclic strain fatigue test to the case of uneven temperature distribution

. i L

10s / O 4


F I G . 1 0 . Thermal fatigue and strain cycling fatigue of 1 8 - 8 C b - M o (AISI 3 4 7 type) stain-less steel (Taira and Ohnami).

Ther

mal

str

ain

ampl

itude

A

€ %

10s


From the results of the experiment described above, it appears that in the case of austenitic steel the equivalent temperature for mechanical fatigue is close to the mean between Ti and T% of the cyclic variation of temperature, while it is rather close to the upper limit T% for the case of ferritic steel. This result is interpreted by the following analysis.

The test condition for some of the experimental points in Figs. 7 and 8 is shown in Tables 1(a) and 1(b), 2 5 respectively. Because of the large linear expan-sion of austenitic steel, the range of plastic strain for the imposed temperature cycling in the test is larger in the case of austenitic steel than in the case of ferritic steel. Referring to the range of plastic strain shown in the table, the equivalent temperature was determined for the austenitic steel and the ferritic

2fCr-lMo steel


F I G . 11. Thermal fatigue and strain cycling fatigue of 2\ Cr-1 M o steel (Taira and Ohnami).

steel by using equations (3.12) and (3.14) and is presented in the table. It is found that thermal fatigue in the case of austenitic steel is rather close to the case covered by equation (3.12), which gives the value of the equivalent tem-perature close to the mean temperature of the cyclically varying temperature. On the other hand, the thermal fatigue for ferritic steel is governed by equation (3.14), which gives the equivalent temperature rather close to the upper tem-perature level in temperature cycling. The experimental findings are thus sup-porting the analysis.

The analysis presented in this section is based on a simple assumption con-cerning large strain cycling fatigue of metals. The actual mechanisms of fracture in cyclic strain fatigue and in thermal fatigue are necessarily very complex and there are numerous factors that influence the fatigue p r o c e s s . 2 9 ' 3 0 ' 3 1 Thus, although it cannot be said that the analysis relating mechanical fatigue to thermal fatigue is complete, it offers some suggestion on the solution of practical problems.


(a) 2i Cr-1 M o steel (Fig. 7)

Analytical Te

Ae 1 T2 Ti AT Eq. (3 .12) Eq. (3 .14) per cent °C °C °C °C °C

0 . 2 5 0 . 0 4 610 10 600 390 540 0 . 2 3 0 . 0 3 575 25 550 380 500 0 . 2 0 0 . 0 0 550 50 500 370 470

(b) 18-8 Cb steel (Fig. 8)

Analytical Te

As 1 Ti AT Eq. (3 .12) Eq. (3 .14 ) per cent °C °C °C °C °C

0 . 6 0 0 . 4 5 700 100 600 0 . 5 0 0 . 3 4 650 150 500 480 575 0 . 3 6 0 . 1 9 600 200 400 450 560 0 . 3 0 0 . 1 0 600 200 400 450 560

4 . T H E R M A L F A T I G U E C O M B I N E D W I T H A L T E R N A T I N G

M E C H A N I C A L S T R E S S

In this section the problem of thermal fatigue combined with cyclic mech-anical stress of equal frequency is discussed. Since the amplitude of total strain Ae, that is, the sum of the thermal AeT and mechanical strain components A €M, is held constant during test, the problem belongs to the category of large strain cycling fatigue. Therefore, the interest in the discussion is focused on the relation of the strength in thermal fatigue combined with cyclic mechanical stress in relation to that in simple mechanical fatigue. In other words, the aim of the discussion which follows is to find the equivalent temperature for a mechanical fatigue test under prescribed strain amplitude in which the material breaks at a number of cycles that is equal to that in thermal fatigue combined with cyclic mechanical stress.

The test condition of synchronized cyclic thermal and cyclic mechanical strain can be realized 3 2 fixing both ends of the specimen tightly so as to restrain the free expansion or contraction caused by the cyclic alternation of temperature between T± and T2; only cyclic thermal strains of amplitude AeT arise therefore

8§

Equation (3 .12)

Equation (3 .14)

TABLE 1. TEST CONDITION FOR EXPERIMENTAL POINTS OF F I G . 7 A N D F I G . 8


in the specimen. If the specimen is now pushed or pulled at both ends when the upper or lower level of temperature cycling is reached, an additional mechanical strain component AeM is superimposed on the thermal strain. The ratio of ther-mal strain to combined strain Ae is denoted as the thermal strain ratio rj and the test condition is characterized by fixing the temperature cycle, that is, T± and T2 (AT= T\ — T2), the thermal strain ratio rj and the strain amplitude.

There are various combinations of synchronized cyclic thermal and mech-anical strain; these combinations are classified approximately according to the value of the thermal strain ratio 77. The case of rj = 0 corresponds to ordinary mechanical fatigue (AeT = 0), and the case of 17 = 1 to the problem of pure thermal fatigue (Ae = AeT). For thermal strain ratios | rj \ < 1, Ae is greater than ACT; in this case cyclic thermal strain and cyclic mechanical strain are of the same sign or of opposite sign. When 177 | > 1, Ae is smaller than AeT; this is the case when both components of strain are of opposite sign. The former two cases have been discussed in the preceding sections; the latter two cases will be discussed in the present section.

F I G . 12. Equipment for the test of thermal fatigue combined with mechanical stress (Taira and Ohnami).

According to the analysis of Section 3, fracture in thermal fatigue occurs when the amount of fatigue damage attains a critical value. The magnitude of fatigue damage that is absorbed during one half cycle of alternating strain is assumed to be given by equation (3.5), which means that the increment of fatigue damage caused by an infinitesimal plastic strain is dependent on the momentary temperature level; since the infinitesimal plastic strain may be brought about by any means, the analysis presented in the preceding section is applicable to the present problem.

Figure 12 3 2 > 3 3 shows the equipment for the test of thermal fatigue combined with cyclic mechanical stress with some change in electric circuit. This equip-ment is also used for thermal fatigue tests combined with steady mechanical stress, which are discussed in the next section. Figure 13 shows a schematic diagram of the equipment with a brief explanation.


Figure 14 3 4 shows the results of the experiment on thermal fatigue combined with cyclic mechanical stress on a 18-8 Cb stainless steel (AISI 347 type) for the case of rj = f, that is, AeM\AeT = } , with the mean of Ti and T2 at 400°C. The equivalent temperature at which the mechanical fatigue test was to be made was determined by analysis and the cyclic strain fatigue tests were per-

IOOV

200V IOOV

© multiplier

© recorder

© strain indicator

© detector of decrease in minimum temperature © level

© relay amplifier

© weighing bar

© variable speed pulley

© dial gage and limit switch assembly-

reversing circuit for motor

( a )

F I G . 13. Schematic view of the apparatus for the test of thermal fatigue combined with mechanical stress and dimension of specimen (Taira and Ohnami).


formed at this temperature level. The results are plotted by full circles on the same figure. As in the preceding case, the line connecting the experimental points is converted to the case of non-uniform temperature distribution and indicated by a dotted line. The numbers put next to the experimental points are the analytically determined values of equivalent temperatures at which the cyclic strain fatigue tests were performed. It can be seen that the experimental results of thermal fatigue combined with cyclic mechanical stress are close to the corrected line of test results for cyclic strain fatigue. Thus also in this case the analysis gives a good correlation.

AISI 347 type stainless steel

^^rr^46o°c ) 450°C)

/ cpm

) 450°C)

*0°C)

420°C)

O test of thermal fatigue combined with alternating mechanical stress (T m= 400°C, 7 = 5 A )

• cyclic strain fatigue test for uniform temperature distribution

conversion of experimental line for cyclic strain fatigue to the case of uneven temperature distribution

10 id1 io* Number of cycles to fracture N

I04

F I G . 14. Thermal fatigue combined with alternating mechanical stress of 18-8 Cb (AISI 347 type) stainless steel (Taira and Ohnami).

5 . T H E R M A L F A T I G U E C O M B I N E D W I T H

S T E A D Y M E C H A N I C A L S T R E S S

The problem of thermal fatigue combined with steady mechanical stress is interesting from a practical point of view because the fracture is accompanied by significant deformation and the phenomenon is therefore directly related to both fatigue and creep rupture.

Engineering problems of creep rupture are generally discussed in terms of applied stress versus life at a given temperature. On the other hand, in the present paper fatigue fracture is treated in terms of plastic strain versus fracture time with temperature as a parameter. Since we are concerned with both fatigue and creep rupture, it is necessary to use either stress or strain as the significant variable in order to treat the combined effect of fatigue and creep by a common term. Since fatigue fracture can be dealt with in terms either of stress or strain,

Com

bine

d to

tal

stra

in

ampl

itude

A

€ %

o


while creep rupture is a phenomenon dependent on stress, it is preferable to base the discussion of the problem of combined fatigue and creep rupture on stress as a variable.

In order to analyze the fracture life in thermal fatigue combined with steady mechanical stress, the same fracture mechanism that has been used in the pre-ceding sections is considered. It is assumed that fracture occurs when the damage </> accumulated in the materials by alternating thermal stress attains a critical value (fro. In the present case, fracture is brought about by the combined effect of fatigue and creep. This damage which determines the life is composed of the damages in fatigue and in creep, denoted as </>/ and </>c respectively.

t = <f>f + <t>c (5.1)

Fracture occurs when

For damage in creep, on the other hand, the life consumption hypothesis is introduced he re , 3 4 by assuming that a material subjected to a stress or at a temperature T fractures in creep rupture at a time tr when the amount of damage in creep <f>c reaches a critical value </>o, so that the amount of damage absorbed by the material during a short time interval is expressed by

(5.2)

From equation (2.2) and equation (3.3)

(5.3)

where Xf(T) is the temperature coefficient of fatigue damage. If the relation of amplitude of thermal stress to that of thermal strain is expressed in an analytical form, such as

Aep= Kf(AaY (5.4)

where Kf and £ are constants, equation (5.3) becomes

(5.5)

(5.6)

This equation is applied to the case of creep rupture under varying stress and temperature by neglecting the influence of load history. Thus the amount of damage which is absorbed until time t is

(5.7)

The life of a material under conditions of creep rupture is a function of


stress and an analytical relation such as equation (5.8) is often introduced on the basis of experimental results:

t r = XC(T) a-y (5.8)

where XC(T) is the temperature coefficient in creep rupture and y is a constant that is determined as the slope of a diagram representing creep rupture tests. Hence equation (5.7) becomes

If we denote the period of cyclic variation of temperature and stress by p, then t = N . p and therefore

Although the values of X(Te) and <£o can be determined from mechanical fatigue tests as described in Section 3, they are easily obtained from thermal fatigue tests with completely reversed strain cycling, which in terms of thermal stress amplitude versus number of cycles to fracture produce a straight line relation in log A a versus log N as is well known. The coefficient [Xf(Te)/(f>o] K2/ is obtained by introducing the value of A a and N obtained from experiments into equation (5.12).

The coefficients y and XC(T) in equation (5.11) can be obtained from creep rupture tests.

Figure 15 3 3 ' 3 5 shows the result of thermal fatigue tests combined with steady mechanical stress on AISI 347 stainless steel. The equipment described in the preceding section was used for this test. It can be seen that the fracture life very much depends on the value of the stress ratio A. The material subjected to

(5.9)

(5.10)

(5.12)

where A a denotes the amplitude of thermal stress and a is the sum of mechanical mean stress and superimposed thermal stress, that is, a = aM + AaT. Equation (5.11) relates the stress amplitude A a, the mechanical mean stress <JM and the number of cycles to fracture AT. Because of its complexity it is difficult to obtain a direct relation of stress condition to fracture life.

Considering thermal fatigue without mean stress, only the first term in equation (5.11) is relevant:

(5.11)

Substituting equations (5.4) and (5.10) in equation (5.1)


cyclic thermal stress is weakened by the existence of superimposed mechanical stress. Figure 16 3 3 > 3 5 shows the deformation of specimens during the tests. In tests at small stress ratios A the fracture is accompanied by considerable permanent deformation. 3 6 > 3 7

AISI 347 type stainless steel

IO2 IO5


F I G . 15. Thermal fatigue combined with steady mechanical stress of 18-8 Cb (AISI 347 type) stainless steel (Taira and Ohnami).

2 0 AISI 347 type stainless steel

'AT » 3 0 0 ' C ACT= 19.5 kg /mm2

(Tm =3 9 .0 kg/mm2

A T *240*C ACT = 17.5 kg/mm2

Om s 35.0 kg/mm2

stress ratio A s 0.5

I cpm

AT s I60'C ACT = 15.0 kg/mm2

am =3 0.0 kg/mm2

2000 4000 6000 Number of cycles of strain cycling N

8000

F I G . 16. Elongation of 18-8 Cb (AISI 347 type) stainless steel during the test of thermal fatigue combined with steady mechanical stress (Taira and Ohnami).

The strength characteristics of the material under the combined conditions is more clearly demonstrated by the stress range diagram, the thermal stress amplitude being taken as ordinate and the mechanical mean stress as abscissa, as shown in Fig. 17 . 3 5 The experimental points in Fig. 17 are obtained by using the experimental line of Fig. 15; the points on the ordinate axis of Fig. 17 are the results of tests in thermal fatigue without mechanical mean stress and those on the axis of abscissae are the results of creep rupture tests under steady load and temperature.


The analytical stress range curves in Fig. 17 are drawn for 10 3 and 10 4 cycles to fracture. The curves are calculated with the aid of equation (5.11), using information from thermal fatigue tests without mechanical mean stress and creep rupture tests under steady load and temperature. It is found that the analytical curves are close to the experimental points.

50 AISI 347 type stainless steel

40

30

20

£ 10

1

O experimental

analytical _^

N = IO3

* 2

A ' 1 /

N = / 0 4 X \ / I '0.5 -/ \ A I '0.5 -

\ 10 20 30 40

Tensile mean stress Cm kgAnm2

50

F I G . 17. Stress range diagram for fracture in thermal fatigue combined with steady mech-anical stress of 18-8 Cb (AISI 347 type) stainless steel (Taira and Ohnami).

6 . C O N C L U D I N G R E M A R K S

The following observations can be listed as the conclusion of the present study.

(1) Plastic strain amplitude plays an important role in mechanical fatigue of metals at elevated temperatures. The discrepancy of test results in mechanical fatigue under different frequencies is well explained by taking plastic strain amplitude as the basis of comparison.

(2) Thermal fatigue and mechanical fatigue at elevated temperature are directly related by considering the plastic strain component as the common variable. An equivalent temperature TE for mechanical fatigue is introduced based on the concept of cumulative damage in fatigue. Fracture life in thermal fatigue can be approximately predicted from mechanical fatigue tests at the equivalent temperature.

(3) In the case of thermal fatigue at rather large strain amplitude the equi-valent temperature is close to the mean of the upper and lower levels of the cyclic temperature, while in the case of small strain amplitudes it is close to the upper level of cyclic temperature.

Ther

mal

st

ress

am

plitu

de

bar

kg

/mm

2


(4) Fracture life in thermal fatigue combined with synchronized cyclic mech-anical stress is also predicted by using the results of mechanical fatigue tests at the equivalent temperature.

(5) Life in thermal fatigue combined with steady mechanical stress producing a combined effect of fatigue and creep rupture can be predicted by combining the basic analyses for fatigue and creep rupture based on the hypotheses of cumulative damage and life consumption, respectively. The result presented as a stress range diagram for fracture compares well with the experimental results.

In the study of correlation of thermal fatigue and mechanical fatigue, it must be considered that thermal fatigue tests are performed by using direct heating, while mechanical fatigue tests are performed by using indirect heating, as de-scribed in the paper. The temperature of the specimen is approximately uniform in the case of indirect heating, whereas it is non-uniform in the case of direct heating. In the comparison of the results of such tests, the number of cycles to fracture is compared by taking strain amplitude as a common variable. However, this mode of comparison is not adequate because of the existence in the specimen of strain concentrations due to non-uniform distribution of temperature.

When a specimen is directly heated and stressed in axial direction beyond its elastic limit, the strain measured at both ends of the parallel length is the average strain e a v . The actual strain, however, is not uniform along the length of the specimen being concentrated in the parts at the higher temperatures. The maxi-mum value of strain that appears at the highest temperature plays an important role in the fracture of the specimen. In other words, if we adopt direct heating, the estimation of the fracture life has to be made on the basis of the maximum strain amplitude Aemax, instead of the average strain amplitude J c a v . An analyti-cal method to determine the maximum strain amplitude Aemax for a specimen of a given temperature distribution under a given average strain amplitude J c a v is therefore required.

Let us denote the ratio of Aemax to Aeay as iff, which may be considered a strain concentration factor. Then

where (A ee)max — elastic strain component at the location of the maximum strain amplitude

(Aev)max— plastic strain component at the location of the maximum strain amplitude

A P P E N D I X

( A . l )


and

/ — parallel length of specimen £ — ratio of (Aep)max to Aen c m a x

It should be noted that the stress is uniform along the length, although tem-perature is not uniform. Since the plastic strain component is more affected by temperature than the elastic component, the influence of concentration of elastic strain is neglected in computing the concentration of strain at the location of high temperature. Thus ^ e is tentatively put as unity and equation ( A . l ) is written as

It is assumed that the temperature dependence of the plastic strain component is represented in the analytical form

Aep= ce~KlT (A. 4)

where c and K are constants and T is the absolute temperature. Then the reci-procal of the plastic strain concentration factor I/JP can be written in the form

(A. 5)

where x is the distance from the end of the parallel length, the temperature of that portion being T. The constant K is determined from tensile tests made at various uniform temperatures. For example, the value of K for a 18-8 M o - C b stainless steel was determined as 2 X 10 4 by the author and his colleague.

The value of tj included in equation (A.3) cannot be obtained directly from experiment but is determined indirectly as follows: Denoting the ratio of the plastic strain component to the elastic strain component as R9 we have

(A. 6)

(A.7)

on the other hand £ is written as

(A.8)

Then the reciprocal of the concentration factor of strain amplitude is written as

(A . 3)

A . 2 )


In equation (A. 10) x is determined by using equation (A.5) and the value of Rav can be determined from the stress-strain relation in tensile tests at high temperatures. Thus the value of R is obtained by equation (A. 10). Substituting the value of R in equation (A. 8) we obtain the value of f. Then the value of the strain concentration can be determined from equation (A.3) .

Figure 18 shows an analytical relation between the reciprocal of the strain concentration factor l/yj and the ratio f for a 18-8 M o - C b stainless steel.

(A. 9)

(A. 10)

K« 2 0 x I0 4"K

0 0-5 10

F I G . 18. Relat ion between strain concentration factor and plastic strain ratio.

The dotted line in Fig. 9 indicates the correction of test results of cyclic strain fatigue under uniform temperature distribution along the parallel length con-verted into non-uniform temperature distribution obtained by using Fig. 18. Figure 9 shows that the analytical method of conversion of test results from uniform temperature distribution to non-uniform distribution produces good agreement with experimental results.

Combining equations (A.3) , (A.8) and (A.9) , we have

In equation (A.7) , ( A e e ) a v / ( J € e ) m a x is assumed as unity. Then


D I S C U S S I O N

E. K R E M P L (Miinchen, Germany): The result shown by Professor Taira indicating that the speed effect on mechanical fatigue at elevated temperatures can be eliminated by plotting the actual plastic strain versus cycles-to-failure can also be found in a similar way with thermal fatigue tests. In our thermal fatigue tests we plot the width of the hysteresis loop at the end of the first cycle versus cycles-to-failure and we get for each material only one cycles-to-failure curve for this cyclic plastic strain. W e tested Nimonic 80A in four different series and P H 15-7 M o , R H and T H in two different series each. This finding again strongly emphasizes Professor Taira's suggestion that the cyclic plastic strain plays an important part in low cycle fatigue.

I should like to ask Professor Taira whether he observed a decay in the mean stress during his thermal fatigue tests.

S. TAIRA (Author's reply) : The author wishes to thank Dr. Krempl for his valuable comment. Dr . Krempl writes that he made thermal fatigue tests on Nimonic 80A, P H 15-7 M o , R H

and T H under various conditions of temperature cycling and plotted the results by relating the width of the hysteresis loop in the first cycle to the number of cycles to fracture. H e mentions that the results of these tests strongly support the author's suggestion that the cyclic plastic strain plays an important role in determining fatigue life in thermal fatigue. Since the width of the hysteresis loop is directly related to the plastic component of the strain amplitude in low cycle fatigue and thermal fatigue, the author fully agrees with Dr. Krempl's statement.

In the author's analysis of the correlation of mechanical fatigue with thermal fatigue, it is assumed that the component of plastic strain in the strain amplitude imposed is constant throughout the lifetime of the material subject to cyclic strain, and that the fatigue damage done to the material is directly related to the magnitude of the plastic strain amplitude. It is very interesting to note that Dr . Krempl obtained the results supporting the author's analysis by taking as the measure of plastic strain amplitude the width of hysteresis l oop in the first strain cycle.

In practical experimentation, the determination of the plastic strain amplitude is a very important problem, both with respect to measuring method and accuracy. In the author's experiments, the value of the plastic strain component was tentatively determined from the stress-strain curves in static tensile tests for all materials tested. The method of experimentation proposed by Dr. Krempl is very interesting in this connection.

In the author's experiments and analysis, concentration of plastic strain was found around the middle of the specimen length because of uneven temperature distribution; a method of correction of test data has been proposed. It is assumed that similar corrections would be needed in Dr . Krempl's tests.

In thermal fatigue tests combined with mechanical stress, it was originally intended to main-tain the level of mechanical mean stress constant. For this purpose, as described in the paper, the test equipment was constructed so as to keep the lower stress level constant, instead of the mean stress level, because of the difficulty of determining the magnitude of the mean stress for every stress cycle from direct reading of the stress indicator. If the stress amplitude were constant during the whole life of the material subject to cyclic temperature variation between two fixed temperature levels, the mean stress would also be constant. However, in the early stage of damage materials are hardened due to the absorption of cyclic plastic strain, and the mean stress tends to increase with the number of cycles. Conversely, with approach to fracture, it tends to decrease slightly because of the softening effect due to minute cracks in the material.

R E F E R E N C E S

1. COFFIN, L. F. and WISLEY, R. P. Trans. ASME 76, 923 (1954). 2. COFFIN, L. F. and WISLEY, R. P. Trans. ASME 76, 931 (1954). 3. COFFIN, L. F. and WISLEY, R. P. ASTM Special Technical Publication, N o . 165, p. 31 (1954). 4. COFFIN, L. F. and WISLEY, R. P. Trans. ASME 79, 1637 (1957). 5. COFFIN, L. F. and WISLEY, R. P. Paper presented at the Fourth Sagamore Ordnance

Materials Research Conference (August 1957). 6. COFFIN, L. F. and WISLEY, R. P. Proc. SESA 15, N o . 2, 1638 (1958). 7. CLAUSS, F. J. and FREEMAN, J. W . NACA Tech. Note, 4160 (1958). 8. C L A U S S , F. J. and FREEMAN, J. W . NACA Tech. Note, H-165 (1958). 9. M A J O R S , H. Trans. ASME 51 , 421 (1951).


10. BLASER, R . U . Proc. SESA 15, 2, 421 (1958).

11. JOHNSTON, J . R . , W E E T O N , J . W . and SIGNORELL, R . A. N A S A Memorandum 4 - 7 - 5 9 E

(1959). 12. HELMS, H . E . and M C L O E D , R . N . Proc. SESA 17, 1, 57 (1959).

13. N . Sulzer Tech. Review, N o . 3, p. 31 (1955). 14. MILLER, D . R . Trans. ASME, Ser. D , 26, 100 (1959). 15. PARKER, E . W . Aircraft Engrg. 26, 402 (1954). 16. M O R R O W , J . D . and SINCLAIR, J . N . ASTM Special Tech. Publication, N o . 237, p. 56 (1958). 17. M O R R O W , J . D . Trans. ASME, Ser. D , 81 , 190 (1959). 18. S W I N D E M A N , R . W . and D O U G L A S , D . A. Trans. ASME, Ser. D , 8 1 , 203 (1959).

19. FORREST, P . G . and TAPSELL, H . J . Proc. IME 168, 763 (1954).

20. N I S H I H A R A , T . , T A I R A , S., T A N A K A , K . and K O T E R A Z A W A , R . Proc. First Japanese Congress

on Testing Materials, p. 34 (1958). 21. T A I R A , S. and K O T E R A Z A W A , R . Bull. JSME 3 , 10, 325 (1960). 22. B A L D W I N , E . E . , S K O L , G . and COFFIN, L . F . Proc. ASTM 57 , 576 (1957).

23. COFFIN, L . F . Proc. Symp. on Internal Stress and Fatique in Metal, p. 363 (1959). 24. T A I R A , S., K O T E R A Z A W A , R . and O H N A M I , M. JSTM 9, 85, 636 (1960).

25. T A I R A , S. and O H N A M I , M. Trans. JSME, 28 , 191, 792 (1962); to be published in English in Bull. JSME 6 (1963).

26. JOHANSSON, A. Proc. Colloquium on Fatigue, p. 112 (1955). 27. T A I R A , S., O H N A M I , M. and K O T E R A Z A W A , R . / . JSTM 10, 89, 96 (1961).

28. T A I R A , S. and O H N A M I , M. Proc. Fourth Japanese Congress on Testing Materials, p . 45 (1961).

29. Proc. Colloquium on Fatigue (Stockholm, 1955). 30. Proc. Symp. on Internal Stress and Fatigue in Metal (1959). 31. ASTM Special Tech. Publication, N o . 237 (1958). 32. T A I R A , S., O H N A M I , M. and SHIRAISHI, T . T O be published in Proc. Sixth Japanese Congress

on Testing Materials (1963). 33. T A I R A , S., O H N A M I , M. and M I N A T A , H . Proc. Fifth Japanese Congress on Testing Materials,

p. 45 (1962). 34. T A I R A , S. and O H N A M I , M. Bull. JSME 5, 17, 45 (1962). 35. T A I R A , S., O H N A M I , M. and SHIRAISHI, T . T O be published in Trans. JSME 12, 114 (1963). 36. T A I R A , S., O H N A M I , M. and K Y O G O K U , T . / . JSTM, 10, 89, 402 (1961).

37. T A I R A , S., O H N A M I , M. and K Y O G O K U , T . Trans. JSME2S, 191, 800 (1962); to be published in Bull. JSME 6 (1963).

THE BEHAVIOR OF MATERIALS IN A

HIGH SPEED ENVIRONMENT

F. P. BOWDEN and J . H . BRUNTON

Cavendish Laboratory, Cambridge, England


I N an age of spacecraft, rockets, and high speed aircraft, there is a considerable need for structural materials which are capable of functioning satisfactorily in a high speed environment. High speeds may also mean high temperatures. I t is now commonplace in the design of vehicle structures to allow for long period working temperatures in excess of 1000°C. The problem here is one of obtaining materials with the necessary high temperature strength and at the same time an economical strength to weight ratio. As a result of recent research into the properties and behavior of the higher melting-point solids, a number of new engineering materials have been proposed for this work. In addition to the high temperature problems, there are also difficulties associated with operating at very low atmospheric pressures. In outer space the gas pressure falls to less than 10~ 1 5 m m Hg. At these pressures protective films of oxides, adsorbed gases, and lubricants cannot be maintained on moving parts, and as a result the friction is high and seizure due to adhesion of the surfaces in contact has to be contended with. Research on the friction and lubrication of clean surfaces in vacuo, once of academic interest only, is now of considerable practical importance in this field. Another somewhat unusual hazard met with in high speed flight is that of the rapid erosion of structures by fine meteoritic particles in outer space, and by rain drops or ice crystals in the lower atmosphere. The impact velocities are extremely high; the impact of a single raindrop can result in appreciable damage or even penetration of the outer skin of an aircraft or rocket. Studies are now being made in the laboratory on erosion under high velocity impact in an attempt to understand the mechanism of damage, and to measure the useful life of materials under this kind of attack.

We hoped it might be of interest to this conference to describe briefly some of the work we are doing in our own laboratory which is relevant to these problems. The object of the various studies has been to examine, using relatively simple physical methods, the basic mechanisms which determine the behavior of solids under these extreme conditions.

2. F R I C T I O N A T H I G H S L I D I N G S P E E D S

A very elegant technique was developed by Professor Beams in 1947 1 for

suspending and spinning rotors up to very high speeds. This same technique has

214

B E H A V I O R OF M A T E R I A L S 215

been used by Bowden and Freitag 2 for studying the friction and the deformation of solids at sliding speeds up to 1000 m/sec. Sliding speeds of this magnitude are well in excess of those met with in normal engineering practice but are ap-proached in certain special cases, as for example by rocket-driven sledges run-ning on rails or again by the driving band of a shell as it moves down the barrel of a gun.

Figure 1 shows diagrammatically the arrangement used. A steel ball is sus-pended in the magnetic field of a solenoid, and vertical stability is achieved by a

F I G . 1 . Schematic drawing of apparatus for studying friction of solids at very high speeds. (Steel sphere S suspended electromagnetically in vacuum chamber and rotated by mag-netic field. Friction measured by deceleration of sphere when the three specimen surfaces

D rub against it.)

photoelectric feed-back system. If the ball sinks, more light falls on to a photo-cell, the current in the electromagnet is increased, and the ball rises. The freely suspended sphere is accelerated by a rotating magnetic field of constant fre-quency (22,500 c/s) in an atmosphere of low pressure (about 10~ 4 m m Hg). The forces opposed to the revolution are so small that the surface speed is limited only by the bursting strength of the sphere. A hard ball-bearing ball can with-stand operating speeds of up to about 1000 m/sec. To measure the speed of rotation, the polished ball is marked by two short lines at opposite ends of a diameter. If fine grooves are produced by grinding or etching, the symmetry of the sphere is altered in such a way that the spinning ball assumes the desired orientation, i.e. one of the marks will always be on top and therefore traverse the area illuminated by the controlling light-beam once every revolution. A photomultiplier cell picking up the reflected light then transmits the signal to a

Electromagnet

Ught-Eeam

216 F. P . B O W D E N A N D J. H . B R U N T O N

frequency integrator whose output—a voltage proportional to the frequency of rotation—drives a pen recorder.

As soon as the rapidly spinning ball touches another surface it slows down and, provided it remains stable, the deceleration can be recorded. If the normal force between the sliding surfaces is known, the coefficient of friction can then easily be calculated. To guard against instability at the instant of contact, the specimens are arranged symmetrically as illustrated by the diagram of Fig. 1. In this way the desired load can be applied at speeds of more than 800 m/sec

F I G . 2 . Suspended ball floating between magnet pole piece and stabilizer cone. (The 4 horizontal accelerating coils can be seen on either side of the ball; the vacuum chamber

and friction pads are not shown.)

and smooth sliding is achieved. Three vertical surfaces are mounted at 120° to each other and in such positions that they cannot interfere with the ball during its acceleration. Two of them are screwed down rigidly while the third one is fastened to a leaf spring. When the spring is released this surface touches the ball first and pushes it a short distance (about 0.001 in.) sideways until the stationary specimens are met. Thus the ball is trapped and cannot escape in the horizontal plane in which the normal loads and the friction forces are acting. Small metal disks (i in. diameter and ^ in. thick) were normally used as specimens and rigidly attached to Tufnol holders. The Perspex cylinder on which these are mounted is hollow and contains the stabilizer. This is an essential part of the apparatus since it counteracts the ball's tendency to precess or oscillate. It con-


sists of a short rubber stem which supports a ferromagnetic cone and is thus magnetically linked to the suspended ball. In addition to an elastic restoring force a certain amount of damping is necessary and this is provided by immersing the cone in glycerine. The suspended ball and driving coils are shown in Fig. 2 together with the stabilizer. The vacuum chamber and friction pads are not shown in this figure.

Some of the values for the coefficient of friction as a function of sliding speed are shown in Fig. 3. The curves show that at high sliding speeds the coefficient of friction in every case drops to a small value. As the speed drops the friction force increases until, at very low speeds, seizure occurs with a violent jerk. This general pattern of behavior occurred in all metals and alloys studied in this way.

O SOO IOOO 1 5 0 0 m.p.h.

SLIDING SPEED

F I G . 3. Coefficient of friction of metals as a function of sliding speed. (Seizure occurs at very low speeds.)

A study has been made of the structure of the wear marks produced during high-speed sliding. Figure 4 illustrates typical changes in appearance of wear marks with sliding speed, the metal in this case being copper. At 1 m/sec (Fig. 4(a)) normal abrasion marks are found. As soon as the oxide film has been removed by this process, the metals adhere together and seizure occurs. Above 150 m/sec the appearance of the wear mark changes (Fig. 4(b) and 4(c)), and large-scale plastic flow is apparent. The comparatively deep layer of metal sheared in this operation will cause the observed coefficient of friction to be high—at these speeds it has a value of 1.5. At much higher sliding speeds (ca. 600 m/sec), Fig. 4(d), the wear marks are quite shiny in appearance and suggest that surface melting has occurred. Further, the metal grains in the deformed area are smaller and randomly orientated compared with the rest of the speci-

218 F . P . B O W D E N A N D J. H . B R U N T O N

men; this change in structure provides further evidence of surface melting. I t is not surprising then that, at high speeds with part of the surface layer molten, the coefficient of friction should be as low as 0 .2 . Calculations show that the temperature gradient will be extremely steep near the surface, and that only a thin surface layer will be molten. In this respect the friction of metals at high sliding speeds is analogous to the friction of ice or snow crystals at ordinary speeds, where a layer of water is produced on the surface of the ice (Bowden3).

In experiments using Duralumin surfaces the temperature at the interface was measured using very fine thermocouples. The arrangement of the thermocouple

F I G . 4. Surface damage of copper, produced by rotating steel balls at various sliding speeds (arrow indicates the direction of sliding).

(a) 1 m/sec abrasion and tearing (180 x ) . (b) 150 m/sec flow of metal (225 x ) . (c) 150 m/sec prolonged sliding flow of deep layers (120 x ) . (d) 600 m/sec surface flow (225 x ) .

is shown in Fig. 5. Temperature flashes in the region of 600 °C were recorded on the screen of a cathode-ray oscilloscope as the steel sphere was decelerated. Since the melting-point of duralumin is 638 °C, it seems likely from these meas-urements that local melting occurs, at least in the surface layers.

The experimental method used above has some limitations if the surface of the specimen breaks up during sliding or if large-scale melting occurs. The metal which is ejected from the area of contact will cut off the beam of light used to measure the friction. Further, a soft metal with a low melting-point will ex-perience a considerable amount of wear although the time of contact may be only a few seconds (this corresponds to a sliding distance of a mile or more).


thermocouple

F I G . 5. Schematic presentation of method used for measuring surface temperatures at the interface.

If excessive wear occurs, the load applied to the specimen changes appreciably during sliding and in an uncontrolled manner.

In such cases, then, a technique is required which does not depend upon a beam of light for the friction measurements, and is suitable for contact of a very short duration so that the wear is small.

A method developed by Bowden and Persson 4 satisfies both these require-ments. The steel ball is first suspended and accelerated in the same way as before and then dropped on to a flat specimen which is set at an angle of 30 degrees with respect to the axis of rotation. The spinning ball bounces off, but not in the plane normal to the specimen surface (Fig. 6). The sliding friction

F I G . 6. Diagram illustrating the measurement of friction at high sliding speeds. (Spinning sphere is dropped onto the inclined plane, deflection is proportional to the friction ju.)


causes a deflection sideways and it is easy to show that the angle of deflection is directly proportional to the coefficient of friction and independent of the load. Essentially the measurement of friction consists simply in finding the spot where the spinning ball hits a certain target after the rebound. The time of contact during which the ball slides on the inclined surface and the load exerted during this time are measured by means of a pressure sensitive piezo-crystal.

The results obtained in this way for the coefficient of friction agree well with those found by the older method. These results were further supported by values obtained for the frictional force using a shear sensitive piezo-crystal. In this latter case a suitably orientated and polarized lead zirconate-lead titanate piezo-electric element was used to measure the shear or friction force exerted by a spinning ball dropped on to it.

i o n

IOO 2 0 0 3 C O 4 0 0 5 0 0 6 0 0 7 C O

S L I D I N G SPEED m/s

F I G . 7. Coefficient [x for steel sliding on Wood' s alloy and on bismuth.

The frictional behavior of a large number of materials has been studied using the deflection method. It has been observed that in all cases the deformation is plastic and not brittle. This is perhaps surprising since the shear rates are very high indeed. Under such conditions it might be expected that a brittle material such as glass would shatter; instead it was found to flow plastically. This and similar observations on lead, tin, bismuth, silver nitrate, and various polymers suggest that thermal softening, together with the large compressive stresses exerted for the duration of the impact, favour plastic rather than brittle failure even in normally quite brittle materials.

In bismuth and the low melting-point Wood's alloy it was found that under the high-impact load and at high rotational speed, large-scale melting occurred within the time of contact ( I O - 4 sec). The friction in cases such as bismuth arises primarily from the shearing of the liquid layer. As this is wiped away by the spinning ball the area of the film grows rapidly leading to an increase in fric-tional drag. The effect on friction is shown graphically in Fig. 7. Analysis has shown that a low melting point and low thermal conductivity are required if


0 200 400 600 sl iding speed (m/sec)

F I G . 8. Coefficient jl for steel sliding on lead and on tin.

melting and large-scale wear is to occur. In this respect it is significant, that the minima in the friction curves for lead and tin (Fig. 8) are noticeably higher than those for bismuth and Wood's alloy (Fig. 7).

Highspeed friction measurements using the dropping-ball method were also made on certain polymers. It was found that in nylon and Terylene, surface melting occurred at moderate sliding speeds. With increasing sliding speeds the coefficient of friction for nylon decreased steadily from 0.23 at 30 m/sec to 0.075 at 675 m/sec (Fig. 9) ; while the diameter of the wear mark was very much less than that produced in metals at equivalent speeds. This behavior is thought to be due to the relatively high viscosity and low thermal conductivity of molten polymers compared with molten metals at their respective melting points. Because of this the molten polymer absorbs a considerable amount of heat, which is removed by the "wiping-off" action of the moving ball and is not available for further melting. The heat generated therefore fails to reach the liquid-solid interface and the molten zone remains quite small.

The experiments which have just been described give some idea of what happens at a frictional contact at extreme sliding speeds. Surprisingly enough

o

0 200 400 600 sl iding speed (m/s)

F I G . 9. Coefficient [i for steel sliding on nylon.


we find the frictional resistance to be very small, even in conditions (high vacuum) where we might expect seizure of the surfaces to occur. On the other hand, wear rates were found in general to be high. Some of the new polymers such as Terylene and nylon behave remarkably well at very high speeds—they have better wear properties than most metals and have a low coefficient of friction.

Although normally a designer aims to avoid solid-to-solid contact at any sliding speed, it sometimes happens that this occurs. The right choice of mater-ials, for example, in the case of bearing alloys, can avoid serious damage. As we move into the range of higher speeds, it becomes even more important to know just how solids behave and what frictional forces are likely to arise when contact occurs. Studies of the kind just described give some idea of what is involved under these unusual conditions and help in deciding on the best material for the work.

3 . T H E B E H A V I O R O F M A T E R I A L S A T H I G H T E M P E R A T U R E S

There is at the present time a great need for solids which possess satisfactory mechanical and frictional properties at very high temperatures. Propulsion units now function at white heat, while the frictional heating during the re-entry of a space vehicle into the atmosphere may lead to temperatures which exceed the melting point of highly refractory solids. Conventional engineering materials cannot be used much above 800 °C. An intensive study is therefore being made of the properties and behavior of the high melting point metals and their oxides, carbides and borides with a view to their eventual application as structural materials operating at elevated temperatures. These investigations of course cover a broad field; we thought it would be of some interest here if we gave a brief account of preliminary studies on the hardness and friction properties of refractory solids at temperatures up to 2000 °C.

3 . 1 . Hardness at Elevated Temperatures

The measurement of hardness, apart from its obvious practical importance, provides a method for studying the strength properties of solids and their mode of deformation over a range of temperature. At elevated temperatures a re-bound method of measurement similar in principle to the Shore Scleroscope 5

has much to recommend it. With this method impact occurs between a cold spherical indenter and the hot specimen, the time of contact being of the order of a microsecond. This eliminates the problem of softening of the indenter and possible indenter/specimen reaction, effects which limit conventional hot-hardness measurements to temperatures below about 1500°C. The measurement itself (the rebound height) is made outside the high temperature zone. L. M. Fitzgerald 6 has constructed a high temperature hardness apparatus of this type and has used it to measure changes in hardness with temperature up to a tem-perature of 2000°C. The apparatus is shown schematically in Fig. 10. The speci-men under test is mounted on a massive graphite anvil inside a carbon tube


resistance furnace. Tungsten carbide indenters ^ i n . in diameter and 0.25 g in weight are dropped from a height of 22 in. through the furnace tube on to the specimen surface. The height of rebound is calculated from the time of suc-cessive transits of the rebounding ball past a photomultiplier tube.

RadioMon shielding

Iropping mechanism

»Water out-

Double walled "cooling jacket

) Lamp

Photomultiplier

Specimen SCALE • INCHlt

0 1 1 » 4

F I G . 10. Apparatus for dynamic hardness measurement at elevated temperatures.

Tabor 7 has shown that the dynamic hardness P is given by

0 )

where hi = height of fall (22 in.)

h2 = height of rebound

n = radius of indenter ( ^ in.)


m = mass of indenter (0.25 g) E\ and E% are the values of Young's modulus for indenter and speci-

men respectively. Poisson's ratio is assumed to be 0 . 3 .

This relation applied only to indentations which produce permanent deformation in the surface. In most of the experiments referred to here the impact pressure was sufficient to cause a permanent depression. The dynamic hardness at room temperature agreed well with hardness values obtained with a conventional hardness machine. At high temperatures it has not been possible so far to cal-culate the absolute hardness values since the variation in Young's Modulus with temperature was unknown for the materials under study. However, it is

201

c r 6

X

4

2[ _____ 0 S00 1000 1500 2000 2500

Temperature (°C)

F I G . 11. Variation of height of rebound with temperature for a fixed height of fall (22 in.) for titanium carbide, tungsten carbide, and boron carbide specimens.

clear from equation (1) that if the value of the term containing the elastic con-stants does not vary greatly, the height of rebound will provide a good measure of the hardness.

Fitzgerald has used the apparatus to obtain the hardness temperature curves for sintered specimens of WC, TiC, TaC, NbC, ZrC, VC, B 4 C, UC, U B 4 , T h B 4 and Z r B 2 up to temperatures of 2000 °C. Curves for TiC, WC, and B 4 C are shown in Fig. 11. These materials were sintered to densities greater than 98 per cent of the theoretical density, the final grain size was of the order of 1 p.

The curves show a continuous decrease in hardness with increasing tempera-ture. The shape of the individual curves varies between that for titanium carbide and that for tungsten carbide. Boron carbide is exceptional. With this material no decrease in height of rebound is observed until approximately 1400 °C. This result for boron carbide is all the more remarkable when seen in relation to its relatively low melting point of 2350°C. By comparison the transition metal


carbides have melting points exceeding 3000 °C and yet show a noticeable decrease in hardness at temperatures well below 1000 °C.

Examination of the metal carbide surfaces after indentation showed that for temperatures above 500 °C a permanent depression remained in the surface. It seems likely that these depressions are due to sliding at the grain boundaries, for similar hardness tests on a single crystal of titanium carbide failed to pro-duce permanent depressions at temperatures up to 1600°C, although depres-sions were later found at 1800°C. Little change in hardness of the single crystal could be detected below 1600°C. These experiments illustrate the important influence exerted by the grain boundaries on the mechanical properties and strength of sintered ceramics.

3 .2. Friction at High Temperatures

The study of the friction and wear of solids as a function of temperature was originally carried out mainly for the academic interest in the subject. At the present time new materials with many unknown properties are being tried out as bearings and seals in mechanisms operating at very high temperatures. As a result the friction characteristics of these materials have assumed considerable practical importance.

Unlike metals, the friction and wear of refractory solids such as borides and carbides are very low even at elevated temperatures. For this reason they have found wide use as machine tools and dies for extrusion and drawing. For un-lubricated surfaces the friction is of the order of / x = 0 .2 , and even when out-gassed j i t does not exceed 0 .6 in a vacuum of 1 0 - 4 mm Hg at room temperature. Frictional properties of polycrystalline refractory materials at temperatures up to 2000°C have been studied by B. L. Mord ike 8 using the apparatus shown in Fig. 12. With this method two annular specimens are rotated relative to one another inside an evacuated carbon tube furnace. The uppermost springs in Fig. 12 apply the normal load while the deflection of the lower springs measures the frictional torque between the two specimens. The deflection is measured with strain gauges.

The results obtained with this apparatus for carbides and borides may be briefly summarized as follows. When the two specimens are of the same material the friction-temperatures curves are all very similar and show initially, with increasing temperature, a slight but continuous decrease in friction until at a characteristic and reproducible temperature (Tm) it increases rapidly—as shown in curve I of Fig. 13. This characteristic temperature ranges from 830°C for tungsten carbide to 1420 °C for tantalum carbide. The slope of the initial portion of the curve varies with the material used but generally describes a decrease of approximately 0 .15 in the value of / x . The exception to the above behavior is boron carbide where JU, remains low throughout but increases steadily from / x = 0 .2 at room temperature to / x = 0.36 at 1850°C (Curve II, Fig. 13).

The curves for the variation of / x with temperature for different carbides and borides are similar to curve I, Fig. 13. The characteristic minimum in the curves

9


W.lood

S«»troin gauges

F I G . 1 2 . Apparatus for measuring the friction of polycrystalline refractory materials up to 2 0 0 0 ° C . (Specimens are in the form of annular cylinders.)

I

' 500 1000 1500 2000

Temperature (°c)

F I G . 1 3 . Effect o f temperature on the coefficient of friction of some "clean" polycrystalline refractory materials.

is now however changed, Tm is generally lower, lying in the region 800 °C to

900°C. The friction of graphite on itself and on the refractory borides and carbides

is illustrated by a typical curve (curve III in Fig. 13). As the temperature is increased n decreases fairly rapidly at first but there is very little further change in fx above 1000 °C.

For most of the refractory carbides and borides the initial decrease in friction


(curve I) is considered by Mordike to be associated with a reduction in the strength of the grain boundaries as the temperature is increased. This decrease in strength will, it is thought, reduce the deformation component of the fric-tional force; the adhesion component meanwhile remains appreciably constant or may even increase slightly. At Tm and above the adhesion component domi-nates the friction behavior. This is due to the increase in diffusion with tem-perature leading to local sintering across the sliding interfaces and consequently a high adhesion and friction. The characteristic temperature Tm is seen then to be the temperature at which appreciable sintering sets in for the solids in-volved.

The curve for boron carbide is of interest. It is known from hardness measure-ments that the strength properties of this material are relatively insensitive to temperature and that very little grain boundary flow is observed. Further, since it is a very difficult material to sinter, it follows that the adhesion is also poor. The total friction force, which is the sum of the deformation and adhesion com-ponents, therefore remains small over the temperature range. The slight increase at high temperatures is probably associated with a small increase in adhesion.

The decreasing friction of graphite with temperature (curve III) is explained on the basis that the frictional behavior is dominated by the unusually low shear fracture strength of graphite. Although the adhesion component of the fric-tional force may increase with temperature, any increase is offset by the decrease in the deformation term due to a lowering of the fracture strength.

C. A. Brookes 9 has recently studied the high temperature friction properties of single crystals of refractory materials. Briefly, his apparatus consists of a hemispherical stylus specimen sliding under load across a flat horizontal surface of a specimen of the same material and in a given crystallographic direction. The specimens are heated directly during the experiment by radio frequency induction.

The friction curves obtained for single crystals of TiO, T i 2 0 3 and M o S i 2 are very similar in appearance to curve I, Fig. 13 for polycrystalline material. The initial decrease in fx with temperature for the single crystals, however, is far less pronounced and in some cases the curve is at first almost horizontal. This difference in the two sets of results provides further evidence for linking the drop in friction with a weakening of the intercrystallite bonding.

In a study of the microstructure of polycrystalline titanium carbide, Brookes has confirmed that nearly all the frictional deformation of the surface at low temperatures is confined to a sliding and tilting at the grain boundaries. F rom the point of view of friction then the strength of the grain boundary in sintered ceramics is the controlling factor at temperatures up to about 1000 °C. Above this temperature diffusion resulting in sintering or welding at the points of contact determines the friction in both single crystal and polycrystalline ceramics.

In his study of the frictional deformation of a single crystal of titanium carbide Brookes found that the coefficient of friction as a function of temperature was surprisingly independent of the crystal face or the direction of sliding. This frictional isotropy is ascribed to the high degree of elastic isotropy shown by


titanium carbide.* Above 1000 °C a change from brittle to ductile deformation during sliding was observed. Slip lines were apparent and corresponded to slip on the (111) planes. The brittle/ductile transition in titanium carbide occurs at a temperature appreciably below the minima (1400 °C) on the friction/tempera-ture curve. The curve is apparently unaffected by this change in the mode of deformation.

3 . 3 . The Friction of Materials in vacuo and the Influence of Oxide Films

One of the many problems accompanying the development of space vehicles is that of providing low friction between the moving surfaces of the control mechanisms. It must be remembered that equipment of this sort has to function at the very low ambient pressures encountered in space. Normal greases and lubricants are of little value because of their high evaporation rates under these conditions. There are, in addition, friction problems arising from the absence in outer space of protective oxide films on metal surfaces. Without these films and without a lubricant, sliding metal interfaces weld together and seize. The pressures in outer space may be as low as 10~ 1 5 mm Hg, this being considerably lower than the best vacuum conditions (about 10~ 1 0 mm Hg) now obtainable in the laboratory. Most of the information we have on friction at low pressures refers to a pressure range of between 10~ 5 and 10~ 7 mm Hg. At these pressures, adsorbed contaminant films still cover the surfaces of solids and considerably influence their friction properties. To obtain a really clean surface the solid must be outgassed by heating in vacuo to just below the temperature at which appreciable vaporization sets in. In this way the adsorbed layers of contami-nants will be driven off and the surfaces will then closely resemble, at least as far as friction measurements are concerned, conditions at much lower pressures.

It will be as well to consider first the origin of the force of friction under atmospheric conditions. The experimental laws governing friction are very simple. I t is usually found that friction is directly proportional to the load and is independent of the size of the surfaces in contact. These laws can be explained on the basis that the true area of contact between solids is only a small fraction of the apparent area of contact (for flat steel surfaces this fraction is less than 1/10,000 of the apparent area). Consequently the local pressure at the few iso-lated points of contact is very high, since they carry the entire load. For most materials plastic flow occurs under the very high pressures, and the asperities are crushed down until their cross-section is sufficient to support the load. The real area of contact is given by A = W/Pm where W is the load and Pm is the yield pressure of the metal. The load distribution for two surfaces in contact is shown diagrammatically in Fig. 14.

where C u , C12 and CAA are the usual elastic constants. For an ideal elastically isotropic solid A is equal to unity. For titanium carbide A has a value of 0.91 indicating a high degree of isotropy.

A s a measure of anisotropy Brookes has used the Z e n e r 1 0 anisotropic factor A given by


There is strong evidence that the friction of metals is due in large measure to adhesion at these contact regions and represents the force necessary to shear the small junct ions . 1 1 As an approximation we may write F = As where F is the friction and s the shear strength of the junctions. Since A = W/Pm we have F = s/Pm . W. The constant s/Pm is seen to be the coefficient of friction / x .

In the presence of oxide films the simple laws of friction are not always obeyed. If the friction between a hard steel slider and a copper surface is measured, it has a value of about n = 0 .4 . Above 1 g the friction rises and reaches an upper steady value of about p = 1.2. There is also a corresponding change in the surface damage. At low loads the surfaces are lightly scratched; at high loads there is marked welding, plucking, and metallic damage. It is evident that at

W

i

F I G . 1 4 . Schematic presentation of two solids placed in contact. (Solid flows plastically until area of contact A supports the load W.)

low loads sliding occurs within the oxide film; at higher loads the oxide film is penetrated, and appreciable metal-metal contact occurs. With aluminum surfaces, on the other hand, the friction and the surface damage are always large, indicating that the oxide film is penetrated even at the lightest loads. This work has been extended by Wilson 1 2 , who has also measured the electrical re-sistance across the sliding surfaces. The breakthrough of the oxide film is then shown by a very marked drop in the resistance. Typical friction-load curves for tin, chromium, and copper surfaces are shown in Fig. 15. With tin the friction is always high, and the resistance measurements indicate appreciable metallic contact at all loads. With chromium the friction is always low, and the resistance very high, showing that even at high loads the oxide film is not ruptured. With copper the results are as described above. An additional factor brought out by these measurements is that even with copper at high loads the contact is not completely metallic; some oxide still remains sandwiched between the sliding surfaces. A further observation of importance is that the frictional measure-ments alone do not always indicate with certainty whether or not the oxide film has been penetrated, for in some cases the friction of oxide on oxide may be as high as that observed when the oxide film is disrupted. Apparently under certain conditions the adhesion between metal oxides may be very strong.


The overriding factor which determines whether an oxide will support a given load appears to be a straightforward mechanical one. If the oxide is hard and brittle and the underlying metal soft and ductile, then the oxide will fracture easily, and metallic contact will occur at the lightest loads. This is the case with tin and, to a lesser extent, with lead and aluminum. If the metal has similar properties to its oxide, as with copper and steel, then the oxide will deform with the metal under the applied load until heavy loads are used and may persist in part at all loads.

12,

0-8

04

r- a

1-2 r -

1 o '£

0-8 -<*. o c 04 -£ 8

12

0 8!

041

10 10 102 KT 1 load (g)

F I G . 1 5 . Coefficient of friction as a function of load:

(a) Tin on tin (oxide film is broken through over the whole load range). (b) Chromium o n chromium (oxide film remains intact over the whole load range). (c) Copper on copper (oxide film is penetrated for loads greater than 1 g).

Although the coefficient of friction of oxide on oxide may well be as high as that of metal on metal, this does not conflict with the widely held opinion that oxides will reduce the wear. When oxide is sliding on oxide, even if the junctions formed are very strong, they will shear in the oxide layer, and material transfer will be small, whereas when metallic junctions are formed, the amount of transfer is considerable.

Since oxide films can exert so marked an influence on surface interaction, it is to be expected that the complete removal of these films would have a profound effect on friction and adhesion. Bowden and Y o u n g 1 3 and Bowden and R o w e 1 4

have studied friction and adhesion of outgassed oxide free metal surfaces using an apparatus of the type shown in Fig. 16. The specimens A and B, held together


under a dead load, are heated by radio frequency induction. A manipulator H is used to apply tangential forces, another manipulator D applies the normal force used in measuring normal adhesion between the specimens. The apparatus

F I G . 16. Apparatus for studying friction and adhesion of outgassed metal surfaces. (Two cylindrical specimens, A and B, are heated by radio frequency induction. A tangential force can be applied by H to the top specimen A, which rests on B under a known dead load; a normal force can then be applied to separate specimens using D. The forces are

measured by the deflection of springs / and F.)

) 400 800 1200

outgassing temperature °C

F I G . 1 7 . Relationship between coefficient of friction measured at r o o m temperature and temperature of preliminary outgassing ( # nickel; O plat inum; A silver).

is of silica and can be baked out at 600 °C to give a working pressure down to 10~ 8 mm Hg.

It was found that the friction between two surfaces measured at room tem-perature, rose slowly at first with temperature (Fig. 17), but beyond a certain


well-defined outgassing temperature the friction increased rapidly with con-tinued outgassing until seizure occurred.

The effect on the friction of introducing contaminant gases to thoroughly cleaned surfaces is illustrated in Fig. 18. In this case a trace of oxygen brings about a marked reduction in the friction between clean iron surfaces. Most gases including water vapor, carbon dioxide, nitrogen, chlorine, and hydrogen sulphide effected a similar reduction in friction. The change in friction is brought about by the formation of a chemisorbed film of the gas which reduces the extent of the metal to metal contact and limits the growth of the junctions under load.

Experiments were carried out on admitting fatty acid vapors to cleaned metal surfaces. 1 3 The friction, although reduced, did not fall to the low values normally

n t oxygen at:

10 4 mm.

10 mm.

Y ' i . Several mm.

rval of 15 hr. j

F I G . 18. Friction of Fe surfaces, cleaned by outgassing in a vacuum, allowed to cool to room temperature, and then placed in contact. (Admission of small quantities of oxygen

lead to a marked reduction in friction.)

associated with fatty acid lubrication. In fact a low friction could only be pro-duced by the fatty acid in the presence of oxygen and water vapor. This is in keeping with earlier work (Bowden 1 5 ) which showed that a lubricant film is most effective in the condensed or "solid" state. The presence of oxygen and water vapor facilitate in the formation of a solid soap film on the surface; it is this metal soap which acts as the lubricant. The lubrication of clean metal surfaces is thus seen to be rather more than simply adding a good boundary lubricant—conditions must favor the formation of a metal soap—and this requires the presence of oxygen and water vapor.

Since the very high friction of clean metals is clearly due to adhesion of the two surfaces at the local regions of contact it is pertinent to ask why strong normal adhesions between metals are not commonly observed. This adhesion force should be comparable with the normal force originally pressing the surfaces into contact. The experiments of Bowden and R o w e 1 4 have shown in

CO

EF

FIC

IEN

T

OF

F

RIC

TIO

N

(ft

beizure Adrr

V

Inte


fact that adhesion is negligibly small for hard metals at room temperature. If however the temperature is raised to a point where annealing can occur and then reduced again to room temperature, the adhesion is very strong. Some typical results for platinum surfaces are shown in Fig. 19. The explanation of adhesion as a result of annealing is that normally when two metal surfaces are pressed together, there is heavy plastic deformation and work-hardening at the junctions. When the load is removed, the elastic stresses in the bulk metal are released and cause a slight change in shape of the surface contour. This slight displacement is sufficient to break the brittle work-hardened junctions one by one so that none are left when adhesion measurements are made. Annealing has the effect of restoring ductility to the junctions such that they can withstand the small

h e a t i n g t e m p e r a t u r e ° C

F I G . 1 9 . Onset o f adhesion with denuded platinum. (Adhesion occurs at temperatures which are high enough to cause annealing of the junctions.)

release strain without fracture, and are still intact when adhesion is measured. It is perhaps interesting to note in this respect that soft metals which do not appreciably work-harden, such as lead or indium, will show a large normal adhesion when pressed into contact with a hard metal surface at room tempera-ture. When the surfaces are pulled apart, the force required is equal to the original load, and fragments of lead or indium are found adhering to the harder metal.

If, while the load is still applied, the surfaces are subject to a tangential stress (without sliding), a large increase in adhesion can be produced. This effect is illustrated for a number of metals in Fig. 20. The increase in adhesion is due to an increase in the real area of contact at the junctions. On release of the load the junctions will remain intact providing they are not stretched by more than a few per cent of their diameter.

The above experiments illustrate the fact that friction and adhesion are likely to be high for metal surfaces in contact in a low pressure atmosphere. Normal boundary lubricants, even if they could be maintained are unlikely to be effective

9 §


in the absence of oxygen and water vapor. One alternative is to consider reducing metal to metal contact with such high vapor pressure lubricants as graphite, molybdenum disulphide and possibly boron nitride. Even with these materials however the friction in vacuo is considerably higher than under atmospheric conditions. R. H. Savage 1 6 and Bowden and Y o u n g 1 3 found an increase from the low value of n = 0 .1 for graphite in air to / x = 0 .6 for out-gassed graphite; there was also an enormous increase in the wear. Seizure was not observed for graphite on graphite though for denuded copper on clean graphite the friction reached values of ^ = 1.5 and there was appreciable transfer of copper to the graphite (Kenyon 1 7 ) . The mechanism of friction of

tangential pre-stressing force g m

F I G . 20. Relationship between normal adhesion of denuded metals and the amount of tangential pre-stressing ( © gold; • nickel; O platinum; A silver).

graphite has been discussed by Savage 1 6 and Goodman and Deacon 1 8 . Savage found that minute amounts of adsorbed gas—sufficient only to cover the edges of the plate-like crystallites composing the bulk graphite—was sufficient to lower the friction of initially clean graphite from very high values to very low values. Goodman and Deacon consider that the friction of graphite is deter-mined primarily by the edge-edge bonding between crystallites. In the atmo-sphere the high energy edges react with oxygen and water vapor to give various oxygenated groups, thus reducing edge-edge interactions between crystallites. By outgassing at high temperature, the gases and vapors are removed, the edge-edge bonding is increased, and there is an accompanying rise in the friction. The flat low energy cleavage faces of the crystallites provide relatively weak intercrystallite bonds, both in air and in the outgassed condition, and therefore do not contribute noticeably to the friction. A similar explanation has been given for the low friction of molybdenum disulphide. 1 8 Experimentally it is found that the friction of outgassed M 0 S 2 remains low at temperatures up to


the decomposition temperature (ca. 800 °C). The difference now is that the edges interact with oxygen as before but the oxide layers are much less volatile than the oxides of carbon and are not removed at temperatures below that at which molybdenum disulphide itself decomposes; the friction therefore remains low throughout. The friction of the lamella form of boron nitride in vacuo has also been studied by Rowe. The friction properties were found to be similar to those of graphite, the coefficient of friction was generally a little higher than that for graphite.

The experiments we have described on the study of friction and adhesion in vacuo have shown clearly that, unless special precautions are taken, friction and adhesion can be very large. The corresponding rates of wear of the surfaces in contact may also be unacceptably large. Further, it has been shown that boundary lubricants are not effective in vacuo but require the presence of oxygen and water vapor. Far better lubrication can be obtained using one of lamella type solids.

4 . H I G H - S P E E D L I Q U I D I M P A C T

A small drop of liquid striking against a solid surface at high velocity can produce considerable deformation and damage. This type of damage has been found to occur in aircraft during high speed flights through rain. Leading edges of the wings and the nonmetallic radomes forming the nose of the aircraft are particularly prone to rain erosion of this kind. At supersonic flying speeds rain-erosion damage may become severe and seriously impair the performance of the aircraft.

In a recent experimental investigation of the mechanism of rain-erosion we have attempted to reduce the problem to a simple form (Bowden and Brunton 1 9 ) . We have confined our study to the single impact of a cylindrical column of liquid against a stationary target specimen. The principle of the apparatus used is illustrated in Fig. 21 . A small volume of liquid is contained in a stainless steel chamber. The chamber is open at one end and converges at the other to a fine orifice. The liquid in the chamber is extruded at a high velocity by firing a flat-nosed slug from a gun into the chamber. A neoprene disk spreads out during the extrusion and effectively prevents leakage of the liquid back along the slug. Jet velocities up to 1500 m/sec have been obtained in this way, the upper velocity being set by the bursting strength of the chamber.

Measurements of the velocity of the jet and its behavior on impact were made using high speed photographic methods. Two systems of photography were used, a six-spark Cranz-Schardin apparatus and a Beckman and Whitley rotating mirror camera. With both systems exposure times per frame of a fraction of a microsecond together with a framing rate of 10 6 pictures per second were necessary to effectively photograph the impact.

A series of pictures taken with the Beckman and Whitley camera and showing a typical water jet striking against a transparent (polymethyl methacrylate) target is given in Fig. 22. The jet diameter in this case is 2 .1 m m and its striking velocity 680 m/sec. It can be seen from this photograph that it is possible to


produce reasonably cylindrical jets moving at very high velocities by this method. Although most of the experimental work has been carried out with cylindrical jets, it has also been possible to produce a large variety of shaped jets by slightly altering the design of the extrusion chambers.

In order to understand the mechanism of the deformation of a solid when struck by a high speed liquid jet, the load exerted during the impact and variation of this load with time must be known. Values for these quantities have been obtained experimentally using a simple piezoelectric pressure transducer. This consisted of a long steel bar 1 cm in diameter with a barium titanate crystal mounted at one end. A thin hardened steel anvil was pressed against the other face of the crystal. The charge developed across the crystal faces when the anvil was struck by the jet was displayed as a voltage on an oscilloscope screen. An

\ TARCET

F I G . 2 1 . Schematic presentation of method for producing high velocity liquid jet.

example of a load-t ime trace obtained in this way is shown in Fig. 23. The trace shown has been redrawn from the original photograph which was too faint for reproduction. The pressure gauge was calibrated by allowing metal bars to fall from different heights on to the crystal.

In every case the oscilloscope traces showed that the load rose to a peak value within a microsecond and then immediately began to fall. The decay time was normally less than 3 /xsec. The rapid decay is perhaps at first a little surprising, particularly in view of the fact that the jet impinges on the surface for something like 20 /xsec with almost a constant striking velocity. However, if the water-hammer effect described by C o o k 2 0 is recalled, this in fact is the sort of decay time to be expected.

If we consider the jet as a column of liquid with a density p colliding end-on with a velocity V against a flat rigid surface, the impact pressure P will be, according to the simple water-hammer equation

P = PCV (2)


where C is the velocity of a compression wave in the liquid. The load L will rise instantaneously to a peak value:

L = pCVa (3)

where a is the area of the jet head. The load will then immediately start to fall since the high pressure at the front of the jet will be relieved by sideways flow

F I G . 22. Impact of a 2 m m diameter water jet against polymethyl methacrylate plate. (Impact velocity 680 m/sec. Time between frames 0 . 8 yusec.)

of the liquid. Release waves will move in from the free boundaries of the jet, in front of the waves the pressure will still be P (equation 2) and behind the wave the pressure will eventually drop to the steady-flow value. The decay time will therefore be the time for release waves to reach the centre of the jet. For the jets used in this work, this time is between 1 and 2 /xsec. The diameter of the jet therefore determines the duration of the load. The reason why the experi-mental curves show slightly larger rise and decay times than those predicted


by the simple theory is thought to be due to the fact that the end of the jet is neither perfectly flat nor parallel to the surface.

