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STRUCTURES AND MECHANICS

35

Technical RepNo 5ICivil Engng. Res. Rep 78-SM-B

.

(,, LINE CRACK SUBJECT TONTIPLANE SHEAR ~

Princeton University

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under

Modification No. P00002

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July 1.978

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A471FICATIaN .. . .. .. ...............

. . . .................. EP 26 1978j.. .. .... ... .. .. .... .... ..

0UiLLUI

LINE CRACK SUBJECT TO

ANTIPLANE SHEAR 1

A. Cemal EringenPrinceton University

/II

ABSTRACT (Field equations of nonlocal elasticity are solved to determinethe state of stress in a plate with a line crack subject to aconstant anti-plane shear. Contrary to the classical elasticitysolution, it is found that no stress singularity is present atthe crack tip. By equating the maximum shear stress that occursat the crack tip to the shear stress that is necessary to breakthe atomic bonds,the critical value of the applied shear isobtained for the initiation of fracture. If the concept ofthe surface tension is usedone is able to calculate thecohesive stress for brittle materials.

1. INTRODUCTION

In several previous papers 1i] - [4] we discussed the state of

stress near the tip of a sharp line crack in an elastic plate subject to

uniform tension and in-plane shear. The field equations employed in the

solution of these problems are those of the theory of nonlocal elasticity.

The solutions obtained did not contain any stress singularity, thus

resolving a fundamental problem that persisted over half a century. This

enabled us to employ the maximum-stress hypothesis to deal with fracture

L problems in a natural way. Moreover it has been possible to predict

the atomic cohesive stresses by introducing the experimental values

of the surface energy.

iThe present work was supported by the Office of Naval Research

The present paper deals with the problem of a line crack in an elastic

plate where the crack surface is subject to a uniform anti-plane shear

load. This problem, classically, is known as the Mode III displacement.

We employ the field equations of nonlocal elasticity theory to formulate

and solve this problem. The solution, as expected, does not contain the

stress singularity at the crack tip and therefore a fracture criterion

based on the maximum shear stress hypothesis can be used to obtain the

critical value of the applied shear for which the line crack begins to

become unstable. If the concept of the surface energy is introduced, it

is possible to calculate the cohesive stress holding the atomic bonds

together. Estimates of cohesive stress are also given for perfect

crystalline solids.

In section 2 we present a resume of basic equations of linear non-

local elastic solids. In section 3 the boundary-value problem is form-

ulated, and the general solution is obtained. in section 4 we give the

solution of the dual integral equations, completing the solution. Cal-

culat:rons for the shear stress are carried out in a computer, and the

results are discussed in section 5.

2. BASIC EQUATIONS OR NONLOCAL ELASTICITY

Basic equations of linear, homogeneous, isotropic, nonlocal elastic

solids, with vanishing body and inerti.a forces, are (cf. [5]):

(2.1) tk 0

(2.2) t [X'(Ix'-xl)e r X +2p'(ix -xl)e (x')ldv(x')

V

(2.3) ekZ = k, +u Z, k

where the only difference from classical elasticity is in the stress

constitutive equations (2.2) in which the stress tk (x) at a point xdepends on the strains ek(x'), at all points of the body. For homo-

geneous and isotropic solids there exist only two material moduli,

X'(Ix'-xl) and 1i'(jx'-xj) which are functions of the distance I-xl.

The integral in (2.2) is over the volume V of the body enclosed within

a surface 3V. 4Throughout this paper we employ cartesian coordinates xk with the

usual convention that a free index takes the values (1, 2, 3), and repeated

indices are summed over the range (1, 2, 3). Indices following a comma

represent partial differentiation, e.g.

