PRINCETON UNIVERSITY

Department of Civil Engineering x

Approved for rub llc release;Distribution Unlimited

ID

"STRUCTURES AND MECHANICS

.- i ~ nn nnnn u . • .,, nmm mm~~II I i i[Hl

"Technical V'epto p. 4

"Civil ,,ngng. No. 77-SM-,J0 - J

LINE CRACK SUBJECT TO SHEAR#

by

( / 6~/A.Cemal/E~ringen".........................................

Research Sponsored by theOffice of Naval Research

("f~.2} under

Cont ract N0l09'I4-76-C-.24/• [ModifitN

AModV IO ffoar o --. I••

IIITISWhite Section

)DC Buff section 0!

UNANNOUNM[ [

UISTRIBIJ1WIAYALA6BItLIY CNES . ": •,i'' "\

Di. .AVAL __:.../.........

Approved for Public Release; Distribution Unlimited

>/,9._/ -

LINE CRACK SUBJECT TO SHEARI

A. Cema! Eringen

Princeton UniversityPrinceton, NJ 08540

ABSTRACT

Field equations of nonlocal elasticity are solved todetermine the state of stress in the neighborhood ofa lin'. crack in an elastic plate subject to a uniformshear at the surface of the crack tip. A fracture criterionbased on the maximum shear stress gives the criticalvalue of the applied shea: for which the crack becomes unstable.Cohesive stress necessary to break the atomic bonds iscalculated for brittle materials.

1. INTRODUCTION

In several previous papers, [11 - [31, we discussed the state of

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

to a uniform tension perpendicular to the line of the crack at Infinity. The

solution of this problem was obtained within the framework of the non-

local elasticity theory. The resulting solution did not contain the

stress singularity present In the classical elasticity solution and

therefore a natural fracture criterion based on the usual maximum stress

hypothesis could be established. This most interesting outcome could

be used to calculate the cohesive stress in various materials.

1 The present work was supported by the Office of Naval Research

--2-

TheL, present paper- dtils with the problem of a line crack in anl

eLa~st ic plate where tile crack is s,?ubject to a uni form shear I oad. We

employ the field equations of nonlocal elasticity theory to formulate

and solve this problem. Crat ifyingly the resulting solution does not

contain the stress singularity at the crack tip and therefore a fracture

criterion based c.i the maximum shear stress hypothesis can be used to

obtain the critical value of the applied sh2ar for which the line

crack begins to become unstable. If the concept of the surface energy

is used It is possible to calculate the cohesive stress holding the

atomic bonds together. For steel (with no dislocations) we give an estimate

for the cohesive stress. In section 2 we give a resum4 of basic

equations of the linear nonlocal elasticity theory. In section 3 the

boundary value problem is formulated and the general solution is

obtair.ed. In section 4 the solution is given for the dual integral

equations completing the solution of the problem of line crack subject to

a shear load. Calculations for the shear stress are carried out on a

computer and restilts are discussed in section 5.

2. BASIC EQUATIONS OF NONLOCAL ELASTICITY

The basic equations of linear, homogeneous, isotropic, nonlocal elastic

sol'ds, with vanishing body and inertia forces, are (cf. [4, 5])

(2.1) t = 0kl.,k

(2.2) tk = f[X'(Ix'-xl)e rr(x') 6 k +2p' (1x'-xJ)ekZ(X') ]dv(x')

V

(2.3) ek 1 " -(U +U,)'t 2 kZ i,k

_________________________ -- -.... O*

--3--

where the only d1fference from classical elasticity is .n the stress

constitutivc equations (2.2) which states that the streqs tkW(X), at a

polkt x, depends on strains ekZ(W), at all points of the body. For

homogeneous and Isotropic solids the nonlocal, elastic moduli )'(Ix'-x!)

and oi'(Ix'-xI) are functions of the distance between the variable point

x' and the fixed point x at which the stress is to be evaluated. The

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

the surface (V.

Throughout this paper we employ cartesian coordinates xk with

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

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

a comma represent partial differentiation, e.g.

k,9- k Z.

In our previ.us work [4, 6, 7] we have obtained the forms of

X'(Ix'-xI) and ij'(Ix'-xj) for which the dispersion curves of plane

waves coincide within the ent 4 'e Brillouin zone with those obtained in

the Born-Von KdrmAn theory of lattice dynamics. Accordingly

(2.4) (A',p') - (ApIc([x'-xj)

a x - {l a (a-lx'-,,I). Ix'-xlSa,

0 IX,'-xl'a

where a is the lattice parameter, X and p are classical Lam6. constants,

and a is a normalization constant to be determined from0

-4-

All

(2.5) J a(Ix'-xl)dv(x') 1.