The load trace in Fig. 23 indicates that only the head of the jet is responsible for failure of the specimen. This has also been observed experimentally. For all but the highest impact velocities, high speed photographs have shown that the fracture process is completed within the first 3 jusec of impact.

The magnitude of the impact load for the case of a water jet with a core diameter of 1.2 m m and a head diameter of 3 mm striking the pressure gage at an impact velocity of 720 m/sec was found to be 630 kg. There is no simple way of determining the distribution of this load over the area of impact. However,

1 — 1

O IO 2 0 T IME IN MICROSECONDS

F I G . 23 . Load time trace for the impact of a water jet (Impact velocity 720 m/sec) .

an average value for the impact pressure over the area of contact of the jet head with the surface (this is also the area deformed by the jet) can be obtained by dividing the load by this area. The average pressure obtained in this way is 94 kg /mm 2 (134,000 psi). This value may be compared with that determined from equation (2). For a water jet with an impact velocity V of 720 m/sec and taking C = 1500 m/sec, P has a theoretical value of 108 kg /mm 2 (153,500 psi). In view of the assumption made about the area over which the load is exerted, the agreement between the two values is good. It may be observed that since the yield strength of a metal such as mild steel is about 25 kg /mm 2 (35,550 psi) plastic deformation will occur at this impact velocity. Examples of deformation of this kind will be given later.

To summarize, then, the impact of a liquid jet against a solid surface sets up

LO

AD


an intense compression pulse which moves off into the solid. The remainder of the jet impinging against the surface exerts a much lower pressure on the surface. The magnitude of this pressure will be given by the usual Bernouilli's equation for incompressible flow.

4 . 1 . The Deformation of Hard Polymers

Polymethyl methacrylate target specimens have been used a great deal in this study because of their transparency and the ease with which they can be prepared. The deformation produced in a 3 m m thick plate of this material by the impact of a water jet is illustrated in Fig. 24 and in cross-section along a diameter in Fig. 25. Very similar failure patterns have been observed in other hard polymers and glasses.

F I G . 2 4 . Ring deformation in polymethyl methacrylate due to impact of a water jet at 950 m/sec. (Mean diameter of ring 3 mm.)

The main feature of the deformation is a ring crack separating a region of intense circumferential fracture from a central unfractured area of the surface. The central star crack can be seen in Fig. 25 to lie below and separate from the surface. A large number of measurements using jets of different sizes have shown that the diameter of the central undamaged area is equal to the diameter of the head of the impinging jet.

The appearance of the ring fracture altered with the impact velocity. For velocities below 450 m/sec the ring fracture was replaced by an annular depres-sion; below this depression large planes of fracture could still be seen. These fractures cut through to the surface to form tiny hairline cracks rather than the wide ring crack shown in Fig. 24. For impact velocities approaching 1000 m/sec, the ring fracture was replaced by a surface pit of approximately the same dia-meter. The pits often extended to a depth of between 0 .5 cm and 1.0 cm below the surface.

It is interesting to note that the higher velocity impacts were always accom-panied by a blue-white flash of light. Photographs of the flash showed that it


occurred on the target surface. It seems likely that the flash is caused by adiabatic compression of the air layers between the jet and the target.

The profile of the surface over the outer fractured zone was obtained using a sensitive multiple beam interferometry method. The profile showed that each fracture formed a very small step in the surface, the average height of a step being only about 2000 A. It was also found that the vertical surface of each step faced in towards the center of impact. These small surface steps produced by the crazing fractures are important in that they obstruct the liquid flowing over the surface. As a result there is a radial force acting on the step face tending to chip material out of the surface along the line of the fracture. The deep annular pit or fracture in Fig. 24 is an example of a pit formed by this process.

F I G . 25. Cross-section through 3 . 5 m m thick polymethyl methacrylate plate. (Fractures at A are shear fractures and lie along shear trajectories; they can be seen in Fig. 23 as the central star crack. Fractures at B are thought to be associated with tensile components o f the expanding compression wave. Fractures at C and D are caused by reflection and interference of initial compression wave in the lower surface. Scabbing fractures at D are

far more extensive than the ring fractures on the impact surface.)

It might be asked why the center of the impact region is undamaged and why there are ring fractures at all. The fracture process may be visualized thus : On impact the region under the jet head is suddenly put into compression, while the surrounding area experiences radially acting tensile stresses. The latter will tend to cause circumferential fractures to form. This fracture process has been observed using high speed photography; the fractures were seen to be formed by the distortion wave as it moved out over the surface. The central surface area in compression will not of course fracture, although the associated shear stresses may cause some plastic deformation.

Failure of the surface is therefore seen to consist of two processes: first, the production of fractures by the radial tensile stresses and, second the removal of material from the fractured regions by the liquid flowing out at high speed over the surface. It should perhaps be mentioned here that high speed photo-graphy has shown the velocity of the liquid flowing out over the surface to be in


most cases far greater than the impact velocity, an effect directly related to the shape of the head of the jet as it strikes the surface.

The fractures below the surface (Fig. 25) have rather a complicated structure. The uppermost of these fractures 04) is the central star pattern fracture seen in Fig. 24. These fracture planes lie along shear trajectories and open under the rapidly applied shear stress. In addition to shear stresses there are tensile components acting across the front of the expanding compression wave as it moves into the material ; these too cause fractures lying perpendicular to the surface (B) and below the main star crack. High speed photographs have shown this fracture region forming as the compression wave moves through and then closing up again but not completely when the wave has passed. Fracture regions C and D, Fig. 25, are related to the reflection in the lower surface of the plate of the initial compression wave set up on impact. The main fracture zone D is a spalling fracture formed under the large tensile stresses associated with the reflected compression wave. Fractures at C form as a result of interference between this wave and the slower travelling outgoing shear wave. In brittle materials fractures due to the reflection and interference of stress waves are usually far more extensive than those at the impact surface itself. High strength ceramic plates for example fail in this way by repeated flaking of the rear surface. Eventually the plate section becomes so thin that it perforates. This type of failure is of considerable importance in connection with the erosion of high speed aircraft.

4 .2 . The Deformation of Soft Polymers and Elastomers

Two characteristic features of the failure pattern found in the impact area are an outer ring of torn surface, and a narrow central penetration or piping which, depending on the polymer, extends more than 1 cm into the block. The outer ring damage is not a fracture as in the case of the hard polymers but a shearing or tearing caused by the liquid flowing out from the center of impact. The central penetration, on the other hand, is a true fracture which occurs after the surface had been depressed several millimeters. This type of failure was found in neoprene, natural rubber, and polyethylene. In unplasticized poly-vinylchloride (a relatively hard polymer), an outer Hertzian ring fracture similar to that found in the hard polymers and a separate inner shear ring characteristic of an elastomer were both produced in the same impact area.

4 . 3 . The Deformation of Metals

In metals large-scale plastic flow is produced for impact velocities greater than 650 m/sec. The depth of the saucer-shaped depression is normally less than 0 .5 mm and the diameter the same as that of the jet head which produced it. An example showing the deformation of aluminum caused by the impact of a water jet at 750 meters (2500 ft) per second is given in Fig. 26. The surface of the depression is smooth near the center, but around the periphery the surface


has been sheared into a wavy series of elevations and depressions by the radial flow of the liquid. Some metal is always torn from the surface by this action. As in the case of polymers, shear deformation was most intense where the surface underwent small but sharp changes in contour. Slip lines, grain boundaries, twin boundaries, and surface scratches were heavily deformed.

Similar deformation patterns have been observed in such metals as copper and its alloys, iron, mild steel, and 18-8 stainless steel. With very hard martenistic steels and with tungsten carbide, surface depressions were not observed for impact velocities below 850 m/sec. The peripheral shear deformation, however, was still very much in evidence.

F I G . 26. Micrograph of deformation of aluminum under liquid impact (55 x ) .

In multiphase alloys it was found that the softest phase was the most heavily deformed by liquid flowing along the surface. In duralumin type alloys and in graphitic cast irons the intermetallic and nonmetallic phases were in many cases sheared from the surface in areas near the rim of the depression.

For impact velocities in the region of 6000 ft/sec and with metals of relatively low yield strength such as lead and tin, the impact stresses are very much greater than the mechanical strength so that it is possible to regard metal target as flowing hydrodynamically. The crater which is produced is hemispherical in shape with a pronounced lip—in appearance similar to a frozen splash. A typical example of a high velocity crater in lead is given in Fig. 27.


4 .4 The Mechanism of Deformation

It has been shown that three general types of failure are possible in a solid as a result of the high speed impact of a liquid mass. The predominating form of failure in any given case depends upon the mechanical properties of the solid.

1. Failure in the impact region due to the rapid application of a large load over a relatively small area. In hard polymers and brittle solids this causes circumferential fractures, ring cracks, and sometimes subsurface fracture. In metals and materials capable of deforming plastically at very high rates of strain a simple surface depression is produced.

F I G . 2 7 . Plane and cross-sectional views of lead block struck by a 2 - 3 m m diameter jet at a velocity of 1500-1800 m/sec. (Diameter of crater 8 m m , depth 5 mm.)

2. Failure due to the shearing action of liquid flowing outward over the surface at very high speeds. This type of failure is localized at surface discon-tinuities. These may already exist in the surface as scratches, pits, blemishes, and inclusions, or they may be put in by the impact load and appear as cracks, slip lines, and grain boundaries. Surfaces which remain smooth during the impact appear to be little affected by this type of deformation. A simple calculation shows that the shear force exerted on a flat surface even by a very fast flowing liquid is very small compared with the force exerted against a surface step. It is

244 F . P . B O W D E N A N D J . H . B R U N T O N

interesting to note that this shearing failure is indeed true erosion since it in-volves removal of material from the surface.

3. Failure caused by the reflection and interference of stress waves in the solid. This type of failure is normally confined to hard brittle solids. When it occurs it is often catastrophic and leads to more damage than is to be found in the region of impact itself. It is, of course, very dependent on the shape and size of the target and is therefore to a large extent controllable.

We have shown that the deformation of a solid by a liquid drop is due to several distinct mechanisms and that it is possible, in a qualitative way, to separate them. It is also apparent that very few solids are able to withstand liquid impact under these conditions without severe deformation or fracture. The damage increases enormously as the velocity is increased. The volume of liquid we have used is comparable with that of a raindrop and the observations have a bearing on the problem of rain erosion of aircraft.

C O N C L U S I O N

We have described a number of experimental investigations (21) into the properties and behavior of materials under the extreme conditions asso-ciated with present-day high speed travel. It is apparent in the cases we have considered that the experiments have not produced any ready-made solutions to the materials problem. However, as we emphasized at the beginning, the purpose of these studies has been to examine the basic mechanisms underlying the behavior of materials in a high speed environment. Given an understanding of the various mechanisms it then becomes easier to choose and even to design new materials which are capable of withstanding these severe conditions more effectively.

R E F E R E N C E S

1 . BEAMS, J . W . / . Wash. Acad. Sci. 3 7 , 2 2 1 ( 1 9 4 7 ) .

2 . B O W D E N , F. P. and FREITAG, E. H . Proc. Roy. Soc. A 2 4 8 , 3 5 0 ( 1 9 5 8 ) .

3 . B O W D E N , F. P. Proc. Roy. Soc. A 2 1 7 , 4 6 2 ( 1 9 5 3 ) .

4 . B O W D E N , F. P. and PERSSON, P. A. Proc. Roy. Soc. A 2 6 0 , 4 3 3 ( 1 9 6 1 ) .

5 . SHORE, A. F. Iron and Steel Institute 9 8 , 5 9 ( 1 9 1 8 ) . 6 . FITZGERALD, L. M . Less Common Metals, April, 1 9 6 3 . 7 . T A B O R , D . T. Proc. Roy. Soc. A 1 9 2 , 2 4 7 ( 1 9 4 8 ) .

8 . M O R D I K E , B . L. Wear 3 , 3 7 4 ( 1 9 6 0 ) .

9 . BROOKES, C. A . Special Ceramics. P. Popper, ed. L o n d o n : Academic Press, 1 9 6 3 . 1 0 . ZENER, C. Elasticity and Anelasticity of Metals. Chicago: University of Chicago Press, 1 9 4 8 . 1 1 . B O W D E N , F. P. and TABOR, D . T. The Friction and Lubrication of Solids, Part I ( 1 9 5 4 ) .

Oxford: The Clarendon Press. 1 2 . W I L S O N , R. Proc. Roy. Soc. A 2 1 2 , 4 5 0 ( 1 9 5 2 ) .

1 3 . B O W D E N , F. P. and Y O U N G , J . E. Proc. Roy. Soc. A 2 0 8 , 3 1 1 ( 1 9 5 1 ) .

1 4 . B O W D E N , F . P. and R O W E , G . W . Proc. Roy. Soc. 2 3 3 , 4 2 9 ( 1 9 5 6 ) .

1 5 . B O W D E N , F. P. / . Inst. Petrol Tech. 3 4 , 6 5 4 ( 1 9 4 8 ) . 1 6 . SAVAGE, R. H . / . Appl. Phys. 1 9 , 1 ( 1 9 4 8 ) .

1 7 . K E N Y O N , D . M . P h . D . Thesis. University of Cambridge. 1 8 . D E A C O N , R. F. and G O O D M A N , J . F. Proc. Roy. Soc. A 2 4 3 , 4 6 4 ( 1 9 5 8 ) .

1 9 . B O W D E N , F. P. and B R U N T O N , J . H . Proc. Roy. Soc. A 2 6 3 , 4 3 3 ( 1 9 6 1 ) .

2 0 . C O O K , S. S. Proc. Roy. Soc. A 1 1 9 , 4 8 1 ( 1 9 4 8 ) .

2 1 . For fuller account see B O W D E N , F. P. and TABOR, D . , The Friction and Lubrication of Solids, Part II ( 1 9 6 3 ) .

MATERIALS ASPECTS OF THE RE-ENTRY

PROBLEM

E. SCALA

Cornell University

Ithaca, N . Y . , U .S .A .


A MISSILE of 2000 lb re-entering the atmosphere at 25,000 ft/sec has a kinetic energy of 20 billion ft lb, most of which is dissipated as heat to the atmosphere with the deceleration. Only a fraction of the equivalent thermal energy of 25,000,000 B.t.u. is absorbed by the vehicle depending to a great extent upon aerodynamic design. The shape, center of gravity, and center of pressure estab-lish the vehicle stability, roll, pitch, and yaw (and the probability of coming in backwards). The weight, drag coefficient, and cross-sectional area are the design variables in ballistic parameter W/CDA. The resultant deceleration and distri-bution of heat transferred to the vehicle are pertinent to this discussion. The allowable deceleration rate and the velocity at impact are normally established by mission requirements. The peak deceleration would be less than 10 g's for manned re-entry and of the order of 100 g's and over for unmanned vehicles. Of the re-entry mechanical forces of prime significance to the heat shield, however, are the stagnation pressure and shear forces at the boundary layer. These factors are superimposed upon the heat fluxes, which for brevity can be given as stagnation point peak fluxes: 1000 to 10,000 B.t.u./ft 2/sec for typical ballistic missiles and up to one to six million B.t.u./ft 2/sec for possible super-orbital vehicles. More impressive are the total heat input of several hundred thousand B.t.u. for current missiles to many millions of B.t.u.'s for the ad-vanced high angle interplanetary recoverable packages. What this alternative means in terms of heat shield materials and their sub-structures is the subject of this discussion.

2. T H E R M A L F A C T O R S

Parametric studies must determine the relative importance of the thermal properties of both heat shield and sub-structure materials. The envelope of re-quirements generally involves an outer aerodynamic configuration, minimum in-side dimensions, and a maximum inner wall temperature and a maximum weight. Ranges of thermal conductivity and density properties often have to be assumed. For heat sink type materials such as copper or beryllium, these properties are

245

246 E. S C A L A

fairly well established and not too difficult to check. For the ablative materials, however, with their chemistry of decomposition and transient property changes, the thermal properties are difficult to obtain. The materials are less well developed or characterized and too often of inconsistent quality.

The materials properties and the environmental conditions are interrelated. 1

The changing thickness of the ablator and the temperature profile of the heat shield and its sub-structure is represented briefly by :

1. Rate of ablation s = q/pq* and 2. Temperature penetration Tx = Ts exp (sx/a)

where Ts is surface temperature, q is heat flux in B.t.u./ft 2/sec, p is density, a is thermal diffusivity k/pCp, x is depth or thickness and q* is the thermochemical heat of ablation in B.t.u./lb (corrected for hot wall and radiation). Ideally, both the ablation rate s and the temperature penetration Tx (or back wall tem-perature) should be minimized. Low ablation or mass loss requires a high q*, while a low Tx requires a high mass loss and low diffusivity. Whereas high density will mean a low Tx and i , the tendency is to keep the heat shield weight to a minimum by considering lower density materials. In addition, the con-ductivity of most ablative materials increases significantly with small density increases. The functions of the components include the combined ablative and insulating characteristics of the heat shield and of the sub-structure acting as a heat sink.

Optimizing by compromising is necessary not only for different missions, but for different parts of the vehicle. Both fabrication and structural analysis, how-ever, are made quite complex by the use of several different materials. In fact, an early review by manufacturing engineers, concurrent with structural analysis is essential.

Among the difficulties encountered at re-entry velocities greater than 25,000 ft/sec is the higher degree of heat transfer by radiation over and above the convective heating. Less than 1 per cent of the total kinetic energy of an ICBM-type vehicle is actually transferred to the nose cone (mostly by convective heating). With the higher velocities, a very large increase would be added by radiation from the ionized gases behind the shock wave. The degree to which most ablating materials can block the radiative heating has not been sufficiently explored experimentally and modifications are being developed that do offer improved radiation blocking, particularly for recovery from trans-lunar mis-sions.

Figures 1 and 2, taken from Winkler and Riddell 1 , compare the three heat transfer regions for re-entry vehicles and the relative heat fluxes involved.

Significant in the comparisons in the first figure are the conditions encountered relative to the equilibrium surface temperature of 6000 °F for a 1 ft nose radius and the velocity-altitude combinations at which the heating radiation becomes equal to that of combined conduction and convection. The ICBM is designed to re-enter the atmosphere and slow down rapidly. The peak de-celerations and heat fluxes are therefore high. The recoverable satellite has a

M A T E R I A L S A S P E C T S OF THE R E - E N T R Y P R O B L E M 247

i . 1 • RE~£NTftY FROM MARS

EXPOSURE TfM£-MJNUT£S

F I G . 2. Stagnation re-entry heat fluxes and exposure time.

much lower performance or ballistic parameter, and hence it must slow down at much higher altitudes with a much lower peak deceleration force. The inter-planetary vehicle, which must travel at much higher velocities to be practical, encounters large radiation heating conditions. The combined effects of velocity and altitude can be best appreciated by comparing the relationships of aero-dynamic heat flux (q^TO) vs. the radiant heat flux ( # r a d ) in the following ap-proximations :

I O 9 i

a^~(ol/2)(V*)(R)-V2 ?rad~0>3/2)(Fi°)OR)

248 E. S C A L A

where p is density, v is velocity and R is nose radius. The heat transfer by con-duction/convection is a function of the cube of velocity, as compared to the tenth power of velocity for radiant heat transfer.

The comparatively high re-entry angles for the ICBM and the space re-entry results in the short time high peak heating that can be solved by ablation. The manned vehicles such as the Dyna-Soar, Gemini and Apollo call for a pre-dominately radiating structure with possible provisions for ablation when maneuvering superimposes heat flux peaks. As a result of materials develop-ments for manned re-entry vehicles, the ablation area in Fig. 2 can be extended beyond 10 minutes, contrasted with that of the ICBM, which is less than one minute.

3. S T R U C T U R A L A N D D E S I G N F A C T O R S

The thermal and stress parameters are not generally coupled, i.e. the system can be analyzed first for the temperature distribution in three dimensions— along the surfaces, thickness and as a function of time. The thermal stress analysis is then based on their profiles. In most instances, uniaxial heat flow is assumed, but in critical areas, biaxial heat flux calculations must be made.

The determination of the temperature profiles assumes known material properties. These often elusive values are not too serious a problem when thermal conductivities in a monocoque structure are involved, but more often the following must be taken into account:

1. Anisotropic properties a. Crystallographic variations b . Process orientation variables

2. Inhomogenieties and defects a. Density variations b . Voids and separations

3. Composites a. Interface properties b . Interacting properties

4. Dynamic vs. static properties a. Decomposing organics b . Rates of loading

5. Surface variations a. Aerodynamic and heat transfer b . Emissivity and transmissivity

6. Sample test data vs. full-scale tests.

Having inherited the temperature profiles, the structural analysis can disclaim responsibility by uncoupling the thermal and stress problems. This is justified in most instances since the amount of energy conversion (increasing the tem-perature by property changes with stress) is negligible.

An example of temperature profiles is illustrated for the Mercury vehicle in


Fig. 3 . 2 This illustrates the use of two of the basic approaches to the thermal stress problem.

1. The after body is shingled to allow for free expansion. 2. The forward heat shield is basically an ablator and insulator, keeping the

temperature rise of the structure to a minimum (and separated deliberately after re-entry to prevent soaking).

3. A third approach would be to design a complete shell that expands as a unit. This may be monocoque or a composite, such as welded honeycomb panels on a supersonic aircraft fuselage and wing structures. Expansion joints are not part of at least the outer shell.

I N N E R S K I N

T E M P E R A T U R E - 1 0 0 ° F

H E A T I N G - A F T E R B O D Y ( 6 0 - 4 0 3 0 )

F I G . 3. Mercury vehicle re-entry temperatures.

4. In the case of ceramics, the single piece design is modified by putting in crack stoppers. Having established the design concept, and having chosen at least a reference material with its calculated temperature profiles, the structural analysis determines the stresses, possible modes of failure and a feedback on the design thicknesses established initially by thermal requirements.

The structural analysis of an elastic system can be carried out qualitatively with considerable success, i.e. defining the probable modes of failure. The non-elastic problems with edge effects and transient conditions, plus changing shapes and mechanical properties, however, is a more typical case. For indeterminate conditions, sufficient assumptions and iterations can usually be made at least to define critical experiments. The experimental simulation, however, is generally necessary to confirm the calculations and to make satisfactory quantitative determinations.

An analysis predicting stresses below the failure levels can usually be con-sidered as encouraging and is therefore grounds for optimizing to shave weights etc. A failure prediction, however, requires considerable re-examination, since it calls either for extensive redesign, added development, or postponement. There is a tendency for superimposing conservative suppositions, pessimistic

2 5 0 E. S C A L A

properties and calculated restrictions, resulting in excessive calculated safety factors on the one hand or delays by further materials developments and testing on the other.

The composites considered for heat shields involve essentially rigid, metal sub-structures with "plastic" shields. The demon in the thermostructural analysis is the thermal expansion, or more precisely, differences in thermal expansion. The mechanical properties of the heat shield such as yield and tensile stress, elastic modulus and Poisson's ratio are generally all that is re-quired on the first round to establish the type of failure, if stress build-up or relief is not possible.

The types and conditions for thermo-stress failures over and above the re-entry heating are :

1. Thermal treatments during processing and manufacturing (annealing, bonding, curing, post-curing, etc.)

2. Environmental tests (cold-chambers, cyclic climate changes, etc.) 3. Ascent heating and free-flight soaking and radiation in outer space (cooling

in shadow). 4. Thermal shock (pre-testing in radiant or rocket motor heat sources). 5. Impact (pressures and particles).

To the list can be added the numerous structural loads independent of thermal effects including launch and ascent vibrations, high g deceleration loads, landing impact and, not least, transportation (and dropping it).

The ablative materials include ceramics, salts, polymers and their composites. The reinforced plastics of phenolics and refrasil are well-proven thermal pro-tection shields. 3 ' 4 The mechanical properties of these materials at room tem-perature exhibit a brittle behavior. The polymer-base materials generally have low elastic moduli and are less subject to thermal shock failure, but their strengths drop off rapidly and they are usually structurally useless above 200 °C. Their problems vary from brittle failure at sub-zero environmental temperatures to creep above 100 °C. Flexibilizers can reduce the former, and reinforcement can decrease both fracture and creep tendencies. There is usually no reliance on the heat shield sharing any loads, but it must transmit the loads to the sub-structure and survive the extremes of vibration and temperatures. The strength at the bond line and method of attachment to the structure are key problems. Adhesives such as epoxy resins are desirable for uniform structural loading, but mechanical fasteners are usually employed as a precaution.

4. S U P P O R T S T R U C T U R E S

Examples of materials selection for structures are the Agena vehicle employing primarily magnesium in Fig. 4 5 with a stainless steel nose cap, and the Mercury capsule in Fig. 5, 2 employing titanium and Rene 41. Aluminum alloys are commonly employed in most of the re-entry vehicle structures.

In a calculated optimum design for a sub-orbital re-entry vehicle, the primary


ROUGH SKETCH OF AGENA STRUCTURE ILLUSTRATING USE OF MAGNESIUM ( 6 0 - 4 0 2 1 )

F I G . 4. Agena magnesium structure.

WELDED Ti

BASIC CONSTRUCTION (60-4026)

F I G . 5. Sample cross-section of mercury vehicle aft structure.

techniques, the practical limits of improvement by materials development have been reached, except for other superimposed problems such as radiation block-ing and particle impact resistance. The play is back on the structure when the back wall temperature is increased. The use of intermediate insulating materials solves the thermal problems and adds a few structural ones.

The ideal, structurally, would be to match coefficients of expansion, and in

structure outweighs the insulation. 6 To maintain an aluminum structure below 500 °F in a super-orbital re-entry vehicle, the heat shield thickness must increase, with an inefficient if not prohibitive design weight.

For short-time ICBM re-entries the material for the supporting structure could be varied, but it has generally been the heat shield that required develop-ment and improvement for optimization. With both ablative and heat sink

252 E. S C A L A

some instances the attempt produces at least a minimum of mismatch. The common use of aluminum and magnesium alloys unfortunately introduces the highest expansion coefficient. A fiberglass cylinder is often better suited in this sense, but the elastic modulus is usually too low. The newer reinforced phenolics can be made with an E of 10 x 10 6 psi or better, but this is of value primarily where a short-time re-entry condition exists (30-60 sec re-entry time). The environmental soak times, however, may still be high enough so that the rapid decrease of modulus with increasing temperature may make the fiber reinforced polymer marginal with ground environments up to 135°F and ascent heating approaching 200 °F.

A second type of problem arises, however; this is the changing expansion coefficients of the resin base heat shields. Although some can be made fairly flat (and less than 5 x 10~ 6 in./in. °F) large variations with temperature and composition are quite common. High soaking temperatures (>300°F) have a second effect, however; namely, shrinkage. A negative a can result in either buckling the thin shell substructure, or cracking the heat shield, whereas the addition of flexibilizers can permit plastic flow; long storage times result in possible creep failure. The compromises result in very specific solutions for particular missions, with the development of heat shield material with limited ranges of suitable strength, conductivity and expansion coefficient, over and above their ablative properties.

The designer's prayer finally is for an adhesive that can be used as a potting compound, capable of absorbing differences in expansion between the inner structural shell and the outer heat shield. But this compound must also have high temperature strength (such as 1000-2000 psi at 500 to 1000 °F) and be applied at room temperature.

5. H E A T S H I E L D M A T E R I A L S

5 . 1 . Arc Testing and Analysis

The early experimental ablative materials for theoretical studies were teflon and silica. These established the relative effects of heat capacity, decomposition and surface temperatures vs. gas enthalpies and the blocking or blowing effect of the evaporating or ablating solids. Initially, these were for laminar stagnation conditions, but the re-entry problem involved a transition to turbulent heating. For engineering data, both the laminar and the turbulent tests were performed by high intensity arc plasmas. The sample sizes on which data is based run from less than 1 i n 2 to 30-40 in 2 . Translating the test results to applications over many square feet on the vehicle itself required a confidence in the result arrived at by many hundreds of sample tests. The difficulty in obtaining a uniform heat flux and truly homogenous materials gives rise to a considerable scatter in data.

Confirmation of the quality of results is in the data obtained in flight tests by recovery (limited to only a few vehicles) and by the translation of telemetered data on both ablation rates and temperature profiles. The temperature profile


data were reliable, and when combined with recovered vehicles and samples, demonstrated the validity of theory and experimental data as described in Fig. 6. These results applied to silica. 7 ' 8

time, sec

F I G . 7. Teflon ablation on a scout test vehicle (telemetered data).

The problem of simulation and prediction is enormous, since the scaling up of the shape and size while retaining the gas enthalpy and the gas flow and pressure characteristics, is a large and expensive facility problem. The most recent evidence of testing and correlation is shown in Fig. 7, the results of a 5-stage Scout missile testing teflon, 9 a " low" temperature ablator, and checking sensor data for a 2-sec period at Mach 13 at 78,000 ft altitude.

254 E. S C A L A

5.2. High Re-entry Angles: The ICBM and the Super-orbital Vehicle

The peak heating fluxes of 1000 to 10,000 B.t.u./ft2/sec for the ICBM with total re-entry heating times of 10-30 sec were solved with ablation techniques, and a number of reinforced ceramics and plastics were successfully tested. The materials problems were those of devising the most thermally efficient com-posites which could be formed with uniform properties into comparatively large diameter cylinders as well as the conical and hemispherically shaped nose sections. Designed originally as ablation materials, the balance of density and tensile strength became a criterion. The lower density favored the thermal properties and was necessary to meet weight restrictions. However, the loss in mechanical strength was critical when accounting for bending loads. The loads encountered upon deceleration called for unusual property measurements. The samples had to be tested dynamically, i.e. under the changing conditions of ablation affecting both the thickness, composition and rapidly decreasing resin properties with temperature. The short-time tensile properties were a good deal higher than those obtained by standard elevated temperature tensile tests.

PURE SILICA ' $ / K * L 4 ' S / « «0.S4 U N F I L L E D "POLYMER %

P L . 7 5 6 8 * •'• • . -

F I G . 8. Arc plasma ablation specimens with varying silica/resin ratios.

The organic base materials have continuing chemical reactions at the higher temperatures—a curing and overcuring, with ultimately the decomposition and charring. Considerable investigation into the process of decomposition that accompanies abla t ion 1 0 has resulted in an understanding of the mechanism and the methods of altering the process.

Figure 8 illustrates the varying appearances of arc plasma test specimens with varying compositions, from pure silica to a straight polymer. Figures 9 and 10 are cross-sections representative of the reactions occurring.

There are three phases involved, the most interesting of which is the in situ formation of a pyrolytic graphite (Fig. 10), i.e. the decomposition of complex organic degradation products on the outer char layer. (Table 1).

The controlled formation of a mechanically strong char permits applications under high shear force conditions such as in rocket nozzles. These particular polymers in Figs. 9 and 10 were treated to achieve high shear strength chars.


T A B L E 1 . 5 STEPS IN C H A R LAYER FORMATION

Phase I. Degradation of the polymer to form residual carbon and volatile products, varying with the original polymer and catalyst.

Phase II. React ion of degradation products with hot char to form pyrolytic graphite and low molecular weight products. Occurs at > 800°C. A m o u n t depends on chemistry and temperature. Char does not shrink appreciably because a fraction of carbon in the volatiles is re-deposited.

Phase III. React ion of pyrolytic graphite and inorganic constituents near surface. Large shrinkage.

SURFACE

»>^0LYTIC GRAPHITE

0*1 CARBONACEOUS

SUBSTRATjE

CARBONACEOUS

RESIDUE

D E C O M P O S I N G POLYMER -SURFACE •

UNCHARRED POLYMER PL. 2399

AS POLISHED

F I G . 9. Photomicrograph of cross-section of an ablated polymer sample.

5.3 . Manned Re-entry

The long-low enthalpy exposures obtained by low angle re-entry and lift (Apollo) has required modifications of the phenolic-silica composites to lower k and to form stronger char layers. The latter tend to be thicker and must adhere reliably for much longer periods. Typical of many chemical processing operations, the uniformity of the product becomes a critical factor preventing severe aggravation or local weak points. The blunt nose vehicles have of the order of 50-100 ft2 of exposed ablative materials in the critical high heat area, as compared to a few square feet for the high performance missiles. The radiative heating from the ionized gases behind the shock wave superimposed on the

256 E. S C A L A

convective heating has become a much greater factor. The testing and simulation requires the combination of a radiating source, such as a high intensity arc projector (with a controlled frequency distribution) plus the conventional arc plasma source. The sample sizes again are small compared to the vehicles and flight testing and instrumentation is the only full-scale test method. The reliability of ablation sensors have often been questionable and calculations of temperature profiles from thermocouple data must be relied upon, as well as recovery. The construction of the sensors is complicated by the composite compared to the pure teflon gauges referred to above.

PL, 2 3 9 9 C AS POLISHED MAS. iOOOX P 0 L A R I 2 E D

LIGHT

F I G . 10. In situ formation of pyrolytic graphite during ablation.

The aft sections with lower heat fluxes depend upon the radiation equilibrium of the hot wall structures (Rene 41 on Mercury, Inconel on X-15, stainless steels on B-70 and coated molybdenum on Dyna-Soar). The low temperature ablator or the high emissivity coating can be superimposed to decrease the total sensible heat absorbed. In comparison, the lower heat flux aft section of a higher performance vehicle can use a low thermal conductivity heat shield for the short re-entry time.

M A T E R I A L S A S P E C T S OF T H E R E - E N T R Y P R O B L E M 257

5.4. Super-Orbital Re-entry

Future space vehicles whose missions require recovery after re-entry into the earth's atmosphere, must either be slowed down to 50,000 or 60,000 ft/sec with maneuvering capability, parking orbits and/or retro-rocket devices, or spend very long periods (beyond 1 year) on their missions at lower velocities. The best of materials conceivable today could not survive without prohibitive amounts of ablation and therefore a severe weight penalty at launching. Figure 11 is an extreme case (from the ablation and materials viewpoint) illustrating the heat fluxes for a re-entry velocity of 140,000 f t / s e c . 1 1 ' 1 2 But the extreme heating conditions when compared to available meteorite statistics indicate little probability of successful re-entry. A velocity of 50,000 ft/sec to 60,000 ft/sec

TIME - SEC

F I G . 1 1 . Heat transfer for a space re-entry vehicle ( 1 4 0 , 0 0 0 ft/sec).

would probably have to be a maximum. Successful re-entry today at 35,000 ft/sec would be a major achievement.

Even modest increases in radiative heating to between 10 per cent and 30 per cent of the total heat absorbed by convection, require some consideration of the absorbtivity by the ablating gases and a high emissivity surface of probably a graphite or liquid material, as well as a very low k and high effective heat of ablation, plus a blocking effect while retaining a high shear strength.

6. S U B - S T R U C T U R E M A T E R I A L S

The greater developments must come in the application of more efficient sub-structures.

The re-entering vehicles more or less inherited the reliance in aluminum and stainless steel structures with conventional or commercially available epoxy bonding materials to hold the heat shields. The limiting inner wall temperature

10

258 E. S C A L A

was either the 500°F for aluminum or the shear strengths of the bonds, when low curing temperature potting compounds were used.

The recent manned vehicles have employed titanium substructures and con-siderable insulation (Mercury). Higher back wall temperatures will, however, be necessary with stainless steels employed such as the P H series. Extensive B-58 and B-70 manufacturing experience is available.

The advanced vehicles will employ more beryllium sheet metal structures and an increasing number of reinforced plastic shells. The improvement of beryllium processing by ring rolling plus a greater confidence in designing with higher ductility (in the plane of the sheet) cross-rolled beryllium, will probably help alleviate at least the most critical weight problems, that is, where the added cost is warranted. Al-Be alloy/composites are interim solutions.

With the steadily increasing tensile and modulus properties of reinforcing fibers, and the commercial availability of resins with stable properties over 700 °F, the reinforced plastics will become increasingly attractive. Perhaps their strongest advantage will be in matching coefficients of expansion through composition and process variables. The highly anisotropic properties will persist as a detriment to generalizations and analysis. The composites will be altered according to the application, which can also be viewed as an inherent advantage of the polymer-base composites, with the usual changing requirements mid-stream in most programs.

7. C O N C L U S I O N

Aside from the higher re-entry velocities, and the accompanying higher gas enthalpies and shear forces, there are increasing mission requirements super-imposing added environments on the basic heat-shield function.

The manned vehicles call for not only long soak times (over \ hr of steady heat input) but a hard vacuum as well. The re-entering space vehicles will experience re-entry heat fluxes after long exposures to space conditions of hard vacuum and ultraviolet, the combination of which will degrade most resins.

The hardening of vehicles to anti-missile devices is another superimposed requirement. The probability is that the most successful and/or expedient means will be the use of specialized protective coatings for both the above problems. The complications of more complex composites makes testing and prior analysis even more difficult, with considerable dependence on high temperature ad-hesives that can be applied without curing. The demand for non-destructive testing for defects and mechanical properties will increase and become more difficult.

Probably the best test is the scheduled exposure of the full-scale device to moderated environmental tests, acting as a sorting procedure to cull out those units most likely to fail, with a calculated risk on the retained properties of the surviving units. In so testing, the glaring errors due to lack of bond, stress-raisers and locally defective materials can be eliminated. More important, the


increased experience with the analysis of failures will permit greater feed-back to both structuial studies and materials developments.

A C K N O W L E D G E M E N T

The reports and figures referred to are through the courtesy of the Research and Development Division of the AVCO Corporation. The review is based on a number of their unclassified, published papers. The author was head of their Materials Department from 1955 to 1961, during which time they developed the Titan and Minuteman nose cones, and the heat shield materials for the Apollo re-entry vehicle.

R E F E R E N C E S

1. R I D D E L L and W I N K L E R . "Super-Orbital Re-entry", presented at the Joint A R S / I R E Meet-ing, Cincinnati, Ohio (April 12, 1960).

2. SANDERS, F. "Structural Analysis of Space Re-entry Vehicles", presented at the A R S (1104-60), Santa Barbara, California (April 1960).

3. SCALA, E. ASM Metals Review 3 3 , 6 (June 1960). 4. SCALA, E. ASM Metals Review 3 3 , 11 (November 1960). 5. STAMBLER, I. Space j Aeronautics 34 , 52 (July 1960). 6. M O R R I S , R. S. and D A V I D S O N , J. R. NASA Tech. Note 2)990 (March 1962). 7. A D A M S , M. and SCALA, E. High Temperature Technology, p. 54. N e w York: McGraw-Hil l

(1960). 8. SCALA, E. ASM Metals Engineering Quarterly 2 , 4, 69 (November 1962). 9. WINTERS, W I T T E , RASHIS and H O P K O . NASA-Langley , T N D-1500 (December 1962).

10. W A L K E R , A. and SCALA, E. Materials in Space Technology Chapter 11. L o n d o n : Iliffe Books .

11. W I N K L E R , H. B. "Some Problems Associated with Re-entry after Planetary Exploration" presented at the Joint A R S / I R E Meeting on Electronic Data Processing and Space Technology, Cincinnati, Ohio (April 12-13, 1961).

12. R I D D E L L , F. R. and W I N K L E R , H. B. "From I C B M Re-entry to Meteoritic Re-entry" presented at the Joint I A S / A R S Meeting (June 13-16, 1961).

THE ANALYSIS OF PROBLEMS OF HEAT

CONDUCTION AND MELTING

BRUNO A. BOLEY

Columbia University, N e w York, N . Y .

A B S T R A C T

A review of the developments of the past few years in the field of heat conduction is presented, including the application to problems of change of phase and moving boundaries. Considered in detail are mathematical studies of properties of solutions of parabolic partial differential equations and approximate methods of solution. N e w techniques for the construction of approximate solutions are indicated, which bracket the exact results between easily calculated upper and lower bounds. Problems in which the properties are temperature-dependent are included. The calculation of thermal stresses in a melting inelastic material is discussed.


The theory of heat conduction in solids sprang into full adulthood with the appearance of Fourier's treatise 1 some 140 years ago; it has been, since that time, the object of hundreds of investigations on the part of engineers, physicists, and mathematicians covering it in its theoretical as well as in its practical aspects. As far as the discovery of exact solutions in heat conduction is con-cerned, however, an examination of the well-known text of Carslaw and Jaeger 2

will show that most of the available methods are either direct descendants of those known and used by Fourier, or stem from the development of the 1890's and of the first few years of this century in the area of integral transforms and operational techniques. All these methods suffer from the disadvantage that they tend to become extremely cumbersome when the geometry of the body in question is at all complicated, and usually fail to be valid when the material properties are temperature-dependent, or when another kind of non-linearity is introduced by the presence of unknown moving boundaries. Because of the current importance of these types of problems, a great deal of research is presently being carried out both in this country and abroad, dealing mainly with three aspects of it, namely (a) the mathematical theory of partial differential equations of the parabolic type (of which the Fourier equation is one of the simplest examples), (b) the establishment of approximate methods of solution, and (c) the development of numerical techniques, particularly to be used in conjunction with high speed computers. A discussion of the developments of the last decade or so in all these aspects of the field, concerning both the linear and the non-linear problems previously mentioned, forms one of the principal objects of this

260

P R O B L E M S OF H E A T C O N D U C T I O N A N D M E L T I N G 261

paper. In this review an effort will be made to given an overall view of the present state of the field, and to indicate some of the most important unsolved problems and some of the areas in greatest need of further study.

In addition to this general review, a number of new approaches and methods will be presented, which appear quite promising at the present time, though of course their full potentialities are yet to be assessed. It is hoped that these new approaches, as well as the various conjectures and suggestions for future work found in this paper, would be of interest in providing at least a start of some useful researches.

Interest in the effects of temperature on solid bodies has increased in recent years because of developments in space and high speed atmospheric flight and in nuclear energy applications; of great importance in these problems are the thermal stresses and thermal deformations, which must often be calculated taking into account inelastic effects. Moreover, the importance of thermal stresses is by no means limited to such advanced technological applications; for example, of direct naval interest are the thermal stresses which arise in ships under many operating conditions. 3 ' 4 For these reasons problems of thermal-stress analysis are on some occasions touched upon in this paper; for a fuller discussion of such problems one may consult Refs. 5 or 6, the former of which also deals with some of the pertinent aspects of heat conduction.

The field of heat conduction is part of the general subject of heat transfer, under which the topics of radiation and convection are also included. These will in general not be treated here, since as a rule only the process of conduction is of importance in the case of solids;* as basic references for heat transfer in general, the reader may take for example the texts of Eckert and D r a k e 1 0 and of Krei th 1 1 . Simultaneous heat and mass transfer is treated in the book of Lykov and Mikhaylov 1 2 . Up-to-date listings of the current literature in all phases of heat transfer are published in the International Journal of Heat and Mass Trans-fer, as well as, of course, in the standard review journals such as Applied Mechanics Reviews.

To facilitate the reading of this paper, it is perhaps useful to outline here briefly its contents by a listing of the titles of its sections:

1. Introduction 2. Thermomechanical coupling; temperature propagation 3. General considerations: problems with prescribed boundary temperature 4. General considerations: prescribed heat-input problems 5. The approximate solution of heat conduction problems 6. Use of upper and lower bounds to obtain approximate solutions 7 . Saint-Venant's Principle in heat conduction 8. Moving-boundary problems: general discussion

* In the case of transparent materials (e.g. glass), or of foam-type materials such as are at times employed in sandwich cores, radiation may become significant; for a discussion of the latter case see for example Swann 7 or Swann and Pit tman 8 . For an analysis of the relative importance of radiation and conduction in a wing structure, see Hoff 9 .

262 B R U N O A . B O L E Y

9. Approximate solutions of moving-boundary problems 10. Upper and lower bounds in moving-boundary problems, and their

application in obtaining approximate solutions 11. Thermal stresses in melting or solidifying bodies 12. Conclusions.

Further details of certain developments of Sections 2, 6, and 7 are collected in three Appendices.

The work discussed in Section 7 and in Appendix C, dealing with the de-velopment of Saint-Venant's Principle in transient heat conduction problems in which the surface heat flux is prescribed, was performed under a project spon-sored by the U.S. Army Research Office (Durham). The author wishes to acknowledge the contributions of Ting-Shu Wu particularly to this phase of the work. The remainder of the research described here was performed under sponsorship of the Office of Naval Research. The assistance of David Sikarskie in the analyses pertaining to the embedding technique described in Section 6 is gratefully acknowledged.

Fourier's law of conduction leads to the well-known Fourier heat conduction equation, which, for the case of constant properties, and in the absence of heat generation, has the form

where k is the thermal conductivity, p the density, and c the specific heat. This is a partial differential equation of the parabolic type and consequently does not yield a finite velocity of temperature propagation; in other words, a thermal disturbance applied to a portion of a body instantaneously affects all points of the body, though by very small amounts at large distances from the disturbance. This behavior is often felt to be physically unreasonable and in contradiction with any microscopic theory of material behavior, whether this describes heat conduction as essentially a flow of electrons or an atomic vibration 1 3 » 1 4 Cor-rections such as that proposed by Roy 1 5 , to include possible effects of large temperature gradients by the addition of terms containing higher space-deriva-tives of the temperature, clearly do not affect the parabolic character of the equation. The introduction into the energy balance represented by equation (2.1) of a term corresponding to the energy of deformation leads to the coupled heat equation*

2 . T H E R M O M E C H A N I C A L C O U P L I N G :

T E M P E R A T U R E P R O P A G A T I O N

(2.1)

(2.2)

* This equation was first obtained by D u h a m e l 1 6 ; a derivation of it and of its counterpart for a linear viscoelastic material may be found in Ref .• 5, while the case of a plastic material has been examined by D i l l o n 1 7 .


where A and /x are the Lame elastic constants, a is the coefficient of linear ex-pansion, r* is an absolute temperature at which the material is stress-free, and e is the dilatation. Since the deformations now appear as dependent variables in this equation (in addition to the temperature), a solution cannot be effected unless equations describing the mechanical behavior of the material (i.e. equa-tions of motion, constitutive or stress-strain laws, and strain-displacement rela-tions) are also introduced. In the one-dimensional elastic case, these relations may be combined into the single equation 5

(2.3)

where the displacement u is related to the dilatation by the relation e = du/dx. Equation (2.3) is of the hyperbolic type, or in other words it possesses real characteristics indicating that discontinuities are propagated with the velocity

(2.3a)

Nevertheless the system formed by the two equations (2.2) and (2.3) still con-tains a parabolic component, and thus any solution of this coupled system still exhibits a portion which is propagated with an infinite velocity.*

Further corrections to the Fourier heat conduction equations, which are necessary for certain applications, have been derived by including electromag-netic terms in the energy balance. Theories of this type have been developed by Mindl in 1 8 and Kaliski and Petykiewicz 1 9 . Examination of the temperature terms in these equations shows that the basic derivatives of equation (2.1) are still dominant and that therefore a finite velocity of temperature propagation cannot be expected.f

In order to arrive at a finite velocity of temperature transmission, it is clearly necessary to change equation (2.1) to a hyperbolic equation; the simplest way to do this is to insert a term proportional to the second time-derivative of the temperature, leading to the following equation

(2.4)

Such an equation is stated for example by Morse and Feshbach 2 0 , on the physical argument that a velocity of propagation a must exist, which is, for a gas, a function of the mean free path of the molecules. A microscopic derivation of this equation, or indeed any derivation for a solid, has not yet been given. In its absence, one may take refuge in the view that, even in a gas, the diffusion equation (2.1) represents a statistical average of molecular motion, and thus finite but very small probabilities at large distances from the disturbance are not a priori unreasonable. In any case, no experimental failure of the diffusion

* A more detailed discussion of the solution of equations (2 .2 ) and (2 .3 ) is given later in this section.

t When the full Maxwell equations are used in the derivation, waves traveling with the velocity of light appear, in addition to those traveling with the velocity of equation (2 . 3a).


(b) When tractions are prescribed on the surface of the half-space in con-junction with homogeneous thermal conditions, no responses are present, in the uncoupled theory, ahead of the wave front traveling with the above velocity v; in the coupled solution, however, they occur instantaneously at all points of the body, though they are quite small at large distances from the surface.

(c) The temperature distribution is a continuous function both in the coupled or the uncoupled theory, but discontinuities in either the first or the second space-derivatives of the temperature are present in the coupled solution; these discontinuities also travel with the velocity v and decrease exponentially with distance.

The formulation of similar boundary value problems is also discussed by Chadwick 2 5 , where further references on this subject will also be found.

Since the derivation of exact solutions to the coupled problem is often quite difficult, it is desirable to devise approximate methods for its analysis. Ap-proximate solutions may of course be obtained by suitable simplifications in the Laplace or Fourier Transform domain (e.g. for short or long times or for small S) as for example in Refs. 21, 22, 24. It is, however, useful to obtain ap-proximate solution directly as was done by Achenbach 2 6 in the following way.

It seems reasonable to seek a first approximation by solving (for the case of a mechanical disturbance, for example) equation (2.2) with the last term cal-culated from the solution of equation (2.3) with the temperature term omitted,

* Earlier results of Par ia 2 4 have been shown by Hetnarski 2 2 to be in error as they predict at discontinuous temperature distribution in contradiction with the results quoted above.

(2.5)

equation has been detected, so that one may ask whether, apart from the obvious theoretical importance of a correct understanding of heat transmission, any engineering problems exist which admit of a macroscopic description and are nevertheless not adequately described by equation (2.1) or one of the cor-rections previously mentioned.

In spite of these unanswered questions, it should be emphasized that many problems exist in which the various corrected forms of the heat equation can be satisfactorily used. For example, a study of thermoelastic damping can be carried out on the basis of equations (2.2) and (2.3), and in fact problems concerned with this coupled system have been studied by a number of authors. The principal boundary-value problem treated is that of a thermoelastic half-space under prescribed traction and temperature variations on its surface, for which the following principal results (Refs. 5 (p. 73), 21-23) are obtained:*

(a) Mechanical discontinuities are propagated with the velocity given by equation (2.3a), as in the uncoupled theory, but their magnitude decreases exponentially with time in the coupled theory while it remains unaltered when coupling is omitted; that is, it decreases as e~8(vx/2K\ where the dimensionless coupling parameter is


i.e. from the wave equation. Note that in this case the term (de/dt) is infinite at the wave front and zero everywhere; else it can therefore be written in terms of the Dirac delta function. This procedure indeed leads to good results for short times; if in addition the term (de/dt) is taken so as to account for the expo-nentially decreasing magnitude of the discontinuity at the wave front,* then excellent agreement with the known exact results is obtained for all times.

Thermoelastic wave-propagation problems have been treated by Deresie-w i c z 2 8 - 3 0 and by Chadwick 2 5 . In an infinite medium these studies predict the existence of two damped, dispersive, and dissipative dilatational waves, one of which is predominantly elastic, and the other predominantly thermal, as well as a purely elastic shear wave.

The thermodynamics of themomechanically coupled systems has been ex-tensively studied by B i o t 3 1 - 3 4 and leads to general variational procedures for solution of both coupled and uncoupled problems, some applications of which will be discussed later in this paper. Biot's work has in general been responsible for much of the recent understanding of irreversible thermodynamics and has established a consistent basis for thermoelasticity and indeed for other coupled problems (cf. Ref. 18). It may be of interest to note also one earlier result of irreversible thermodynamics in the field of heat conduction, namely an ex-planation (on the basis of Onsager's reciprocal relations) of the symmetry of the conductivity tensor in anisotropic cases for which it cannot be derived from considerations of crystalline symmetry. 3 5

3 . G E N E R A L C O N S I D E R A T I O N S :

P R O B L E M S W I T H P R E S C R I B E D B O U N D A R Y T E M P E R A T U R E

We now turn to an examination of the Fourier heat conduction equation, which for an isotropic medium, has the form:

(3.1)

where the k, p, and c are (positive) functions of x, y, z, t9 and T in the general case of an inhomogeneous material with temperature-dependent properties, and where / represents a source of heat distributed throughout the body. The bulk of this section will be taken up by a discussion of certain basic theorems con-cerned with properties of solutions of this equation under certain boundary and initial conditions, several of which will be used later for the construction of exact or approximate solutions to specific problems. On occasion (as in Theorem I) a somewhat more general equation than (3.1) will be used, but, more fre-quently, special cases of (3.1) will be considered. Proofs will be given only where they cannot be found in the literature, with references to the appropriate litera-ture in other cases.

* The magnitude of the discontinuities may be readily found, for example, by the method of Ref. 27.

io§


Clearly equation (3.1), with temperature independent properties, is a special case of (3.2), obtained with

Atj = 0 for i j , i.e. for an isotropic medium; (3.3)

Consider a (twice continuously differentiable) solution of (3.2) in a domain D in the space of points P with coordinates* xi, x2, . . . , xn, t. Let B be the boundary of D, and, in particular, let B-t be that part of B which includes all points P such that (a) the interior normal to the boundary exists and is directed in the negative ^-direction, and (b) all such points are interior to B-t. Thus, for example, in the one-dimensional case of (3.1), the boundary B is A\A2AzA±A\ and B-t is A3A4. Note that conditions specified on portions such as A\A2

(whose interior normal is in the positive ^-direction) are normally referred to as "initial conditions", while those specified on other portions,t such as A\A± and A2A3, are usually called "boundary conditions"; no conditions are pre-scribed on B-t. If the portions of the boundary A1A4 and A2A3 are composed exclusively of segments parallel to the t axis, the domain D is said to be one with fixed boundaries, while if they are not so composed, the domain D is said to have moving boundaries; these terms clearly refer (like the number of dimen-sions discussed earlier) only to the behavior of the boundaries in the physical space. We can now state Theorem I

Theorem I (Picone's Theorem?®)

(a) If u > 0 on B — B-t and / > 0 in D, then u > 0 throughout D; similarly, if u < 0 on B — B-t and / < 0 in D, then u < 0 throughout D. This means,

* This is therefore an (n + l)-dimensional space; however in order to conform to usual practice, we will refer to it as describing an ^-dimensional problem, so that equation (3 .1 ) then pertains to three-dimensional and equation ( 4 . 1 ) to one-dimensional heat conduction.

f If the portion A1A2 is denoted by Bt, then we can write (B — Bt — B-t) for the portions A1A4 and A2A3.

K, the thermal diffusivity:

Consider the equation

(3.2)

(3.2a)

where

Aij = Ajt, Bt, C, a n d / a r e real finite continuous functions of xi (i = 1, 2, . . . , n)

but not of u, and where the quadratic form with coefficients Atj is positive definite, i.e.


for example, that it is impossible to lower the temperature anywhere in a body by raising the boundary temperature or by a positive distributed heat generation. It follows that

(b) if u = 0 on B — B-u and f=0, then u = 0 throughout D. Hence, from the linearity of the problem, it follows that

(c) if u is prescribed throughout B — B-t, and / is prescribed throughout D, then u is uniquely determined throughout Z), and that

(d) if two solutions u\ and U2 of equation (3.2), corresponding respectively to / = fi and f = fz where fi > fa, are such that u\ > U2 on B — B-t, then m > W2 throughout Z). It can also be proved that

(e) if C < 0 and / = 0 the maximum value of | u | occurs on B — B-t (i.e. either initially or on the physical boundary) and, if C — / = 0, then both the maximum and the minimum values of u occur on B — B~t-

Much of Picone's Theorem is readily acceptable on intuitive physical grounds, though of course this does not eliminate the necessity for proof. Theorem II, which is stated next, is on the other hand less obvious on such a basis. State-ment (e) above is known as the "weak" maximum principle for parabolic differential equations, as compared to the following "strong" principle which has been proved by Nirenberg 3 7 .

Theorem II

If u assumes its maximum value in the interior of D, then u is a constant throughout D. It can easily be seen that the strong principle implies the weak one, but the weak principle does not imply the validity of the strong one.

A generalization of this theorem was obtained by Fr iedman 3 8 , who also proves a uniqueness theorem under certain non-linear boundary conditions including the ones of principal importance in heat conduction; further exten-sions and applications were given by Pucci 3 9 and by Velte 4 0 . A uniqueness theorem for linear parabolic differential equations of orders higher than the second was presented by Lax and Milgram 4 1 , and, for the case of constant coefficients, by Rosenbloom 4 2 .

We wish to consider next the extension of Theorem I to the case in which the coefficients depend on the temperature, as well as explicitly on the space co-ordinates. Clearly this makes the problem a non-linear one, but any portions of the proof of Picone's theorem which do not make use of the linearity of the equation will still be valid here. The proof of part (a), and its corollary (b), rest on the fact that a negative minimum or a positive maximum occurring in any solution of equation (3.2) would be in contradiction with relation (3.2a) and hence are not possible; since linearity is not involved here, we can state as follows.

Theorem III

Consider a solution u equation (3.2), with finite continuous coefficients Aij = An (xi, X 2 , . . . , xn, u) = AJU Bi = Bt (xi, . . . , xn, w), but still with


/ = f(x±9 X2, . . . , xn) and with relation (3.2a); in this case

(a) if u > 0 on B — B-t and / > 0 in D9 then u > 0 throughout D

if u < 0 on B — B-t and / < 0 in D, then u < 0 throughout D

(b) if u = 0 on B - a n d / = 0 in Z>, then u = 0 in Z>

The proof of parts (c), (d), and (e) of Picone's Theorem cannot be so readily extended to the non-linear case. We restrict ourselves now to the class of prob-lems in which the properties are functions of the temperature but not explicitly of the space coordinates (that is to homogeneous materials) and to the heat-conduction equation proper, equation (3.1), rather than the more general case (3.2).* Here k, p, and c are taken to be positive single-valued functions of T, and we introduce the heat content H(T), namely a non-negative monotonic function of T defined as follows:

so that equation (3.1) becomes:

(3.4)

(3.5)

It will be useful to note that integration of (3.5) over a volume of space V (which may vary in time) with bounding surface S gives, after use of the divergence theorem:

(3.6)

where m denotes the direction cosines of the outward normal to the surface S. Another result which will be useful concerns the addition of a constant

N to a solution T of equation (3.5). The function T* = T + N satisfies the equation

(3.7)

(3.7a)

Hence the addition of a constant to any solution of (3.5) yields a solution of another equation, which is however of the same form as (3.5), provided the quantities k, />, and c are suitably defined over the required range.

We shall now prove the following.

* A general treatment for domains with fixed boundaries, for boundary conditions including those of prescribed temperature, heat flux and radiation is given in a recent paper by K a p l a n 1 5 8

which came to the author's attention after the completion of this work.

where

k*(T*) = k(T* - AO; p*(r*) - p(r* - N); c*(r*) = c(r* -


Theorem IV

(a) if a solution T of (3.5) is prescribed throughout B — B~t, then it is uniquely determined throughout D

(b) if two solutions J i and T2 of equation (3.5), corresponding respectively to / — fi and / = / 2 where / 1 > fa are such that T\ > T2 on i? — 2?-*, then 7 i > T 2 throughout D

(c) i f / = 0, then the maximum and minimum values of T occur on B — B-t To prove part (a), assume that two solutions Ti and T2 of equation (3.5)

exist, such that Ti = T2 on B — B~t, but which are not identical throughout D. If this is the case, there must be a subdomain of D (say Di) such that Ti < T2 in the interior of Di and Ti = T2 on the portion (Bi — B±t -t) of its boundary B\. Write equation (3.6) for each of these solutions, for the domain Di, and subtract the results noting that

The left-hand side of this equation is clearly non-negative, while (Hi — H2) is zero initially* and non-positive thereafter; hence the right-hand side cannot be positive and thus the only possibility is Ti = Ti. A different type of proof, analogous to the classical uniqueness proof valid for the constant-properties case, is given in Appendix A for the special case in which the diffusivity is constant.

To prove part (b), assume that there exist two solutions Ti and T2, satisfying the hypotheses of the theorem, but such that one subregion at least of D exists within which Ti < T2. We can certainly choose this subregion so that it is surrounded by another subregion (whose outer boundary may be identical with that of D), in which Ti > T2; but then Ti =-- T2 on the boundary of the former subregion, and this, as has been seen in the proof of part (a), leads to a contra-diction. Hence Ti > T 2 throughout.

To prove part (c), we note that, if M and m are respectively the maximum and minimum values of T on B — B-t, then the functions TM = T — M and Tm = T — m are non-positive and non-negative, respectively on B — B-u and satisfy equations of the form (3.5) ; hence, by Theorem III, they maintain (if / = 0) this character throughout D, and the proof is completed.

Properties of Green's Functions

Several important properties of the Green's functions pertaining to the linear transient heat conduction equationf can be derived from Picone's Theorem. The principal ones are as follows 4 3 J

* This implies that, under certain conditions, its integral over Vis zero; cf. Ref. 2. f The Green's function G(Pi\P2\t1 — ti) is the temperature at a point Pi in a body at a

time ti > H due to a unit source released at P2 at a time / 2 , the boundary and initial tempera-tures being zero.

t In the steady-state case these properties are of course wel l-known; see for instance Ref. 44.

k(Ti) = k(T2) and H(Ti) = H(T2) on S (3.8)

The result is (3.8a)

270 B R U N O A . BOLEY

(a) G(Pi; P2, h — ^2) > 0, or, the temperature due to a source in a body, whose initial and boundary temperature is zero, cannot be negative; hence

(b) dG/dn(Pi) > 0, or, the inward normal derivative of G cannot be negative

(c) if Gi(Pi; P2; h — t2) and G2CP1; P2I ti — £2) are the values of the Green's functions respectively for two domains D± and D2 such that D2 completely encloses D±9 then at all points P± and P2 in Di, G2 > Gi. As a consequence

(d) if the above domains D± and D2 have a portion of their boundary in common, over this portion dG2ldn(Pi) > dGijdn(P\)9 where n(Pi) is the inward normal at the boundary there.

Properties such as the ones just enumerated are important because they allow (through the well-known integral formula giving the solution corresponding to a prescribed surface temperature explicitly in terms of G), the construction of bounds and of approximate solutions to many problems. Unfortunately, the Green's function is known only for a limited number of domains, 2 and it would therefore seem desirable to devise methods for the construction or for the calculation of G, either exactly or approximately, for arbitrary domains. Methods for the construction of Green's function have been discussed in the l i t e r a t u r e 4 4 ' 4 5

but appear to be in general rather cumbersome for analytic derivations; their use for approximate or numerical analyses has however not been fully explored.*

The entire discussion of this section has been concerned with problems in which the boundary temperature is prescribed, and it is now necessary to extend them to include more general boundary conditions, such as those of prescribed flux or of linear or non-linear radiation into a medium at a given temperature. Generally speaking, less work has been done on these types of problems than on those with prescribed surface temperature, although they are often technically more important. Some work pertaining to one-dimensional problems under heat-flux conditions is presented in the next section; clearly further generalizations would seem to be very desirable. Similarly, all the results of this section and the next should be extended to the more general case of equation (3.2), with coefficients dependent on both the Xi's and T. Such an extension would include the case of anisotropic solids, for which the general heat conduction equation is indeed 2 of the form (3.2).

Some of the published work concerning general results in problems with boundary conditions other than that of prescribed surface temperature may be listed here. Evans 4 7 has considered the problem of a slab insulated on one face and under a radiation condition on the other, with heat generation permitted over a fixed portion of the thickness. Uniqueness theorems have been established by Fr iedman 3 8 for the case of the linear equations, but allowing certain non-linearities in the boundary conditions.!

* A different approach which in effect consists of a method for the construction of Green's function has been introduced in elasticity theory by Massonnet ; 4 6 its application to heat con-duction has not yet been attempted.

f N o t e K a p l a n ' s 1 5 8 paper mentioned earlier, which also contains a bibliography of recent Russian work on this subject.


4 . G E N E R A L C O N S I D E R A T I O N S :

P R E S C R I B E D H E A T - I N P U T P R O B L E M S

We turn now to an examination of the one-dimensional conduction equation in a simply-connected domain in the .xt-plane (Fig. 1), under boundary condi-tions specifying a heat-input history. Curves AiA^ and A2A3 are taken to be defined by Lipschitz continuous single-valued functions x = Fi(t) and x = 7*2(0

t i

x

F I G . 1 .

respectively; in some cases curve A2A3 will be a fixed boundary, i.e. x = X2, a constant. For any solution u of the following special case of (3.2)

(4.1)

where the coefficients A and B are finite continuous functions either of u or of x and t (unless otherwise indicated), we then have 4 8 *

Theorem V

(a) if du/dx = 0 on x = Fi(t) and x = F2(t),f = f(t), and u = uo (a constant) on the line t = to, then

(4.2)

and, in particular, if / = = 0, then u = uo throughout D (b) let A = A(x, r), B = B(x, t), and / = f(x, t); it follows from part (a) that

if du/dx is prescribed on x = F±(t) and x = Fz(t), then u is uniquely determined throughout D

(c) consider now the special case in which! x = X29 a constant, i.e. a fixed

* The statements given here are slight generalizations of those actually proved in that work; no proof is however presented here since it would be entirely analogous to that given there. See also Ref. 49.

t In this case we may extend the solution by reflection about x = X 2 , so that u(x,t) = u{2x2 — x,t) and thus u is defined throughout the domain Z>* bounded by x = Fi(t) and x = 2x2- Fi(t).


boundary, and A and B are again functions either of x and t or of u, and let

du —- — 0 on x = X2 (i.e. an insulated edge) ox

u = UQ(X) on t = ft (i.e. a prescribed initial temperature) (4.3)

and either du/dx = — cf>(t) or u = <j>(t) o n x = Fi(t). Then, in either of these cases, if (f>(t) > 0 and uo(x) > 0, u > 0 throughout D, and if </>(£) < 0 and uo(x) < 0, u < 0 throughout Z>.

