The influence of non-singular terms on the precision of stress description near a sharp material inclusion tip

A theoretical elastic stress field in the vicinity of a sharp material inclusion tip has a singular character. The power of a stress singularity, characterized by the exponents of the singularity, is different from the power of a singularity in the case of a crack in homogenous media. Stress distribution near the singular point can be described by an asymptotic expansion. The series consists of singular and non-singular terms, depending on the eigenvalue 2 in the exponent of each term. The singular terms are characterized by 0 < R(lambda) < 1, while the relation 1 < R(lambda) applies to non-singular terms. A sharp material inclusion is modelled as a special case of a multi-material junction, a bi-material junction. A method to calculate eigenvalues for a problem of a bi-material junction of given boundary conditions is described. Then based on the knowledge of eigenvectors, the eigenfunctions can be formed and Generalized Stress Intensity Factors (GSIFs) obtained by the Overdeterministic Method (ODM). The ODM returns GSIFs as a least square solution of a system of linear equations, which consists of analytical relations and a large amount of Finite Element Method (FEM) displacement results. In the majority of fracture mechanics analyses of cracks and notches, only the singular terms are considered to assess the stability of these general singular stress concentrators. This article presents results for four bi-material combinations of a sharp inclusion and a matrix in the form of stress plots. Stress plots show an analytical solution with the use of (i) singular terms, and (ii) singular and non-singular terms. Analytical solutions are then compared to pure FEM results. Further, the absolute and relative errors between both cases of analytical description and an FEM solution are calculated and plotted. The effect of an elastic moduli mismatch and the distance from the inclusion tip is investigated and quantified by mean absolute error of tangential stress. (C) 2017 Elsevier Ltd. All rights reserved.