Uk, 9, k zU/X

In our previous work [6, 7] we obtained the forms of o'(ix ssXl)

and p'(Ix-xI) for which the dispersion curves of plane elastic waves

coincide with those known in lattice dynamics. Among several possible

curves the following has been found very useiul

0"" (X'1j)a(fx-xI)

(2. 4)

a(lx-xl)= a exp[- ($/a)2 (x'-x)-(x'-x)] ,

where S is a constant, a is the lattice parameter, and a0 is determined

by the normalization

(2.5) f c(Ix'-xl)dv(x') 1

V

"•,1

•' :,•_ •,:! ,,.•• ,' ,•,'',,'" ,..',• , ••: , •: ••,• •' ' '• ' • ',J• ... -• •,- , : 'Z:• • • •: ````I `I ••.•• •`::• `` :L•` `.• ...- ,.-;,':;-•• : ,'-'. '- '•' '•': '

-4-

In the preseat work we employ the nonlocal elastic moduli given

by (2.4)2. Carryin- (2.4)2 into (2.5) we obtain

(2.6) 0/ (fla)2.

Substituting (2.4)1 into (2.2) we write

(2.7) tk (x) dv(x') ,JZ a(1x.tY'-x)OkV

where

(2.8) aki (x') e , e x) 6ki + 2Vl ek£ (x')

SUr,r (x) Sk +)j [Uk, (xt) + uk (x')]

is the classical Hooke's law. Substituting (2.8) into (2.1) and using

Green-Gauss theorem we obtain:

(2.9) W a(jx-xf) 0ktk (x') dv (x') - a(Ix'-xI) "kL (x')dakIx')= 0-

The contribution to the surface integral from the parts of the

surface at infinity would be dropped since the displacem2nt field

vanishes' at infinity.

- ~ -

-5-

3. CRACK UNDER ANTI-PLANE SHEAR

We consider an elastic plate in the (Xl-x, x2 -Y) - plane weakened

by a line crack of length 2Z along the x-axis. The plate is subjected

to a constant anti-plane shear stress tyz-T along the surfaces of the

crack, Fig. 1. For this problem we have

(3.1) u 1-u 2-O , u3 "w(y,y),

(3.2) ax-" x ay w , all other OkaUO 0

so that the only surviving member of the field equations (2.9) is

(3.3) a(Ix'-xIIy'-yl)V' 2w(x',y')dx'dy'- (xtO)]dx'mO

where the integral with a slash is over the two-dimensional infinite

space excluding the line of the crack (I xl <k, y-0). A boldface bracket

indicates a Jumip at the crack line.

When an undeformed and unstressed body is sliced to create a free

surface, it will in general be deformed and stressed on account of the

long-range interatomic forces. Thus if we are to consider that the

plate with a crack is undeformed and unstressed in its natural state

then we must apply the boundary conditions on the unopened crack surfac,.

Under the applied anti-plane shear load on the unopened surfaces

of the crack,the displacement field possesses the following symmetry

regulations

(3.4) w(x,-y) - -i(x,y, •

I I~ I I I.. F>

-6-

Using this in (3.2) we find that

(3.5) [a (x,O)] - 0-yz

Hence the line integral in (3.3) vanishes. By ta'ting the Fourier

transform of (3.3) with respect to x', we can show that the general

solution of (3.3) is identical to that of

d2- 2-y)

(3.6) w2 .-•(C ,y)_0dy

almost everywhere. Here a superposed bar indicates the Fourier

transform e.g.

Sf(•,y) ff(xy) exp (i~x) dx

The boundary conditions are

w(x,o) - 0 for jxI>t

(3.7) t (X,O) - T for Ix1< ,

"W(X,y) - 0 as (x 2+y2)1/2..-.D

The general solution of (3.6) (for yj0) satisfying (3,7)3 is

(3.8) v(x~y) - (2/w)*f A(Q) eEY coo (Ex) twdonay odtos

where A) is to be determined from the remaining two boundary conditions.

For the non-zero components of the stress tensor we have

iI

-7-1

( .) t ~ -- (2/7w ) ji vf 'A( &) 4d & Icy ~ x -l 'y -l

yz

+-a(xIx-xIIy'+yI)] e- coo (tx') dx'.

