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Structured Deformations in Linearized Elasticity

Manuel Friedrich, José Matias, Elvira Zappale

Abstract

We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a direct approach by means of a global method for relaxation in BD and via an approximation from nonlinear elastic energies associated to {nonsimple} materials.

Structured Deformations in Linearized Elasticity

Abstract

We extend the theory of structured deformations to the setting of linearized elasticity by providing an integral representation for the underlying energy that features bulk and surface contributions. Our derivation is obtained both via a direct approach by means of a global method for relaxation in BD and via an approximation from nonlinear elastic energies associated to {nonsimple} materials.
Paper Structure (11 sections, 23 theorems, 225 equations)

This paper contains 11 sections, 23 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Definition of structured deformations in $BD$ and approximation theorem
  4. Integral representation
  5. Relaxation
  6. Linearization of energies associated to a structured deformation
  7. The global method in $StBD^p$ and $StSBD^p$
  8. Proof of the relaxation results
  9. Proof of the linearization result
  10. BV and BD fields
  11. Auxiliary statements

Key Result

Theorem 2.1

Let $(g,G) \in StBD^p(\Omega)$. Then, there exists $\lbrace u_n\rbrace \subset SBD(\Omega)$ such that $u_n \to g$ strongly in $L^1(\Omega;\mathbb R^{N})$, $\mathcal{E}u_n = G$, and Moreover, there exists $C(N)>0$ such that

Theorems & Definitions (44)

  • Theorem 2.1: Approximation
  • proof
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Remark 2.6: Invariance
  • Corollary 2.7
  • Theorem 2.8: Relaxation
  • Remark 2.9
  • ...and 34 more