We now restrict ourselves (just as in Section 3, cf. Theorem IV) to the heat conduction equation (3.5), or, for the one-dimensional case, to

for a domain D bounded by a curve x = F±(t) and by x = X2; then we have the following uniqueness and comparison theorem

Theorem VI

(a) if a solution T of (4.4) corresponds to a prescribed heat input on its boundary and is initially zero, i.e. (with n the outward normal)

(4.5)

then 7 \ > T2 throughout D. To prove part (a),* assume that two solutions 7 i and T2 of (4.4) exist, each

of which satisfies (4.5), and consider the expanded domain D* of Fig. 2. If T± = T2 on x = Fi(t) for a period to < t < h, then Theorem III insures that Ti = T2 throughout D* in this period. Suppose then that a time h ( > ft) exists such that on x = F±(t), T\ = T2 for to< t < ti and, without loss of generality, T±> T2 for h < t < h + 8, 8 > 0; then Theorem III insures that 7 \ > T2

throughout D* within this period. Write equation (3.6) for each solution, and

* A s will be seen, the general proof of part (a) follows that of part (b); the present limited proof is nevertheless of interest and is therefore included anyway.

(4.4)

T(x, to) = 0

(4.6b)

and are such that the heat input corresponding to T\ is never less than that corresponding to T2, i.e.

(4.6a)

then T is uniquely determined throughout D. (b) if two solutions T± and T2 of equation (4.4), corresponding respectively

to / = fi and / = / 2 where f± > fa are both initially zero, i.e.


subtract to get, with the first of (4.5),

(4.7)

Now, if dFjdt > 0, clearly neither of the terms of the right-hand side can be negative and hence in this case T± == T2. To remove this restriction on F(t), we must prove part (b) first as is done below, and part (a) then follows as a corollary.

J A \

x - x 2 y c - 2 x 2 - F , ( t )

\

to

F I G . 2.

To prove part (b), we may assume without loss of generality that a time

h (> to) exists such that

Ti > T2 for t 0 < t < h

Ti < T2 for h<t<h + 89 8 > 0 on x = F\(t) (4.8)

Clearly T± > T2 in the first of these periods throughout D*. At the point [x = F(h), t{\9 however, we have T\ = T2 and thus, from equation (4.6b), dTi/dx < dT2/dx; but this would give a temperature T\ lower than T2 some-where along the line t = t\9 and hence cannot occur, unless at that point it also happens that dTi/dx = BT2/dx. But then differentiation along x = Fi(t) gives*

or further, with equation (3.5),

(4.9a)

(4.9b)

* This portion of the proof bears great resemblance to the second part of the proof of Theorem IV of Ref. 48, or of Theorem V of Ref. 49. A more general proof, namely one eliminating the assumption of the existence of all the higher derivatives involved, has not been given.


This, however, again leads to a contradiction, unless of course the equality sign holds. Further differentiation shows that this too leads to a contradiction and the same is true of all subsequent special cases of this type; the theorem is thus proved.

Use of the theorems of this and of the preceding section will be made later in this paper in the calculation of approximate solutions and particularly of bounds to solutions, both with and without change of phase. The remarks contained at the end of the previous section, concerning the desirability of generalizing all the present results in various directions, should again be emphas-ized. It may be mentioned here, for example, that no roster of properties such as were listed in Section 3 for the Green's function is known for the Neumann or Robin functions, and consequently the construction of bounds and the establishment of Saint-Venant's principle is very difficult in these cases.

5. T H E A P P R O X I M A T E S O L U T I O N O F

H E A T C O N D U C T I O N P R O B L E M S

We now turn to an examination of approximate methods for the solution of heat conduction problems, and restrict ourselves in this section to problems pertaining to bodies with fixed boundaries, moving boundaries being examined in Sections 8-11.

Numerical and computational techniques have been considered in several papers, of which a small number of recent ones will be listed here. The general problem of the replacement of a one-dimensional linear (or quasi-linear) para-bolic differential equation in an infinite domain by an appropriate finite-difference equation has been considered by John 5 0 . The question of the stability of finite-difference approximations to the linear partial differential equation has been treated by Lax and Richtmyer 5 1 and T o d d 5 2 ; a good bibliography is included in the latter paper. The convergence of an explicit finite-difference scheme is studied by Douglas 5 3 and that of an implicit scheme with variable time increments by Douglas and Gallie 5 4 , Varga 5 5 ; see also Dusinberre 5 6 . A one-dimensional treatment, including the effect of temperature-dependent pro-perties and a study of the error involved in the calculations, was undertaken by Chu and Abramson 5 7 . The above problems, as well as that of composite slabs, were also studied by Lotkin 5 8 . Other treatments of stability and convergence criteria were given by O'Brien, Hyman, and Kaplan 5 9 , DuFor t and Frankel , 6 0

and Douglas 6 1 ; the procedure of Crank and Nicolson 6 2 was further studied by Juncosa and Young 6 3 . Relaxation methods were treated by Allen and Severn 6 4 . The use of finite-difference approximations in the space domain, but with retention of continuity in the time variation, has been extensively used by Paschkis 6 5 ' 6 6 , particularly in conjunction with the use of analog computers.

The major portion of this section will be taken up with approximate methods

(4.9c)

so that finally, at the point in question,


of solution which are generally designed to yield answers in an analytical form. The best known approximate methods of this type are based either on the weighting-function approach or on Biot's Lagrangian formulation. In all these methods the first step is the choice of an analytical expression for the tempera-ture, satisfying the boundary conditions of the problem, and containing a number of possibly time-dependent parameters; the methods differ in the manner in which these parameters are obtained.

The weighting-function approach consists in replacing the differential equa-

* Use of the Galerkin technique in conjunction with the Laplace Transformation for the solution of equation (5 .2 ) has been introduced by G r e e n 6 8 , Weiner 6 9 , and Dicker and Fried-m a n 7 0 ; a discussion of the latter paper is given further on in this section.

f See also the papers by Biot cited earlier for further details for the equations pertaining to the case of variable properties, and for further examples. For a derivation of a variational principle in heat conduction, see also Chambers 7 6 .

(5.1)

by the following n definite integrals over the volume V of the body

(5.2)

where the functions Ft are to be suitably selected. However these functions are taken, it is clear that each of equation (5.2) results in an expression containing the parameters and their first time-derivatives; n is thus taken equal to the number of parameters and so a system of n first-order ordinary differential equations in n unknowns is obtained. We distinguish two principal cases, as follows: let y, z, t; qi, q2, . . . , qn) be the chosen expression for the tem-perature in terms of the n parameters qt; then if

(5.3a)

the procedure reduces to the Galerkin method 6 7 * and if

Ft = xni yN2 zns; m, n2, n% = 0, 1, 2 . . . (5.3b)

the procedure reduces to the method of moments (Refs. 71, 20 (p. 94)). In the latter case this procedure has been widely used with the special restriction that n = 1 and Fi = 1, and is then known as the "heat-balance" method. 7 2 ' 7 3

Biot's method 7 4 ' 7 5 | requires the determination of the n parameters from the n Lagrangian equations

(5.4)

where the thermal potential U, the dissipation function D and the thermal


forces Qt are defined as

(5.4a)

in which HX, HY, HZ are the components of a heat-flow vector H such that

The choice of method to be used in any one problem must be made on the basis of accuracy and ease of application, but unfortunately it is very difficult to establish any general criteria to facilitate this choice. The heat-balance method often leads to integrations which are simpler than those of the other methods, but this advantage may disappear if further approximations are sought by its natural extension, the method of moments. On the other hand, if the temperature is expressed as a sum of orthogonal functions (e.g. in a Fourier series), then the Galerkin or Biot approaches are certainly preferable. The possibility of obtaining, by any of these methods, not only one approximate solution but also further corrections is not to be overlooked, since often a comparison between two successive approximations is the only available criterion for the accuracy of a given solution,* bearing in mind, of course, that the inclusion of more parameters in the analysis does not guarantee greater accuracy uniformly in the region for which it is sought. The possibility exists, at least in theory, of deriving exact solution by choosing the temperature as an infinite series whose terms form a complete set of (if possible orthogonal, in the Galerkin or Biot methods) functions; if this is done, some comfort may be derived from the fact that the exact solution can be approached as closely as one wishes by the inclusion of more and more terms. A slight modification of Biot's method, such that each successive approximation corresponds to a mini-mum of a certain functional (which is zero if and only if the exact solution is used) has been given by Ci t ron 7 7 , and may be useful in this connection.

In using all the preceding methods it has often been found useful, when a thermal disturbance is applied to the surface of a body, to assume that the temperature is zero beyond a distance q(t) from the exposed surface. This distance (Biot's "penetration depth") is then used as one of the parameters of the solution. A new temperature function must of course be used after the time tT (Biot's "transit time") at which q(tT) is large enough to reach another surface of the body. Only one-dimensional problems, either in rectangular or cylindrical coordinates, have thus far been solved in the literature by these methods,! perhaps because of the difficulty of choosing a suitable temperature function.

(5.4b)

* Sometime upper and lower bounds can be used for this purpose as discussed in Section 6. t See however a two-dimensional application of the penetration depth concept, although in

a different way, in Problem (c) of Section 6.


It would seem therefore desirable to show how they can be practically used in two- or three-dimensional cases.

These one-dimensional solutions may be used for a further remark on the relative accuracy of these methods. Consider for example the case of a half space (or a slab before the transit time), heated in some manner on its surface; then it is reasonable to suppose that the solution in a region adjacent to the surface is more crucial than that near the penetration depth, or in other words that a more accurate solution will be obtained if the weighting functions are such as to give more importance to the former than to the latter of these regions. This may be verified by actual calculation; for example if the half-space x > 0 is initially at zero temperature, and its surface x = 0 is raised to a temperature 7b, the choice

(5.5)

gives the surface heat flux as*

[0 .578 = l/\/3, by the heat-balance method: q=^(\2Kt)

0.632 = 2/^/10, by the Galerkin method: \ (5 . 5a)

q = V ( 1 0 * 0 [0 .564 = 1/yV (exact solution)

Similarly, if the surface of the half-space is exposed to a constant heat-input go, the surface temperature i s |

1.23 = \ / 3 / 2 by the heat-balance method:

q = V ( 6 l c 0 1 . 1 2 = V (5) /2 by the Galerkin method:

q = V O 0 1.13 = 2 / ^ / 7 7 (exact solution)

(5.6)

as obtained from a temperature of the form

(5.6a)

In these two cases the Galerkin method employs the weighting functions Fi = 2x/q2(l — x/q) and F± = \ [1 — (x/q)2] respectively. The first one of these is zero at x = 0, q and reaches a maximum at x = q/2; the second is zero at x = q and reaches a maximum at x = 0. According to the preceding dis-

* The corresponding entry for Biot's method would be 0 .595 = 2 \ / (13 /147) , from Q

- V(147K / /13) . f The corresponding entry for Biot's method would be 1 .32 = \/(J)j2, from Q = \/(LKT).


cussion therefore one would expect that the heat-balance method would give more accurate answers than the Galerkin method in the first example, and less accurate in the second; this is borne out by the numerical values listed above.*

The heat-balance method has been used a good deal in the literature. The accuracy of solutions for slabs obtained on the basis of polynomials of various degrees and of exponential variations of temperature has been studied by G o o d m a n 7 8 and by K o h 7 9 respectively. These papers, as well as that of Yang and Szewczyk 8 0 , consider the material properties to be temperature-dependent. The application of the method to problems with cylindrical symmetry was carried out by Lardner and Poh le 8 1 and by Veinik 8 2 .

Further work on the Galerkin method, in particular as concerns its use in two- or three-dimensional transient heat-conduction problems, was done by Dicker and Fr iedman 7 0 . These authors point out that the discovery of a suitable set of functions to be chosen for the proper description of the temperature field in Galerkin's method is very difficult when the shape of the boundary is not a simple one, and even when such a set has been found, integrations which cannot be carried out in closed form are usually encountered. To overcome these diffi-culties they devise a general transformation which maps a large class of hexa-hedral domains! into rectangular parallelepipeds (or rectangles for two dimensions) for which the solution is then readily obtained. These authors also use Laplace transforms to take care of the time-dependence of the temperature, and study the questions of separability of the equation in certain domains, of convergencej and of error estimates; detailed solutions for a number of two-dimensional problems are included.

One might mention here another type of approximate analysis, which was studied by Brull and Vinson 8 4 for the case of bodies having the shape of a shell of revolution; this approach is approximate in that it is based on assump-tions similar to those of the elastic theory of thin shells. These authors then show that the resulting differential equations are separable for all shapes, and that furthermore, for a broad class of problems, the solution may be expressed as combinations of a one-dimensional slab solution and a suitable correction function. Approximate one-dimensional solutions for thin-walled structures have been studied by Hoff 8 5 and Pohle and Oliver 8 6 ; for other references see (Ref. 5, p . 160 ff and p. 377). The application of this method to analyze the effect of joint contact resistance was carried out in Ref. 87. Other theoretical analyses of contact resistance are those of Gatewood 8 8 , Seide 8 9 , and F r a n k 9 0 . Recent ex-perimental work has been reported by Barzelay 9 1 .

Before turning, in the next section, to a method of deriving approximate solutions with the aid of upper and lower bounds, it may be worthwhile to remark briefly on the error which an approximate expression for the tempera-

* A s an alternative possibility, arbitrary weighting functions may be used to emphasize whatever region is intuitively felt to be of greatest importance.

f A ny closed two-dimensional domain bounded by four arbitrary curves or any three-dimensional domain bounded by six arbitrary surfaces is included in this term.

J With the aid of Michlin's w o r k 8 3 on elliptic equations.


ture may introduce in the corresponding thermal stresses. It is clear from the well-known result 5 that a temperature distribution which is linear in cartesian rectangular coordinate system produces no stress (and hence can be added or subtracted from any other temperature distribution without altering the stresses) that even extremely accurate temperature solutions can produce sig-nificant errors in the stresses.

As an example,* consider the example of a beam of height 2c, occupying the region — c < y < c, initially at zero temperature, under a step heat input on one face and insulated over the other. The stresses are 5

(5.7)

2CQA ' !t-c

y

EX4CT

•5

*

* 1 I

(COMPP.)

F ^ / / II

( COA/PFF.)

7(T£MS/OA/)

II/

F I G . 3. Comparison of thermal stresses in a slab as obtained from the exact and an approximate temperature distribution.

* See L e v i n s o n 9 2 for an analysis of errors of thermal stress calculations in a thin-walled structure.


The exact temperature distribution is well-known; 2 ' 5 the approximate one will be taken as that calculated by Galerkin's method, which has already been seen, in equations (5.6), to agree well with the exact one before the transit time. Plot of the approximate and exact temperatures and of the approximate and exact stresses are presented in Fig. 3. It is evident that errors in the temperature approximations are much smaller than those in the stresses; for example, the error in the approximate maximum temperature is an extremely small fraction of the exact maximum temperature, while the error in the approximate maximum stress is about 10 per cent of the exact maximum stress. No general criteria for the determination of the error induced in the stresses by an error in the tem-perature appear to have as yet been proposed.*

6. U S E O F U P P E R A N D L O W E R B O U N D S T O

O B T A I N A P P R O X I M A T E S O L U T I O N S

The estimate of the error incurred in the approximate solution of any problem is obviously very important, and except in the few cases mentioned above, it has not been very thoroughly studied in conjunction with analytical solutions. A somewhat different viewpoint on this question may be arrived at by trying to determine first whether a certain approximation is higher or lower than the exact solution, and then to proceed to seek, if possible, to bracket the solution between upper and lower bounds. In this manner an error estimate can always be obtained; lacking any better information one may always write the exact temperature TE as the average of the upper and lower bounds Tu and 7L, with bounded error e, namely

TE = ${Tu + TL) + e; \ e\<i(Tu- TL) (6.1)

The construction of upper and lower bounds can be carried out by utilizing some of the theorems of Sections 3 and 4, as will be indicated in the following three examples, dealing respectively with: (a) a one-dimensional slab problem with constant properties, (b) a one-dimensional problem with variable properties, and (c) a two-dimensional problem with constant properties. In all these cases we will be dealing with bodies initially at zero temperature, and with either the temperature or the heat flux specified on the boundary; in other words the problem to be solved is formulated mathematically as

®(T) = 0

T(PB, t) = F(PB, t) or k — (PB, t) = F(PB, i) for points PB

n or the boundary T(P, 0) = 0 J

(6.2)

where the operator 3) is defined in equation (5.1) and where n is the outward normal. The procedure consists in constructing two functions Tu and TL such

* For a discussion of the aspects of Saint-Venant's Principle which are related to this question see Section 7.


that

(6.3a)

(6.3b)

so that, according to the theorems of Sections 3 and 4*

TV>T and TL < T (6.4)

at all points P of the body. Thus, if the error indicated by equations (6.1) is not too large, it may be considered that a solution adequate for the purposes at hand has been obtained.

In general, not all three of the inequality signs in equations (6.3a) or (6.3b) will prevail, but it will be more convenient usually to deal with functions which satisfy the initial and boundary conditions, say, but not the differential equation, or vice versa. A certain amount of practice is no doubt necessary in order to find suitable functions for the construction of 7V and Tu and it seems a good idea to choose expressions containing some arbitrary parameters or functions, which can then be adjusted so as to satisfy the appropriate inequalities. For the construction of lower bounds in problems with positive boundary tem-perature, for example, it is quite reasonable to use the penetration-depth concept, and again use this time-dependent depth as a parameter; since the temperature is assumed zero beyond the penetration depth, it is certainly lower than the actual one.

The various bounds derived for the three problems mentioned above will now be summarized and briefly discussed; further details will be found in Appendix B.

Problem (a). The slab 0 < x < L, constant properties

prescribed constant heat flux at x = 0 (— k BT(0, t)/dx = Qo] and perfect insulation at x = L[dT(L, t)/dx = 0). Three solutions for this problem are outlined in Table 1 and lead to the plots of Fig. 4. The expression chosen for the temperature in the first of these solutions was motivated by the known steady-state solution for this problem; the second expression is simply the solution for a half-space, and the third was suggested by the leading terms of the exact solution of the problem. The temperature on the exposed face is plotted in Fig. 4 as detained from these three solutions and from the exact

2 = k(d*/dx*) - Pc d/Bt;

* It should be recalled that only one-dimensional proofs were given in Section 4 for the prescribed heat flux case.

either TL(PB, t) < T(PB, t)

TL(P, 0) < 0 or

2>{TL) > 0;

either TV(PB, t) > T(PB, t)

TV(P, 0) > 0

or

&(Tu) < 0;

TABLE 1. UPPER A N D L O W E R B O U N D S FOR THE TEMPERATURE IN A H A L F - S P A C E ; PROBLEM (a), SECTION 6 .

r) satisfies:

Sol. N o .

Selected (kT/LQo) Parameters

Diff. eq.

Init. cond.

Bound x = 0

Cond. x =L

Lower bound (kTL/LQo)

Upper bound (kTu/LQo)

1 Yes N o Yes Yes * - z O - e ) ^ 2 L 2

2 none Yes Yes Yes N o 2 2 V T . i e r t c 2 i v V

none Yes Yes Yes N o v 2 Z V T

3 9 - \ / T 1/7I icrfc ^ # 1 , # 2 Yes Yes N o N o / ^ \l~ierfc * 3 z v 1 ? 1 i e n c 2 L - v / T

+ < ? 2 i e r f c - 2 i v J

# 1 , # 2 Yes Yes N o N o

+ e r f c 2 r „ i e r f c 2 L v J

0 <^ T < _ T 0

282 B

RU

NO

A

. B

OL

EY


solution; it can be seen that the bounds given by the first solution are quite inaccurate (though their average is not), those of solutions 2 and 3 are quite satisfactory.

Problem (b). The half-space x > 0, with temperature-dependent properties given by*

k = 241 - 0.019 T 1 (6.5)

Pc = 50 + 0.0086 T

F I G . 4. Upper and lower bounds for the temperature in a half-space; Problem (a), Section 6.

where Tis in °F, k in B.t.u./hr ft°F, c in B.t.u./lb °F, and p in lb/ft 3. Temperature at x = 0 prescribed as T(0, t) = 600°F = T0. Let

[1 -x/q(t)]*; x<q(t) (6.6a)

0 ; x > q(f)

where the penetration depth q(t) is taken as a parameter to be determined from the condition 9(T) > 0, since all the other conditions of the problem are ful-filled if #(0) = 0. It is found (Appendix B) that q(t) = 6.02 y'V will insure a lower bound. For an upper bound, let

(6.6b)

where the constant KQ is a parameter to be determined from the condition

* These values correspond to c o p p e r ; 9 3 this reference also gives the variation of properties with temperature for graphite Iconel-X and beryllium. Another approximate solution of this problem was given by Y a n g 9 4 , w h o also starts from a temperature of the form (6. 6b) and then proceeds by means of an iterative scheme.


@(T) < 0, all other conditions of the problem being fulfilled. Appendix B shows that KQ = 4 .82 will insure this. Hence

(6.7)

Plots of these bounds are shown in Fig. 5 and show a reasonably close bracketing of the exact solution for the entire half-space and for all time.

T(x,t)

1.0-

0.8-

0.6-

0 .4

0.2-

-T o=600°F

k-k(T)

c = c(T)

^ , 1 6 6

7 Vs F I G . 5. Upper and lower bounds for the temperature in a half-space with temperature-

dependent properties; Problem (b), Section 6.

/ ( 0 ) = 0 (6.8)

Problem (c). The region y > ax2 bounded by the parabola y = ax2 (Fig. 6 inset), in two dimensions, with constant properties; temperature on y = ax2

prescribed as To, a constant. Let

T = jexp{-i /»/[( l + a f 2 ) / ( r ) ] } ; 0 < u < W I ( T ) B

To \ e x p { - ^ / [ ( l + aa/(r)]}; U > Ui(r)

where u = t\ — £2, u = ay, £ = ax, r = 4f<ta2; the constants n and a, and the functions f(f) and ui(j) were taken as parameters. All conditions of the problem are satisfied except the differential equation. Only a statement of the principal results will be given here, since their derivation is rather lengthy. It can be shown that

and that, for short times,

(6.8a)

w i < 1; a > 1. (6.8b)


With n = 2, and with the values of a appropriate to each w, the upper bounds plotted in Fig. 6 were obtained.

A lower bound is now needed; for this, the penetration depth concept was again used, by letting

0 ; q(0) = 0

(6.9)

where the constant n and the function q(t) were taken as parameters to be adjusted so as to make ®(T) > 0. It can be shown that the choice n = 3 is suitable, and that q satisfies the equation

(6.9a)

or, for short times, q = <\/2(n — 1) r. The corresponding lower bounds are plotted in Fig. 6. It can be seen that in the present two-dimensional problem the bounds are not quite as satisfactory as those in the one-dimensional examples, and that further studies are needed to assess the usefulness of the present approach in such cases.

1.0

z-o 05

y:ax s

10

F I G . 6. Upper and lower bounds for the temperature in a two-dimensional problem; Problem (c), Section 6.

It appears from the preceding examples that, particularly in one-dimensional problems, one can construct upper and lower bounds to the exact solution by application of one of the appropriate results of Sections 3 and 4. Of course, this is not a "method" of solution in the strictest sense, since no procedure or formula is presented which will automatically lead to the desired result, and a certain amount of ingenuity will often be required in the choice of the functions Tu and Ti. This drawback, however, is present in all analytical methods which


require, as in Section 5, the choice of an expression describing the temperature; on the other hand, the advantage of an immediate error estimate is evident. Furthermore, the few solutions discussed here may provide some general suggestions on the procedures to be followed: for example, the penetration depth concept appears useful in the derivation of lower bounds. As another example, the determination of the parameters need not be carried out for the optimum case, but often conditions merely sufficient (and not necessary) insure the non-negative or non-positive character of the quantity in question may be quite satisfactory. Another approach which may be at times useful for the con-struction of approximate solutions and of bounds is described next in this section. It would seem that the establishment of various rules or techniques such as have been mentioned above is desirable, in order to facilitate the efficient use of this type of approach, especially in two-dimensional cases.

The Embedding Technique

In one of the methods which will be discussed in the section for problems of melting and solidification the solution is obtained by considering the actual body mathematically as part of a large fictitious body which completely en-velops it. The temperature (or heat flux) on the surface of the fictitious em-bedding body is unknown, but is determined from the known conditions at the surface of the actual body. The simplifications which occur in this type of analysis arise from the fact that the fictitious body may be chosen as geo-metrically very simple, even if the actual one has a very complicated shape. This approach can be used to construct approximate solutions and bounds to solutions of heat conduction problems also when no change of phase occurs; exact solutions cannot be always so obtained because of the well-known pro-perty of solutions of the heat equation, that even with abrupt surface temperature changes, the interior temperature always varies smoothly; that is, the temperature and all its time-derivatives are always continuous functions of t ime . 9 5 Thus initially, for example,

As a simple example of a solution obtained by this "embedding" technique, let it be desired to determine the temperature in the half-space x > c under a constant flux Qo on its surface, namely

k 2 y(Kt)

Take as the fictitious body in question the half-space x > 0 and consider temperatures in it which are simple linear combinations of known solution of the heat conduction equation, are zero initially, and correspond for large times to a constant heat-flux Qo on x = 0. Such a temperature is

(6.12)

(6.10)


where T = 4Kt/c2 and where A, B are arbitrary constants. The first term is the exact solution due to a unit flux at x = 0, the second term is the exact solution due to a constant temperature at x = 0, and the third term is the solution due to a doublet. The heat flux at x = 0 is

and at x = c, the actual body surface, it is

(6.12a)

(6.12b)

The constants A and B must now be adjusted so as to make the right-hand side of (6.12b) conform as closely as possible to the desired step function; alterna-

F I G . 7. Upper and lower bounds obtained by the "embedding" technique for a half-space.


lively, they may be chosen so as to lie either wholly below or wholly above the step function, thus leading respectively to lower or upper bounds. By trial and error it was found that A = 0.975, B = 0.451 lead to a lower bound. It is easily seen, however, that no upper bound can be thus constructed; to obtain such a bound, the expression (6.12) for the temperature was still used, but the origin of time was shifted by an amount T O , i.e. the temperature is T\(x, r + r 0).

An upper bound now results with ro = 1, A = 1, B = 0.581. Plots of these bounds (and, for comparison, of the exact solution) are shown

in Fig. 7. They are in reasonable agreement, and in particular the average of the upper and lower bounds [cf. equation (6.1)] is very close to the exact solu-tion except for very short times. Certainly this type of approximation is adequate except for extremely short times, even in the present very severe case of a step heat-input. It would therefore seem desirable to devise some more general techniques for the application of approaches involving the embedding concept. The work which has been done on the numerical solution of the inverse problem (i.e. the determination of the surface flux history corresponding to a given interior temperature) may be useful in this connection; see for example Stoltz 9 6 , M i r s e p a s s i , 9 7 ' 9 8 or Shumakov 9 9 .

7 . S A I N T - Y E N A N T ' S P R I N C I P L E I N H E A T C O N D U C T I O N

Saint-Venanfs Principle, well known in elasticity and structural theory, plays an important role in heat conduction as well. It may be stated, in a general way, as follows: 5 if a body is subjected either (a) to a temperature T(Po, t) at points P within a small portion So of the total surface area S and to a zero temperature over the remainder of S (i.e. S — So), or (b) to a non-zero heat input Q(Po, t) over .So such that*

J Q(Po,t)dS = 0 (7.1)

and is insulated over S — So, then

(7.2)

where P is a point of the body at a distance x from So and h is the maximum linear dimension of So (which can often be taken as h = -y/So).

If the problem is linear then, whenever the above principle holds, it follows that if over So

T(P0, t) = 7i(/> 0, 0 + W o , t); T2(Po, i) < 7 i (P 0 , 0

then T\ is a good approximation to T, or

(7.3a)

(7.3b)

* Such a heat input is termed self-equilibrating.


Similar results hold for the self-equilibrating heat input condition. The simplest application of equation (7.3) occurs for example in the problem of a bar of which one end is held at a temperature which is nearly, but not quite, uniform; Saint-Venant's Principle assures us that a uniform temperature distribution will effectively prevail at distances from the end larger than the height of the bar. Hardly a practical problem of heat conduction can be conceived whose solution can be directly applied without the implied use of Saint-Venant's Principle.

The preceding statements of Saint-Venant's Principle are not very precise; to achieve a more satisfactory statement of the principle, one capable of proof, it has been suggested 1 0 0 that a reformulation of it be given in terms of upper bounds. In other words, if in a given problem one can prove that

and one finds that

(7.4)

then obviously the principle will hold. This approach has been very successful in problems of prescribed surface temperature, 4 3 since in such cases rather general upper bounds can be established by means of the properties of Green's functions outlined in Section 3. The extension to other boundary conditions has been

F I G . 8. Upper bounds for the steady-state temperature in a two-dimensional wedge under local heating.

T(P, 0 | < | TV(P, t)


hampered by the fact that similar properties are not available for the Neumann function (i.e. for the heat flux condition) or for the Robin function (i.e. for the radiation or convection condition). A method of circumventing this difficulty for all types of boundary conditions has been devised for the steady-state p rob lem, 1 0 1 by using certain fundamental integral equations which were pre-viously established. 4 5 An example of the bounds obtained there is shown in Fig. 8 for the case of a wedge under a known heat input over a small portion of its surface, under steady-state conditions.

The fundamental integral equations just referred to are available in the transient as well as the steady-state case, and therefore it is possible to extend all the results referred to in the last paragraph to the transient case.* Such an extension has been indeed carried out (Appendix C) but, as will be seen, leads to bounds which are not always satisfactory, i t is shown in Appendix C that

for the case in which

T(Po, t) 7(Po, 0 for Po on So

0 for Po on S — S0

Here

B(P, t) =

(7.5)

(7.5a)

(7.5b)

where Go is the Green's function for a domain which completely encloses the given body, and where g * and £* are respectively a point of the body and a value of time (0 < £* < t) chosen so as to maximize the upper bound B(P, t). Inequality (7.5) holds only if the denominator of the right-hand side of (7. 5b) is positive.

As an example of the use of equation (7.5) consider the half-space x > 0, with

\T0 0<t<t0

f(Po,t)= \ (7.6) [0 t< 0, t> t0

Taking Go as the effect of a source in an infinite domain, i.e.

* Such integral equations exist also in other fields, such as for example e lastostat ics ; 1 0 2 the derivation of bounds in that case has thus far proved practically intractable, though certainly possible in principle.

[7.7)

P R O B L E M S OF H E A T C O N D U C T I O N A N D M E L T I N G

and thus

The exact solution of the problem is

E X A C T S O L U T I O N

291

(7.7a)

(7.8)

(7.9)

2 0 4 < c t 2 4

F I G . 9. Upper bounds for a transient problem of local heating; general plot.

It can be seen that both the bounds and the exact solution are exclusively functions of the dimensionless quantities r = 4f<t/x2 and T O = ; plots of equations (7.8) and (7.9) in terms of these quantities are shown in Fig. 9, and generally show that the upper bounds are satisfactory only for not too large values of r and r 0 . A more meaningful plot, which can be constructed from Fig. 9, is that of temperature decay with distance, for various values of time and for a specific value of to, as shown in Fig. 10. The calculated bounds became closer and closer to the exact solution for large x.

one obtains here the value of \ for the denominator in (7. 5b) and


The establishment of an overall Saint-Venant's Principle in thermal-stress problems has been explored in Ref. 103, that is of a principle which would allow conclusions on the rapidity of decay of the stresses to be reached on the basis of the thermal conditions alone, without the double use of the principle. General conditions for the validity of such a principle, or indeed further special examples illustrating its use, are not yet available.

UPP£& BOU/VO

F I G . 10. Upper bounds for a transient problem of local heating; t0 = 0.5.

8 . M O V I N G - B O U N D A R Y P R O B L E M S : G E N E R A L D I S C U S S I O N

An important class of problems in heat conduction is concerned with the determination of the temperature in a body whose boundaries are not fixed in space. We may distinguish two types of problems here, namely those in which the motion of the boundary is due to rigid-body motions of the entire body, and those in which it is due to local conditions near the boundary. In the former case, we usually speak of a moving body, rather than of a moving boundary; solutions for several problems of this type, for bodies moving with a constant velocity, are discussed by Carslaw and Jaeger 2. In the latter case, we can again distinguish two cases, namely those in which the motion of boundary is pre-scribed, and those in which it must be determined as part of the solution of the melting problem. The first of these cases is a little simpler than the second, since it is a linear problem, although with variable coefficients; a recent treat-ment of this type of problem has been given for example by G i b s o n 1 0 4 ; the second of these cases, often referred to as a "floating-boundary" problem, will be the subject of the remainder of this paper.


The importance of floating-boundary problems in heat conduction arises primarily from the problems of solidification and melting, in which the position of the interface between the two phases of the material is not known beforehand and must indeed be determined in the solution. These problems are of great importance in many applications of the theory of heat conduction, as for example in the solidification of castings (cf. Rudd le 1 0 5 ) , in the design of heat-shields for re-entry vehicles on the basis of aerodynamic a b l a t i o n , 1 0 6 - 1 1 6 and in geological p r o b l e m s . 1 1 7 ' 1 1 8 It should also be noted that the problems to be discussed here are mathematically analogous to many which arise in other fields such as that of soil consolidat ion 1 1 9 or that occurring in a superconductor with the transition from the superconducting to the normal s t a t e . 1 2 0 ' 1 2 1

The conditions to be satisfied on the moving front between a liquid and a solid phase, in a typical problem, stipulate that the temperature there must be equal to the melting (or solidification) temperature Tm, and that a heat balance condition must be satisfied. Denoting by the subscripts L and S quantities pertaining to the liquid and to the solid respectively, we then have, on the interface,

TL = Ts — Tm (8.1) and

ks ^ ~ kL ^ = plvn (8.2) on on

where / is the latent heat of melting, n is the normal direction to the interface, positive as interior to the liquid phase, and vn is the velocity of advance of the interface in n direction. These equations must be modified if the densities of the solid and the liquid are not the same, because there will be then motion of the liquid, and the ensuing convective heat transfer* must be taken into account (Chambre 1 2 2 ) .

The only known exact closed form solutions of such problems are those listed by Carslaw and Jaeger 2 , and are all obtained by essentially the same approach which was introduced by Neumann in the 1860's.f Neumann treats the case of an initially liquid half-space at a constant temperature higher than Tm, with its surface x = 0 maintained at a constant temperature lower than Tm. As may be shown by dimensional analysis, this solution is characterized by similarity, that is, it is exclusively a function of (x\<\Jt), and by the fact that, as a consequence, the distance traveled by the solid-liquid interface is proportional to <\/t. The other exact solutions listed in Ref. 2, all have these same characteristics; they refer to various problems such as that of an infinite solid, one half of which is initially liquid and the other solid, of solids with a melting range or with a change of volume upon solidification, of multiphase materials, and of some of the corresponding problems in cylindrical and spherical coordinates. Various

* A problem in which this is taken into account has been worked out by Carslaw and Jaeger (Ref. 2 , pp. 290-1) .

f This solution was given by Neumann in his lectures, but the first published solution is that of S t e f a n 1 1 7 and for this reason this problem is often referred to as Stefan's problem.


studies of similarity solutions have been published in the last few years, as for example that of Grigor ian 1 2 3 , who considers a solid in contact with a moving viscous incompressible fluid, that of Andr iankin 1 2 4 , who examines some solu-tions for the temperature-dependent properties case, and that of Miranker and Keller 1 2 5 , who establish an existence theorem for a solid with temperature-dependent conductivity.

Similarity conditions cease to hold under more complicated boundary or initial conditions, e.g. a prescribed heat-flux on the moving boundary or an arbitrary initial temperature. Existence and uniqueness theorems for a solid initially at the critical temperature, under a prescribed heat input were treated by E v a n s 1 2 6 and, under general heating conditions, by Kolodner 1 2 7 , who also studied a number of applications by setting up a functional equation for the position of the liquid-solid interface. A similar approach was also used by Miranker 1 2 8 and, under somewhat less restrictive conditions, by Kyner 1 2 9 . In all the above papers, the problems considered are one-dimensional, and (except as noted) the thermal properties are constant. The latter restriction is not maintained by Kyner 1 3 0 , but no work of this nature appears to have been done in two- or three-dimensional problems.

Comparison theorems for melting slabs with variable thermal properties under arbitrary heat-flux conditions were proved by the au thor ; 4 8 ' 4 9 they state the intuitively reasonable conclusion that higher temperatures and faster melting rates will always result from higher heat inputs; these theorems will be discussed further later (Section 10) since they form the basis of one of the available ap-proximate methods of solution of this type of problem.

Analytical and numerical solutions to many one-dimensional floating bound-ary problems have been obtained; these will be indicated here (except for approximate analytical solutions analogous to those of Sections 5 and 6 which will be discussed in the section which follows).

The principal problems studied are those of a half-space or a slab, initially solid,* heated in some prescribed manner on one of its surfaces, so that the temperature rises and eventually melting begins at this surface; the melted portion is then either instantaneously removed or remains stationary in most of the published solutions. For a half-space under a constant heat input the first of these problems was solved numerically by Landauf 1 3 1 and for the second of these problems (the body being initially at its melting temperature) by Evans, Isaacson, and MacDona ld 1 3 2 , who also obtain a series solution for short times. Results similar to those of the latter paper, but for the radiation boundary condition, were derived by Kreith and Romie 1 3 3 . Short-times series solutions for a half-space initially at zero temperature, and with the melted portion either instantaneously removed or stationary have been derived by the au tho r ; 1 3 4 as may be intuitively expected, for very short times after the start of

* It is easily verified that all the solutions discussed apply equally well to the melting of an initially solid body and to the solidification of an initially liquid mass. For brevity, only one of these possibilities will be mentioned in each case.

t Analytical solutions are given for the limiting cases of zero and infinite latent heat.


* For a discussion of the application of Riemann's method to parabolic equations see for example G o u r s a t 9 5 and M i l e s 1 3 5 , who devotes special attention to the melting problem.

melting the two solutions are the same, and the melting rate is higher in the first than in the second problem. It may be mentioned here that some of most important methods for developing exact series or approximate solutions of this type of problem are those involving the use of integral equations.* These may be set up in a variety of ways, as for example in Ref. 135 by Riemann's method, in Ref. 136 by superposition of sources at the interface, in Ref. 132 by finding first with the aid of the Laplace transform a solution satisfying all but one of the desired boundary conditions, and in Ref. 134 by embedding the given body into a fictitious one of simpler geometry under unknown surface heating condi-tions (cf. Sections 6 and 10).

For the case of a slab of finite thickness, Landau's problem has been solved by C i t ron 1 3 7 by an iterative procedure and numerically by Dewey, Schlesinger, and Sashkin 1 3 8 . A simpler method of solution has been subsequently devised by Ci t ron 1 3 9 , in which a series expansion in space about the melting face is used; the coefficients of such a series can be readily derived in succession. The only published solution pertaining to a problem which is not one-dimensional is the numerical one of Springer and Olson 1 4 0 , in which case the axisymmetric condi-tion is considered. The extension to such problems of the method of Ref. 134 is now being studied at Columbia University, but it is too early to present any definite results.

It is evident from the preceding list of published works that a great deal remains to be done in the way of deriving solutions of floating-boundary problems. While purely numerical solutions are useful, and will probably appear with increasing frequency in the future, the need for analytical solutions re-mains; not only can they be used for comparison purposes, but they provide accurate information for example on questions concerning the onset of melting which cannot be otherwise answered.

We might close this section by referring to some of the researches into the fundamental aspects of numerical analysis in floating-boundary p r o b l e m s . 1 4 1 - 1 4 7

A different approach, employing the concept of a "weak solution", i.e. one defined in terms of certain integral conditions, has been proposed by R o s e 1 4 8 .

9. A P P R O X I M A T E S O L U T I O N S O F M O V I N G - B O U N D A R Y P R O B L E M S

As has been remarked in the preceding section, very few exact solutions are available for problems of melting and solidification, and indeed these problems tend to become rather difficult except in the few cases which have been noted. As a consequence, approximate methods of solution are even more important here than they were in the absence of melting. The approaches discussed in Sections 5 and 6 are still applicable in the melting problem, as a number of published papers have shown.

296 B R U N O a . B O L E Y

The variational method was applied to a moving-boundary problem by Ci t ron 7 7 and, more recently by Biot and Daughaday 1 4 9 . The latter authors obtain much better agreement than Citron with the exact solution, and showed that their approach (which employs the penetration-depth concept) is generally not too sensitive to changes in the assumed temperature distribution.

The heat balance approach was applied to this type of problem by Goodman 7 2 , and by Goodman and Shea 1 5 0 .* In the first of these papers several problems of the melting of a half-space are solved, including the cases studied by L a n d a u 1 3 1

and by Evans, Isaacson, and M a c D o n a l d 1 3 2 ; in both the approximate solution is in excellent agreement with the previous results, and in the second of these cases it predicts correctly the first few terms of a series for the exact short-time solution. This approximate theory provides a way of generalizing the exact results, since it gives this same series for all heat input histories, while the exact one predicts it only for a step heat input. The second of the two above papers considers a slab under a prescribed heat input on one face, and under either isothermal or insulation conditions on the other; the solution is obtained by means of an expansion in terms of a physically small parameter, and is based on the assumption of a quadratic temperature distribution both in the liquid and in the solid phase.

A comparative analysis of the variational and heat-balance methods for the solution of melting problems was undertaken by Lardner 1 5 2 , who bases his work both on the Neumann problem and the half-space under constant heat input with instantaneous melt removal. He finds that while the approximate solutions agree well with the exact ones for certain ranges of the governing parameters, there are important ranges where disagreement is evident, as for example at short times in the second of the above problems.

A different type of approximate solution was employed by Pekeris and Slichter 1 5 3 , who solve the problem of the region r > a initially at the melting temperature, while the surface r = a is kept at T = 0, by assuming that the temperature distribution has the same shape as that in the corresponding steady-state problem.

10. U P P E R A N D L O W E R B O U N D S I N M O V I N G - B O U N D A R Y

P R O B L E M S , A N D T H E I R A P P L I C A T I O N S I N

O B T A I N I N G A P P R O X I M A T E S O L U T I O N S

The method of Section 6, i.e. the calculation of approximate solutions by means of upper and lower bounds, has been found very successful in certain types of melting problems, on the basis of the following theorems:!

* See also P o o t s 1 5 1 . f The proof of these are given in the stated references for the case of thermal properties

dependent on position but independent of temperature. They are however valid in the temperature-dependent properties since they require the fundamental theorems which were extended in Section 2 of the present paper to this case as well.


Theorem VII: Comparison Theorem for Melting Slab, with Instantaneous Melt

Removal**

Consider a slab 0 < x < L, initially (7 = 0) at zero temperature, heated on x = 0 by an arbitrarily varying heat input Q(t) and insulated on x = L. The mathematical formulation of this problem is as follows:

(10.1)

where t m is the time of start of melting and s(t) is the melt thickness. Consider now two solutions 7 \ and T2 of equations (10.1), corresponding respectively to Qi(t) and Q2(t) such that

Qi(t)> 22(0 (10.2a) Then

Ti(x, t) > T2(x, t) and s±(t) > s2(t) (10.2b)

Theorem VIII: Comparison Theorem for Melting Slab, Melt Remaining Stationary*9

Consider the same slab as in the preceding problem, but with the heat input Q(i) always applied at x = 0, and assume that an additional heat input 2*(0 is applied on the moving front x = s(t). The mathematical formulation of this problem is as follows: for the temperature Ts(x, t) < Tm in the solid:

s(i) < x < L, 0<t<tL

0<x<L

0<t<tL

0<t<tm

for the temperature TL(x, t) > Tm in the liquid:

(10.3b)

(10.3a)

0 < x < s(t), t m < t < t L

tm < t < th

s(t) = 0 for t < t m

T[s(t\ t] = Tm

T(x, 0) = 0


with the following interface conditions:

Ts[s(t)91] = TL[s(t\ t] = Tm

tm<t<tL (10.3c)

and with s(t) = 0; 0 < t < t m (10.3d)

Consider two solutions of this problem, denoted by the subscripts 1 and 2, corresponding respectively to heat inputs Q*(t) and Qt(t) such that

Qt > Qf (10.4a) Then

1 (10.4b) T2(x, t) > Tx(x9 t); 0 < x < L J

It is understood, in equation (10.3b) and elsewhere, that for the temperature T one must read the pertinent one of the functions Ts and TL.

As has been mentioned, these theorems are useful for constructing upper and lower bounds to solutions corresponding to given heat inputs. For example, in the second of these cases, the procedure used for estimating the solution to an actual problem in which Q* = 0, consists essentially of constructing a solution of equation (10.3), with a heat input at x = 0 equal to the prescribed input Q(0> disregarding however the last of equation (10.3c). The latter equation is then used to calculate a n d Theorem VIII insures that either an upper or

a lower bound has been found in a range 0 < t < ti, according to whether the relation

6 * ( 0 > 0 (10.5)

is always satisfied or always violated in that range. An analogous procedure is used to construct bounds in the first of the above

problems (melt instantaneously removed), and is as a matter of fact somewhat easier than the one just outlined. The reason for this is that, as may be readily observed, any solution valid within the original slab thickness 0 < x < L is a solution of the melting problem for a particular set of functions Q(t) and s(t). Indeed, such a solution satisfies all the equations of the formulation of equation (10.1) with the exception of the third and forth of these equations: the former may be used to determine s(t) and the latter to calculate Q(t). This observation forms in fact the basis for the integral-equation method devised in Ref. 134 for the solution of this melting problem, in which the given body is embedded in a fictitious solid subjected to a fictitious heat input Q* on its surface. Let the temperature in the fictitious body be T*; then the procedure consists of cal-culating T* for suitable choices of 2*, and evaluating Q(t) and s(t) for each after melting has begun; Theorem VII then insures that the melting rate s so


calculated is lower than the actual one in the range tm t <^ ^* if 2* ^ Q in that range, and higher if g * > g in that range. An example of the results of this procedure may be found in Fig. 11 in which upper and lower bounds so calculated bracket Landau's exact solution with reasonable accuracy. Exten-sions of this approach to include approximations which (as in Section 6) do not satisfy the differential equation are also possible and may be convenient in some cases.

o»(y)'-4

AT X = 0 i

r~

£"/7Tktm

F I G . 11. Upper and lower bounds in a melting problem.

As a further study of bounds, we can compare the solution for the instan-taneously removed melt with that for stationary melt, as follows : 4 9

Theorem IX: Comparison of Solutions of the Problem of Equations (10.1) with

those of the Problem of Equations (10.3)

Consider two pair of functions Ti(x, t), s±(t) and T2(x, t), s2(t), such that the

pair T i , s± is a solution of equation (10.1) and the pair T2, s2 satisfies equation


(10.3) with g * = 0, where the same function Q{t) is used in both cases; then

si(t) > s2(t), t > t m (10. 6a) and

T±(x, t) > T2(x, t), t > tm, s2(t) < x < L (10. 6b)

This theorem therefore states that, under the same heat input history, a more rapid advance of the interface occurs when the melted material is instantaneously removed (problem 1) than when it remains stationary (problem 2): This con-clusion was already reached in Ref. 134 for the special case of Q = constant.

Problem 1 is thus seen to provide an upper bound to the solution of problem 2; a fortiori, upper bounds to the solution of problem 1 are also upper bounds to the solution of problem 2.

Between the two extremes of instantaneous removal and of stationary melt there may be defined intermediate problems corresponding to finite rates of ablation. It may be conjectured, as an extension of the last theorem, that a monotonic relationship exists, in these problems, between the rate of ablation and the rate of advance of the solid-liquid interface; this question has however not yet been examined.

11. T H E R M A L S T R E S S E S I N M E L T I N G O R

S O L I D I F Y I N G B O D I E S

The calculation of thermal stresses in melting bodies has received little attention, though it is obviously important in the prediction of failure and in the calculation of deformation. To begin with, consider an elastic half-space x > 0 which melts under some heating condition applied uniformly over x = 0. The temperature is then a function of x alone, and all field equations and boundary conditions of static thermoelasticity (i.e. neglecting inertia effects) are satisfied with

uy = uz = 0; ux = u(x)

°xx Jyz Gxy — 0; (Jyy (JzZ

( 1 1 . 1 )

where w, e, and a denote displacement, strain, and stress components respectively, E is Young's modulus, a the coefficient of thermal expansion, and v Poisson's ratio. For a slab under the same conditions, the only non-zero stress com-ponent would be that given by equation (5.7) in which of course the time dependence of the thickness c ( t ) must be included. Thus if the slab occupies initially the region — Co < y < Co, and the face originally at y = Co moves to y = co — s ( t ) after melting, the other face remaining stationary at y = — CQ9

€yy — €zz — €yz — €zx — €xy — 0; e x x

z


(11.2)

The above analysis is open to two objections: first, that both inertia and coupling effects have been neglected, and, second, that an elastic material behavior has been assumed.

The importance of inertia and coupling can be assessed by means of the same general considerations that are used in thermal-stress problems without moving boundaries (Ref. 5, Chapter 2). It is known, for example, that coupling effects will be negligible if inertia effects are, and therefore the latter should be examined first. The problem contains three principal characteristic times, namely the mechanical time tM, the thermal time IT, and a new characteristic time tF connected with the moving front, as follows:

(11.3)

and this will in general be very small compared to unity since the velocity s is of an order of magnitude comparable with that of the motion of isotherms. Some exceptions to this can be expected, however, namely those in which sudden changes in temperature occur, as for example in Neumann's solution. Indeed there we obtain s = k t 1 ! 2 and therefore s is initially infinite. We may thus conclude that in this problem inertia (and probably coupling) will be important at very short times, while in the problems in which the heat flux is prescribed (and in which for short t imes 1 3 4 s = kit3?2) no such difficulty will arise. These general conclusions are borne out by the work of Tadjbakhsh 1 5 4 , who solved for the stresses in both the problems mentioned above including the effect of inertia.

then

where v is the velocity defined by equation (2. 3a), and s is the distance traveled by the front moving with a velocity s. In general tT/tM <^ 1; if further tFJtM <^ 1 then inertia effects will be negligible. Written at the moving front, the latter condition gives

(11.3a)


The second objection is much more serious in melting problems, since one then deals with materials close to their melting temperature; then certainly inelastic effects will be of paramount importance. Any solution which includes such effects must of course be based on an idealized type of inelasticity suitable to the material in question. The behavior of polyemethyl methacrylate was assumed by Rogers and L e e 1 5 5 to be essentially viscoelastic, while that of a solidifying metal was taken to be essentially plastic by Weiner and Boley 1 5 6 .

The first of these two works consider the stresses in a sphere with a con-centric spherical cavity, under conditions of constant surface temperature. It is found that the influence of viscosity is concentrated in the vicinity of the moving boundary, while the remainder of the solid responds essentially in an elastic manner.*

FIG. 12. Plastic and elastic zones in a solidifying slab.

The problem considered in Ref. 156 is an idealization of the early stage of solidification of a metal casting in a square mold. Initially, the metal is com-pletely in the molten state and at a uniform temperature, while the mold is at a uniform temperature sufficiently lower than the solidification point so that solidification begins immediately. At early times, the temperature distribution in the solidified and in the liquid portions is closely approximated (except near the corners) by that given by Neumann's solution for a semi-infinite slab.

The idealization used for the material behavior corresponds to an elastic-perfectly plastic body with a yield stress which decreases to zero at the melting temperature. A similarity solution is found to hold, which predicts the presence of two plastic zones and one elastic (unloading) zone in the solid at all times (Fig. 12); the stresses are compressive at the surface, and are approximately proportional to the surface temperature drop, regardless of the value of the initial temperature in the liquid. A typical plot of stress distribution is shown in Fig. 13.

* Uniqueness of solution for this type of problem formulation has been established by Sternberg and G u r t i n 1 5 7 .


- . 0 1

-.02

-.03

-.04

-.05

-.06

| |

X = -4 / * > = 2 X / ^ T

_ (I-V)Y.

ELASTIC UNLOADING : £ < — < £ ,

F 1 G . I 3

FIG . 13. Thermal stresses in an elastoplastic solidifying slab.

12. C O N C L U S I O N S

The brief survey of heat conduction which has been presented in this paper indicates that work is currently being done principally in four aspects of the field, namely (a) the discovery of new exact analytical solutions,* (b) the cal-culation of solutions by high speed digital and analog computers, (c) the de-velopment of approximate methods of solution, and (d) fundamental work in the determination of general properties of solutions of parabolic differential equations. The first two of these items are self-explanatory, and their value needs no elaboration. The last two topics offer a wide horizon of possibilities, and will undoubtedly be the subject of much further research. In particular, the use of upper and lower bounds for the actual calculation of approximate solutions, while of course evident in principle, has been shown in Sections 6 and 10 to be practically feasible in a variety of situations. Such calculations cannot be carried out without the fundamental theorems on properties of solution which would belong, in a strict classification, to the realm of mathe-matical theory rather than to that of engineering analysis. The present dis-cussion would seem to indicate, however, that there is a great deal to be gained

* A s well as of methods of extending known analytical solutions; cf. Refs. 5 and 159.


by pursuing lines of research which draw upon the most appropriate segment of knowledge, regardless of such rigid classifications. Such joint approaches should indeed be further explored, and their interests would be served by the availability of a book which combined the rigorous mathematical developments with an engineering treatment in a manner of which only a mere indication could possibly be given in this paper. The field of heat conduction seems to be a particularly fruitful one for such a broad-based attack; perhaps this is the field in which a wedding of theory and practice can be most readily carried out, without the supremacy of either, and to the general advantage of both.

(picik2~P2C2ki)2>0 (A. 4)

APPENDIX A

U N I Q U E N E S S P R O O F F O R C O N S T A N T D I F F U S I V I T Y

Consider two solutions (distinguished by the subscripts 1 and 2) of equation (3.5), and form the expression (using indicial notation and the summation convention):

( A . l )

This expression may be transformed by means of the divergence theorem, leading to the left-hand side of (A. 2), and by means of equation (3.5), leading to the right-hand side of (A. 2):

(A. 2)

Assume now that, on the bounding surface S, 7 i = T2 and thus Hi = H2; then the surface integral in equation (A.2) vanishes, and the right-hand side becomes

It is thus obvious that the classical uniqueness proof [Refs. 2, 5] will hold if the integrand of the remaining volume integral is positive definite. This will be case if and only if

picikiTi, iTi, i — (picik2 + p2C2ki) Ti, tT2, i + P2C2IC2T2,1T2, i>0 (A. 3)

or in other words when (since p\c\k\ > 0 and p2C2k2 > 0


or in other words if KI = K2 (A. 5)

Hence for the case of constant diffusivity the classical uniqueness proof holds. This is not surprising because in this case equation (3.5) can be reduced to an equation of the form (2.1) by a suitable transformation. 2 Note that for the case of bodies with fixed boundaries the above proof can be extended to the case of prescribed surface flux.

APPENDIX B

B O U N D S O F S O L U T I O N S T O T H E

P R O B L E M S O F S E C T I O N 6

This Appendix contains some details of the calculations of the bounds for some of the examples which were formulated in Section 6, where the principal results were also given.

Problem (a). Three solutions for this problem are summarized in Table 1. Of these, the first presents no difficulty since it satisfies all conditions of the problem with the exception of the initial condition, which gives

(B.2)

Here the smallest values of the right-hand sides occur at the largest value of r, and so if results are desired for a period 0 < r < T 0 , T = T 0 should be used in solving for q± and q2\ once again the equality signs are to be used for best results. This leads directly to the entry in Table 1 for an upper bound.

Problem (b). The choice of temperature expression of equation (6.6a) gives

( B . l )

Hence T(x, 0) > 0 or T(x, 0) < 0 according to whether A > 0 or A < — (1/2), corresponding respectively to an upper and a lower bound, best results of course being given by the equality signs. Solution 2 contains no free parameters, and it is easily verified that k(BT/dx) < 0 at x = L.

In the third solution, the only conditions of the problem which are not ful-filled by the chosen expression give (for an upper bound)

(A. 4a)

But this can only occur if


The second term in the bracket is certainly negative, and the first will certainly be if K 0 > (241/50) = 4.82.

APPENDIX C

B O U N D S F O R T H E S T U D Y O F S A I N T - V E N A N T ' S

P R I N C I P L E I N T R A N S I E N T P R O B L E M S

The solution of the heat conduction equation (2.1), initially (t = 0) zero and with prescribed boundary temperature f(Q, t), namely

T(Q, t) =f(Q, t) Q on boundary S, t >0 (C. 1)

for x < q(t)

(B.3)

for a lower bound, and with the numerical values of equation (6.5). With u = 1 — xla, this can be written as

(B.3a)

Numerical calculations showed that the least value of the right-hand side is approximately 18.12 (for u = 0.5). Thus

(B.4)

For the upper bound, from equation (6. 6b) one obtains

(B.5)


is (C.2)

The analogous solution with prescribed boundary heat input q(Q, t), namely

(C.3)

is

(C.4)

where G and N are the Green's function and Neumann's function respectively, and n is the inward normal. Derivations of the above fundamental relations may be found in Refs. 2 or 5.

Let Go(P, Q, t — T) be the Green's function of a domain larger than D and g(P, Q, t — r) be a regular function such that

G(P, Q9t-r) = G0(P, Q,t- T)+ g(P, Q,t-r) (C.5)

W e c a n e x n r e s s p- in t e r m s o f G a n d Gn (cf. R e f . 45"> a s

(C.6)

Let Qi, T I be the boundary point and the time, respectively, at which

reaches its maximum; then by differentiating equation (C.5) and using the expression of equation (C. 6) for g, we will have

C.7)

provided that the denominator on the right side of equation (C. 7) is positive. The details of this derivation are analogous to that of Ref. 101.

Let the right side of equation (C. 7) have its maximum at the time r* and at the boundary point 2*; then

(C.8)

BG(P, t]


and the corresponding bound for temperature, from equation (C.2) is

( C 9 )

Through a similar procedure, the bound on the temperature for Neumann's problem, from equation (C.4), is

( C I O )

wher<

( C . l l )

As an example, consider a semi-infinite space x > 0 with boundary condition prescribes either as

(C.12a)

T0; 0<t<t0

[ 0 ; t>t0

Qo; 0 < t< to

(a)

or as

[ 0 ; t>t0

(C.12b)

Let our choice for either Go or No be the solution for a source in an infinite space, i.e.

( C . 1 J

(C.14)

Then

Since XQ = 0 on the boundary. Here we see that the denominators y, equations (C.7) and ( C . l l ) are positive, and thus we have BG(P . i) and BN(P . t) as

(C.15)

then through equations (C.9) and (C.10) the bounds for the temperature for


cases (a), (b) are

(C.16a)

and

(C.16b)

For purposes of reference, the exact solutions may be noted here; they are

; 0 < t < t0

and

0 < t < to

t> to

respectively for the problem of equations (C.12a) and (C. 12b). The results of Section 7 are easily obtained from the formulas derived

above.

R E F E R E N C E S

1 . FOURIER, J. Theorie analytique de la chaleur. Paris (1822); transl. by A . Freeman, N e w York: D o v e r Publications (1955).

2. CARSLAW, H. S. and JAEGER, J . C. Conduction of Heat in Solids. 2nd ed. Oxford: Claren-don Press (1959).

3. BROWN, J . A . "Problems Related to the Des ign of Structures for Ships of the U .S . N a v y " Structural Mechanics: Proc. First Symp. on Naval Structural Mechanics, p. 11. J . N . Goodier and N . J . Hoff, (eds.). N e w York: Pergamon Press (1960).

4. MERIAM, J . L . , LYMAN, P. T., STEIDAL, R. F . , and BROWN, G. W. "Thermal Stresses in the SS. Boulder Victory" / . Ship Res. 2 , 2, 55-71 (October 1958).

5. BOLEY, B . A. and WEINER, J . H. Theory of Thermal Stresses. N e w York: J . Wiley and Sons (1960).

6. PARKUS, H. Instationdre Warmespannungen. Vienna: Springer-Verlag (1959); Melan, E . and Parkus, H . Warmespannungen infolge stationdrer Temperaturfelder. Vienna: Springer-Verlag (1953).

0 < t< t

t> to

V<t<t0

t> to

t> to


7. SWANN, R. T. "Heat Transfer and Thermal Stresses in Sandwich Panels" N.A.C.A. Tech. Note 4349 (September 1958).

8. SWANN, R. T. and POTMAN, C. M. "Analysis of Effective Thermal Conductivities of Honeycomb-Core and Corrugated-Core Sandwich Panels" NASA Tech. Note D-llA (April 1961).

9. HOFF, N . J. "Comparison of Radiant and Conductive Heat Transfer in a Supersonic Wing" / . Aeron. Sci. 23 , 694-696 (1956).

10. ECKERT, E. R. G. and DRAKE, R. M. Heat and Mass Transfer. N e w York: McGraw-Hill

(1959). 11. KREITH, F. Principles of Heat Transfer. Chicago: Intern. Textbook (1959). 12. LYKOV, A. V. and MIKHAYLOV, Y. A. Theory of Energy and Mass Transfer. Englewood

Cliffs: Prentice-Hall (1961). 13. SPROULL, R. L. "The Conduction of Heat in Solids" Scientific American, 207, 6, 92-104

(December 1962). 14. ESHELBY, J. D . "The Fundamental Physics of Heat Conduction" Proc. of the General

Discussion on Heat Transfer, pp. 267-270. L o n d o n : The Inst, o f Mech. Eng. (September 1951).

15. ROY, M. "Not ions d'effluence et de production d'entropie et sur la loi de Fourier" La Recherche Aeronautique, O.N.E.R.A., N o . 75, p. (March-April 1960).

16. DUHAMEL, J. M. C. "Memoire sur le calcul des actions moleculaires developpees par les changements de temperature dans les corps solides" Memories . . . par divers savants 5 S 440-498 (1838). See [5], p. 31 , for further work of Duhamel .

17. DILLON, O. W. "Coupled Thermoplasticity" / . Mech. Phys. Solids, 1 1 , pp. 21-33 (1963). 18. MINDLIN, R. D . "On the Equations of Mot ion of Piezoelectric Crystals" Problems of

Continuum Mechanics, pp. 282-290. Philadelphia: Soc. for Industrial and Applied Mathematics (1961).

19. KALISKI, S. and PETYKIEWICZ, J . "Dynamical Equations of Mot ion Coupled with the Field of Temperatures and Resolving Functions for Elastic and Inelastic Anisotropic Bodies in the Magnetic Field" Proc. of Vibration Problems, N o . 3, pp. 81-94 (1960).

20. MORSE, P. M. and FESHBACH, H. Methods of Theoretical Physics, p. 865. N e w York: McGraw-Hill (1953).

21. BOLEY, B. A. and TOLINS, I. S. "Transient Coupled Thermoelastic Boundary-Value Problems in the Half-Space" / . Appl. Mech. 29, Series E, 4, 637-646 (December 1962).

22. HETNARSKI, R. "Coupled One-Dimensional Thermal Shock Problem for Small Times" Archiwum Mechaniki Stosowanej, 13, 295-300 (1961).

23. MUKI , R. and BREUER, S. "Coupling Effects in a Transient Thermoelastic Problem" Oest. Ing. Archiv. 16, 349-368 (1962).

24. PARIA, G. "Coupling of Elastic and Thermal Deformations" / . Appl. Sci. and Res. Sect. A7. Kharagpur, India: Indian Institute of Technology (1959).

25. CHADWICK, P. "Thermoelasticity, the Dynamical Theory" Chapter VI of Progress in Solid Mechanics, Vol. I, pp. 265-328. I. N . Sneddon and R. Hill (eds.). Amsterdam: North Holland (1960).

26. ACHENBACH, J. D . "Approximate Solutions of Coupled Thermoelastic Problems" Tech. Rep. , Dept . of Civil Engineering and Engineering Mechanics, Columbia University (forthcoming, 1963).

27. BOLEY, B. A. "Discontinuities in Integral Transform Solutions" Quart. Appl. Math. XIX, 4 , 273-284 (January 1962).

28. DERESIEWICZ, H. "Plane Waves in a Thermoelastic Solid" / . Acoustical Soc. A. 29, 2, 204-209 (February 1957).

29. DERESIEWICZ, H. "Solution of the Equations of Thermoelasticity" Proc. Ill U.S. Nat. Congress of Applied Mechanics, A S M E (June 1958).

30. DERESIEWICZ, H. "Effect of Boundaries on Waves in a Thermoelastic Solid: Reflection of Plane Waves from a Plane Boundary" / . Mech. Phys. Solids, 8 , 3, 164-172 (1960).

31. BIOT, M. A. "Linear Thermodynamics and the Mechanics of Solids" Proc. Ill U.S. Nat. Congress of Applied Mechanics (June 1958). For reference to earlier work of Biot see also [5].

32. BIOT, M. A . " N e w Thermomechanical Reciprocity Relations with Applications to Thermal Stress Analysis" / . Aero/Space Sci. 26, 1, 401-408 (July 1959).

33. BIOT, M. A. "Thermodynamics and Heat F low Analysis by Lagrangian Methods" Proc. VII Anglo-American Aeron. Conf. N e w York: Inst, of the Aero Sci. (1959).


34. BIOT, M. A. "Lagrangian Thermodynamics of Heat Transfer in Systems Including Fluid Mot ion" J. Aero Space Sci. 29, 5 (May 1962).

35. D E GROOT, S. R. and MAZUR, P. Non-Equilibrium Thermodynamics. Amsterdam: North-Holland. (1962).

36. PICONE, M. "Sul problema della propagazione del calore in un mezzo privo di frontiera, conduttore, isotropo e omogeneo" Mathematische Annalen, 101, 701 (1929).

