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Curvature-dependent Eulerian interfaces in elastic solids

Katharina Brazda, Martin Kružík, Fabian Rupp, Ulisse Stefanelli

Abstract

We propose a sharp-interface model for a hyperelastic material consisting of two phases. In this model, phase interfaces are treated in the deformed configuration, resulting in a fully Eulerian interfacial energy. In order to penalize large curvature of the interface, we include a geometric term featuring a curvature varifold. Equilibrium solutions are proved to exist via minimization. We then utilize this model in an Eulerian topology optimization problem that incorporates a curvature penalization.

Curvature-dependent Eulerian interfaces in elastic solids

Abstract

We propose a sharp-interface model for a hyperelastic material consisting of two phases. In this model, phase interfaces are treated in the deformed configuration, resulting in a fully Eulerian interfacial energy. In order to penalize large curvature of the interface, we include a geometric term featuring a curvature varifold. Equilibrium solutions are proved to exist via minimization. We then utilize this model in an Eulerian topology optimization problem that incorporates a curvature penalization.
Paper Structure (9 sections, 5 theorems, 71 equations)

This paper contains 9 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Piecewise constant functions of bounded variation
  4. Oriented curvature varifolds
  5. Model
  6. States
  7. Energies
  8. Existence of equilibrium states
  9. Topology optimization

Key Result

Theorem 1

Let $U\subset\mathbb{R}^3$ be an open, bounded Lipschitz domain. Let $(\phi_n)_{n\in\mathbb{N}}\subset SBV(U)$ be piecewise constant functions, satisfying Then there exists a piecewise constant function $\phi\in SBV(U)$ such that after passing to a subsequence, we have $\phi_n\to \phi$ in $L^1(U)$ and $D\phi_n \rightharpoonup^* D\phi$ in $\mathcal{M}(U;\mathbb{R}^3)$ as $n\to\infty$.

Theorems & Definitions (14)

  • Theorem 1: Compactness of piecewise constant SBV-functions
  • Theorem 2: Compactness of oriented curvature varifolds without boundary
  • Definition 3: Coupling
  • Definition 4: Admissible set
  • Lemma 5: Properties of admissible triples
  • proof
  • Remark 6
  • Theorem 7: Existence
  • proof
  • Remark 8: Multiple phases
  • ...and 4 more