Singular integral equations of two-dimensional problems of the theory

4. 5. 6. 7. 8. 9.

V. V~ Panasyuk, "Deformation criteria in fracture mechanics," ibid., No. i, 7-17 (1986). P . M . Vitvitskii, V. V. Panasyuk, and S. Ya. Yarema, "Plastic deformations in the vicinity of cracks and criteria of failure (a review)," Probl. Prochn., No. 2, 3-18 (1973). V . V . Panasyuk, The Limiting Equilibrium of Elastic Bodies with Cracks [in Russian], Naukova Dumka, Kiev (1968). N . I . Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow (1966). M . P . Savruk, Two-Dimensional Problems of Elasticity for Bodies with Cracks [in Russian], Naukova Dumka, Kiev (1981). S. Pashkovskil, Computer Applications of Polynomials and Chebyshev Series [in Russian].

SINGULAR INTEGRAL EQUATIONS OF TWO-DIMENSIONAL PROBLEMS OF THE THEORY OF ELASTICITY FOR BODIES WITH EDGE SLITS UDC 539.375

M~ P. Savruk

The basic boundary plane and antiplane problems of the theory of elasticity for a multiply connected area containing curved slits and holes of arbitrary form have been reduced [1-3] to a system of singular integral equations of the first order for the closed (contours of the holes and external boundary) and open (slits) contours. It was assumed that the contours of the slits and holes do not intersect each other. Edge cracks were considered only in certain particular cases of the boundary contour (circle, straight llne) when it was possible to construct modified singular integral equations not containing the sought-for functions on this contour [i, 4]. Recently problems for an arbitrary symmetric area with an edge crack located on the axis of elastic and geometric symmetry have also been studied [5-7]. Below the results of [1-3] are extended to the case of a multiply connected area with slits and holes when one or both ends of the slits may enter onto the outer boundary and the contours of the holes. Numerical solutions were obtained for the integral equations constructed in uniaxial tension of an infinite plane with one or two round holes into the contours of which enter radial cracks. Formulation of the Problem and Reductinn of It to Integral Equations. Let the area S (Fig. i), which is occupied by an elastic isotropic body, be bound by one or more contours LI, L2, ..., LM, Lo, where the first M contours are located outside of one another and the last envelops all of the others. In this case the contour L k (k = 0, M) consists, generally speaking, of the contours L k' (contour of the hole from which the edge cracks originate or the combination of contours of the holes connected by the slits) and L k" (combination of the contours of the edge slits and/or of the slits connecting the contours of the holes). In addition, in the area S there are N-M curved isolated slits L k (k = M +i, N). The positive direction of by-passing of the contours Lk(k = i, M)and Lo will be assumed to be that with which the area S remains to the left (Fig. i). Let L = L' +L", where L' is the combination of contours of the holes and the outer boundary (without the edge slits) and L" is the combination

Fig. 1 G. V. Karpenko Physicomechanical Institute, Academy of Sciences of the Ukrainian SSR, Lvov. Translated from Fiziko-Khlmicheskaya Mekhanika Materlalov, Vol. 23, No. i, pp. 61-67, January-February, 1987. Original article submitted November 26, 1985.

58

0038-5565/87/2301-0058512.50

9 1987 Plenum Publishing Corporation

of .... the isolated and edge slits and also of the slits connec_ting the holes, thatis, L = U L ~ O, N), L'=UL~' ( k = O .~I) a n d L " = U L ~ (k=O, M ) f f - U L ~ ( k = M + I , N ) .

(k=

Let us consider the first basic problem of the plane theory of elasticity, when on the boundary of the area L are given the stresses

N-+-iT=p*(t),

t~L';

N +iT

=p*(t)-4-q(t),

t~L'.

(l)

Here N and T are the normal and tangential components of the external forces and the superscript "+" ("--") indicates the limiting value of the function in approaching the slit from the left (right). In this case there occur the relationships = O, L'

L"

(2)

L'

which express equality to zero of the main vector and the main moment of the external forces acting on the contour L. Let X k and Yk be the projections on the Ox and Oy axes of the main vector of forces applied to the contour of the hole (or holes) L~, that is,

X ~ + i F k = i ~ p ; ( t ) dt. ,J L~

Then the complex potentials of the Kolosov-Muskhelishvili represented in the form

9 , (z)

1

-

2~(1 H-x)

;r., (=~ -

[8] in the area S may be

a~ Xk + iy~ v +- r (z),

~1

z--z~

(3)

.u X~, iYk v + ~" (z), ~ Z--Z~ -

" - ( 1 --' "z)

stresses

-

where the functions r and ~(z) are holomorphic in the area S and z k are points arbitrarily located within the contours L~(k = i, M). We will search for the solution of the boundary problem (!) in the form [1-3]

q, (z) = - - I i _ _llt) Q ~t, .... ,y t - - z 1 ~~

.~.k'