IIf1

While this simpie and elegant result is useful in many calculations,

it is not the or.ly one that approximates the dispersion curves of

lattice dynamics. In fact a very useful one is

(2.6) ct\jx'-.xj) M a 0exp[-(63/a) 2 (X'k- Xk)(X'k- Xk)]

where B is a constant. For the two-dimensional case (2.6) has the

specific form

(2.7) 0(lx'- X) - (0/1)2exp (/a )2 [(X' -Xl2 + (x' 2 -x 2

Employing (2.4), in (2.2) we write

> (.8) tkj = a(jY,'-xi) Ck (X')dv(x')

(2.8) t ki - k

V

where

(2.9) OkG(X) Xe (x')6k + 2 1jek WX)

= XU (x')6 +k+U (X,)+u (x')]r,r - k9. k,R. Z,k-

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

the Green-Gauss theorem we obtain

(2.10) ra(Ix'-xI)Gktk(x')dv(x') - cl(Ix'-xl)ok (x')daW 0v 2.o av

_____ ____ ____ k. ' k.. . . - -- .

Here The surface integral may be dropped If the only surface of the

body Is at Infinity.

3. CRACK UNDER SHEAR

Consider a plate in (Xl=X, x2 •Y) - plane weakened by a line crack

of length 29 along the x-axis. The plate is subjected to a constant

shear stress T0 along tht" surfaces of the crack, Fig. 1. For the

plane strain problem (2.10) takes the form

(3.1) j c(Ix'-xl)Gk kzk(xv ,y )dx'dy' - a(Ix'-xI)[0T2 Z (x',0)]dx'- 0

R -Z

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

space excluding the crack line (Ix1<Z, yi0). A bold-face bracket indicates a

Jump at the crack line.

When an incision is made in an undý,!ormed body, the body will

in general be deformed and stressed because of the long-range inter-

atomic attractions. Thus if we are to treat the problem of a plate

with crack, undeformed and unstressed in the natural state, we must

consider that after an incision is made the crack is not opened, i.e.

the boundary conditions are to be applied to the plate in the natural

state.

Under the applied uniform shear load on the unopened surfaces of

the crack the displacement field possess the following symmetry regulations

(3.2) u(x,-y) -u(x,y) , v(x,-y) - v(xy)

f.A

-6-

Emplo•ying this in (2.9) we see that

(3.3) [02 (x 0)1 ] 0 , [x[>k

Hence the limits (-9•,9) in the second integral of (3.1) may be

replaced by (

The Fourier transform of (3.1) with respect to x' gives

(3.4) { (t,jy'-yIl[-i0o lz(,y') + Tyy,(&,y ]dy

-cz(&,Iyl)j C2i(&,0) 1 - 0

where a superposed bar indicates the Fourier transform, e.g.,

f(g,y) = (27T) f(x,y)exp(i,x)dFCO

If we take the Fourier transform of (3.4) with respect to y, and solve

for the fact- of a in the integrand of (3.4), upon inversion we obtain

d -(2Tt) •(xO) I {exp(-iny)dndyl,( ,y) + •y 292. ",0 2t 1.

since the left hand side is defined for all y except y=O. Thus we

obtain

(3.5) _i&G d+29. 0 Z - 1,2it dy

-7-

Substituting this into (3.4) we have

(3.6) o (XO) 1 = 0 , = 1, 2

Hence we have shown that the general solution of (3.4) is the

same as that of the system (3.5) and (3.6).

The Jump condition (3.6) for JxI>Z is satisfied identically. For

JxI<Z we also have [o 2 1 (x,O) I = 0, on account of (3.2). Since

2(X,-y) = -o 2 2 (x,y) we also see that t 2 2 (x,O) - 0 for IxI<k. Thus

the normal stress condition on the crack surface is satisfied identically.

Considering also the continuity requirement of the displacement field

satisfying (3.3) we find that the boundary conditions at x=O are:

a (x,O) = 0 , t yx(X,O) - T , lxl<z(3.7) YY oX

0 (x,O) - 0 u(x,O) = 0 ,]xlj>YY

In addition we must have

(3.8) u = V = 0 , as y-+

Consequently we must obtain the solution of (3.5) subject to (3.7) and1

(3.8).

1Even though some authors feel that v(x,O)=O, lxj>t should also besatisfied for this (so-called Mode II) problem (cf. [81), the resultsbased on the boundary conditions (3.7) and (3.8) are accepted byworkers on the theory of fracture. The physical reasoning indicatesthat constant shear load if not balanced by an opposing couple, shouldgive a rotation to the whole body so that v(x,O)=O for JxI>£ appearsto be in contradiction with the expected displacement field.

-.- 1..

