Effective properties of power-law solids containing elliptical inhomogeneities. Part II: Voids

Abstract

The plane strain deformation of a power-law material containing a dispersion of randomly oriented elliptical voids is investigated. Constitutive relations are established first for solids containing a dilute concentration of voids. The essential step in the analysis is the solution for an isolated void, and this solution is obtained using a Ritz procedure developed in Part I of this investigation. In the procedure, trial displacement fields are derived using a displacement potential in elliptic-cylindrical coordinates, which allows the full range of void shapes from circular cylinders to cracks to be treated both accurately and efficiently. Approximate constitutive relations are then developed for nondilute concentrations of voids. For linearly elastic matrix materials, these relations are established using a differential self-consistent scheme, while for nonlinear matrix materials they are constructed using a procedure recently introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)). The accuracy of the approximate relations obtained with the latter procedure are assessed by specializing them to dilute concentrations of voids and comparing the resulting predictions with the accurate results obtained here.