~!" (r) =Or.,"__. ~ ; ( Z - - G ) "

Q (:) -- e' U) - - -2i - - , (t), Iq-x

._;. 1 ' ."--', ,,

Q(t)-giq{o t" Z

(4)

dt ,dt

,

L

where k =- 1, .;r; r'

q(t) = 0 with t~L', G is the shear modulus, • 3-4~ for plane strain and • = (3-~)/(i+~) for the plane stresses state, ~ is Poisson's ratio, g(t) is the sought-for function expressed for the slits L therough the vector of the jump in displacements u and v, and

g(t) = - - 2 i G [ ( u + i v ) + - - ( u + i v ) - ] / ( l + • Having substituted the potentials the singular integral equation

t~L'.

(3) and (4) in the boundary condition

(i), we obtain

59

~

L Fig.

/

2

/p."

/

(,r

2t

7--//,Z--/-Fig. 3

.1 3" {2

t \~ "~ A~Mk d~ + ~ . a . ds= Q(t~+_i._q(t)dl+k~(t,~)[Q(t)+2iq(t)]dt+k2(t,~)Q(t)dt}---

L

~L,, n=O,N

p (z),

- --

(5)

for determination of the unknown function g'(t). Here ~n = i with n = , ~ M , 6n = 0 with n = M--~I, N, Akn = i + (Sn--l)Sko; 8kn is the Kronecker symbol, s is the arc abscissa corresponding to the point r, zo = 0 E S ,

k,(t,")=(d/d0 {In [(t -- =)/~---~)]}; 1 and

(d/dt~{(t -- "O.(t -- ~)} ; ~ {2ReXk+iYk

k2 (r =) = --

P (x) = p* ('=) ~ 2= (1 + z)

~,

" = - - z~

(6)

- [~ (x~ - iyO,.'(': - z~) ~-+ ,, (x~ + i r O / ( ~ - z f i l ( ~ . a ~ ) To the left portion of Eq. (5) are added the functionals

,u,,=-2lm

t, i g ' ( t ) dt

a.=

'

i'

i' g ' ( t ) . J t

(n=~,M

o/

(7)

L,;

Then for any right portion Eq. (5) has a single solution for fulfillment of the additional conditions

~ r (t)dt=,O,

k=Al"[- I,N,

L~

resulting from uniqueness of the displacements in by-passing of the contours of the isolated slits. With fulfillment of conditions of equilibrium (2) the functionals (7) are equal to zero, so that the Eq. (5) obtained gives the solution of the problem formulated. Equation (5) was constructed by passage to the limit from the solvable singular integral equation, which was written fro a multiply connected area containing isolated slits and bound by the smooth contour Le [1-3], when Lg tends toward the initial boundary contour L. Let us illustrate this passage to the limit in the case of the simply connected area S with

60

one edge slit (Fig. 2). Let us assume that on the external boundary L' and on the edges of the slit L" are given the stresses (i) and q(t) = 0 with t ~ L ~. Let us consider the auxiliary boundary problem for a simply connected area bound by the smooth contour Ls (Fig. 2), on which acts the self-balanced load:

N+iT=p(t), Such a p r o b l e m r e d u c e s to t h e e q u a t i o n 1 S {[

t~L~.

[1-3]

2 .+k,(/,~)]g;(t)dt+k2(t,~)g',(t)dt}--

(8)

Lt -- M, (dZ/d~.),.'(2~ 7) + a, (ds/d~) = p (~), : ~ Z,, where the regular nuclei kx(t, r) and kz(t, z) are given by Eqs. (6) and

"'

....

=--21m

\ g:(O tdt

t'

a, =

- ,

g ; ( t ) at.

tJ L~

tJ L~.

E q u a t i o n (8) i s s o l v a b l e f o r any r i g h t p o r t i o n p ( t ) and t h e f u n c t i o n a l s Me and a~ a r e e q u a l to zero when the external load p(t) is self-balanced. It II We will assume Chat L~---~L'-~-L~+L"__ where L+ and L_ are the left and right edges of the slit L". Then the integrals in Eq. (8) are transformed in the following manner:

il

L~

+ .i I... L

L'

L'+

f ll

"

1

+i i I

L'

~

L"