Using (2.4) for C6 1x'x), we carry out integrations on x' and y'. To2 -

this end we note the following integral&, [8]:

I L e- r~ (-I) t2) ain&(x '+x)l x TP ex -~~ 2/AP) fsin(Cx)F P cos&(x +x) (Cos (&x)

~0

O(z) 2ffir* 2 t

Hence

-(27r 11 &A(&% [e erfctibZ - e~y erfc uuin(&x)d&

oz 2 r 2/

(3.11) --( F&() e& erfc( + et'erf c /?0 o&x)d&

p (0/a) 2 , erfc(z) -1- O(z).

The boundery conditions (.)and (37 now read1 2

I

fC(4) K(cz) cos(zý) dr. -(n/2 T O T0 O ,0o

(3.12) 1-½C(M) cos(zW) dU - 0 z>l

where we set

z = x IC , 1 - . , c - a/23i ,

(3.13) K(cW) * erfc (W)

A()T -T oT2 1

To determine the unknown function A(M), we must solve the dual integral

equations (3.12).

4. THE SOLUTION OF THE DUAL INTEGRAL EQUATIONS

Recalling the eApresaion

cos (zO) - (wz;/2)i J. (zr),

where J (z) is the Bessel function of order v, we write the system

(3.12) in the form

C1 -T z-i O<z<l

I ~(4.1) ;C(1) J)i (z4) d; 0 z>l

- To

-9-

The kernel function k(cC) is given by

(4.2) k(c;) - K(cC) -1 - (cC)

The solution of the dual integral equations (4.1) is not known. However,

it is possible to reduce the problem to the solution of a Fredholm

equation (cf. [91)

1

(4.3) h(x) + o h(u)L(x,u)du - -4(wx)IT°

for the function h(x), where

(4.4) L(x,u) - (xu)4Ftk(ct)Jo(xt)Jo(ut)dt

When (4.3) is solved, then C(;) is calculated by

(4.5) C(W) - (2C) JxjJ 0 (Wx) h(x) dxfI

As discussed in a previous work [4], if we note that E is extremely small,

k(ec) may be neglected an compared to unity in (4.1), (see Fig. 2). In

this case the zeroth order soluticn of (4.3) namely h (x) - -To0 (x)f/2

suffices for the calculations when the crack size is larger than 100 atomic

distances. In such a case we have

(4.6) Co (C) N -(r/2)4ToC-•rJ() 1

-10-

and therefore

(4.7) A 0 -((/2)To0Ji(M)/

The shear stresses are then calculated by (3.11). Interesting among

these is the shear stress t along the crack line y - 0. For thisyz

we obtain

(4.8) ty (Z,O)IT° - K() Jl(0) cos(z;) dý

As observed before, this integral converges for all z provided K(c;)

is not approximated by unity for c small. For c - 0 at z 1 1 we have

the classical stress singularity. However, so long as E 0, (4.8)

gives a finite stress all along y - 0. At O<z<l, t z/T0 is very close

to unity, and for z>l, t /T possesses finite values diminishing from ayz o

maximum value at z=1 to zero at z=-.

For c>>1/100 the approximate solution given by (4.7) is not very

good. However, further improvements can be achieved by the iterative

solution of (4.3) with the use of Co(). Since c>1/100 represents a

crack length of less than 10- 6 cm, and at such submicroscopic sizes other

serious questions arise regarding the interatomic arrangements and force

laws, we do not pursue solutions valid at such small crack sizes.

5. NUMERICAL CALCULATIONS AND DISCUSSION.

Calculations of the shear stress t , given by (4.8) along the crack

line, were carried out on a computer. The results are plotted for c - 1/20,

1/50, 1/100, 1/200, in Figures 3 to 6. For a crack length of 20 atomic

distances (c 1/20) the result is not very good in that the boundary

-11-

condition at Ixl<Z, y - 0 is satisfied only very roughly. However, for a

crack size of 100 atomic distances (Fig. 5) the shear streas boundary

condition io fulfilled in a strong approximate sense. The relative error

in this case is less than 110/o. Hence we conclude that the classical

Ao(t) given by (4.7) gives satisfactory results for crdck lengghs

greater than 100 atomic distances.

