Theory and numerics of geometrically non-linear gradient plasticity

This work presents the theory and the numerics of a thermodynamically consistent formulation of geometrically non-linear gradient plasticity. Due to the lack of the classical local continuum formulation to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. Especially within the framework of crystal plasticity we resort to physically motivated arguments in terms of geometrically necessary dislocations densities that imply the incorporation of higher gradients. In a first simplified approach presented here we adopt the gradient of the internal hardening variable as a provision for geometrically necessary dislocations. We start from a thermodynamic formulation within a geometrically non-linear setting including the additional contribution of the gradient of the internal history variable. This introduces e.g. the vectorial hardening flux and the quasi-nonlocal drag stress. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non-linear programming. Finally, some demonstrative numerical examples complement the presentation.