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The evolution of an isolated elliptical–cylindrical void contained in a power-law viscous matrix is investigated. For linearly viscous solids, the void evolves through a sequence of (rotating) ellipses, and Eshelby's equivalent inclusion method is applied to exactly determine the history of shape, orientation and volume of the void as a function of time or remote strain. For non-linearly viscous matrix materials, the void evolution is idealized as proceeding through a sequence of elliptical shapes, and a numerical procedure developed by Lee and Mear (Lee, B.J., Mear, M.E., 1992. Mechanics of Materials 13, 337) is utilized to simulate finite deformations of the void. Detailed studies are presented first for the growth and collapse of initially circular voids, after which the role of initial void shape is investigated by considering the evolution of initially elliptical voids. Results are also presented for the evolution of the void volume fraction for solids containing a dilute concentration of randomly oriented elliptical voids. The findings of the study have relevance to modeling ductile fracture as well as to modeling the final stages of the densification of powder metal compacts.

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.

The plane strain deformation of a power-law material containing a dispersion of rigid elliptical inclusions is investigated. Accurate constitutive relations are established for dilute concentrations of inclusions over a wide range of martrix hardening exponents and inclusion aspect ratios. The essential step in the analysis is the solution of the kernel problem for an isolated inclusion, and this is obtained using a Ritz procedure in which trial displacements are derived from a displacement potential in elliptic-cylindrical coordinates. Approximate constitutive relations for nondilute concentrations of inclusions are then established using a differential self-consistent scheme. The study is primarily concerned with randomly oriented inclusions, but limited results are also presented for aligned inclusions. As part of the investigation, the procedure introduced by Ponte Castaneda (J. Mech. Phys. Solids (1991)) is evaluated. This is a variational procedure which exploits information about a linear solid to obtain bounds or estimates for the overall response of nonlinear solids with the same microgeometry of second phase. Finally, connection is made with recent results for spheroidal inclusions given by Lee and Mear (J. Mech. Phys. Solids (1991); Int. J. Solids Struct. (1991)), and an ad hoc estimate for the overall response of solids containing randomly oriented rigid, spheroidal inclusions is proposed.