Inspection of free energy functions in gradient crystal plasticity

The dislocation density tensor computed as the curl of plastic distortion is regarded as a new constitutive variable in crystal plasticity. The dependence of the free energy function on the dislocation density tensor is explored starting from a quadratic ansatz. Rank one and logarithmic dependencies are then envisaged based on considerations from the statistical theory of dislocations. The relevance of the presented free energy potentials is evaluated from the corresponding analytical solutions of the periodic two-phase laminate problem under shear where one layer is a single crystal material undergoing single slip and the second one remains purely elastic.