A thermodynamically consistent framework for the gradient approach to plasticity was proposed. The second-order gradient of kinematic, and isotropic, hardening measures were introduced via the Helmholtz free energy and the plastic potential function. The macroscale gradients permitted consideration of the non-local behavior of materials and of the interpretation of the collective behavior of defects such as dislocations. The macroscale internal state variable and the corresponding gradient term were assumed to be independent internal state variables with respect to each other. This permitted these 2 different physical phenomena to be identified separately. The Aifantis–Walgraef dislocation model was used to give micromechanically based definitions for the non-local coefficients.

Non-Local Dislocation-Based Plasticity Incorporating Gradients of Hardening. R.J.Dorgan, G.Z.Voyiadjis: Mechanics of Materials, 2003, 35[8], 721-32