Journal of Elasticity, Vol. 8, No. 3, July 1978 Sijthott & Noordhoff International Publishers Alphen aan den Rijn Printed in The Netherlands

Asymmetric problem .of a Grittith crack in a nonhomogeneous medium under sheart

BRIJ M O H A N SINGH* and RANJIT S. D H A L I W A L

The University o]: Calgary, Calgary, Canada; and S. Singh, Regional Engineering College, Kurushetra, India

(Received March 18, 1977)

1. Introduction

Solutions to some mixed boundary value problems of elasticity for a nonhomogene- ous, isotropic medium have been obtained recently by Kassir and Sih [1], Kassir [2], Kassir and Chuaprasert [3], Popov [4] and Singh [5]. All the above authors have dealt with undirectional nonhomogeneity. Singh [6] has solved the Reissner-Sagoci problem for an infinite elastic solid whose shear modulus ~ is a function of r and z referred to a cylindrical polar coordinate system.

Most natural or rolling materials are nonhomogeneous in both x and y directions

if one takes xy-plane as the cross-section of a three dimensional medium. Therefore, it would be worthwhile to consider a crack problem in this type of medium. In this paper, we consider the problem of determining the state of stress in an infinite, nonhomogeneous, isotropic elastic solid containing a Griftith crack under shear. The shear modulus of the material is assumed to vary symmetrically, about the xz-plane, in the vertical direction and nonsymmetrically, about the yz-plane, in the horizontal

direction. The plane of the crack is assumed to be the xz-plane. The formulation of the problem leads to the solution of two sets of dual integral equations which are further reduced to the solution of two Fredholm integral equations of the second kind in two unknown functions tk(t) and ~(t), respectively. The expressions for the stress intensity factors K1 (at x = 1) and K_I (at x = - 1) are obtained in terms of qS(t) and t~(t) when the crack is located at - 1 < - x - 1 , y = 0 and - o o < z < o o . The integral equations are solved numerically and the numerical values of K 1 and K_I are displayed graphically to demonstrate the effect of nonhomogeneity of the material.

t This work was supported by National Research Council of Canada through NRC-Grant No. A4177. * Permanent Address: 30/3 Gandhi Colony, Muzaffar Nagar, U.P., India.

Journal of Elasticity 8 (1978) 313-318

314 B. M. Singh and R. S. Dhaliwal

2. Formulation of the problem and derivation of the sets of dual integral equations

We consider the problem of a crack located at - 1 - x---1, y = 0 in an infinite solid under shear. Assuming the problem under consideration to be symmetric about the plane y = 0 the displacement field in the half-space y - 0 is given by

ux(x, y, z) = O, Uy(X, y, z) = O, uz(x, y, z) = w(x, y). (1)

The only non-zero stress components are given by

o'~,z(X, y, z) = IX Ow/Ox, O'yz(X, y, z)= Ix ~w/ay, (2)

where Ix, the modulus of rigidity of the material, is assumed to be a continuous and twice differentiable function of x and y. The equation of equilibrium can be written in the form

O/ Ow\ O/ Ow\ (3)

The boundary conditions for the problem may be written as:

rAx, o, z)-= o Ixl<l, (4)

uz(x,O,z)=O Ixl>l, (5) o'yz(x, y , z ) ~ S as y~oo . (6)

Now, if we take the shear modulus IX of the form

Ix(x, y )= Ixo exp (ax+/3 ]Yl), (7)

where Ixo, a and/3 are constant, the solution of equation (3) for y >- 0 and satisfying the condition (6) may be taken as

w(x, y) = Ixo----/3 e - ~ 1 - e-~Y+ ~ o / 3e(~-~y)/2 [A(~) cos (~:x)

+ B(~)sin (~x)]e-~(e)'d~}], (8)

where A(~) and B(~) are arbitrary functions of ~ and

s(~) = [~2 + (a2 +/32)/411/2" (9)

From (2) and (8), we find

o'yz(x,y,z)=S[l-~oe('~+c3Y~/z{I ° [s( ,) +/3/2]

x[A(~)cos(~x)+B(~)sin(~x)]e-~(')Yd~}]. (10)

The above expression for the shearing stress satisfies the boundary condition (10) and the boundary conditions (4) and (5) now lead to the following dual integral

A s y m m e t r i c problem of a Griffith crack 315

equations for the determination of A(¢) and B(~):

Io ° e - a x/ 2 [s(~) +/3/2][A(~) cos (~x) + B(¢) sin (~x)] d~ = x/~o ' Ix] < 1, (1 1)

Io =[A(~) (~x) + B(~) sin (¢x)] d~ = 0, Ix[ > 1. (12) cos

The dual integral equations (11) and (12) may be split into the following two sets of dual integral equations for the determination A(¢) and B(¢), respectively:

Io ~[S(~) +/3/2]A(~) = cosh (o~x/2)/~/~o, 0 < < (~x) d~ cos x 1, (13)

Io =A(~) cos d~ 0, x 1, (14) (~x) >

and o~

Io [s(~) /312]B(~) (¢x) d~ < x 1, (15) ( ,~xlZ )lx/-~o, + s in ~ s i n h 0 <

Io °~B(~) (~x) de 0, x (16) sin > 1.

