UNCLASSIFIED

AD 435688

DEFENSE DOCUMENTATION CENTERFOR

SCIENTIFIC AND TECHNICAL INFORMATION

CAMERON STATION, ALEXANDRIA, VIRGINIA

wUNCLASSIFIED

NOTICE: When government or other drawings, speci-ficati ons or other data are used for any purposeother than in connection with a definitely relatedgovernment procurement operation, the U. S.Government thereby incurs no responsibility, nor anyobligation whatsoever; and the fact that the Govern-ment may have formulated, furnished, or in any waysupplied the said drawings, specifications, or otherdata is not to be regarded by implication or other-wise as in any manner licensing the holder or anyother person or corporation, or conveying any rightsor permission to manufacture, use or sell anypatented invention that may in any way be relatedthereto.

00 ASD TR 8-111 (X) ASr[ INTERIM REPORT 8-111 (X) III (N)

0APRIL 1964

ELECTRO - SPARK EXTRUDING

J.H, WAGNER

"| REPUBLIC AVIATION CORPORATIONJ MANUFACTURING RESEARCH

*" CONTRACT: AF 33(657)11265

CD ASD PROJECT: 8-111

|r3 INTERIM PROGRESS REPORT

I JANUARY 1964 to I APRIL 1964

Q .J The effects of shock loading on a thick walled cylinder at an internal nal

pressure of 230, 000 psi hydrostatic pressure are analysed. A container *r

geometry it proposed. DDC

BASIC INDUSTRY BRANCHMANUFACTURING TECHNOLOGY LABORATORY

Co AERONAUTICAL SYSTEMS DIVISIONUNITED STATES AIR FORCE

WRIGHT- PATTERSON AIR FORCE BASE , OHIO

A

ASD TR 8-111 ASD INTERIM REPORT 8-111 (III) 3-111 (111)April, 1964

ELECTRO-SPARK EXTRUDING

J. Wagner

Republic Aviation CorporationManufacturing Research Department

Contract AF33(657) 11265ASD Project 8-111

Interim Technical Engineering Report No. 3

1 1 January 1964 - 1 April 1964

Republic Aviation Corporation Report No. RAC ZZ88

The effects of shock loading on a thick wailed cvl,.xder at an internal internalI pressure of ?C , 000 psi hydrostatic pressure are analysed. A container rntainer

geometry is proposed.

U

I BASIC INDUSTRY BRANCHMANUFACTURING TECHNOLOGY LABORATORY

Aeronautical Systems Division (AFSC)United States Air Force

Wright-Patterson Air Force Base, Ohio

3I

IABSTRACT-SUMMARY ASD INTERIM REPORT 8-111 (I1) -'ORT 8-Ill (I11)Interim Technical April, 1964Progress Report No. 3

ELECTRO-SPARK EXTRUDING

J. Wagner

IRepublic Aviation Corporation

IThe spark gap discharge phenomena as applied to experimental extrusion ental extrusionare reviewed. The equations derived for a thick walled cylinder under ,linder undertransient internal loading are applied to check the stress levels encountered evels encountered3 in two proposed experimental extrusion devices. The st'ength requireatie.Ongth requirementsof the equipment are reviewed and arah.yseu with both static and dynamic c and dynamic

I Jloaaug. A container concept is presented.

IIII

NOTICE

I

When Government drawings, specifications or other data are us

for any purpose other than in connection with a definitely related Gove

ment procurement operation, the United States Government thereby in

no responsibility nor any obligation whatsoever; and the fact that the

Government may have formulated, furnished, or in any way supplied t

said drawings, specifications, or other data, is not to be regardee' by

im-lication or otherwise as in any manner licensing the holder or any

other person or corporation or conveying any ights or permission to

manufacturc, use or sell any patented invention that may in any way b

related thereto.

The work reported in this document has been made possible thrc

the support and sponsorship extended by the Air Materiel Command ur

I Contract No. AF33(657)11265. It is published for technical inforrmatic

only and does not necessarily represent recommendations or conclush

of the sponsoring agency.

I Oualified requesters may obtain copies of this report from Defei

Documentation Center, Cameron Station, Alexandria, Virginia.

I Copies of ASD Technical Reports should not be returned to fl-.

Aeronautical Systems Divisio ' -.nless return is required by security

considerations, contractual obligations, or notice on a specific docum

II!I

I

NOTICE

F

When Government drawings, specifications or other data are used

for any purpose other than in connection with a definitely related Govern-

ment procurement operation, the United States Government thereby incurs

no responsibility nor any obligation whatsoever; and the fact that the

Government may have formulated, furnished, or in any way supplied the

said drawings, specifications, or other data, is not to be regarded by

implication or otherwise as in .ny manner licensing the holder or any

other person or corporation or conveying any rights or permission to

manufactuie, use or sell any patented invention that may in any way bez elated thereto.

The work reported in this document has been made possible through

the support and sponsorship extended by the Air Materiel Command under

i Contract No. AF33(657)11265. It is puzlished for technical information

only and does not necessarily represent recommendations or conclusions

of the sponsoring agency.

