Procedure of the overdeterminisctic method for finding the field expansion coefficients at the crack tip based on a finite element solution for the stress tensor components

Abstract

The article proposes and implements a procedure for reconstructing the asymptotic series expansion of stress, strain and displacement fields in anisotropic materials, generalizing the Williams solution for linearly elastic isotropic materials, based on a finite element solution to the problem of deforming a sample with a defect in an anisotropic orthotropic material in the approximation of a plane problem of elasticity theory. The stress field expansion coefficients near the crack tip in an anisotropic material are determined using an overdeterministic method originally proposed to reconstruct the asymptotic expansion from experimental data of a photoelastic study. In this paper, this method is extended to anisotropic materials with various types of symmetry and the novelty of the proposed approach lies in the reconstruction of the asymptotic expansion from the finite element solution for the stress tensor components in the nodes of the finite element grid, which allows us not to exclude their displacement fields components corresponding to the displacement of a body as an absolutely solid body. In the proposed approach, it is possible to use data from finite element calculations directly in the scheme of the overdeterministic method. It is shown that the coefficients of higher approximations are reliably determined by an overdeterministic method based on the stress field found from finite element analysis.