3. Solution of the dual integral equations

To solve the dual integral equations (13) and (14), let us assume

i 1 A ( ~ ) = (a(t)Jo(~t) dr, (17)

vchere J,( ) denotes the Bessel function of the first kind and n-th order. For A(~) given by (17), we find that equation (14) is identically satisfied and the equation (13) will also be satisfied if ~b(t) satisfies the Fredholm integral equation

Io' 4 , ( t ) - t 4 ) ( u ) K ( u , t ) d u = O < t < l , (18) /~o

where

I; g ( u , t) = [(; - s(~) - ~/2]Jo(r;U)Jo(¢t) d(; (19)

and I,( ) denotes the modified Bessel function of the first kind and n-th order. Similarly, assuming

B(t~) = ttO(t)Jl(,~t) dr, (20)

316 B. M. Singh and R. S. Dhaliwal

we find that the solution of the dual integral equations (15) and (16) is reduced to the solution of the following Fredholm integral equation of the second kind in qJ(t):

~b(t)+ u@(u)G(u, t) du= - LIx(at/2), 0 < t < l , (21) x/ l*o

where

io G(u, t) = [-~ + s(~) +/3/2]Jl(~u)J~(~t) d~. (22)

4. Expressions for the stress intensity factors and numerical results

From equations (10), (17) and (20), we find that the expression for the shear stress component o'yz at y = 0 may be written in the form:

tryz(x,O,z)=S[1-~-~o~oe~x/2d{IolCh~x~t)dt}-F(x)] , [x l> l (23)

where 1 oo

F(x)=,ff-~oe~X/2{Io 4)(t) dtlo [S(,)- ,+ /3/2]Jo(,t'cos (~x) d'

io io } + tO(t) dt [s(~)- ~+/3/2]Jl(~t) sin (~x) d~ , Ixl> 1. (24)

The expression (23) may be put in an alternative form:

°yz(x,O,z)=S[l+~-~o e~/z[dO(1)+xql(l)[ xxZff~--{-1 ~lt&'(t)+xO'(t)dt}-F(x)],xx2vr~-~_t2 Ix' >1-

(25)

where prime denotes the derivative with respect to the argument of the function. It is of interest to workers in fracture mechanics to calculate the stress intensity

factors at the ends of the crack. In this problem we find that the stress intensity factors at x = 1 , - 1 are respectively given by:

K1 = lim [~l)l~yz(X,O,z)}x>i]=~oe'~/2[4)(1)+t)(1)l, (26) x--~ 1+ L

lim [x/2(-x-X){o'yz(X,O,z)}~<_ll=ff-~oe-'~/z[ch(X)-@(1)]. (27) K _ l = x - -~ - 1 - t J

We solve the integral equations (18) and (21) numerically for the values of ~b(t) and t~(t) respectively for t = 0.0 (0.05) 1.0. The numerical values of ~(1) and tp(1) are then used to obtain the values of the stress intensity factors for all combinations of the values of c~ = 0.0, 0.05, 0.1, and/3 = 0.0 (0.1) 0.5. The numerical values of the stress intensity factors are displayed graphically in Figure 1 to demonstrate the effect

Asymmetr ic problem of a Griffith crack 317

0 ~1 I 0-0 0 I'̀ 1.2 01.3 01-4 0.5

B

Figure 1

of n o n h o m o g e n e i t y of the elastic material. I t is not iced that the stress intensity

factor K1 (at x = 1) is very nearly 1 for 0 - <o~<-1 and /3 = 0 where as the stress

intensity factor K_I (at x = - 1 ) is 1 for ~ =/3 = 0 but decreases considerably for

/3 = 0 as ~ increases. For /3 > 0.2 and 0 - - ~ -< 1, we notice that K1 and K_a vary almost linearly.

REFERENCES

[1] Kassir, M.K. and Sih, G. C., Three dimensional crack problems, Noordhoff International Publishing, Leyden 1975.

[2] Kassir, M. K., Boussinesq problems for a nonhomogeneous solid, Proc. ASCE, Jour. Engng. Mech. Div. 98 (1972) 457-470,

[3] Kassir, M. K. and Chuaprasert, M. F., A rigid punch in contact with a nonhomogeneous elastic solid, Jour, Appl. Mech., 42 (1974) 1019-1024.

318 B. M. Singh and R. S. Dhaliwal

[4] Popov, G. Ia., Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion, Jour. Appl. Math and Mech., 37 (1973) 1052-1059.

[5] Singh, B. M., The Reissner-Sagoci problem for a nonhomogeneous solid, Defence Sci. Jour., 22 (1972) 81-86.

[6] Singh, B. M., A note on Reissner-Sagoci problem for a nonhomogeneous solid, ZAMM, 53 (1973) 419-420.