The mixed-type integral equations for transient elastodynamic plane crack problems

Abstract

Abstract. In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems. I n t r o d u c t i o nIn fracture dynamics, the impact or transient loading problems possess high value of practice and theory. But due to mathematical difficulties, the solution of problems in classical elastodynamics remains an extremely complicated and difficult task. Only limited solutions in a closed form have been obtained for infinite domain problems. To obtain solutions for finite and irregular domains, there is a clear need for developing powerful theory and effective numerical techniques which can model arbitrary time-dependent loads and geometry.In special transient crack problems, the Griffith crack problem may be the most basic one. This problem was firstly considered by Sih and Chen [1 ]. They applied the Laplace-Fourier integral transforms and obtained a system of dual integral equations. In 1980's the method was further applied to some other special problems [2][3][4][5]. In general ones, the pure numerical methods, finite difference method (FDM) and finite element method (FEM) have been applied with some success to solve dynamic problems of cracks. But some difficulties in using FDM and FEM for fracture dynamics have been pointed out [6]. The boundary integral equation method (BIEM) provides an efficient numerical approach towards crack analysis. However the conventional displacement BIE formulation gives rise to degenerate integral equations for crack problems and is not suitable for numerical solutions. For dynamic crack problem, the ordinary BIEM was firstly used by Fan and Hahn [7], Sladek and Sladek [8], Chirno and Dominguez [9], where the domain has to be divided into subdomains by means of a cut along the crack. To circumvent the difficulties in using the ordinary integral equation method, several other approaches have been proposed; see for example, the papers by Cruse [10], Sladek and Sladek [11 ], Nishimura and Kobayashi [ 12]. Most of these studies first reduced the high order singularities to integrable ones, and then solved the modifi...