5 Oualified requesters may obtain copies of this report from Defense

Documentation Center, Cameron Station, Alexandria, Virginia.

I Copies of A3D i'echnical R~ports should no. be returned to the

Aeronautical Systems Division unless return is required L; security3 considerations. contLactual obligations, or notice on a specific document.

IIII

I

I

FOREWORD

This Interim Technical Progress Report covers the work performedunder Contract AF33(657)11265 from January I, 1964 to April 1, 1964.It is published for technical information only and does not necessarilyrepresent the recommendations, conclusions or approval of the Air Force.

This contract with Republic Aviation Corporation of Farmingdale, NewYork, was initiated under ASD Manufacturing Technology LaboratoryProject 8-Ili, "Electro Spark Extruding." It is administered under thedirection of Mr. T.S. Felker of the Basic Industry Branch MATB,Manufacturing Technology Laboratory, Aeronautical Systems Division,Wright-Patterson Air Force Base, Ohio.

Mr. J. H. Wagner of the Manufacturing Research Department, RepublicAviation Corporation is the engine,,r in charge of the project. Mr. GunthertPfanner is coope rating ip the res(-arch.

The transient wave analysis presented in this report was done by Mr. B. P.l.eftheris of Re-entry Simulation Laboratory, Research Division.

Th( primary objective of the Air Force Manufacturing Methods Programis to increase producibility, and to improve the quality and efficiencyof fabrication of aircraft, missiles and components ther(of. This reportis being disseminated in order that methods and/or equipment developedmay be used throughout industry, thereby reducing costs and giving"MORE AIR FORCE PER DOLLAR."

Your comments are solicited on the potential utilization of the informationcontained herein as applied to your present or future production programs.Suggestions concerning additional manufacturing methods developmentr(quired on this or other subjects will be appreciated.

PUBLICATION REVIEW /

j Approved by-&Robert W. Hussa, Ass't. ChiefManufacturing Rsch. Engineer

Approved by:T. F. Irriholz, Chief -

Mfg. Rsch. Engr. "x

I

II

I

]I CONTENTS

SECTION Page

INTRODUCTION 1

BASIC EQUIPMENT DESIGN 5

1. The Spark Discharge Extrusion Concept 5

2. Effect of Shock Loading in ContainerCavity Dimensions 7

3. Combined Effect of Internal Detonativeand Hydrostatic Loading 11

4. Replaceable Liner Construction 13

STATUS OF THE WORK 16

REFERENCE 17

APPENDIX 19

Calculation for Strain Deformation Dueto Transient Wave 19

Calculations Showing Combined Effect ofImpulse and Static Loading 35

DDISTRIBUTION 41l!

IUII

i

INTRODUCTION

Electro-spark extrusion is a metalforming concept involving re-

covery of the energy derived from a rapid discharge of capacitor stored

electrical power by reduction of a metal billet through an appropriate die.

One method, the electrohydraulic reduction of the billet, places the

extrudable material in an ultra-high pressure fluid environment with the

rapid energy lischarge occurring across a suitable electrode gap. This

short time duration gap discharge in a properly designed high strength

container cai.ses a shock-pressure wave to strike the billet face and,

upon reflection, to initiate parti-kle acceleration in the billet material

and to release to the billet face some of the intrinsic kinetic and pressure

energy available as a consequence of known wave reflection parameters.

The sum of these phenomena when added to the potential energy present

j in the high fluid pressure environment have been shown by mathematical

analysis to be ample under carefuDy controlled conditions to cause a

metal billet to yield and to extrude a given length for the time duration

of one electrical pulse. The problem of maintaining extrusion then

properly becomes one of providing sufficient discrete pulse discharges

with the correct time interval to allow a constant velocity extrusion.

In addition, the capacitor discharge equipment can be alternately

employed to produce impulsive electromagnetic forces upon the billet in

the hydrostatic container. In this method, the rapid discharge of the

-zc-d electrical ererg , produced high magnetic fields about an appro-

priately constructed conductive coil. By induction, to a con.luc,-Ve

j pistoi.-disc i.terposed ietween the coil and the billet, epulsive magnetic

forces rapidly accelerate the disc and thereby drive a pressure wa -

against the billet.

As an alternative, where the billet material is sufficiently conductive.3 the electromagnetic repulsion may be coupled to the billet face directly in

order to induce the ac,.eleration forces required to extrude the billet.

A third method of capacitor stored energy utilization is to recover

the energy created by the electrohydraulic discharge and to transfer it

!

!

by mechanical means to the billet. Displacement of the billet is accom-

plished mechanically by the use of a piston-ram on the billet material in

a suitable container. The pressure pulse or electromagnetic ener y

created by a series of discharges impinges on the face of the piston causing

it to move forward to extrude the billet.

