Determination of stress distribution due to a crack with the help of dislocation layers

FatgmectingFractureMechanics Vol.15,No. I-2,pp.73-75,1981

0013-7944/81[06~073-03502.00/0

Printed in Great Britain.

© 1981Persmon Press Ltd.

DETERMINATION OF STRESS DISTRIBUTION DUE TO A CRACK WITH THE HELP OF DISLOCATION LAYERS

M. MAITI and R. PARAMGURU Department of Mathematics,Indian Institute of Technology,Kharagpur, India Aima'act--Dislocationlayers have been exploited in this note to determine the stress distribution in the n¢ighbourhoodof a crack openedby a non-uniformpressure. 1. INTRODUCTION LOWEN~RUB[1] has determined the stress distribution in the neighbourhood of an external crack opened by a non-uniform pressure. The crack extension condition has been derived recently for this problem in [2]. The problem, though three-dimensional in nature, has been reduced to a two-dimensional one with the assumptions pertinent to plane strain. Indeed, the authors have assumed the crack to occupy the region Ix[ >~a, y = 0 in an isotropic infinite plane. In this note we determine the stress distribution in the vicinity of a crack occupying the region x I> 0, y = 0, where the crack surfaces are subject to a non-uniform pressure. Thus it is, as in the earlier case, a mixed boundary value problem, where the normal stress is specified over a part of the x-axis and the displacement is specified over the rest. However, the present problem has been considered earlier by Barenblatt[3], who has based his analysis on complex variable techniques. The novelty lies here in the exploitation of dislocation layers to derive the solution. The formulations in terms of dislocation layers, when fitted into the boundary data, leads to an integral equation of Cauchy type whose solution is well known; see, e.g. Gakhov[4]. The present approach is straightforward and closed form solution has been derived without undue labour. Further, crack extension condition has been derived using Irwin's criterion[5]. Singular integrals appear very often in this note and are to be understood in the sense of the Cauchy principal value. 2. DISPLACEMENTS AND STRESSES IN A HALF-PLANE

If U(x) is the normal displacement specified along the line y = 0 in the absence of shear traction, then it has been shown in [6] that the displacements ui in the upper half-plane y > 0 are given by -

ux(x, y ) = - f : ® U'(x')[~log{(x-x')2+

u,(x,y)=f: U,(x,)[ltan_,(x-x'~+ \ y /

1 y2 y2}+ 2.(1-,)(x-x')'+

1.

y(x-x') ]

y2] dx',

2~r(1- v) (x - x') 2 + y2j dx',

(2.1) (2.2)

where z, is the Poisson ratio and U'(x) = dU/dx. The above displacements may be considered as due to a continuous distribution of edge dislocation of density 2(d Uldx) along the line y = 0. The corresponding stresses cri~in the upper half-plane are given by Orxx(X,

O'yy(X,

Y)

Y) -

~.(#_ ,,)._f-= utx)[ ...... r!x-{(x-x') x')/y~-(xx')~}ljdx', 2+y2}2 ~.(1¢.,- ,,)___:-:..,. ,. r(x- x'){(x- x?+ 3~,~1.1., u tx ) /

{(x - x') 2 + y2}2 2

(

P2

J ox,

o-.(x, y)= .(1 ~_ ,,)f_= ...... ryly - x- xT;Lr u tx J[{(x-xT+ y ~'JI dx', 73

(2.3) (2.4)

(2.5)

74

M. MAITI and R. PARAMGURU

where # is the shear modulus. In the limit as y~O, we obtain from eqn (2.4)

#

o'.(x. 0) = ~'(1 - v)

f ~ U'(x') dx' x-TZx '

(2.6)

which is a standard result in dislocation theory, see Bilby and Eshelby[7].