37. NIRENBERG, L. "A Strong Maximum Principle for Parabolic Equations" Comm. on Pure and Appl Math. V I , 167-177 (1953).

38. FRIEDMAN, A . "Remarks on the Maximum Principle for Parabolic Equations and Its Applications" Pacific J. Math. 8, 201-211 (1958).

39. PUCCI, C. "Proprieta di massimo e minimo delle soluzioni di equazioni a derivate parziali del secondo ordine di tipo ellittico e parabolico" Consiglio Nazionale delle Ricerche, Publ. dellTstituto per le Applic. del Calcolo, N o . 517, pp. 1-10 (1958).

40. VELTE, W. "Eine Anwendung des Nirenbergschen Maximumprinzips fur parabolische Differential gleichungen in der Grenzschichttheories" Arch, for Rational Mechanics and Analysis, 5, 420-431 (1960).

41 . LAX, P. D . and MILGRAM, A . N . "Parabolic Equations", Ch. 9 in Contributions to the Theory of Partial Differential Equations, Annals of Math. Studies, N o . 33, pp. 167-190. Princeton: Princeton University Press (1954).

42. ROSENBLOOM, P. C. "Linear Equations of Parabolic Type with Constant Coefficients", Ch. 10 of the book cited in Ref. 39, pp. 191-200.

43. BOLEY, B. A. "Upper Bounds and Saint-Venant's Principle in Transient Heat Conduc-t ion" Quart. Appl. Math. X V I I I , 2, 205-207 (July 1960).

44. BERGMAN, S. and SCHIFFER, M. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. N e w York: Academic Press (1953).

45. BOLEY, B. A. "A Method for the Construction of Green's Functions" Quart, of Appl. Math. X I V , 3, 249-257 (October 1956).

46. MASSONNET, C. "Un appareil nouveau pour determiner les efforts dans les pieces elasti-ques planes" Mem. de VAssoc. Int. des Fonts et Charpentes, 7, 321-340 (1949).

47. EVANS, G. W. II, "Applications of the Mauro Picone Theorem for Heat Conduct ion" Bull. Am. Math. Soc. 59, 82 (1953) and Proc. Am. Math. Soc. 4 , 6, 961-968 (December 1953).

48. BOLEY, B. A. "Upper and Lower Bounds for the Solution of a Melting Problem" Quart. Appl. Math. 21 , 1. 1-11 (April 1963); also Tech. Rep. N o . 20, same author and title, Columbia University (February 1962).

49. BOLEY, B. A. "Upper and Lower Bounds in Problems of Melting or Solidifying Slabs" Tech. Rep. No. 23, Columbia University (October 1962), Quart. J. of Mech. and Appl. Math. (1963).

50. JOHN, F. "The Integration of Parabolic Equations by Difference Methods" Comm. on Pure and Appl. Math. V , 155-211 (1952).

51. LAX, P. D . and RICHTMYER, R. D . "Survey of the Stability of Linear Finite Difference Equations" Comm. on Pure and Appl. Math. I X , 267-293 (1956).

52. TODD, J. "A Direct Approach to the Problem of Stability in the Numerical Solution o Partial Differential Equations" Comm. on Pure and Appl. Math. I X , 597-612 (1956).

53. DOUGLAS, J., JR. "On the Numerical Integration of Quasi-Linear Parabolic Differential Equations" Pacific J. Math. V I , 3 5 ^ 2 (1956).

54. DOUGLAS, J., JR. and GALLIE, T. M., JR. "Variable Time Steps in the Solution of the Heat Flow Equation by a Difference Equation" Proc. Am. Math. Soc. 6, 787-793 (1955).

55. VARGA, R. S. "On Higher Order Stable Implicit Methods for Solving Parabolic Partial Difference Equations" / . Math. Phys. 40, 220 (1960).

56. DUSINBERRE, G. M. "A N o t e on the Implicit Method for Finite Difference Heat Transfer Calculations" / . Heat Transfer, Trans. ASME, 83 , 94-95 (1961).

57. CHU, W. H. and ABRAMSON, H. N . "Transient Heat Conduction in a R o d of Finite Length with Variable Thermal Properties" / . Appl. Mech. 27, Series E, 4, 617-622 (December 1960).

58. LOTKIN, M. "Numerical Integration of Heat Conduction Equations" / . Math, and Physics, 37, 178-187 (1958).

59. O'BRIEN, G. G., HYMAN, M. A. , and KAPLAN, S. "A Study of the Numerical Solution of Partial Differential Equations" / . Math, and Phys. 29, 223-251 (1951).


60. DUFORT , E. C . and FRANKEL, S. P. "Stability Conditions in the Numerical Treatment of Parabolic Differential Equations" Math. Tables and Other Aids to Comput. 7 , 135-152 (1953).

61. DOUGLAS, J., JR. "On the Relation between Stability and Convergence in the Numerical Solution of Linear Parabolic and Hyperbolic Differential Equations" J. Soc. Industrial and Appl. Math. 4 , 20-37 (1956).

62. CRANK, J. and NICOLSON, P. "A Practical Method for the Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type" Proc. Cam-bridge Phil. Soc. 4 3 , 50-67 (1947).

63. JUNCOSA, M. L. and YOUNG, D . "On the Crank-Nicholson Procedure for Solving Para-bolic Equations" Proc. Cambridge Phil. Soc. 5 3 , Part 2, 448^161 (1957).

64. D E G. ALLEN, D . N . and SEVERN, R. T. "Application of Relaxation Methods to the Solution of Non-Elliptic Partial Differential Equations" Quart. J. Mech. and Appl. Math. 4 , 209-222 (1951); and 5 , 447-454 (1952) (the latter part including the effect of solidi-fication).

65. PASCHKIS, V. "The Heat and Mass F low Analog Assimilator for the Study of Heat Conduction" Ann de VAssoc. Intern, pour le Calcul Analogique 5 , 1 , 4 -18 (January 1963).

66. PASCHKIS, V. "Combined Geometric and Network Analog Computer for Transient Heat F low" / . Heat Transfer, ASME, 8 1 , Series C, 2, 144-150 (May 1959).

67. KANTOROVICH, L. V. and KRYLOV, V. I. Approximate Methods of Higher Analysis, 3rd ed. N e w York: Interscience (1958).

68. GREEN, J. W . "An Expansion Method of Parabolic Partial Differential Equations" /. Res. Nat. Bureau Stand. 5 1 , 127-132 (1953).

69. WEINER, J. H. "A Method for the Approximate Solution of the Heat Equation" W.A.D.C. Tech. Rep. 54-427 (March 1955).

70. DICKER, D . and FRIEDMAN, M. B. "Non-Separable Solutions of the Transient Heat Conduction Equation" Tech. Rep. , Department of Civil Engineering and Engineering Mechanics, Columbia University (March 1961).

71. SHOHAT, J. A. and TAMARKIN, J. D . "The Problem of Moments" Am. Math. Soc, Mathematical Surveys, N o . 1 (1943).

72. GOODMAN, T. R. "The Heat-Balance Integral and Its Application to Problems Involving a Change of Phase" Trans. ASME, 8 0 , 335-342 (1958).

73. GOODMAN, T. R. "The Heating of Slabs with Arbitrary Heat Inputs" J. Aero j Space Sci. 2 6 , 187-188 (1959).

74. BIOT, M. A. "Further Developments of N e w Methods in Heat F low Analysis" / . Aero/ Space Sci. 2 6 , 6, 367-381 (June 1959).

75. BIOT, M. A. "New Methods in Heat F low Analysis with Application to Flight Structures" /. Aero. Sci. 2 4 , 12, 857-873 (December 1957).

76. CHAMBERS, L. G. "A Variational Principle for the Conduction of Heat" Quart. J. Mech. and Appl. Math., IX, Part 2, 234-235 (June 1956).

77. CITRON, S. J. "A N o t e on the Relation of Biot's Method in Heat Conduction to a Least-Squares Procedure" / . Aero/Space Sci. 2 7 , 4, 317 (April 1960).

78. GOODMAN, T. R. "The Heat-Balance Integral—Further Considerations and Refinements" Trans. ASME, Series C, 8 3 , 81-85 (1961).

79. KOH , J. C. Y. "One-Dimensional Heat Conduction with Arbitrary Heating Rates and Variable Properties" / . Aero/Space Sci. 2 8 , 989-990 (1961).

80. YANG, K . T. and SZEWCZYK, A . "An Approximate Treatment of Unsteady Heat Con-duction in Semi-Infinite Solids with Variable Thermal Properties" / . Heat Transfer, Trans. ASME, Series C, 8 1 , 251-252 (1959).

81. LARDNER, T. J. and POHLE, F . V. "Application of the Heat Balance Integral to Problems of Cylindrical Geometry" / . Appl. Mech. 2 8 , 2, 310-312 (June 1961).

82. VEINIK, A . I. "Approximate Calculation of Heat Conduction Processes" Gosud. Energe-tich. Izdat., N o . 24, Moscow (1959).

83. MICHLIN, S. G. "Some Sufficient Conditions for Convergence of Galerkin's Method" translated from Uzhjonyje Zapiski LGU, N o . 135, 1950, by the Inst, of Math. Sciences, N e w York University.

84. BRULL, M. A. and VINSON, J. R. "Approximate Three-Dimensional Solutions for Tran-sient Temperature Distributions in Shells of Revolut ion" / . Aero/Space Sci. 2 5 , 12, 742-750 (December 1958).


85. HOFF, N . J. "Structural Problems of Future Aircraft" Third Anglo-Amer. Aero. Conf. Proc. pp. 103-110 Brighton, England (September 1951).

86. POHLE, F. V. and OLIVER, H. "Temperature Distribution and Thermal Stresses in a Model of a Supersonic Wing" / . Aero. Sci. 21 , 1, 8 -16 (January 1954).

87. BARBER, A. D . , WEINER, J. H. , and BOLEY, B. A. "An Analysis of the Effect of Contact Resistance in a Sheet-Stringe Structure" / . Aero. Sci. 24 , 3, 232 (March 1957),

88. GATEWOOD, B. E. "Effect of Contact Resistance of Joints upon Thermal Stress" J. Aero. Sci. 24 , 152 (1957).

89. SEIDE, P. "An One-Dimensional Temperature Distribution in Two-Layered Slabs with Contact Resistance at the Plane of Contact" J. Aero. Sci. 25, 523 (1958).

90. FRANK, I. "Transient Temperature Distribution in Aircraft Structures" / . Aero. Sci. 25, 265 (1958).

91. BARZELAY, M. E. "Range of Interface Thermal Conductance for Aircraft Joints" NASA Tech. Note £ - 4 2 6 (May 1960).

92. LEVINSON, M. "Thermal Stresses in an Idealized Wing Structure" / . Aero)Space Sci. 28 , 899-901, 55 (1961).

93. STALDER, "The Useful Heat Capacity of Several Materials for Ballistic Nose -Cone Construction" NACA Tech. Note 4141 (November 1957).

94. YANG, K. T. "Transient Heat Conduction in a Semi-Infinite Solid with Variable Thermal Conductivity" / . Appl. Mech., Trans. ASME. 80, 146-147 (1958).

95. GOURSAT, E. Cours d? analyse mathematique. Fifth Ed., Vol . 3, Chapt. 29, Paris: Gauthier-Villars (1942).

96. STOLTZ, G., JR. "Numerical Solution to an Inverse Problem of Heat Conduction for Simple Shapes" / . Heat Transfer, 20 -26 (February 1960).

97. MIRSEPASSI, T. J. "Heat-Transfer Charts for Time-Variable Boundary Condit ions" British Chemical Engineering (March 1959).

98. MIRSEPASSI, T. J. "Graphical Evaluation of a Convolution Integral" Math. Tables and Other Aids to Computation (July 1959).

99. SHUMAKOV, N . V. "A Method for the Experimental Study of the Process of Heating a Solid Body" Soviet Physics—Technical Physics, transl. by the A m . Inst, of Physics, Vol. 2, p. 771 (1957).

100. BOLEY, B. A . "Some Observations on Saint-Venant's Principle" Proc. HI U.S. Nat. Congress of Appl. Mech., ASME, pp. 259-264 (June 1958).

101. BOLEY, B. A. "Some Bounds for Steady-State Temperature Distributions" Proc. IV U.S. Nat. Congress of Appl. Mech., ASME (June 1962).

102. BOLEY, B. A . " A Method for the Construction o f Fundamental Solutions in Elasticity Theory" / . Math, and Physics, X X X V I , 3, 261-268 (October 1957).

103. BOLEY, B. A . "Thermal Stresses" paper in "Structural Mechanics" Proc. First Conf. on Naval Structural Mechanics, pp. 378-406. N e w York: Pergamon Press (1960).

104. GIBSON, R. E. "A Heat Conduction Problem Involving a Specified Moving Boundary" Quart. Appl. Math. 16, 426-430 (1959).

105. RUDDLE, R. W. The Solidification of Castings. 2nd ed. L o n d o n : Institute of Metals (1957).

106. TRUITT, R. W. Fundamentals of Aerodynamic Heating, p. 193 ff. N e w York: Ronald Press (1960).

107. SIMKINS, P. G. "An Introduction to Hypersonic Ablat ion" / . Roy. Aeron. Soc. 66, N o . 618, pp. 387-393 (June 1962).

108. SUTTON, G. W. "The Hydrodynamics and Heat Conduction of a Melting Surface" /. Aero. Sci. 25, 1, 29 -32 , 36 (January 1958).

109. SUTTON, G. W. "Ablation of Reinforced Plastics in Supersonic F low" J. Aero/Space Sci. 27, 377-385 (May 1960).

110. HIDALGO, H. "Ablation of Glassy Material Around Blunt Bodies of Revolut ion" ARS J. 30 , 9, 806-814 (September 1960).

111. BETHE, H. A. and ADAMS, M. C. "A Theory for the Ablation of Glassy Materials" / . Aero/Space Sci. 26, 6, 321-328, 350 (June 1959).

112. ROBERTS, L. O. "A Theoretical Study of Stagnation Point Ablat ion" NACA Tech. Note 4392 (September 1958).

113. ADAMS, M. C , POWERS, W. E. and GEORGIEV, S. "An Experimental and Theoretical Study of Quartz Ablation at the Stagnation Point" / . Aero/Space Sci. 27, 7, 535-543 (July 1960).


114. STEG, L. "Materials for Re-Entry Heat Protection of Satellites" ARS J. 3 0 , 9, 815-822 (September 1960).

115. SCALA, S. M. "A Study of Hypersonic Ablat ion" Proc. Tenth Int. Astron. Congress. Vienna: Springer-Yerlag (1959).

116. ABRAHAM, L. H. Structural Design of Missiles and Spacecraft, p. 250 ff. N e w York:

McGraw-Hill (1962). 117. STEPHAN, J. "Ueber der Theorie des Eisbildung, insbesondere ueber die Eisbildung im

Polarmeere" S. B. Kais Akad. Wissenschaften, Vienna, 9 8 , part II, 965-983 (1889); and "Ueber einige Probleme der Theorie des Warmeleitung" S. B. Kais. Akad. Wissen-schaften, Vienna, 9 8 , part II, 473-484 (1889). See also Annalen Phys. and Chem. (Wiede-mann), 4 2 , 269-286 (1891).

118. LARSEN, E. S. "Time Required for the Crystallization of the Great Batholith of Southern and Lower California" Amer. J. Sci. 2 4 3 , Series A , 399-416 (1945).

119. GIBSON, R. E. "A One-Dimensional Consolidation Problem with a Moving Boundary" Quart. Appl. Math. 18 , 123-129 (1960).

120. COHEN, H. and MIRANKER, W. L. "Boundary-Layer Behavior in the Superconductor Transition Problem" J. Mathematical Physics, 2, 4 , 575-583 (July-August 1961).

121. LINIGER, W, "The Solution of a Transition Problem in a Superconducting Strip" IBM Research Report RC-502 (July 20, 1961).

122 CHAMBRE, P. L. "On the Dynamics of Phase Growth" Quart. J. Mech. and Appl. Math. IX, Part 2, 224-233 (June 1956).

123. GRIGORIAN, S. S. "On Heating and Melting of a Solid Body Owing to Friction" Appl. Math, and Mech. 22, N o . 5, 815-825 (1958). (Transl. from the Russian P. M. M.).

124. ANDRIANKIN, E. I. "Propagation of Thermal Waves from the Boundary of T w o Media" Appl. Math, and Mech. 2 3 , 5, 1420-1423 (1959). (Transl. from the Russian P. M. M.).

125. MIRANKER, W. L. and KELLER, J. B. "The Stefan Problem for a Non-Linear Equation" /. Math, and Mech. 9 , 1, 67 -70 (January 1960).

126. EVANS, G. W. II. "A N o t e on the Existence of a Solution to a Problem of Stefan" Quart. Appl. Math. IX, 2, 185-193 (July 1951).

127. KOLODNER, I. "Free Boundary Problem for the Heat Equation with Applications to Problems of Change of Phase" Comm. on Pure and Appl. Math. IX, 1, 1-31 (February 1956).

128. MIRANKER, W. L. "A Free Boundary Value Problem for the Heat Equation" Quart. Appl. Math. XVI, 121-130 (1958).

129. KYNER, W. T. "On a Free Boundary Value Problem for the Heat Equation" Quart. Appl. Math. XVII, 3, 305-310 (October 1959).

130. KYNER, W. T. "An Existence and Uniqueness Theorem for a Non-Linear Stefan Problem" J. Math, and Mech. 8 , 483-498 (1959).

131. LANDAU, H. G. "Heat Conduction in a Melting Solid" Quart, of Appl. Math. 8, 81-94 (1950).

132. EVANS, G. W., II, ISAACSON, E. , and MACDONALD, J. K . L. "Stefan-Like Problems" Quart Appl. Math. VIII, 312-319 (1950).

133. KREITH, F. and ROMIE, F. E. "A Study of the Thermal Diffusion Equation with Boundary Conditions Corresponding to Solidification or Melting of Materials Initially at the Fusion Temperature" Proc. Phys. Soc. London, Series B, 6 8 , 377-291 (1955).

134. BOLEY, B. A. "A Method of Heat Conduction Analysis of Melting and Solidification Problems" / . Math, and Physics, XL, 4, 300-313 (December 1961).

135. MILES, J. W. "A N o t e on Riemann's Method Applied to the Diffusion Equation" Quart. Appl. Math. VIII, 95-101 (1950).

136. LIGHTFOOT, N . M. H. "The Solidification of Molten Steel" Proc. London Math. Soc. 3 1 , 2 , 97 -116(1929) .

137. CITRON, S. J. "Heat Conduction in a Melting Slab" / . Aero/Space Sci. 27 , 3, 219-228 (March 1960).

138. DEWEY, C. F. , JR., SCHLESINGER, S. I., and SASHKIN, L. "Temperature Profiles in a Finite Solid with Moving Boundary" / . Aero/Space Sci. 27, 59-64 (1960).

139. CITRON, S. J. "On the Conduction of Heat in a Melting Slab" Proc. IV U.S. Nat. Congress of Appl. Mech., ASME (June 1962).

140. SPRINGER, G. S. and OLSON, D . R. "Method of Solution of Axisymmetric Solidification and Melting Problems" Heat Transfer Division A S M A , paper N o . 6 2 - W A - 2 4 6 , pre-sented at the A S M E Annual Meeting, N e w York (November 1962).


141. DOUGLAS, J., JR. and GALLIE, J. M. JR. "On the Numerical Integration of a Parabolic Differential Equation Subject to a Moving Boundary Condit ion" Duke Math. J. 22, 557-571 (1955).

142. CRANK, J. "TWO Methods for the Numerical Solution of Moving-Boundary Problems in Diffusion and Heat F low" Quart. J. Mech. and Appl. Math. 10, 220-231 (1957).

143. EHRLICH, L. W . "A Numerical Method of Solving a Heat Flow Problem with a Moving Boundary" J. Assoc. of Comput. Machin. 5 , 161-176 (1958).

144. FRIEDMAN, A . "Free Boundary Problems for Parabolic Equations: I Melting of Solids" J. Math, and Mech. 8 , 499-518 (1959).

145. TRENCH, W . "On an Explicit Method for the Solution of a Stefan Problem" / . Soc. Industr. and Appl. Math. 7 , 184-204 (1959).

146. LOTKIN, M. "The Calculation of Heat F low in Melting Solids" Quart. Appl. Math. X V I I I , 1, 79-85 (1960).

147. TING, T. C. T. "On the Solution of a Non-Linear Parabolic Equation with a Floating Boundary Arising in a Problem of Plastic Impact of a Beam" Brown University, Tech. Rep. No. 78, Contract N o . N o n r 562(10) (May 1962).

148. ROSE, M. E. "A Method for Calculating Solutions of Parabolic Equations with a Free Boundary" Math. Tables and Other Aids to Comput. X I V , 71, 249-256 (July 1960).

149. BIOT, M. A. and DAUGHADAY, H. "Variational Analysis of A b l a t i o n " / . Aero/Space Sci. 2 9 , 2, 228 (February 1962).

150. GOODMAN, T. R. and SHEA, J. J. "The Melting of Finite Slabs" / . Appl. Mech. 2 7 , 1, 16-24 (March 1960).

151. POOTS, G. "An Approximate Treatment of a Heat Conduction Problem Involving a Two-Dimensional Solidification Front" Int. J. Heat and Mass Transfer, 5 , 339-349 (May 1962).

152. LARDNER, T. J. "Approximate Solutions for Melting and Ablation Problems" Polyt. Inst, of Brooklyn, PIBAL Rep. No. 654 (June 1962).

153. PEKERIS, C. L. and SLICHTER, L. B. "Problem of Ice Formation" / . Appl. Phys. 10, 135-137 (1939).

154. TADJBAKHSH, I. "Thermal Stresses in an Elastic Half-Space with a Moving Boundary" IBM Research Report RC-616 (January 1962), and AIAA J. 1, 1, 214-215 (January 1963).

155. ROGERS, T. G. and LEE, E. H. "Thermo-Viscoelastic Stresses in a Sphere with an Ablating Cavity" Tech. Rep. No. 6, Brown University, Contract NOrd-18594 (August 1962).

156. WEINER, J. H. and BOLEY, B. A. "Elastic-Plastic Thermal Stresses in a Solidifying Body" J. Mech. and Physics of Solids, 1 1 , pp. 145-154 (1963).

157. STERNBERG, E. and GURTIN, M. E. "Uniqueness in the Theory of Thermo-Rheologically Simple Ablating Viscoelastic Solids" Tech. Rep. No. 16, Brown University, Contract N o . Nonr-262(25) (September 1962).

158. KAPLAN, S. "On the Growth of Solutions of Quasi-Linear Parabolic Equations" Report 1MM-NYU 305, Courant Institute of Math. Sci., N e w York University (November 1962).

159. BOLEY, B. A. "The Determination of Temperature, Stresses, and Deflections in T w o -Dimensional Thermoelastic Problems" / . Aero. Sci. 2 3 , 1, 67-75 (January 1956).

METHODS OF SOLUTION OF

THERMOELASTIC BOUNDARY VALUE PROBLEMS

H E I N Z PARKUS

Technische Hochschule, Vienna, Austria

I N T R O D U C T I O N

Although interest in the field of thermoelasticity dates as far back as 1837, when Duhamel published his famous Memoire sur les phenomenes thermo-mecaniques, it was only during the last two decades that active and systematic research has been conducted. Approximately 850 papers and a number of books have appeared during that period. The flood is abating now. Linearized theory has practically been exhausted while nonlinear thermoelasticity quite naturally presents considerable mathematical difficulties.

In the following an attempt is made to present a concise survey of the various methods employed in attacking thermoelastic problems. Depending on their mathematical and physical character these problems may be divided into many groups: linear and nonlinear, stationary and nonstationary, isotropic and anisotropic, deterministic and random, and so on. The equations governing these problems are all special cases of the general equations of thermoelasticity which, therefore, will be presented first.

Four groups of equations constitute the basis of the theory of thermoelasticity. Using cartesian coordinates xt(i = 1, 2, 3) in a Newtonian frame of reference and adopting the summation convention, these equations may be written as follows (see Refs. 1, 2, 3) :

Equations of motion

1 . C O N S T I T U T I V E E Q U A T I O N S

(1.1)

Kinematic relations

(1.2)

Conservation of mass

(1.3)

317

318 H E I N Z P A R K U S

Conservation of energy (1st law of thermodynamics)

(1.4)

where Kis any arbitrary volume of the deformed* body, and S the corresponding surface; d/dt means differentiation with respect to time, keeping the Xi constant.

The xt are the initial coordinates of a particle. All quantities are taken as functions of these coordinates and of the time t. The displacement vector is u%, Fi is the body-force vector per unit mass and Pi the applied surface stress; p denotes density and po its initial value; sy and ey are the stress tensor and the strain tensor, respectively, and U denotes the internal energy per unit volume of the undeformed body.

The last two integrals on the right side of equation (1.4) represent, respec-tively, the rate of work of the surface forces and the heat transported per unit time into the volume by conduction; here m is the unit outward normal and Qi is the heat-flux per unit area of the deformed body.

The stress vector acting at a point of a surface with normal initially in the direction of xi has the three physical components! measured per unit area of the undeformed body

a» = ^ A / [ 1 +em] (1.5)

They are in the three oblique directions which, before deformation, were parallel to the three coordinate axes.

Measured per unit area of the deformed body the same components are

= <jyh{i) (1.6)

The quantities

hi = V K 1 + ^ 2 ) (1 + £ 3 3 ) - 4 d , e t c - 0 . 7 )

are the ratios of the deformed to the undeformed areas. While sn is a symmetric tensor, cr?. and ay are not tensors at all. The law of conservation of energy (1.4) may be transformed into a more

convenient form by introducing the principle of rate of work which states that the rate of change of kinetic energy equals the rate of work of all forces, external and internal. Since that of the internal forces is equal to

* Existence of an equilibrium state without external forces and without stress is postulated. Deformations are measured from this state.

t Indices appearing in parenthesis are not summed.

S O L U T I O N OF T H E R M O E L A S T I C B O U N D A R Y V A L U E P R O B L E M S 319

one finds from (1.4)

Applying now Gauss's theorem to the surface integral and then transforming to the initial volume Vo one obtains, with pdV = podVo,

where q\ represents the heat-flux vector per unit area of the undeformed body, q% = Qi\/g- The above relation must hold for any arbitrary volume VQ. Hence

(1.8)

Equation (1.8) may be transformed further by introducing the free energy 0 per unit volume of the undeformed body,

(1.9) 0= U-TS

where T(x, t) is absolute temperature and S is entropy. It has been shown 4

that, in the elastic solid, 0 is a function depending only on the quantities etj and T. Hence

Substitution into equation (1.8) renders

Choosing e%j and T for independent variables of state, one finds from the above equation

(1.10)

(1.11)

(1.12)

Equation (1.10) is the stress-strain law while equation (1.11) defines entropy. Substituting S from equation (1.11) into equation (1.12) gives

(1.13)

This is the equation of heat conduction.

H E I N Z P A R K U S

The heat-flux vector qi will, in general, be a function of the temperature gradients dTjdxi and of the strain components (see Ref. 2, section 8.5). The elastic properties of the material are mirrored in the free energy function & 5 .

The 16 scalar relations (1.1), (1.2), (1.10) and (1.13) form the basic equations of thermoelasticity. They contain ui, ey, and T as 16 unknown functions of space Xi and time t.

(2.8)

320

2. L I N E A R I Z E D E Q U A T I O N S

If displacements are small in comparison with the dimensions of the body, if displacement gradients are small compared with 1, and if, upon putting T = To + 0, the temperature increase 0 is small in relation to the uniform reference temperature To, then all nonlinear terms in the basic equations may be neglected and no distinction need be made between deformed and undeformed body in setting up equations of motion. Free energy function 0 and heat-flux vector qt will have the form, with e# denoting the linearized strain tensor,

(2.1)

(2.2)

For a homogeneous body all coefficients are constants. 0 in equation (2.1) has been referred to the "natural" or "undeformed"

initial state of the body where 0 = 0, e# = 0, <T# = 0. N o linear terms can then be present and we have 0 = 0 and S = 0 in the undeformed state. Equa-tions (1.1), (1.2), (1.10) and (1.13) now take on the form

(2.3;

(2.4)

(2.5) aij = bijO + djjci ejd

(2.6)

Putting a = —cp introduces the specific heat c at fixed deformation. For the homogeneous and isotropic thermoelastic solid equations (2.2) and

(2.5) simplify to Hooke's law and to Fourier's law, respectively,

(2.7)

3 . B O U N D A R Y C O N D I T I O N S . U N I Q U E N E S S

Associated with the basic equations (1.1), (1.2), (1.10) and (1.13) are initial and boundary conditions.

Initial conditions specify the state of the body at time / = 0 by giving its initial position, initial velocity distribution and initial temperature. It will, in general, be assumed in the following that the body is initially in its "unformed" state with given velocity u% and, hence, given strain rates ey.

Boundary conditions prescribe temperature or heat-flux and displacement or stress on the surface of the body for all values of time t > 0. The most general type of boundary value problem will occur if the stress vector is pre-scribed on a certain part Si and the displacement vector on the remaining part S2 of the surface S while, independently, temperature is given on part .S3 and heat-flux on the remaining part S4 of the same surface S.

The question now arises as to whether the boundary value and initial value problem thus stated has a unique solution—the existence of at least one solution being assumed.

In the general dynamic case as presented in the preceding paragraphs, unique-ness of solution may be expected as a consequence of the deterministic structure of macroscopic physics: if the initial state of a system is known exactly together with all external agencies acting on it in the course of time, the future of the system is uniquely determined. A mathematical proof, however, has so far been given only for the linear isotropic body by Weiner 6 .

The situation is different in the static case where the body is considered in its state of equilibrium when forces and temperature have reached constant values. Since in this case the time history of the loading process is not known or is not considered there may, and in fact in general do, exist more than one position of equilibrium under given loads and temperature. The solution of the static problem is therefore, in general, not unique. However, uniqueness exists in the linearized case provided the possibility of discontinuities in displacements and stresses is properly taken into account. This has been proved by Boley and Weiner 7 for the homogeneous isotropic body.

Static Problems

All quantities are independent of time. If, in addition, the heat-flux vector

qi does not depend on the components of strain, the equation of heat conduction

12

(2.9)

(2.10)


where G is the shear modulus, v is Poisson's ratio and a is the coefficient of linear thermal expansion, e = en is the first invariant of the strain tensor while k represents thermal conductivity. Equations (2.1) and (2.6) now become


(1.13) will separate from the remaining equations and can be solved indepen-dently. The corresponding boundary value problem will not be considered here (see Carslaw and Jaeger 8), but the temperature distribution will be assumed to be known.

has to be superposed such that m = u% + ut satisfies the given boundary conditions.

* For a survey see Refs. 7, 9, 10, 11.

4 . S M A L L D I S P L A C E M E N T S . H O M O G E N E O U S A N D

I S O T R O P I C B O D I E S

This is the "classical" case of thermoelasticity. It has been the subject of extensive studies, and a vast literature exists in this field.* The basic equations are, from Section 2, equations (2.3) and (2.10),

V20 = O (4.1)

(4.2)

together with equations (2.4) and (2.7). Upon eliminating atj and e# one finds

(4.3)

as the extension to thermoelasticity of Navier's equations. Associated with these equations are boundary conditions as stated in Section 3, with stress or dis-placement vector being prescribed on the surface of the body.

Body forces have been assumed absent, Ft = 0. If necessary, a solution with Fi T £ 0 and 0 = 0, with zero stress or zero displacement on the surface, may always be superposed.

It has been noted by Good ie r 1 2 that the linear equations of thermoelasticity may be reduced to the equations of isothermal elasticity by introducing a par-ticular solution of equations (4.3) in the form

(4.4)

Substitution into (4.3) renders

(4.5)

The scalar function 0 is termed thermoelastic potential. Since in determining 0 no regard need be given to the boundary conditions of the problem, any par-ticular solution of equation (4.5) may be used. Then a solution ut of the homo-geneous equations

(4.6)


Equation (4.6) is, in general, not very convenient to handle. As is well known, however, its general solution may be expressed either in terms of harmonic functions or in terms of biharmonic functions. 1 3 The first method* leads to the Neuber-Papkovich representation of u\

* A third method developed by V. M . Maysel , which is analogous to the Green's function method in potential theory (see Ref. 13, Section 61), will not be discussed here. For details see Ref. 10.

t Under certain restrictions one of the four functions may be set equal to z e r o . 1 4

(4.7)

which involves four independent harmonic functions ^ and i/ro-t The second method leads to the Galerkin-Westergaard form of solution

(4.8) V2V2^ = 0

which contains three biharmonic functions xi- Since every biharmonic function can be represented in terms of two harmonic functions it follows from the representation (4.7) that two of the six harmonic functions involved in (4.8) are not independent.

Combining equations (4.4) and (4.7) one has

(4.9)

as the general solution of the thermoelastic problem. The corresponding stresses follow from equations (2.4) and (2.7), with e = V2& + (1 — 2v) VW,

(4.10)

Similarly, upon combining equations (4.4) and (4.8), one finds for the stresses

(4.11)

A different approach to the solution of the linear thermoelastic problem results if one eliminates ut and e # from the basic equations. Upon taking com-patibility relations into account, one finds (cf. Boley and Weiner 7 )

(4.12)

where s = an and E = 2(1 + v) G. The solution of equation (4.12) has to be


statically admissible, i.e. it has to satisfy equation (4.2) in the interior B of the body. It is unique provided the body is simply connected and the boundary conditions are of the form

aijnj = Fi(P) (4.13)

at all points P of the surface S of the body. Following the procedure of Maxwell and Morera the solution of equation

(4.12) may be represented in terms of certain stress functions. In general, how-ever, no particular advantage is to be gained from this.

Putting j = i in equation (4.12) and summing one obtains

(4.14)

If a is constant (independent of 9) one has V 2? = 0 from equation (4.1). We consider now some special cases. (a) Problems with shearing stress vanishing in a plane. If the shearing stress is

zero in all points of the plane z = 0 the four Neuber-Papkovich functions can be reduced to o n e . 1 5 Putting

(1 — 2v) (f>, 0 i = = 0, 03 = —

in equation (4.9), where <f>(x,y,z) is a harmonic function, and writing Xi = (x, y, z), Ui = (w, v, w), one obtains

dx

Equation (4.10) renders, with V 2 ^ = - 2d2<f>/dz2,

(4.15)

(4.16)


If the temperature field is symmetric with respect to the plane z = 0 then d&/dz and hence the shearing stress are zero in that plane. If, therefore, the normal stress a z z or the normal displacement w and the temperature distribution 0 are given on z = 0, the problem reduces to the determination of a potential function d(f>/dz with prescribed values for dcf>/dz, or d2(f>/dz2, on z = 0.

A general problem of this type—the half-space with stress-free surface and arbitrarily prescribed surface temperature 6(x, y, 0)—has been treated by Sternberg and McDowel l 1 6 . They find the following expressions for the two functions 3> and <f>:

where R = | T — fs | and the integral is to be extended over the plane z = 0. Substitution of 0 into equations (4.16) reveals the remarkable fact that the stress field, in this case, is plane, i.e. a z x = o z y = o z z = 0 in the entire half-space.

(b) Axisymmetrical problems. The three Galerkin-Westergaard functions xi reduce here to a single biharmonic function, xi = X2 = 0, %3 = x> known as Love's displacement function (Ref. 17, Section 188). Using cylindrical coordinates r, 0, z, with djdd = 0, and writing u% — (u, 0, w), equations (4.4) and (4.8) are transformed to

(4.18)

while equations (4.11) go over into

(4.19)

X satisfies the equation

It frequently happens that the problem, besides exhibiting axial symmetry, is such that the shearing stress vanishes in the plane z = 0. In that case the representation of the solution in terms of a single harmonic function appears to be more advantageous. Transforming equations (4.15) and (4.16) to cylindrical

(4.20)

(4.17)

326

coordinates one finds

(4.21)

(4.22)

A great number of solutions of axisymmetrical problems has been published, for which the reader is referred to the l i te ra ture . 7 » 9 > 1 0 A few examples are given below.

An effective method for solving the potential (or bipotential) problem con-nected with the preceding equations is the application of the Hankel transform in case of the semi-infinite solid, and of the Dim series expansion in case of the semi-infinite cylinder. The method has been used extensively by Sneddon and his co-workers. As an example consider the following mixed boundary value problem. An infinite solid contains a crack in the shape of a thin coin of radius 1 lying in the plane z = 0. The temperature is symmetric with respect to that plane and the surfaces of the crack are free from stress. The boundary condi-ditions in z = 0 are then

a z l = 0, 0 < r < 1

w = 0, r > 1 and or* = 0

Upon application of a Hankel transform of order zero, i.e.

(4.23)

the heat equation (4.1) transforms into

For z > 0 its solution is e*(p9z)=pf(p)e-P*

where f(p) is determined by the given temperature in z = 0. Equation (4.5)

H E I N Z P A R K U S

Using equations (4.19), the two remaining boundary conditions, azr = azz = 0

S O L U T I O N OF T H E R M O E L A S T I C B O U N D A R Y V A L U E P R O B L E M S 3 2 7

then gives

where the symmetry condition d®jdz = 0 in z = 0 has already been incor-porated. The transform of the harmonic displacement function <f> is written in the form

Using the inversion formula

(4.24)

substituting for 0 and cj> in equations (4.21) and (4.22), and utilizing the bound-

ary conditions one has in z = 0

where

is a known function. For some special cases the pair of dual integral equations for ifi(p) has been

solved in closed form by Olesiak and Sneddon 1 9 . A discussion of more general types of dual integral equations may be found in Sneddon 2 0 .

The solution for the stress field in a semi-infinite body due to a point source of heat located at a distance £ below the surface z = 0 has been given by N o w a c k i 1 0

with the aid of the Love displacement function. If the surface is kept at constant temperature To the following boundary conditions have to be satisfied o n z ^ O

(9 = 0, (JZr = 0, (TZZ = 0

The temperature distribution is then given by

where W is the heat produced by the heat source per unit time, k is the thermal conductivity and Rx = [r 2 + (z - £) 2] 1 / 2> ^ 2 = [r2 + (z + £ ) 2 F 2 . F rom equa-tion (4.5) one finds immediately


may now be put in the form

where R = ( r 2 + t2)1'2. The Hankel transform of the biharmonic displacement function must satisfy the transformed biharmonic equation

and must remain bounded for z -> oo. Hence, it will have the form

with A(p) and B(p) as arbitrary constants. Multiplying now the first boundary condition by rJo(pr) and integrating over r from 0 to oo one finds, with z = 0,

A + (1 - 2v) B = 0

Multiplying the second boundary condition by rJ\(pr), integrating and using the relat ions 2 0

one gets, with z = 0,

With A and B known x*(P> z) m a y t>e inverted to give x0% z). For the half-space with axially symmetric temperature prescribed on the

surface the solution (4.17) by Sternberg and McDowel l 1 6 simplifies to

(4.25)

where R2 = r2 + p2 — 2rp cos OJ. If the surface temperature 0(r, 0) is uniformly or hemispherically distributed over a circular region and is zero outside, the solution can be given in closed form.

(c) Plane strain and plane stress. Temperature, stress and strain are inde-pendent of coordinate z, and a z x — o z y = 0. In addition, one has € Z Z = 0 in plane strain and a z z = 0 in plane stress. The formulas of plane strain represent an exact solution of the thermoelastic equations while those of plane stress are only approximations.


The equations of plane strain may easily be obtained by specializing equation (4.9) to two dimensions. Connecting the two harmonic functions 0i and 02 in

V = xfi + yfa + fa (4.26)

by choosing them as real and imaginary parts of a complex analytic function,

<Kz) = Mx,y) + ih(x,y) (4.27)

and utilizing the Cauchy-Riemann equations one has

Equation (4.10) then reads

(4.28)

The function 2GW is known as the Airy stress function. Substituting equation (4.27) into equation (4.26) we obtain the complex representation due to Goursat

(4.29) 2W = z<j>(z) + z<f>(z) + y(z) + x ( z )

where is the real part of x(z) and a bar denotes the conjugate complex quan-tity. Equations (4.28) may then be combined into the two complex equations

(4.30)

(4.31]

where

Similarly, upon combining equations (4.9),

[4.32)

It is now a simple matter to formulate the two fundamental boundary value problems of plane thermoelasticity. In the first the stress vector X(s), Y(s) is prescribed on the boundary B of the plane region R. Then

12§


or, in complex form, using equations (4.28)

Substitution from equation (4.29) and integration along the boundary renders

(4.33)

on B. The integration constant C may have different values on each contour of a

multiply connected region. On one of these contours it can be fixed arbitrarily. In the second boundary value problem the displacement vector u, v is pre-

scribed on the boundary B. Hence, from equation (4.32),

(4.34)

on B. The same relations are true for a body in the state of plane stress provided A

in equation (4.30) is taken to be

(4.35)

and 3 — 4v in equations (4.32) and (4.34) is replaced by (3 — v)/(l + v ) . Equation (4.5) has to be replaced by

V2<2> = (1 + v)a6 (4.36)

The methods for solving equations (4.33) and (4.34) are due to N . I. Musk-helishvili. The reader is referred to the l i t e r a t u r e . 2 1 ' 1 5 The problem may, of course, be attacked directly without recourse to complex function theory by using equations (4.28) in connection with suitably chosen biharmonic functions ^ ( s e e Refs. 7, 9, 10, 13).

5 . S M A L L D I S P L A C E M E N T S . H O M O G E N E O U S

N O N I S O T R O P I C B O D I E S

While the static thermoelastic response of a linear isotropic body is character-ized by three constants G, v9 a the number of elastic and thermal coefficients is much larger in the general nonisotropic case. Taking into account symmetry of the stress and strain tensor in equation (2.5)

°ij = byO + djki ejci (5.1)

and interchangeability of the derivatives in equation (2.6)

(5.2)

and noting from equation (2.1) that cyu = cjcitj, one finds that there are al-together 33 independent constants: 6by9 2\cyu and 6ay.


The values of the constants and hence the relations (5.1) and (5.2) vary with the choice of the coordinate frame. If they remain invariant under certain transformations of coordinates the body exhibits properties of elastic symmetry and the number of constants is reduced. A thorough discussion of the various classes of symmetry (crystal classes) may be found in Ref. 2. We restrict our attention here to the so-called orthotropic body.

An orthotropic body possesses three mutually orthogonal planes of thermo-elastic symmetry. Hence, reflecting the coordinate system in these planes, i.e. performing any one of the three transformations

I = — Xl, * 2 = X2, * 3 = X3

II x\ = Xl, = — X2, * 3 = * 3

III = Xl, x% = * 2 , x 3 = — * 3

I G12 = — CT12, a 1 3 = — ^13,

II °"23 = — cr23, a21 = — C721,

III CT31 = — 0-31, G32 = — 0"32,

(5.3)

will leave the coefficients unchanged. Using the relations

x\ = atkXjc, v'ij = <*>ik<*>]l<*kh €ij = aik^l€kl

one finds

all others unchanged



The same holds true for the Substituting into equation (5.1) and (5.2) it follows then that only 15 of the 33 constants are different from zero. We have therefore in the orthotropic case

<*XX = Cl €XX + C\2 €yy + ClS €ZZ ~ filO ]

°yy ~ C21 *xx + C2 tyy + C23 *zz — P%0

v z z = C31 € X X + <?32 €yy + C3 €ZZ — 03 #

<*xy — gl2 *xy, °~yz = g23 *yz, <*zx = gZl *zx

with ctj = CJU ga = gtf- The equation of heat conduction simplifies to

(5.4)

A further simplification occurs in the case of a transversely isotropic body. Such a body shows thermoelastic symmetry with respect to a plane, the x, y-plane say, and is isotropic in that plane, i.e. its equations are invariant under a rotation about the z-axis. Consequently c2 = ci, C23 = C 3 1 , £ 2 = £ 1 , £ 2 3 = £ 3 1 and & 2 = ki.

As with the isotropic body the equations of the nonisotropic body may be reduced to three generalized Navier's equations by substituting the stress-strain relations together with equation (2.4) into the equations of equilibrium (4.2).


The solution of equations (5.5) is cumbersome. Depending upon the geometry of the body series expansions or, in case of the infinite or semi-infinite body, integral transforms may be used. A few examples are available in the literature (see Refs. 22 and 23).

The case of plane strain or plane stress in an orthotropic body can be treated in a manner analogous to that of the isotropic body by using a generalized Airy stress function. Putting, for plane stress, a z z = 0 in the stress-strain relations one has

*xx = ai<jXx H

€yy — ci2\oxx

*xy — bi20xy

ayiVyy + a i #

(5.6)

where, in a more familiar notation,

(5.7) Substituting equations (5.6) into the equation of compatibility

For the orthotropic body there results

(5.5)

Equations of equilibrium (4.2) are identically satisfied by introducing repre-sentation (4.28) for the components of stress

utilizing equations (5.7) and choosing the thermoelastic potential 0 as a par-


The general solution of this equation may again be represented in complex form (cf. Ref. 15, chapters 6 and 9, and Ref. 24). Problems of the infinite and semi-infinite plate have been treated in the l i t e i a t u r e . 2 5 ' 2 6

6. S M A L L S T R A I N . N O N H O M O G E N E O U S B O D I E S

The main source of inhomogeneity in a body under non-uniform temperature is the variation of thermoelastic properties with temperature. The functional dependence of the coefficients in equations (2.5) and (2.6) as obtained from experiments is, in general, too complicated to make analytic solutions possible. Apart from a few special cases , 1 0 one has therefore to resort to approximations or numerical procedures. A frequently used approximation consists in replacing the variable coefficients by their mean values in the temperature range under consideration. However, it has been shown by Hi l ton 2 7 and others that this procedure may lead to considerable errors in the peak values of stress and strain. Fortunately, the variation of properties is usually small in the elastic range of the body.

7. S M A L L S T R A I N . N O N L I N E A R S T R E S S - S T R A I N

R E L A T I O N S

Elastic materials exist which deviate from Hooke's law even for small strain. An attempt has been made by Di l lon 2 8 to consider "mild nonlinearities" by expanding the free energy function 0 in a power series of the strain invariants and breaking off after the fourth power. N o applications to static problems are given.

The nonlinear stress-strain relations introduced by Kauderer have been used by J indra 2 9 in connection with thermal stress problems. A perturbation procedure is employed in the solution. Dependence of strain on temperature is—somewhat inconsistently—assumed to be linear.

A nonlinear stress-strain law suggested by Reiner 3 0 must also be mentioned here. N o application to thermoelastic problems has as yet been attempted.

8. L A R G E S T R A I N

If displacements are "finite" and/or temperature changes 6 are not small in comparison with the reference temperature some or all of the nonlinear terms in the basic equations have to be retained and equations of equilibrium and

ticular solution of

(5.8)

one is led to the following "quasi-biharmonic" equation for the Airy function W

(5.9)


heat conduction have to be referred to the deformed body. Exact solutions can then be found only in the very limited number of cases where marked simpli-fications are possible. Unfortunately those cases are mostly of a purely academic nature.

Apart from the Ritz-Galerkin method or from purely numerical procedures the only method of a sufficiently general nature that has been developed so far seems to be the method of successive approximations.2

The basic assumptions of the method a re : (a) the displacement vector u% is an analytic function of some parameter e of the problem and can be expanded as an absolutely convergent series in e with some radius of convergence c 0 > 0, (b) the series is at least twice termwise differentiable with respect to the co-ordinates xt, the resulting series being absolutely convergent with the same radius of convergence.

Let

Then, from equation (1.2)

(8.1)

(8.2)

where

(8.3)

Equation (1.10) goes over into

(8.4)

In particular, for a body initially homogeneous and isotropic and obeying Hooke's law we have in analogy to equation (2.7)

Practical applicability of equation (8.5) is limited to small components of strain (but not to small deformations!). Hence temperature increase 6 has to be small too. Expansion (8.4) reads then, with 6 = €0i,

(8.6)

(8.5)

S O L U T I O N OF T H E R M O E L A S T I C B O U N D A R Y V A L U E P R O B L E M S 3 3 5

Equations of equilibrium with zero body forces are from equation (1.1)

(8.7)

From the fact that the equations in each of the approximation steps are identical with the equations of classical linear thermoelasticity it follows that their solution is unique. Hence the series solution, if it exists, is unique. 2

As a simple example consider a cube initially of length a and exposed to a temperature rise S. The cube is free to expand in the y- and z-direction while the distance between the two surfaces x = 0 and x = a is kept fixed. The exact solution of the problem is easily obtained.

Letting

u = 0, v = fiy, w = fiz one finds

fl2

eXx = 0, eyy =• ezz = ft + 1—

From equation (8.5) there follows

$xy — S y z — SZx — 0

Since all stress components are constant the equations of equilibrium are satisfied and one finds from the boundary condition syy = 0 for the unknown quantity 0

0 = ~ 1 + Vi1 + 2(1 + v) aS]

In the approximation procedure outlined above we choose equation (8.1),

= y9 w*1) = z, all others zero,

and from equations (8.3) and (8.6)

e = 0. Then, from

The stress vector fh measured per unit area of the undeformed body and acting on a surface in the deformed body whose unit normal in the underformed state was ru is

(8.8)


All terms of higher order vanish. Putting / fo^ + fPs®* = 0 renders j8 which, in this particular case, agrees with the exact solution.

Green and Adkins 2 have applied the method to problems of plane stress and plane strain using the complex function representation. In that case, however, it is more convenient to take x9 y as being cartesian coordinates in the body after deformation. A generalized Airy stress function can then be introduced and expanded into a power series in the parameter e together with the complex displacement vector u + iv.

9. S T A B I L I T Y

If sufficiently large compressive thermal stresses appear in thin-walled struc-tures such as slender bars, thin plates or shells, equilibrium may become unstable, and buckling may occur. A reduction in Young's modulus E with elevated temperature may also be present and may contribute to the phenomenon.

Buckling and postbuckling behavior are due to nonlinear, second-order effects. The linearized thermoelastic equations, therefore, are inadequate for treating problems of this type.

Gossard, Seide and Rober t s 3 1 have been the first to study thermoelastic buckling of a thin plate exposed to a temperature distribution 9{x9 y) constant over the thickness of the plate. Let oy{x9y) be the corresponding thermal stresses, equation (5.7). If 9 is now increased to a value X6(x9y) the plate will buckle provided A corresponds to an eigenvalue of the homogeneous equation for the finite deflection w(x9 y) of a thin plate

(9.i;

where h is the plate thickness. The associated homogeneous boundary conditions are either of the kinematic type (involving only displacement and rotation) or of the dynamic type (involving bending moment and Kirchhoff shearing force).

In contrast to isothermal buckling problems the coefficients a y in the plate equation are, in general, not constants. Only in very rare cases is an exact solu-tion therefore possible. As an approximation procedure the method of Ri tz -Galerkin may be used. The deflection pattern w(x9 y) is assumed in the form

(9.2)

where the suitably chosen functions wt(x,y) satisfy at least the kinematic boundary conditions. A set of r linear and homogeneous equations in the co-efficients a% is then obtained

(9.3)

where m„ and q are unbalanced bending moment and shearing force, respec-


tively, along the edge of the pla te :

(9.4)

Putting the determinant of these equations equal to zero yields upper bounds for the desired eigenvalues A. Examples may be found in the literature, e.g. Refs. 7 and 32, chapter 13; postbuckling behavior is also briefly discussed in these references.

Bimetallic strips or plates of small initial curvature "snap" , at a certain critical uniform temperature, into a new position of equilibrium. 7 ' 3 3 In thermostats this phenomenon has found its engineering application.

In equation (9.1) as well as in other related problems treated in the literature it has been assumed that deformations before buckling are sufficiently small to be described by the linearized thermoelastic equations. This is not always true. In such cases the method has to be based on the so-called theory of "small deformations superposed on large ones" as outlined in Ref. 15, chapter 4, and Ref. 2, chapter 9. A description of the method with particular reference to thermoelasticity has been given by England and Green 3 4 for the initially isotropic body and by Green 3 5 for the initially nonisotropic body.

Quasistatic Problems

We consider now temperature fields which are non-stationary. In such fields not only temperature but also stress and strain are functions of time. Hence the problem is, in principle at least, one of dynamics. However, heat propaga-tion within a body is a relatively slow process, and inertia effects may there-fore, in general, be neglected. This has already been conjectured by Duhamel and has been born out more recently by a number of researchers for a wide class of problems (Ref. 7, chapter 2). The corresponding theory which is termed "quasi-static" will be treated in this chapter. "Dynamic" problems will be dis-cussed in the last section of this survey.

In a non-stationary field the equation of heat conduction (1.13) is no longer independent of the remaining equations. The first term on the right-hand side of this equation—the "coupling term"—shows that variations of strain within a body are accompanied by variations of temperature. This phenomenon is known as thermoelastic dissipation. But again, as with inertia effects, it appears plausible to disregard coupling for all problems of small strain except where thermoelastic dissipation is of primary interest, see Section 13. It m a y b e stated quite generally that inertia effects and coupling effects are closely related and neglection of the one usually justifies neglection of the other.

10. U N C O U P L E D Q U A S I - S T A T I C T H E O R Y

The only difference between this theory and the static case is in the equation

of heat conduction which now reads for a homogeneous and isotropic linear solid,


equation (2.10),

(10.1)

Equations (4.7) to (4.11) remain unchanged. Application of a Laplace trans-form, therefore, leads to a boundary value problem of thermoelastic statics in the image space. Hence there is no difference, from a mathematical point of view, between static and uncoupled quasi-static thermoelasticity, at least for homogeneous bodies. All complications introduced by non-stationary characters are shifted into the inversion problem.

A great variety of uncoupled quasi-static thermoelastic problems have been treated in the literature (see Refs. 7, 10, 11).

The method of the Laplace transform fails in the case of large deformations. The second step in the procedure of successive approximations already leads to differential equations with time-dependent coefficients. Moreover, inertia and coupling effects are, in general, no longer negligible in this case.

11 . S T O C H A S T I C F I E L D S

Not always is the temperature field known in a deterministic sense. It may happen for instance that only statistical information is available about the

Using this equation a particular solution of equation (4.5) for the thermo-elastic potential may be found immediately in the form

(10.2)

#o is the initial value of 0, corresponding to the initial temperature To, and 0± is an arbitrary harmonic function. As in the static case a solution in terms of harmonic or biharmonic functions has to be superposed on the solution corresponding to 0 in order to satisfy the boundary conditions.

As long as temperature change 6 and deformation remain small, non-stationary thermoelastic problems are linear initial-value problems. A very effective tool for attacking problems of this kind is the Laplace transform. Denoting the trans-form of a function F(t) by an asterisk, i.e.

and assuming temperature change, stresses and displacement initially zero one obtains from equations (10.1) and (10.2)

(10.3)

(10.4)

and

where ptj(P. Q. f) is the stress at point P produced by a dipole of strength a at point Q. The dipole appears at time f = 0 and its axis coincides with the surface normal n. From equation (11.3) there follows

£ { a y } = 0, E{6} = 0 (11.5)

Forming now the correlation function of cry one finds from equations (11.1), (11.2) and (11.4)

R** (P, si, s2) = R*r* (si, s2) A\P, Sl) A*(P, s2) (11.6)

R** denotes the two-dimensional or double Laplace transform of the cor-relation function Rx(h, t2) of a function x(t)

one has

F{Q)P%(P,Q,s)dS A*(P,.

For the particular case where G{t) is a white noise random function

(11.7)

(11.4) Q*(Q,s)Pfj(P,Q,s)dS

The corresponding thermal stress field can be represented in the form

(11.3) £ { G ( 0 } = 0

where (7(f) is a random function with zero expectation

(11.2) £>(Q, t) = F{Q) G(t)

Q(Q, t) is the surface temperature for t > 0 at a generic point Q of the surface S of the body, and T is the Green's function of the temperature field. Let Q be given in the form

(11.1)


temperature distribution on the surface of the body. Temperature and thermal stress fields are then random solutions of the differential equations of thermo-elasticity.

Consider the linear case of small deformations. 3 6 The Laplace transform of the temperature at a generic point P within the body is


and hence from equation (11.6)

Other examples may be found in Ref. 36.

Dynamic Problems

In contrast to static and quasi-static problems dynamic problems require the solution of the equations of motion (1.1). Methods of some generality are available only in the linear theory. In the nonlinear case where finite deforma-tions occur the method of successive approximations, section 8, may be used.

1 2 . L I N E A R U N C O U P L E D E Q U A T I O N S

After elimination of ay and ey one obtains for the equations of motion in terms of displacements

(12.1)

as generalization of the static equation (4.3). The equation of heat conduction remains in the form (10.1) and may be solved independently.

In order to solve equation (12.1) one proceeds in a manner analogous to the static case 1 1 > 3 7 First the thermoelastic potential is introduced with the aid of equation (4.4). Substitution into equation (12.1) renders

(12.2)

(12.3)

where

ci represents the speed of propagation of isothermal dilatational waves. Upon the particular solution of equation (12.1) represented by & the general solution of the homogeneous equation

(12.4)

has to be superposed. As in the static case this solution may be expressed in terms of three functions xt in the form

; i2.5;

However, the xt a r e now solutions of the repeated wave equation

(12.6)


C2 = \/(G/p) is the speed of the shearing waves. The functions xt may t>e con-sidered as generalizations of the Galerkin-Westergaard functions of equi-librium. Sternberg and Eubanks 3 8 have proved the completeness of this solution and, in addition, have shown that every solution of the repeated wave equation (12.6), provided c2 c±, admits the representation

Xt = fa + °>i (12.7) where

A1<f>i = 0, A2a>i = 0 (12.8)

The general solution of equation (12.1) may therefore be written

(12.9)

A third form is obtained by using generalizations of the Neuber-Papkovich functions. Writing as in equation (4.9)

one finds that the functions fa and W = xtfa + now satisfy the equations

(12.11)

(12.10)

î0o + xiAxfa = 0, A2fa = 0

Instead of expressing the equations of motion in terms of the displacement components one may write them in terms of the stress components , 3 9 by eliminat-ing m and €y from the basic equations.

The solution of equations (12.2), (12.6) or (12.11) is greatly facilitated by employing the Laplace transform. For instance, equation (12.2) transforms into

Putting

<P* = CO*

and utilizing equation (10.3) one has at once the solution

(12.12]

With #* known this can be transformed back into the original space. Another method of solving equation (12.2) makes use of the retarded poten-

tial.^ The "classical" problem in uncoupled dynamic thermoelasticity—the semi-

space z > 0 exposed to a uniform thermal shock on its surface—was first solved by Danilovskaya 4 0 . Let the temperature on the surface z — 0 be given by

6 = H{t)

where H{t) is the Heaviside step function. Equation (10.1) then has as solution


for / > 0

The equation for uniaxial motion of the solid may be written in terms of the normal stress azz

Applying a Laplace transform and utilizing the boundary condition azz = 0 on z = 0 leads to the solution

where

One notices a stress wave travelling from the surface into the body with speed d. After the wave front has passed a generic point in the body the stress at this point decreases rapidly to its quasi-static value

vzz = 0, axx = (jyy = Eadj{\ — v)

The case of plane strain has been studied by R a d o k 4 1 and Nowacki 4 2 . The equations of motion are satisfied by introducing a generalized Airy stress function W such that

(12.13)

W now satisfies the repeated wave equation

(12.14)

with d/dz == 0. As in the static case W may be expressed in terms of two complex analytic functions. However, these functions are now of a type similar to that occurring in the plane theory of orthotropic bodies . 4 1

If the temperature field varies periodically stress and strain will, after a suffi-ciently long time, also assume periodic values. Putting (as in Ref. 11, p . 71),

d = Oi cos cot + 62 sin at, <P = 0i cos cot + 02 sin cot (12.15)


and substituting into equation (12.2), one obtains for the two components of the thermoelastic potential

(12.16)

The functions \% and fa may be represented in a similar form. Since solutions of the wave equation are now involved it should be remembered that, in the case of an infinite body, the Sommerfeld radiation condition has to be satisfied at infinity (Ref. 11, p . 103).

Periodic solutions for plain strain have been obtained by Ignaczak and Now-acki 4 3 .

1 3 . L I N E A R C O U P L E D E Q U A T I O N S

The coupling term is now retained in the equation of heat conduction (2.10). Together with the equation of motion (12.1) we have then the following set of two equations in the three displacement components u% and the temperature change 6,

(13.1

(13.2)

To is the uniform reference temperature. Equations (13.1) and (13.2) may be solved with the aid of displacement

functions in the same manner as equation (12.1). A better insight into the structure of the solution, however, is gained by following the theory of elastic waves and resolving the displacement vector into an irrotational and a solenoidal component:

(13.3)

Equations (13.1) and (13.2) then render a set of three equations

(13.4)

u = V 0 + V x ip

(13.5)

(13.6)

A2fa = 0

The thermoelastic potential 0 describes dilatational waves with particle motion parallel to the direction of propagation. The vector potential fa corresponds to shearing waves which produce no volume change. Equations (13.5) and (13.6) show that the shearing waves are independent of the heat-conduction properties of the body.

On combining equations (13.4) and (13.6) it is found that both 0 and 6


(13.11)

where fx± and M2 are obtained from Ai and A2 by replacing s by ito. Putting Mi, 2 = qi, 2 + i(<*>/vi, 2) we see that the real parts of MI, 2 represent the attenu-ation constants of the two constituent waves while the reciprocals of the imaginary parts are proportional to the corresponding phase velocities vit 2.

are solutions of the same differential equation

(13.7)

The simplest case is that of a plane wave where u% and 0 depend on x and t only. Omitting the independent shear waves w2 and t/3 from the investigation equations (13.1) and (13.2) reduce to

(13.8)

(13.9)

The same equations hold for the propagation of thermal stress waves in a thin rod (uniaxial stress) provided c\ is replaced by E/p and v is replaced by zero. The two cases have been studied by Deresiewicz 4 4 , Chadwick and Sneddon 4 5 , Sneddon 4 6 and Ignaczak 4 7 for various initial and boundary conditions. A thorough discussion of the salient features of the solution is given by Chad-wick 4 8 and by Nowack i 1 0 .

If a Laplace transform is applied to equations (13.8) and (13.9) one finds for the transforms w* and 0* in a semi-infinite body or a semi-infinite rod x > 0

(13.10)

0*(x, s) = Ae~\x + Be~\x

A and B are integration constants to be determined from the boundary con-ditions at x = 0, and

when

The real parts of Ai and A2 are bo th taken to be positive. In the case of time-periodic boundary conditions the periodic part of the

solution is given by

9(x, t) = (Ae~Hx + Be~V) eio)t


Solutions of equations (13.4) to (13.6) are now sought in the form

{0, 0, ifj} = { 0 ( z ) , *(z), 0(z)} exp i(vx + tot)

which, on the boundary z — 0, satisfy the condition of vanishing surface stresses and the linearized condition of heat radiation. All quantities are to approach zero as z -> oo.

A similar form of the solution with boundary conditions prescribed in z = =b a has been used by Nowacki and Sokolowski 4 9 in a study of thermo-elastic waves in plates.

Thermoelastic waves due to the action of heat sources in an infinite and semi-infinite body have been studied by Eason and Sneddon 5 0 , Nowack i 5 1 and Locket t 5 1 . References to other problems may be found in the paper by Chad-wick 4 8 . The infinite body with cylindrical and spherical cavities has been in-vestigated by Ignaczak and Nowack i , 5 3 who also studied periodic solutions for plane s t ra in . 4 3

The generalization of Danilovskaya's problem to include coupling has been given by Hetnarsk i 5 4 with the aid of the Laplace transform and, independently, by Boley and Tol ins 5 5 using Fourier integral representation of discontinuous functions.

Plane thermoelastic waves superimposed on large uniform extensions in a homogeneous isotropic solid have been treated by Flavin and Green , 5 6 while Raleigh waves have been studied by F lav in 5 7 .

R E F E R E N C E S

1 . KAPPUS, R. "Zur Elastizitatstheorie endlicher Verschiebungen" Z. ang. Math. Mech. 19, 2 7 1 ( 1 9 3 9 ) .

2 . GREEN, A . E. and ADKINS, J. E. Large Elastic Deformations. Oxford: Clarendon Press ( 1 9 6 0 ) .

3 . BIOT, M. A . "Thermoelasticity and Irreversible Thermodynamics" / . Appl. Phys. 2 7 , 2 4 0 ( 1 9 5 6 ) .

4 . ADKINS, J. E. "Large Elastic Deformations" Progress in Solid Mechanics, Vol. II. Amster-dam ( 1 9 6 1 ) .

5 . GRIOLI, G . Mathematical Theory of Elastic Equilibrium. Berlin: Springer-Verlag ( 1 9 6 2 ) . Chapter III.

6 . WEINER, J . H . "A Uniqueness Theorem for the Coupled Thermo-elastic Problem" Quart. Appl. Math. 15, 1 0 2 ( 1 9 5 7 ) .

7 . BOLEY, B. A . and WEINER, J . H . Theory of Thermal Stresses. N e w Y o r k : J . Wiley ( 1 9 6 0 ) .

Explicit formulas for q±t 2 and vi, 2 in the form of series expansions have been given by Chadwick 4 8 .