L

Here is introduced the designation g'(t) = g . i ( t ) - - g ' - ( t ) , tEL". Consequently in the passage to the limit we obtain a solvable singular integral equation corresponding in form to Eq. (8) if in the latter we replace L E by L and g~(t) by g'(t). In the case when q(t)#0 with t~L", to the equation obtained are added the known terms containing q(t) (Eq. (5)), which obviously does not disturb its solvability. The integral Eq. (5) constructed differs from the known equation in the absence of edge slits [1-3] only in the fact that in the functlonals M n and a n (n = 0--~M) to the contours of the holes are added the contours of the corresponding edge slits. The singular integral equations of other boundary problems may be extended in a similar manner, including of the second boundary problem of the plane theory of elasticity [i-3], of basic analytical problems of the theory of elasticity [i, 3], and of plane problems of thermal conductivity and thermal elasticity [I, 3, 9 ] . On the basis of Eq. (5) below are obtained numerical solutions of problems in the case of an infinite plane weakened by one or two round holes onto the contours of which enter radial cracks. TABLE i

0,5 FI [ FII ~/12 ~/6 ~[4 n/3 5n/12 n/2

--0,115 0,247 0,741 1,234 1.596 1,728

0,276 0,475 0,547 0,475 0,276 0

FI

1,0 [ FII

0,012 0,266 0,613 0,950 1,213 1,306

0,300 0,519 0,599 0,519 0.300 0

2tlR 2,0 FI i FII 0,059 0,250 0,510 0,771 0,961 1,031

0,281 0,487 0,552 0,487 0,281 0

1 4,0 ! FI i FII 0,063 0,223 0,44t 0,660 0,819 0,878

0,247 0,428 0,494 0,428 0,247 0

FI

I0,0 [ FI[

0,~55 0,197 0,391 0,585 0,727 0,779

O,211 0,355 0,422 0,365 0,211 0

61

O~ ~o

L

2,0 2,5 3,0 3,5 4,0

hIR

TABLE

2

Fig. 4

1,5 2,0 3,0 4,0

hlR

10.4

1,844 1,786 1,752 1,746

0,5

[

[

i

L414 1,365 1,333 1,322

1.0

1,922 1,591 1,400 1,277 1,191

1,134 1,092 1,062 1,051

2,0

0,984 0.917 0,917 0.904

4,0

1,861 1,532 1,349 1,233 1,154

[ 0.5 10.6

2t/R

2,460 2,206 2,034 2,140 1.869 1,698 1,920 1,654 1,496 1,757 1 , 5 0 5 1,361 1,627 1,394 1,265

I 0,2 10,3

TABLE 3

2,110

2,242

2,819 2,581 2,394

o.,

2t/(h- R)

03 97 0,866 0,837 0,822

1o,o

1,856 1,520 1,338 1,226 1,149

10,7

1,933 1.574 1,384 1,269 1,190

[ o,8

Fig. 5

1,793 1,571 1,437 1,347

2,221

0,9

Uniaxial Tension of a Plane Weakened b~ a Round Hole with an Edge Radial Crack. Let the contour of the hole and the edges of the crack be free of load and at infinity the plane is in tension by the forces p directed at an angle y to the line of the crack (Fig. 3). Since the area is infinite, the contour Lo is absent. To the potentials (4) must be added the functions ~o(z) = p/4 and ~o(z) =--p exp(--2iy)/2, which determine the stressed state in the continuous plane. Then the problem reduces to Eq. (5) with n = i (M = N = i) and q(t) = O, p(r) = --(p/2)[l--exp(2iy)(d~/dr)]. We obtain the numerical solution of this equation by the method of mechanical quadratures [i]. As the result we arrive at a system of no +n~ linear algebraic equations

~=0

k=l

m---1,no; p----l, r e = l ,

(p=O,

.4,

,=~

relative to no +nl unknowns signations

=p~,r

p,

(-

~)~.,(~)t~

4n~

/

nt--1),

I

~ --o

Uo(~o) (k=l, no) and u1(~)

(9)

(k=l, nO ~

Here are introduced the de-

= -- (1/2) [1 -- e x p ( 2 i 7) ~,(~t,)/ p(~qp)l, Ao--'2/no, A~ = 1/nL,

~vt,.. (o~. -.~. a) = (1/2) {1/(o~ - ~,) + [1/(~. - .~) - o , / ( ~ , - a)~l ,.'p ( ~ " ) / ~ t~") + !~j, ('~')I/to'p (.~')}.

N.. (O.. ~ . a) = (112) { l f(~. - u - [(O. -- .~)I(~. - - ' ; , ) ' -e,l (s ~ (~,;')I~'.(,,;)>, o~ = ,~ ( ~ ) + b~, "b = ",, ( ~ ' ) + O~, a = bo = O, b, = 1 + ,q/Z, mo (~) = (R/l) exp (i~), m, (~) = L ~ = 2ink/no, rtg'= (2m - - 1) ~:/no, ~ = cos ((2k - I) r."2n~), "qf = cos (,':mint) (no=2, 4, 6 . . . . ; r e = l , 2, 3 . . . . ). The stress intensity factors at the crack tip are determined from the equation n

K,-it