In this paper, the mathematk: al problem of the second fundamental problem of composite materials with a doubly periodic set of arbitrary shape cracks are investigated, and the interface are arbitrary smooth closed contours. At first, we establish mathematical models by using Muskhelisvili complex variable methods, change the primitive problems into searching complex stress functions whwh satisfy four boundary value problems and construct forms of the solution. then, under some general restrictions it is reduced to normal type singuiar integral equation, the unique olvability is proved mathematically.

本支利用复势方法讨论三维各向同性不同材料带双周期孔洞的弹性焊接混合边值问题。将三维应力系统分解为线性独立的两组二维应力系统。然后把复Airy函数法用于推广IIIepMab方法，将寻求复应力函数的问题归结为求解正则型的奇异积分方程，并证明了其解的存在唯一，由于所用方法是构造性的，故有利于数值计算。

The effect of a holnomeneous cvlindrical inlay Oll cracks in the doubly-periodic complete. 1ane strain (CPS)problenl is investigated in this paper By employing the solutions of doubly quasi-periodic and doubly-l：eriodic Rielllann boundary value problelllS for analytic thnctions we obtain the general solution in closed f01T11 And apl：rroximate analytical expression of the stress intensity factors which are identical with the known results are obtained when there are no holes or gaskets in the periodic parallelogram and e3=0，Fk=0，Tk=0，k-1,2 with constantload al：r01ied atthe edges ofthe crack

In this paper, we wish to use complex potential methods to solve the fundamental complete plane strain (CPS) problems of a three-dimensional nonhomogeneous elastic body with a doubly-periodic set of cracks in, the x1, x2plane. We resolve the complete plane strain state, which is a special three-dimensional elastic system, into two linearly independent two-dimensional (plane) elastic systems by the superposition principle of force. Based on a suitable modifieation of Cauchy-type integrals, which is defined by the replacement of the Cauehy kernel 1/(t-z) by the Weier.stra.s.s zeta function S (t-z), the general representation for the solution is constructed, under some general restrictions the boundary value problem is reduced to a normal type singular integral equation with a Weierstrass zeta kernel, and the existence of an essentially unique solution is proved.

The problem of embedding a periodic continous elastic gasket in a periodic cracked body is investigated by approaches using the singular integral eqution (SIE) with the Hilbert kernel, the periodic solution in closed form of the SIE and analytical experession of the stress intensity factors are obtained. Finally, some special examples are given.