Volume 24, Number 1, January–March 2018
|Page(s)||153 - 176|
|Published online||17 January 2018|
Stochastic homogenization of plasticity equations
1 Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany.
2 TU Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, 44227 Dortmund, Germany.
Received: 27 April 2016
Accepted: 31 January 2017
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 ε > 0 denotes the typical length scale of oscillations. We derive effective equations that describe the behavior of solutions in the limit ε → 0. The homogenization procedure is based on the fact that stochastic coefficients “allow averaging”: For one representative volume element, a strain evolution induces a stress evolution . Once the hysteretic evolution law Σ is justified for averages, we obtain that the macroscopic limit equation is given by −∇·Σ(∇su) = f.
Mathematics Subject Classification: 74C05 / 35R60 / 74Q10
Key words: Small strain plasticity / stochastic homogenization
© EDP Sciences, SMAI 2018
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