Second-order and affine estimates for the behavior of viscous polycrystals

The recently proposed second-order and affine estimates for the effective behavior of nonlinear heterogeneous materials rely on the same linearization procedure and refer to some "thermoelastic comparison composite". But both approaches differ in the way the overall nonlinear answer is defined, either through average stress-strain relations or energy considerations. When combined with the linear self-consistent model, they provide new tools for the predictions of the behavior of viscoplastic polycrystals (steady state creep). They are compared to each other and to other more classical estimates (incremental and tangent) in the following situations: a 2D antiplane model problem for which efficient rigorous upper bounds are available, the creep of isotropic aggregates of FCC crystals and HCP crystals with different critical stresses on the basal and prismatic slip systems.