Variational self -consistent estimates for the effective behavior of viscoplastic polycrystals

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) was used to compute “variational” self-consistent estimates for the effective behavior of different types of viscoplastic polycrystals, including a two-dimensional model, as well as cubic and hexagonal polycrystals. In contrast with the “incremental” and “tangent” self-consistent estimates, the new results are found to satisfy all known bounds, even in the strongly nonlinear, rate-insensitive limit. The new results also exhibit a more realistic scaling law for the macroscopic tensile flow stress at large grain anisotropy, depending only on the number (less than five) of independent slip systems, but not on the strain rate sensitivity. The predictions of other nonlinear extensions of the self-consistent method were found to be inconsistent with this scaling law, suggesting that they may be less accurate than the variational self-consistent estimates proposed here. One additional advantage of the variational procedure over earlier self-consistent models is that it is not restricted to pure power law behavior and can be applied to polycrystals with various hardening rules. In this context three cases of hexagonal polycrystals in which slip systems obey power laws with different exponents were considered: creep of ice, recrystallized Zircaloy-4, and the simultaneous thermal and irradiation creep of Zr-2.5Nb. The macroscopic loading curves, predicted by the variational procedure, suggest that the slope transition associated with different values of creep exponent occurs only if the grains have at least four independent systems available for each exponent value. Also, unlike the Taylor and Reuss upper and lower bounds, the new self-consistent estimates 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. For example, for ionic and hexagonal polycrystals with highly anisotropic “flat” grains the new self-consistent estimates can be less than half of the corresponding Taylor predictions