Analysis of plastic stress singularities of cracks and wedges under the plane stress conditions

Abstract

The aim of this paper is to analyze the plastic stress singularities of the V-shaped notches and cracks under the plane stress conditions and mixed mode. Based on the formulation of assuming an asymptotic displacement field with respect to the radial coordinate centered at the notch tip, the stress components near the notch tip are expressed as the asymptotic series expansions in term of the radial coordinate by satisfying the plastic yield criterion and the nonlinear stress-strain relation. Then, the series expansions of the stress and displacement components are substituted into the governing differential equations of the plastic theory. Hence, the nonlinear second-order system of ordinary differential equations (ODEs) with the plastic stress singularity orders and the associated eigenfunctions are deduced. The interpolating matrix method is applied to solving the nonlinear eigenvalue problems of the ODEs by an iteration procedure. Thus, the leading plastic stress singularity orders and the associated eigenvectors of the displacement and stress fields in the notch tip region are achieved simultaneously. In addition, the numerical examples are given to demonstrate the accuracy and the effectiveness of the present approach to solve the plastic stress singularities of the plane V-shaped notches and cracks. Furthermore, some mechanical behaviors related to the plastic stress singularities in the notch tip region are investigated.