Analytical determination of cyclic hydrostatic stress-strain relations for a composite sphere with a soft inclusion and a hard bilinear, isotropically hardening matrix

Abstract

This paper deals with the analytical determination of cyclic hydrostatic stress-strain relations for an inclusion-matrix concentric sphere. Both phases are taken to be elastically isotropic, and the inclusion is taken as elastically softer than the matrix. The matrix is taken to be bilinear, and isotropic hardening is assumed. Yielding is assumed to occur in the matrix by the von Mises' criterion. Using Hill's [1] approach as a starting point, the exact solution is first determined for the first five sequences of loading (i.e., alternate tensile and compressive loadings). Based on the developed equations for the first five sequences and an inductive approach, the analytical relation for the overall hydrostatic stress and strain for the Nth loading sequence issuggested. With the developed equations the Bauschinger effect for the composite sphere is studied. Interestingly, it is seen that irrespective of the inclusion volume fraction, the relative stiffness of the soft inclusion/hard matrix or the work-hardening nature of the matrix, the composite response is initially governed by isotropic hardening, whereas an asymptotic response is approached where both kinematic and isotropic mechanisms play equally important roles. Such an evolution in the composite response is attributed to the evolution in internal stresses of the composite sphere.