-8--

Equations (3.5) arenorx other than the Fourier transforms of the

Navier's equationsIn two dimensions, namely

- 2-O - (X~+211)& U-Iý()+Pi)V, = 0,yy

(3.9)

-i$(X+O)uy, +(X+ 2)v, yy-2t 2 = 0

The general solution of this set (for y>O ), satisfying (3.8) is.

u = -(2ff)-2 1 [ICIA( + (I&Iy- ~X+3p)B(F)]exp(-Ily-i~x)d&,-(2•)-I12 f "0- [ I()+ (a y

(3.10)

v (2Tr)1/2 {i[A() + yB(&)]exp(-jtIy-itx)dF

where A(O) and B(&) are to be determined from the boundary conditons

(3.7). Using (2.9) we calculate

(3.11) 2y(p,y) 2ii[-I[IA(t) + (0y) B(&)-exp(-I~Iy)yy A1

According to (3.7) and (3.7) this must vanish at y-0. Hence

'. (3~~.12) B(ý) ÷. ._ 1 A(F,)

Noting that A(-C) - A(&), on account of symmetry u(x,y) u(-x,v),

the displacement field may be put into the form

u(x,y) - 1/2 X J A(F Ax4-- w y e &coB (&x)d&,

(3.13)-;Y ( 2)12 k_+ A(F) e e Ysin (&x)dF,

0 /)--v~~y .. -,- o .'_

-9-

For Coe, I hrough (2.9) and (3.12) we obtain (y>O)

0 (x'y) =-0~ (x.-y) -2(2/7r) /(X.4+) jAQ()(2r-t 2Y~e,- Y n(FxWdr0

(3.14) 0 o (',.,V) = -0i Yyx,-Nv) -2(2)!rr) 1 /2 (,A-J) ft A (F)'y-Ysrý~0

ci y(x,y) - Y (x,-y) =-2(2/nt) t/2 0+0 F (F) Fl-ýy)e y cos(~xWd0

The stress field according to (2.8), is then given by

M

(3.15) t yy(x,y) = fdyl fm o (x',Y') Lc(xIx'-xI,Iy'-yj) -ct(Ix'-xI~ly'+yl)]dx'0

t (x,y) - f ""dy' f OO(xW ,y') [a (x'-x ,y '-y) +iaIx '-xjy '+ y I)d x

Substituting for a from (2.7), the Integrations may be performed with

respect to x' and y' by noting the integrals (cf.. [91).

f ex(ý- ,2 in ýx+x')? dx' = (T/)112 cp- 2 A)sinti xI~ ~ JCPIX ois r(x+x')I (I/p Cos(- Fx

(3.16) 12 = fexp(_py,2--yy, )dy' . 1 /P1/2k~pY 24)1ý12p]0

a3 Jyexp(_,py,2_yydy 1n 1/ ýx _Y

3 O y)y 2p 4 p P 4)lý,/'Pj

Hence,

-10-.

t =x -(2/70(0+11 fA) sin(rx) 1'e-t'Y[2ý+(& /2p)(ý-2py)I

0

yx 0

0.17 -e 2p 2vp 2 d&

/~T,..i exp[-py -(p)/4p)]}dj

0

whe remnn two/ bound ar condiis t ieerrfntons aefnd d ( b74my nwb

expressed as

CiCOcos(ýz) dý 0 z>10

where we have introduced

-11.

×/Z z k C F () • -C(O)

2 2 2 2(3.19) K(r,p) [l+(t /2pt )l[l-i(•/2 )l-(t/9¢'Wexp(-•2/4pt2),

T 0 02 /2(X+p) , p = (a/a) 2

To determine the unknown function AM) we must solve the dual integral

equations (3.18) for C(O).

4. THE SOLUTION OF DUAL INTEGRAL EQUATIONS

By introducing

cos(Cz) = (n~z/2) J_()

where J (z) is the Bessel's function of order •, we write the systemV

(3.18) in the form

i4(ý)[l+k(ECj)J_J(Cz)dý = -TO z , O~z<l0

(4.1)

fC(O)J I(Cz)dý - 0 , z > 1

00

The kernel function k(6 ) is given by

(4.2) k(cc) = K(C,p)-l = 2 C[I- (E )]-2()-2 -½cexp(- 2• )

where

(4.3) E l/2tpi- a/20k

-12-

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 (see [10] § 4.6).

I

(4.4) h(%)+ h(u)L(x,u)du -1(7x)½To0

for the function h(x), where

(4.5) L(x,u) = (xu:)' tk(et)Jo (xt) Jo(ut)dt

0

Once (4.4) is solved then C() is calculated by

(4.6) C(4) (2C)l Jl x'jo(Wh(x)dx

We observe that for c=O we have k=O, and the dual integral equations

(4.1) reduce to those obtained in the classical elasticity. For this

case from (4.4) we have h (x) = -T (Tx)½/2,and (4.6) gives the classical0 0

result:

(4.7) C0 (0) -(0/2)1ToC-½Jl( W

or

(4.8) Ao0 () = -0/2) ToJ1 (L) /&t

The next iteration of the integral equation (4.4) with the use of

ho(x) gives

(Li.) l'(x)W -0n/2) ~T 0.[l~(

where

(4.10) S(')=f J(xix&d

Jo f-2 kRJj~t (t)J (0)-tj (0i ( ~ d

In which f steHankel transform of t1 k(cEt)j (t), i.e.,

0

f(t,E) t-ke: l)

* . If we write

X/E - 11 C

in (4.10),we will have

But since c fi(ey,c) is the Hankel transform of f(n/c,c), we see that

lrn I(CE) lin f(t,) - i ur [ (O)k(EC)IC-+0 C4-0 -

-14-

Hence (4.10) vanishes as k(cý) with c-O, i.e.,

(4.12) C(() + 0(ý)

Thus for small c the difference between C (0) and C will be of order

c. Since any number of iterations will contribute higher order terms

in c,the solution for C(O) will also be in the form (4.12) as c-O.

