Variational self-consistent estimates for cubic viscoplastic polycrystals: the effects of grain anisotropy and shape

Abstract

A fundamental problem in mechanics of materials is the computation of the macroscopic response of polycrystalline aggregates from the properties of their constituent single-crystal grains and the microstructure. In this work, the nonlinear homogenization method of deBotton and Ponte Castañeda (1995) [deBotton, G., Ponte Castañeda, P., 1995. Variational estimates for the creep behavior of polycrystals. Proc. R. Soc. Lond. A 448, 121–142] is used to compute “variational” self-consistent estimates for the effective behavior of various types of cubic viscoplastic polycrystals. In contrast with earlier estimates of the self-consistent type, such as those arising from the “incremental” and “tangent” schemes, the new estimates are found to satisfy all known bounds, even in the strongly nonlinear, rate-insensitive limit, and to exhibit a more realistic scaling law at large grain anisotropy. Also, unlike the Taylor and Reuss estimates, they are able to account for grain shape in a rigorous statistical sense. For these reasons, they can be shown to be significantly more accurate than earlier estimates. Thus, for example, the new self-consistent estimates can be less than half the corresponding Taylor estimates for ionic polycrystals with highly anisotropic “flat” grains.