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A homogenization result in finite plasticity

Elisa Davoli, Chiara Gavioli, Valerio Pagliari

Abstract

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the $Γ$-convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to $\mathsf{SL}(3)$ poses the biggest hurdle to the analysis, and we address it by regarding $\mathsf{SL}(3)$ as a Finsler manifold.

A homogenization result in finite plasticity

Abstract

We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we establish the -convergence of the energies in the limiting of vanishing periodicity. The constraint that plastic deformations belong to poses the biggest hurdle to the analysis, and we address it by regarding as a Finsler manifold.
Paper Structure (9 sections, 9 theorems, 91 equations)

This paper contains 9 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Mathematical setting and results
  3. Preliminaries
  4. Localization and integral representation
  5. Finsler structure on $\mathsf{SL}(3)$
  6. Homogenization in finite plasticity
  7. Integral representation
  8. Characterization of the limiting energy density
  9. Convergence of minimum problems

Key Result

Theorem 2.2

Let $\mathcal{F}_\varepsilon$ be the functionals in Feps, which we extend by setting If $W$ and $H$ satisfy E1--E-lip and H0--H2, respectively, then for all $y \in L^2(\Omega;\mathbb{R}^3)$, $P \in L^q(\Omega;\mathsf{SL}(3))$ the $\Gamma$-limit exists. We also have that where $W_\mathrm{hom} \colon \mathbb{R}^{3 \times 3} \times K \to [0,+\infty)$ and $H_\mathrm{hom} \colon K \to [0,+\infty)$

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Theorem 3.1
  • Definition 3.2
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Example 3.6: Von Mises plasticity
  • ...and 12 more