PMM U.S.S.R.,Vol.46,pp. 116-124 Copyright Pergamon Press Ltd.1983.Printed in U.K. 0021-8928/83/l 0116 $7.50/O UDC 539.383 zyxwvutsr ASYMPTOTICSOLUTIONSOF CONTACT PROBLEMSOF ELASTICITY THEORY FOR MEDIA INHOMOGENEOUS IN DEPTH* S.M. AIZIKOVICH There are considered the dual integral equations generated by contact problems for half-spaces and half-planes inhomogeneous with depth. There is extended the method from /l/ for the construction of asymptotic solutions of the problems under consid- eration. Correctness of classes and the solvability of equations are established, and the approximate method proposed for their solution is given a foundation. 1. Contact problems for half-spaces and half-planes ix-homogeneous with depth reduce, in a number of cases /i-44/, to finding the solution of a dual integral equation of the form (h is a geometric parameter) ~*(a)P(a)L~).)LJ(a,;c)~~=-f(:c), jz[<l (1.1) r T(a)B(a, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG z)da=O, Ix]> 1 In particular, in the problem of shear of an inhomogeneous half-space by a stamp /2/ (problem 1) and in the problem of impression of a stamp in an inhomogneous half-plane (problem 2) p (a)= 1 a I-l, B (a, r) = Ziax, c = -d = 00 where (1.1) is considered with the additional condition ? m s r(Z)dr=P, T(Z)=& s t 7'(a) e-iaxd~, s ~(E)e'akdf= T(a) zyxwvutsrqponmlkjihgfedc (1.2) -1 -02 -1 and P is the shearing force (problem 1) or the impressing force (problem 2) acting on unit length of stamp. Find T(X). For problems on the torsion by a circular stamp /3/ (problem 3) and on the impressionof a circular stamp in an elastic half-space that is inhomogeneous with depth /4/ (problem 4) p (a) = a-', B (a,5) = aJk (ax), c = zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO 0, d = 00 k = 1 for problem 3, andk = Ofor problem 4. Find T(X). Here T(z)=fZ'(a)aJ,(a.r)dc, T(c)=ir(p)Jl(ap)pdp 0 0 Upon satisfying the conditions min G(y)>cl>O, max G(y).<c<-~ uao, =) l/E(O> a) lb G(y)=const (problems 1 and 3) U-m min 13 (Y)> cl 0, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA max 8 00 baa. -1 > (y) <c < uao. -) 0 6) = 2P (I/)P (I/)+ CL zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE (Y)l [h(Y) + 2P (Y)l limCl(Y)=const (problems 2 and 4) I-" where G(y) is the shear modulus and p(Y) and h(y) are Lan& coefficients of the half-space,and Y is the distance from the surface of the medium, it can be shown /2-4/ that the transforms of the kernel ~(u)possess the following properties (B and D are constants): *Prikl.Matem.Mekhan.,46,No.1,148-158,1982 116