ON CRACK PROBLEMS IN AN INFINITE PLANE CONSISTING OF THREE DIFFERENT MEDIA

Abstract

The mathematical problem of an infinite elastic plane consisting of three different media with an arbitrary number of cracks is considered. It is reduced to singular integral equations along the interfaces and the cracks by a constructive method. Those along the interfaces are further reduced to Fredholm ones.There were many works on crack problems in an infinite plane which is composed of two half—planes with different materials, for example, [1, 2]. The most general case of such problems (with cracks not intersecting to the interface) was considered in [3] theoretically and with applications and numerical examples in [4]. In practice, we are often confront with the similar problems when the plane consists of three different materials. The method proposed by us in [3, 4] is extended here to solve such problems. They may be reduced to certain boundary value problems of sectionally holomorphic functions and then be reduced to certain singular integral equations along the cracks as well as along the interfaces. Using the results for solving a special kind of singular integral equations, we finally reduce the problem to solve singular integral equations, singular on the cracks and Fredholm on the interfaces.