本文给出了一类R iemann边值逆问题的正则型与非正则型情况的数学提法和解法1MR（1991）主题分类45E

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.

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.

The numerical solutions of the periodic crack problems of anisotropic strip are investigated in this paper. Using the complex variable method the mathematical model, boundary value problems for analytic functions, is established. After constructing appropriate integral transformations the boundary value problems are transformed into integral equations. Employing Lobotto-Chebyshev quadrature formulas and Gauss quadrature formulas, the approximate analytical expressions of the stress intensity factors are obtained. And some numerical results are performed for several special cases.