We observe that in general B is in the neighborhood of 1, [6).

The ratio of the atomic distance to the crack length a/29, is, however,

extremely small for even microscopic cracks. Thus c is very small

and k(cý) is in general negligible as compared to unity for all values

of cC. As can be seen from Fig. 2, k(eC) gogs uniformly from 0 to

-I as cC varies between 0 and -. Thus we expect that the solution of (4.1)

for C(ý) will be almost the same as the classical solution C (&)o

corresponding to c=0. Some differences are of course expected

for crack lengths close to atomic distances.

Substituting A (0) for A(&) in (3.13) and (3.17), we obtain the

displacement and stress fields. It is of interest to calculate the

shear stress along the x-axis. This is given by

(4.13) tyx/T f"[l+k(cC)]Jl(c)cos(ýz)dc , z x/z

This integral for O<z<l gives t yx/.t 1 since in this interval the

integral converges even for E-O. For z>1 the integral again converges

for all E>O and it is permissible to ignore k(cC) as compared to unity.

However for z-1 this is no longer the case and we cannot ignore k(cý)

as compared to unity. To see this we write t at z-1 as:yx

ill i I-i-----i--l----------------------------.......-----

-1.5-

tyx (X,) - t + t +3

where

t T 0 (n) [l-$(rn)]cosndn,

t2- 2T o 2n2Ji(n) [1-0(En)]cosndn,

0

t 3 -2To r-½ enJl(n)exp(-c2n2)cosndn"0

Since 1-0(cn)zO and T ,0 it is clear that0

[t11 _• (o f) [l-O(y) 1dy -T/,½

t 2 .(2-c 0 ,f noy 2 [1-4(y)Jdy - 4T /3nrrc0

t 31 (2TO/•½) f exp(-y 2 )dy = /n C0

Hence

Ity (x,L) 0 1 0 .-I) C-1

Thus we see that t yx(x,) has finite value for c#O. In the evaluation

of the stress field near z-14-. therefore we cannot ignore the function

k(cO) in (4.13).

-16-

5. NUMERICAL ANALYSIS AND DISCUSSION

Calculations of the shear stress t , given by (4.13) along the

crack line, were carried out on computer. The results

are plotted for E-1/20, 1/50, 1/100 and 1/200 in Figures 3 to 6. For

a crack length of 20 atomic distance (c=1/20) the rebults are not very

good. However for a cra hl00 atomic distance (Fig. 5) we can see

that the shear stress boundary condition ty (x,O))° for ]xI<z is

satisfied in a strong approximate sense. The relative error is less than

l½ 0/o. Hence we conclude that the use of the classical A0 () given

by (4.8) gives satisfactory results for crack lengths greater than 100

atomic distances.

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

by

(5.1) tyx (,O)/T 0 c/ dE , =-a/22Z

where c converges to about -0.30, i.e.

(5.2) c Z -0.30

We now make the following significant observations:

(M) 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-+O. This is the classical continuum limit

of square root singularity.

-17-

(iii) When t (ZO) = t ( cohesive shear stress),fracture willyx c

occur. In this case

(5.3) 2 z - G0 c

where

(5.4) G M (Ba/2c 2)t 2

Equation (5.3) is non other than the expression of the Griffith criterion

for brittle fracture. Note that we have arrived at this criterion via

the maximum sheai- stress hypothesis. The present criterion of fracture

not only unifies the fracture mechanics at the macroscopic and microscopic

scales,but also employs the natural concept of bond failure in the

atomic scale.

(IV) The cohesive stress t may be estimated i;" one employs thec

Griffith definition of surface energy y and writes

S~2(5.5) t a K Xc c

where

(5.6) K - 8c2W/rB(1-v)c

Employing the values of y and the elastic constants for steel

y , 1975 CCS , - 6.92x101 CGS

v - 0.291 a 2.48 A° 1.65

S"r

- 8

we find that

t - 1.04xlO1 3 CGS , t /p 0.• 1 '•c c

This result is in the right range and well accepted by metallurgists

based on other considerations. For example Kelly [III gives

t • 6.OxlO10 CGS and t /W - 0.11.c C

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

the computations.

