Numerical homogenization of nonlinear viscoplastic two-dimensional polycrystals

In this article, we numerically determine the effective stress-strain relation of some two-dimensional polycrystals. These are aggregates of a few tens of perfectly bonded single-crystal (hexagonal atomic lattice) grains, with varying orientations. Each grain obeys a given nonlinear viscoplastic stress-strain relation, which depends on the orientation of the grain. Precise calculations performed with this microscopic model are compared with calculations done with a macroscopic approximate model (in which matter has no microstructure) in order to determine the macroscopic constitutive law. We find an effective behaviour for the stationary response which appears to be also consistent for the transient response. The influence of the number of grains as well as that of the distribution of grain orientations are investigated.