Non-periodic homogenization of infinitesimal strain plasticity equations

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