Modeling yield stress scaling and cyclic response using a size-dependent theory with two plasticity rate fields

This work considers a recently developed finite-deformation elastoplasticity theory that assumes distinct tensorial fields describing macro-plasticity and micro-plasticity, where the latter is determined by a higher-order balance equation with associated boundary conditions. Specifically, micro-plasticity evolves according to a contribution to the Helmholtz free-energy density that depends on a Nye–Kröner-like dislocation density tensor and is referred to as the defect energy. The theory is meant to set the onset of micro-plasticity at a stress level lower than that activating macro-plasticity, such as micro-plasticity aims at explaining and characterizing the increase in yield stress with diminishing size. Additionally, the formulation relies on smooth elastic–plastic transitions for both plasticity fields, even if focusing on rate-independent response. This investigation demonstrates the capability of the proposed theory to predict size-effects of interest in small-scale metal plasticity by focusing on multiple loading cycles and, prominently, on the scaling of the apparent yield stress with sample size, the latter being a crucial open issue in the recent literature on modeling size-dependent plasticity. To this end, this work considers the specialization of the theory to small deformations and proposes a finite element implementation for the constrained simple shear problem. Importantly, it is shown that the simplest treatment of plastic strain gradients, which consists of adopting a quadratic defect energy, can be conveniently used to predict reliable size-effects, although in the literature on strain gradient plasticity quadratic defect energies have always been associated with a relatively poor description of size-effects. In fact, in the present theory the limits of the quadratic defect energy are overcome by leveraging on the complex interplay between micro- and macro-plasticity fields. The capability of the proposed theory is quantitatively demonstrated by predicting results from the literature that are obtained from discrete dislocation dynamics simulations on planar polycrystals of grains with variable size subjected to macroscopic pure shear.