Eulerian rates of elastic incompatibilities applied to size-dependent hardening in finite torsion

Measures of rates of elastic incompatibilities are developed within an Eulerian framework for finite-deformation response of anisotropic elastic-inelastic materials. Such framework relies on the evolution of microstructural vectors. It is emphasized that the rates of incompatibilities, here denoted as R-ij, depend on the constitutive equation for the rate of inelasticity. For small strains and rotations, R-ij are equal to the negative of the components of the rate of NyeKroner's dislocation density tensor. In contrast to these small strain components, each R-ij is invariant under superposed rigid body motions such that it can be used independently in the constitutive equations to describe the material behavior. Specifically, in this work, R-ij provide a size-dependent enhancement to hardening that can increase or decrease during loading history, modeling the generation and annihilation of densities of geometrically necessary dislocations in metal plasticity. The application to the finite-deformation cyclic torsion of thin wires demonstrates the potential of this approach and the importance of the constitutive equation for the plastic spin rate both on the rotations of the microstructural vectors and on the predicted size-effect.