IThe altenuation constant B-1.65, used in these calculations, makesthe Fourier transform of a coincide with the dispersion curve ofelastic waves In Botn-Von K'rman model of lattice dynamics (cf. 161).

-• --- ... .... .~ ~..-. - -..- . ..-. -. .. . . .... • f-l•* .".,,-,, .•-,•,•..•• I ,.:LXVS- =mm m ----

REFERENCES

[1] A. C. Ei~ngen and B. S. Kim, "Stress Concentration at the Tip ofCrack," Mech. Res. Comm. 1, 233-237. 1974.

[2] A. C. Eringen, "State of Stress in the Neighborhood of a SharpCrack Tip," Trans. of the twenty-second conf. of Army mathematicians,1-18, 1977.

[3] A. C. Eringen, C. G. Speziale and B. S. Kim, "Crack Tip Problem inNonlocal Elasticity", to appear in J. Mech. and Phys. of Solids.

[41 A. C. Eringen, "Linear Theory of Nonlocal Elasticity and Dispersion

of Plane Waves", Int. J. Engng, Sci. 10, 233-248, 1972.

[5] A. C. Eringen, Continuum Physics Volume IV Acad. Press, §7, 1976.

[6] A. C. Eringen, "Continuum Mechanics at the Atomic Scale", to appearin Cryst. Lattice Defects.

[7) A. C. Eringen, "Nonlocal Elasticity and Waves" Continuum MechanicsAspects of Geodynamics and Rock Fracture Mechanics, (ed. P. Thoft-Christensen) Dordrecht, Holland, D. Reidel Publishing Co., pp. 81-105.

[8] I. N. Sneddon and M. Lowengrub, "Crack Problems in the MathematicalTheory of Elasticity", John Wiley & Sons, New York, 1969, pp. 32-37.

[91 I. S. Gradshteyn and I. W. Rhyzhik, "Tables of Integrals, Series andProducts", New York; Acad. Press, 1965, pp. 489, 307, 338.

[10] I. N. Sneddon, "Mixed Boundary Value Problems in Potential Theory",North Holland Publ. Co., Amsterdam, 1966, pp. 106-108.

[il] A. Kelly, "Strong Solids", Oxford, 1966, p. 19.

y

TO

LINE CRACK UNDER SHEAR

FIGURE I

1.0

.8

.6

.4-

*i,!

.2

00 1.0 2.0

K THE BEHAVIOR OF K VSe) \s e

FIGURE 2

O. ....__I ___I __ ___ __ II

1.0

.4-

,yx T .0 2.0 x/4

-I.0

tyx/ To VS. x/i FOR E 1/20

FIGURE 3

1.0

.8

.6-

.4

.2-.21.0 2.0tyx To " 0 0 . . .4li i iX0

-21 -. 4'

-1.8

tyx/To VS. x/e FOR C -1/50

FIGURE 4

1.0

.6-

.4-

tyx/TLo 0 i -- F x/.

"1.0 2.0.2-

.4-

.6-

.8-

1.0-

1.2-

1.4-

1.6-

1.8-

2.0-

2.2-

2.4-

2.6-

2.8rtyx/To VS. x/' FOR E 1/100

FIGURE 5

,4 1.0

.8

.6

.4

.2- 1.0 2.0

-. 2

-. 4

-2.8

-1.0-

-1.2

-1.4-

-1.6-

-1.8-

-2.0-2.2-

-2.4-

-2.6

-2.8

-3.0-

-3.2

-3.4

-3.6-

-3.8-

-4.0-

tyx/0To VS. xK/ FOR E 1/200

FIGURE 6

SECURITY CI.ASSIýIICATION OF THIS PAGE (When Date Entered)REPORT DOCUMENTATION PAGE READ INSTRUCTIONS

REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORMSREP)RT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

lTechni ca l Report L40_.TITLE (and Subtitle) S, TYPE OF REPORT 6 PERIOD COVERED

.j.ne Crack Subject to Shear Technical Report6. PCAFOMINo ONG. REPORT NUMBER

7 AUTHOR(*) S. CONTRACT ON GRANT NUMBIER(e)

A. Cemal Eringen N000 4-76-C-0240

9. PERFORMING ORGANIZATION NAME AND ADDRESS tO. PROQRAM ELEMENT, PROJECT, TASKAREA A WORK UNIT NUMBERS

Princeton Universitv P00002

Princton, New Jersey 08540II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Office of Naval Research (Code 471) July 1977

Arlington, VA 22217 1 BER OF PAGES

4iMO.II'ORING AGENCY NAME. A ADDRESS(if different from Controlling Office) 1S. SECURITY CLASS. (of til' report)

Unc lass if Ied

IS&. PECLASSIPiC ATION/DOWNGRADINGSCHEDULE

16 DISTRIBUTION STATEMENT (of thh, Report)

Approved for public release; Distribution Unlimited

17 DISTRIBUTION STATEMENT (of the abltract entered In Block 20, If different from Report)

In SUPPLEMENTARY NOTES

19 KEY WORDS (Continue on reverse aide If neceeaery asd Identify by block number)

line crack

('rack under shearnonlocal elasticity/theoret cal shear stress

0. ABSTRACT (Continue on reverse aide If neceseary and Identity by block number)

Field equations of nonlocal elasticity are solved to determine the state ofstress In the neighborhood of a line crack in an elastic plate subject to atiniform shear at the surface of the crack tip. A fracture criterionbased on the maximum shear stress gives the critical value of the ,:I''',dshear for which the crack becomes unstable. Cohesive stress necessary tobreak the atomic bonds Is calculated for brittle materials.