3. THE CRACK PROBLEM AND ITS SOLUTION

Consider an infinite isotropic plane containing a crack, which occupies the region x/> 0, y = 0 and whose surfaces are subject to non-uniform pressure p(x). Thus in this case the region x > 0, y = 0 is under normal (compressive) stress, whereas there is no displacement over its elongation, i.e. over the region x ~<0, y = 0. The stress distribution due to this crack may be determined by considering the following boundary value problem of the upper half-plane:

uy(x,O)=O,

x~
o'yy(x, O) = - p ( x ) ,

(3.1)

x > O,

(3.2)

Crxy(X,O) = O, - z < x < :c,

(3.3)

where p(x)> 0. Substituting from (3.1) into (2.6) we get

~,

~ryy(x,O) = rr(1 - u)

fo~ U'(x'_) d_x'

(3.4)

x' - x

whence, by applying (3.2), we derive the integral equation 1 / ' ~ U'(x') dx' _ J0 X'--X

1- v

¢3.5)

p(x)

in U'(x) for x > 0. Thus our problem reduces to that of solving integral eqn (3.5) which is of Cauchy type. The physics of the situation demands that we must seek a solution of this equation which is bounded at infinity but is unbounded at x = 0, i.e. at the crack tip. The integral eqn (3.5) is not one of the known types, but it can be reduced to a known one by suitable transformations. Setting x' = (a - t)/(a + t) and x = (a - s)/(a + s) in (3.5) we derive the integral equation 1

g,

/

~b(t) dt

-~ j _ o

t-s

=

~(s),

(3.6)

where 1

_ ,~a-t\

~O(t)=~--~-~ U ~--~--~),

1-v

[a-s~

~o(s)=#-(a--~-sipl-~--~Ts).

(3.7)

The solution of the integral eqn (3.6) is well known ([4], p. 428) and is given by

l (a+s~l/2 f

~I(S) = - - - ~ \ a -- S,I

a

( a - t ~ m~o(t)dt, \-d-~ / -~- s

(3.8)

which is correspondingly bounded at s = - a and unbounded at s = a. Returning to the original variables we now write (3.8) as

1 - v fo ~ k/(x')p(x') dx' U'(x) = zr~X/x x'- x

(3.9)

Determination of stress distribution

75

for x > 0. Substituting from (3.9) into (3.4) we derive ,r,,(x,O) =

1 fox V'(x')p(x') dx' ~rV- x x'- x

(3.10)

for x <0. From (3.10) we observe that

1

~r.(x, O) ~rV- x

fo ° p(x') dx' Vx'

(3.11)

near the crack tip on the negative x-axis. This behaviour has also been observed by Barenblatt[3]. The stress intensity factor ko at the crack tip is given by ko = X/(2~') lira X/(- x)tryy(x, 0) x-,O

,/(}) r

Jo

x/x'

"

(3.12)

Irwin's crack extension condition is given by ko2(1 - v2) = 2yE,

(3.13)

where 3' is surface energy and E is the Young's modulus. Then substituting from (3.12) into (3.13) we derive

[ Jo 7 ;

J:i-P'

(3.14)

which is the required crack extension condition for this problem. Finally, we may note that not all p(x) would ensure the convergence of the integral in (3.11), (3.12) and (3.14), e.g. if p(x)=p (a constant), the integral is not convergent. For the convergence of the integral a plausible necessary condition on p(x) might be that p(x) is bounded near the crack tip and p(x) ~ 0(x-~), a/> 1, at infinity.

REFERENCES [1] M. Lowengrub, Some dual integral equations with an application to elasticity. Int. J. Engng Sci. 4, 69-79 (1966). [2] M. Maiti, On the extension of an external crack under non-uniform pressure. J. Engng Fracture Mech. 11,603-605 (1979). [3] G. I. Barenblatt, Advances in Mechanics (Eds. H. I. Dryden, T. Von Kfirm~in and G. Kuerti), Vol. 7. Academic Press, New York (1%2). [4] F. D. Gakhov, Boundary Value Problems. Pergamon Press, Oxford (1%6). [5] G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate. J. Appl. Mech. 24, 361-364 (1957). [6] M. Maiti, B. Das and S. S. Palit, Somigliana's method applied to plane problems of elastic half-spaces. J. Elasticity 6, 429-439 (1976). [7] B. A. Bilby and J. D. Eshelby, Fracture (Ed. H. Liebowitz), Vol. 1. Academic Press, New York (1%8).

(Received 8 December 1980; received for publication 20 January 1981)