Thermoelastic Rayleigh waves, i.e. surface waves propagated in the x-direction in the half-space z > 0, have been investigated by Chadwick 4 8 . Putting 0 X = 03 = 0, 02 = *A in equation (13.3) and assuming all quantities inde-pendent of y one has


8. CARSLAW, H. S. and JAEGER, J . C. Conduction of Heat in Solids. 2nd ed. Oxford: Clarendon Press (1959).

9. MELAN, E. and PARKUS, H. Warmespannungen zufolge stationdrer Temperaturfelder. Vienna: Springer-Verlag (1953).

10. NOWACKI, W. Thermoelasticity. Warsaw and London (1962). 11. PARKUS, H. Instationdre Warmespannungen. Vienna: Springer-Verlag (1959). 12. GOODIER, J . N . "On the Integration of the Thermoelastic Equations" Phil. Mag. VII,

23, 1017 (1937). 13. SNEDDON, I. N . and BERRY, D . S. "The Classical Theory of Elasticity" Encyclopedia of

Physics, Vol . VI. Berlin (1958). 14. EUBANKS, R. A . and STERNBERG, E . "On the Completeness of the Boussinesq-Papkovich

Stress Functions" / . Rational Mech. Anal. 5 , 735 (1956). 15. GREEN, A . E. and ZERNA, W. Theoretical Elasticity. Oxford: Clarendon Press (1954). 16. STERNBERG, E. and MCDOWELL, E . L. "On the Steady-State Thermo-elastic Problem for

the Half-Space" Quart. Appl. Math. 1 4 , 381 (1957). 17. LOVE, A . E. H. Mathematical Theory of Elasticity. 4th ed. N e w York: Dover (1944). 18. SNEDDON, I. N . "Solutions of the Equations of Thermoelastic Equilibrium" Arch. Mech.

Stos. 1 4 , 113 (1962). 19. OLEASIAK, Z . and SNEDDON, I. N . "The Distribution of Thermal Stress in an Infinite

Elastic Solid Containing a Penny-Shaped Crack" Arch. Rational Mech. Anal. 4 , 238 (1960).

20. SNEDDON, I. N . Fourier Transforms. N e w York: McGraw-Hil l (1951). 21 . MUSKHELISHVILI, N . I. Some Basic Problems of the Mathematical Theory of Elasticity.

Groningen, Hol land: P. Noordhoff Ltd. (1953). 22. SHARMA, B. "Thermal Stresses in Transversely Isotropic Semi-Infinite Elastic Solids"

J. Appl. Mech. 2 5 , 87 (1958). 23. MOSSAKOWSKA, Z . and NOWACKI, W. "Thermal Stresses in Transversally Isotropic Bodies"

Arch. Mech. Stos. 1 0 , 569 (1958). 24. SAVIN, G. N . Stress Concentration around Holes. N e w York: Pergamon Press (1961). 25. MOSSAKOWSKI, J . "The State of Stress and Displacement in a Thin Anisotropic Plate

due to a Concentrated Source of Heat" Arch. Mech. Stos. 9 , 565 (1957). 26. NOWACKI, W. "Thermal Stresses in Orthotropic Plates" Bull. Acad. Pol. 7 , 1 (1959). 27. HILTON, H. H. "Thermal Stresses in Bodies Exhibiting Temperature-Depending Elastic

Properties" / . Appl. Mech. 1 9 , 350 (1952). 28. DILLON, O. W. "A Nonl inear Thermoelasticity Theory" / . Mech. Phys. Solids, 1 0 , 128

(1962). 29. JINDRA, F . "Warmespannungen bei einem nichtlinearen Elastizitatsgesetz" Ing.-Archiv,

2 8 , 109 (1959). 30. REINER, M. "Rheology" Encyclopedia of Physics, VI, 507. Berlin (1958). 31. GOSSARD, M. L., SEIDE, P. and ROBERTS, W. M. "Thermal Buckling of Plates" NACA

T. N. 2771 (1952). 32. VON DER NEUT, A . "Buckling Caused by Thermal Stresses" High Temperature Effects in

Aircraft Structures. London (1958). 33. BAUMANN, W. "Beitrag zum Durchschlagsproblem kreisformiger Bimetallplattchen"

Oesterr. Ing.-Archiv, 1 4 , 161 (1960). 34. ENGLAND, A . H. and GREEN, A . E. "Steady-State Thermoelasticity for Initially Stressed

Bodies" Phil. Trans. Roy. Soc. Lond. (A), 2 5 3 , 517 (1961). 35. GREEN, A . E. "Thermoelastic Stresses in Initially Stressed Bodies" Proc. Roy. Soc. (A) ,

2 6 6 , 1 (1962). 36. PARKUS, H. "Warmespannungen bei zufallsabhangiger Oberflachentemperatur" Z. ang.

Math. Mech. 4 2 , 499 (1962).

37. NOWACKI, W. "The Three-Dimensional Problem of Thermoelasticity" Bull. Acad. Polo-naise Sc., s. techn. 9 , 419 (1961).

38. STERNBERG, E. and EUBANKS, R. A . "On Stress Functions for Elastokinetics and the Integration of the Repeated Wave Equation" Quart. Appl. Math. 1 5 , 149 (1957).

39. IGNACZAK, J . "Direct Determination of Stresses from the Equations of Mot ion in Elasti-city" Arch. Mech. Stos. 1 1 , 671 (1959).

40. DANILOVSKAYA, V. I. "Thermal Stress in the Elastic Semi-Space D u e to a Sudden Heating of the Surface" (in Russian), Prikl. Math. Mekh. 1 4 , 316 (1950).


4 1 . RADOK, J. R. M. "On the Solution of Problems of Dynamic Plane Elasticity" Quart. Appl. Math. 1 4 , 289 (1956).

42. NOWACKI, W. "The Plane Non-Coupled Dynamic Problem of Thermo-elasticity" Bull. Acad. Polonaise Sc., s. techn. 9 , 427 (1961).

43. IGNACZAK, J. and NOWACKI, W. "The Plane Dynamic Problem of Thermoelasticity" Proc. Vibration Problems 2 , 355 (1961).

44. DERESIEWICZ, H. "Plane Waves in a Thermoelastic Sol id" / . Acoust. Soc. Amer. 2 9 , 204 (1957).

45. CHADWICK, P. and SNEDDON, I. N . "Plane Waves in an Elastic Solid Conducting Heat' /. Mech. Phys. Solids, 6, 223 (1958).

46. SNEDDON, I. N . "The Propagation of Thermal Stresses in Thin Metallic R o d s " Proc. Roy .Soc. Edinburgh Sec. (A), 65, 121 (1959).

47. IGNACZAK, J. "Note on the Propagation of Thermal Stresses in a Long Metallic R o d " Bull. Acad. Polonaise Sc., s. techn. 7, 309 (1959).

48. CHADWICK, P. "Thermoelasticity. The Dynamica l Theory" Progress in Solid Mechanics, 1 , 265. Amsterdam (1960).

49. NOWACKI, W. and SOKOLOWSKI, M. "Propagation of Thermoelastic Waves in Plates" Arch. Mech. Stos. 1 1 , 715 (1959).

50. EASON, G . and SNEDDON, I. N . "The Dynamic Stresses Produced in Elastic Bodies by Uneven Heating" Proc. Roy. Soc. Edinburgh, 65, 143 (1959).

51. NOWACKI, W. "Some Dynamic Problems of Thermoelasticity" Arch. Mech. Stos. 1 1 , 259 (1959).

52. LOCKETT, F . J. "The Propagation of Thermal Stresses in a Semi-Infinite Medium" Proc. Edinburgh Math. Soc. (2), 1 2 , 75 (1961).

53. IGNACZAK, J. and NOWACKI, W. "The Sommerfeld Radiation Conditions for Coupled Problems of Thermoelasticity" Arch. Mech. Stos. 1 4 , 3 (1962).

54. HETNARSKI, R. "Coupled One-Dimensional Thermal Shock Problem for Small Times" Arch. Mech. Stos. 1 3 , 295 (1961).

55. BOLEY, B. A . and TOLINS, I. S. "Transient Coupled Thermoelastic Boundary Value Problems in the Half-Space" / . Appl. Mech. 2 9 , 637 (1962).

56. FLAVIN, J. N . and GREEN, A . E. "Plane Thermoelastic Waves in an Initially Stressed Medium" / . Mech. Phys. Solids, 9 , 179 (1961).

57. FLAVIN, J. N . "Thermo-elastic Rayleigh Waves in a Prestressed Medium" Proc. Cambridge Philos. Soc. 58, 532 (1962).

ON THE ANALYSIS OF THERMAL STRESSES IN

VISCOELASTIC SOLIDS*

E L I STERNBERG

Brown University Providence, R.L


The study of thermal stresses and deformations in viscoelastic solids has, for various technological reasons, attracted growing attention during the past few years. This paper, which is devoted exclusively to solids that under isothermal conditions and for infinitesimal strains exhibit linear viscoelastic behavior, is intended to serve a dual purpose. First, an attempt is made to sketch a systematic account of relevant recent theoretical developments; second, we hope to supply a helpful, if necessarily incomplete, guide to available specific results and to the literature on the subject at hand.

The present treatment is largely confined to the quasi-static analysis of thermal stresses in homogeneous and isotropic, linear viscoelastic media. Questions regarding inhomogeneous and anisotropic solids, inertia effects, and thermo-mechanical coupling effects, are touched upon only at the end of the paper (Section 7).

The first part of the paper (Sections 2, 3, 4) deals with thermo-viscoelasticity theory within the assumption that the mechanical response of the material is temperature-independent. Since the rate processes of viscoelasticity are known to be highly sensitive to temperature changes, this assumption is remote from physical reality. The theory of temperature-independent materials would never-theless appear to merit attention as a useful preliminary to a more realistic treatment of the problem and because it has in fact been taken as the basis of numerous particular investigations.

Subsequent portions of the paper (Sections 5, 6) deal with temperature-dependent viscoelastic solids, particular emphasis being placed on the theory of thermo-rheologically simple materials. This theory rests on the hypothesis that a constant temperature change of the entire solid affects its mechanical response merely within a uniform distortion of the time scale. It should be apparent that the smaller amount of space devoted here to the influence of temperature-dependent response characteristics does not reflect the relative importance of

* This paper was prepared under Contract Nonr-562(25) of Brown University with the Office of Naval Research in Washington, D . C . , for the International Conference on High Temperature Structures and Materials at Columbia University, N e w York, 1963.

348

A N A L Y S I S OF T H E R M A L S T R E S S E S 349

this influence but rather the increased complexity of the problem once the temperature-dependence of the material is taken into account.

Although throughout the following discussions some attention is given to stress-strain relations in differential-equation form, our primary preoccupation is with the more general integral laws appropriate to solids with a continuous spectrum of relaxation or retardation times. Such a preference is motivated by the well-known fact that the finite models governed by the differential operator law (despite their heuristic value and traditional popularity) provide an inade-quate basis for the description and prediction of actual viscoelastic behavior over any appreciable range of time or frequency. This shortcoming is even more pronounced in temperature-dependent models.

The following treatment of thermo-viscoelasticity theory is strongly influenced by, and draws heavily on the results obtained in an earlier study of the iso-thermal theory contained in Ref. 1. In particular, we make extensive use of the properties of Stieltjes convolutions established in Ref. 1, which enable us to cope economically with field histories that are discontinuous in time. As emphasized in Ref. 1, the algebra and calculus of convolutions furnish the natural basic tool of linear viscoelasticity theory. In contrast, an excessive reliance on integral transforms has occasionally tended to obscure the issues under consideration. Some of the theoretical results included in this paper, though elementary in character, seem to be unavailable elsewhere. In this sense the present paper is not wholly expository.

2 . T E M P E R A T U R E - I N D E P E N D E N T M E C H A N I C A L R E S P O N S E :

F O R M U L A T I O N O F B O U N D A R Y - V A L U E P R O B L E M S

The fundamental system of field equations governing the quasi-static linear theory of viscoelastic solids, in the absence of thermo-mechanical coupling effects, consists of the linearized displacement-strain relations, the stress equa-tions of equilibrium, and the appropriate linear hereditary stress-strain law. We refer this system of equations to rectangular cartesian coordinates xi and adopt the usual indicial notation.*

Let ut(x, r), €tj(x, t), oij(x, t) be the component values of the displacement, infinitesimal strain, and stress field histories at a material point with the position vectorf x and at the time t. The displacement-strain relations then become

= i(ut, j + Uj, i) (2.1)

whereas the stress equations of equilibrium assume the form

a i J , j + Ft = 0, (Tji = (Ttj (2.2)

provided Ft denotes the components of the body-force density field history.

* Latin subscripts, unless otherwise specified, have the range of the integers (1, 2, 3 ) ; sum-mation over repeated subscripts is implied and subscripts preceded by a c o m m a indicate partial differentiation with respect to the corresponding cartesian coordinate.

t Letters in bold face designate vectors.

350 ELI S T E R N B E R G

With a view toward a convenient statement of the relevant constitutive rela-tions for isotropic viscoelastic solids we introduce the deviatoric components of strain and stress by means of

(2.3)

in which Sy is the Kronecker delta. Next, we designate by T(x, t) the local instantaneous temperature and let To be an arbitrarily chosen reference tem-perature. We then call the function 0 defined by

(2.4) 0 = T - To

the temperature field history. Finally, we write a for the coefficient of thermal expansion, which we shall suppose to be constant. If instead a is a function of the temperature, all of the succeeding considerations undergo an entirely elementary generalization: in this instance one need merely replace a by a 0 = a(7o) and (2.4) by

(2.5)

The appropriate linear isotropic hereditary stress-strain relations in the form

of the relaxation integral law now appear as

(2.6) Sij = etj*dGi, Vkk = (*kk — 3aO)*dG2

Here Gi and G2 designate the respective relaxation moduli in shear and iso-tropic compression. These moduli are at present functions of the time exclusively since we confine our attention to homogeneous solids and assume until later on that the material's mechanical response is temperature-independent. In writing (2.6) we have employed a notation for Stieltjes convolutions introduced previously 1 in connection with the isothermal theory. Thus, if / and g arc functions of position and time, a> = /*dg stands for the function defined by the Stieltjes integral

(2.7)

provided this integral is meaningful. We observe that according to (2.6), the local stress tensor at each fixed

instant is a linear, continuous, and isotropic functional of the entire preceding local strain and temperature histories; moreover, this mapping of the strain and temperature histories into an associated history of stress is invariant under a translation of the time scale and has the property that the stresses induced by a free thermal expansion vanish identically. One can show* (under suitable regularity assumptions) that (2.6) is the most general stress-strain law con-forming to the foregoing requirements.

* See Ref. 1, Section 2, for the corresponding result in the isothermal theory.


The field equations ( 2 . 1 ) , ( 2 . 2 ) , ( 2 . 6 ) must hold throughout the space-time domain* 0t x (— oo, oo), i.e. for all (x, i) such that x is in the open region occupied by the interior of the body and t lies in the interval (— oo, oo). To these field equations we may, without essential loss of generality, adjoin the requirement that the body is originally undisturbed in the sense of the initial conditions

U i = eij = ay = Fi = G = 0 on R X ( - oo, 0) ( 2 . 8 )

where JR stands for the closure of @t, i.e. for the union of & and its boundary. Finally, in the case of the standard mixed boundary-value problem, the boundary conditions become

ut = u\ on B\ X (— oo, oo)

St = Gijnj = 5? on ft X ( - oo, oo)

Here Bi and B2 are complementary subsets of the boundary B of 0t, nj is the outward unit normal of B, whereas u\ and S\ are prescribed surface displace-ments and surface tractions, respectively.

In the absence of an explicit statement to the contrary, the point sets R, B±, B2

will henceforth be assumed to be independent of the time. Further, we shall suppose from here on that R is a bounded regular region of spacef and that Bi, B2 are both integrable.

The problem under consideration thus consists in determining field histories uu c y , a y which—for given R, B\, B2, known Gi, G2, a , and prescribed Ft, @, u\, Sj—satisfy the field equations ( 2 . 1 ) , ( 2 . 2 ) , (2 .3) , ( 2 . 6 ) in ® x ( - oo, oo) and meet the initial conditions ( 2 . 8 ) , as well as the boundary conditions ( 2 . 9 ) . The temperature field history © may, in particular, be specified as the solution of an independent heat-conduction problem.

For future economy we now adopt the following definition of a

Viscoelastic State

We say that the ordered array of field histories [/x̂ , a y ] , belongs to the class of viscoelastic states on Rx(— oo, G O ) corresponding to the data Gi, G2, a, Fi, ©, and write

[m, €tj, a y ] e V[Gh G2, a, Fu @] on Rx ( - oo, oo) ( 2 . 1 0 )

if: (a) Gp(fi = 1 , 2 ) vanishes on ( — oo, 0), is twice continuously differentiable

on [0, oo), and G / 0 ) > 0;

(b) Ui, € y , a y , Fi, S vanish on R x (— oo, 0) and are continuous onRx [0, oo), 8 being once—and u% three times continuously differ entiable on R X [0, oo);

* We use the conventional notation A x B for the cartesian product of a set A and a set B.

t By a "regular region of space" we mean a region whose boundary consists of a finite number of non-intersecting "closed regular surfaces", the latter term being used in the sense of Kel logg 2 .

( 2 . 9 )


(c) equations ( 2 . 1 ) , ( 2 . 2 ) , ( 2 . 3 ) , ( 2 . 6 ) hold in St x (— oo, oo).

If in particular, 0 = 0 on Rx(— oo, oo), we say that the state \u%, <r#] is isothermal and write

[ut, €ij9 an] e V [Gi, G 2 , Ft] on R x (— oo, oo) ( 2 . 1 1 )

Requirements (a), (b), (c) are clearly partly redundant but mutually con-sistent. Also, the smoothness assumptions contained in (a), (b) could be relaxed (especially as far as the time-dependence is concerned) at the expense of more elaborate regularity hypotheses. Such refinements would, moreover, necessitate occasional supplementary smoothness assumptions in subsequent theorems and would merely tend to detract from the main purpose of the present paper.

It is essential to observe that the field histories involved in the preceding definition of a viscoelastic state may evidently exhibit finite jump discon-tinuities at t = 0. Although such singular field histories are physically un-realistic—particularly within the context of the quasi-static theory—they play an important part in the theory of integration of the governing field equations because of Duhamel's principle. One of the advantages derived from the for-mulation of the constitutive relations ( 2 . 6 ) in terms of Stieltjes convolutions lies in the fact that it permits a systematic treatment of the relevant discon-tinuities and enables one to avoid the usual purely formal manipulations with Dirac's delta function.

If ( 2 . 1 0 ) holds, the stress-strain law ( 2 . 6 ) admits the conventional Riemann integral representation (see Ref. 1, Theorem 3 . 4 ) , valid for (x, t) in R X [0, oo),

°kk(x, t) = G2(t)[ikk(x) - 3aS(x)] ( 2 . 1 2 )

Here, as in the sequel, / stands for the first time derivative of a function / of position and time, whereas

/ ( x ) = / ( x , 0 ) ( 2 . 1 3 )

We consider next the case of an elastic solid. To this end let h henceforth denote the Heaviside unit step function defined by

h(t) = 0 for - oo < t < 0 1 h(t) = 1 for 0 < t < oo j

It is immediate from ( 2 . 1 2 ) that for the particular choice

G! = 2fJLh, G2 = 3xh ( 2 . 1 5 )

where ^ and K are constants, the stress-strain relations ( 2 . 6 ) pass over into

Sij = 2jieij, ajck = 3x(ekk ~ 3a(9) ( 2 . 1 6 )

G2(t - t')[ikk(x, tf) - 3a@(x, O] dt'

A N A L Y S I S O F T H E R M A L S T R E S S E S 353

Equations (2.16), in view of (2.3), are equivalent to Hooke's law (modified to account for thermal expansion), provided /x and K are the shear modulus and the bulk modulus of the elastic material, respectively. This remark reveals the role of the classical quasi-static theory of thermoelasticity as a special case of the theory of thermo-viscoelasticity under consideration and motivates the following definition.

Quasi-static Elastic State

If (2.10) holds and the relaxation functions obey (2.15), in which \i and K are (positive) constants, we say that [in, <ry] belongs to the class of quasi-static elastic states on Rx(— oo, oo) corresponding to the data /x, K, a, Ft, @, and write

[ut, etj, on] e & [ f t , K , a, Ft, 0] on Rx(— oo, oo) (2.17)

It is clear from the above definition that a quasi-static elastic state which fails to vanish identically on Rx(— oo, oo), cannot be independent of time on this entire space-time domain. For, (2.17) implies that m, ay must meet the initial conditions (2.8). Since we shall need to deal also with purely position-dependent solutions of the fundamental field equations of thermoelasticity theory, we find it convenient to introduce the notion of a

Stationary Elastic State

We say that \u%, o-#] belongs to the class of stationary elastic states on R corresponding to the data /x, K, a, Fi, 0, and write

[ui, €ih utj] eS [fi9 K, a, Fi, 0] on R (2.18)

if:

(a) /x and K are (not necessarily real) constants;

(b) Ui, €ij, an, Fi, 0 are functions of position continuous on R, S being once— and Ui three times continuously differentiable on R;

(c) equations (2.1), (2.2), (2.3), (2.16) hold in 0t.

The reason why we do not restrict /x and K to be real constants in the present instance will become apparent at the end of Section 4. Meanwhile we note that (2.17) implies, for each fixed t in (— oo, oo),

[«<(•, 0, t)9 <Jij{-,t)] efi[fi, K, a, Fi(-, 0, ©(•, 01 on R (2.19)*

so that every quasi-static elastic state on jRx (— oo, oo) may be regarded as a one-parameter family (with time as the parameter) of stationary elastic states on R corresponding to the same elastic constants and to the appropriate family of body-force and temperature distributions. This observation reflects the fact

* I f / i s a function of position and time defined on R x(— G O , G O ) , we w r i t e / ( • , t) for the function of position defined on R that results from the m a p p i n g / b y holding the time fixed in ( - oo, oo).

13


fin) = &nf (n = 0, 1, 2, . . . ) (2.28)

that the time in quasi-static elasticity theory plays the role of a mere parameter, whereas the quasi-static theory of viscoelasticity is genuinely time-dependent.

We have so far based the formulation of boundary-value problems in thermo-viscoelasticity theory upon the relaxation integral law (2.6). On the assumption that Gp (jS = 1, 2), €y9 and ay meet the conditions set down under (a), (b) in the definition of a viscoelastic state, the stress-strain relations (2.6) may be inverted. This leads (Ref. 1, Theorem 3.3) to the equivalent creep integral law

€y = Sy*dJi, ejtfc = ajck*dJi + 3a 0 (2.20)

where J\ and J2 are the creep compliances in shear and isotropic compression, respectively. Moreover, the two pairs of response functions Gfi and Jfi (p = 1,2) are linked by the relation

Gfi*dJp = h on (— oo, oo) (2.21)

or, using a notation adopted in Ref. 1 (Theorem 1.3) for the "Stieltjes inverse" of a function of time,

J/t = Gp1 on ( - oo, oo) (2.22)

Equation (2.21) implies further that

(2.23)

The familiar physical significance of the relaxation moduli and of the creep compliances is immediate from (2.12) and its creep counterpart. Thus

ey = h on Rx{— oo, oo) implies sy = Gi on i ? x ( — oo, oo) (2.24)

sy = h on Rx{— oo, oo) implies ey = J± on Rx(— oo, oo) (2.25)

Analogous interpretations apply to G2 and J2. Iff is a function of position and time that possesses a Laplace transform with respect to time, we write

(2.26)

in which rj designates the transform parameter. Accordingly, supposing G# and Jfi (P = 1, 2) to be of exponential order as t - > oo, one draws from (2.23) in conjunction with the convolution theorem for the Laplace transform that

(2.27) (fi = 1, 2)

We turn finally to stress-strain relations in differential equation form, i.e. to what is traditionally referred to as the differential operator law of linear visco-elasticity. For this purpose we first make the following notational agreements. If / i s a (suitably smooth) function of position and time defined on R x (— oo, oo), we denote its «th partial time derivative by

A N A L Y S I S OF T H E R M A L STRESSES 355

o D being the time-derivative operator. Further, we write for the function of position defined by

o

/ ( n ) ( x ) =f{n){x, 0 + ) for x in R ( 2 . 2 9 )

The differential operator law now becomes

Px(D) Si, = Qi(D) eih P2(D) okk = Q2(D)[ekk - 3a 0] ( 2 . 3 0 )

provided Pp(D), Qp(D) (/3 = 1 ,2 ) are the linear differential operators

( 2 . 3 2 )

As is clear from an elementary extension of Theorem 4 . 1 in Ref. 1, equations ( 2 . 3 0 ) , ( 2 . 3 2 ) are implied by ( 2 . 8 ) and the relaxation integral law ( 2 . 6 ) or the creep integral law ( 2 . 2 0 ) provided z%h GIU a n d 0 are sufficiently smooth and according as the relaxation moduli or the creep compliances exhibit the de-generacies characteristic of a finite spectrum of relaxation or retardation times. In the first casep n .p 0 for n = N$ (j8 = 1, 2), whereas in the second instance Qn\p ^ 0 for n = Np (/? = 1, 2). The physical significance of the initial condi-tions ( 2 . 3 2 ) was established in Ref. 1 (Theorem 4 . 2 ) for the isothermal case.* Thus suppose 0 vanishes identically and hold x fixed. Then, roughly speaking, ( 2 . 3 2 ) are necessary and sufficient that every pair of local histories ey(x , •)> atj(x, •) which vanishes on (— oo, 0), satisfies ( 2 . 3 0 ) on (0, oo), and exhibits finite jump discontinuities at t = 0, be the limit of a pair of history sequences which vanishes on (— oo, 0), obeys the same stress-strain relations ( 2 . 3 0 ) , and has time derivatives of the orders entering ( 2 . 3 0 ) that are continuous on (— oo, oo).

If an integral law ( 2 . 6 ) is reducible to a differential operator law ( 2 . 3 0 ) , then the Laplace transforms of the relaxation functions exist, are necessarily

* See also Boley and Weiner 3 , Art. 15 .6 , for a related discussion.

Here Nfi (j8 = 1, 2) is a non-negative integer and the coefficients pp. n , n are constant response parameters for a given (temperature-independent) material. Moreover, we may evidently assume that either pn;fi ^0 or qn-,p 0 when n = Nfi (P = 1, 2) so that, for fixed /?, at least one of the operators in ( 2 . 3 1 ) has the degree Nfi. The differential equations ( 2 . 3 0 ) are to be met in ^ x ( 0 , oo) and must be accompanied by the following initial conditions, which are valid on R:

( 2 . 3 1 ) 0 8 = 1 , 2 )


rational, and are given by (Ref. 1, Theorem 4.8)

0 8 = 1,2) (2.33)

Analogously, the reducibility of a creep integral law (2.20) implies

03 = 1, 2) (2.34)

Additional results concerning the transition from an integral to a differential operator law, and vice versa, may be found in Ref. 1 (Section 4).

Since materials with a finite relaxation or retardation spectrum admit the familiar model representation in terms of easily visualized finite networks of springs and dashpots, the bulk of the existing literature on viscoelasticity has favored the differential operator law over stress-strain relations in integral form. It is well to bear in mind, however, that such a preference is not warranted on the grounds of theoretical generality; nor is it justified from a practical point of view since an adequate description of the behavior of actual visco-elastic solids over any appreciable range of time (or frequency) ordinarily necessitates the use of differential operators of a comparatively high order.


We turn next to general results concerning the fundamental field equations and boundary-value problems discussed in the preceding section. In this con-nection we shall limit our attention to materials governed by the relaxation integral law (2.6); analogous conclusions apply to the creep integral law (2.20) and to the differential operator law (2.30). To facilitate the task at hand we first establish a link between the theory of thermo-viscoelasticity under present consideration and the corresponding isothermal theory, which is supplied by the

Body-force Analogy

Suppose

ut = u\ on BiX(— oo, oo), Si = Sf on B%x(~ oo, oo) (3.2)

Further, let u%, e y , a y , Fi be field histories defined on Rx(~co, oo) by means

G E N E R A L I M P L I C A T I O N S

[Ui, ey, ay] E V [G±, G 2 , a, Fi, 0] OH RX{— GO, oo) (3.1) and

of Ui = Ui,

Ft = Ft-a9, t*dG2 j (3.3)

5 y = <*y + aSy @*dG2, Then

[ui, ey, ay] Eu [Gi, G 2 , Fi] on Rx(— oo, oo) (3.4)

a n a l y s i s o f t h e r m a l s t r e s s e s 357

and

ui = u\ on BiX(— oo, oo), §t = Sf + am@*dG2 on B2x(— oo, oo) ( 3 . 5 )

Conversely, if 0 has the same properties as in the definition of a viscoelastic state, then ( 3 . 3 ) , ( 3 . 4 ) , ( 3 . 5 ) imply ( 3 . 1 ) and ( 3 . 2 ) .

The truth of the foregoing theorem is readily inferred from the underlying definitions of viscoelastic and isothermal viscoelastic states with the aid of a known result (Ref. 1, Theorem 1 .6) on the space differentiation of Stieltjes convolutions. The theorem yields as a special case the well-known body-force analogy of thermoelasticity* if the relaxation functions obey ( 2 . 1 5 ) . Although the generalized analogy under discussion, which reduces the mixed boundary-value problem of thermo-viscoelasticity to the standard mixed problem in the isothermal theory, is of no practical usefulness as far as the actual solution of specific problems is concerned, its theoretical interest is considerable. For, the analogy enables one to obtain economically extensions to non-isothermal con-ditions of theorems available in the isothermal theory. We now cite some of the more important conclusions reached in this manner.

Volterra's 4 theorem! concerning the uniqueness of the solution to the mixed isothermal problem, in conjunction with the body-force analogy, at once furnishes the

Uniqueness Theorem

Suppose

[uu e y , cry] eU[Gi, G2, a , Fi, ©] on Rx(— oo, oo), 1 , , , ( 3 . 6 )

lt*i> €tp £V[Gi, G2, a, Fi, ®] on Rx(- oo, oo), J and let

Ui = u\ on BiX(— oo, oo), Si = S't on B2x(— oo, oo) ( 3 . 7 )

Then [Ui, e y , ay] = [ll], €fj, o^] + [wU 0, 0] OH Rx{~ 00, oo) ( 3 . 8 )

where wt = 0 on Rx(— oo, 0) and Wi represents an (infinitesimal) rigid motion of the entire body on Rx [0, oo).

According to (a) in the definition of a viscoelastic state, the relaxation func-tions entering ( 3 . 6 ) are required to satisfy the inequalities

Gfi>0 ( 0 = 1 , 2 ) ( 3 . 9 )

i.e. must possess positive initial values. The striking fact that no restrictions (apart from a smoothness assumption) need be imposed on the subsequent behavior of G$ in order to insure uniqueness, will be made plausible shortly.

A characterization of the initial field distributions appropriate to a visco-

* See, for example, Ref. 3, Art. 3 . 3 . t See Ref. 1, Section 8, for a slightly more general version and a more detailed proof of

Volterra's result in the special case of an isotropic solid.


elastic state is supplied by the following theorem, which is an elementary consequence of Theorem 6.1 in Ref. 1, and of the body-force analogy.

Initial Response

Let [uu meet (2.10). Then

[uu iij9 ay] e*[n, K, a, Fu 4] on R (3.10) with

M = ± < ? i , K = iG2 (3.11)

This result lends a precise meaning—within the context of thermo-visco-elasticity theory—to the familiar assertion that the initial response of a visco-elastic solid is elastic. Moreover, the preceding theorem enables one to deter-mine directly the initial displacements, stresses, and strains belonging to the solution of a mixed boundary-value problem in thermo-viscoelasticity theory by solving a steady-state thermoelastic problem; the latter is governed by the initial body-force, temperature, and surface data of the original problem, as well as by the elastic constants (3.11). Similarly, Theorem 6.2 in Ref. 1 furnishes an analogous characterization of the initial (right-hand) time derivatives of all existing orders belonging to the desired viscoelastic state. These initial time rates u\n\ off (n = 1, 2, . . . ) may be found directly from the correspond-ing initial data by solving successively a sequence of steady-state thermoelastic problems, each of which is governed once again by the elastic constants defined in (3.11). Since, according to the uniqueness theorem of the classical (un-coupled) theory of thermoelasticity, the inequalities ^ > 0, K > 0 are sufficient to guarantee the uniqueness of a suitably regular solution to the mixed problem, Volterra's uniqueness criterion (3.9) is not so surprising.

An additional conclusion regarding the time-dependence of viscoelastic states, that permits one to infer the smoothness of the state with respect to time from the corresponding smoothness of the field and surface data, is de-ducible from Theorem 6.4 in Ref. 1. We proceed now to a theorem on the

Position-dependence of Viscoelastic States

Let [ui, etj, uij] meet (2.10) and assume that

V - F = 0 , V A F - 0 , V 2 < 9 = 0 on &x(— oo, oo) (3.12)*

Then throughout &x(— oo, oo),

V2(V • u) = 0, V2(V A i/) = 0 (3.13)

= 0, V *6t f = 0, V*cry - 0 (3.14)

Thus in the present circumstances, as in thermoelasticity theory, (3.12) imply that the dilatation and the rotation are harmonic, while the cartesian compo-

* Here, as in the sequel, V is the usual spatial gradient operator, whereas V, VA , and V 2

designate the divergence, curl, and Laplacian operators, respectively.


nents of displacement, strain, and stress are biharmonic functions of position. It follows that all of these field histories possess continuous space derivatives of all orders. The above theorem is a consequence of Theorem 6.6 in Ref. 1.

The counterpart of Betti's elastostatic reciprocal relations in the isothermal theory of viscoelasticity given in Ref. 1 (Theorem 7.4), in view of the body-force analogy, the divergence theorem, and by virtue of Theorems 1.2, 1.6 in Ref. 1, leads to the following

Reciprocal Theorem

Suppose [m, €^7, dij] eU[Gi, G2, a, Fu ®\ on Rx(~ oo, oo), )

(3.15)

[ui9 eip atj] eV[Gu G2, a, Fi9 0 ' ] on Rx(- oo, oo). J

Then, on (— oo, oo),

(3.16)

It should be emphasized that the reciprocal relations obtained from the present theorem by specializing Gp (p = 1, 2) in accordance with (2.15) and by setting S = 0 on Rx(— oo, oo), are distinct from the quasi-static adaptation of Betti's reciprocal relations in isothermal elastostatics. This distinction disap-pears if the body force and temperature field histories, as well as the surface data, of the two states in (3.15) are separable functions of position and time that share a common time-dependence. In this particular instance the con-volutions entering (3.16) (e.g. S^du^) can be shown* to be reducible to ordinary products (e.g. S^), and the resulting reciprocal relations are identical with those arising from the extension of Betti's theorem to quasi-static thermo-elasticity theory.

The reciprocal theorem stated above may be used to derive formulas for the average strains induced in a (temperature-independent) viscoelastic solid by given body-force and temperature field histories and by specified surface trac-t ions.! We include here merely a formula for the total volume change, which can be deduced directly from Theorem 7.6 in Ref. 1 with the aid of the body-force analogy and the divergence theorem.

* Cf. Theorem 7 . 5 in Ref. 1. f See Ref. 5 for a derivation of analogous results in the isothermal theory of anisotropic

viscoelastic solids.


The same conclusion was also reached by Nowacki (Ref. 7, Chapter XI) on the basis of a less direct argument. Thus, in the absence of surface tractions and body forces, the total volume change is the same as that which would arise from uninhibited thermal expansion. This well-known result of classical thermo-elasticity theory! is accordingly seen to survive without modification under the present more general hypotheses.

The next theorem, which also pertains to vanishing surface tractions and body forces, follows from the appropriate specialization of a result established in Ref. 6 for a class of temperature-dependent viscoelastic materials.

Stress-Free Temperature Field Histories

Assume (2.10) holds and suppose St = 0 on Bx [0, oo), Ft = 0 on Rx [0, oo). Then ay = 0 on Rx(— oo, oo) if and only if

6>(x, 0 = 0O(O + at(t) Xi for (x, t) in R x [0, oo) (3.21)

where ao, at are functions continuous on (0, oo). An indication of the proof of the necessity of (3.21) if the body is to be free

from stress, appeared prior to Ref. 6 in a note by Hilton 9 . Analogous two-dimensional theorems on temperature distributions that fail to induce thermal stresses (applicable to conditions of plane strain and generalized plane stress) are also to be found in Ref. 6. For the familiar corresponding three-dimensional

* See Ref. 6, where such an alternative derivation is given for the special case of a material with a purely elastic dilatational response.

f See, for example, Ref. 3 , Art. 9 . 1 5 . The result was reached by Hieke 8 , who based his proof on an extension to thermoelasticity theory of Betti's reciprocal relations.

(3.20)

It is essential to observe that (3.17) may also be derived* solely from (3.18), (2.2), and the second of (2.6) by recourse to (2.21) and Theorems 1.2, 1.6 in Ref. 1. Consequently, formula (3.17) is valid for any solid that is in a state of infinitesimal deformations and has a temperature-independent linearly visco-elastic dilatational response, regardless of the nature of its response in shear. If, in particular, Si = 0 on Bx(— oo, oo) and Fi = 0 on Rx(~ oo, oo), (3.17) implies

provided J2 is the creep compliance corresponding to the relaxation modulus G2

and where

(3.18)

(3.19)

(3.17)

Volume Change

From (2.10) follows, for t in (— oo, oo),


and two-dimensional results in thermoelasticity theory, which are implied by the theorems under present consideration, reference may be made to Ref. 3, Arts. 3.9, 4 .9 .

Before concluding this section we merely mention several additional implica-tions of the body-force analogy. Thus, one may extend Duhamel's principle of the isothermal theory (see Ref. 1, Section 5) to the analysis of thermal stresses and deformations. Similarly, one may obtain analogous extensions of the varia-tional principles of isothermal viscoelasticity theory.* Finally, the integral representations (in terms of Green's functions) of the solution to isothermal boundary-value problems established in Ref. 11 are readily generalized to include temperature effects within the current theoretical framework.

Some further consequences of the body-force analogy will be encountered in the succeeding section, which deals with results that have a direct bearing on the solution of boundary-value problems of the type formulated in Section 2.


M E T H O D S O F I N T E G R A T I O N , A V A I L A B L E S O L U T I O N S

With a view toward alternative formulations of the boundary-value problems stated in Section 2, we now cite the appropriate displacement equations of equilibrium and stress equations of compatibility, which could be arrived at by a respective elimination of either the stresses and strains, or of the displace-ments and strains, among the governing fundamental system of field equations. Actually, these desired generalizations of the Cauchy equations of equilibrium and of the Beltrami-Michell equations of compatibility are immediate from their isothermal counterparts in Theorem 5.5 and Theorem 5.7 of Ref. 1 because of the body-force analogy.

Displacement Equations of Equilibrium

From (2.10) follows, on &x(— oo, oo),

(4.1)

(4.2)

or, equivalently,

Stress Equations of Compatibility

From (2.10) follows, on Rx(— oo, oo),

where J $ is the creep compliance corresponding to G $ {$ = 1, 2),

(4.3)

(4.4

* See G u r t i n 1 0 for a generalization to isothermal viscoelasticity theory of the classical, and of more recent, elastostatic variational principles.

1 3 §


m d Q = Ji*d(J2 - Ji)*d(Ji + U2)-1 (4.5)*

The mixed problem, when cast in terms of displacements alone, consists in finding a solution of (4.1) on St x (— 00 , 0 0 ) , such that u% = 0 on R x (— 00 , 0) and subject to the boundary conditions

(4.6)

which follow from (2.9) by virtue of (2.1), (2.3), (2.6). On the other hand, if R is simply connected and B2 = B (surface tractions prescribed over the entire boundary) the unknown stresses are completely characterized by (2.2), (4.3), and the second of the boundary conditions (2.9).

Our next objective is to exhibit a particular solution of (4.2), corresponding to vanishing body forces and to a given temperature field history, by generalizing the familiar thermoelastic potential, which appears to have been discovered originally by Borchardt 1 2 .

Thermo-Viscoelastic Potential

Let Gp (fi = 1, 2) and G meet conditions (a) and (b) in the definition of a viscoelastic state. Suppose 0 is a (sufficiently smooth) real-valued function defined on & X (— 00 , 00 ) that vanishes on St X (— 00 , 0) and satisfies

V20 = 3ae*dG2*d(2G! + G2)^ (4.7)

in & X ( — 00, 00 ) . Then throughout that domain the function u defined by

u=V6 (4.8)

satisfies (4.2), provided F = 0 on & X ( — 00, 00) To confirm this assertion one need merely substitute from (4.8) into (4.2),

use (4.7), and invoke the algebra and calculus of Stieltjes convolutions.! A suitably regular solution of the Poisson equation (4.7) may, in turn, be generated with the aid of a Newtonian potential, as is apparent from Lemma 9.1 in Ref. 1. Indeed, if p designates the right-hand member of (4.7), such a solution is given by

* Here, as on subsequent occasions, we use the notation for the Stieltjes inverse introduced in Ref. 1 (Theorem 1.3). Cf. (2 .21) , (2 .22) .

t For related, though less explicit, particular solutions of (4 .2 ) see Parkus 1 3 (Chapter VI) and N o w a c k i 7 (Chapter XI) .

(4.9)

The last theorem evidently supplies yet another means of reducing the boundary-value problem of Section 2 to a mixed problem in the isothermal theory. Further, the present theorem, in conjunction with Theorem 9.2 in Ref. 1, leads to the


following generalization to thermo-viscoelasticity theory of the Papkovich-

Neuber stress functions in classical elastostatics.

Generalized Papkovich-Neuber Solution

Let G# (P — \, 2), 0, 9 meet the hypotheses of the preceding theorem and let F conform to (b) in the definition of a viscoelastic state. Suppose </> and u> are (sufficiently smooth) functions defined on & X (— oo, oo), both of which vanish on & x (— oo, 0) and satisfy

V ^ = - ± x - H , V2u> = i H (4 .10) with

H - ¥*dG1

1*d(2G1 + G2)- 1 (4 .11)

in & X (— 00 , 0 0 ) . Then throughout that domain the function u defined by

u = V0+V(<f> + x- ty)*d(Gi + 2G2) - 4uW ( 2 G i + G2) (4.12)

satisfies (4 .2).

In view of Theorem 9.4 in Ref. 1, the foregoing solution of the displacement equations of equilibrium is complete in the sense that every suitably regular solution of (4 .2) admits the representation (4.7), (4 .10) , (4 .12). Consequently, the introduction of the displacement potentials (stress functions) 6, </>, and <|> reduces the mixed boundary-value problem to the determination of appropriate solutions of Poisson's equation. On the other hand, since </> and i|> enter (4.12) under Stieltjes convolutions and because of the structure of the second of (4.6), the application of the boundary conditions to the generating stress functions will ordinarily lead to a system of simultaneous integral equations.

We turn now to the well-known correspondence principle that links the linear theories of viscoelasticity and elasticity. This principle, which is of paramount importance to the solution of the class of boundary-value problems with which we are concerned, may—with the aid of the state definitions adopted in Section 2 —be phrased in the following concise form.

Correspondence Principle

Suppose

[ut, e i j 9 ay] eV[Gi, G2, a, Fu 0] on Rx ( - 00 , 00) (4.13) and

ut = u\ on Bi X ( — 00, 0 0 ) , Si = Sf on B2 X ( — 00 , 00) (4.14)

Assume that Gfi(i) (P = 1, 2), Ui(x9 t)9 e^(x, t)9 Fi(x9 t)9 and 0(x9 t),for every x in R9 are all of the exponential order o ( e x P (̂ oO) a s t - > 00 , where so is a (real) constant. Then, for each rj with Re(r)) > SQ9

[Ui( ' , rj), hj % ( ' , rj)] G < f [ / x (rj), K(rj),

a, Fi ( • , rj), 0( • , rj)] on R

* Recall the footnote on p. 353 and the notation for the Laplace transform introduced in (2 .26) .

j ( 4 . 1 5 ) *


as is suggested by (2.33). A direct proof of the correspondence principle applic-able to differential operator laws may be conducted by means of an argument that is strictly analogous to the one we have outlined for the relaxation integral law. Note, however, that operating on (2.30) with the Laplace transform one a r m / A O a t

(4.20)

on R9 for all rj with a sufficiently large real part, only if the initial conditions (2.32) are met.* Thus (2.32), the physical significance of which was discussed

* For details see the proof of Theorem 4 . 7 in Ref. 1.

(4.19)

which, by virtue of (2.16), account for the "elastic constants" (4.16). Suppose now that the formulation of the original (thermo-viscoelastic)

boundary-value problem is based on the differential operator law (2.30) to-gether with the initial conditions (2.32), rather than on the relaxation integral law (2.6). In this case the conclusions (4.15), (4.17) continue to hold true (in the presence of adequate regularity assumptions on the field histories involved), provided (4.16) are replaced by

(4.18) stj( ' >V) = vGiiv) ' , rj),

°kk( ',*!) = r)(j2W[€kk{ ' , rj) — 3a&( • , rj)\

The theorem just stated evidently implies that if the solution to the thermo-viscoelastic problem governed by (4.13), (4.14) is Laplace-transformable, it must coincide with the inverse transform of the solution to the one-parameter family of stationary thermoelastic boundary-value problems characterized by (4.15), (4.16), (4.17). Accordingly the correspondence principle yields a reduc-tion of the original problem to one in steady-state thermoelasticity theory.

To confirm the principle one need merely remove the time-dependence from the field equations and boundary conditions appropriate to a viscoelastic state by applying the Laplace transform to (2.1), (2.2), (2.3), (2.6), (2.9) and com-pare the resulting system of equations with (2.1), (2.2), (2.3), (2.16), and (2.9). In particular, since the stress-strain relations (2.6), because of (2.8), imply (2.12), the convolution theorem for Laplace transforms furnishes

where

and furthe

Ui ( • ,rj) = u»(- ,rj) on BL9 S< ( • ,yj) = ( • 9rj) on B2 (4.17)

(4.16)


in Section 2, are precisely the conditions needed to justify the usual formal application of the Laplace transform to differential operator laws, in which jump discontinuities in a y , ey (and in their relevant time derivatives) at t = 0 are either tacitly ruled out or else (2.32) are tacitly assumed to hold. This issue was apparently first recognized by Corneliussen and Lee 1 4 and was further clarified by Boley and Weiner 3 (Article 15.6); it is treated rigorously in con-siderable detail in Ref. 1.

The preceding analogy between boundary-value problems in the linear theories of viscoelasticity and elasticity has its origins in a paper by Alfrey 1 5 . Alfrey's form of the correspondence principle, which is based on the differential operator law, is confined to incompressible solids, and to isothermal conditions. A limited extension of Alfrey's analogy to compressible solids is due to Tsien 1 6 , who abandoned the assumption of incompressibility and postulated instead an artificial (physically unrealistic) connection between the deviatoric and the dilatational response of the material. Integral transforms were first brought to bear on the question raised in Ref. 15 by Read 1 7 , who reduced the viscoelastic problem for a compressible solid to an elastostatic problem with the aid of the Fourier transform. Although the stress-strain law employed in Ref. 17 involves only three independent differential operators, this superfluous restriction does not affect the generality of Read's argument. Brul l 1 8 arrived at an analogous correspondence principle for a medium with a continuous spectrum of relaxation times by means of the Laplace transform and on the basis of the relaxation integral law; in Ref 17 the volumetric response is (unnecessarily) assumed to be purely elastic. L e e 1 9 deduced the counterpart of Brull's version of the cor-respondence principle for the general differential operator law. Hilton, Hassan, and Russell 2 0 allowed for thermal expansion within the framework of Alfrey's analogy,* whereas Lee's scheme was similarly extended to thermo-visco-elasticity theory in Ref. 21. Additional related results will be referred to later. We note, in passing, that the initial conditions (2.32), which must supplement the differential operator law (2.30), are not mentioned in Refs. 15, 16, 17, 19, and 20, while the particular initial conditions presupposed in Ref. 21 are unduly stringent and hence misleading.

The extended correspondence principle was applied in Ref. 21 to the solution of a particular spatial boundary-value problem and has since been further exploited in the quasi-static analysis of thermal stresses and deformations for linear viscoelastic solids with temperature-independent properties. Specific examples and pertinent references may be found in Refs. 3, 7, and 13. Nowacki 's 7

monograph supplies a convenient guide to recent Polish papers on problems of the type formulated in Section 2. Among these we cite investigations by N o w a c k i 2 2 ' 2 3 and by Sokolowski 2 4 . A study by Shaffer and Levitsky 2 5 should also be referred to in the present context. Additional studies, which aim primarily at dynamic thermal effects in temperature-independent linear viscoelastic solids, will be mentioned in Section 7.

* The inclusion of the thermal expansion term is, strictly speaking, inconsistent with the accompanying assumption of incompressibility.


5. T E M P E R A T U R E - D E P E N D E N T R E S P O N S E : F O R M U L A T I O N

O F B O U N D A R Y V A L U E P R O B L E M S F O R T H E R M O -

R H E O L O G I C A L L Y S I M P L E M A T E R I A L S

It is evident from the preceding section, that the quasi-static analysis of thermal stresses and deformations within the linear theory of homogeneous, isotropic viscoelastic solids, and in the absence of thermo-mechanical coupling, presents no essential difficulties, provided the mechanical response of the material is assumed to be independent of the temperature. Unfortunately, as was pointed out in the Introduction, such a treatment of the problem is remote from physical reality unless the temperature range involved is exceedingly small. The remainder of this paper is chiefly concerned with the modifications arising in the theory discussed so far if the relaxation moduli in (2.6), the creep compliances in (2.20), and the material response parameters in (2.30), (2.31) are themselves temperature-dependent.

The importance of effects that stem from the temperature-dependence of the viscoelastic properties was emphasized and illustrated relatively early by F r e u d e n t h a l 2 6 - 2 9 . Special problems concerning temperature-dependent linear viscoelastic solids were also considered by Hilton, Hassan, and Russell 2 0 , as well as by Hil ton 3 0 , Rongved 3 1 , Weiner and Mechanic 3 2 , Landau, Weiner, and Zwicky 3 3 , and Aggarwala 3 4 . All of the problems treated in the publications just listed concern either infinite plates or (solid or hollow) spheres and circular cylinders, the field data and boundary conditions being such as to preclude the presence of more than a single space coordinate. Further, the underlying stress-strain relations correspond to degenerate cases of the differential operator law. In fact, the solids considered exhibit Maxwell, Kelvin, or Standard Linear behavior in shear, while the volumetric response is ordinarily taken to be either elastic or incompressible. Finally, the temperature-dependence of the material is in all cases, with the exception of Ref. 34, confined to the viscosity parameters and its specific choice is usually in part a matter of analytical expediency. Of particular interest is Rongved 's 3 1 explicit solution (in closed elementary form) for the transient thermal stresses in an elastically compressible Maxwell sphere since it accommodates an arbitrary radially symmetric temperature field history and presupposes no restrictions upon the temperature-dependence of the shear viscosity.

A systematic scheme for including the influence of temperature on the response characteristics of a viscoelastic solid in the analysis of thermal stresses was initiated by Morland and Lee 3 5 , who took as their point of departure the temperature-time equivalence hypothesis originally proposed by Leaderman 3 6 and subsequently introduced in a slightly different form by Ferry 3 7 . According to this postulate the mechanical response of the material is affected by a uniform temperature change only within a uniform change of the time scale; the response is uniformly accelerated or slowed down depending on whether the temperature is increased or lowered. Materials obeying the temperature-time equivalence postulate are said to be "thermo-rheologically simple" in the terminology of Schwarzl and Staverman 3 8 who, in their expository article, 3 9 cite experimental


evidence in support of the postulate, which appears to be in remarkably good agreement with tests performed on a variety of high polymers over considerable temperature ranges. The analytical framework developed in Ref. 35 was further explored in Ref. 40.

We proceed now to a brief resume of the theory of thermo-rheologically simple viscoelastic solids and in this connection consider first the required generalization of the relaxation integral law (2.6). Thus, let Gfi(t) (/3 = 1, 2) henceforth denote the values of the relaxation moduli at time t measured at the base temperature 7b and denote by ^ 7 , T) the corresponding values measured at the (fixed) temperature T. Consequently

^(t,To) = Gp(t) (j8= 1,2) (5.1)

The temperature-time equivalence hypothesis may then be expressed analytic-ally by

T) = G , ( 0 , f = tcf>(T) for (/, 7 ) in ( - oo, oo) x [Tu T2] (5.2)

where [7i, T2] is the temperature range for which the postulate of thermo-rheologically simple behavior is presumed to hold. Here f is the "reduced time", while <j> represents the characteristic "shift function" of the material, which governs the contraction (or the stretching) of the time scale that results from the temperature change T — To. Clearly,

# 7 b ) = 1, $(T) > 0 (Ti < T < T2) (5.3)

and (f> is evidently a steadily increasing function. The constitutive law (2.6) must at present be confined in its validity to a

material that is permanently maintained at the uniform temperature 7b. In these circumstances one gathers from (2.6) on setting O = 0, recalling (2.7), and making use of the commutativity of Stieltjes convolutions, that

t

sv(x, t)= j G±(t — t') deij(x, tf)9

(5 .4 ,

akk(x9 i)= J G2(t - t') dekk(x, tf). t'=-co

If the material is instead permanently at the uniform temperature 7, the values Gp(t — t') in (5.4)—according to (5.2)—are to be replaced by G^(f — f ) , where = tf<j>(T). If, finally, the material is under the influence of a variable (time and position-dependent) temperature distribution T with values in [T±, T2], (5.4) are in need of a two-fold additional amendment: first, the definition of the reduced time f must now be generalized to account for the cumulative effect of successive temperature changes; second, one must allow once again for the influence of thermal expansion. This leads to the modified relaxation integral


law t

stfa t)= J Gi(f - f ) deif(x91')

a**(x, 0 = / G 2(f - n dejck(x9 t') -t'= — oo

- 3 a / - O d@(x, /') r = - c o

(5.5)

with

f - p(x, 0 = / <f>(T(x91')) dt'9 f = P (x , O (5.6)* o

We note that the temperature enters (5.5) both through @ and through f, f. In contrast to (2.6), the stress-strain relations (5.5) imply a nonlinear dependence of the local instantaneous stresses upon the local temperature history. It is apparent from (5.6), (5.3) that />(x, • ) is a monotone increasing function of the time on (— oo, oo), the inverse of which we denote by OJ(X9 • ),

whence

t = o>(x9£) (5.7)

If / is a function of position and time, we shall consistently write / for the function defined by

/ ( x , 0 = / ( x , " ( x , 0 ) (5.8)

By subjecting the variable of integration in (5.5) to the transformation t' = OJ(X9 f ) one may eliminate from (5.5) any explicit dependence upon the physical time in favor of the reduced time. In this manner, using the notations adopted in (2.7), (5.8) and invoking once more the commutativity of Stieltjes convolutions, one arrives at the following particularly convenient version of the modified relaxation integral law:

Sij = ea*dGi )

Okie = (ekk — 3a @)*dG2 j

As is apparent from (5.8), the convolutions in (5.9) are taken with respect to the reduced time rather than the physical time. We note that (5.9) have other-wise the same structure as (2.6).

Strictly analogous considerations apply to the generalization of (2.20) and ultimately lead to the modified creep integral law in the form

e^s^dJu _ | ( 5 i Q )

*kk = <?kk*dJ2 + 3a 0 j

which may also be deduced directly from (5.9). Finally, the same process of

* Although this generalization of the reduced time, introduced in Ref. 35, is convincing on physical grounds, a rigorous deduction of (5 .5 ) from (5 .4 ) and the temperature-time equi-valence postulate, appears to be lacking.


specialization that leads from the integral laws (2.6) or (2.20) to (2.30) now furnishes the modified differential operator law

P!0) stj = Q!0) eih P2(D) &kk = Q2(D)[ikk - 3a 9] (5.11)

where D is the reduced-time derivative operator, i.e.

(5.12)

while the polynomial operators PQp (j8 = 1, 2) retain their previous meaning. To (5.11) one must adjoin the initial conditions (2.32), which remain unaltered.

On referring (5.11) to the physical time t by means of (5.6), (5.7), (5.8), one is evidently led to a pair of differential equations which has the same structure as (2.30), except that the previously constant response parameters pp.n, qP;n

are now functions of the temperature. It is also clear that these functions cannot be prescribed independently for a thermo-rheologically simple viscoelastic solid since the entire temperature-dependence of such a solid is governed by the single shift function </>. Moreover, an arbitrary disposition over the tem-perature-dependence of the response parameters in (2.30) is not only incom-patible with the temperature-time equivalence hypothesis but is also readily seen to be inadmissible on energetic grounds.

The implications of the equivalence postulate for elastic, Maxwell, and Kelvin solids were examined in Ref. 40. As is at once apparent from (2.15), elastic materials with temperature dependent moduli do not belong to the class of thermo-rheologically simple viscoelastic solids. On the other hand, the shear response of a thermo-rheologically simple Maxwell solid is found to be charac-terized bv

(5.13)

in which fi and r are the shear modulus and the relaxation time of the solid; whereas /x remains constant in the present instance, r obeys

(5.14)

and is thus a monotone decreasing function of the temperature. Analogous conclusions apply to the dilatational response and to solids of the Kelvin type. It is interesting to note that the ad hoc assumptions regarding the temperature-independence of certain response parameters made in Refs. 26 to 33 are in fact precise consequences of the temperature-time equivalence hypothesis. Before leaving the present subject we observe that, as pointed out by Lee and Rogers 4 1 , the shortcomings of the differential operator law (i.e. of finite-spectrum models) are apt to be even more pronounced once the (thermo-rheologically simple) temperature dependence of the material is taken into account, in view of the concomitant contraction of the time scale.

We are now in a position to state the standard mixed boundary-value problem


in the quasi-static linear theory of thermo-rheologically simple viscoelastic solids. A formulation based on the modified relaxation integral law may be phrased as follows. One is to find field histories Ui, a y which—for given R, Bi, B2, known Gi, G2, a, To, cf>, and prescribed Fu T, u\, Sf—satisfy (2.1), (2.2), (2.3), (2.4), (5.5), (5.6) on 0t x ( - oo, oo) and meet the initial conditions (2.8), as well as the boundary conditions (2.9). This problem suggests the subsequent generalization of the definition of a viscoelastic state introduced in Section 2.

Thermo-Rheologically Simple Viscoelastic State

We say that [m, atj] belongs to the class of thermo-rheologically simple viscoelastic states on Rx(— oo, oo) corresponding to the data Gi9 G2, a, To, <f>9

Ft, T for the temperature range [7\, T2], and write

[uu €ih ai,] e^[Gi, G2, a, T0, cf>, Fi, T] on Rx ( - oo, oo) (5.15)

if:

(a) G±, G2, Ui, eij, a y , and Fi meet requirements (a), (b) in the definition of a viscoelastic state;

(b) <j> is continuous on [7i, T2] and obeys (5 .3) ;

(c) T = To on R X (— oo, 0), is continuous on R X [0, oo), and has values in Pi, T2);

(d) equations (2.1), (2.2), (2.3), (2.4), (5.5), (5.6) hold on 9t x ( - oo, oo).

Evidently, (5.15) in conjunction with <f>(T) = 1 for T in [7i, T2] implies (2.10), whence this specialization of the shift function <f> yields a reduction of the present theory to the theory of temperature-independent materials treated in the preceding sections.

6 . I M P L I C A T I O N S O F T H E R M O - R H E O L O G I C A L L Y S I M P L E

B E H A V I O R , A P P L I C A T I O N S

Our current objective is the compilation of a few general conclusions per-taining to the theory of thermo-rheologically simple viscoelastic solids. Most of the theorems about to be cited are extensions of, and upon the appropriate specialization of </> reduce to, analogous propositions in Section 3. We state first the

Uniqueness Theorem Suppose

[ut, €ij, an] ef[Gl9 G2, a, To, <f>, Fu T] on R X ( - oo, oo) 1

\u\, e i p atj] ef[Gi9 G2, a, To, Fu T] on Rx (- oo, oo) J

and let

ut = u't on B± x (— oo, oo), Si = S't on B2 x (— oo, oo) (6.2)


* See the remarks following (3 .19) .

Then ( 6 . 3 ) N, ey, <Jij] = [ui9 eip at1] + [wi9 0, 0 ] on R X ( - oo, oo)

where wi = 0 on R X (— oo, Q) and wt represents an (infinitesimal) rigid motion of the entire body on R X TO, oo).

This result is implied by a more inclusive uniqueness theorem established in Ref. 42, the latter being applicable also to ablating viscoelastic solids. It is interesting to observe that no restrictions on the shift function </>, beyond continuity and ( 5 . 3 ) , are needed to assure uniqueness in the present circum-stances. The next theorem may be inferred directly from the definitions of stationary elastic and thermo-rheologically simple viscoelastic states with the aid of ( 5 . 8 ) , ( 5 . 9 ) , and Theorem 1.2 in Ref. 1.

Initial Response

Let [ui9 ey, ay] meet ( 5 . 1 5 ) . Then o o

[ut9 eiJ9 Sy] eS[^ K9 a, Fi9 S] on R ( 6 . 4 ) with

( 6 . 5 )

Consequently the initial state is again a stationary elastic state and may be found directly from the initial field and surface data. We turn now to a propo-sition which follows from a result cited* in Section 3.

Volume Change

Let [ut9 ey9 ay] meet ( 5 . 1 5 ) and suppose

( 6 . 6 ;

where K is a constant (elastic dilatational response). Then the volume change is given by ( 3 . 1 7 ) , ( 3 . 1 9 ) .

The subsequent result was deduced in Ref. 6 and extends an earlier observa-tion due to Hil ton 9 .

Stress-Free Temperature Field Histories

Assume ( 5 . 1 5 ) holds and suppose St = 0 on B X [0, oo), Fi = 0 on R X [0, oo). Then ay = 0 on R X (— oo, oo) if and only if

6>(x, t) = a0(t) + at(t) xt for (x, t) in R x [0, oo) ( 6 . 7 )

where ao, ai are functions continuous on [0, oo). Related theorems concerning stress-free temperature fields, applicable to plane

strain and generalized plane stress, are given in Ref. 6, where the two-dimensional theory of thermo-rheologically simple viscoelastic solids is explored in some detail. The two-dimensional considerations in Ref. 6 also led to a connection


between the plane-strain and the generalized plane-stress solution associated with one and the same plane problem.*

The chief analytical complications introduced by the departure from tem-perature-independent viscoelastic behavior arise in the actual solution of boundary-value problems. Unfortunately, the theory of integration presented in Section 4 does not admit a useful extension to thermo-rheologically simple viscoelastic media. This is true in particular of the powerful correspondence principle discussed at the end of Section 4, which yields a reduction of the viscoelasticity problem (for a solid with a temperature-independent response) to a problem in steady-state thermoelasticity theory.

To bring the difficulty to which we have just alluded into focus, we observe first that the integrals appearing in the modified relaxation law (5.5) are no longer of the convolution type. Consequently, an application of the Laplace transform (with respect to the physical time) to (5.5) fails to furnish algebraic stress-strain relations in the transform domain. The alternative version (5.9) of the modified relaxation law, which has the requisite convolution structure, suggests the possibility of referring also the remaining field equations, as well as the boundary conditions, to the new independent variables (x, f) with a view toward a subsequent elimination of the reduced time by means of the Laplace transform. Such a procedure in general does not result in a worthwhile simplification of the problem. Indeed, let us use (5.6), (5.8) to effect the pro-posed change of variables and let us agree to write / i and / ' for the space derivatives and for the reduced-time derivative of a function / of position and of the reduced time. Then the displacement-strain relations (2.1) become

= Wi, + Uj, i) + Xfi'iP, i + ujp, i) (6.8)

whereas the stress equations of equilibrium (2.2) pass over into

°iU 1 + °iiP, i + F% = 0, &ji = ay (6.9)

Because of the terms involving pt u the transforms of (6.8), (6.9) (taken with respect to the reduced time) no longer possess the desired structure of (2.1), (2.2), unless p, % vanishes. Such will be the case, as is clear from (5.6), if T is a function of the time alone. In this special event one is accordingly led to a sig-nificant generalization of the correspondence principle. To avoid unduly cumbersome notation in the subsequent statement of this extension we shall write

* For the analogous connection in two-dimensional thermoelasticity theory, see Mindlin and Salvadori 4 3 , p. 762.

(6.10)

for the Laplace transform with respect to the reduced time of a function / with values / ( x , £).


Correspondence Principle for Purely Time-Dependent Temperature Histories Suppose

[ut, €ij9 atj] er[Gi, G2, a, To, </>, Fi, T] on Rx ( - go , oo) ( 6 . 1 1 )

with T a function of the time alone, and let

m = u\ on Bi X (— oo, oo), Si = Sf on B2 X (— oo, oo) ( 6 . 1 2 )

Assume that Gfi(g) (/3 = 1, 2), m (x, £), %(x, f ) , Fi(x, £), and f{£), for every x R, are all of the exponential order 0 (exp (sog)) as £ - > oo, w/zere j o w (re#/)

constant. Then, for each rj with Re(rj) > so,

[ U i ( • , rj), ' , T ? ) , &a{ ' , ^ ) ] G t f [ ^ ) , K(rj), a, F<( • , rj), 0{rj)] on R ( 6 . 13)

where

K V ) = \ vGi(r)), K(rj) = i rjG2(v) ( 6 . 1 4 ) and further

H ' , r j ) = fi?(- , r j ) on Bi, §i( • , r j ) = S»(- , v) on B2 ( 6 . 1 5 )

The preceding analogy becomes trivial if B = B2, Fi = 0 on x (— oo,oo), and & = 0 on i? x (— oo, oo). In this instance the analogy merely confirms our previous conclusion* that ay = 0 on R x (— oo, oo), in view of the well-known theorem on stress-free temperature fields in thermoelasticity theory (Ref. 3, Art. 3 . 9 ) .