The advantages offered by the foregoing approach to electrohydraulic

extrusion are: (1) elimination of billet container wall friction, (2) re-

duction of billet die surface friction, (3) creation of pressures consider-

ably beyond those obtained hydrostatically, (4) reduced container size and

cost. 1

To determine the production potential for extruding a steel alloy by

the use of capacitor stored energy, the Aeronautical Systems Division iof Wright Patterson Air Force Base has awarded Contract No. AF33(656)-

11265 to Republic Aviation Corporation. Extrusion will be attempted in

the course of the program by conducting experiments with capacitor dis-

charge equipment to develop the most suitable techniques for metal reduc-

tion by several electrohydraulic methods. The program consists of two

phases as follows:

Phase I - Design of the Extrusion Equipment

Phase!! - Development of the Extrusion Process

The approaches mentioned in the foregoing are to be Thoroughly

investigated for suitability t'ward attaining the objectives of the program; Ii. e., extrusion of a 2 inch diameter to a 1/4 inch round.

Mathematical treatment of the electrohydraulic pulse extrusion case I(Reference 1) produned a descriptive equation containing a combinatiot, of

dynamic and static variables generally similar to the well known equation

used for static extrusion ( P = K In -A- ) . In the static extrusion case,

the pressure is raised until extrusion starts; while in the p.lse extrusion

case, the pulse propagates through the billet imparting a motion to the

billet. The extrusion produced for one pulse depends npon the pressure

amplitude, while the final extruded length depends upon the rate thepulses are generated.

1

From the foregoing, it is apparent that achievement of the totalenergy necessary to extrude a target section requires multiple electrical

discharges. The time between discharges should be minimum to reduce

the overall extrusion time and avoid stop-start extrusion. In order to

optimize these requirements, it is necessary to modify the 156, 000 joule

Republic Aviation capacitor discharge facility to include the necessary

equipment to allow faster charging rates and to design a switching

arrangement and duty cycle timer to favor the rapid cyclical release ofstored energy at precisely timed intervals.

The theory of unsteady waves, (Reference 1) used to develop workingequations for computing the radial and hoop strains in hollow cylindersunder transient internal loads indicated that pressure vessel design

requirements to contain dynamic forces superposed over high hydrostaticpressures was possible and practicable. This analysis pointed up desirable

container dimensional geometry and indicated an order of magnitude of

mechanical properties for the materials in the critical areas of the chamber.1 These criteria have been formulated into general design requirements andare presented in this report.

I3

!|!

II!1!

uI

At 0

E-43U)

i14

u

0

I1

DESIGN OF A PRESSURE VESSELFOR ELECTROHYDRAULIC EXTRUSION

A. Basic Equipment Design

1. The Spark Discharge Extrusion Concept

To accomplish extrusion, power is initially supplied to a capa-

citor bank and the stored charge held off by a suitable switch until re-

leased to the high pressure device containing an open gap electrode, an

extrusion die and a metal billet. The billet is maintained under high

hydrostatic pressure defined by the mechanical properties of the part-

icular extrudable material used and the degree of metal reduction required.

Once the storage of energy is completed, the discharge event begins when

a trigger switch (vacuum gap) is ionized and current flows into the circuit,

thereby ionizing the high fluid pressure gap between the electrodes (refer

to Figure 1). A spark channel is abruptly created and the vapor products

in the channel expand into a spherical "gas bubble." The inertia of the

circumjacent fluid coupled with the extremely high velocity of expansion

of the "gas bubble" produces a compression shock wave in the surrounding

liquid. This compressed water layer then travels through the liquid as

a shock pressure front at approximately acoustic velocity with a time

duration of approximately 20 microseconds.

The energy relea d by the gap discharge consists of kinetic energy

plus the energy due to the increased pressure in the traveling wave. At

the billet face the pressure approximately doubles because of the rein-

forcement of the incident and reflected waves. The shock presoure wave

splits into a reflected pulse back into the fluid and a transmitted pulse*

forward into the billet material. The billet attains, by the same phenomena

as the fluid, a particle velocity sufficient to cause longitudinal movement

for a length proportional to the energy released.

The very benefits derived from the shock pressure pulse in extrusion(i. e., particle velocity sufficient to cause plastic strain and movement of

the billet)become a limiting factor when applied directly to the structure3 in the pressure vessel walls. As the cylinder is initially under high

5

) 1

U)

0 1

Lo

ca)

U)

6

Iinternal static pressures the effects of the superposed dynamic loading

become critical. Results of the analysis completed in subject Reference I,

indicated a practical container geometry is attainable to withstand the

explosive forces from an electrical gap discharge event.

The illustration (Figure 1) shows the spark discharge extrusion

process and equipment in a conceptual manner. A capacitor bank to

supply the energy, a high pressure chamber, pumping system and asso-

ciated controls are shown.

2. Effect of Shock Loading in Container Cavity Dimension

Equations have been previously derived and calculations prepared

in Reference 1 for the general problem of a thick-walled cylinder under the

influence of a transient internal load where the unbalanced forces from a

detonative capacitor discharge set up velocity gradients in part of the

cylinder while the remainder is completely at rest. The problem has

been formulated and explicit solutions given for elastic and elasto-plastic

j cases. These specialized interpretations are presented in Appendix I

for reference.