DD O 1473 EDITION OF 1 NOV65 IS OBSOLETES/N 0 102-o014-6601

SECURITY CLASSIFICATION OIF THIS AGE (When Dote Entefed)

r - -. t~~...-

I -1 GOVERNMENT Commanding OfficerAdministrative & Liaison Activitie,, AMXMR-ATL

Attn: Mr. R. Shea

Chief of Naval Research U.S. Army Materials Res. AgencyDepartment of the Navy Watertown, Massachusetts 02172Arlingon, Virginia 22217Attn: Code 474 (2) Watervliet Arsenal

471 (AGGS Research Center222 Watervliet, New York 12189

Attn: Director of Research

DirectorX-'R 3ranch Office Technical Library•,95 Survier Street6oston, Massachusetts 02210

Director Redstone Scientific Info..CenterONR Branch Office Chief, Document Section2'9 S. Dcarborn Street U.S. Amy Missile CommandChicago, Illinois 60604 Redstone Arsenal, Alabama 35809

Director Army R&D CenterNaval Research Laboratory Fort Belvoir, Virginia 22060Attn: Code 2629 (ONRL)Washington, D.C. 20390 (6) Nav

U.S. Naval Research Laboratory Commanding Officer and DirectorAttn: Code 2627 Naval Ship Research & Development CenterWashington, D.C. 20390 Bethesda, Maryland 20034

Attn: Code 042 (Tech. Lib. Br.)Oirector 17 (Struc. Mech. Lab.)ONR New York Area Office 172715 Broadway- 5th Floor 172New York, N.Y. 10003 174177Director 1800 (Appl. Math. Lab.)ONR Branch Office 5412S (Dr. W.D. Sette)1030 E. Green Street 19 'Dr. M.M. Sevik)Pasadena, California 91101 1901 (Dr. M. Strassberg)

1945Defense Documentation Center 196 (Dr. 0 Fait)Cameron Station 1962Alexandria, Virginia 22314 (12) Naval Weapons Laboratory

Arry Dahlgren, Virginia 22448

Commanding Officer Naval Research LaboratoryU.S. Army Research Office Durham Washington, D.C. 20375Attn: Mr. J. J. Murray Attn: Code 8400CRD-AA-IP 8410Box CM, Duke Station 8430Durham, North Carolina 27706 2 8440

630063906380

:-;,ISl'r i ,, J1o.,ien Retx.arch Div. Chief of Naval Op,'raltir)i'

;,,.val Shiii R&D Center Dept. uf the Navy.,rf•lk Naval Shipyard Washin~iton, D.C. 20J30Pot,,mouth, Virginia 23709 Attn: Code OpO7T.ttn: Dr. E. Palmer

Code 7.n Strategic Systems Project OfieDepartment of tne Navy

,aval Ship Retearch & Development Center Washington, D.C. 20390fnnap)olis Division Attn: NSP-O01 Chief Scientist

Annapolis, Maryland 21402Attn: Code 2740 - Dr. Y.F. Wang Deep Submergence Systems

?8 Mr. R.J. Wolfe Naval Ship Systems Cnnmand181 Mr. R.B. Niederberger Code 39522

"P',14 Or 11. Vanderveldt Department of the NavyWashington, D.C. 20360

7c,-hriLal Library Engineering Dept..•1,il Underwater Weapons Center U.S. Naval AcademyPasadena Annex Annapolis, Maryland 214023202 E. Foothill Blvd.Pasadena, California 91107 Naval Air Systems Command

Dept. of the NavyU.S. Naval Weapons Center Washington, D.C. 20360China Lake, California 93557 Attn: NAVAIR 5302 Aero & StructuresAttn: Code 4062 - Mr. W. Werback 5308 Structures

4520 - Mr. Ken Bischel 52031F Materials604 Tech. Library3208 StructuresCommanding Officer Director, Aero Mechanics

U.S. Naval Civil Engr. Lab. Naval Air Development CenterCode L31 JohnsvillePort Hueneme, California 93041 Warminster, Pennsylvania 18974

Technical Director Technical DirectorU.S. Naval Ordnance Laboratory U.S. Naval Undersea R&D CenterWhite Oak San Diego, California 92132Silver Spring, Maryland 20910

