Differential variational estimates for the macroscopic response and field statistics of elasto-viscoplastic polycrystals

Abstract

This paper presents analytical homogenization estimates for the elasto-viscoplastic response of polycrystals with Maxwellian single-crystal constitutive behavior. For this purpose, use is made of the incremental variational procedure (Agoras et al., 2016) for dealing with long-term memory effects, together with the fully optimized second-order method (Ponte Castañeda, 2015) for handling the viscous nonlinearity. By taking the limit as the time increment tends to zero, a new formulation is obtained consisting of a system of nonlinear ordinary differential equations for the time evolution of the stress averages and fluctuation variances in the grains. Comparisons with the Maxwellian approximation for the overall response, obtained by separately homogenizing the elastic and viscoplastic components, serve to characterize the long memory effect. It is found that the long memory effects increase with the anisotropy and nonlinearity of the viscoplastic response of the grains, and that these effects are correlated with rapid changes in the inter- and intra-granular field fluctuations. Under creep loading, ice-like hexagonal close-packed (HCP) polycrystals deform with significantly larger strains than those estimated by the Maxwellian approximation, while for linear face-centered cubic (FCC) polycrystals, the long memory effect is rather minimal. For constant overall strain-rate loading, significant improvements over the Maxwellian approximation are also observed for the HCP polycrystals, especially in the transient regime. To assess their accuracy, the new estimates are compared with available full-field and experimental results from the literature and good agreement is generally found even for polycrystals with highly nonlinear or anisotropic grains.