Stochastic homogenization of plasticity equations

Heida, Martin¹ ; Schweizer, Ben²

¹Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany.
²TU Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 153-176

Résumé

In the context of infinitesimal strain plasticity with hardening, we derive a stochastic homogenization result. We assume that the coefficients of the equation are random functions: elasticity tensor, hardening parameter and flow-rule function are given through a dynamical system on a probability space. A parameter $\epsilon >0$ denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit $\epsilon \to 0$. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution $[0,T]\ni t\mapsto \xi \left(t\right)\in$ induces a stress evolution $[0,T]\ni t\mapsto \Sigma \left(\xi \right)\left(t\right)\in$. Once the hysteretic evolution law $\Sigma$ is justified for averages, we obtain that the macroscopic limit equation is given by $-\nabla ·\Sigma \left(\nabla {}^{s}u\right)=f$.

Détail

DOI : 10.1051/cocv/2017015

Classification : 74C05, 35R60, 74Q10
Keywords: Small strain plasticity, stochastic homogenization