Non-periodic homogenization of infinitesimal strain plasticity equations. (English) Zbl 1335.74015
Summary: 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 \(u^\varepsilon\), we ask whether we can characterize weak limits \(u\) when \(u^\varepsilon\to u\) as \(\varepsilon\to 0\). 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 \(\Sigma\) that maps strains to stresses; more precisely, in each spatial point \(x\), given the evolution of the strain in the point \(x\), the operator \(\Sigma\) provides the evolution of the stress in \(x\).
MSC:
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
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