A thermodynamic theory of elastoplasticity was developed which included kinematic hardening and dislocation density tensors. The theory was self-consistent and was based upon 2 fundamental principles of thermodynamics: entropy increase and maximum entropy production rate. Thermodynamically consistent governing equations were rigorously derived for the plastic spin and back stress. An expression for the plastic spin tensor was derived from the constitutive equation of the dislocation drift rate tensor, and an expression for the back-stress tensor was given as a balance equation which expressed an equilibrium between the internal stress and a micro-stress which was conjugate to the dislocation density tensor. It was shown that, in order to obtain a thermodynamically consistent theory of kinematic hardening, the free energy density tensor should have the dislocation density tensor as one of its arguments.

A Thermodynamic Theory of Plastic Spin and Internal Stress with Dislocation Density Tensor. K.Shizawa, H.M.Zbib: Journal of Engineering Materials and Technology, 1999,121[1], 247-52