An Eulerian thermodynamical formulation of size-dependent plasticity

The objective of this work is to develop a finite-deformation, thermodynamically-consistent theory to model the size-dependent irreversible behavior of metals at the micron-scale. In contrast with other approaches, the proposed Eulerian formulation introduces an evolution equation directly for an elastic distortional deformation measure without the need for defining total or plastic deformation measures. Motivated both by experimental observations and by the literature on generalized continua, the proposed theory assumes tensorial fields separately describing macro-plasticity and micro-plasticity. An additional evolution equation defines a Nye-Kroner-like tensor that depends on the curl of the non-symmetric micro-plasticity distortional rate. By drawing inspiration from higher-order strain gradient plasticity, the micro-plasticity distortional rate is determined by a micro-plasticity balance equation with associated boundary conditions. A quadratic function of the Nye-Kroner-like tensor contributes to the Helmholtz free energy, which leads to an energetic stress entering the micro-plasticity balance that controls the material size-dependent response. The theory is designed to allow the onset of micro-plasticity to occur at a stress level below that triggering macro-plasticity and, regarding size-effects, the response characterized by macro-plasticity alone is the strongest one. In addition, the formulation uses a smooth elastic-plastic transition model for rate-independent response. The theory is specialized to the case of small strains and rotations and applied to the constrained simple shear problem, which allows the derivation of an analytical solution able to demonstrate the predictive capabilities ensuing from the rich interplay between micro- and macro-plasticity.