Suppose next that for non-negative time the temperature T, is a function of position alone. In this second degenerate case the modified relaxation integral law ( 5 . 5 ) may once again be written in terms of convolutions with respect to the physical time, as is apparent from ( 5 . 6 ) . In fact one now has

sy = Hi*dea, GJCJC = H2*d(€jcjc — 3a S) ( 6 . 1 6 )

provided

H^x, t) = Gfj(tcf>(T(x))) for (x, 0 in R X [0, oo) ( 6 . 1 7 )

Upon taking the Laplace transform of ( 6 . 1 6 ) one finds that

Sij(x, rj) = 2/x(x, rj) eij(x, rj),

Gkk(x, rj) = 3#c(x, rj)[ekk(x, rj) — 3a@(x, rj)]

for each fixed rj with a sufficiently large real part and for all x in R, if

/*(*, v) = i vHi(*> v)> *(x> y) = I -nH2(x, rj) ( 6 . 1 9 )

But ( 6 . 1 8 ) are the stress-strain relations appropriate to an inhomogeneous linear elastic solid. Consequently, if the temperature depends on position alone, the original boundary-value problem in thermo-viscoelasticity theory is reducible to a steady-state thermoelastic problem for an inhomogeneous medium. Unhappily, this generalization of the correspondence principle is of very limited practical

* N o t e that 0 conforms to (6 .7 ) since T is at present independent of position.

( 6 . 1 8 )


interest since the complexity of the reduced problem is apt to be comparable to that of the original one.

For the differential operator law an analogous extension of the correspondence principle to temperature-dependent materials was given earlier by Hilton and Russell 4 4 in a paper which is based on their previous repor t 2 0 with Hassan. The analysis in Ref. 44 is confined to differential operator laws with temperature-dependent material parameters and thus presupposes a medium with a finite relaxation or retardation spectrum. The analogy deduced in Ref. 44 for purely time-dependent temperature distributions is limited to mechanically incom-pressible solids (although thermal expansion is included) and is an extension of Alfrey's 1 5 principle.* On the other hand, no restriction is placed in Ref. 44 on the linear viscoelastic volumetric response for the case of a purely position-dependent temperature field; the correspondence principle obtained in this instance is an elementary generalization of Read ' s 1 7 work in the isothermal theory. Finally, Ref. 44 contains a sketch of an approximative approach to the general case of a position and time-dependent temperature field history, based on the assumption that the material may be regarded as stepwise temperature-independent (in time). The extent to which such an approximation scheme is computationally feasible remains to be assessed.

We turn now to available applications of the quasi-static linear theory of thermo-rheologically simple viscoelastic solids. As was mentioned before, all but one of the special investigations listed at the beginning of Section 5 are based on temperature-dependent viscoelastic models that fall within this cate-gory, although the problems treated were not intentionally selected to illustrate the general theory under present consideration.

The temperature-time equivalence hypothesis was applied by Morland and Lee 3 5 to the plane-strain analysis of an incompressible hollow circular cylinder which is exposed to a stationary radial distribution of temperature and is sub-jected to a suddenly applied uniform internal pressure.

Implications of the theory in circumstances that involve temperature dis-tributions depending on both position and time were studied in Ref. 40, which contains exact solutions to two space problems of this type. The first of these concerns an infinite slab which, in the absence of loads, is confined to a purely transverse motion, the stress-inducing temperature field history being permitted to vary arbitrarily with the thickness coordinate and the time. The second problem aims at the thermal stresses and deformations generated in a sphere by an arbitrary transient radial temperature distribution. Both solutions accom-modate an unrestricted (thermo-rheologically simple) temperature dependence of the material and apply to solids with a continuous relaxation spectrum,

* In connection with this extension of Alfrey's scheme the authors remark that the effect of body forces and surface tractions may be determined separately from, and may be super-posed upon, the ensuing thermal stresses and deformations. Such a superposition is in fact not permissible for a temperature-dependent material. Indeed, as has been pointed out already, a purely time-dependent temperature field would, in the absence of loads, always induce vanishing thermal stresses.


except that in the second problem the volumetric response is assumed to be elastic*

We return here briefly to the example of the slab treated in Ref. 40 since certain aspects of this problem have a wider significance. Thus consider an infinite plate of constant thickness 2a and choose the coordinate frame in such a way that X3 = 0 coincides with the middle plane of the plate. Then R is the region characterized by — a < X 3 < a and B consists of the pair of bounding planes x% = ± a. We now seek a thermo-rheologically simple viscoelastic statej [ut, €ij, Gfj] on R X (— oo, oo) corresponding to the data Gi , G2, a, 7b, <f>9

Ft, T9 subject to the boundary conditions

aw = 0 on Bx ( - 00 , 00 ) ( 6 . 2 0 )

We assume further that T, U3 are functions of (xs9 t) alone and

Fi = M I = u2 = 0 on R X ( — 00 , 00 ) ( 6 . 2 1 )

The foregoing constraining assumptions are consistent with the governing field equations and boundary conditions, which at present are readily found to imply! that all field histories depend solely on ( X 3 , t)9 while

€11 = ^22 = 0, €ij = 0 (1 # j) on R x ( — 00 , 00 ) ( 6 . 2 2 )

0 1 1 = 0-22, cr 3 3 = 0, ay = 0 (*' j) on R x (— oo, oo) ( 6 . 2 3 )

whereas

G = — i*dGi, 2a = (e — 3a@)*dG2 on ^ X (— oo, oo) ( 6 . 2 4 )

if we conveniently write a and e for a n and € 3 3 , respectively. The elimination of 2 among the pair of integral equations ( 6 . 2 4 ) , in view of the algebra of Stieltjes convolutions (Section 1 in Ref. 1), now leads to

a = — 3aA*d@ on R X (— 00 , 00 ) ( 6 . 2 5 )

where A is the auxiliary response function defined by

A = Gi*dG2*d(2Gx + G 2 ) - 1 on ( - 00 , 00 ) (6.26)§

Bearing in mind ( 2 . 7 ) , ( 5 . 6 ) , ( 5 . 7 ) , ( 5 . 8 ) , and putting x% = x, we draw from ( 6 . 2 5 ) that the desired stress a = a n = a 2 2 admits the integral representation

* Although this assumption, which is c o m m o n in the stress analysis of viscoelastic solids, is in general agreement with experimental findings, the available quantitative information regarding volumetric viscosity effects appears to be inadequate.

t N o t e that, contrary to our previous assumption. R is not bounded in this instance. t See Ref. 40 for details. § The same response function is also found to play a crucial role in the problem of the

sphere . 4 0

( 6 . 2 7 )


Formula (6.27) was obtained in Ref. 40 by applying the Laplace transform to the pair of integral equations (6.24) and by subsequently inverting the solution of the resulting pair of algebraic equations. As was noted later by Lee and Rogers 4 1 and as is clear from the preceding derivation, there is no need to involve the transform calculus. A similar observation is relevant to the numerical evaluation of the solution under discussion, which was carried out in Ref. 40 for a commercial polymethyl methacrylate on the basis of available relaxation data and test results for the shift function In this connection the auxiliary response function A was determined from (6.31). The laborious procedure adopted in Ref. 40 consisted in first devising a convenient analytical approxima-tion to the relaxation data that permitted the analytical computation of A; the required values of A were then found with the aid of two asymptotic inversions— one applicable to large, the other to small values of the time. In contrast, Lee and Rogers 4 1 subsequently re-computed A by solving the integral equation (6.29) directly on an electronic computer. Their procedure, which stays closer to the actually measured data, is evidently more direct and also more accurate.

The values of A obtained by the two alternative methods described above are compared in Fig. 1, in which E is Young's modulus of the initial elastic response. Figure 2, which is taken from Ref. 40, displays the time-dependence of the normal stress a in the middle plane of the slab for polymethyl metha-crylate. In the underlying computations it was assumed that a2/k = 1 hour, h being the thermal diffusivity of the material; this choice corresponds to a slab thickness 2a of approximately 5.7 cm. The temperature field history used in these calculations is that supplied by the solution to an elementary transient heat-conduction problem: the entire slab is initially at the uniform base tem-perature To = 80 °C and its faces at t = 0 are suddenly raised to the temperature T± = 110°C, which is steadily maintained thereafter.

* Recall (2 .13) .

so that A satisfies a linear integral equation of Volterra's second type. Finally, if the requisite Laplace transforms exist, (6.29) and (6.30) furnish

(6.31)

(6.30)

(6.29)

where

Moreover, (6.26), by virtue of Theorems 1.2, 1.3 in Ref. 1, for t in [0, oo) is equivalent to

with

(6.28)


log t

FIG. 1. A{t) for polymethyl methacrylate.

CD Temperature-dependent response: TQ = 8 0 ° C , T =IIO°C

© Temperature-independent response based on behavior at 9 5 ° C

® Elastic response based on initial behavior

FIG. 2. Slab problem. Time-dependence of a in middle plane for polymethyl metha-crylate.

Curve 1 in Fig. 2 shows the stress values obtained from (6.27). Curve 2 exhibits the behavior predicted if the temperature-dependence of the response is disregarded and the analysis is based on relaxation data appropriate to the average surface temperature of 95 °C. Curve 3, finally, shows the results obtained


if one neglects all viscosity effects as well and determines a on the supposition of purely elastic behavior, using the elastic constants characteristic of the initial viscoelastic response. As is apparent, the three curves almost coalesce for an initial period of about 10 min, during which the "actual" response is nearly elastic and hence temperature-independent. For approximately the first 20 min Curve 2 slightly underestimates the stress values predicted by Curve 1 since during this stage the temperature of the middle plane is less than the average temperature of 95 °C and consequently the actual relaxation process is slower than that underlying Curve 2. Later on, however, Curve 2 progressively over-estimates the stress values depicted by Curve 1 until, after a two-hour period, the actual stress is less than 15 per cent of the corresponding value in Curve 2. These results demonstrate once again the unrealistic character of any visco-elastic thermal-stress analysis that leaves out of account the temperature-dependence of the material properties.

The relative merits of the direct numerical integration of integral equations over the use of integral transform techniques in viscoelastic stress analysis were further discussed and illustrated by Lee and Rogers in Ref. 45. Apart from the fact that this approach makes more efficient use of experimental data, it obviates the need for long-time extrapolations from the data beyond the time range which the desired solution of the problem is to cover. Further, such a treatment extends the scope of the analysis to problems which are not amenable to a Laplace-transform solution. Among these are mixed problems in which the surface tractions and displacements are prescribed over time-dependent subsets of the boundary and problems in which the boundary itself is a function of the time (surface ablation). An example of the latter type is dealt with numerically in Ref. 45, where the solution to the problem of the sphere obtained in Ref. 40 is generalized to accommodate an ablating spherical body. In addition, Ref. 45 contains a formal solution to the problem of an ablating spherical shell; for fixed boundaries this problem was also treated previously in Ref. 6.

Although the developments just described are promising, it is well to keep in mind that boundary-value problems within the theory under consideration are reducible to the solution of independent integral equations only in highly exceptional circumstances. Furthermore, all of the applications available to date depend on the fortuitous possibility that the required space and time integra-tions can be carried out separately. While certain less degenerate problems may nowadays be accessible to a purely numerical treatment, such prospects should not detract from the need for a systematic theory of integration.

7. C O N C L U D I N G R E M A R K S

Throughout the preceding developments the material has been considered to be isotropic with regard to both its mechanical and its thermal response. For an anisotropic linear viscoelastic material the constitutive relations (2.6) give way to

(Jij = (etj — aij@)*dGijki (7.1)


in which Gyjci and ay are the components of the tensor of relaxation functions and of the thermal-expansion tensor, respectively. Furthermore, a n = a y and

The first of (7.2) follows from the symmetry of the stress tensor, while the second entails no loss in generality because of the symmetry of the strain tensor. In addition, it is usually assumed that

This last set of symmetry relations, which is implied by (7.2) in the special case of isotropy, expresses an independent requirement. The available theoretical support for the validity of (7.3) depends on thermodynamic arguments involving an appeal to Onsager's reciprocity relations.*

If (7.3) are presumed to hold, the generalization to homogeneous anisotropic solids of most of the theoretical results discussed in this paper, presents no difficulties. This is true, in particular, of the correspondence principle discussed in Section 4 and Section 6, which now leads to a tie between the linear theories of anisotropic viscoelastic and elastic solids, as suggested by Biot 4 9 . An extension of this principle to inhomogeneous (isotropic or anisotropic) viscoelastic solids is equally elementary,! though of little practical consequence.

Next, we have so far considered only the quasi-static analysis of thermal stresses in viscoelastic solids. Thermoviscoelastic inertia effects have been the object of several recent special investigations. Among these we cite publications by Katasonov 5 1 , N o w a c k i 2 3 ' 5 2 > 5 3 > 5 4 , and Z o r a w s k i 5 5 - 5 8 . The physical signifi-cance of these investigations is difficult to assess since they invariably pre-suppose a temperature-independent material, do not include quantitative discussions of the results obtained, and usually refer only to instantaneous temperature changes.}

Further, nothing has been said as yet about thermo-mechanical coupling effects, which have been consistently disregarded. An interesting study of such effects is due to Hun te r 6 0 , whose analysis is set within the framework of the theory of thermo-rheologically simple viscoelastic solids. As suggested by the unassuming title of Ref. 60, this subject is in need of further consideration.

Finally, it may be well to emphasize once more that the present paper is confined essentially to linear thermal stress analysis for viscoelastic solids. Thus no attention has been given to the influence of nonlinear viscosity effects and to finite deformations, which haunt the realistic treatment of thermal stresses in metals at elevated temperatures.

* See, for example, Biot 4 6 * 4 7 . A discussion of this issue may be found in a recent note by Rogers and P ipkin 4 8 , as well as in Ref. 5. N o t e that the counterpart of (7 .3 ) in elasticity theory is implied by the existence of an elastic potential.

t In this connection see Refs. 20, 44, as well as Hilton and D o n g 5 0 . % Related studies in thermoelasticity theory (see, for example, Ref. 59), indicate that the

size of such inertia effects is drastically reduced once the fiction of sudden temperature changes is abandoned.

Gyjci = Gjikl = Gyiic (7.2)

Gym = Guy (7.3)


A C K N O W L E D G M E N T

The author is greatly indebted to M. E . Gurtin, who read the manuscript and offered numerous helpful criticisms and suggestions.

R E F E R E N C E S

1. GURTIN, M. E. and STERNBERG, ELI. "On the Linear Theory o f Viscoelasticity" Arch. Rat. Mech. Anal. 1 1 , 4, 20 (1962).

2. KELLOGG, O. D . Foundations of Potential Theory. Berlin: Springer (1929). 3. BOLEY, B. A. and WEINER, J. H. Theory of Thermal Stresses. N e w York: Wiley (1960). 4. VOLTERRA, V. "Sulle equazioni integro-differenziali della teoria dell' elasticita" Atti R.

Accad. Line. 1 8 , 2, 295 (1909). 5. GURTIN, M. E. and STERNBERG, ELI. A Reciprocal Theorem in the Linear Theory of Aniso-

tropic Viscoelastic Solids. Report N o . 17, Contract Nonr-562(25) , Brown University (September, 1962).

6. STERNBERG, ELI and GURTIN, M. E. Further Study of Thermal Stresses in Viscoelastic Materials with Temperature-Dependent Properties. Report N o . 2, Contract Nonr-562(30) , Brown University (October 1961). T o appear in Proc, I.U.T.A.M., Symposium on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, Haifa (1962).

7. NOWACKI, W. Thermoelasticity. L o n d o n : Pergamon Press (in press). 8. HIEKE, M. "Eine indirekte Bestimmung der Airyschen Flache bei unstetigen Warmespan-

nungen" Z. angew. Math. Mech. 3 5 , 285 (1955). 9. HILTON, H. H. "Thermal Distributions without Thermal Stresses in Nonhomogeneous

Media" ./. Appl. Mech. 2 6 , 1, 137 (1959). 10. GURTIN, M. E. Variational Principles in the Linear Theory of Viscoelasticity. Report N o .

18, Contract Nonr-562(25) , Brown University (January 1963). 11. A L KHOZAIE, S. and STERNBERG, ELI. On Green's Functions and Saint Venanfs Principle

in Linear Viscoelasticity Theory. To appear in Arch. Rat. Mech. Anal. 12. BORCHARDT, C. W. "Untersuchungen iiber die Elasticitat fester isotroper Korper unter

Berucksichtigung der Warme" Monatsber. Akad. Wiss. Berlin, 9 (1873). 13. PARKUS, HEINZ. Instationdre Warmespannungen. Wien: Springer (1959). 14. CORNELIUSSEN, A . H. and LEE, E. H. "Stress Distribution Analysis for Linear Visco-

elastic Materials" Proc, I.U.T.A.M., Berlin: Springer (1962). 15. ALFREY, T. "Non-Homogeneous Stresses in Visco-Elastic Media" Quart. Appl. Math.

2 , 2, 113 (1944). 16. TSIEN, H. S. "A Generalization of Alfrey's Theorem for Viscoelastic Media" Quart.

Appl. Math. 8 , 1, 104 (1950). 17. READ, W. T. "Stress Analysis for Compressible Viscoelastic Materials" Appl. Phys. 2 1 ,

7, 671 (1950). 18. BRULL, M. A. "A Structural Theory Incorporating the Effect of Time-Dependent

Elasticity" Proc. First Midwestern Conf Solid Mech. Urbana (1953).

19. LEE, E. H. "Stress Analysis in Visco-Elastic Bodies" Quart. Appl. Math. 1 3 , 2, 183 (1955).

20. HILTON, H. H., HASSAN, H. A . and RUSSELL, H. G. Analytical Studies of Thermal Stresses in Media Possessing Temperature-Dependent Viscoelastic Properties. Technical Report 53-322, Wright Air Development Center (September 1953).

21. STERNBERG, ELI. "On Transient Thermal Stresses in Linear Viscoelasticity" Proc. Third U.S. Nat. Cong. Appl. Mech. (1958).

22. NOWACKI, W. "Thermal Stresses D u e to the Action of Heat Sources in a Viscoelastic Space" Arch. Mech. Stos. 1 1 , 1, 111 (1959).

23. NOWACKI, W. "Transient Thermal Stresses in Viscoelastic Bodies (I)" Arch. Mech. Stos. 1 1 , 5, 649 (1959).

24. SOKOLOWSKI, M. Thermal Stresses in a Viscoelastic Sphere (in Polish). W. Wierzbicki Anniversary Volume. Warsaw (1959).

25. SHAFFER, B. W. and LEVITSKY, M. "Thermal Bond Stresses in Case-Bonded Viscoelastic Propellant Discs" / . Aerospace Sci. 29, 7, 827 (1962).

26. FREUDENTHAL, A . M. "Effect of Rheological Behavior on Thermal Stresses" / . Appl-Phys. 2 5 , 9, 1110 (1954).


27. FREUDENTHAL, A . M. On Inelastic Thermal Stresses. V o n Mises Anniversary Volume. N e w York: Academic Press (1954).

28. FREUDENTHAL, A . M. "On Inelastic Thermal Stresses in Flight Structures" / . Aero. Sci. 2 1 , 11, 772 (1954).

29. FREUDENTHAL, A . M. "Problems of Structural Des ign for Elevated Temperatures" Trans. New York Acad. Sci. Ser. II, 19, 4, 328 (1957).

30. HILTON, H. H. "Thermal Stresses in Thick-Walled Cylinders Exhibiting Temperature-Dependent Viscoelastic Properties of the Kelvin Type" Proc. Second U.S. Nat. Cong. Appl. Mech. (1954).

31. RONGVED, L. Residual Stress in Glass Spheres. Report N o . 16, Contract Nonr-266(09) , Columbia University (July 1954).

32. WEINER, J. H. and MECHANIC, H. Thermal Stresses in Free Plates under Heat Pulse Inputs. Technical Report 54-428, Wright Air Development Center (March 1957).

33. LANDAU, H. G., WEINER, J. H. and ZWICKY, E. E. , Jr. "Thermal Stress in a Viscoelastic— Plastic Plate with Temperature-Dependent Yield Stress" / . Appl. Mech. 2 7 , 2, 297 (1960).

34. AGGARWALA, B. D . "Thermal Stresses in Spherical Shells of Viscoelastic Materials" Z. angew. Math. Mech. 4 0 , 482 (1960).

35. MORLAND, L. W. and LEE, E. H. "Stress Analysis for Linear Viscoelastic Materials with Temperature Variation" Trans. Soc. Rheology, 4 , 233 (1960).

36. LEADERMAN, H. Elastic and Creep Properties of Filamentous Materials. Washington, D . C . : Textile Foundation (1943).

37. FERRY, J. D . "Mechanical Properties of High Molecular Weight" J. Amer. Chem. Soc. 7 2 , 3746 (1950).

38. SCHWARZL, F. and STAVERMAN, A. J. "Time-Temperature Dependence of Linear Visco-elastic Behavior" J. Appl. Phys. 2 3 , 838 (1952).

39. STAVERMAN, A . J. and SCHWARZL, F. "Linear Deformation Behavior of High Polymers" Chapter 1 in Die Physik der Hochpolymeren. Berlin: Springer (1956).

40. MUKI , R. and STERNBERG, ELI . "On Transient Thermal Stresses in Viscoelastic Materials with Temperature-Dependent Properties" / . Appl. Mech. 2 8 , 2, 193 (1961).

41. LEE, E. H. and ROGERS, T. G. Solution of Viscoelastic Stress Analysis Problems Using Measured Creep or Relaxation on Data, Interim Technical Report N o . 1, Grant D A - A R O ( D ) - 3 1 - 1 2 4 - G 5 4 , Brown University (August 1961). T o appear in / . Appl. Mech.

42. STERNBERG, ELI and GURTIN, M. E. Uniqueness in the Theory of Thermo-Rheologically Simple Ablating Viscoelastic Solids. Report N o . 16, Contract Nonr-562(25) , Brown University (September 1962).

43 . MINDLIN, R. D . and SALVADORI, M. G. "Analogies", in Handbook of Experimental Stress Analysis. N e w York: Wiley (1950).

44. HILTON, H. H. and RUSSELL, H. G. "An Extension of Alfrey's Analogy to Thermal Stress Problems in Temperature Dependent Linear Viscoelastic Media" / . Mech. Phys. Solids, 9, 152 (1961).

45. LEE, E. H. and ROGERS, T. G. Nonlinear Effects of Temperature Variation in Stress Analysis of Isothermally Linear Viscoelastic Materials. Report N o . 3, Contract Nonr-562(30) , Brown University (May 1962). T o appear in Proc, I.U.T.A.M., Symposium on Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, Haifa (1962).

46. BIOT, M. A . "Theory of Stress-Strain Relations in Anisotropic Viscoelasticity and Re-laxation Phenomena" / . Appl. Phys. 2 5 , 11, 1385 (1954).

47. BIOT, M. A. "Linear Thermodynamics and the Mechanics of Solids" Proc. Third U.S. Nat. Cong. Appl. Mech. (1958).

48. ROGERS, T. G. and PIPKIN, A . C. Asymmetric Relaxation and Compliance Matrices in Linear Viscoelasticity. Report N o . 83, Contract Nonr-562(10) , Brown University (July 1962). T o appear in Z. angew, Math. Phys.

49. BIOT, M. A. "Dynamics of Viscoelastic Anisotropic Media" Proc. Fourth Midwestern Conf Solid Mech., Lafayette (1955).

50. HILTON, H. H. and DONG, S. B. An Analogy for Anisotropic, Nonhomogeneous, Linear Viscoelasticity Including Thermal Stresses. T o appear as Aerojet-General Corp. Technical Report TP120SRP.

51. KATASANOV, A . M. "Propagation of Spherical Thermal Viscoelastic Excitations (in Russian)" Vestnik M. G. U, ser. Mekh. Matem. Nauk, 3 , 39 (1957).

52. NOWACKI, W. "Thermal Stress Propagation in Visco-Elastic Bodies (I)" Bull Acad. Polonaise Sci., Ser. Sci Tech. 7 , 4, 257 (1959).


53. NOWACKI, W. "Thermal Stress Propagation in Visco-Elastic Bodies (II)" Bull. Acad. Polonaise Sci., Ser. Sci. Tech. 7 , 7 -8 , 459 (1959).

54. NOWACKI, W. "Ausbreitung der Warmespannungen in viskoelastischen Korpern" Osterr. Ing. Arch. 15, 1 - 4 , 1 1 5 ( 1 9 6 1 ) .

55. ZORAWSKI, M. "States of Stress Generated in a Viscoelastic Semi-Space by a Flat Heat Source" Bull. Acad. Polonaise Sci., Ser. Sci. Tech. 8 , 4, 161 (1960).

56. ZORAWSKI, M. "Determination of Stresses Generated in a Layer and a Viscoelastic Closed Spherical Shell" Bull. Acad. Polonaise Sci., Ser. Sci. Tech. 8 , 10, 557 (1960).

57. ZORAWSKI, M. "Moving Dynamic Heat Sources in a Visco-Elastic Space and Correspond-ing Basic Solutions for Moving Sources" Arch. Mech. Stos. 2, 13, 257 (1961).

58. ZORAWSKI, M. "Dynamic Nucleus of Thermoelastic Strain in Viscoelastic Space" Bull. Acad. Polonaise Sci., Ser. Sci. Tech. 9 , 2, 77 (1961).

59. STERNBERG, ELI and CHAKRAVORTY, J. G. "On Inertia Effects in a Transient Thermo-elastic Problem" / . Appl. Mech. 26, 503 (1959).

60. HUNTER, S. C. "Tentative Equations for the Propagation of Stress, Strain and Temperature Fields in Viscoelastic Solids" / . Mech. Phys. Solids 9 , 39 (1961).

ANALYTICAL METHODS OF CREEP DESIGN,

ESPECIALLY WITHIN THE NONLINEAR RANGE

L . M . KACHANOV

Mathematical Faculty, Leningrad State University, U.S.S.R.

I N T R O D U C T I O N

This paper deals with problems of creep in metals; however, some of the results presented are also applicable to nonmetallic materials.

Problems of creep of metals are best handled by applying a comparatively simple model of quasi-viscous flow, and attempting to find simple approximate solutions. While it is true that the creep of metals is governed by nonlinear laws, such nonlinearity in the flow equations of the creep theory creates difficult mathematical problems which can be solved only by extremely complicated numerical methods; in such cases even the most modern procedures of num-erical analysis are not always effective. Moreover, creep deformations are very sensitive to small changes in stress, temperature, grain structure and heat treatment, which cause wide scatter in experimental data. Under such circum-stances, simple theoretical formulations and approximate solutions are prefer-able to complicated ones, even though the latter may be somewhat more exact, a fact that has been emphasized by many authors.

This paper deals only with variational methods of solution of nonlinear creep problems. The author, whose attention has been centered on this field, believes that such a limitation is justified by the fact that some interesting results of practical value have been obtained with the aid of these methods; moreover, it does not seem expedient to deal with the various aspects of the solution of nonlinear creep problems in a single paper.

Therefore the application of the methods of successive approximations, of finite differences, of perturbation, and that of Galerkin will not be dealt with in this paper and only a few words will be said about them. Thus, it should be noted that the method of successive approximations has often been applied to the problem of steady creep of discs but is seldom applied to other problems. Because of numerical difficulties the method of finite differences has not been widely applied to problems of creep. Galerkin's method coincides in fact with the variational method as shown by Michlin and Gagen-Torn 1 . The method of perturbations is applicable only to a few specific problems, such as creep of slightly elliptical tubes and of tubes of variable thickness.

We shall not present a discussion of experimental data or of the analysis of the various creep theories. Such analysis has been presented in numerous papers

383

384 L . M. K A C H A N O V

and books on the theory of creep, such as Marin 2 , Finnie and Heller 3, Rabot-nov 4 , Kachanov 5 , Odqvist and Hull 6 , and Johnson 7 .

where a = \ u u , 8tj is Kroneker's delta, G the elastic shear modulus, v Poisson's ratio and t the time from the moment of loading. The square root of the second invariant of the stress deviation ("shear stress intensity")

where s t j denotes the stress deviation tensor. The intensity of the creep rate

y* = 2 V / 2 = (2etj ^ ) V 2 (1.3b)

depends on r* in the form y * = f ( r * , t ) T * (1.4)

If all creep curves are similar

f(r*,t) = B(t)fi.(r*) (1.5)

In the particular case of the assumption of a simple power function

f(r*9t) = B(t)(r*)^-l (1.6)

where m is a constant which, as a rule, is much larger than unity and B(t) is a positive, monotonically decreasing function of time. The creep curves in simple tension (Fig. 1) can then be represented as follows:

e n = C i ( f ) * ? l (1 .7)

where the function C±(t) is proportional to some of the observed creep curves. For long periods of time Ci(t) is approximately linear (Fig. 2 ) . It is easy to see 4 .1. „ A.

The form of equation ( 1 . 2 ) , including also the Maxwell continuum, has been suggested by Jeffreys8, Odqvist 9 , Davenpor t 1 0 , Mar in 2 and others. They fairly

( 1 . 2 )

1. C R E E P E Q U A T I O N S

The deformation components are made up of the elastic deformation component e t j and the creep component e # :

€a = €ij + €ij (l •!)

The elastic deformation is governed by Hooke's law. Since we shall deal only

with slow monotonic changes in time of the stress component o i p creep under

such conditions can be considered as a process of nonlinear viscous flow. The

strain rates e t j are defined by the relation

A N A L Y T I C A L M E T H O D S OF CREEP D E S I G N 385

well describe the significant features of creep under monotonically varying stress and under conditions of simple loading. Some authors replace the function B(t) by the constant B = lim B(t); however, this substitution, which neglects

the period of primary creep, is hardly admissible since it invalidates the theo-retical solution for creep problems within the range of nonsteady creep.

FIG. 1. Creep curves in tension.

FIG. 2. Parameters B{t) and Ci(t).

If we introduce the elastic potential

(1.8)

where K = 3(1 — 2v)jE denotes the bulk compliance, and the "complementary rate of energy dissipation"

(1.9)

the creep equations (1.2) can be represented in the form

(1.10)

The function E = A + d<P/dt will be called the "complementary power". In the case of similarity of the creep curves we have A = B(t) AI(T*), where

Ai does not depend on time. For AI(T*) we introduce a power function so that

(1.11)

14

386 L. M. K A C H A N O V

The above equations are easily generalized for the non-uniformly heated body for which / = / ( T * , t, 6) where, however, the temperature 6 is assumed not to depend on time. Moreover, under fairly simple conditions of loading, plastic deformations can be described by the equations of the theory of elastic-plastic deformation in which case instead of equation (1.10) we have the equation

(1.12)

where II denotes the complementary energy; for unloading II = 0.

2. S T E A D Y S T A T E C R E E P

In steady state creep the stress components as well as the components of strain velocity do not depend on time. Therefore instead of equation (1.2)

while 2 ^ - = / ( T * ) stj or = 2g(y*) ii:

7* = / ( T * ) T * o r T * = g(y*) 7*

Introducing the specific dissipation potential

g(y*) y* dy*

(2.1)

(2.2)

FIG. 3. Relation y* = / ( T * ) T * .

and the specific complementary dissipation potential (Fig. 3) according to equation (1.9), the relations (2.1) can be written in the form

(2.3)

Introducing the power function for the relation between the invariants

y* = B(r*)m and r* = B(y*y (2.4)

where /x = 1/ra and B = B~^, the following expressions are obtained for the potentials:

and therefore

(2.5)

(2.6) L = mA


where Pi is a generalized force and vi is a corresponding velocity of the point of application of this force.

The above results are analogous to the results obtained by Kachanov in the theory of elastic-plastic deformation or in the theory of nonlinear elastic bodies . 1 3 The analogy between the solutions of the corresponding problems for a nonlinear elastic body and the body under steady state creep has been empha-sized by Kachanov 1 1 and by Hoff 1 4 .

Transforming W with the aid of the Gauss-Ostrogradsky equation we obtain the energy theorem 1 1

W=$r*y*dV (2.9)

which, under the assumption of a power function, takes the form

(2.10)

We consider now a body of finite volume V and surface S under the action of stationary volume forces Xi and stationary surface tractions ti that are prescribed over a certain part St of the total surface S of the body, while the velocity vi

is prescribed over the rest of the surface Sv, so that under a general velocity variation 8vt for all points of the body, hvi = 0 over the surface Sv. Applying the divergence theorem

and using equation (2.3) the following relation is obtained

S(L* — W) — 0 (2.7)

where W denotes the sum of the rate of work of the surface forces and of the body forces and L* the rate of energy dissipation L* = J LdV. It is easy to see

V

that 82L > 0 as a result of the condition that dr*jdy* > 0. Hence the real velocity vi is associated with a minimum of total rate of energy dissipation or

(L* - W) -> min

If we follow Hill's reasoning 1 2 we see that this is an absolute minimum. In order to derive the principle of a minimum rate of complementary dissi-

pation energy we compare the real state of stress with an infinitely close statically admissible state of stress + where vi = 0 on Sv. Hence

(2.8)

Under the same conditions as above dr*/dy* > 0, we have SM > 0 so that L* is a minimum.

It is not difficult to prove the following generalization of Castigliano's theorem

388 L . M. K A C H A N O V

It should be mentioned that if St. Venant's polygonal yield surface and associated flow ru l e 1 5 is applied, it is possible to prove variational principles analogous to those of equations (2.7) and (2.8).

With the aid of the principles derived above, one can conveniently study the creep of trusses, beams and f r a m e s . 1 1 ' 1 6 In more complicated problems the finding of the minimum presents considerable difficulties. Under zero conditions on the surface Sv, only the first approximation can be defined by the Ritz method in the form vi = cv* if equation (2.7) is used, where c is a constant and is a suitable velocity field. The solution for the corresponding linear ("elastic") problem is usually taken for v*. These methods of solution are frequently used for obtaining approximate solutions of engineering problems, for instance, that of creep of turbine diaphragms solved by Rosenblum 1 7 .

The example of bending of a cantilever beam of span / by a force applied to its free end is evidence of the unreliability of this procedure. The ratio of the approximate velocity of deflection v*(l) to its exact value is, in the case of a power function for the stress-strain relation e n = Bio™,

For m = 7, for instance, this ratio is 0.285. This difference is a result of a dis-turbance of the equilibrium equations. In other problems, however, for in-stance in problems of the bending of plates, the differences are probably smaller.

This method with one arbitrary constant is more effective when the unknown value is proportional to the value to be minimized. The problem of bending of a curved tube (Fig. 4) under conditions of c reep 1 8 may serve as an example. We are looking here for the minimum of the complementary rate of energy dissipation.

3. O N V A R I A T I O N A L M E T H O D S O F

S O L U T I O N O F P R O B L E M S

FIG. 4. Curved tube.


while the coefficient of flexibility of the curved tube A = A*/A*0, where A*0 is a complementary rate of energy dissipation for the straight tube. The stress components o^ , are connected by a differential relation. The solution is found in the form

FIG. 5. Coefficeint of flexibility A(X2).

We shall now describe ways for the successful use of variational me thods 1 9

which may also be applied when looking for minima in other nonlinear prob-lems. Taking, for instance, the variational equation (2.8), we construct a suc-cessive approximation in the form

o /*

FIG. 6. Sheer moduli G(T*) .

given conditions on St9 while Gtj represents solutions of the equilibrium equa-tions which satisfy the zero boundary conditions on Stl Cjcs are arbitrary constants. Assuming now that / ( T * ) = const. = I/Go (Fig. 6) we find the zero approximation a[? } which corresponds to the elastic problem and compute

where o^-o is a particular solution of the equilibrium equations which satisfies

( 3 . i ;

where the constant c is obtained from an algebraic equation when m is set equal to m = 1, 3, 5, 7, . . . . The curves of the coefficient A are shown in Fig. 5. With m = 1 the problem corresponds to Karman's problem; A2 = a2/Rh.

[2 sin <f> + 6c sin 3(/>)

390 L . M. K A C H A N O V

T*(°). We now assume that Gi = g(r*^/Go) and determine the first approxi-mation o\p from the condition of a minimum of the quadratic function

For the kth approximation

The solutions of problems in the theory of elasticity based on variational methods are easily extended to corresponding creep problems. In this case it is reasonable to keep the number of terms in equation (3.1) unchanged. When determining the secant modulus Gk we start directly from the experimental relation between the stress intensity T* and strain rate intensity y*. Keeping the same form of the solution in every approximation and changing only the co-efficient Ck8, the computation can be simplified a great deal; in contrast to other methods of successive approximation the results obtained are less complicated. The convergence of this method was proved by R o s e 2 0 and the method has been successfully used in the solution of a number of problems, such as the torsion of a bar of square cross-section 2 1, the tension of a square plate under non-uniform loads, etc.

Let us now say a few words about simple approximate solutions which are characterized by a certain similarity 2 2, and restrict ourselves to the case when the velocity vi = 0 on Sv. We believe that the power function (2.4) is applicable here. The state of stress of the body does not depend on the coefficient B, while the exponent m influences the state of stress considerably. When m = 1 the state of stress obviously coincides with the state of stress in the corresponding problem of an elastic body. The author and o the r s 1 4 * 1 6 have considered the analogy between the state of stress in the nonlinear medium when m tends toward infinity and the state of stress in the corresponding ideal plastic body, and we can assume that the limiting state of stress a?, with m - > oo is developing in the "ideal creeping body".

In general the question of finding the limiting state is not a simple one. In order to determine certain properties of this state let the tractions specified on St change in proportion with the parameter A, i.e. ti = Xtio, and let the creep law be specified in the form y* = fi(j*lT*\)m where r\ is a certain characteristic value of the stress intensity; the coefficient fi has a specific value when m is finite and may have any finite value when m -> oo.

According to Ilyushin's theorem 2 3

Vi = vi0Xm

where vi0 is a vector proportional to fi and depending on the coordinates and the parameters T\ and m. Whenever m is finite, vi is determined and limited; when m - > oo we consider vi to be finite but containing an arbitrary factor.


According to equation (2.10) we have

W = A^+i J ti0 vi0 dS = A^+i Wo

where Wo is finite. From the energy theorem

Am+i w0 = (m+ 1) A*

or

From the fact that Wo is finite we arrive at the conclusion that in the limiting state there are regions of two types in the body:

for type (1): T*/AT* = 1; for type (2): T*/AT* < 1. There must always exist regions of the first type because Wo > 0; in these

regions the intensity T * is constant

r* = AT* = const.

This value is proportional to the parameter associated with the external loads. Because of the intensive flow taking place in these regions, y * W > 0. From the relation

it follows that y*< 2 ) ->0 and that therefore e<2> 0 ; hence regions of the second type are undergoing only rigid body displacement. Thus if in a certain problem rigid body displacements do not arise, the limiting state of creep co-incides with the ideal elastic state. The only difference is that there is no fixed yield stress; it is expressed by the load.

Considering, for instance, the creep of a hollow sphere subject to internal pressure, the rate of velocity vr = C/r2, where C is an arbitrary constant. The strain rate intensity y* = (2\/3jr) vr. Regions of the second type do not exist; hence the state of stress converges to the ideal plastic state when m - > oo. This conclusion can be verified by introducing m ^ oo into the solution.

The considerations can be easily extended to the torsion problem, which is not included in the boundary value problems discussed above. Using the energy theorem equation (2.14) let us assume that in the torsion problem regions of the second type exist; then in these regions

V 4 + 4 o

and therefore

where a> is a rate of torsion per unit length and the function cf>(x, y) character-izes the rate of warping of the cross-section. Since these relations lead to con-tradictions the limiting creep state coincides with the ideal plastic state.

y*


When m = 1 the factor K = 1; as m grows larger the state of stress of the body changes from the state atj to the state 4- If the limiting state 4 does not coincide with the state of stress for a finite value of m then it may be assumed that K > 0 for any finite value of ra, while (dK/dm) < 0 (Fig. 7(a)).

FIG. 7. Parameter K as function of m.

For the determination of K(m) the following method is expedient: let m be an integer, for example, m = 3, 5, 7, 9, . . . then the expected value K(m) can be roughly estimated from the ratio l/m. By giving K suitable values larger and smaller than 1 \m for every given value of m, Pm(K) for the selected m and K(m) are computed and the unknown root determined graphically. Having obtained K(3), K(5), K(l), . . . the curve K(m) can be drawn. Comparing the approximate solution equation (3.2) with the exact solutions obtainable in the cases of torsion of a round bar, pure bending or of thick-walled spheres and thick-walled cylinders under internal pressure, it can be seen that this solution is practically adequate.

If the limiting creep state is unknown, the state of stress for a large enough value of m can be introduced instead of the limiting state o-^.

It should be mentioned that in certain exceptional cases involving simple systems with a finite number of generalized coordinates (trusses, frames) the

The simple approximate solution of the steady state creep problem when tractions are given on St and vi = 0 on Sv can be obtained by means of the variational principle (2.8). If the solution is specified in the form

(3.2)

the factor K = K(m) can be determined from the equation

o{j = 4 + K(atj - a« )

or

(3.3)

A N A L Y T I C A L M E T H O D S OF C R E E P D E S I G N 393

limiting state of stress for m - > oo may coincide with the state of stress for a certain finite value of m. In such cases the diagram K = K(m) has the form shown in Fig. 7(b).

Finally it should be pointed out that the methods described above can be used for extrapolation of the results of a recently developed method of photo-creep. 2 4 For instance, take the results of the solution of the problem of torsion

FIG. 8. Parameter K(m), 0(m) and J*(m) in torsion of bar of square cross-section.

of a bar of square cross-section of length 2a. Because of symmetry only one-quarter of the square between the diagonals need be considered. We have

where M is the torque. The stress components

where 0 = <P(x, y) denote the stress function of the elastic problem. The co-efficient K(m), the ratio 6(m) of the maximum shear stress in creep to the maxi-mum shear stress in an elastic bar, and the value J*(m) all of which are shown in Fig. 8, determine the rate of twist:

With the aid of this method some simple solutions of problems of creep of circular plates have been obtained. 5

4. P R I N C I P L E O F M I N I M U M C O M P L E M E N T A R Y E N E R G Y

D I S S I P A T I O N R A T E I N P R O B L E M S O F

T R A N S I E N T C R E E P

For a statically admissible variation of the state of stress which does not change with time

14§


Hence

while

Let us distinguish three boundary value problems for which the minimum principle (4.1) is valid, assuming for the sake of simplicity that body forces are zero.

4 . 1 . The Fundamental Problem

Tractions that remain constant in time are given over the surface S of the body.

4 .2 . The Relaxation Problem

The tractions ti = 0 on St, while on St the displacements that are constant in time are given, or vi = 0.

4 . 3 . The Mixed Problem

Tractions that are constant in time are given on St and displacements con-stant in time are given on Sv.

5. T H E A P P R O X I M A T E S O L U T I O N O F

T R A N S I E N T C R E E P P R O B L E M S

The exact solution of transient problems is analytically difficult even in simple cases such as pure bending. As mentioned before, it seems hardly worthwhile to look for exact solutions with the aid of complicated methods, particularly since a process of transient creep can be presented in a rather simple form. Let us consider some approximate solutions obtained by Kachanov 1 1 .

5 . 1 . The Fundamental Problem

At initial time t = 0 the state of stress and deformation is described by the elastic equations. The corresponding values will be marked by primes (o-^, e'^); this solution is assumed to be known. The stresses and rates of deformation due to steady state creep are marked by double primes (o-^., <Q and are also

* Similar variational principles can be developed for other creep theories. For the theory of work hardening this was shown in the article by Schesterikov 2 5 .

t The generalization of the minimum principle (4 .1 ) has been used for the solution of the problem of creep buckling by R o s e n b l u m 2 6 .

Therefore, and because it is evident that S2E > 0, the true stresses are associated with the minimum value of the rate of complementary energy dissipation in the body

( 4 . 1 ) * |


assumed to be known. The process of transient creep is now investigated with the aid of the variational principle (4.1). By looking for approximate solutions of this equation in the form

(5.1)

where the factor k(t) is an arbitrary function of time, and by assuming for the sake of simplicity that the creep curves are similar so that A* = B(t) A* and

o l k

FIG. 9. Q(k) for the fundamental problem.

introducing the value from (5.1) into <£* and A*, it is evident that &* and A* are functions of k only. Thus (4.1) leads to

(5.2)

The values referring to stress differences will be marked by a bar (a{j

a i j ~ G'ip e t c - ) - Since &*(k) is a positive quadratic function of k

(5.3)

It is evident that <f>* does not depend on k; it is also assumed that dtj =?k 0 Integrating (5.2) with the initial conditions

we obtain t = 0 ; k = 0 (5.4)

(5.5)

where

The stresses a % i are the solution of the variational principle (2.8). If we look for the solutions of this equation in the form (5.1) the factor k is

obtained from the equation Q(k) = 0 which has a solution k = 1; hence


<2(1) = 0. It is possible to prove that

Q(k) is thus a monotonically decreasing positive function in the interval be-tween 0 < k < 1 (full line in Fig. 9). The integral in (5.5) diverges when k = 1. The function Q{t) increases monotonically and tends toward infinity in a linear relation. Thus equation (5.5) determines k{t) as a monotonically increasing time function which asymptotically approaches k = 1 as t - > oo (Fig. 10).

On the ordinate axis the function Q(k) cuts off the segment Q(0) > 0. For a power function with an odd value of m, Q(k) is a polynomial of degree m of a form

Q(k) = 2(0) (1 - k) Qm_x (k)

where the polynomial Qm-i(k) is positive in the interval between 0 < k < 1,

while 2™-i0) = 1. It is recommended to construct the integral curve of equation (5.2) by num-

erical methods. However, computations have shown that the linear approxi-mation (dotted line in Fig. 9)

Q(k) = 2(0) (1 - k)

is quite adequate. Then from (5.5)

k=l - e x p ( - ; * ) (5.6)

where t* is a non-dimensional time parameter

= Q(t) 2(O)/20*

It should be mentioned that the above method can be easily generalized for a non-uniformly heated body, for the case of plastic deformation and for the case when the creep curves are not similar. It has been used in the solution of many practical problems such as tubes, discs, plates and bars. Compared to known exact solution, it gives good approximations for the principal stress component.


where o-f) is the solution (5.1) described above, offi are the particular solutions of the homogeneous differential equilibrium equation under zero boundary conditions on St, and Cn(t) are arbitrary time functions which are determined by a system of ordinary differential equations. It is possible to show that the functions Cn(t) tend asymptotically toward zero as t - > oo.

Applications of this method to problems of transient creep of plates and shells have recently been described in a paper by Teregulov 2 8 .

In this approximate solution the relaxation curve for a fixed value of m is computed once for any form of the body; only the time scale changes for different problems. In some problems certain parts of the body are relaxing independently of each other. The simplest examples of such bodies are trusses with fixed hinges. In dealing with such "disintegrating systems" the approxi-mate method of solution is applied separately to every part of the system. More exact solutions can be constructed in the form (5.7).

Finally, it should be mentioned that approximate solutions of mixed problems could be formulated with the aid of equation (5.1).

5.2. The Relaxation Problem

At initial time t = 0 the stress components are those of the elastic state a\y Because of creep these stresses gradually decrease in intensity, a process that can be approximately described by a solution of the form

°ij = p(t)°t, (5.8)

where the relaxation factor p(t) is an arbitrary function of time with p(0) = 1. The minimum principle (4.1) provides a differential equation for p(t). Its solution is

(5.9)

since d2(P*ldp2 = 2<P*' > 0, and does not depend on p. It is easy to see that

T * = p T * ' and dA\jdp > 0. In the case of a power law with m > 1 we have

P = [1 + (m - 1) /*JV(m-l)

where /* = f<Q(t) is a non-dimensional time parameter in which

A method for finding more exact solutions has been obtained by Kachanov 2 7 . Let us look for the solution of the minimum principle equation (4.1) in the form

398 L . M. K A C H A N O V

R E F E R E N C E S

1. GAGEN-TORN, N . and MICHLIN, S. G. Doklady Acad. Sci. USSR, Vol. 1 3 8 , N o . 2 ( 1 9 6 1 ) . 2 . MARIN, J. Mechanical Properties of Materials and Design. N e w York and London:

McGraw-Hil l ( 1 9 4 2 ) . 3 . FINNIE, I . and HELLER, W . Creep of Engineering Materials. N e w York: McGraw-Hill

( 1 9 5 9 ) .

4 . RABOTNOV, Y . N . Vestnik Mosk. Univ. N o . 1 0 ( 1 9 4 8 ) . 5 . KACHANOV, L. M. Theory of Creep. M o s c o w : Fismatgiz ( 1 9 6 0 ) . 6 . ODQVIST, F . K . G. and HULT, J . Kriechfestigkeit metallischer Werkstoffe. Berlin: Springer-

Verlag ( 1 9 6 2 ) . 7 . JOHNSON, A . Complex Stress, Creep, Relaxation, and Fracture of Metallic Alloys.

Edinburgh ( 1 9 6 2 ) . 8 . JEFFREYS, H. Proc. Royal Soc. London ( A ) , 1 3 8 ( 1 9 3 2 ) . 9 . ODQVIST, F. K . G. Proc. Fourth Int. Congress of App. Mechanics ( 1 9 3 4 ) .

1 0 . DAVENPORT, C. C. J. Appl. Mechanics, 5 , 2 ( 1 9 3 8 ) . 1 1 . KACHANOV, L. M. Prikl. Matem. Mechan. 1 3 , 4 ( 1 9 4 9 ) . 1 2 . KACHANOV, L. M. Some Questions of the Theory of Creep. M o s c o w : Gostekhizdat ( 1 9 4 9 ) . 1 3 . KACHANOV, L. M. Prikl. Matem. Mechan. 6 , 2 - 3 ( 1 9 4 2 ) . 1 4 . HOFF, N . J . Quart. Appl. Math. 1 2 , 4 ( 1 9 5 4 ) . 1 5 . PRAGER, W . / . Appl. Mechanics, 2 0 , 3 ( 1 9 5 3 ) .

1 6 . HODGE, G. and VENKATRAMAN, B. Mem. Symp. La Plasticita Nella Scienza Delle Cos-truzioni in onore di A. Danusso. Bologna ( 1 9 5 6 ) .

1 7 . ROSENBLUM, V. I . Injenern. Sbornik, 2 0 ( 1 9 5 4 ) . 1 8 . KACHANOV, L. M . Isvestia Acad. Sci. U.S.S.R., O T N , 5 ( 1 9 5 7 ) . 1 9 . KACHANOV, L. M. Prikl. Matem. Mechan. 2 3 , 3 ( 1 9 5 9 ) . 2 0 . ROSE, S. N . Vestnik Leningrad Univ. 1 9 ( 1 9 6 1 ) . 2 1 . KACHANOV, L. M. Analysis of Elasticity and Plasticity I. Leningrad: Leningrad University

( 1 9 6 1 ) .

2 2 . KACHANOV, L. M. Isvestia Acad. Sci. U.S.S.R., O T N , ser. Mechanica a. Maschin. N o . 3 ( 1 9 5 9 ) .

2 3 . ILYUSHIN, A . A . Plasticity. M o s c o w : Gostekhizdat ( 1 9 4 8 ) . 2 4 . BUGAKOV, I . I . Analysis of Elasticity and Plasticity I. Leningrad: Leningrad University

( 1 9 6 1 ) .

2 5 . SHESTERIKOV, S. A . Isvestia Acad. Sci. U.S.S.R., O T N , 2 ( 1 9 5 7 ) . 2 6 . ROSENBLUM, V. I . Injenern. Sbornik, 1 8 ( 1 9 5 4 ) . 2 7 . KACHANOV, L. M. Prikl. Matem. Mechan. 2 4 , 1 ( 1 9 6 1 ) . 2 8 . TEREGULOV, I . G. Prikl. Matem. Mechan. 2 6 , 4 ( 1 9 6 2 ) .

ASPECTS OF RELIABILITY UNDER CONDITIONS OF

ELEVATED TEMPERATURE CREEP AND FATIGUE

A. M. FREUDENTHAL

Columbia University, N e w York


AMONG the many problems facing the designer of structures operating at ele-vated temperatures, the most serious appears to be that of attaining adequate safety and reliability. "Elevated temperature" conditions arise only for com-binations of structural material and temperature for which the repeated tem-perature and loading cycles produce significant changes in the deformational (elastic and inelastic) response or in the mechanical strength of the material, or appreciably affect the stress distribution within redundant structures. Thus for instance, for the structural alloys used at present, "elevated temperature" conditions arise in a supersonic transport above Mach 2 if constructed of light-weight alloys, or above Mach 3 if constructed of steel or titanium alloys.

In the design of power generators and jet engines, "elevated temperature" conditions are usually met by the use of alloys of relatively high time-tempera-ture strength combined with extensive testing of full-scale parts under critical conditions, so as to permit statistical evaluation of the test results as a basis for reliability analysis. Moreover, in such structural elements the creep de-formation tolerances are rather high, the expected service lives relatively short and replacement of critical parts rather easy so that testing is possible within the operational conditions and reliability analysis requires only moderate extrapolation.

For large structures statistically adequate test replication is neither tech-nically nor economically feasible, so that reliability analysis which cannot be based on the evaluation of statistical data must be developed from physically relevant probability concepts. This is a procedure similar to that suggested for the analysis of fatigue sensitivity of subsonic aircraft structures at normal atmospheric temperatures, 1 with the significant difference that under elevated temperature conditions the damaging effects of fatigue and creep must be com-bined.

An additional difficulty arises from the fact that in the design for a life range of 10,000 to 100,000 hr the double aspects of design for failure and design for a tolerable deformation requires that the design creep rate per hour be extremely low while the time to fracture becomes extremely long. A survey of the existing creep data will, however, show that observations within the struc-

399

400 A . M. F R E U D E N T H A L

turally relevant range of low creep rates are almost non-existent; also the recorded times to creep fracture are too short to be relevant in design, the design life being too long to be used as actual testing time. The discouraging conclusion of such survey is, therefore, that data for elevated temperature design of large long-life structures for the relevant range of service lives are not available. They could be obtained from the available data only by extrapola-tions, the reliability of which both in the direction of very low creep rates and of very long creep-rupture lives is dubious. 2

With most creep and creep-rupture tests performed within the 300 hr range, extrapolation over two orders of magnitude would be an inadmissibly wide range even for thermally and chemically stable metals, which most of the alloys developed for high temperature service are neither. The use of various t ime-temperature parameters suggested by different authors, based essentially on the assumption that a single rate-equation covers the whole range of behavior, can therefore hardly be expected to produce reliable data by extrapolation, unless the validity of this assumption has been clearly established. This, however, cannot be done directly without extending the testing time towards the actual service time.

The known lack of reproducibility of creep observations within the (primary) micro-creep range makes prediction on the basis of a small number of tests unreliable: the scatter associated with (primary) micro-creep rates is wider than the scatter associated with (secondary) minimum steady-state creep rates and creep fracture data, so that designs for long lives in which a small total creep deformation (0.5-1 per cent) represents a critical design criterion are subject to a high degree of uncertainty.

While actual data for the establishment of design procedures under conditions of elevated temperature are lacking in spite of many years of creep research, it seems desirable to establish a general scheme for such a procedure in the light of which the current and future research work in creep and creep fracture for the purposes of structural design should be re-evaluated so as to provide in-formation that is relevant to design problems rather than to materials evalua-tion, as for the latter purpose relatively short-time tests are usually considered sufficient. Without such a scheme the haphazard approach to the study of mechanical properties of metals at elevated temperatures will continue to remain unrelated to the needs of the designer.

It is, in general, not understood by metallurgists that for the designer creep and creep fracture are not metallurgical problems alone, but problems of reli-ability of a structure under highly variable conditions of stress, temperature and environment. Many of the materials involved are known to have a structure that tends to become unstable under critical combinations of time, temperature and environment. But even under stable conditions the scatter of the test results is to the designer not less important than their trend. Since even a small number of full-scale tests of large structures cannot be performed, the scatter of small specimen tests represents the only information on the basis of which procedures of reliability analysis can be devised.

E L E V A T E D T E M P E R A T U R E C R E E P A N D F A T I G U E 401

Therefore, an urgent task of engineering research is the formulation of an integrated, schematic approach to the mechanical design of structures under elevated temperature service which, by combining the safety factor approach with the reliability approach, would make it possible to estimate the design life of a structure and to assess the effect on this service life of modifications of the design, of design criteria and of changes in operating conditions.

2 . A S P E C T S O F D E S I G N

The main aspects of design for elevated temperature service are the interactions of sustained loads, cyclic loads and temperature in producing within the service life of the structure:

(a) an acceptable limiting, relatively small amount of peimanent deformation; (b) accumulated damage in the form of micro- or macro-cracks that reduces

the resistance of the structure to "ult imate" loads; (c) modification of the structural action of redundant structures by the inter-

action of stress-relaxation, temperature and cyclic stress. Because of the usually highly variable character of the loads, it is reasonable

to assume that elevated temperature creep-fatigue failure is more likely to occur under an excessive load of short duration and relatively low probability of occurrence after the structural resistance has been partially reduced by creep-fatigue damage, than under normal service conditions involving only loads of relatively low intensity occurring under service conditions. Hence the critical design conditions for elevated temperature service are:

(a) ultimate load failure of the undamaged structure under a load intensity of very low probability of occurrence and thus of a "return period" that is a large multiple of the design life;

(b) ultimate load failure of the structure damaged by fatigue and creep and therefore failing under an "ultimate load" of lower intensity but higher prob-ability of occurrence or shorter "return period";

(c) limiting permanent deformation under operating conditions specified in terms of load spectra and temperature history.

The principal creep design problem therefore is that of structural damage by crack initiation and propagation under relatively low-intensity loads of long duration, reducing the resistance to loads of high intensity and short duration. For this purpose tests under combinations of sustained and cyclic loads should be performed not to failure but to a specified level of reduced resistance meas-ured either by a high strain rate test to failure or by a specified crack length. Results of tests to failure under stationary conditions of sustained and cyclic loading can provide information concerning the general trend of the pheno-menon of damage; however, the significant aspect for design is the fact that actual failure is likely to be "premature", because of the possible coincidence of a partially damaged (cracked) structure with a load-time-temperature com-bination which is just high enough to make this damage critical; the more severe the preceding damage, the lower the intensity of the critical load com-


bination and the higher therefore its probability of occurrence or the shorter its "return period".

The results of conventional creep tests are not relevant to the problem of creep design, since the very limited amount of operationally permissible total creep deformation makes the region of primary micro-creep of principal im-portance, while the emphasis in conventional creep tests is on the steady-state (secondary) macro-creep. Not only is the primary micro-creep deformation of polycrystalline metals highly statistical in itself, but the accuracy of the measure-ment required in this range is usually far beyond that of conventional creep tests. The experimental difficulties arising in observations within this region are clearly demonstrated by the lack of reproducibility in stress-relaxation tests of the initial short-time range which depends primarily on the application of the initial strain which, being of the order of magnitude of the elastic strain, falls into the range of micro-creep. Since this lack of reproducibility cannot be overcome even by careful experimental technique, stress relaxation tests of metals are unreliable within this, foi the designer very important, range.

The experimental difficulties are increased by the fact that, phenomenologic-ally, primary creep in polycrystalline metals is the result of the combined action of an irrecoverable component of strain hardening associated with transcrystal-line slip and a (delayed recoverable) anelastic component associated with grain-boundary motion; the former is non-linear (with respect to stress), the latter linear. Hence creep tests without unloading periods with the aid of which the anelastic component can be separated from the strain hardening component provide inadequate information for the designer, since he is unable to arrive at the analytical characterization of the material required for stress analysis without such separation. Moreover, it appears that even the highest accuracy of direct observation of micro-creep and creep recovery cannot cover the short range relaxation to separate the linear anelastic from the non-linear components making up the total creep; the direct creep measurements should therefore be supplemented by observation of the mechanical response of the material under low-frequency small-amplitude stationary oscillations, by which the relevant frequency dependent visco-elastic parameters (storage and loss moduli) can be obtained. Conversion of these data into creep and relaxation functions repre-senting the anelastic component of the creep deformation is permissible if the time scales are comparable.

In the interpretation of creep deformation of a particular metal for the pur-poses of design two basically different conditions defined in terms of stress-level and temperature must be distinguished:

(a) conditions that produce the limiting creep deformation essentially in the form of primary creep;

(b) conditions that produce the limiting creep deformation essentially in the form of stationary (secondary) creep.

In the first case a significant part of the limiting creep will be reached rela-tively early in the service life with little subsequent permanent deformation, while in the second case creep will accumulate rather uniformly over the service life.


For both types of creep thermally unstable conditions of the periodically stressed material produce more rapid creep accumulation than is to be expected from tests under thermally stable conditions, because of the effect of recovery on primary creep. Such increase in creep accumulation is, however, not neces-sarily accompanied by creep-damage accumulation, since the mechanism of primary creep is one associated with strengthening by work hardening rather than weakening of the structure.

The methods of stress analysis and the resulting stress intensities and distri-butions will differ significantly for both conditions and so will the damage accumulation due to combined steady and cyclic loading: in the first case damage is predominantly due to cyclic loading and concentrated in the fatigue striations resulting from reversed slip, while the effect of primary creep on damage is likely to be minor; in the second case damage accumulation due to creep is concentrated in the grain boundaries while transcrystalline fatigue-damage, when the cyclic stresses are high enough to produce it, proceeds independently until, at a rather advanced stage of propagation and coalescence of micro-cracks, the coexistence of both intercrystalline and transcrystalline microcracks is likely to facilitate the propagation of the macrocrack producing final failure through both crystal grains and grain boundaries. Any effect that weakens the grain boundaries, such as increasing temperature, structural in-stability, or chemical attack facilitates intercrystalline cracking and thus reduces the time to creep failure at a specified stress level or reduces the stress required to produce such failure at a specified time.

Indirectly those effects also facilitate the initiation and propagation of (trans-crystalline) fatigue cracks by reducing the restraints on slip initiation and slip propagation imposed by grain boundaries. Such indirect action is, however, less significant in its results on fatigue strength or fatigue life than in its direct effect on creep-rupture strength and life. Therefore, creep-damage accumulation as the critical failure mechanism may grow so fast in relative importance with increasing temperature and service life in comparison to fatigue damage ac-cumulation that beyond a certain combination of temperature and service life the weakening of the structural resistance due to creep becomes the principal design criterion, while fatigue and fatigue cracking become less critical, as the working stresses of the structure are reduced to meet the limiting creep and creep-rupture criterion.

3 . I N T E R A C T I O N C U R V E S

Because of the absence of damage accumulation under very small sustained stresses the effect of creep damage on fatigue damage can be neglected for small ratios of sustained to cyclic stress; on the other hand, the effect of low-amplitude cyclic stresses on damage under sustained stress must be expected to vary with amplitude and frequency. High frequency cyclic stressing, by its interaction with micro-structural processes may significantly affect the creep rate and associated damage accumulation either by accelerating the rate of recovery and of thermal


FIG. 1. Elliptic interaction curves for failure at various temperatures.

mixture of intergranular and transgranular. By relieving grain-boundary stress concentrations and reducing the associated cracking, a cyclic stress of moderate amplitude may even increase the creep-rupture strength, a phenomenon that has actually been observed. 3 On the other hand by substructure formation it may accelerate the grain dissociation under sustained load and thus increase the creep rate and decrease the life. 4 In the absence of low-amplitude high-frequency interaction, "interaction curves" representing the combined creep damage under a sustained load Sm and superimposed load cycles of amplitude ±Sa

at a certain combination of time and temperature should, in the rectangular Sm-Sa coordinate system, be tangent to the rectangle Sa = Sao and Sm = Smo where Smo and Sao denote, respectively, the sustained stress and the stress amplitude applied at frequency oo, producing failure or a specified amount of damage (crack length or percentage loss of "ult imate" strength) at a certain time t or at a number of cycles N = cot and at temperature T (Fig. 1). In general the condition of the existence of a vertical tangent at Smo is not physically justified; the interaction curve may intersect the Sm-axis at an acute (thermal softening) or obtuse (hardening) angle.

softening in a work-hardened metal, or by accelerating metallurgical transfor-mations such as precipitation, or other forms of atomic migration leading to hardening. Hence the result of a superimposed high frequency low-amplitude oscillation may be either a significant decrease or an increase of the creep-strength, depending on the microstructure. Many creep-resistant alloys, such as cobalt and nickel alloys, achieve this resistance by some form of atomic migra-tion; in that case acceleration of the migration by superimposed cyclic stressing may considerably increase this resistance.

On the other hand, the imposition of a cyclic stress of low or moderate fre-quency that does not interact with microstructural processes will not signi-ficantly affect the creep damage unless the stress amplitude is high enough to change the character of the damage mechanism from purely intergranular to a

s 0


Since the existing experimental evidence suggests 5 that for relatively short times and moderately elevated temperatures the ratio Smo/Sao > 1.0, while it decreases to Smo/Sao < 1.0 as the temperature and time increase, a damage or fracture surface at temperature T in a rectangular Sa-Sm-t coordinate system can be constructed by assuming elliptic "interaction curves" and connecting the elliptic interaction curves for different times t = N/OJ by a continuous surface (Fig. 2). The intersection of this surface with the Sm-t plane is the creep-rupture curve (for failure) or the locus of points of specified constant damage for various combinations of stress and time (at constant temperature); the intersection of the surface with the Sa-t(=NjoS) plane is the SV-iV-diagram for failure or specified damage converted to the equivalent time scale at the specified frequency.

>0 T > T

2 1

FIG. 2. Interaction surfaces for failure.