With the aforementioned analysis in mind, a tentative pressure

vessel construction (Figure 2) using a single shrink shroud of 4340 steel

(24 inch 0. D.) over an 18% Ni. maraging steel liner (2 inch I. D.) with a static

internal pressure of ZOO 000 psi was derived using the customary Lame

equations. A desirable residual compressive hoop stress resulting from

the -nmbination of shrin!. .it assembly and cold work fabrication proced-

ures in the most highly stressed portion of the apparatus, (th,, inside

diameter; r o = I inch) wa-: raquired to withstand safely hydrostatic internal

pressures LO the maximum amount of 200, 000 psi.

The question of a serviceable design to contain the forces re-

leased by the discharge of electrical capacitor stored energy in addition

to the loading imposed on the equipment by the high order of hydrostatic

"ressure presented a problem to which no exact solution could be found.

For this reason, the mathematical analysis, Reference (l),was preparedto predict the effects of detonative shock loads on a container using normal

I

I

construction materials of significant mechanical strength values.

This analysis indicated that residual hoop stresses result in

the container walls from reaction of the metal to the passing of a maximum

detonative shock pressure pulse. The maximum response occurs at the

fluid metal interface at the internal radius (r o = I inch) where the pressure

pulse is greatest. The material has been shown by the analysis to exper-

ience first plastic flow in the outward direction and then inward, thus T

giving rise to 'he possibility of secodary yielding. |

Reference to Case I in Anpendix I, the hoop strain for zero Istress (permanent set) is 0. 0036. The residual strain after unloading

is 0. 00463 (com)ression). The diagram for this set of conditions indicates

that a permanent set to a residual compressive stress level at the bore

-.05 \60a.0 0 38

--.0046/ .065 .01 .015

1I

- 253,000 PSI HOO

of 53, 000 psi. This is a value greater than the ultimate strength of the

liner material (250, 000 psi) being considered without the addition of Ihydrostatic pressure. Therefore, the assembly shown in Figure Z is

proven unsatisfactory, because of the high degree of deformation present

-t the critical location r o 1 inch from the action of the passing of the

shock wave. IAs the reverse yielding at r o 1 inch has been shown to be beyond

8

iI

the mechanical property values of 18% Ni. maraging steels, it is of interest

to find a radius where the severity of the shock wave has been sufficiently

diminished so as to present a less drastic effect on the walls of the pressure

container. A plot, Figure 3 of the particle velocity against the radius in

the container wall indicates that the particle velocity attenuates rapidly from

its initial critical value of u = 2864 inches per second. It is reasonable to

assume that a wall radius can be found where the effect of the particle

velocity on the container walls is reduced to the degree that severe deform-

ations do not occur from the loading-unloading cycle of the trnsient wave.

The curve changes slope between the values r = 2 and r = 3 inches. At

r = 3 inches, a 40% attenuation has diminished the particle velocity to a

value of 1654 inches per second. This value of r o = 3 inches was selected

for analysis.

The calculations for an r o = 3 inches are outlined in Appendix

I, Case (2).

The hoop strain imposed at r o = 3 inches from the action of the

loading wave where the pressure has attenuated to 248, 280 psi has been

jshown using the concept of elastic recovery to be equal to zero, thus

indicating no permanent set in the hoop direction from the loading wave

action. Due to the unloading there is elastic recovery strain in the amount

of 0. 00368 (compression). The equivalent stress for this strain value is

SI 10, 000 psi (compression). The significance of this is that a permanent

-009 -+.005

100,000'1lOtOOOPSi

200,000

residual hoop stress (compression) due to the action of the wave remains

at the inside diameter of the chamber (r o = 3 inc;hes). If there were no

9

1U!

r- I inch

3000. r any radius

u= particle velocity at r,

2800.

I

2400--

2000

0

0

> 1600

800-

II II" t

02 3 4 5

Container Radius, r (Irtches)

FIGURE 3

p)I

!considcra'ions, other than hoop stress, the pressure vessel at ro = 3 inches

I could elastic-lly withstand dynamic stresses from -110, 030 psi to the yield

point of the material, 250, 000 psi or a total of 360, 000 psi in the hoop

direction.

It must be remembered that the analysis outlined in the Appendix

I indicates the condition in the pressure vessel walls at ro = 3 inches. This

case is for the same wave (at a later period in time) as was analyzed for

ro = 1 inch. The particle velocity and the pressure, of course, are atten-

uated from'passage through the material of the pressure vessel walls.

Although the calculations show a small residual compressive stress of

30, 000 psi, this value can be disregarded if we consider r, = 3 inches to

be the container cavity dimension at the fluid-metal interface. (This will

be made further evident in the following section.1

3. Combined Effect of Internal Detonative and Hydrostatic Loading

Because the shock wave loading by itself has been previously

jshown to behave elastically between 110, 000 and the yield point of the

material in the wall at r o = 3 inches, it becomes of interest to subject

1' the stresses produced from a combination of the dynamic and static

loading to analysis to see if the coupling can be accomplished without

i failure in the wall. The calculations for this are outlined in Appendix Ir.

