WIAS Preprint No. 1583, (2010)

An evolutionary elastoplastic plate model derived via Gamma convergence



Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35J85 35Q72 49J45 74C05 74K20

Keywords

  • Linearized elastoplasticity, rate-independent system, Gamma-convergence, Mosco convergence, hysteresis, generalized Prandtl--Ishlinskii operator

Abstract

This paper is devoted to dimension reduction for linearized elastoplasticity in the rate-independent case. The reference configuration of the three-dimensional elastoplastic body has a two-dimensional middle surface and a positive but small thickness. Under suitable scalings we derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations (linear Kirchhoff--Love plate), which are coupled via plastic strains. We establish strong convergence of the solutions in the natural energy space. The analysis uses an abstract Gamma-convergence theory for rate-independent evolutionary systems that is based on the notion of energetic solutions. This concept is formulated via an energy-storage functional and a dissipation functional, such that energetic solutions are defined in terms of a stability condition and an energy balance. The Mosco convergence of the quadratic energy-storage functional follows the arguments of the elastic case. To handle the evolutionary situation the interplay with the dissipation functional is controlled by cancellation properties for Mosco-convergent quadratic energies.

Appeared in

  • Math. Models Methods Appl. Sci., 21 (2011) pp. 1961--1986.

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