The stress concentration occurs at the crack tip, and this is given by

(5.1) tyz (,0)/t° = 31re , e a/20t

where c 3 converges to about

(5.2) c3 0.4

The following observations are very significant:

(i) The maximum shear stress occurs at the crack tip, and it is

finite \eq. 5.1)

(ii) The shear stress at the crack tip becomes infinite as the atomic

distance a-0. This is the classical continuum limit of square

root singularity.

(iii) When tyz(Z,0) - tc ( - cohesive shear stress), the plate will fail.

In this case

2(5.3) -E CQ03 G

where

(5.4) CG (a/26c 3 t 2

G .3 c

- j

Equation (5.3) is the Expression of the Griffith -racture criterion

for brittle fracture. We have arrived at this result via the maximum

shear-stress hypothesis, rather than the surface energy concept used

by Griffith and his followers. The sigrificance of this result is that

the fracture criteria are unified at both the macroscopic and zhe

microscopic scales and that the natural concept of bond failure is

employed.

(iv) The cohesive shear stress tc may be estimated if one employs

the Griffith's definition of the surface energy y and writes

2-(5.5) t a-K yc c

where

2(5.6) K 8pc

c 3

Since some measurements exist on y, by employing these values we

can calculate the cohesive shear stress. For steel we have

Y =1975 CGS , P=6.92 x 1O CGS

S= 0.291 , a =2.48 A0

(5.7) t /i = 0.2568 a1/2C

-13-

Let at a n atomic distance the nonlocal effects attenuate to T% of its

value at x=O. Using (2.4) we find that

(5.8) = 2.1•6/n

(5.9) tc /I = 0.3764 n-1/2

For n 6 this gives

(5.10) tc/P = o.14

which is in the right range and well accepted by metallurgists. For

example Kelly [10] gives t /P = 0.11. There is however a question ofC

in-plane versus antiplane shear failure which need special examination.

Compared to the results obtained in our previous work on the in-plane

shear [4), the cohesive stress seems 30% higher. However, this is somewhat

artificial since it is necessary to know the value of $ or n in either

case more precisely. This of course requires at least one experiment.

Acknowledgement: The author is indebted to Dr. Balta for carrying out

computations.

?II

REFERENCES

[1] A. C. Eringen and B. S. Kim, "Stress Concentration at theTip of' Crack," Mech. Res. Comm. 1, 233-237, 197h.

[2] A. C. Eringen, "State of Stress in the Neighborhood of aSharp Crack Tip," Trans. of' the twenty-second conferenceof Army mathematicians, 1-18, 1977.

[3] A. C. Eringen, C. G. Speziale and B. S. Kim, "Crack TipProblem in Nonlocal Elasticity", J. Mech. and Phys. ofSolids, 25, 339-355, 1977.

[4) A. C. Eringen, "Line Crack Subject to Shear", to appear inInt. J. Fracture.

[5) A. C. Eringen, "Linear Theory of Nonlocal Elasticity andDispersion of Plane Waves", Int. J. Enpng. Sci. 10, 233-248, 1972.

[61 A. C. Eringen, "Nonlocal Elasticity and Waves," ContinuumMechanics Aspects of Geodynamics and Rock Fracture Mechanics,(,e•d. P. Thoft-Christensen) Dordrecht, Holland, D. Reidel

Publishing Co., pp. 81-105, 197h

17] A. C. Eringen, "Continuum Mechanics at the Atomic Scale",Cryst. Lattice Defects, 7, Do. 109-130, 1977.

18] I. S. Gradshteyn and I. W. Rhyzhik, "Tables of Integrals,Series and Products", New York; Acad. Press, 1965, pp. I480, 307, 338. •

[9] I.N. Sneddon, "Mixed Boundary Value Problems in Potential Theory",North Holland Publishing Co., Amsterd !1, 1966, pn. 106-108.