The work indicates that a compressive pre-stress in Lhe hoop

direction of approximatrly 133, 000 psi must be incorporated in the pressure

vessel cunstruction by some method to insure that failure will not occur

at r o = 3 inches as a result of the combining of the hydrostatic and dynamic

loading. This order of compression stress is to be obtained by thp, use of

carcful shrink fit procedures in addition to appropriate autofrettage of the

inner liner prior to assembly. As an interference shrink fit constructionwill be used. The interference must be calculated commensurate with

ordinary assembly procedures. In addition, the residual stress level

after shrink and at pressure must be maintained so that this area xill not

3 be subject to failure during usage.

II

P 430, 000 %

u = 2864 11 - P~ 248, 240

u 1654

r = Iinch

r~p

r =3 inches/

FIGURE 4

121

1

4. Re-,Aaceable Liner Construction

The analyses and calculations presented for r o 1 ir.h and

ro = 3 inches container dimensions have considered the decay of one

transient detonative shock wave in a cylinder from a point at the pressur-

ized fluid-metal interface outward for a distance of three inches. Figure 4

shows the respective condition of one wave at r o = 1 inch and r o = 3 inches

at two different instants of time. If we should consider a chamber only of

cavity diameter of 6 inches (r o = 3 inches) the pressure upon reflection

would again be 430. 000 psi and the effect on the container wall would be

too severe to contemplate for th- proposed experiments. If we, however,

take the maximum pressure (430, 000 psi) at the 2 inch diameter and allow

the effects (particle velocity and pressure) to attenuate by some means,

the consequences of the event at the 6 inch diameter interface can be

I tolerated.

The means proposed to mitigate the severity of the shock pulse1 on the wall material is the insertion of a removable liner . inch L D. x

6 inch 0. D. in the pressurized container. This liner will be completely

restrained by the hydrostatically pressurized fluid in the radial, tangential

and axial direction. Such a construction is shown in Figure 5.

SIt is anticipated that over a period of time after a specific

number of maximum duty cycles, the replaceable liner should fail. When

this occurs, the liner c" 1 be removed and a replacement provided.

The initial static and dynamic loading cycle is dopicted graphically

in Figure 6. -3 is the initial prestress at P = 0. Upon the insertion of

preseure into the cavity, the stress is raised ho q 3. The first ,nax-mum

3 discharge resets this stress pattern to the value at cband upon release of

pressure, the residual hoop stress returns to the point of % This is then

3 the permanent working stress in the cavity of the pressure chamber.

1

3 13

z

0

tn t

144

1

Note: Dotted line indicatesloading due to impulse.Solid line indicatesloading due to staticpressure

300-

2-00--

100.O + b =84,000 psi- 3-S Pressure x 10 3 psi' 0

Us-26, 000 psi

100U)

*4) a3= -133,000 psiEn

U 0-243,000 psi

I

STRESS DISTRIBUTION DUE TO STATIC ANDDYNAMIC LOADING

I FIGURE 6

I

j 15

STATUS OF THE WORK

The analyses derived for the effect of detonative shock loading in a

thick walled cylinder hydrostatically loaded have been completed and the

conclusions derived therefrom have been applied to the design of a suitable

piece of equipment to be used in the extrusion experiments.

All vendor proposals for equipment have been reviewed for conform-

ance to specification and suitability of costs, and based upon this review

an agreement has been entered into with Harwood Engineering Company,

Walpole, Massachusetts to develop with Republic Aviation Corporation,

and to manufacture the equipment required for the proposed experiments. 1"ar d _s hac started fabrication of a ZOO, 000 psi high pressure pumping

system to displace 20 in 3 /min. Engineering has been completed in the fdesign of the pressure vessel and work has commenced with the assembly

drawings. These should be ready for detailing early in the next reporting

period.

Work has proceeded with the electrical considerations necessary to

alter the Republic Aviation Corporation 156, 000 joule capacitor bank to

attain suitable charging rates and to release energy to the extrusion

chamber at the repetition rates that are anticipated to be necessary to

accomplish extrusion. This work will continue for the next reporting

period.

1!III

J6

I

REFERENCE

I. "Electra -Spark Extruding," Republic Aviation CorporationReport No. 2027, (ASD TR 8-111 (11), 1 January, 1964

I1

1APPENDIX I

CALCULATION FOR STRAIN DEFORMATIONDUE TO TRANSIENT WAVE

I Statement of the Problem

Consider a thick-walled cylinder with a y= 250,000 psi and the following

dimensions (the cylinder is filled with water):

I INCH

3 INCHES" .. ... ' FIGURE 1

;riangular wave of the following form is transmitted along the cylinder.

i / / 215,000 PSI

2_ISTATIC PRESSURE

This wave reflects upon arrival on a closed end. Find the resulting strains

r ° = I inch and r° = 3 inche-.

The pressure of the rcf.Atd wave front is approximately twice the pressure

of the inciduta. wave front. This is considered the highest possible internal p, ,-

sure; thus, the calculations will be performed for the reflected region only.