It is important to note that the elliptic or quasi-elliptic shape of the interaction curves can be justified only under elevated temperature conditions under which creep damage actually accumulates under the sustained stress Sm. These inter-action curves should not be confused with the fatigue-interaction curves at normal temperatures (modified Goodman diagrams) in which no different damage mechanisms are involved. Indeed, such interaction diagrams are fre-quently linear or even convex towards the Sm axis. The time-cycle conversion t = N/co is, in this case, purely fictitious since, in the absence of creep, time itself does not contribute to damage accumulation. Therefore constant creep-fatigue damage surfaces cannot be extended to conditions of zero or non-damaging (first stage) creep.

The creep strain attained at a certain time under a combination of sustained stress Sm and cyclic stress amplitude Sa is, in general, only slightly and indirectly affected by the cyclic stress. Hence the "interaction surfaces" for constant creep


So

FIG. 3. Interaction surfaces for limiting creep deformation.

t=N/cj

FIG. 4. Intersection of interaction surfaces for failure and limiting creep deformation (design ranges).

strain are either parallel to the S a-axis or slightly inclined to it if the stress cycles are associated with a permanent strain or, as is usually the case, because of the plastic deformation accompanying the first few stress cycles (Fig. 3). The inter-section of the constant creep surfaces at the same temperature with the constant damage surfaces delimits the validity of the two design criteria (Fig. 4). It appears that for short or moderate times and small or moderate ratios of the "fatigue factor" tan a = Sa/Sm the creep-deformation criterion will be relevant,


while for other conditions damage accumulation is more significant. Obviously, the limit varies with time and temperature.

On the basis of the constant damage surface f(Sm, Sa, t) = 0 damage accumulation at various combinations (Sm, Sa) could be estimated, in first approximation, by finding the time ta at which the point of intersection of the radius vector of slope tan a = Sa/Sm in the plane t = ta parallel to the Sm-Sa

plane with the damage surface has the required combination (Sm, Sa) as co-ordinates, and by assuming a linear rule of damage accumulation with respect to this time. Under this assumption the expression

(3.1;

where tdkfa) are the times to constant damage or to failure at the various com-binations (Sm, Sa)k, and pic the percentage of total time ta during which they are applied. If interactions between the various combinations exist the times tdk should be reduced with the aid of interaction factors wjc to (tdk/wjc) if the combination (Sm, Sa)k forms part of a load spectrum instead of being continued to failure.

In the absence of surfaces of failure or of constant damage the times tdk are unknown and must be estimated from the time tc to creep fracture under Sm and the number of cycles N or time t = N/OJ to fatigue fracture under ±Sa. On the basis of an assumed independence between the two damage pro-cesses a tentative rule of addition of linear damage rates produces the relation

(3.2)

which could be improved and adjusted to experimental results by the intro-duction of a factor wc(a) expressing the relative weight of creep damage in relation to fatigue damage, which is necessarily a function of a = arctan (Sa/Sm): hence,

(3.3)

The interaction surface associated with the assumption of linear addition of the constant damage rates co/Aând l/t can be obtained by assuming the relations between Sa and N and Sm and t in the usual forms

Sa = SaoN-P and Sm = Sm0fK (3.4)

where p < 1, K < 1 and Sao and Smo are the values of Sa and Sm for N = 1 and t = 1 respectively, so that

N=(Sao/Sa)1tP and t = (SmolSm)W< Hence

(3.5)


which for constant times (ta = constant) defines a family of interaction curves of pseudo-elliptic type:

With increasing temperature p increases slowly while K increases rapidly; hence while 1/p remains almost unchanged 1/K decreases, increasing the effect of creep.

From the designer's point of view the constant damage and failure surfaces f(Sm, Sa, t) = 0 as well as the constant creep surfaces represent the trend or "central tendency" (mean, mode or median) of the test results which should be represented by families of such surfaces (Fig. 5) each associated with a different

FIG. 5. Interaction surfaces for failure at various probabilities of survival reliabilities R.

probability of failure P or of survival (reliability) R = 1 — P. What the designer needs is the surface associated with a specified, very small value of P (or a large value of R) rather than the surface of central tendency at which P ~ R. From the family of failure surfaces fii(SM, SA, t) = 0 a four-dimensional reliability function R(a, s91), where a = arctan (SA/SM) and S = VC5™ + $1) could theoretically be constructed point by point by determining for values 0.01 < R < 0.99 the values t associated with the points of intersection with all those surfaces of a parallel to the £-axis through any combination ( a , S); for a = 0 it degenerates into the three-dimensional reliability surface RC(S, t) for creep rupture under S = SM, for a = TT/2 into the reliability surface for fatigue RF(S, N) under S = ±SA for a specified frequency of oscillation. For intermediate constant values 0 < a < T T / 2 , as well as for specified values S = const, the reliability function can be represented in the form of reliability surfaces RA(S, t) for constant a and Rs(a, t) for constant values of 0 < S < SMAX,

(3.6)

4 . R E L I A B I L I T Y A N D R I S K O F F A I L U R E

R*O.OI =0.10 1 = 0 . 5 0 — I = 0 . 9 0 N

I 3 "


FIG. 6. Reliability surface Ra(S, t) at constantangle a = arctan (Sm/Sa).

R.

FIG. 7. Reliability surface / ) at constant stress intensity S = \/(Sm + 5 ^ ) .

constructed through the family of reliability functions RJt), obtained by varying S at constant a and Rs(t) by varying a at constant S (Figs. 6 and 7).

If it is assumed that, in first approximation, a and S are independent statistical variables with a bi-variate frequency density p(a, S), the reliability function R*(t) under random application of combinations (a, S) is obtained by per-


and can be obtained if R*(t) can be specified by an analytic function or vice versa. However, while extensive information is available on the propagation of fatigue cracks at room or low temperatures, little or no information exists on crack propagation at elevated temperatures under the joint action of creep and fatigue on which a quantitative relation could be based that would express the reduction of resistance with time of a cross-section stressed at elevated tempera-ture by the combined action of sustained and cyclic stresses. Thus no more can be done here than to point to this lack of vital information from which the risk function ru(t), expressing the deterioration with time of the ultimate strength due to combined creep and fatigue cracking, could be obtained in order to establish the total risk of failure considering all conditions:

r*(0 = ru + ru(t) + n(t) + r2(t) (4.3)

where r\(t) is the risk of fatigue failure, r2{t) the risk of creep failure under service conditions and ru is the (constant) risk of ultimate failure of the undamaged structure

ru-kPf (4.4)

V

Pf is obtained from the distribution function of the safety factor P(v) = J p(v) dv o

at the abscissa v = 1, while k 1 is the ratio of the expected number of in-tensities within an "ult imate" load intensity spectrum (above limit load) to the total number of applied loads.

A measure of elevated temperature fatigue-creep sensitivity, similar to the proposed measure of fatigue sensitivity 6 can be defined as the sum of the ratios

(4.5)

provided analytic expressions for p(a, S) and R(a, S, t) can be established. The reliability function R*(t) for the distributions pi(S) and p2(a) is based

on the tacit assumptions of independence of S and a and linear damage accu-mulation under variable stress intensity S. It represents essentially the reliability function for the mean stress intensity S and the mean fatigue factor a. As such it is only an approximation to reality, but without it not even the order of magnitude of the service life can be predicted.

Such prediction could now be based on the procedures proposed for fatigue 1

and particularly on the concept of the addition of risk functions developed there. The risk function r*(t) associated with the reliability function R*(t) is defined by the relation

(4.2)

forming the integration

(4.1)

E L E V A T E D T E M P E R A T U R E C R E E P A N D F A T I G U E

where the first ratio can in general be assumed to be small in relation to the second. Values 10 > f{t) > 1 would designate structures of high elevated temperature sensitivity for which special design and testing procedures are to be developed, while structures with f(t) < 1 may be designed by standard pro-cedures for ultimate load only. Structures with f(t) > 10 would be classified as inadmissible.

For known shapes of the reliability and risk functions diagrams for the pre-diction of service life t or reliability at a specified life t, similar to those computed for fatigue 6 could be constructed, or the diagrams directly used replacing the variable N by f, provided the reliability and risk function, are of Extreme Value type, an assumption for which plausible physical-probabilistic arguments can be put forward. Considering two extremal risk functions, one for fatigue and one for creep rupture

and

(4.6)

(4.7)

where Vs and ts are, respectively, the characteristic fatigue life and the charac-teristic creep life, and y and S are the scale parameters of the associated reli-ability functions

R'(N) = exp[(-N/Vs)y] = exp[(-iot/Vs)y] (4.8)

and

* " ( / ) = exp[(-f/f,)*] (4.9)

which are inversely proportional to the standard deviations <?(log N) and

a(log t), the combined reliability function

R*(t) = R\N) R"(t) = exp (4.10)

The effect of the interaction can be illustrated by comparing the modes or another suitable measure of central tendency of the probability densities p\N) and p"(t) with the respective measure of the combined function /?*(/)• Using for the sake of expediency the medians t\t" and /* rather than the modes, the following expressions are obtained for these values: from

from

and from

exp

exp

exp

(4.11)

(4.12)

(4.13)

411


the superposition rule

(4.14)

from which r* can be evaluated. Introducing the ratio t'jt" = n, equation (4.14) can be written in the form

which indicates that for n ^> 1: i* -> i", while for n <^ 1: i* --> V. If V and t" are of the same order of magnitude (n ~ 1) an estimate of t* is obtained from

Hence i* < t" or thowever, the reduction is rather small, being a maximum for y = 8 for which t* = i"joy/2. With the usual range of values 3 < 8 < 5:

a result that suggests a relatively weak interaction if the independence of the failure mechanisms implied in equation (4.10) is a valid assumption. With y = 8 = 1, equation (4.14) degenerates into the assumption of linear interaction which is therefore compatible only with pure chance failures with constant risk functions r\N) = V~x and r"(t) = t^1; these are incompatible with the physical character of the fatigue and creep-rupture processes.

The conclusion to be reached from equation (4.15) is that whenever the creep life under Sm and the fatigue life under ± 5 ^ are significantly different, the critical life is equal to the shorter of the two; when they are of the same order of magnitude there is slight interaction which reduces the critical life moderately. Since this reduction is well within the scatter range, it appears that existing creep data and existing alternating stress fatigue data might provide sufficient inform-tion for design under combined action of creep and fatigue, provided the validity of equation (4.15) is experimentally established.

A C K N O W L E D G M E N T

The paper is based on studies supported by the Metals and Ceramics Laboratory, Directorate of Materials and Processes, Aeron, Systems Division, Wright-Patterson Air Force Base.

1. FREUDENTHAL, A . M. WADD Technical Report 61-53 (July 1961). 2. SHIMMIN, K . D . WADD Technical Report 60-523 (October 1960). 3. LAZAN, B. "High Temperature Effects in Aircraft Structures", Chapter 9 of Agardograph

N o . 28 (1958). 4. KENNEDY, A . J. Rheologica Acta, 2, 22-26 (1962). 5. ALLEN, N . P. and FORREST, P. G. Session 4, Paper 1, International Conference on the

Fatigue of Metals (London, 1956). 6. FREUDENTHAL, A. M. Reference 1, pp. 20, 50-53 .

(4.15)

(4.16)

0.8?" < i* < 0 .87 ; " ,

R E F E R E N C E S

ASPECTS OF ELEVATED TEMPERATURE DESIGN AND

DESIGN CRITERIA FOR SUPERSONIC AIRCRAFT

STRUCTURES

M. O. W . WOLFE

Royal Aircraft Establishment, Farnborough, England


The design of aircraft structures and the choice of the materials of which they are composed is now well recognized by its practitioners as being as much of an art as it is a science. It is not perhaps without some significance that the first great designer of flying machines, Leonardo da Vinci, happened also to be a great artist.

The aircraft designer's task is always to seek the best compromise between the frequently conflicting requirements of the aerodynamicist, the power plant de-signer, the structure's engineer, the aeroelastician, the certification authority and the customer in terms of structural and aerodynamic efficiency, structural integrity and economy. The first major compromise which sets the stage for all that is to follow is that of reconciling the aerodynamic shape with the most efficient structure. Little in the way of detailed structural analysis can be under-taken at this stage and the designer must rely to a large extent on past ex-perience and the use of criteria. This is followed by the detailed design of the structure, an evolutionary process involving increasingly more detailed analysis and supporting tests as it approaches the final prototype stage of full scale proving tests for aeroelasticity, static strength and fatigue.

Once the loading spectra have been established in terms of the performance role and lifetime expectancy of the structure and appropriate factors have been applied, the detailed design consists essentially of an optimization procedure for minimum structural weight where stress, stability of structural components and stiffness requirements are the main ingredients. However, this process is so closely interwoven with the choice of the component materials in terms of their optimum combination of properties that the two processes merge into one. The shape, the environment, the loads and the integrity requirements largely determine the particular type of structure adopted and the choice of materials. However, the shape alone may in many cases have a decisive effect in the matter of material selection. Strength and stiffness in tension, compression and shear, and buckling stability in struts, plates and shells are the primary design para-

413

414 M. O. W . W O L F E

meters. Very little of an aircraft structure is in fact designed with tensile strength as the sole requirement. Hea th 1 points out that more than half the structure has stability in either compression or shear as its criterion and the requirements of stiffness, fatigue and creep play such an important part in the design of the remaining structure that tensile strength of itself may be of little importance.

The designer must therefore evolve his structure and choose his materials so as to exploit the best forms to suit the complex of properties of the materials he has available having regard to time dependence and other phenomena.

The new factor which has been ushered in by the advance from subsonic to supersonic flight is the profound influence of kinetic heating. The designer must now have regard to the influences of temperature and time at temperature.

Notation

b critical typical dimension of element (either length or width)

C2 0.2 per cent proof stress in compression

fe ultimate compression stress

ft ultimate tensile stress h effective thickness (Section 3.1) c specific heat (Section 3.2)

h 0.2 per cent proof stress in tension E Young's Modulus I second moment of area per unit width of element K constant depending on edge conditions (Section 3.1) thermal con-

ductivity (Section 3.2) L length of web P critical load per unit width of element T time taken to reach the maximum temperature W weight per unit area of element a coefficient of thermal expansion

V plasticity reduction factor temperature difference between the middle of the web and the skin

G maximum temperature K diffusivity

P density (text)

2 . T H E S T R U C T U R E

The enormous and continuous increase in flying speeds which has taken place since the earliest days of flight has been accompanied by corresponding changes in the shape of the airplane and in the magnitude, distribution and character of the applied loads. The supporting role of the structural designer is always to try to provide the optimum structure although he is invariably denied the time to allow any one design to achieve a state of stable maturity.

In the subsonic era wings underwent an evolution from high aspect ratio forms of moderately large thickness-to-chord ratios to thin wings of low aspect

E L E V A T E D T E M P E R A T U R E D E S I G N A N D D E S I G N CRITERIA 415

ratio. As a result, the traditional design associated with predominantly rect-angular planforms and consisting of two spars with thin covering skins sup-ported by stringers and closely spaced ribs tended to be replaced by thicker skins, possibly of sandwich construction, with more spars and fewer ribs. This, coupled with the introduction of swept and delta planforms, has brought many new problems, particularly in the stress analysis field. Conventional methods of stress analysis, based on simple beam theory, are now of limited value and resort to plate theory is necessary in order to deal with the closely interwoven torsional and flexural properties.

In our emerging supersonic age there is already interest in the design of a civil transport to fly in the Mach 2 to 3 range. A considerable amount of study has been devoted, in this connection, to the slender delta configuration for its advantageous aerodynamic characteristics. This presents the structural designer with additional problems of strength, stiffness and aeroelastic stability in the chordwise direction as well as along the span. To meet these new conditions multi-cell forms of construction of high redundancy are being evolved. A feature of significance is the comparatively great depth of the wing—a conse-quence of the very low aspect ratio and of the shape. This implies relatively low spanwise loadings in the skin so that a large proportion of the panels are designed on the basis of elastic buckling considerations; the weight of the necessary stabilizing members precludes the development of high stresses and this fact is important in determining the choice of material in relation to the effects of kinetic heating.

The point that needs to be emphasized is that, in addition to the effects of heating, the approach to the design of supersonic aircraft structures is also greatly influenced by the shape of the airplane.

The kinetic heating of the structure produces effects which must be con-sidered in relation to the elastic and aeroelastic behaviors, static strength, fatigue and creep. In what follows an attempt is made to describe briefly a few of the related theoretical contributions made at the R.A.E. concerning mostly the elastic behavior.

Considering first the stiffness effects, it is clear that loss of stiffness may be due to either the effects of temperature on material moduli or the effects of thermal stress. Of these, the latter is likely to be more important in practice and has been examined, in the case of solid wings, by many investigators. The possibility of the loss of torsional stiffness was recognized at an early stage in the United S t a t e s ; 2 ' 3 more recently, it has been shown that loss of flexural stiffness can also occur. 4 However, analysis and, to a very limited extent, flight experience have shown that although loss of stiffness may be significant in thin solid wings, the built-up wing is less susceptible, mainly because of its more uniform chordwise temperature distribution. 4

The accurate analysis of transient heat flow through a complex structure generally presents formidable difficulties. Solutions of unidimensional flow through a composite slab or along a reinforced thin plate have been obtained for the transient heat flow through an insulated slab 5 and approximate methods

416 M. O. W . W O L F E

have been developed for the transient temperature distributions in an insulated multi-spar wing. 6 A digital computer program, using finite difference tech-niques, has been developed for the solution of general two-dimensional heat flow problems. 7 Further difficulties are presented by the estimation of joint conductivities, effects of heat sinks, internal convection and radiation.

Assuming that the time-temperature history of every element is known, one can then proceed with the analysis of the associated elastic stress system. At this state, use is often made of matrix methods of analysis such as have been de-veloped in the United States and by Argyris in the United Kingdom. The accu-racy of the analysis depends, among other things, on the legitimacy of the structural idealization which is necessary to formulate the problem within a framework suitable for digital computation. The determination of the thermal stresses near the roots of the wing is an important problem and an analysis, using conventional elasticity theory, has been performed for the particular case of a rectangular wing with two continuous spars passing through the fuselage and with an abrupt change of temperature at the junction. 8

Leading edge buckling, due to stresses arising from thermal gradients, is a phenomenon which may be instrumental in promoting transition to turbulent flow in the affected areas thereby creating further temperature gradients to aggravate the failure process. The prediction of leading edge buckling has been studied for thin solid wings 9 and for built-up wings with a leading edge fillet. 1 0

A special phenomenon, arising from the combined effects of heat and load cycling, is that of incremental "set" of the structure as a whole as studied by Parkes 1 1 . The thermal stresses in a supersonic aircraft may well be significant and, taken in combination with the mechanical stresses, there may be areas where the elastic limit (in itself temperature dependent) is exceeded. Parkes showed that under repeated cycles one of four conditions may develop: (i) per-manent elasticity, (ii) shakedown to an elastic state, (iii) alternate plasticity, and (iv) incremental collapse. The first two conditions are safe, while the fourth is unlikely to occur in practice since the permanent deformations of the wing, observable at an early stage, would lead to other unacceptable effects. The third condition is probably best considered in conjunction with the ever-present problems of fatigue due to the combined effects of thermal and load cycling. Nevertheless, the potential permanent structural deformation must be estimated in design to ensure that unacceptable deformations, leading to gross changes in the distribution of aerodynamic load or some form of aeroelastic instability, do not occur. Whereas Parkes considers a perfectly plastic material, Mansfield 1 2

considers linear work-hardening characteristics for a two-bar tie under various sequences of loading, heating, cooling and unloading. In a particular example the applied load is just sufficient to reach the elastic limit and successive cycles of heating and cooling introduce plastic deformations which increase with each cycle; the increase, however, becomes progressively smaller and virtually reversible conditions are soon approached. Although the two-bar element is too simple to represent an aircraft structure it can be considered as one repre-

E L E V A T E D T E M P E R A T U R E D E S I G N A N D D E S I G N C R I T E R I A 417

sentation of a continuous sandwich construction in which the outer skin attains a higher temperature than the inner skin.

Finally, the basic structural element in an aircraft is the plate, whether flat or curved, and the new geometries of wing planform have produced a number of significant plate shapes. For this reason, attention has been given to the solution of a wide range of buckl ing 1 3 and b e n d i n g 1 4 ' 1 5 problems for plates of parallelo-gram and trapezoidal shape. Allied problems in which the thickness of the plate is tapering have also been cons ide red . 1 4 ' 1 6 The advent of integral machining techniques for wing surfaces, which make minimum weight surfaces of varying thickness a practical proposition, has emphasized the need for further analytical studies of plate problems.

It is apparent from this brief account that the process of optimum design also requires further theoretical investigations. It may also be said that the in-vestigator needs practical experience in the behavior of these new forms of construction, under their new conditions of loading and heating, in order to arrive at a better formulation of some of the requirements.

3 . M A T E R I A L S

The choice of material with the correct balance of properties from materials having different relative properties can only be made for a specific design; there is no universal criterion.

The primary properties governing structural efficiency are density, tensile and compressive strength and elastic moduli. Other properties, for example fatigue resistance, elongation and creep behavior, are also important but their signifi-cance as compared with that of the primary properties is extremely difficult to assess. For those components which are designed on strength the usual criterion of structural efficiency is defined as a strength to weight or stress to density ratio and the parameters c2\p, fc/p, H\p9 and ft/p are normally employed to express the specific strengths. For those portions of the structure which are designed on the basis of stiffness and structural stability the following para-meters are commonly employed: E/p, E^2/p and E1^^. The quantity E/p gives a measure of stiffness efficiency from the aeroelastic standpoint. The quantity Eli2lp gives a measure of the efficiency of the material with regard to buckling of solid struts and the quantity E^/p relates to the buckling of panels. The range of applicability of this last criterion requires some qualification and of this more will be said later. For present purposes consideration of a few of these parameters will suffice to show how the relative merits of different materials are influenced by temperature and how significantly these comparative figures are influenced by the particular criterion chosen.

The variation with temperature of the 0 .2 per cent specific tensile proof strength for seven selected materials over the temperature range 0° to 1500°C is shown in Fig. 1. Their chemical composition and some of their salient physical properties are given in the Appendix. If we consider the materials appropriate

15

418 M. O. W . W O L F E

to the lower end of the temperature scale we find that this method of evaluation places them in the following order of merit: titanium alloy, stainless steel alloy and aluminum alloy, with the titanium alloy nearly twice as effective as the aluminum alloy. The effect of temperature is most marked in the aluminum alloy. Its efficiency declines rapidly above 150°C.

O 2 0 0 4 0 0 6 0 0 BOO IOOO 1 2 0 0 M O O 1600

TEMPERATURE °C

FIG. 1 . Specific tensile strength of several materials (British Crown copyright).

In the range 400°C to approximately 800 °C these materials are replaced by the nickel based alloys two of which, one a high and the other a lower strength alloy, are shown. Above this range there is a considerable fall in the specific strength achievable. To give some perspective of the possibilities at the higher temperatures two high density alloys of molybdenum and tungsten respectively are shown. Evidently, for those portions of an aircraft structure designed on the basis of tensile strength the prospects at the higher temperatures are at present not encouraging.

In Fig. 2 a comparison is presented of the stiffness efficiencies of the same group of materials and here the situation is evidently quite different. At room temperature there is very little difference between the titanium, steel and alu-minum alloys. A point to notice is that the relative order of merit differs from that for the tensile strength. Here again there is a decline in efficiency with increasing temperature. However, in passing from one material to the next over the whole range of temperature the decline in efficiency is far less marked than for tensile strength.

Figure 3 shows a comparison of these materials on the basis of the criterion E^z/p which gives a broad indication of their relative merits from the point of view of the compressive buckling of panels. Here the differences are more

2 0 0 4 0 0 6 0 0 8 0 0 IOOO 1 2 0 0 M O O 1600

TEMPERATURE °C

F IG. 2. Specific stiffness of several materials (British Crown copyright).

I O

•8

•6

O

D T D 5 0 7 0 A D T D 5 0 7 0 A

^ T , - 4 A . - 3

P H I 5 - 7 M o

M » - 0 - 5 T I

TUN< /

* S T E N - 2 ° / o T h o

1—

2 0 0 4 0 0 6 0 0 B O O IOOO 1 2 0 0 1 4 0 0 1600

T E M P E R A T U R E ° C .

FIG. 3 . Specific stability criterion for several materials (British Crown copyright).

marked; of particular interest is the considerable superiority of the aluminum alloy over the titanium and steel alloys in the temperature range 0° to 200°C. However, as has been observed, the application of this criterion must be qualified by some other structural and loading considerations related to the geometry and boundary conditions of the structural elements concerned and the nature of the loading.

The few criteria we have chosen are by no means exhaustive; however, they

P

420 M. O. W . W O L F E

If we examine this expression we find that the structural efficiency depends on three quantities:

(i) the structural index P/b

(ii) the parameter (rjE^/p

(hi) the parameter lx^\h

The first of these, the structural index, is simply a function of load and size of the structural element to which it is applied.

The second is a parameter of material properties. Its value depends on the stress-strain relationship for the material and the stress level, which accounts for the presence of the quantity 77, the plasticity reduction factor. An important point to notice is that the optimum stress is also a function of the structural index.

The third quantity is a non-dimensional shape parameter which depends on the kind of structural element considered; for whether it is a skin stringer, honeycomb or corrugated sandwich construction or of some other form. Again the optimum value of this parameter is a function of the structural index.

The last two parameters are dependent variables and the only independent variable, the structural index, is the quantity which determines the criteria to be applied.

At small values of the structural index the optimum stress is low and the value of 77 is unity; in these circumstances the quantity Ex^\p is an appropriate criterion. At higher values of the structural index the stresses are higher and in the extreme the optimum efficiency is determined by the permissible material compressive stress, so that the approximate criterion is the ratio of the com-pressive proof stress to density for the materials considered.

The relative significance of the material properties is therefore very intimately

do serve to show that the selection of material and type of structure are inter-dependent processes and cannot be separated.

3 . 1 . The Stability Criterion

A very large part of an aircraft structure is designed to be stable under com-pression and shear loads. Stability criteria are therefore important and merit a closer examination. The E^/p criterion provides a convenient broad basis for initial assessment. It is, however, valid only at low values of a parameter known as the Structural Index. For detailed analysis of the behavior of materials, for example, in skin stringer panels, a more elaborate criterion is required. The structural efficiency of such an element of a compression structure is given in slightly different forms by Bur t 1 7 and Heath 1 . Both are derivatives of the funda-mental Euler buckling formula. The more convenient for our purposes is that given by Heath, which is as follows:


related to the type of construction employed, particularly for compression structures. Heath 1 , for example, compares the relative structural efficiencies of a titanium alloy, a stainless steel alloy and an aluminum alloy for two similar constructions—honeycomb sandwich and corrugation sandwich—over a range of values of the structural index after short time soaking at 250 °C. His results are shown in Fig. 4, taken from his paper. This demonstrates the influence these variables have over the range of the structural indices considered. It should, however, be remembered that the inclusion of rib weights will influence the comparison in such a way that assessments based on the structural index may well be invalidated.

FIG . 4. Structural efficiency of sandwich construction at 250 °C (British Crown copyright).

3.2. The Effects of Diffusivity

During transient heating temperature gradients arise in a structure, for instance, between the outer skin and the center of a supporting web. If the materials are the same, the thermally induced stress is proportional to the temperature difference A 6, the modulus of elasticity E and the coefficient of thermal expansion a. The quantity Ea is sometimes termed the "thermal stress modulus" . This quantity is itself temperature dependent for which due allowance must be made.

A low modulus of elasticity has the effect of reducing thermal stresses; on the other hand, a high value is desirable for stability and rigidity requirements. Thus, in most cases E is unlikely to be reduced. If the thermal conductivity is increased one might expect the thermal stresses to be reduced. This property by itself does not, however, give the complete picture; specific heat and density are also important. The significant parameter is the thermal diffusivity K which is the thermal conductivity divided by the specific heat and the density.

422 M. O. W . W O L F E

Under certain conditions, other things being equal, the induced stresses for a given heat input are inversely proportional to the value of the diffusivity and directly proportional to the thermal stress modulus. Hence the intensity of the thermal stress Eac/k. M u r p h y 1 8 quotes typical values of this parameter for several alloys. These are shown in Table 1.

TABLE 1

Thermal stress modulus

Thermal diffusivity

Material c.g.s. units x 1 0 3

Titanium alloys 147 Magnesium alloys 16 Aluminum alloys 27 Al loy steels 350 Nimonics 860

The superiority of the magnesium and aluminum alloys is striking, by contrast to the nimonics. The steels and titanium are comparatively poor.

3 .3 . Thermal Stress in a Web

To illustrate the influence of these quantities in an idealized case let us con-sider the variation of the thermal stresses in a skin web configuration. We will assume the joint resistance between skin and web to be negligible and the temperature of both upper and lower surfaces to rise uniformly with time to a fixed temperature and thereafter to remain constant. The heat flow is assumed to be by pure conduction; the effects of radiation and convection are ignored. The temperature difference between the middle of the web and the outer skin

I O I ' U l

O 0

O- 6

3> /

° 4 / /

O 2 / 0 2 / /

/ a l

O 5 IO IS 2 0 25 3 0 35 4 0 45 S O

kT

FIG. 5. Thermal stress parameters (British Crown copyright).

The quantities zivmSij& ana L°\KL are piottea in tig. D . i t win oe seen tnai 1 1 the web is very deep, or the diffusivity very low, or the time short, the value of ^ m a x is almost equal to 0. In these circumstances the thermal stresses will be maximal. For short webs, long heating periods and high diffusivities the value of A 0 m a J 6 approaches the value of L2/KT. In this special region the thermal stresses are significantly influenced by the diffusivity, whereas at large values of the parameter L2/KT its influence is comparatively slight. The diffusivities at 200°C for the seven materials considered are shown in Table 2.

TABLE 2. DIFFUSIVITY AT 2 0 0 ° C

Diffusivity Material in 2 / sec x 1 0 - 3

Rene 41 4 . 4 0 Inconel X 6 .11 M o - 0 . 5 T i 7 3 . 0 0 Tungsten 7 1 . 9 0 R R 58 ( D T D 5070A) 100 .00 PH15-7MO 8 .55 T i - 4 A l - 3 M o - I V 5 .23

A typical example of the variation of the temperature distribution in a web as a function of time is shown in Fig. 6 for the case where the parameter L2/KT has the value of 24.

3.4. Time and Temperature-dependent Effects

The three major temperature and time-dependent phenomena of interest are creep, fatigue and degradation of the primary properties of static strength and stiffness. It is not possible within the scope of this paper to deal with each of these phenomena and their interaction in more than a cursory way. Moreover, the problem of fatigue in aircraft structures was considered at a previous conference held at Columbia University in 1956. So far as the material itself is concerned, the effect of elevated temperature is ultimately to reduce the fatigue life. However, in a structural element subject to kinetic heating the influence of thermal stresses and creep must be considered in combination with the stresses arising from the purely mechanical loads. While a substantial body of evidence exists for isothermal fatigue tests at elevated temperatures, there is at present very little published information available for the case where the temperature


rises progressively until the skin temperature reaches its maximum value and thereafter decreases asymptotically to zero. It may be shown that the maximum value of the temperature difference to which the thermal stress is proportional can be expressed by the relationship

424 M. O. W . W O L F E

as well as the mechanical loads are cycled. The designer's problem, when he attempts to determine the life of his structure in terms of gust and manoeuvre loads combined with heat cycling, is therefore an extremely difficult one. In this, as indeed in most other aspects of stressing, the alleviation offered by the degree of ductility of the material is of importance. Very high strength materials of low ductility are by no means the obvious choice.

MAXIMUM

SKIN MIDDLE OF SKIN WED

FIG . 6. Variation of temperature distribution in a web with time L2/kT = 24 (British Crown copyright).

The interactions between creep and fatigue are so complex that it is difficult to generalize. In some cases the failure process associated with load and heat cycling may be essentially one of creep, in others one of fatigue and sometimes a combination of the two processes may occur.

Very few data exist on the long time effects of creep. Curves for the aluminum alloy D.T.D. 5070A showing the stress required to produce 0 .2 per cent plastic deformation in 1000 and 30,000 hr under load at temperature are given in Fig. 7; the 30,000 hr curve is an extrapolation. To illustrate the importance of the creep results the 0 .2 per cent proof stress in tension is also shown; it is


determined after soaking at temperature but without load for 20, 1000 and 30,000 hr respectively; again the 30,000 hr curve was derived by extrapolation. Cumulative creep and interaction between creep and static deformations are not well determined yet but it appears likely from Fig. 7 that creep at the steady 1 g loads at temperature will be an important factor and will probably make

the proof loading case critical in limiting the allowable temperature for a supersonic civil transport, unless the form of construction is such that the airworthiness authorities will permit larger permanent deformation strains than heretofore.

4 . T H E C A S E F O R E X T E R N A L T H E R M A L P R O T E C T I O N

It will be evident from our considerations of the complex interactions between structural forms, material properties and thermal and loading environments that somewhere between the Mach numbers of 2 and 3, depending on the role

15§

426 M. O. W . W O L F E

of the aircraft, a change from aluminum alloy structure to one of steel or titanium alloy seems inevitable.

This is not, however, necessarily so. A basically aluminum alloy structure, thermally protected to reduce temperatures and temperature gradients to acceptable levels, may still be competitive over the whole range of supersonic Mach numbers and indeed at hypersonic speeds. One significant reason for this is that quite apart from the influence of heat on the metallic materials its effects on non-metals, fuels and passengers must also be taken into account. For an externally protected aircraft some proportion of the weight of the internal protection otherwise required will be unnecessary. Some weight will thus be saved for which due allowance must be made in assessing the weight penalty of external protection.

4 . 1 . The Thermally Unprotected Structure

Let us first, therefore, consider the plausibility of an externally unprotected structure from the standpoint of the load carrying metallic materials.

Several authors have suggested the existence of hypothetical flight corridors extending through the Mach number and altitude ranges to quite high Mach numbers and altitudes within which flight appears feasible. These have been determined primarily by estimating an upper temperature limit for the particular materials chosen at which they may still be expected to behave effectively. More recently some broad estimates have been made at the R.A.E. of the structure weight of a fairly large aircraft designed to cruise at high Mach number. It is assumed to be constructed of nickel alloy and limited to a maximum skin temperature of 700 °C. The upper limit of flight speed is taken as 400 knots E.A.S. and the manoeuvring factor is assumed to be 2 .5 . From statistical data for existing aircraft of similar size and with empirical factors applied to allow for the effects of heat on the material properties, first estimates gave the struc-tural weight as 20-30 lb/ft 2 of wing area and 30-40 per cent of the over-all weight; the wing loading being taken as about 75 lb/ft 2.

We may now consider Fig. 8 which is a plot of E.A.S., Mach number and height; the various boundaries defining the practical limits for cruising flight are plotted. Two lines of constant E.A.S. are shown. For the chosen wing loading of 75 lb/ft 2 these are also lines of constant lift coefficient. The 200 knots line is considered to be a lower limit below which there is insufficient lift to provide a reasonable manoeuvre margin with a usable CL of 0 .8 to 1.0. The upper limit of 400 knots is that beyond which the structure is considered to become too heavy for aeroelastic reasons.

The line defining the upper skin temperature limit of 600 °C depends, among other things, on the value assigned to the surface emissivity coefficient. Emissivi-ties as high as 0 .8 are possible if stable oxide films or other special surface coatings can be used in a form durable under operating conditions. As this is somewhat conjectural, lines are shown for the two values 0 .3 and 0 .8 , the former of which is probably unduly pessimistic.


In order to determine the local heat transfer rates it is necessary to choose a chordwise datum. A station 10 ft aft of the leading edge is chosen. This deter-mines the position of both the Reynolds number lines and the equilibrium

FIG. 8. Boundaries defining high-speed flight corridor (British Crown copyright).

temperature lines shown on the chart. It can be shown that at high Mach numbers the equilibrium temperature is increased by approximately 10 per cent for turbulent flow and 20 to 30 per cent for laminar flow if an alternative datum 1 ft from the leading edge is chosen. This is roughly equivalent to a 15 per cent increase in Mach number for turbulent flow or a 40 per cent increase for laminar flow at the higher altitudes.

428 M. O. W . W O L F E

The conditions at which laminar flow may be assumed to occur are evidently a matter of prime significance. It is considered reasonable to assume that at the higher altitudes and Mach numbers it will occur somewhere in the range of Reynolds number between 2 x 10 6 and 50 x 10 6 , and Reynolds number lines for these limiting values are shown on the chart.

If we assume turbulent flow conditions throughout and an emissivity value of 0 .8 it will be seen that a corridor exists in which flight would seem to be feasible extending through the whole range of Mach numbers. It is bound on the one side by the minimum lift limit and for part of the range on the other by the maximum permissible E.A.S. line beyond which the structure becomes too heavy. This boundary intercepts the temperature limit line at an altitude of approximately 115,000 ft above which maximum permissible temperature becomes the limiting factor.

If we now consider the lower value of emissivity we find that for similar conditions the 400 knot E.A.S. line intercepts the temperature limit line at 96,000 ft altitude and the upper portion of the corridor is correspondingly attenuated. If the laminar flow is assumed to occur somewhere in this region the corridor is again opened up and the boundary determinant then becomes the Reynolds number line defining laminar flow. However, an emissivity of 0 .3 is undoubtedly far too pessimistic a value to assume and it is probable that the 400 knot E.A.S. line would, in fact, define the upper boundary over most of the ranges of altitude and Mach numbers of interest.

Although the structural and other assumptions made for the nickel alloy aircraft are strictly not appropriate to one constructed of aluminum alloy to operate at the lower Mach numbers, it may nevertheless be of some interest to see where the temperature limits would roughly lie in this case, assuming, an upper temperature limit of 150°C and an emissivity value of 0 .8 . A line on the chart indicates the limit for turbulent flow conditions. In this case the temperature barrier very nearly coincides with the Mach number of 2 .5 over the altitude range 65,000 to 94,000 ft. It is interesting to note that it happens to lie approximately in the region of Mach number and cruising altitude associated with British and French Supersonic Airliner proposals.

It must, however, be emphasized that the bounds of these hypothetical corridors are necessarily based on very crude data, and moreover, structural loading and metallic material considerations constitute the sole criteria. It is very likely that when more precise data become available on aerodynamic and propulsion characteristics and on the weights of ancillary systems and internal thermal insulation, the final balance sheet may show that some proportion of our hypothetical corridor may vanish.

4 .2 . Thermally Protected Structures

The case for external thermal protection of a basically aluminum alloy structure is evidently worthy of careful consideration. If the temperature of such a structure can be maintained at a reasonable level and the rate of heat


input reduced so that the temperature gradients within it are minimized, many of the superior qualities of aluminum alloys may still be exploited. It is indeed possible that a structure including its external protection may be competitive on a weight basis with an externally unprotected structure constructed of the more heat resistant alloys.

Thermal protection may take the form of either insulation, cooling or a combination of both. Schmitt, Brull and Walke r 1 9 have examined the weight of insulation and internal cooling required to limit the temperature of a structure under various slight conditions. Tay lor 2 0 considered radiation and insulation

160 ,000

140 ,0001

120,0001

100 ,000FT

ALTITUDE

8 0 . 0 0 0 F T .

6 0 , 0 0 0 F T

4 0 , 0 0 0 F T

2 q p O O F T

O

O 2 4 6 8 IO MAC H. N U M B E R

FIG. 9. Relationship between altitude Mach number and E . A . S . (British Crown copy-right).

effects separately without the restriction of a fixed wall temperature. M c C u e 2 1

has extended Taylor's work to include the radiation from the surface together with the variation with temperature of the thermal conductivity of the insulant material and has optimized the combination of insulation and cooling to give the minimum weight penalty. He based his estimates on steady state cruising conditions over distances of 2500 and 5500 miles respectively.

The aircraft is assumed to be of an aluminum alloy whose surface temperature is limited to 150°C. Emissivities of 0 .3 and 0 .8 and heating rates appropriate to Mach numbers ranging from 2 .5 to 10.0 respectively are considered. Turbulent flow is assumed and the representative position on the surface is taken to be 10 ft from the leading edge.

430 M. O. W . W O L F E

The value chosen for the efficiency of the insulation material is 400 °C k W - 1

ft 4 l b - 1 for a specific gravity of 0 .5 . This value may seem unduly pessimistic since forms of glass fibre insulations now being developed have nominally much higher efficiencies. However, several practical problems have still to be overcome in the evolution of suitable methods of attachment for these insulations and in rendering them adequately resistant to their flight environments. The efficiency assumed is therefore probably not unrealistic.

The cooling system envisaged consists of a suitable fluid, glycol for example, circulated in a system of channels attached to, or preferably forming an integral load bearing part of the outer surface of the structure, by means of which heat is removed and transferred in a heat exchanger to a coolant, which is water

7 0 0 ° C

6 0 0 ° C

SOO°C

EQUILIBRIUM

TEMPERATURE

4 0 0 ° C

3 0 0 ° C

2 0 0 ° C

lOO°C

O e C

O 2 4 6 8 IO MACH NUMBER

FIG . 1 0 . Relationship between equilibrium temperature, Mach number and E . A . S . (emis-sivity = 0 . 8 ) (British Crown copyright).

heated from 15°C and converted to steam at 100 °C. A cooling rate of 1 kW is taken to require a coolant disposal rate of 3 lb/hr. N o allowance is made at this stage for the dead weight of the circulating fluid and heat exchanger system because of the difficulty of making realistic estimates from existing data, though some allowance is made later for purposes of comparison with unprotected structures. Allowance is made for the heat capacity of the skin and adjacent structure for which a reasonable value of the distributed weight is taken to be 5 lb/ft 2.

McCue's work has been extended here to give the minimum weights of cooling alone, insulation alone and cooling and insulation combined for four fixed values of E.A.S. over a range of Mach numbers for the two flight ranges


2500 and 5500 nautical miles respectively. The speeds chosen are 200, 300, 400 and 500 knots. For convenience of reference the relationships between E.A.S., Mach number, altitude and equilibrium temperature are shown in Figs. 9 and 10.

4 . 3 . Cooling Alone

The values for cooling alone for each of the ranges are shown in Fig. 11. It will at once be noticed that the weight required is approximately doubled when the range is increased from 2500 to 5500 miles. In both cases the weights are not excessive at the lower value of E.A.S. At higher values, particularly for the longer range, the weights are evidently impractically high. For all values of E.A.S. the weights increase initially with Mach number, reaching maxima beyond which they decrease markedly as the Mach number is further increased.

1 »SOO KNOTS

\ O KNOTS

/ ///

OO KNOTS

I 2C OKNOTS V 2 4 6

MACH NUMBER 4 6 8 MACH NUMBER

FIG . 1 1 . Min imum weight of cool ing required for a structural temperature of 1 5 0 ° C (British Crown copyright).

This effect is observed also in the cases of insulation alone and insulation and cooling combined. The reason for this is the following: Since the E.A.S. is fixed, an increase in Mach number implies a corresponding increase in altitude and consequently a reduction in air density. The air heat transfer coefficient decreases as the density is reduced, but the adiabatic wall temperature rises because of the increase in Mach number. The combined effect gives a slight increase in heat input. However, since the speed is increased, the flight time is decreased so that the time available for heat to be absorbed is reduced. On balance the overall effect is to reduce the total quantity of heat absorbed above the critical values of Mach number.

432 M. O. W . W O L F E

4 .4 . Insulation Alone

The curves for insulation alone are shown in Fig. 12. Again the same broad trends are observed. In all cases, with the exception of the highest value of E.A.S. for the longer range, cooling is more effective than insulation and becomes increasingly more so as the E.A.S. is reduced. This comparison is, of

RANGE 2 5 0 0 MILES RANGE SSOO MILES

MACH NUMBER MACH NUMBER

FIG . 12. Minimum weight of insulation for a structural temperature of 150°C (British Crown copyright).

course, very sensitive to the efficiency of the insulant and the value of surface emissivity coefficient, particularly at the higher values of Mach number and E.A.S. Nevertheless, at the lower values of these quantities the superiority of the coolant alone is sufficiently well marked to be of significance. If we consider for example the 300 knot E.A.S. curve for the coolant for the 2500 miles range,

LB/FT*

MACH NUMBER MACH NUMBER

FIG . 13. Minimum weight of insulation plus cooling for a structural temperature of 150°C (British Crown copyright).


the maximum weight occurs between the Mach numbers of 5 and 6 and has the value 2 .3 lb/ft 2, at Mach 4 it is 2 lb/ft 2 and at Mach 3 it is less than 1 lb/ft 2. The corresponding figures for insulation are more than double these values.

4 . 5 . Cooling and Insulation Combined

We now consider cooling and insulation combined. The minimum optimized weights for this case are shown in Fig. 13 for conditions identical to those in the previous cases. Again the broad trends are similar, but the most noticeable feature is that at all conditions the weights are considerably less than in the cases of either insulation or cooling alone. To illustrate these effects a comparison of all three methods of thermal protection is shown in Fig. 14 for a Mach number of 4. The weights are compared at each value of E.A.S. for the two ranges.

8

• . WEIGHT LB/FT* 4

EAS. 2COKNOTS EAS. 300KNOTS

RANGE

2 5 0 0 MILES 2

EAS. 400KNOTS

# EST ii

EAS. SOOKNOTS

WEIGHT L B / F T 4

RANGE

5 5 0 0 MILES

12 r

IO

8

KEY COOLING = ^

INSULATION - ( ^

6 "

4 V \ \

2 " \ N

1 ^ 1

1 /// \ 4 \

V O

FIG . 14. Comparison of weights of cooled insulated and combined insulation and < for a Mach number of 4 (British Crown copyright).

For both ranges the rate of increase of weight with E.A.S. for combined cooling and insulation is less than for either alone. However, by far the most notable feature is the very much greater effectiveness of combined insulation and cooling, particularly at the higher values of E.A.S.

Consider, for example, the 400 knots E.A.S. case for the shorter range. The weight of protection is rather less than 2 lb/ft 2. Something must be added to this figure to allow for the dead weight of the coolant and the heat exchanger system. This weight is extremely difficult to estimate. However, on available evidence the figure of 0 .5 lb/ft 2 does not seem to be unreasonable. The weight of protection required would therefore seem to lie somewhere in the region of 2 .5 lb/ft 2.

The structural weight which would be saved by changing from a nickel alloy or steel structure at 350°C to an aluminum alloy structure at 150°C is again

8

* t WEIGHT

4 L B / F T 2

2

O

WEIGHT L B / F T *

12

IO

B

V

6 L

4

2

O

434 M. O. W . W O L F E

very difficult to estimate. Recent British studies suggest that a saving of perhaps 2 to 3 lb/ft 2 of total surface area may be possible.

However, there will be additional savings in weight in the case of the pro-tected aircraft of the amount of internal cooling that would otherwise be re-quired for the crew, passengers and equipment. So that it is possible that an externally protected aluminum alloy aircraft may be competitive with its un-protected steel or nickel alloy rival for the case we have considered. For the longer range the time of exposure to temperature is increased and at the higher Mach numbers the surface temperatures are greater. The structural weight and the weight of internal protection required must therefore also be expected to rise for an externally unprotected structure. On the other hand for the longer range aircraft the weight of external protection required never rises above 4 lb/ft 2 over the whole range of Mach numbers considered if the cruising speed is limited to say 400 knots E.A.S., so that under these conditions the protected aircraft may well turn out to be the lighter.

While the uncertainties of the assumptions on which these studies are neces-sarily based are admittedly great, there would nevertheless seem to be a fairly strong prima facie case for considering external protection systems of the com-bined cooling and insulation type for aircraft designed to cruise at Mach numbers of 4 and above. It is also possible that some form of protection may also be justified at lower Mach numbers, but this could only be established by detailed studies of particular cases.

Apart from questions of comparative weights, another cogent reason for considering external protection arises from the need to conduct full-scale strength and fatigue tests with similated heating on unprotected aircraft. The difficulty and expense of such tests would clearly be many times greater than it would be for similar tests without heat. By choosing to protect the structure, many of these difficulties of testing would be substantially reduced, thus effecting savings on overall cost and aircraft operating costs.

We may therefore reasonably conclude that there is a strong case for further research and development of thermal protection systems despite the additional systems complexity and the many problems to be overcome in evolving satis-factory insulants and methods of attaching them to the surface.

C O N C L U S I O N S

In this short survey it has not been possible to give more than a cursory outline of the problems involved in the design of supersonic aircraft. Kinetic heating is a dominating factor, although the shape of the structure as dictated by the requirements of the aerodynamicist and the operational role of the aircraft are also important considerations.

In the new environment the choice of materials becomes a much more difficult matter and must be based on an optimum balance of qualities in relation to the loading and heating spectra, the shape and the life expectancy of the aircraft. Aircraft designers have had little previous experience with heat resistant


materials, particularly as regards production and manufacturing techniques, whereas a vast reservoir of knowledge exists on the use of aluminum alloys. For certain supersonic projects which are intended to be operated at thermal conditions that would preclude the use of unprotected aluminum alloys, we therefore seem to be faced with an important decision—whether to build the aircraft in heat resistant alloys and face all the problems this would entail as regards internal heat protection systems, manufacturing difficulties, and the extra time and cost of more elaborate proof testing procedures, or avoid some of these problems by constructing the aircraft in aluminum alloy protected externally against the effects of heat. On the basis of existing knowledge it is impossible to make this decision positively in a general way. Indeed such a decision will always depend to a very large degree on the operational role of the aircraft.

However, estimates based on existing evidence do show that there is at least a substantial case for further research and development work to be done on thermal protection systems generally in order to provide the data in which to make this evidently very crucial decision.

A C K N O W L E D G E M E N T S

The author wishes to express his indebtedness to several of his colleagues in the Structures and Chemistry, Physics and Metallurgy Departments of the Royal Aircraft Establishment for their advice and assistance in the preparation of this paper, and to the Royal Aeronautical Society for giving their permission to reproduce Table 1 and Fig. 4 from the Society's Journal, and also to the Chief Scientist, Ministry of Aviation, for permission to publish the paper.

The opinions expressed are the author 's . The illustrations are British Crown Copyright Reserved. The permission of the Controller, Her Britannic Majesty's Stationery Office, has been received for their reproduction.

APPENDIX. MATERIAL PROPERTIES

Precipitation Titanium hardening

Aluminum alloy stainless steel Nickel Nickel Molybdenum Commercially alloy T i - 4 A 1 - PH15-7MO alloy alloy alloy pure

Material D T D 5070 3 M o - l V R H 9 5 0 cond. Rene 41 Inconel X M o - 0 . 5 T i tungsten

Chemical composit ion (of element)

Aluminum Al Remainder 3 . 7 5 - 4 . 7 5 0 . 7 5 - 1 . 0 1 . 5 - 1 . 8 0 . 4 - 1 . 0 Titanium Ti 0 . 2 max Remainder 3 - 3 . 3 2 . 2 5 - 2 . 7 5 0 . 5 Iron F e 0 . 9 - 1 . 4 0 . 2 5 max Remainder 5 . 0 max 5 . 0 - 9 . 0 N o m i n a l Nickel N i 0 . 8 - 1 . 4 6 . 5 - 7 . 7 5 Remainder 70 min N o m i n a l Molybdenum M o 2 . 5 - 3 . 5 2 . 0 - 3 . 0 9 - 1 0 . 5 9 9 . 5 Tungsten W 100 Boron B 0 .01 max Carbon C 0 . 0 8 max 0 . 0 9 max 0 . 0 6 - 0 . 1 2 0 . 0 8 max Columbium Cb 0 . 7 - 1 . 2 Cobalt C o 10-12 1 .0 max Chromium Cr 1 4 . 0 - 1 6 . 0 18-20 1 4 . 0 - 1 7 . 0 Copper Cu 1 . 8 - 2 . 7 0 . 5 max Hydrogen H 0 . 1 5 max Magnesium M g 1 . 2 - 1 . 8 Manganese M n 1.0 max 0 . 0 5 max 1 .0 max Nitrogen N 0 . 0 5 max Phosphorus P 0 . 0 4 max Sulphur S 0 . 0 4 max 0 .01 max Silicon Si 0 . 2 5 max 1 .0 max 0 . 0 5 max 0 . 5 max Vanadium V 0 . 7 5 - 1 . 2 5

436 M

. O

. W

. W

OL

FE

EL

EV

AT

ED

T

EM

PE

RA

TU

RE

D

ES

IGN

A

ND

DE

SIG

N

CR

ITE

RIA

4

37

0.693

19.0

3 . 3 at 200°C 3.0 at800°C 2.7atl600°C

4.3 20-100 4.6

20-800

8 8

rt a

s (N

No data

0.369

10.2

3.4 at20°C 2.9 at430°C 2.6 at 1090°C

5.5 20-100

6 .1 20-1000

80 at 20 °C 59.5 at 1090°C

No data

0.300

m

oo

0.37 at 50°C 0.64 at 500°C 0.95 at 900°C

12.1 30-90 16.2

30-370

12.8 at 900°C

24 at R.T. 30 at430°C 4.5 at760°C

12 at870°C

0.296

<N

OO

27 at 200°C 42 at 650°C 55 at 980°C

9 at 90°C Oat 65°C 7 at 1090°C

5 at 200°C 1 at 980°C

0 at R.T. 0 at 650 °C 6 at 980°C

O

O

O

Tt r-

Tt

OS

O

o

o

uo

U

o uo

m

rt O

u

o ©

VJ

o

o

o

o O

O

O

O

U

O

<N r

t 0

0

U

o uo

m

rt O

u

o ©

O

as

ON

i

uo

oo

VJ

o

o

o

o O

O

O

O

U

O

<N r

t

©

U

o uo

m

rt O

u

o ©

o

uo

oo

o o o

o

UO

T

fr U

O T

j"

0.163 \

4.51

0.17at20°C 0.33 at 540°C

9.0 20-90 9.9

20-540 &

O

00

(N

(N

rt rt (N

uo

5 .5 at R.T. 6.2at200°C 5.0at400°C

0.099

2.75

3 .6 at50°C 4.2 at200°C

22.1 20-100

25.1 20-400

86.0 at 50°C 100 at 200°C

6.5 at R.T. 7.2atl50°C

11.5 at 200°C

f Expansion \ Range °C /Expansion \ Range °C

/—s

CO

a >>

f Expansion \ Range °C /Expansion \ Range °C

l 2 in. G

Density (lb/j

Specific gra\

Thermal coi (watts/in.

11

-a a _

x *

X

O

"35

S X

CO

O

Elongation (oi

438 M. O. W . W O L F E

R E F E R E N C E S

1 . HEATH, W . G. / . Roy. Aero. Soc. 6 3 ( 1 9 5 9 ) . 2 . DRYDEN, H . L. and DUBERG, J. E. Paper presented to the Fifth General Assembly,

A . G . A . R . D . (June 1 9 5 5 ) . 3 . BISPLINGHOFF, R. L. / . Aero. Sci. (April 1 9 5 6 ) . 4 . MANSFIELD, E. H . A . R . C . , R. and M. N o . 3 1 1 5 ( 1 9 5 7 ) . 5 . CAPEY, E . C. and MCKENZIE, K . I . A . R . C . Current Paper N o . 4 1 2 . 6 . CAPEY, E. C. Aircraft Engineering (October 1 9 6 1 ) . 7 . MCKENZIE, K . I . A . R . C . Current Paper N o . 4 1 7 ( 1 9 5 8 ) . 8 . POPE, G. G. A . R . C , R. and M. N o . 3 2 3 6 ( 1 9 6 0 ) . 9 . MANSFIELD, E. H . A . R . C , R. and M. N o . 3 1 9 7 ( 1 9 5 9 ) .

1 0 . MCKENZIE, K. I. Unpublished R .A .E . Report. 1 1 . PARKES, E. W . Aircraft Engineering, 2 6 , 3 1 0 (December 1 9 5 4 ) . 1 2 . MANSFIELD, E. H . Unpublished R . A . E . Report (to be published in / . Roy. Aero. Soc). 1 3 . MORLEY, L. S. D . Quarterly Journal of Mechanics and Applied Mathematics, 1 5 ( N o -

vember 1 9 6 2 ) . 1 4 . POPE, G. G. "The Buckling of Plates Tapered in Thickness", R. and M. N o . 3 3 0 9 (October

1 9 6 1 ) .

1 5 . POPE, G. G. "The Buckling of Plates Tapered in Planform", R. and M. N o . 3 3 2 4 (April 1 9 6 2 ) .

1 6 . POPE, G. G. "The Bending under Normal Loading of Plates Tapered in Planform", R. and M. N o . 3 3 2 5 (April 1 9 6 2 ) .

1 7 . BURT, M. E. / . Roy. Aero. Soc. 6 6 ( 1 9 6 2 ) . 1 8 . MURPHY, A . J. / . Roy. Aero. Soc 6 1 ( 1 9 5 7 ) . 1 9 . SCHMITT, A . , BRULL, M. A . and WALKER, H . S. Symposium on Structures for Thermal

Flight. A .S .M.E. Aviat ion Divisional Conference, March 1 9 5 6 {A.S.M.E. Transactions, January 1 9 5 7 ) .

2 0 . TAYLOR, J. A . R . C . Current Paper N o . 5 4 5 (July 1 9 6 0 ) . 2 1 . MCCUE , D . J. Unpublished R .A.E . Report (April 1 9 6 2 ) .

PANEL DISCUSSION

T H E Session convened at 9 a.m., Walter J. Trapp (Materials Laboratory, ASD, Wright Patterson Air Force Base, Dayton, Ohio) presiding. Members of the panel were Messrs. W. Boccius (Lockheed Missiles and Space Company, Sunnyvale, California), A. M. Freudenthal (Columbia University, New York, New York), J. M. Hedgepeth (Martin Company, Baltimore, Maryland), H. Lowndes (ASD, Wright-Patterson Air Force Base, Dayton, Ohio), and M. O. W. Wolfe (Royal Aircraft Establishment, Farnborough, Hants, England).

CHAIRMAN T R A P P : Gentlemen, I should like to open the panel discussion. Dr . Hedgepeth will present a short introduction.

D R . HEDGEPETH: Unfortunately, I wasn't able to hear Professor Freudenthal 's or Mr. Wolfe's talk, but I have a few things to say which I thought would be pertinent at a session on Design Criteria. They have to do with the connection between what we use as design criteria and reliability.

If one were to design a vehicle, he would have to establish design criteria for the staff members who were actually doing the detailed design. Normally, in the past, we have constructed what are called "limit loads" ; now that we are in the realm of elevated temperatures, we find "limit temperatures"; and then we put on factors of safety, which are supposed to take care of all sorts of things— fatigue, long-time exposure, ignorance, etc. We then hope that the factor of safety gives us the reliability we want.

U p until the last few years, the design process has taken place without any definite regard to the question of reliability. The structures man has always said, "This structure is not going to fail."

On some of the more recent structures, however, we have had to start thinking about their structural reliability. Calculations have been made after the fact, to obtain a reliability figure and to see that this number has more nines in it than the budget assigned to the structure by the reliability people.

I feel that in the future, particularly for single-use vehicles, such as launch vehicles and, perhaps, re-entry vehicles, we will have to introduce reliability earlier in the design phase, which means that we will have to go through the process of selecting factors of safety that will give us the kind of reliability we want.

When you have a single type of loading, the problem is fairly well under-stood. One imagines the statistical probability distribution of applied loads and the probability distribution of allowables, or structural failure loads, plots these and notices where the two probability distributions overlap.

One thereby makes a calculation of reliability. Curves have been constructed from which one can read the required factor of safety to give a certain reliability

439

440 P A N E L D I S C U S S I O N

when you know the standard deviations of the load distribution and of the allowable distribution.

This procedure can be used to come up with factors of safety to give to the designers, because it will be a number of years before the designers themselves will be able to think on their own about statistics and reliability.

As I said before, the procedure is fairly well understood in principle, although we do not know well enough what the shapes of the distribution functions are.

There is, however, another problem which is not even well understood in principle, and that is what happens when our structure is subjected to a multi-plicity of inputs, in particular, inputs of loading and temperature. The question arises of how one handles this problem in a practical way. Here one is involved in interaction problems, with a multivariate statistical distribution. I know of no way that has yet been found to put this problem in a practical framework which the designer can use.

CHAIRMAN T R A P P : I think we may possibly have some questions on Professor Freudenthal's paper and Mr. Wolfe's paper. I should like to open the discussion on these first.

QUESTION: Referring to Professor Freudenthal's paper, I should like to hear your comments on extending your analysis to the case of elevated temperature conditions under combined alternating and direct stress.

D R . FREUDENTHAL: The extension would face the same difficulties that we face in the extension of the fatigue analysis to variable loads, for which we have, or at least are trying to develop, a procedure of damage accumulation involving random loads. We would have to develop a procedure involving damage accu-mulation under load at temperature, which is going to be much more difficult. However, I believe that if we can first verify experimentally the tentative suggestion that at elevated temperature we do not have to worry too much about the com-bination of alternating and mean loads, but can study alternating loads separ-ately and mean loads separately, and if we also know from fatigue analysis that the shape of the distribution function and, therefore, the shape of the risk func-tion due to random loading does not differ significantly from the shape under constant loading, we might have a key to the reliability analysis.

CHAIRMAN T R A P P : I should like to open the discussion on Mr. Wolfe's paper. Are there any questions or comments ?

QUESTION: In connection with Mr. Wolfe's paper, I can only say that I agree with him on the points made in the first part of his paper. I should like to point out that it does not necessarily follow that maximum or critical load stresses act in direct combination with thermal stresses. These are two separate and distinct critical conditions. In the transonic Mach number range, which produces a large number of the critical load stresses, the structure is relatively cool; whereas,

P A N E L D I S C U S S I O N 441

in the high Mach number range with operations at a higher altitude, and conse-quently lower stresses, the thermal stresses are of significance. However, in the realm which we are now considering, principally that of aircraft rather than missiles and rockets, thermal stresses are in themselves not prohibitive. They may vary from 15 to 20, rarely to 25 per cent of the yield stress in the material. This, combined with the load stresses, results in gross stress intensities which you can live with. Thus, all things do not add up adversely; however, we may have to apply different factors of safety to each of the different conditions for which we have to design, and to the different parts of the airplane to which the particular condition would be applicable.

We should, therefore, be careful of extreme generalizations in this matter, without due consideration to the specific design at hand.

M R . W O L F E : I agree in general with what has been said, but in my opinion the thermal stresses can, in certain circumstances, be higher than the values which have been mentioned.

I should like to revert to some points I mentioned in my paper, which also have relevance to matters referred to by Professor Freudenthal.

The first is concerned with the extreme difficulty of conducting full-scale approval tests under heating and mechanical loading conditions. Such tests are very complicated and expensive; moreover, it is impractical for economic reasons to test more than one full-scale specimen for static strength and one for fatigue. In these circumstances the statistical significance of the tests from a reliability aspect is debatable, to say the least.

The second point is that from an airworthiness point of view, what is im-portant is the strength of the structure throughout its useful life. It is generally true that the strength of a structure declines during its service life due to attrition caused by fatigue, corrosion and other phenomena. This is recognized and is normally taken account of by applying safety factors. However, the determina-tion of adequate factors in a thermally significant environment is extremely difficult because of degradation of material properties and thermal fatigue. We are therefore faced with the problem of whether to perform a static test on a virgin structure and apply what factors we can, or alternatively test one which has been subjected to a combined heat/load/time history representative of its life cycle. If we are considering the reliability aspect, we must equate the statis-tical indeterminancy of a single test on a full-scale specimen, albeit with factors, to a larger number of component tests with less representative boundary conditions.

In this context the great cost in effort and money of full-scale testing, where controlled heat as well as load is applied, must be borne in mind.

CHAIRMAN T R A P P : H O W about the testing time involved?

M R . W O L F E : This of course raises the whole question of how we may apply factors for thermal effects so as to accelerate them.


There is at present insufficient information on metallurgical and other aspects to indicate how this might be achieved. In these circumstances it may be per-tinent to suggest that if we can avoid some of these difficulties by thermally protecting the structure, so much the better.

CHAIRMAN T R A P P : I think that if we do not protect the system, we need accelerated tests. I was just wondering whether anybody has any ideas on per-forming such tests on structures.

M R . LOWNDES: I think I can only agree to and emphasize the points that have been made by almost everyone. We know we will have to test, and yet what is, I think, often overlooked, is the tremendous cost of testing as an eco-nomic factor when you get into simulating the thermal environments. One must also consider the difficulties involved in simulating these environments that appear to be required when one wants to determine reliability with respect to life, as opposed to static strength reliability.

I can foresee the situation in which, for large vehicles, particularly large re-entry vehicles, it would be totally impractical to build a test facility for a full-scale certification program.

I think we are rapidly approaching an area where we will have to find a way of higher analytic reliability—if you want to use this terminology—and the best techniques we can use to extrapolate component test results into a full vehicle reliability.

As Mr. Wolfe pointed out, the costs, just for static strength certification with elevated temperature simulation, are high; and we certainly don't know how to handle the reliability from a life standpoint, where you cannot compress time, as we do in reliability testing for life with load alone.

This is an area in which work must be done, whether we end up with an arbitrary approach or whether there is some possibility of artificially produced damage that will give us the same effect as a true time simulation procedure.

M R . BOCCIUS: I want to make a few comments on Mr. Wolfe's presentation of the material evaluation. It was noticed that preference has been given to titanium. There are two other materials which also might be worthwhile to be considered.

The first one is magnesium. It has very good sink capabilities, therefore pro-vides a restricting effect on the increase of the skin temperatures; it is weight-efficient at the same time. However, it is sensitive to corrosion. But, if treated carefully and protected accordingly, magnesium presents a good choice of an applicable structural material, including its weight advantage. I noticed the superiority of this material on the temperature stress parameter, as was shown. In addition, magnesium structures result in increased skin thicknesses affecting the buckling instability such that it becomes less essential in consideration. These are some of the reasons why Lockheed Missiles and Space Company has chosen magnesium to be the major material of the Agena vehicle.