I[10] A. Kelley, "Strong Solids", Oxford, iCK6, p. 19.

I.I

SI

x

000 000-

TO

I

LINE CRACK SUBJECT TO ANTIPLANE SHEAR

FIGURE I

r I

If:

UL.

0.8-

0.6-

.0.4-

0.2-

00 0.5 1.0 1.5 2.0

I, ~KERNEL FUNCTION Met)I

FIGURE 2

I

1.0

0.5-

0.5 1.0 1.5 2.0 x/f0 T . . i, -••-

-0.5-

-I.0-

-1.5-E=1/20

-2.0-

•. -2.5"1

-3.0-

-3.5

-4.0-

-4.5-

-5.0 -

-5.5

1

ANTIPLANE SHEAR STRESS

FIGURE 3

I2

1.0

__ __ _ 0.5 _ __ _ 1.0 1.5 2.0 x/d

Eu 1/50

4-40

-4-5

ANTIPLANE SHEAR STRESS

FIGURE 4

1.0

0.5__1.0_1.5 2.0 x/dl

00

0.5

4-,Nlo

-2.0IANIPAE HERSTES

FIGURE

-3.7]

1.0

0.5-

0.5 1.0 1.5 2.0 X/.

-0.5-

-I.0-

-I.5

E 1/200-2.0-

,o-2.5-N

-3.0-

-3.5-

-4.0

-4.5 -

-5.0-

-5.5-

ANTIPLANE SHEAR STRESS

FIGURE 6

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IREPORT NUMBER GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

Princeton Technical Rev'. 53

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Line Crack Subject to Antiplane Shear TcnclRpr

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A. Cemal Eringen N00014-76-C-0240

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fracture mechanics, line crack, antiplane shear, nonlocal theory,crack tin problems, cohesive stress

20 AUSTR4ACT (Continuean revrs s~O*id.i ecam Edwiibp&.Aub)Field equations of nonlocal elasticity are solved to determine the state

of stress in a olate with a line crack sublect to a constant anti-plane shear.Contrary to the classical elasticity solution, it is found that no stresssingularity is present at the crack tip. By equatinq the maximium shearstress that occurs at the crack tip to the shear stress that is necessaryto break the Atomvic ) 'nds, the cr .'cal value of the applied shear is

obtained for the mui ition -)f fi ure. If the concept of the surfacetension is used, one is able to c,.- iilate the coh-e~eMre-i.ý- for brilýttle

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Dept. of Civil EngineeringProf. E. Reissner College Station, Texas 77040University of California, San DiegoDept. of Applied Mechanics Prof. W.D. PilkeyLa Jolla, California 92037 University of Virginia

Dept. of Aerospace EngineeringProf. William A. Nash Charlottesville, Virginia 22903University of MassachusettsDept. of Mechanics & Aerospace Engng, Dr. H.G. SchaefferAmherst, Massachusetts 01002 University of Maryland

Aerospace Engineering Dept.Library (Code 0384) College Park, Maryland 20742U.S. Naval Postgraduate SchoolMonterey, California 93940 Prof. K.D. Willmert

Clarkson College of TechnologyProf. Arnold Allentuch Dept. of Mechanical EngineeringNewark College of Engineering Potsdam, N.Y. 13676Dept. of Mechanical Engineering323 High Street Dr. J.A. StricklinNewark, New Jersey 07102 Texas A&M University

Aerospace Engineering Dept.Dr. George Herrmann College Station, Texas 77843Stanford UniversityDept. of Applied Mechanics Dr. L.A. SchmitStanford, California 94305 University of California, LA

School of Engineering & Applied ScienceProf. J. D. Achenbach Los Angeles, California 90024Northwestern UniversityDept. of Civil Engineering Dr. H.A. KamelEvanston, Illinois 60201 The University of Arizona

