Homogenization estimates for the average behavior and field fluctuations in cubic and hexagonal viscoplastic polycrystals

The “second-order” homogenization procedure (J. Mech. Phys. Solids 50 (2002) 737) is used to compute estimates of the self-consistent type for the effective response of cubic and hexagonal viscoplastic polycrystals with isotropic textures. The method, which requires the computation of the averages of the stress field and the covariances of its fluctuations over the various grain orientations in an optimally selected “linear comparison polycrystal,” is also used to generate information on the heterogeneity of the stress and strain-rate fields within the polycrystals. In contrast with earlier estimates of the self-consistent type, such as those arising from the “incremental” and “tangent” schemes, the new estimates for the effective behavior are found to satisfy all known bounds, even in the strongly nonlinear, rate-insensitive limit. In addition, they are found to satisfy a recently proposed scaling law at large grain anisotropy. The fluctuations of the stresses and strain rates, which are nonzero for all grain orientations, are found to generally increase with decreasing strain-rate sensitivity (i.e., increasing nonlinearity) and with increasing grain anisotropy (which is typically higher for lower-symmetry systems).