A nonlinear finite elment eigenanalysis of singular stress fields in bimaterial wedges for plane strain

Abstract

A displacement-based finite element formulation for the analysis of singular stress fields in power law hardening materials under conditions of plane strain is presented. The displacement field within a sectorial element is quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. A hydrostatic pressure variable, which is linear in the angular coordinate, is introduced to account for the incompressibility of the material. The Newton method is combined with matrix singular value decomposition to iteratively solve the resulting nonlinear homogeneous eigenvalue problem where the eigenvalues and eigenfunctions are obtained simultaneously. The examples considered include the single material wedge, the bimaterial interface crack and the bimaterial wedge. In particular, the case of a single material wedge bonded to a rigid material along one edge is examined to study the possibility of the existence of mixed mode solutions for arbitrary wedge angles, including the important case of an interface crack when the wedge angle is 180$\circ$. This behavior is distinctly different from that of plane stress where a complex singularity is obtained. The possibility of the existence of nonseparable solutions is also discussed.