Engineering DepartmentTechnical Director U.S. Naval Academy

Naval Undersea R&D Center

San Diego, California 92132 Annapolis, Maryland 21402

Naval Facilities Engineering CommandSupervisor of Shipbuilding Dept. of the Navy

Newport News, Virginia 23607 Washington, D.C. 20360Attn: NAVFAC 03 Research & Develoi-

men tTechnical Director 04 "1

Mare Island Naval Shipyard 14114 Tech. LibraryVallejo, California 94592

Naval Sea Systems CommandU.S. Navy Underwater Sound Ref. Lab. Dept. of the NavyOffice of Naval Research Washington, D.C. 20360P.O. Box 8337 Attn: NAVSHIP 03 Res. 6 Jechnoloq•yOrlando, Florida 32806 031 Ch. Scie-ltist for R,0

03412 Hydromechanics037 Ship SilencIng Div.035 Weapons Dynamics

Nav•l Ship rngineering Center terr'-46vernn-n~tk XI Jv 'f,PrinLe Georqe's Plaza4ý •ittsviIle, Maryland 20782 CommandantAttn: NAVSEC 6100 Ship Sys En(r & rDes Dep Chief, Testing & Development ryv.

6102C Computer-Aided Ship Drs U.S. Coast Guard6105G 1300 E. Street, NI.W.6110 Ship Concept Design Washinqton, D.C. ?02?66120 Hull Div.61200 Hull Div. Technical Director6128 Surface Ship Struct. Marine Corps Dev. & Educ. Comnai'd6129 Submarine Struct. Quantico, Virginia 22134

Air-Fo-rce DirectorNation&l Bureau of Standards

Commnander WADD Washington, D.C. 20234Wright-Patterson Air Force Base Attn: Mr. B.L. Wilson, EM (I9Dayton, Ohio 45433Attn: Code WWRMDD Dr. M. Gaus

AFFOL (FDDS) ;,ational Science FoundationStructures Division Engineering DivisionAFLC (MCEEA) Washington, D.C. 20550

Chief, Applied Mechanics Group Science & Tech. DivisionU.S. Air Force Inst. of Tech. Library of CongressWright-Patterson Air Force Base Washington, D.C, 20540Dayton, Ohio 45433

DirectorChief, Civil Engineering Branch Defense Nuclear AgencyWLRC, Research Division Washington, D.C. 20305Air Force Weapons Laboratory Attn: SPSSKirtland AFB, New Mexico 87117

Commander Field CommandAir Force Office of Scientific Research Defense Nuclear Agency1400 Wilson Blvd. Sandia BaseArlington, Virginia 22209 Albuquerque, New Mexico 87115Attn: Mechanics Div.

Director Defense Research & EngrgN-ASA Technical Library

Room 3C-128Structures Research Division The PentagonNational Aeronautics & Space Admin. Washington, D.C. 20301Langley Research CenterLangley Station Chief, Airframe & Equipment BranchHampton, Virginia 23365 FS-120

Office of Flight StandardsNational Aeronautic & Space Admlin. Federal Avalation AgencyAssociate Administrator for Advanced Washington, D.C. 20553

Research & TechnologyWashington, D.C. 02546 Chief, Research and Development

Maritime AdministrationScientific & Tech. Info. Facility- Washington, D.C. 20235NASA Representative (S-AK/DL)f.Q. 9ox 6700 Deputy Chief, Off.Ce of Ohij Cinst-.

i~c

)I umi. Enerriy (ommi!rsion Prof. R.D. TestaDIv. of Reactor Devel. & Tech. Columbia Universityb, ,rnmantown, Maryland 20767 Dept. of CIvil Engineerinq

S.W. Mudd Bldg.ýI'ip Hull Research Comnittee New York, N.Y. 10027National Research CouncilNational Academy of Sciences Prof. H. H. Bleich2101 Constitution Avenue Columbia UniversityWashington, D.C. 20418 Dept. of Civil EngineeringAttn: Mr. A.R. Lytle Amsterdam & 120th St.

New York, N.Y. 10027

, - CONTRACTORS AND OTHER Prof. F.L. DiMaggioTECHNICAL COLLABORA7ORS Columbia University

Dept. of Civil Engineering616 Mudd BuildingNew York, N.Y. 10027

Uni versi t3esProf. A.M. Freudenthal

Dr. 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 D. C. Evans

California Institute of Technology University of UtahDiv. of Engineering & Applied Sciences Coopter Science Division

Pasadena, California 91109 Salt Lake City, Vhsh 84112

Dr. Harold Liebowitz, Dean Prof. Norman JonesSchool of Engr. & Applied Science Massachusetts Inst. of TechnologyGeorge Washington University Dept. of Naval Architecture 6725 - 23rd St., N.W. Marine EngrngWashington, D.C. 20006 Cambridge, Massachusetts 02139