Aerospace & Mech. Engineering Dept.Director, Applied Research Lab. Tucson, Arizona 85721Pennsylvania State UniversityP. 0. Box 30 Dr. B.S. BergerState College, Pennsylvania 16801 University of Maryland

Dept. of Mechanical EngineeringProf. Eugen J. Skudrzyk College Park, Makar.lAnd Z2742Pennsylvania State University Prof. . .Applied Research Laboratory Prof. G. R. IrwlnDept. of Physics - P.O. Box 30 Dept. of Mchanical Engrg.State College, Pennsylvania 16801 University of MarylandCollege Park, Maryland 20742

! Atomic Ener,' -ommirsion Prof. R.i. TestaDiv. of Rea n r Devel. h Tech. Columbia University

Germantown, Maryland 20767 Dept. of Civil EngineeringcS.W. Mudd Bldg.

Sh:p Hull Research Committee New York, N.Y. 10027y National Research Council

National Academy of Sciences Prof. H. H. Bleich2101 Constitution Avenue Columbia UniversityWashington, D.C. 20418 Dept. of Civil EngineeringAttn: Mr. A.R. Lytle Amsterdam h 120th St.

New York, N.Y. 10027

PART 2 - CONTRACTORS AND OTHER Prof. F.L. DiMaggloTECHNICAL COLLABORATORS Columbia University

Dept. of Civil -Engineering616 Mudd Building

_________tie New York, N.Y. 10027Ut

Prof. A.M. FreudenthalDr. J. Tinsley Oden - George Washington UniversityUniversity of Texas at Austin School of Engineering &345 Eng. Science Bldg. Applied ScienceAustin, Texas 78712 Washington, D.C. 20006

Prof. Julius Miklowitz 1). C. evansCalifornia Institute of Technology Univer oe U tah

P, Div. of Engineering & Applied Sciences Computer Science DivisionPasadena, California 91109 Salt Lake City, %sh 84112

; I. Dr. Harold Liebowitz, Dean Prof. Norman JonesSchool of Engr. & Applied Science Massachusetts Inst. of TechnologyGeorge Washington University Dept. of Naval Architecture I725 - 23rd St., N.W. Marine EngrngWashington, D.C. 20006 Cambridge, Massachusetts 02139

Prof. Eli Sternberg Professor Albert I. KIingCalifornia Institute of Technology Biomechanics Research CenterDiv. of Engr. & Applied Sciences Mmne State UniversityPasadena, California 91109 Detroit, Michigan 148202

Eth Prof. Paul M. NaghdiS ertUniversity of CaliforniaDiv. of Applied Mechanicsil ; Etcheverry Hall Dr .R. Hodgson

Wh•n State UniversityBerkeley, California 94720 Sooi of MecineProfessor P. S. Symonds Detroit, Michigat 48202Brown University

PrDivision of EngineeringProvidnece, R.I. 02912 Dean B. A. Boley

i E e o2Northwestern UniversitySProf. A. J. Ourelli Technological InstituteThe Catholic University of Tmercca 2145 Sheridan RoadCivil/Mechanical Engineerinij Evanston, Illinois 60201Washington, D.C. 20017

•ndstry and Research- Institutes

Library Services Department Dr. R.C. DeHartReport Section Bldg. 14-14 Southwest Research InstituteArgonne National Laboratory Dept. of Structural Research9700 S. Cass Avenue P.O. Drawer 28510Argonne, Illinois 60440 San Antonio, Texas 78284

Dr. M. C. Junger Dr. M.L. BaronCambridge Acoustical Associates Weidlinger Associates,129 Mount Auburn St. Consulting EngineersCambridge, Massachusetts 02138 110 East 59th Street1 Dr. L.H. Chen New York, N.Y. 10022

General Dynamics Corporation Dr. W.A. von RiesemannElectric Boat Division Sandia LaboratoriesGr-oton, Connecticut 06340 . Sandia Base

Albuquerque, Ney Mexico 87115Dr. J.E. 3reensponJ.G. Engineering Research Associates Dr. T.L. Geers3831 Menlo Drive Lockheed Missiles & Space Co.Ba'.timore, Maryland 21215 Palo Alto Research Laboratory

3251 Hanover StreetDr. S. Batdorf Palo Alto, California 94304The Aerospace Corp.P.O. Box 92957 Dr. J.L. TocherLos Angeles, California 90009 Boeing Computer ServiLes, Inc.