To illustrate the wave decay, two cases are analyzed; Case 1, where

r = inch, and Case 2, where r0=3 inches. Hence P=-2(215,000) -430,000

0vsi at to= 1 inch. Using equation (23), we haveI

19

u = 430,00g 00 2864.6 in./sec

I o 0C o0

where po = 0.29 lb/in. 3

Co = 200,000 in./sec

g = 32.2 ft/sec2

From equation (34), the velocity at ro = 3 inches (chosen arbitrarily) is

u3 1 r* 2, 864.0 6 - 1654.0 in./sec 1Thus the pressure at ro = 3 inches is given by

P3 = P = -248,280 psi

A. CASE 1

1. Loading Wave

From equation (28) the density ratio at t = 0 is given by

S 0 Co 2.00 x 105 . 98568P Co- u 2.00x105 - 2864.68

The maximum a1 at the start of plastic deformation is given by

SL= 250, 000 =_Il 1-M = 0;7 -357,143psi

where a = - 250,000 psi

= 0.7

Thus = 1 - 357,143 (1-2x0.3 2 ) = 1- .0119x.82 = .9904230 x 106

at Ory = - 357,143 psi

As soon as the material begins stretching out, the loading becomes plastic.

Considering E = 0 from equation (38), we have

20

, . /00833 .98568 ./0082106 .09061

where y 250,000 = 00833ay =3X 630x 106

Also,2864.6 =. 01432

C 0 2x10 5

Substituting in equation (39), we have

cosh .09061t .,1432 Binh .,061- 1.09061

Since u = d , we have

= .09061 (sinh .09061 .15800 cosh .09061t]

First, the time lt, where fl= 0, is found by plotting 5 versus t.

("ie Figure 2). The 4f versus i plot is shown in Figure 3: it is shown that as

G approaches zero, - approaches a constant value of 0.0125. The radial strain

is given by C1 = - 1: The 1 versus t plot is shown in Figure 4: C, ap-proaches the value of .0264.

2. Unloading Wave

I The duration of the triangular wave is 20 Msec: its equivalent

rectangular profile will have a duration of 10 psec. Hence,

= . 2.0X lO 5 xl x 10- 6 = 2.t ---- =2.0

r 1

It is shown In Figre 2, hc t',cr, that the loading process terminates at 1.75.

Thus, whor, the unloading wave occurs, the material is in equilibrium.

I For the unloading wave, we have the following constants (equation (53)):

= 1: The material returns to its initial volume.

1!

21

I

.0161I

.015

.0140 :5,000p.s.I.

.013

.012

.011 1.0101

.007

.006

.005

.003

.0021

.0011

00 1.0 To 2.01

22 FIGURE 2

I

.016

.015

.014 to: I*_ry- 250,000 p.s.l.

.013

.012

.011 -

.010

.009 -

£ 008

.007 -

.006

i .005 -

1 .004

I .oo /

.003

.02.0

II Z3 FIGURE 3

.03 IIoy 3250,000 P.S.i.

.021

.01

0 1.0 2.0 i

24 FIGURE 41

|I

Hence Of + 4-05+0.303 803

where u 0.303. N.B. The value of 01 after unloading is zeo

= 1 -A= 0.697

y= 0.303

Uo =.01432

= - 1. 0000 - o - 1.0000 - .01432 = - 1.01432

= +4A = 1.71697

.85852

Thus = 303 .43472 and 0 = 0.40150.697 *2

Hence + 9Z(4)= - 1.01432 + 0.4015(.43472)

= - 1.01432 +.17454 = - .83978

and Y ... 83978 - .978195!8.853 2

Also, = .8585 x .43472 = .37321

x =.43472 x .5 = .21736

I Y -. 4015 x .978195 = - .392745

x =, =.43472 x.4015 =.17454

- .978195=- .489097

and 1 - 1=-1+.21736=-.78284

II Z5

T

Hence u = .2"686 sinh .8585t - .79696 cosh .8585t + . 7 8 264 e 4015t

and = e" 4015 t. 43472 cosh .8585 F - 1. 01432 sinh .8565 F J -. 43472

In the plot of U versus , Figure 5, the time when the velocity becomes zero

is shown as io = 0.026 (t= .26x 10-6 sec). Substituting in the equation of e we

find the residual hoop strain, 4f = .01713. Equations (51) and (52) were derived,

however, with ?o = 0 at t 0 (with reference to the equilibrium conditions after

the loading). L

To find the residual hoop strain after unloading, we have: 1S eunloading -9loading

=.01713 - .0125 = .00463 (compression) iIn order to find the residual hoop stress we use the principle of

elastic recovery. Thus, the volumetric strain ev is given by:

ev = (1 + I ' unloading T

where c1 is the radial linear strain change upon relief and 4' is the corresponding

hoop strain. We know, however, the radial unloading stress: i.e., &ao = 430, 000

psi. Hence, c1 = 430,000 0143.30x 106 0

The value of ev Is also known from the loading conditions:

S-v -P- + 1=+.01813

V O PO

Thus, the hoop strain for zero stress (permanent set) is given by

o = ev " Cl = .01813 - .0143 = ,00. I

The corresponding permanent set in the radial direction is

to= .026- .0143 = .012. In the calculations for the unloading wave, it was Jassumed that upon relief the recovery was instantaneous: thus L = 1 at t= 0.

