Constitutive Modeling of Viscoplastic Porous Single Crystals and Polycrystals: Macroscopic Response and Evolution of the Microstructure

Porosity can have a significant effect on the overall constitutive behavior of many materials, especially when it serves to relax kinematic constraints imposed by the underlying matrix behavior. In this study, we investigate the multiscale, finite-strain response of viscoplastic porous single crystals and porous polycrystals. For these materials, the presence of voids leads to highly nonlinear dilatational behavior for loads with a large hydrostatic component, even though the matrix material itself is essentially incompressible. In this study, we employ the recently developed fully optimized second-order homogenization approach, along with an iterated homogenization procedure, to obtain accurate estimates for the effective behavior of porous single crystals and porous polycrystals with fixed states of the microstructure. The method makes use of the effective properties of a linear comparison composite, whose local properties are chosen according to a suitably designed variational principle, to generate the corresponding estimates for the actual nonlinear porous materials. Additionally, consistent homogenization estimates for the average strain-rate and spin fields in the phases are used to develop approximate evolution equations for the microstructures. The model is quite general, and applies for viscoplastic porous single crystals and polycrystals with general crystallographic texture, general ellipsoidal voids, and general ellipsoidal grains, which are subjected to general loading conditions. The model is used to study both the instantaneous response and the evolution of the microstructure for porous FCC and HCP single crystals and polycrystals. It is found that the intrinsic anisotropy of the matrix phase—either due to the local crystallography in single crystals or to the texture of polycrystals—has significant effects on the porosity evolution, as well as on the overall hardening/softening behavior of the porous materials. In particular, the predictions of the model for porous single crystals are found to be in fairly good agreement with the full-field, numerical results available in the literature. The results for porous polycrystals suggest that the macroscopic behavior is controlled by porosity growth at high stress triaxialities, while it is controlled by texture evolution of the underlying matrix at low triaxialities