P.O. Box 24346Dr. K.C. Park SE.Sttle, Washington 98124Lockheed Palo Alto Research LaboratoryDept. 5233, Bldg. 205 Mr. Willian Caywood3251 Hanover Street Code BBE, Applied Physics LaboratorPalo Alto, CA 94304 8621 Georgia Avenue

Silver Spring, Maryland 20034LibraryNewport News Shipbuilding & Mr. P.C. DurupDry Dock Company Lockheed-California CompanyNewport News, Virginia 23607 Aeroinechanics'Dept., 74-43

Burbank, California 91503Dr. WV.F. BozichMcDonnell Douglas Corporation5301 Bolsa Ave.Huntington Beach, CA- 92647

Dr. H.N. AbramsonSouthwest Research InstituteTechnical Vice PresidentMechanical SciencesP.O. Drawer 28510San Antonio, Texas 78284

""• iI

Dr. S.J. Fenves Prof. S.B. DongCarnegie-Mellon University University of CaliforniaDept. of Civil Engineering Dept. of MechanicsSchenley Park Los Angeles, .California 90024Pittsburgh, Pennsylvania 15213 fVf. Burt Paul -

University of PennsylvaniaDr. Ronald L. Huston Towne School of Civil & Mech. Engrg.Dept..of Engineering Analysis Rm. 113 - Towne BuildingM ail Box 112 220 S. 33rd StreetUniversity of Cincinnati Philadelphia, Pennsylvania 19104Cincinnati, Ohio 45221 Prof. H.W. Liu

Dept. of Chemical Engr. A Metal.Syracuse University

Prof. George Sth Syracuse, N.Y. 13210Dept. of Mechanics Prof. S, Bodn#~rLehicgh UniversityBethlehem, Pennsylvania 18015 Technion R&D Foundation

Prof. A.S. Kobayashi Prof. R.J.H. BollardDUnepst. of Mechasicangto n g Chairman, Aeronautical Engr. Dept.Dept. of Mechanical Engneer0ing 207 Guggenheim HallSeattle, Washington 98105 University of Washington

Librarian Seattle, Washington 98105Webb Institute of Naval Architecture .................. .Crescent Beach Road, Glen Cove Prof. G.S. HellerLong Island, New York 11542 Division of Engineering

Brown UniversityProf. Daniel Frederick Providence, Rhode Island 02912Virginia Polytechnic (nstituteDept. of Engineering Mechanics Proi'-"Wrner GolsmithBlacksburg, Virginia 24061 Dept. of Mechanical Engineering

Div. of Applied MechaicsProf. A.C. Eringen University of CaliforniaDept. of Aerospace & Mech. Sciences Berkeley, California 94720Princeton UniversityPrinceton, New Jersey 08540 Prof. J.R.Rice

Division of EngineeringDr. S.L. Koh Brown UniversitySchool of Aero., Astro.& Engr. Sc. Providence, Rhode Island 02912Purdue UniversityLafayette, Indiana 47907 Prof. R.S. Rivlin

Center for the Application of Mathem,Prof. E.H. Lee Lehigh UniversityDiv. of Engrg. Mechanics Bethlehem, Pennsylvania 18015Stanford UniversityStanford, California 94305 Library (Code 0384)

U.S. Naval Postgraduate SchoolProf. R.D. Mindlin Monterey, California 93940Dept. of Civil Engrg.Columbia University Dr. Francis CozzarelliS.W. Mudd Building Div. of InterdisciplinaryNew York, N.Y. 10027 Studies & Research

School of EngineeringState University of New YorkBuffalo, N.Y. 14214