Prof. Eli Sternberg Professor Albert I. KingCalifornia Institute of Technology Biomechanics Research CenterDiv. of Engr. & Applied Sciences Wiyne State UniversityPasadena, California 91109 Detroit, Michigan i48202

Prof. Paul M. NaghdiUniversity of CaliforniaDiv. of Applied Mechanics Dr. V. R. HodgsonEtcheverry Hall Wayne State UniversityBerkeley, California 94720 School of Medicine

Professor P. S. Symonds Detroit, Michigan 4820?Brown UniversityDivision of EngineeringProvidnece, R.I. 02912 Dean B. A. Boley

Northwestern UniversityProf. A. J. Durelli Technological InstituteThe Catholic University of America 2145 Sheridan RoadCivil/Mechanical Engineering Evanston, Illinois 60201Washington, D.C. 20017

reI. P ,G. •4(o(]Ce, Jr. Prof. J. KmnipnerUniv¶,rsity of Minnesota Polytechnic Institute (if 11rockly¶

',pt. of Aerospace [ngng & Mechanics Dpt. of Aero.Engrg.t Applied MechNinneapnlis, Minnesota 55455 333 Jay Street

Brooklyn, N.Y. 11201Dr. D.C. DruckerUniversity of Illinois Prof. J. KlosnerDean of Engineering Polytechnic Institute of BrooklynUrbana, Illinois 61801 Dept. of Aerospace & Appi. Mech.

333 Jay StreetProf. N.M. Newmark Brooklyn, N.Y. 11201

,niiversity of IllinoisDept. of Civil Engineering Prof. R.A. SchaperyUrbana, Illinois 61801 Texas A&M University

Dept. of Civil EngineeringProf. E. Reissner College Station, Texas 77840University 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 Taxas ANM 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, Artiona 85721Pennsylvania State UniversityP. 0. Box 30 Dr. B.S. BergerState College, Pennsylvania 16801 University of Maryland

Dept. of Mechanical EngineeringProf. Eugen J. Skudrzyk Col.1epe Park, Maryland 20742Pennsylvania State University Prof. G. R. IrwinApplied Research Laboratory Dept. of Mechanical Engrg.Dept. of Physics - P.O. Box 30 University of MarylandState College, Pennsylvania 16801 College Park, Maryland 20742

_ollegParkMarylad 2074

or'. ,. J. FI nv,, ; Prof. S.B. DongCirneq'ie-Mel lon University University of CaliforniaDept. of Civil Engineering Dept. of MechanicsSchenley Park Los Angeles, California 9002.4Pittsburgh, Pennsylvania 15213 Prof. Burt Paul-

University of Pennsylvania)r. Ronald 1,. }lurton Towne School of Civil & Mech. Engrg.*I ýI.. or' En-fnvorlnjinr Am-lyL s.,; Rm. 113 - Towne Building.,iI �ox .14, 220 S. 33rd Street1Jniv !rr;1tIy of' CGin.rinati Philadelphia, Pennsylvania 19104

"",ui~nlt, , n i,', I~c??tProf. H.W. Liu

Dept. of Chemical Engr. ý Metal.Syracuse University

Prof. George Sih Syracuse, N.Y. 13210Dept. of MechanicsLehigh University Prof. S. BodnerBethlehem, Pennsylvania 18015 Technion R&D Foundation

Hahii, IsraelProf. A.S. Kobayashi Prof. R.J.H. BollardUniversity of Washington Chairman, Aeronautical Engr. Dept.Dept. of Mechanical Engineering 207 Guggenheim HallSeattle, Washington 98105 University of Washington

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

Brown UniversityProf. Daniel Frederick Providence, Rhode Island 02912Virginia Polytechnic InstituteDept. of Engineering Mechanics Prof."Werner GoldsmithBlacksburg, Virginia 24061 Dept. of Mechanical EngineeringProf. A.C. Eringen Div. of Applied MechanicsDept. of Aerospace & Mech. Sciences University of CaliforniaPrinceton University Berkeley, California 94720Princeton, 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 fe. the Application of MathematicsProf. 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

ktf r--and Research institutes

Library Services Department Dr. R.C. DeHartReport Section Bldg. 14-14 Southwest Research InstituteArqonne 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 Street

New York, N.Y. 10022Dr. L.H. ChenGeneral Dynamics Corporation Dr. W.A. von RiesemannElectric Boat Division Sandia LaboratoriesGroton, Connecticut 06340 Sandia Base

Albuquerque, Ney Mexico 87115Dr. J.E. GreensponJ.G. Engineering Research Associates Dr. T.L. Geers3831 Menlo Drive Lockheed Missiles & Space Co.Baltimore, Maryland 21215 Palo Alto Research Laboratory

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

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

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

Burbank, California 91503Dr. W.F. BozichMcDonneWl Douglas Corporation5301 Bolsa Ave.Huntington Beach, CA 92647

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