T

i

.015 T - ' I "

UNLOADING

o x 250,000 p. s.i

.01

II.005

I

0 .001 .005 .008 .01 .02 .025

II

27 FIGURE 5

I

IT

Then, for F > 0, r- decreases, changing both hoop and radial strains according toto the following equations:

ro r

Hence, the material will be under the following strain condctios:

HOOP-. 0046 .00)5 .01 .015 HO

-253,000 PSI

Without pursuing this case any further, it Is easily concluded that secondaryyielding will take place. This is undesirable for most designs.

B. CASE 2

The velocity at ro = 3 inches was given as:

u =- 1654.0 and =U 6o 1654 00827

C u .250 000 37 4 s

= C!S- = I1- O= .99173, and % = =367,143 PatCo ~~ 99 7 , (elastic) 0.7

hence, W .00833 x.99173 = 0082611 = .0908906 -_

28

I

Uo .00827 = - .09098851W .0908908

Hence 1 = cosh .0908906t - .0909885 sinh .0908906i - 1

F and a = .0908906 (sinh .0908906t -. 0909885 cosh .0908906 1

The duration of the wave, on the other hand, is given by

= lx10-6 x2x10 5 23 3

where

t = lox 10- 6

C0 = 2x105

r = 3 inches

The u versus t and 4 versus t plots are shown in Figures 6 and 7,

respectively. Another case of Y = 150,000 psi is also shown for comparison.

With the value of f at ro = 3 inches being smaller than at ro = 1 inch,

it is shown that the unloading wave occurs before the material achieves equilibrium.

12.* Unloadling Wave

I Following the same procedure as for r o = 3 inches, with

Co = 0.00827 - .00275* = .00552

4= e" (. 43472 cosh .8585t - .96794 sinh .M5 .43472

and U - .27793 sinh .8585 t + .78816 cosh .8585 t - .78264 e' 4015

The plot of It versus is shown in Figure 8. Comparing this un-loading with that of r o = 1 inch, we see that, at ro = 3 inches, equilibrium is

achieved in about half the time.

S* velocity from the loading process

2| 29

LI

[ ... .... 1 .7.

.008 LOADINGro =3 "

.007-

.006 ___2_-1

.005

.004

.003 M'Y 150,000 p. S i.

.002 _y2 000 .i. /a'y:29,OO i

!.0- O'y :250,000 p s,i.

DOI"

0 1.0 2.030 I

30 FIGURE 6

,009 - - LOADING .ro=3"

.008

.007 -

- 17_ ___ __

e .006 _ _ _ _ _

ycr =50,000 psi.

.007

Ii

1 .5 1.0

it

31 FIGURE 7

.006 "

.005 = 150 000

UNLOADIN I , 000

.004 y 250,0

.004

.002 I

.001

0 -.001 .005 .01 .011 1

I32 FIGURE 8 I

I

The corresponding value of 4 at 1 .01 is =.00727. The i valueat t 0.667 for loading is 4f = .00365. Hence, the resulting strain is (. 00733- .00365) or U" = .00368 (compression).

To find the resulting stresses, the permanent set in both the hoopand radial directions must be found. Employing the concept of elastic recoveryagain, we have

ev = o+1l

C = 2 .0082761 30x10 6=

Furthermore, - ev = + V=- - + 1 =-.99173 + 1 =.00827.

Th11us, eo = 0: This result indicates that there is no permanent set in the hoopdirection. In the radial direction, L = 1+ 1 .= 1+ 00365 = 1.00365 from

loading. Hence - = - - 99636. Also, - =.99173.

c = .99636 x .99173 - 1 = .98812 - 1 = -. 01188 from loading.

1 During the unloading, there is the elastic recovery where c=. 00827:

then, there is the further change of c due to the Lo change.r o o

.::;ric €1 = -- •1 .00827

r 1- 1-.00727=.99373

re 1.0063r

and C1 =.0063 + .00827 = .01457

I

! 3

I

Tbus, change in = .01457 - .0119 = .00265 (tension). The valueof f I when the radial stress is zero is -. 01190 + .00327 . -. 0036.

Residual radial stress = - .0036 + .0026 = - .001 or - .001 x 30 x 106- -30, 000 psi (compression).

The residual hoop stress = .00368 x 30 x 106 = 110,400 psi(compression). From the secondary yielding criterion, we have T

a2 - Ol < cy

or - 30,000 + 110,000 = 80,000 psi. Since Oy was assumed at 250, 000 psi, thematerial will not go through secondary yielding.

1I

I!1

34

1

APPENDIX II

CALCULATIONS SHOWING COMBINED EFFECTOF IMPULSE AND STATIC LOADING

I. Introduction

The combined effect of the action .f a transient shock wave over an

internal hydrostatic pressure in thick walled cylinder is analysed in the

following to verify the structural integrity of a container. The method of

superposition of stress is used.

