Nonlocal Gradient-Dependent Thermodynamics for Modeling Scale-Dependent Plasticity

Abstract

This article is concerned with formulating the thermodynamics of nonlocal gradient-dependent plasticity based on the nonlocality energy residual introduced by Eringen and Edelen (1972). A thermodynamic based theory for small strain gradient plasticity is developed by introducing gradients for variables associated with kinematic and isotropic hardening. This theory is a three-nonlocal-parameter theory that takes into consideration large variations in the plastic strain, large variations in the accumulated plastic strain, and accumulation of plastic strain gradients. It is shown that the presence of higher-order gradients in the plastic strain enforces the presence of a corresponding history variable brought by the accumulation of the plastic strain gradients. Gradients in the plastic strain introduce anisotropy in the form of kinematic hardening and are attributed to the net Burgers vector, whereas gradients in the accumulation of the plastic strain introduce isotropic hardening attributed to the additional storage of geometrically necessary dislocations. The equilibrium, or so-called microforce balance, between the internal Cauchy stress and the microstresses that are conjugates to the higher-order gradients turns out to be the yield criterion, which can be simply retrieved from the principle of virtual power. The classical macroscopic boundary conditions are supplemented by nonclassical microscopic boundary conditions associated with plastic flow. The developed nonlocal theory preserves the classical assumption of the local plasticity theory such that the plastic flow direction is governed by the deviatoric Cauchy stress. However, it is also argued here that plastic flow direction is the same as if it is governed by the nonlocal microstress. This is not in line with Gurtin (2003), who argued that the plastic flow direction is governed by a microstress and not the deviatoric Cauchy stress. Some generalities and the utility of this theory are discussed, and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.