Non-periodic homogenization of infinitesimal strain plasticity equations

Abstract

We consider the Prandtl-Reuss model of plasticity with kinematic hardening, aiming at a homogenization result. For a sequence of coefficient fields and corresponding solutions u ε , we ask whether we can characterize weak limits u when u ε u as ε → 0. We assume neither periodicity nor stochasticity for the coefficients, but we demand an abstract averaging property of the homogeneous system on reference volumes. Our conclusion is an effective equation on general domains with general right hand sides. The effective equation uses a causal evolution operator that maps strains to stresses; more precisely, in each spatial point x, given the evolution of the strain in the point x, the operator provides the evolution of the stress in x.

Stochastic Unfolding and Homogenization of Spring Network Models

“…where c2 → denotes "cross" two-scale convergence, which is explained in Section 4.3. In the continuum case, similar results have been obtained, for deterministic periodic materials in [37,20] (via periodic unfolding), and recently for random materials in [23] (using quenched stochastic two-scale convergence) and [25,26]. We discuss the literature on problems involving discrete-to-continuum transition in more detail in Section 4.…”

Stochastic homogenization of plasticity equations

“…Our stochastic homogenization result follows by applying the main theorem of [9]. Essentially, we only have to verify that, if the coefficient functions of system (1.1) are given by an ergodic stochastic process, then the coefficients "allow averaging": In the limit ε → 0, averages of the stress (for a homogeneous plasticity system on a simplex with affine boundary data ξ) are given by the operator Σ.…”

“…In a second step, one can realize that the dependence on x can be disintegrated: The two-scale system can be written in the form (1.4), if the hysteretic stress operator Σ is defined through a stochastic cell problem in the variables (t, ω). In the needle-problem approach, we keep these two aspects separated: The abstract result "averaging property for Σ implies homogenization" of [9] is independent of the stochastic description. The stochastic analysis concerns only the operator Σ and its properties (the work at hand).…”

“…The main result of [9] is a homogenization theorem. Under the assumption that causal operators Σ and Π satisfy certain admissibility and averaging properties, we obtain the convergence of the ε-solutions u ε to the solution u of the effective problem (1.4).…”

On Friedrichs Inequality, Helmholtz Decomposition, Vector Potentials, and the div-curl Lemma

“…The crucial step in this approach is to find, given a sequence of functions u ε on a domain Ω, a triangulation of Ω such that the global div-curl lemma can be applied on every simplex of the triangulation. The method was applied to perform the homogenization of plasticity equation in [HS15,HS14].…”