WIAS Preprint No. 1457, (2009)

Homogenization in gradient plasticity


  • Hanke, Hauke

2010 Mathematics Subject Classification

  • 35B27 74C05 74Q15


  • two-scale convergence, folding and unfolding, elasto-plasticity, gradient plasticity, Γ-convergence




This paper yields a two-scale homogenization result for a rate-independent elastoplastic system. The presented model is a generalization of the classical model of linearized elastoplacticity with hardening, which is extended by a gradient term of the plastic variables. The associated stored elastic energy density has periodically oscillating coefficients, where the period is scaled by ε > 0 . The additional gradient term of the plastic variables z is contained in the elastic energy with a prefactor εγ (γ ≥ 0) . We derive different limiting models for ε → 0 in dependence of &gamma ;. For γ > 1 the limiting model is the two-scale model derived in [MielkeTimofte07], where no gradient term was present. For γ = 1 the gradient term of the plastic variable survives on the microscopic cell poblem, while for γ ∈ [0,1) the limit model is defined in terms of a plastic variable without microscopic fluctuation. The latter model can be simplified to a purely macroscopic elastoplasticity model by homogenisation of the elastic part.

Appeared in

  • Math. Models Methods Appl. Sci., 21 (2011) pp. 1651--1684.

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