Torsional loading of elastoplastic materials leads to size effects which are not captured by classical continuum mechanics and require the use of enriched models. In this work, an analytical solution for the torsion of isotropic perfectly plastic Cosserat cylindrical bars with circular cross-section is derived in the case of generalized von Mises plasticity accounting solely for the symmetric part of the deviatoric stress tensor. The influence of the characteristic length on the microrotation, stress and strain profiles as well as torsional size effects are then investigated. In particular, a size effect proportional to the inverse of the radius of the cylinder is found for the normalized torque. A similar analysis for an extended plasticity criterion accounting for both the couple-stress tensor and the skew-symmetric part of the stress tensor is performed by means of systematic finite element simulations. These numerical experiments predict size effects which are similar to those predicted by the analytical solution. Saturation effects and limit loads are found when the couple-stress tensor enters the yield function.

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.