Linearization for finite plasticity under dislocation-density tensor regularization

Finite-plasticity theories often feature nonlocal energetic contributions in the plastic variables. By introducing a length-scale for plastic effects in the picture, these nonlocal terms open the way to existence results (Mainik and Mielke in J Nonlinear Sci 19(3):221–248, 2009). We focus here on a reference example in this direction, where a specific energetic contribution in terms of dislocation-density tensor is considered (Mielke and Müller in ZAMM Z Angew Math Mech 86:233–250, 2006). When external forces are small and dissipative terms are suitably rescaled, the finite-strain elastoplastic problem converges toward its linearized counterpart. We prove a Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence result making this asymptotics rigorous, both at the incremental level and at the level of quasistatic evolution.