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Measure structured deformations

stefan Krömer, Martin Kružík, Marco Morandotti, Elvira Zappale

Abstract

Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.

Measure structured deformations

Abstract

Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.
Paper Structure (10 sections, 15 theorems, 141 equations)

This paper contains 10 sections, 15 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Setting and the definition of the energy in $mSD$
  3. Auxiliary results
  4. Equivalent characterizations of the relaxed energy densities
  5. Nonlinear transformation of measures and area-strict convergence
  6. Proof of Theorem \ref{['thm:representation']}
  7. Upper bound
  8. Lower Bound
  9. Relaxation under trace constraints
  10. Further properties and examples

Key Result

Theorem 2.1

Let $G \in L^1(\Omega; \mathbb{R}^{d{\times} N})$. Then there exist a function $f \in SBV(\Omega; \mathbb{R}^{d})$, a Borel function $\beta\colon\Omega\to\mathbb{R}^{d{\times} N}$, and a constant $C_N>0$ depending only on $N$ such that

Theorems & Definitions (37)

  • Theorem 2.1: Alberti Alberti1991, ChoFo97
  • Lemma 2.2: ChoFo97
  • Theorem 2.3: approximation theorem
  • proof
  • Theorem 2.4: integral representation
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8: Instability of the contribution of $G^s_g$ in $I=J$
  • Proposition 3.1
  • ...and 27 more