Size Effects in Cosserat Crystal Plasticity

Abstract

Some Cosserat elastoplasticity models for single crystals are reviewed in the present chapter. Their size-dependent response is evaluated in the case of a simple one-dimensional shear test involving one single slip system and vanishing microrotation prescribed at the boundaries of a material strip of width 2L. The inhomogeneous distribution of slip in the channel mimics the piling-up of dislocations against the boundaries. The free energy density function depends on the elastic strain and Cosserat curvature tensors. Two types of potentials are examined with respect to the curvature tensor, namely a quadratic function of its norm, on the one hand, and the norm itself, on the second hand. The first model is very often used but turns out to be non-physical since, according to physical metallurgy, the stored energy is proportional to the dislocation density (here the density of geometrically dislocations) rather than its square. The scaling laws predicted by these models are shown to be L −2 or L −1 , respectively. The latter scaling is reminiscent of Orowan's law of yielding [17]. The chapter ends with the combination of both quadratic and rank one contributions in a unified formulation applied to grain boundary modelling.