If we are given a cylinder with the following dimensions:

a =3".'b = 1511

; _a w =b / a = 5 "1IP =internal pressure

(200, 000 psi

I Fo the purpose of discussion, we define the following variables atan internal radius of r. = a = 3 inches:

a, = hoop stress under internal hydrostatic

pressure alone

i b = radial stress under internal hydrostaticpressure alone

aa = residual hoop stress in bore at zero pressure

Sa4 = residual hoop stress resulting from an impulse

C% = 03 + a4 = resultant hoop stress after release ofinternal pressure due to passage of

first maximum shock impulse

I1 35

I

C-- al + Ch + C4 = combined hoop stresses understatic pressure after passage ofimpulses

II Calculation for Stress from Internal Pressure Alone

The equations of Lame may be used to calculate the elastic stresses

in a thick walled cylinder, internally loaded. The relationship for the

hoop stress at the bore states that, T

71 =P w9+T

Then, for a wall thickne:s wv = 5 and an internal pressure of 1P 200, 000 psi

01 =ZOO, 000 x = 17, 000 psi (tension) I

By definition, .1

2= -P - 00, 000 psi (compression)

From the secondary yielding criterion, we have

Oy > C,2 - al

or -Z00, 000 - Z17,000 = -417,000 psi. Since ay was assumed at

250, 000 psi, the material will go through secondary yielding.

However, to reduce the stress to a point under the 250, 000 psiyield stress of the material, we must impose a negative hoop stress in

the bore prior to loading. To bring the bore to a zero hoop stress at an

b-,trr pressure of 230, 000 psi let,

a3 = -Z17,000 psi

Then uzing the Tresca Criterion, we haveOy > O2 -a I

or -200; 000 + 0 = -ZOO, 000 psi. This value is suitably below the

required 250, 000 yield stress. Therefore, the material will not fail andwe can superpose the dynamic loading to predict the final state.

III Continued Dynamic and Static Load -

Previous analysis (Appendix I) has shown that the residual hoop

36 [

I

stress at after discharge (peak pressure, P = 248, 280 psi and time

duration t = 20 1isec is,

O4 = -110,000 psi

The combined hoop stresses ( (7) under static pressure after the

initial capacitor discharge will then be

= (0o1 + o'3) + 04

= 217, 000 -217,000 + (-10, 000)

= -110,000 psi (compression)

the radial stress,

-P = 200, 000 psi (compression)

From Tresca's Criterion, we have,

ay > a 2 - 01

1or -200, 000 - (-110, 000) = -90,000 psi. Since ay is again 250, 000 psi

yielding will not occur in the bore from the combined stresses due to

dynamic aind static forces.

IV Release of Pressure

I Upon the release of the static pressure, the permanent hoop stress

imposed by the first discharge event will reverse by the amount uf .

j Then the -esidual stress with internal pressure P = 0 will be,

a s =(a 3 + N

= -217,00U -110,000

= -317,000 psi (compression)

Since os exceeds the 250, 000 psi yield strength of the material

under the condition of no internal pressure, the bore will yield radially

I inward. This is an unsatisfactory stress condition for cylinder design.

V Solution of the Problem

I Because yielding has occurred at the bore with an initial hoop stress,

a, = 217,000 psi (compression) a cb must be found that will allow elastic

I behavior of the material within the 250, 000 psi yield strength wall material

37

being considered.

Let us calculate a new residual hoop stress % in the bore at no

pressure that will allow the desired elastic action within the limits of

the 250, 000 psi yield strength material. Then

US = or + cb + a4

240,000 = 217,000 + q3 + (-110,000)

a3 = -133,000 psi.

Therefore, the desired value of pressures at the bore o, = 133, 000 psi

(compression) 1'

The combined stress (s) at a pressure P = 200,000 after a maximum

energy impulse is then

as =o +s U + N

= Z17, 000 + (-133, 000) + (-10,000)

= -26, 000 psi (compression)

% = -P = -200, 000psi (compression)

Then using the Tresca Criterion,

ay > (72 - a,

or -200, 000 -(-26, 000) -174, 000. With the c y = 250, 000 psi material,

yielding will not occur in the bore.

To f .d the residuai hoop stress upon release of the 200. 000 psi

static pressure P goes to zero, and the hoop stress is din.-hed by the

amount a, = 217,000 psi. Then

06 aa + 4

b= 133,000 + (-110,000) o

= 243,000 psi

Therefore the bore diameter at r o = a - 3 inches has attained aresidual hoop stress (compression) of 243, 000 psi which is within the

required 250, 000 psi yield strength of the material. The pressure vessel

can be operated safely in the prescribed limits of 200, 000 psi static

pressure and a peak shock pulse of 248, 280 psi for 20 &Jduration.

38

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1 0 0 1 0 9:S1-4 9) 0

0 o< > _ 0.4 1- t - bu- a* 4J

(n.w~ E fo

Z~. -

(o .04 s.... E-4v(. V-m~. ". 0iV UU:

LID 0 bo

~FJ 4

-d P4

tic l v U) - 3 (4

0 v V0~ 0I

U) E E U)

U 04 lz w0.1 IN I

S.. to

4:I. 1~4 04

W 0~ > 0 0 v o UoCC) ZA

00U- 0 ~~O t~~ o

Id r

to 4, . ')

- '5 o

It f

1.1<4) 44 4CZ - 0 9- __ __

r. r to u.

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~ 4: J~~O