Numerical methods for extracting edge stress intensity functions in anisotropic three-dimensional domains

Abstract

The solution to elasticity problems in three-dimensional (3-D) polyhedral domains in the vicinity of an edge is represented by a family of eigen-functions (similar to 2-D domains) complemented by shadow-functions and their associated edge stress intensity functions (ESIFs), which are functions along the edge. These are of major engineering importance because failure theories directly or indirectly involve them. In isotropic materials one may compute analytically the eigen-functions and their shadows [Z. Yosibash, N. Omer, M. Costabel, M. Dauge, Edge stress intensity functions in polyhedral domains and their extraction by a quasi-dual function method, Int. J. Fract. 136 (2005) 37–73], used in conjunction with the quasi-dual function method [M. Costabel, M. Dauge, Z. Yosibash, A quasi-dual function method for extracting edge stress intensity functions, SIAM J. Math. Anal. 35 (5) (2004) 1177–1202] for extracting ESIFs from finite element solutions. However, in anisotropic materials and multi-material interfaces the analytical derivation becomes intractable and numerical methods are mandatory. Herein we use p-finite element methods ( p-FEM) for the computation of the eigen-pairs and shadow functions (together with their duals). Having computed these, the p-FEM is used again to obtain a FE solution from which we extract approximations of the ESIFs based on a family of adaptive hierarchical Jacobi polynomials of increasing order. Numerical examples for 3-D isotropic and anisotropic materials are provided for which the eigen-pairs and shadow functions are numerically computed and ESIFs extracted. These examples show the efficiency and high accuracy of the numerical approximations.