Homogenization of plasticity equations with hardening using Finite-Element approaches

Plasticity equations describe the deformations e.g. of metal [1, 3]. As in elasticity, one describes the body at rest with a domain Ω ⊂ Rn, the deforma-tion of the material point x ∈ Ω by u(x) ∈ Rn, uses the symmetric gradient ∇su(x) = (∇u(x) + ∇u(x)T)/2 to describe local deformations, and the stress tensor σ(x) to describe inner forces. The balance of linear momentum is as in elasticity, (1a) with density % and load f. In contrast to elasticity, the stress tensor is not in a linear relation with ∇su(x). Instead, the deformation is decomposed (here: additively) into two parts, an elastic strain and a plastic strain, ∇su(x) = e(x) + p(x), such that with e(x) Hooke’s law is satisfied, σ(x) = De(x) for some elasticity tensor D. The plastic strain can be considered as a component of a possibly larger vector of interior variables ξ ∈ RN; with a linear operator B: RN → Rn×ns that maps into the space of symmetric matrices, we write p = Bξ and obtain (1b) as stress-strain relation. Finally, in order to close the system, we have to introduce a flow-rule in (1c). It provides an ordinary differential equation for the internal variables ξ(x, t) ∈ RN