P A N E L D I S C U S S I O N 4 4 3

Another choice of criteria points to beryllium. When I say beryllium, prob-ably the answer will be "The cost is prohibitive." However, the price per pound may be well spent for the gains. Our organization has followed the manufacturing and development of this material and its parts through frequent contacts with the beryllium companies of this country, and in addition has contributed to the state of the art by our own material and structural investigations and studies. At first, there were only smaller items which had to be pressed, forged and machined, but now we have available sheets of beryllium on a larger scale, which should be seriously considered for a very worthwhile weight saving. Naturally, with aircraft of relatively large dimensions, the cost will be quite high despite some price changes.

Since the supersonic aircraft is five or more years off, I believe that the de-velopment of large beryllium sheets will probably be perfected, and that the cost of beryllium will be reduced further.

Therefore, in conclusion, I would appreciate learning about the views of Mr. Wolfe on the applicability of these two materials.

M R . W O L F E : Several of my colleagues in England who have read my paper have also called attention to the omission of these two interesting materials.

The reason I chose the materials I did was simply that they were the ones we knew most about, and in any case my purpose was rather to show how the particular criterion chosen influenced the choice of materials in relation to their properties, the loading environment and the nature of the structural element concerned. I was not, in fact, aware of the full extent of the activity in the United States on beryllium development.

With regard to the five-year period which you mentioned, this would be too long to permit the incorporation of beryllium in the British and French Super-sonic Transport project, having regard to the time scale envisaged. The property of beryllium which makes it so interesting to the structural designer is its very high specific modulus. However, its comparative lack of ductility is the real problem, and I believe there is also a difficulty associated with toxicity.

M R . BOCCIUS: The toxicity is present only when beryllium dust is inhaled during machining. Proper safety measures, applying dust removal equipment and cleanliness, will eliminate this hazard.

CHAIRMAN T R A P P : The difference between Mr. Wolfe's and Mr. Boccius' estimate of the time when the supersonic transport will be finished is the difference between Mach 2 and Mach 3 .

QUESTION: I should like to go back to Professor Freudenthal 's discussion. I understand the implication of your conclusion regarding the independent testing for the various conditions. I wonder whether you would clarify the relative importance of mean stress on fatigue and its importance on creep.

I can very well understand that if you want to determine the effect of mean


stress on creep, you could perform a creep test, and that if you wanted to deter-mine the effect of alternating stress in fatigue, you could determine a fatigue test independently.

However, the mean stress is so important in affecting the fatigue charac-teristics, without any regard whatsoever for creep, that I don' t see how you can separate it and, by running independent fatigue tests, come to any conclu-sion regarding the combined behavior.

D R . FREUDENTHAL: It is quite true that at normal temperatures there is a very significant effect of mean stress on fatigue. However, in my presentation I attempted to stress the fact that the usual Goodman diagram in which the effect of mean stress on fatigue is presented should not be confused with the interaction diagram in which the mean stress itself produces creep damage.

What I was trying to suggest on the basis of equation (4.15) is that if the mean stress is high enough to produce failure due to creep, it is likely that the fatigue behavior itself will be of minor significance as the main damage mechanism will be due to creep. On the other hand, for relatively low mean stress we may find that its effect is of minor significance since the main damage mechanism will be that due to fatigue. It is to be expected that the damage process and the fracture appearance will be different in the two cases. In other words, it is not unlikely that if the mean stress is at all significant in its effect on fatigue, it will be the significant failure stress in creep and thus be more important in producing failure directly than in its influence on fatigue behavior. This implication derived here from theoretical considerations should obviously be tested by experiment.

CHAIRMAN T R A P P : Mr. Boccius has some remarks to make on general design philosophy.

M R . BOCCIUS: If you look at the mathematics and the complexity of the concepts presented, it scares you as a structural engineer involved in day-by-day decisions concerning design criteria, their structural applications, and their importance upon the development of modern structures.

I myself happen to be involved with missiles and space structures; in this respect, we are fortunate not to have to go through this very complicated mathe-matical exercise to judge the reliability of our structures.

In our missiles we are faced with temperature problems. Normally, those problems do not disturb us with regard to their creep effects although they are mostly in the elevated temperature range.

We have a clear separation of loads in the cool region where the missile is flying through gusts or when it is standing on the launch pad, while the higher temperature effects are experienced only when we are mostly out of the at-mosphere and have perhaps only large accelerations. These temperatures, how-ever, do not produce too much degradation in the material. Therefore, in establishing design criteria we follow the procedure which has been used over the past decade on structures in the cool range.


We have slightly manipulated the safety factors. These safety factors are still put on top of the limit design condition. Reliability enters into our structure not to such a large extent as it enters into the component testing of electronics and electrical systems. Here lie, in many cases, the weaknesses of certain missile designs. The reliability of those elements has to depend strongly on testing.

In the re-entry phase of certain missiles, we have to face a range of hot tem-peratures, but in this field we do not have all the necessary structural informa-tion and flight data. We have material strength available to resist the external loads imposed upon the structure under very hot conditions on the outside and cooler conditions on the inside. Only a portion of the material can, therefore, carry the load.

One can analyze the structure using the most elaborate approach and come up with values for the stresses. When one tries to justify this analysis, however, there is no proof. One can test the structure, one can put strain gauges on it, but one still has no proof for the strain and stresses experienced in the hot material. Therefore, the only final proof would be the actual cracking or failing of the material.

I also want to point out that in the missile field we are operating very much with the pressure vessel concept in which we do not have to rely on compressive stresses but mainly on tension. I might mention the Atlas which is designed in many cases for tension, this being a limiting factor in its design phase.

We are also relying very much on testing. However, when we go to larger and larger diameters, it is very doubtful that such missile design configurations can be tested. To give an example: If you have a Saturn and you have certain stages put on top of the Saturn, then the height of this total vehicle would exceed the height of the Times Square Building. In this case, it would be impos-sible to have full-scale testing, even of large missile components. Therefore, reliability analysis, which was touched upon here, is an impossibility since you cannot test even one of these items. You have to rely on small component testing and the successful incorporation of those results into the full-scale vehicle.

CHAIRMAN T R A P P : I think there is even a difference of opinion with respect to aircraft, about whether it is sufficient to test components or whether one should test whole structures.

D R . HEDGEPETH: I think it is important for all of us to realize that when we talk about structures and structural reliability, we are not only talking about the load that the structure can carry, or about whether it fails in fatigue or creep, but also about whether the thermal protection system on the structure behaves properly.

In a system that has to stay in space for some time, the question arises whether meteorites will damage the material to an extent that interferes with a good re-entry. This, of course, also holds true for a radiation protection system.

The question of testing, then, is not just a question of applying loads and


temperature to a structure, and seeing whether this structure will hold together. It is a question of seeing whether there is an active system, whether the calculated pumping rates are those actually needed. It is a question of whether the surfaces maintain the required absorptivity capacity during long-time exposure.

We are talking about a couple of million dollars for an airplane; we are talking about two hundred million or three hundred million dollars in facilities to perform tests of space vehicles. These are essentially enormous vacuum chambers with helium-cooled walls and means for applying loads and applying simulated solar radiation and other effects. The question of testing is one of enormous complexity.

With respect to the testing of components versus full-scale testing, I am a believer in the concept that one can test components and thereby assure the reliabilities of the system. There are lots of people who do not believe in this concept. The problem here, of course, is the interaction between the components that is not taken into account.

I do not think we have ever made a complete test in the history of the airplane structures' field. I do not know of any airplane on which a complete test was performed, considering the entire kind of exposure that the airplane might see during its lifetime. We have been able to get away with this because in some instances we have been smart and in some other instances we have been lucky. But we have to make sure that in the future we put as much smartness into the design as we can and reduce the element of luck as much as possible.

QUESTION: I don ' t think you can possibly argue that. I t would be better to test the entire structure under the right environment, if you could do it. In the overall testing of the structure we do not necessarily test the structure in all the possible environments it may be subjected to . Also, as structural engineers we should not lose track of the fact that all the other components in a system are not tested in all the possible environments to which they are subject.

I think we have been forced into an area of reliability which may be difficult to attain. There may be a psychological effort required to accept a structure with less than one hundred per cent reliability, just as we have already accepted various components having less than one hundred per cent reliability.

With respect to Mr. Wolfe's paper, I should like to say that the stress levels which we use for various materials, even for a given alloy, are rather contro-versial. Various strengths curves should be shown, while the stresses may be left out. By varying the stress level we get different results.

QUESTION : In connection with Mr. Wolfe's paper, I was wondering about the cooling system, the cooling weights and the weights of the associated equip-ment and the associated structure needed to circulate the cooling.

M R . W O L F E : I found it extremely difficult to obtain realistic weights. My estimates for the various items of the cooling system were, in fact, based mainly on American sources. On this subject we have as yet done little work in England.

P A N E L D I S C U S S I O N 4 4 7

To answer your question more specifically, I envisaged a system in which a liquid is circulated through channels in the structure for the purpose of trans-ferring heat to a heat exchanger in which water is converted to steam at sea level pressure. The steam produced might profitably be used in the engine for increasing the combustion efficiency. The channels in the structure should pre-ferably be designed to contribute to structural strength. For example, a form of corrugated sandwich might be employed, although this would probably entail the use of a gas as the heat transferring agent because of the large volumes involved.

My estimates included the system weight, with the heat transfer fluid as well as the weight of water required.

QUESTION: I should like a chance to comment on the remarks which Mr. Boccius made relative to hot thermal testing and methods of analysis.

It is true that you cannot get good results when trying to test nose cones or similar structures at very high temperatures. We are designing a radiation-cooled re-entry nose cone at some very high temperatures. We tried to do some testing and, as Mr. Boccius said, it was meaningless. However, we have found that you can make some use of the theory of thermal stress and of solutions of thermo-elastic boundary value problems and get some realistic results.

M R . BOCCIUS: I do not want to discredit the more elaborate analyses. We had our own development in re-entry bodies; as the development went on, we improved and refined our methods. With the material and the concepts we used, we should have enough reserve strength to carry the applied loads, but we do not have the final proof whether the analysis was correct, or whether it should be changed.

In essence, when a certain design is developed, it is first analyzed and checked, then tested, and then re-done until a closer agreement between analysis and test is reached. However, sometimes a considerable gap remains between the results of the analysis and those of the actual tests, the latter offering only the deflection pattern. Some explanation of your analysis might be derived from physical phenomena inside and outside of the body; but, in many respects, you are left on your own judgment.

QUESTION: Mr. Chairman, I should like to refer to the question of testing components versus testing of the complete structure. I might refer to some complicated tests which were conducted in Australia. Although these were done at room temperatures, they may be interesting since we tested some 1 5 0 struc-tures of aircraft wings.

One of the conclusions that came out of this program was that the load distribution in the redundant structure is a function of the fatigue life, and, in fact, varies throughout fatigue life. We have found cases where a wing taken from service and rolled into a new one has shown differences of 4 0 per cent in strain in the critical area. I don ' t think this means that you have to test the complete structure, but you should try to test a big enough part of it.


When the structure is under elevated temperature conditions, I would imagine that the correlation between component tests and tests on structures would become still more difficult.

M R . W O L F E : On this question of full-scale testing, the matters referred to in Professor Freudenthal 's paper prompt me to emphasize again the difficulties of full-scale testing with combined heat and mechanical load and the fact that, in general, for economic and other reasons, only one specimen can be tested.

I would submit that the time has come for a searching reappraisal to be made of the philosophy and procedures of full-scale testing, having regard to the enormous costs involved in man-power, time, and money.

We have to compare the value of a full-scale test, on the one hand, giving consideration to the difficulties of defining the accuracy of load and heat simula-tion, with that of component testing, where larger numbers may be tested but where uncertainties exist with regard to the boundary conditions.

There are, of course, other matters to be considered; designers like to know where their structure is likely to fail in order to bring to light unforeseen weak-nesses of design, and this is quite an important consideration.

Testing procedures for full-scale static tests have become a bit ossified over the years. The applied loads are idealized to represent optimal flight conditions. The distribution pattern is fixed and in the test the overall load is varied in increments. In flight, the overall load and its distribution varies in time. Where heat is not significant this procedure is reasonably satisfactory; but where the heating effects are significant, that is to say, where the thermal and mechanical load strains are acting in combination, it is necessary to vary simultaneously the overall load and its distribution as well as the heat input in their correct phase relationships as functions of time.

This is a complicated procedure and the method presently adopted in the United Kingdom is to apply a fixed load distribution pattern and then to apply the heat, varying as a function of time, the procedure being repeated at increasing increments of the fixed load pattern.

This gives a conservative answer, but at least in the case of static testing it does, in theory, provide an answer which can be compared with the results o design calculations.

However, in the case of a fatigue test with thermal cycling this argument is no longer valid. In this case there seems to be little possibility of escaping the unpleasant necessity of having to apply the heat and mechanical loads in their correct relationships if we are to obtain meaningful results.

To summarize, I would submit that there is a strong case for further research to ascertain the extent to which we may be able to avoid some of these formid-able difficulties by relying more on component tests.

A point which should not be overlooked is that modern flight vehicles are becoming more and more dependent on the reliability of their "black boxes", and this aspect must, in the final analysis, be harmonized with the reliability of the structure itself.


D R . FREUDENTHAL: There seems to be a real danger that the probability statisticians are moving into positions where they try to direct what the structural designer is supposed to do. I believe that as structural designers we ought to put them back where they belong, into electronics testing, "black box" testing and testing in general, in which a sufficient number of tests can be run to make the numerical statistical reliability statement meaningful.

There seems to be a danger that the methods which have been accepted as feasible in electronics testing and in the testing of small component parts will be taken over without criticism, particularly by certifying authorities and the government, and applied to structures where such methods are completely meaningless, since on the basis of a single test you cannot make any statement concerning statistical reliability and having as many nines as the "reliability engineer" would desire. In this case we are in fact in a region where no statistic-ally valid prediction can be made. Therefore, the application of standard statistical reliability techniques to structural design does not make sense.

When we can afford to test full-scale structures, particularly under combined mechanical-thermal spectra, we obviously should do so because there are important advantages to be gained, the main advantage being that the weak-nesses of the structure, which cannot be discovered by analysis, will appear in the course of the full-scale test.

However, whenever we find that we cannot easily afford to test the structure itself because the expense involved becomes a significant part of the whole de-velopment of this structure, the decisions will have to be made, from case to case, whether full-scale testing is worth the effort. The example which was brought up in the discussion is not quite applicable to our present conditions because to test 99 wings of a discontinued little fighter is quite a different proposition from that of testing a single item of one of the newer aircraft or of one of the new space structures under development. I do not believe that but for the special post-war conditions anybody would ever have been able to test 99 wings under normal conditions of development.

With respect to the probability aspect of the single structure test, it is obvious that, statistically, it has a particular attraction; it eliminates all scatter. If, nevertheless, we want to make a statement concerning the range of scatter, the best thing we can do under the circumstances is to make component tests to the largest extent feasible; but even for the interpretation of component tests we have to use some physically relevant types of distributions, in order to extrapolate the results which we can obtain even from component testing, because of their limited number, to the range which is significant for design; even in the 0.99 range we would have to have 100 specimens to make any statistically reliable statement concerning an unknown distribution.

However, as even in component testing we are usually happy to have five or six specimens, it is absolutely necessary to fall back on physical-probabilistic arguments in order to make a meaningful reliability statement on the basis of a physically relevant distribution function.

CHAIRMAN T R A P P : I think we have time for just one more question. 1 6


D R . HEDGEPETH: I just want to say a few words in defense of reliability. I believe we must determine structural reliability. It is not going to be satis-factory for most vehicles for which we are accustomed to do this calculation, to come out with a reliability figure of 0 .9 . I think we have to come out with three or four nines or even more.

We have had structural failures in actual flight. There should be some way of characterizing how these structural failures are going to occur. I agree that we are not going to find out structural reliability by means of a test run. I do believe, however, that one can estimate the structural reliability by means of analysis.

I may give an example of this. We are making a very simple structure, such as a rocket motor ; during the design stage a calculation was made of its re-liability, or more precisely, of how many motors were going to fail in manu-facturing proof tests. The calculations made at that time showed that about one out of every two hundred motors would probably fail in manufacturing; and in fact, one failed after 180 of them were made.

Let me give you another example. The Titan missile reliability calculations have been made and provide a figure of about 10~ 8 ; the main source of unreliability in this missile is in the weldments that are used to put the domes and the tanks together. An allowable stress of 25,000 psi is used there for the weld of the 20-14 material, which is less than half the ultimate strength of the parent material. Something like 18 tanks were pressure-tested and all of these tanks failed above ultimate pressure, but there was a considerable amount of scatter. Some tanks failed at 110 per cent, some at 210 per cent of ultimate load. The fact that these high percentages of failure occurred is obviously hurting the reliability.

As a matter of fact, even during manufacturing proof tests in which tanks are subjected to 90 per cent of limit pressure, a few failures occurred. These failures came about because of flaws in the structure that were undetected, but they were there and they were predicted.

I think that the philosophy of starting with the data on elements and small structural pieces and building up to components and then to the full-scale structure has to be pursued. Reliability cannot be established by tests but only by analysis.

M R . BOCCIUS: Let me make a few comments on the reliability concept. I agree with this approach when you can evaluate the structure after it has been tested. The vehicle reliability rests upon the total structure.

I have seen many cases in which a failure or damage of a certain structure was not expected but nevertheless occurred, resulting ultimately in a total failure of the vehicle. It has been observed in aircraft as well as in missiles. Consequently, it is not possible to design for all eventualities. We must choose between two alternatives. One must either design the whole vehicle for a stress, so as to make it a little heavier and thus give it a higher probability of survival; or one can weight-optimize the structural design of the whole vehicle, a pro-


cedure which will lead at first to some setbacks from unexpected weaknesses, with later modifications resulting from additional testing.

One can design right from the start with the concept that one wants to have no failures and thus assume conservative load conditions, later followed by low stress levels experienced in flight. For instance, the Thor missile, a vehicle which was designed to be rugged, has not suffered much from structural inadequacies. On the other hand, the Atlas, the booster of which was optimized for minimum weights at limit load capabilities, has failed occasionally during the past several years. However, after a modification period the Atlas is now one of the vehicles which show very high reliability.

To come back to the controversy of component testing versus full-scale testing: One cannot completely rely on component testing as a structural proof for the overall vehicle, even if there is no other choice. If one tries to wring out every ounce of material of a structure, inadequacies will develop when perform-ing component testing only, and later modifications will be required.

CHAIRMAN T R A P P : I think we have used up all of our time. I should like to close this session and to thank the panel members and the audience for their lively participation in the discussion. I should now like to give the floor to Dr. Liebowitz.

D R . LIEBOWITZ : We have gone through three stages of this conference. The first stage has been concerned with mechanisms and microscopic phenomena, the second with methods of analysis, and the third with design applications and procedures. There have been many disciplines represented. Significant opinions and thoughts were expressed during the course of this meeting. The results arrived at should have far-reaching influence in science and engineering. Many of us who have been undergoing some soul-searching with respect to these three different stages should find that Professors Freudenthal and Boley suc-ceeded in bringing together the talents required to solve the different problems in this diverse and significant field of technology.

On behalf of the Office of Naval Research, I would like to conclude this excellent meeting by thanking all those who have participated in this con-ference. In particular, I would like to thank the co-chairmen, organizing com-mittee, discussants, and also to give special thanks to those who have come from overseas to be here to contribute to a successful symposium,

I call this meeting adjourned.

AUTHOR INDEX

ABRAHAM, L . H . 3 1 5 ABRAMSON, H . N . 2 7 4 , 3 1 2 ACHENBACH, J . D . 2 6 4 , 3 1 1 ACHTER, M . R . 1 7 8 , 1 8 6 ADAMS, E . N . 5 5 ADAMS, M . C . 2 5 9 , 3 1 4 ADKINS, J . E . 3 3 6 AGGARWALA, B . D . 3 6 0 , 3 8 1 AHMADIEH, A . 1 6 7 AKHEIZOR, A . 1 2 , 1 3 , 1 5 , 5 5 ALFREY, T . 3 6 5 , 3 7 4 , 3 8 0 ALKHOZAIE, S . 3 8 0 ALLEN, D . N . 2 7 4 , 3 1 3 ALLEN, N . P . 4 1 2 ANDREATCH, P . 5 5 ANDRIANKIN, E . I . 2 9 4 , 3 1 5 ANSELL, G . S . 1 3 5 , 1 3 7 , 1 3 8 , 1 6 7 ARGYRIS 4 1 6 AUSKERN, A . B . 1 6 5 AUSTERMAN, S . B . 1 6 5

BAKER, G . S . 5 2 , 5 6 BALDWIN, E . E . 2 1 2 BARBER, A . D . 3 1 4 BARMORE, W . L . 1 0 5 , 1 0 7 , 1 6 5 , 1 6 8 BARZELAY, M . E . 2 7 8 , 3 1 4 BATEMAN, T . B . 5 4 , 5 5 BAUMANN, W . 3 4 6 BAYCE, A . E . 1 1 3 , 1 6 6 BEAMS, J . W . 2 1 4 , 2 4 4 BELLE, J . 1 6 5 BENNEWITZ, H . 1 , 5 4 BERGMAN, S . 3 1 2 BERRY, B . S . 8 , 5 5 , 3 4 6 BETHE, H . A . 3 1 4 BETTI 3 5 9 BILBY, B . A . 1 5 1 , 1 5 6 , 1 6 7 BIOT, M . A . 2 6 5 , 3 1 1 , 3 1 3 , 3 1 6 , 3 7 9 , 3 8 1 , 3 4 5 BIR , G . 3 9 , 5 5

BlRCHENALL, C . E . 1 6 6 BISPLINGHOFF, R . L . 4 3 8 BLASER, R . U . 2 1 2 BLOUNT, E . I . 5 5 BOCCIUS, W . 4 3 9 , 4 4 2 - 4 4 , 4 4 7 , 4 5 0 BOLEY, B . A . 2 6 0 , 2 9 4 , 3 0 2 , 3 1 0 , 3 1 1 , 3 1 4

3 1 5 , 3 1 6 , 3 2 1 , 3 2 3 , 3 4 5 , 3 4 7 , 3 6 5 , 3 8 0 , 4 5 1

BOMMEL, H . E . 1 3 , 5 5 BORCHARDT, G . W . 3 6 2 , 3 8 0 BORDONI, P . 5 2 , 5 6 BOWDEN, F . P . 2 1 4 , 2 1 5 , 2 1 8 , 2 1 9 , 2 3 0 , 2 3 2 ,

2 3 4 , 2 4 4

BRAILSFORD, A . D . 5 6 BREWER, S . 3 1 1 BROOKES, C . A . 2 2 7 , 2 4 4 BROWN, G . W . 3 1 0 BROWN, J . A . 3 1 0 BRUGGER, K . 5 5 BRULL, M . A . 2 7 8 , 3 1 3 , 3 6 5 , 3 8 0 BRUNER, L . J . 4 2 , 5 5 BRUNTON, J . H . 2 1 4 , 2 4 4 BUDWORTH, D . W . 9 4 BUGAKOV, I . 3 9 8 BURKE, J . E . 1 6 5 BURMEN, R . 5 4 BURT, M . E . 4 2 0 , 4 3 8 BUTCHER 1 1 3 , 1 6 6 BUTTENER, F . H . 1 6 5

CAPEY, E . C . 4 3 8 CARNAHAN, R . D . 8 5 , 9 4 CARSLAW, H . S . 2 6 0 , 2 9 2 , 2 9 3 , 3 1 0 , 3 2 2 , 3 4 5 CHADWICK, D . 2 6 4 , 2 6 5 , 3 1 1 , 3 4 4 , 3 4 5 , 3 4 7 CHAMBERS, L . G . 3 1 3 CHAMBRE, P . L . 3 1 5 CHANG, R . 1 2 8 , 1 6 5 CHAUDHURI, A . R . 5 5 , 1 8 6 CHRISTY, R . W . 1 3 1 , 1 6 6 CHU , W . H . 2 7 4 , 3 1 2

CITSON, S . J . 2 7 6 , 2 9 5 , 2 9 6 , 3 1 3 , 3 1 5 CLAUSS, F . J . 2 1 2 COCHARDT, A . 1 5 1 , 1 6 7 COFFIN, L . F . 1 8 7 , 1 8 8 , 1 8 9 , 2 1 2 COHEN, H . 3 1 5 COHN, R . W . 1 6 3 , 1 6 8 COLL, J . A . 1 6 3 , 1 6 8 COOK, S . E . 2 3 6 , 2 4 4 COPLEY, G . J . 1 6 6 COPLEY, S . M . 5 7 , 9 4 CORNELIUSSEN, A . H . 3 6 5 , 3 8 0 COTRELL, A . H . 1 6 6 , 1 6 7 COURSAL, E . 3 1 4 COWLEY, J . M . 1 5 8 , 1 6 8 CRANK, J . 2 7 4 , 3 1 3 , 3 1 6

DANILOVSKAYA, V . I . 3 4 6 DAUGHADAY, H . 2 9 6 , 3 1 6 DAVENPORT, C . C . 3 8 4 , 3 9 8 DAVIDSON, J . R . 2 5 9 DEACON, R . F . 2 3 4 , 2 4 4 DEBYE, P . 2

453

454 A U T H O R I N D E X

D E GROOL, S . R . 312 D E LAUNAY, J . 55 DERESIEWICZ, H . 265, 311, 344, 347 DEWEY, C . F . Jr. 295, 315 DICKER, D . 2 7 8 , 3 1 3 DILLON, O . W . 311 DOETSCH, G . 346 DONG , S . B . 381 DONTH, H . 56 DORN , J . E . 95, 103, 1 0 8 , 1 1 0 , 1 1 2 , 1 1 3 , 1 1 6 ,

125, 136, 138, 164, 165, 166, 168, 170, 186 DOUGLAS, J. Jr. 274, 312, 313, 316 DRAKE, R . M . 261, 311 DRANFELD, K . 13, 55 DRESSELHANS, G . 55 DRYDEN, H . L . 438 DUBERG, J. E . 438 DUFORT, E . C . 274, 313 DUHAMEL, J. M . C . 311, 317, 337 DUSINEBERRE, G . M . 274, 312 DYKSTRA, L . J. 7, 54

EASON, G . 345, 347 ECKERT, E . R . G . 2 6 1 , 3 1 1 EHRENREICH 55 EHRLICH, L . W . 316 ELBAUM, C . 5 3 ELKINGTON, W . E . 94 ENGLAND, A . H . 337, 346 ESHELBY, J. D . 56, 311 EUBANKS, R . A . 341, 346 EVANS, G . W . I I . 270, 294, 296, 312, 315

FELDMAN 5 4 , 2 6 0 , 3 1 0 FELTHAM, P . 1 6 6 FERRY, J. D . 3 6 6 , 3 8 1 FESHBACH, J. 2 6 3 , 3 1 1 FINNIE, I . 3 8 1 , 3 9 8 FISHER, J. C . 1 3 4 , 1 3 8 , 1 6 7 FITZGERALD, L . M . 2 2 2 , 2 2 4 , 2 4 4 FLAVIN, J. N . 3 4 5 , 3 4 7 FLINN, P . A . 1 5 8 , 1 6 8 , 1 6 7 FOLWEILER, R . C . 1 0 4 , 1 0 5 , 1 0 6 , 1 6 5 FORMAR, M . H . 1 6 5 FORREST, P . G . 1 8 9 , 1 9 0 , 1 9 1 , 1 9 3 , 2 1 2 , 4 1 2 FOURIER, J. T . 9 4 FRANK, I . 2 7 8 , 3 1 4 FRANKEL, R . E . 1 1 6 , 1 6 6 FRANKEL, S . P . 2 7 4 , 3 1 3 FREUDENTHAL, A . M . 3 4 4 , 3 6 5 , 3 8 0 , 3 8 1 ,

3 9 9 , 4 1 2 , 4 3 9 - 4 1 , 4 4 3 , 4 4 9 , 4 5 0

FREITAG, E . H . 2 1 5 , 2 4 4 FRIEDEL, J. 1 1 8 , 1 2 1 , 1 2 2 , 1 3 0 , 1 6 6 FRIEDMAN, A . 2 6 7 , 2 7 0 , 3 1 2 , 3 1 6 FRIEDMAN, M . B . 2 7 8 , 3 1 3 FUNK, E . R . 1 6 5

GAGEN-TORN 3 8 3 , 3 9 8 GALERKIN 3 8 3 GALLIE, T . M . Jr. 2 7 4 , 3 1 2 , 3 1 6 GARETH, T . 1 0 9 GAROFALO, F . 1 0 9 , 1 6 5 GALIWOOD, B . E . 2 7 8 , 3 1 4 GEMMELL, G . D . 1 7 2 , 1 8 6 GEORGIER, S . 3 1 4 GIBSON, R . E . 2 9 2 , 3 1 4 , 3 1 5 GIEDT, W . H . 1 3 6 , 1 6 7 GILMAN, J. J. 6 8 , 7 8 , 8 6 , 9 4 , 1 6 4 GLENN, J. 1 6 4 , 1 6 8 GOBLE, R . L . 1 6 5 GOODIER, J. N . 3 2 2 , 3 4 6 GOODMAN, J. F . 2 3 4 , 2 4 4 GOODMAN, T . R . 2 7 8 , 3 1 3 , 2 9 6 , 3 1 6 GORUM, A . E . 7 4 , 9 4 GOSSARD, M . L . 3 3 6 , 3 4 0 GOURSAT, E . 3 2 9

GRANT, N . J. 1 3 4 , 1 3 8 , 1 6 5 , 1 6 6 , 1 6 7 , 1 6 9 , 1 7 6 , 1 7 8 , 1 8 6

GRANATO, A . 5 1 , 5 5 GREEN, A . E . 3 3 6 , 3 3 7 , 3 4 5 , 3 4 6 GREEN, J. W . 3 1 3 GREENOUGH, A . P . 1 6 5 GRIGORIAN, S . S . 2 9 4 , 3 1 5 GROVES, G . W . 9 4 GRUSSARD, C . 1 5 2 , 1 6 7 GRUZIN, P . L . 1 6 6 GUARD, R . 1 3 4 , 1 6 7 GURTIN, M . E . 3 1 6 , 3 8 0 , 3 8 1

HALE, K . F . 1 1 3 , 1 1 6 , 1 6 6 HALL, A . R . 1 6 5 HAPP , M . B . 1 8 6 HARPER, J. 1 0 3 , 1 2 5 , 1 6 5 HARRISON, W . A . 5 5 HART, E . W . 1 3 4 , 1 3 8 , 1 6 7 HASSAN, H . A . 3 6 5 , 3 6 6 , 3 7 4 , 3 8 0 HAZLETT, T . H . 1 6 7 HEATH, W . G . 4 1 4 , 4 2 0 , 4 2 1 , 4 3 8 HEDGEPETH, J. M . 4 3 9 , 4 4 5 , 4 5 0 HELLER, W . 5 5 , 3 8 4 , 3 9 8 HELMS, H . E . 2 1 2 HERRING, C . 9 7 , 9 8 , 1 0 0 , 1 6 5 HERRMANN, G . 5 6 HETNARSKI, R . 3 1 1 , 3 4 5 , 3 4 7 HIDALGO, H . 3 1 4 HIEKE, M . 3 8 0 HILL 3 8 7 HILTON, H . H . 3 3 3 , 3 4 6 , 3 6 0 , 3 6 5 , 3 6 6 , 3 7 1 ,

3 7 4 , 3 8 0 , 3 8 1

HIRSCH, P . B . 1 2 1 , 1 2 3 , 1 3 8 , 1 6 6 , 1 6 7 HODGE, G . 3 9 8 HOFF, N . J. 1 7 0 , 1 7 1 , 1 8 6 , 2 7 8 , 3 1 0 , 3 1 1 ,

3 1 4 , 3 8 7 , 3 9 8

HONEYCOMBE, R . W . K . 1 6 7 HOPKO 2 5 9

A U T H O R I N D E X 455

HOWARD, E. M . 1 6 8 HOWIE, A . 1 4 4 , 1 6 7 HULSE, C . O. 7 4 , 9 4 HULT, J . 3 8 4 , 3 9 8 HUNTER, S . C . 3 7 9 , 3 8 2 HYMAN, M . A . 2 7 2 , 3 1 2

IGNACZAK, J . 3 3 4 , 3 4 5 , 3 4 6 , 3 4 7 ILYUSHIN, A . A . 3 9 8 ISAACSON, E . 2 9 6 , 3 1 5 ISHIDA, Y . 1 8 6

JAEGER, J . C . 2 6 0 , 2 9 2 , 2 9 3 , 3 1 0 , 3 2 2 , 3 4 5 JEFFREYS, H . 3 8 4 , 3 9 8 JINDRA, F . 3 3 3 , 3 4 6 JOHN, F . 2 7 4 , 3 1 2 JOHNSON, A . 3 8 4 , 3 9 8 JOHNSTON, J . R . 2 1 2 JOHNSTON, T . L . 9 4 JOHNSTON, W . G . 6 8 , 7 2 , 7 4 , 7 9 , 8 1 , 9 0 , 9 4 JUNCOSA, M . L . 2 7 4 , 3 1 3

KACHANOV, L . M . 3 8 3 , 3 8 4 , 3 8 7 , 3 9 4 , 3 9 7 , 3 9 8

KALISKI, S . 2 6 3 , 3 1 1 KANTOROVICH, L . V . 3 1 3 KAPLAN, S . 2 7 4 , 3 1 2 , 3 1 6 KAPRUS, R . 3 4 5 KARMAN, T . 3 8 9 KARNEV, U . V . 1 6 6 KATASONOV, A . M . 3 7 9 , 3 8 1 KAUDERER, H . 3 3 3 KEB , A . S . 1 0 9 , 1 6 5

KELLER, J . B. 2 9 4 , 3 1 5 KELLOGG, O. D . 3 8 0 KELLY, A . 6 4 , 9 1 , 9 2 , 9 4 KENNEDY, A . J . 4 1 2 KENYON, D . M . 2 4 4 KEYES, R . W . 4 2 , 5 0 , 5 5 KINGERY, W . D . 1 6 5 K I P , A . E. 5 5

KITTEL, C . 5 5 KLEIN, M . V . 2 6 , 2 7 , 2 9 , 5 5 KLEMENS, P . G . 5 5 KOEHLER, J . S . 5 6 KOH , J . C . Y . 2 7 8 , 3 1 3

KOLODNER, I . 2 9 4 , 3 1 5 KOTERAZAWA, R . 1 9 2 , 2 1 2 KREMPL, E. 2 1 1 , 2 1 2 KREITH, F . 2 6 1 , 2 9 4 , 3 1 1 , 3 1 5 KRYLOV, V . I . 3 1 3 KURDIMOV, G . V . 1 6 6 KYNER, W . T . 2 9 4 , 3 1 5 KYOGOKU, T . 2 1 3

LAKS, H . 1 1 5 , 1 6 6 LANDAU, H . G . 2 9 4 , 2 9 6 , 3 1 5 , 3 6 6 LARDNER, T . J. 2 7 8 , 2 9 6 , 3 1 3 , 3 1 6 LARSEN, E . S . 3 1 5 LAX, P . D . 2 6 7 , 2 7 4 , 3 1 2

LAYMAN, P . T . 3 1 0 LAZAN, B. J. 5 2 , 5 6 , 4 1 2 LEADERMAN, H . 3 3 6 , 3 8 1 LEIBFRIED, G . Z . 5 6 LEVTNSON, M . 3 1 4 LEVITSKY, M . 3 6 5 , 3 8 0 Li, C . H . 7 4

Li, E. H . 3 0 2 , 3 1 6 , 3 6 5 , 3 6 6 , 3 6 9 , 3 7 4 , 3 7 6 ,

3 7 7 , 3 7 8 , 3 8 0 , 3 8 1

Li, J. C . M . 1 6 6

LIEBOWITZ, H . 4 5 1 LIGHTFOOT, N . M . H . 3 1 5 LINIGER, W . 3 1 5 LOCKETT, F . J. 3 4 5 , 3 4 7 LOTKIN, M . 2 7 4 , 3 1 2 , 3 1 6 LOUT, N . 1 5 3 , 1 6 7 LOVE, A . E. H . 3 4 6 L o w , J. R . Jr. 9 4

LOWLEY, A . 1 6 3 , 1 6 8 LOWNDES, H . 4 3 9 , 4 4 2 LURKE, K . J. 4 , 5 1 , 5 4 , 5 5 , 1 5 1 , 1 6 7 LUDEMAN, W . D . 1 1 3 , 1 6 6 LUHMAN, W . J. 7 4 , 9 4 LUNDIN, S . T . 9 4 LYKOV, A . V . 2 6 1 , 3 1 1 LYTTON, J. L. 1 6 4

MAITA, J. P . 5 5 MAJORS, H . 2 1 2 MANFIELD, E . H . 4 1 6 , 4 3 8 MARIN, J. 3 8 4 , 3 9 8 MASON, W . P . 1 , 5 2 , 5 4 , 5 5 , 5 6 MASSONNEL, C . 3 1 2 MAXWELL, J. C . 3 2 4 MAZUR, P . 3 1 2 MCCUE , D . J. 4 2 9 , 4 3 0 , 4 3 8 MCDONALD , J. R . L . 2 9 4 , 2 9 6 , 3 1 5 MCDOWELL, E. L . 3 2 5 , 3 2 8 , 3 4 6 MCKENZIE , K . I . 4 2 8 MCLEAN , D . 1 0 9 , 1 1 3 , 1 1 6 , 1 6 4 , 1 6 5 , 1 6 6 MCLOED , R . N . 2 1 2 MCSKIMIN, H . J. 5 4 , 5 5 MEAD, H . W . 1 6 6 MEAKIN, J. D . 1 6 6 MECHANIC, H . 3 6 6 , 3 8 1 MELAN, E . 3 4 5 MERIAM, J. L . 3 1 0 MERKEDOV, L . G . 2 2 , 5 5 MEYERS, C . L . Jr. 1 3 8 , 1 6 7 MICHLIN, S . G . 3 1 3 , 3 8 3 , 3 9 8 MIKHAYLOV, Y . A . 2 6 1 , 3 1 1 MILES, J. W . 3 1 5 MILGRAM, A . N . 2 6 7 , 3 1 2


MILLER, D . R . 2 1 2 MINATA, H . 2 1 2 MINDLIN, R . D . 2 6 3 , 3 1 1 , 3 8 1 MIRANKER, W . L . 2 9 4 , 3 1 5 MIRSEPASS , T . J. 2 8 8 , 3 1 4 MITCHELL, J. B . 1 3 8 , 1 6 7 MITRA, S . K . 1 3 8 , 1 6 7 MORDIKE, B . L . 2 2 5 , 2 2 7 , 2 4 4 MORERA, Z . 3 2 4 MORIN, F . J. 5 5 MORLAND, L . W . 3 6 6 , 3 7 4 , 3 8 1 MORLEY, L . S . D . 4 3 8 MORRIS, R . S . 2 5 9 MORROW, J. D . 1 8 8 , 2 1 2 MORSE, P . M . 2 6 3 , 3 1 1 MOSSAKOWSKA, Z . 3 4 6 MOTE, J. D . MOTT, N . F . 1 2 1 , 1 6 6 MUKI , R . 3 1 1 , 3 8 1 MULLENDORE, A . W . 1 6 9 , 1 8 6 MULLER, H . 9 4 MURPHY, A . 4 2 2 , 4 3 8 MUSKHELISHVILI, N . I . 3 3 0 , 3 4 6

NABARRO, F . R . N . 9 7 , 9 8 , 1 0 0 , 1 0 2 , 1 0 3 , 1 0 4 , 1 6 5

NADEAU, J. 1 1 6 , 1 6 6 NEUMANN, 2 9 3 NIBLETT, D . H . 5 3 , 5 6 NICHOLSON, P . 2 7 4 , 3 1 3 NIRENBURG, L . 2 6 7 , 3 1 2 NISHIHARA, T . 1 9 1 , 2 1 2 NORTON, F . H . 1 0 4 , 1 6 5 NORTON, J. T . 1 6 5 NOWACKI, W . 3 2 7 , 3 4 2 , 3 4 5 , 3 4 6 , 3 4 7 , 3 6 0 ,

3 6 5 , 3 7 9 , 3 8 0 , 3 8 1 , 3 8 2

NOWICK, A . S . 8 , 5 5 NUTTING, J. 1 3 8 , 1 6 7

O'BRIEN, G . G . 2 7 4 , 3 1 2 ODQVIST, F . K . G . 3 8 4 , 3 9 8 OHNAMI, M . 1 9 9 , 2 0 0 , 2 0 2 , 2 0 4 , 2 0 7 , 2 1 2 ,

2 1 3 , 2 5 8

OLESSIAK, Z . 3 2 7 , 3 4 6 OLIVER, H . 2 7 8 , 3 1 4 OLSON, D . R . 2 9 5 , 3 1 5 OROWAN, E. 1 3 4 , 1 3 5 , 1 3 7 , 1 6 7

PARIA, G . 3 1 1 PARKER, A . E. 9 4 PARKER, E. R . 1 6 7 PARKES, E . W . 2 1 2 , 4 1 6 , 4 3 8 PARKUS, H . 3 1 0 , 3 1 7 , 3 4 6 , 3 8 0 PASCHKIS, V . 2 7 4 , 3 1 3 PASK, J. A . 5 7 , 7 4 , 9 4 PATEL, J. R . 5 5 PEARSON, K . 5 4

PEKERIS, C. L . 2 9 6 , 3 1 6 PELLOUX, R . N . M . 1 8 6 PERRYMAN, E. C. W . 1 5 7 , 1 6 7 PERSSON, P . A . 2 1 9 , 2 4 4 PETYKIEMITZ 2 6 3 , 3 1 1 PFAFF, F . 5 6 PIKUS, G . 3 9 , 5 5 PIPKIN, A . C. 3 8 1 PITTMAN, C. M . 3 1 1 POHLE, F . V . 2 7 8 , 3 1 3 , 3 1 4 POLLACK, M . 5 5 POOTS, G . 3 1 6 POPE, G . G . 4 3 8 POUND, G . M . 1 6 5 POWERS, W . E. 3 1 4 PRAGER, W . 3 9 8 PRANTIS, A . L . 1 6 5 PRESTON, O . 1 3 4 , 1 3 8 , 1 6 7 PUCCI, C. 2 6 7 , 3 1 2

RABOTNOV, Y . N . 3 8 4 , 3 9 8 RADOK, J. R . M . 3 4 2 , 3 4 6 RASHIS 2 5 9 RAYMOND, L . 1 2 7 , 1 6 6 RAYNOLD 4 2 7 , 4 2 8 READ 5 4 , 3 6 5 , 3 7 4 , 3 8 0 REINER, M . 3 3 3 , 3 4 6 RICHTMYER, R . D . 2 7 4 , 3 1 2 RIDDELL, F . R . 2 4 6 , 2 5 9 ROBERTS, L . O . 3 1 4 ROBERTS, W . M . 3 3 6 , 3 4 6 ROETGER, H . 1 , 5 4

ROGERS, T . G . 3 0 2 , 3 1 6 , 3 6 9 , 3 7 6 , 3 7 7 , 3 7 8 , 3 8 1

ROMIE, F . E. 2 9 4 , 3 1 5 RONGRED, L . 3 3 6 , 3 8 1 ROSE, M . E. 3 1 6 ROSE, S . N . 3 9 0 , 3 9 8 ROSENBLOOM, P . C. 2 6 7 , 3 1 2 ROSENBLUM, V . I . 3 8 8 , 3 9 8 ROWE, G . W . 2 3 0 , 2 3 2 , 2 3 5 , 2 4 4 ROY, M . 2 6 2 , 3 1 1

RUDDLE, R . W . 3 1 4 RUSSELL, H . G . 3 6 5 , 3 6 6 , 3 7 4 , 3 8 0 , 3 8 1

SALVADORI, M . G . 3 8 1 SANDERS, F . 2 5 9 SANDERS, T . M . Jr. 4 3 , 5 5 SASHKIM, L . 2 9 5 , 3 1 5 SAVAGE, R . H . 2 3 4 , 2 4 4 SAVIN, G . N . 3 4 6 SCALA, E. 2 4 5 , 2 5 9 SCALA, S . M . 3 1 5 SCHIFFER, M . 3 1 2 SCHLESINGER, S . I . 2 9 5 , 3 1 5 SCHILLER, P . 5 6 SCHMITT, A . 4 2 9 , 4 3 8 SCHOECK, G . 1 3 9 , 1 4 2 , 1 5 1 , 1 6 7

A U T H O R I N D E X 457

SCHVECK, G . 1 6 7 SCHWARZL, F . 3 6 6 , 3 8 1 SCOTT, R . 1 6 5 SCOTT, W . D . 9 4

SEEGER, A . 5 4 , 5 6 , 1 2 1 , 1 4 4 , 1 6 6 , 1 6 7 SEIDE, P . 2 7 8 , 3 1 4 , 3 3 6 , 3 4 6 SERVI, I. S . 1 6 5 , 1 6 6 SEVEREN, R . T . 2 7 4 , 3 1 3 SHAFFER, B. W . 3 6 5 , 3 8 0 SHARMA, B. 3 4 6 SHANINIAN, P . 1 6 6 , 1 7 8 , 1 8 6 SHEA, J. J. 2 9 6 , 3 1 6 SHEPARD, L . A . 1 1 0 , 1 1 3 , 1 6 4 , 1 6 5 , 1 6 6 SHERBY, O. D . 1 0 8 , 1 1 2 , 1 1 3 , 1 1 4 , 1 1 6 , 1 3 6 ,

1 3 8 , 1 4 3 , 1 6 5 , 1 6 6 , 1 6 7

SHESTERIKOV, S . A . 3 9 8 SHIMMIN, K. D . 4 1 2 SHIRAISHI, T . 2 1 2 , 2 1 3 SHOHAT, J. A . 3 1 3 SHORE, A . F . 2 4 4 SHUNAKOV, N . V . 2 8 8 , 3 1 4 SHYNE, J. C . 1 3 8 , 1 6 7

SlGNORELL, R . A . 2 1 2 SlKARSKIE, D . 2 6 2 SKOL, G . 2 1 2 SLICHTER, L . B. 2 9 6 , 3 1 6 SMIKINS, P . G . 3 1 4 SMITH, C . S . 5 5 SNEDDON, I. N . 3 2 6 , 3 2 7 , 3 4 4 , 3 4 5 , 3 4 6 , 3 4 7 SNOEK, J. L . 5 4 , 5 6 , 1 6 8 SOKOLNIKOFF, I. S. 3 4 6 SOKOLOWSKI, M . 3 4 5 , 3 4 7 , 3 6 5 , 3 8 0 SOMMERFELD, M . 3 4 3 SOUTHGATE 5 5 SPRINGER, G . S . 2 9 5 , 3 1 5 SPROULL, R . L . 3 1 1 STALDER 3 1 4 STAMBLER, I. 2 5 9 STARERMAN, A . J. 3 6 6 , 3 8 1 STEG, L . 3 1 5 STEIDAL, R . F . 3 1 0 STEPHAN, J. 3 1 5

STERNBERG, E . 3 1 6 , 3 2 5 , 3 2 8 , 3 4 1 , 3 4 6 , 3 4 8 , 3 8 0 , 3 8 1 , 3 8 2

STOKES, R . J. 7 4 , 9 4 STOLTZ, G . Jr. 2 8 8 , 3 1 4 SULLY, A . H. 1 6 4 SUTTON, G . W . 3 1 4 SWANN, P . R . 1 4 4 , 1 6 7 SWANN, R . T . 3 1 1 SWINDEMAN, R . W . 2 1 2 SZEWCZYK, A . 2 7 8 , 3 1 3

TABOR, D . T . 2 2 3 , 2 4 4 TADJBAKHSH, I. 3 0 1 , 3 1 6 TAIRA, S . 1 8 7 , 1 9 1 , 1 9 2 , 1 9 9 , 2 0 0 , 2 0 2 , 2 0 4 ,

2 0 7 , 2 1 2 , 2 1 3 , 2 5 8

TAMARKIN, J. D . 3 1 3 TANAKA, K. 1 6 8 , 2 1 2

TAPSELL, H. J. 1 8 9 , 1 9 0 , 1 9 1 , 1 9 3 , 2 1 2 TAYLOR, G . 5 8 , 6 4 , 9 1 , 9 2 , 9 4 TAYLOR, J . 4 2 9 , 4 3 8 TEREGULOV, I. G . 3 9 7 , 3 9 8 TETALMAN, A . S . 9 4 THOMAS, G . 9 4 , 1 3 7 , 1 3 8 , 1 6 7 THOMSON, R . 1 5 1 , 1 6 7 TIETZ 1 7 0 , 1 8 6 TING-SHU-WU 2 6 2 TING, T . C . T . 3 1 6 TODD , J . 2 7 4 , 3 1 2

TOLINS, I. S . 3 1 1 , 3 4 5 , 3 4 7 TRAPP, W . J. 4 3 9 , 4 4 0 - 4 5 , 4 4 9 , 4 5 1 TRENCH, W . 3 1 6 TROZERA, T . A . 1 1 2 , 1 6 4 , 1 6 5 TRUELL, R . 5 3 , 5 5 TRUITT, R . W . 3 1 4 TSIEN, H. S . 3 8 0 TURKALO, A . M . 9 4

UDIN , H . 1 6 5

VANDERSLICE, T . A . 5 5 VANDERVOORL, R . R . 1 0 5 , 1 0 7 , 1 6 5 VARGA, R . S . 2 7 4 , 3 1 2 VEINIK, A . I. 2 7 8 , 3 1 2 VELTE, W . 2 6 7 , 3 1 2 VENKATRAMAN, B. 3 9 8 VINSON, J. R . 2 7 8 , 3 1 3 VOELKER, D . 3 4 6 VOLTERRA, V . 3 8 0

VONDERNEUT, A . 3 4 6 VONMISES, R . 5 8 , 9 4

WALKER, A . 2 5 9 WARRINGTON, D . H. 1 2 1 , 1 6 6 WARSHAW, S . I. 1 0 4 , 1 6 5 WASHBURN, J. 7 8 , 9 4 WEBB, M . B. 5 5 WEERTMAN, J. 1 3 1 , 1 3 5 , 1 3 6 , 1 3 7 , 1 3 8 , 1 6 3 ,

1 6 6 , 1 6 7 , 1 6 8

WEETON, J. W . 2 1 2 WEINER, J. H. 3 0 2 , 3 1 0 , 3 1 3 , 3 1 4 , 3 1 6 , 3 2 1 ,

3 2 3 , 3 4 5 , 3 6 5 , 3 6 6 , 3 8 0 , 3 8 1

WEINREICH, G . 4 3 , 5 5 WEISSMAN, S . 1 0 9 , 1 6 5 WHETTEN, N . R . 5 5 WHITE, H. G . 4 3 , 5 5 WIDMER, R . W . 7 8 , 1 7 6 , 1 8 6 WIEDERSICH, H. 1 5 1 , 1 6 7 WILKS, J. H . 5 3 , 5 6 WILLIAMS, J. 1 6 5 WILLIAMSON, G . K. 9 4 WILSDORF, H. G . H. 9 4 WILSON, R . 2 2 9 , 2 4 4 WINKLER, H. B. 2 4 6 , 2 5 9 WINTERS, 2 5 9


WISLEY, R . P . 2 1 2 WITTE 2 5 9 WOLFE, M . O . W . 4 1 3 , 4 3 9 - 4 3 , 4 4 6 , 4 4 8 WOLKO, H . S . 4 2 9 , 4 3 8 WOODREIFF 5 5

YANG, K . T . 3 1 3 , 3 1 4 YIM , W . M . 1 7 8 , 1 8 6

YOUNG, D . 2 7 4 , 3 1 3 YOUNG, J. E. 2 3 0 , 2 3 4 , 2 4 4

ZENER, C . 1 , 7 , 5 4 , 5 5 , 9 1 , 9 4 , 2 4 4 ZERNA, W . 4 3 6 ZORAWSKI, M . 3 7 9 , 3 8 2 ZWELL, L . 1 0 9 , 1 6 5 ZWICKY, E. F . Jr. 3 6 6 , 3 8 1

SUBJECT INDEX

Ablation of heat shield, 253 of polymers, 255

Acoustoelectric effect, 40 Activation energies

for ceramic materials, 104 for creep, 111 for high temperature creep and diffusion,

115 Ageing and creep

effects of on aluminum alloy D . T . D 5070A, 425

Akheiser effect, 12, 13, 14, 15 Anelastic measurements, 32 Arrhenius equation, 6, 11, 22 , 32, 53

activation energy, 6, 7, 11, 26, 28, 3 0 , 4 9 , 53 of Si, 47

Attenuation, 4, 5, 9 -17 , 21 , 22, 25 -28 , 32, 33, 4 1 - 4 3 , 45, 50 -54

and internal friction, 16 in doped silicon, 4 6 - 4 9 in magnetic field, 33 in NaCl , 21 in pure and doped Ge , 44 in Si and G e , 14, 15 measuring techniques, 9, 10 of longitudinal waves, 48 peak temperatures, 25 ratio of to acoustoelectric field, 40 ultrasonic, 4 , 14

in NaC l , 21 zinc crystals, 4

Band theory, 33 conduction band, 34, 35 valence band, 34, 35

Beryllium as structural material, 443

Boltzmann equation, 32 ratio, 38 statistics, 33

Bonding nature of, 60

Bulk modulus, 3, 13

Coefficient o f friction as function of load, 230 as function of outgassing temperature for

N i , Pt, A g , 231

Coefficient of friction (contd.) as function of sliding speed

in metals, 217 for steel on w o o d s alloy and on bismuth,

220 for steel on lead and on tin, 221 for steel on nylon, 221

as function of temperature for some car-bides, 225-27

high speed, 220, 221 of F e surfaces, 232 of graphite, 234

Complementary rate of energy dissipation, 385L, 393f.

minimum, 388 Conductivity, 36, 39

of semiconductor, 36, 40 Constant damage surface, 407 Coupled heat transfer equation, 262f. Crack

growth and propagation in LiF and M g O , 91 stabilization, 177-8

in 80 N i - 2 0 Cr, 178 Creep

activation energy for, 111, 115, 137 for ceramics, 104

analysis, variational methods in, 388f. and ageing, effect of temperature on, 425 at high temperature

nabarro, 97-105 of Al , Mg , Zn, 97 of Cu, Ag , A u , 102 of ceramics, 105-6 of polycrystalline materials with high

stacking fault energies, 117f. Castigliano's theorem for, 387 Cottrell interaction in, 151f. damage accumulation

at cyclic stress, 404 at elevated temperature, 403

design criteria for, 402 dissipation potential for, 386 equations, phenomenological , 384 hardening in, 145 in torsion of a square bar, 393 integral law, modified, 368 intercrystalline cracking in, 180f. locking in, 145-6 , 149f. microstructure in third stage, 184f. of dispersions of C u A l 2 in A l solid solution

matrix, 136-7

459

460 S U B J E C T I N D E X

Creep (contd.) of polycrystalline materials, 117f. of sap alloy, 137-9 primary, 138

in design, 400f. secondary, effect of stress on , 125 solute atoms in, 142f., 154f. stacking fault energy in, 132, 144f., 150 transient, 393f. unstable elongation in, 170f. void formation in, 181

Creep rate effect of carbon on in gamma iron, 114 effect of low stress on secondary creep rate

in Al , 125 effect of stress on in Al , 112, 115-16 effect of stress on , 116, 172 in Al , 128 in F e - A l alloys, 163 minimum as function of stress, 170-2

for A l - M g , 172-3 Creep rupture

crack formation in, 176f. of A l - C u alloys, 182-4 of udimet alloy, 182-3 , 185

Creep strain as function of temperature compensated

time, 114 Crystal structure

of rock salt, 59, 60

Differential operators in viscoelasticity, 355 temperature modified, 369

Diffusion interstitial, 7 stress induced, 7

Diffusivity effect o f design of, 421 , 422 of N i - A l alloy, 142 of several materials at 200°C, 423

Dis locat ion, 9, 10, 11, 22 density with strain in pure Al , 127 effect on

distribution of solute atoms in M g A l solution, 153-5

elastic and anelastic properties, 50 -52 and Bordoni peak, 53

mot ion, 4 , 51 substructure in rock salt type single crys-

tals, 66f. velocity in anorganic crystals, 68, 69

Duhamels principle, 361

Elastic modulus, 2, 3, 11, 12, 13, 15-18 and dislocations, 51 -52 and poping, 4 2 change as function of temperature, 29, 32

Elastic modulus {contd.) change in, 12, 16, 18, 21 , 29, 40, 4 1 , 42 change in for separated energy surfaces, 50

Electron mean free path, 28, 29, 33, 40

in NaCl (Harshaw), 29 velocity, 35

mobility, 35, 36 unstrained, 39; for G e and Si, 39

Energy b o n d , 3 0 internal, 32

of unstressed solid, 16 ionization, 34

of impurity atoms, 34 levels, 30, 31

separation of, 31 -32 Stark effect, 31

thermal, 14, 17, 18, 21 , 33 Energy surfaces

electronic surfaces, 32 Fermi surfaces, 32, 33, 40 hole energy surfaces, 32, 33 of constant energy, 36 of germanium, 38 of silicon, 35, 36, 37 separation of, 39, 50

Fatigue at elevated temperatures, 189f.

effect o f speed on , 190-2 cyclic strain, 191-3 in stainless steel, 199-201 thermal cyclic, 200-1

stainless steel, 2 0 4 - 7 fundamental equation, 193 mechanical equivalent temperature in, 193,

195, 197 under non-uniform temperature, 198-9

Fatigue-creep sensitivity at elevated temperature, 410-11

Flight regimes for re-entry vehicles, 247 Fracture

in inorganic crystals, 86 in inorganic polycrystals, 91 strain of for anorganic crystals, 77-78

Free electron model , 32 Friction

at high temperatures, 225f. effect of adhesion on , 233, 234 effect of oxide film on, 229, 230 in vacuo, 228f. o f refractories, 227 plastic flow in, 229 surface damage at high temperature by,

218 surface temperature at high speed due to ,

219 Full scale testing, 441, 446-8

S U B J E C T I N D E X 461

Galerkin functions, 325 Galerkin method, 334

in heat transfer problems, 276f. Grain junctions

in inorganic crystal, 89, 90 Grueneisen constant, 13, 21

Half-space temperature distribution in, 2 8 3 - 4

Hardness, as function of temperature at elevated temperatures, 222f. for carbides, 224

Heat conduction problems approximate solutions of, 274f. moving boundary, 292f. upper and lower bounds for, 280f.

Heat shield materials, 252 Heat transfer equation

boundary condit ion for, 266 Heat transfer problems

uniqueness theorems, 267f. High speed flight

boundaries defining corridor, 427-8 cool ing system, 430, 431 design criteria, 444, 445 insulation, 430, 432, 433 material properties for, 436 materials evaluation for, 442-3 materials for, 442 thermal protection, 425f.

Impurities, 27 Insulators, 4 , 8, 21 Interaction curves

for creep and fatigue, 403f. Interaction surfaces

for creep and fatigue, 405f. for failure, 405 for failure at various probabilities of sur-

vival, 408 for limiting creep deformation, 406

Interaction of sustained loads, cyclic loading and temperature, 401

Internal friction, 1, 4 , 5, 6, 7, 11, 16, 26 and dislocations, 51 and point defects, 7 Bordoni peak in Cu, 53 in copper, 53 in German silver, 1, 2 in germanium, 5, 12, 13 in lead, 52 in silicon, 12, 13 in zinc, 4

Interstitials, 5, 7, 21

Kinetic heating, 415

Leading edge buckling, 416 Liquid jets

effect of surface on , 235f. pressure due to , 238 surface cracks due to , 239, 240 surface deformation due to , 241 f.

Local heating transient problem, 291

Longitudinal waves in silicon, 48

Mach number relationship between altitude and, 429

Magnesium as structural material, 442

Melt ing slabs with variable thermal properties, 294

Microstructure of magnesium, 58 of porcelain, 57

Moving boundary, 295f. melting and solidification, 295f. thermal stresses in, 300f.

in elastoplastic solidifying slab, 303

Orthotropic body, 331 -2

Paramagnetic resonance, 30 Peierl's stress

in A g - A l , 161 in Mg, 145

Phonon branches, 12, 13, 16 phonon interactions, 5, 8, 10, 11, 16, 22

Piezoresistive effect, 32, 33, 38, 39, 40, 45 and stress, 40 t ime constant of, 32

Plastic deformation of inorganic crystals, 65f. of rock salt type single crystals, 65f.

Polyphase alloys creep of, 133f.

Rebound as function of temperature, 224

Reduced time in viscoelasticity, 367f.

Re-entry temperatures, 249 Relaxation, 7, 8, 16-18 , 2 1 , 22, 26 -29 , 32

due to substitutional atoms, 7 frequencies, 2 1 , 22 , 26 -28 integral, temperature modified, 367-8 , 372 problem analysis of, 397f. strength, 1-8

as function of crystal orientation, 8 in German silver, 2

462 S U B J E C T I N D E X

Relaxation {contd.) time, 3 , 8, 12-16 , 2 1 , 4 0 , 49

and doping, 4 4 - 4 6 and frequency, 2 , 3 intervalley, 4 1 , 45, 47, 49, 50

ultrasonic in NaCl , 30 Zener type, 8

Reliability functions for creep and fatigue, 411 in aircraft design, 449, 450, 451 in creep design, 408f. surface, 409

Resistivity, 49 Risk function, 410

for creep and fatigue, 410-11

St. Venant's principle in heat conduction, 288f.

Sandwich construction structural efficiency of, 421

Semiconductors, 4 , 8, 33, 36, 38 degenerate, 40 doping, 35, 36, 40, 4 1 - 4 9 doping and attenuation, 41 intrinsic, 34

Shear modulus of germanium, 42

Shear wave velocity for silicon, 47

Shear waves in silicon, 10

Shift function, 367, 376 Silicon band, 34 Skin-web configuration

temperature distribution in, 424 thermal stress in, 422

Slip bands, 71

in inorganic crystals, 87 in inorganic polycrystals, 84, 85

in rock salt crystals, 66f. prismatic, effect of temperature on , 161 systems, in inorganic crystals, 6If. thermally activated, 162

Snoek effect, 5, 6 Solidifying slab, 302, 303 Specific dissipation potential, 386 Specific stability

effect o f temperature on, 419 Specific stiffness

effect of temperature on, 419 Stability criterion, 420f.

of several materials, 419 Stiffness criterion

of several materials, 419 Strain hardening coefficient

determination of, 174 Stress

alternating, 16, 32

Stress {contd.) and subgrain diameter

in A l alloys, 110 axis orientation, 62, 63 diagram for {100}<100> slip system, 65 free temperature fields, 371 sensitivity, 30 strain behavior

effect of loading rate on in M g O , 79 in AgCl and N aC l , 85 temperature dependence of, 80

for polycrystalline M g O and LiF, 8If. strain relation

in thermal cycling, 194 of metals at elevated temperature, 189-91

in low carbon steel, 190 strain diagram of anorganic crystals, 73f. yield, temperature dependence of, 76, 77

Stress-strain diagrams for anorganic polycrystals, 8If.

Structural efficiency ratios, 417 Structural index, 420-1 Super-orbital re-entry, 257 Surface adhesion

as function of tangential pre-stressing, 234 in platinum, 233

Tensile strength effect of temperature on , 418f.

Thermal activation of dislocations, 121f. buckling, 336f. coefficient, temperature dependence of in

austenitic and ferritic steel, 196 conductivity, 21 , 27, 28, 32

in German silver, 2 in N aC l , 26-27

resistance, 29 tests, accelerated, 441 time constant, 13

Thermal fatigue and cyclic mechanical stress, 201f. and steady stress, 209f., 212 combined with mechanical stress, 2 0 2 - 3 ,

208 relation to mechanical fatigue, 193f.,

198-99, 200, 209, 211 Thermal stress

cycling, 194 error introduced in by approximate tem-

perature expression, 278 -80 in a slab, 279 in a viscoelastic slab, 377 in a web, 4 2 2 - 4 in melting bodies, 300f. of several materials, 422 parameters, 422 problems, uniqueness in, 304f. random, 339

S U B J E C T I N D E X 463

Thermo-mechanical coupling effects, 362, 379 Theologically simple behavior, 370f. Theologically simple materials, 366f.

Thermo-elastic effect, 1, 3 - 5 , 12-15 , 21 equations, 343 half space, 264 solid, 320f. wave equation, 340f. wave propagation, 265

Thermo-viscoelasticity correspondence principle in, 363f. inertia effects in, 379 Pepkooitch solution in, 363 potential, 362 -4 temperature dependence in, 366f. uniqueness in, 357f.

Ultraviolet absorption band, 27

Vacancies, 98, 99, 101, 102 Variation of temperature distribution in a

web, 424 Viscoelastic state, 357f.

anisotropic linear material, 378 position dependence of, 358 relaxation functions, 357

Wave velocity, 13, 14, 21 , 26 as function of temperature, 26 shear, 26

Ultimate load design for, 401

Yielding behavior effect of impurities on in M g O , 74-76 of LiF single crystals, 70 -73 , 74

Documents

High Temperature Structures and Materials. Proceedings of the Third Symposium on Naval Structural Mechanics Held at Columbia University, New York, N.Y., January 23â€“25, 1963