Analysis of Elastic Media with Voids Using a Mixed-Collocation Finite-Element Method

Abstract

In this paper, a recently developed type of lower-order mixed finite elements is extended to model porous materials based on the microdilatation theory. These mixed finite elements are based on assuming independent linear generalized strain fields and collocating them with the generalized strains derived from primal variables (mechanical displacements and change in matrix volume fraction) at some cleverly chosen points within each element. This mixed formulation is very effective in alleviating the shear locking problem that regular lower-order finite elements suffer from. Hence the accuracy of the predicted mechanical fields (such as displacements and stresses), as well as the fields coupled with them (such as change in matrix volume fraction, which is also called microdilatation), is improved over regular finite-element formulation. The mixed-collocation formulation is also superior over other types of previously published hybrid-mixed finite-element formulations in that it avoids the Ladyzenskaja–Babuška–Brezzi (LBB) stability conditions completely because it does not include any Lagrange multipliers. The paper also presents some numerical examples that help in providing more insight on the effect of porosity-related parameters used in microdilatation theory on the behavior of porous materials. Finally, the paper defines two limits on the coupling number; the first considers the positive definiteness of the stored energy density, whereas the second sets the limit between auxetic (having negative Poisson’s ratio) and nonauxetic material behavior.