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Core-radius approximation of singular minimizers in nonlinear elasticity

Marco Bresciani, Manuel Friedrich

TL;DR

This work develops a rigorous core-radius (perforated-domain) regularization for cavitation in nonlinear elasticity, proving Γ-convergence of a cavity-augmented energy to a limit model that includes both cavity volume and surface-perimeter terms. It introduces an extended distributional determinant to handle Cavitation on perforated domains and establishes compactness, a liminf inequality, and recovery results (under a perimeter-regularity assumption), with full Γ-convergence available when the perimeter penalty vanishes. The analysis generalizes prior core-radius results to Cavitation-Perimeter energies and connects to phase-field relaxations and Šilhavý-type theories, while highlighting limitations and avenues for numerical discretization. The results provide a solid variational and measure-theoretic foundation for numerically simulating cavitation via perforated-domain approximations in nonlinear elasticity, with explicit examples clarifying when the perimeter condition is satisfied or violated.

Abstract

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation (in the sense of Γ-convergence) of the energy by means of functionals defined on perforated domains. Perforations are introduced at flaw points where singularities are expected and, hence, the corresponding deformations do not exhibit cavitation. Notably, those points are not prescribed but rather selected by the variational principle. Our analysis is motivated by the numerical simulation of cavitation and extends previous results on models which solely accounted for elastic energy but neglected contributions related to the formation of cavities.

Core-radius approximation of singular minimizers in nonlinear elasticity

TL;DR

This work develops a rigorous core-radius (perforated-domain) regularization for cavitation in nonlinear elasticity, proving Γ-convergence of a cavity-augmented energy to a limit model that includes both cavity volume and surface-perimeter terms. It introduces an extended distributional determinant to handle Cavitation on perforated domains and establishes compactness, a liminf inequality, and recovery results (under a perimeter-regularity assumption), with full Γ-convergence available when the perimeter penalty vanishes. The analysis generalizes prior core-radius results to Cavitation-Perimeter energies and connects to phase-field relaxations and Šilhavý-type theories, while highlighting limitations and avenues for numerical discretization. The results provide a solid variational and measure-theoretic foundation for numerically simulating cavitation via perforated-domain approximations in nonlinear elasticity, with explicit examples clarifying when the perimeter condition is satisfied or violated.

Abstract

We study a variational model in nonlinear elasticity allowing for cavitation which penalizes both the volume and the perimeter of the cavities. Specifically, we investigate the approximation (in the sense of Γ-convergence) of the energy by means of functionals defined on perforated domains. Perforations are introduced at flaw points where singularities are expected and, hence, the corresponding deformations do not exhibit cavitation. Notably, those points are not prescribed but rather selected by the variational principle. Our analysis is motivated by the numerical simulation of cavitation and extends previous results on models which solely accounted for elastic energy but neglected contributions related to the formation of cavities.

Paper Structure

This paper contains 24 sections, 27 theorems, 204 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Relevant literature
  4. Contributions of the paper
  5. Discussion and outlook
  6. Structure of the paper
  7. Notation
  8. Setting and main results
  9. Variational model and its regularization
  10. Main results
  11. Preliminaries
  12. Approximate differentiability
  13. Topological degree
  14. Invertibility conditions
  15. Distributional determinant
  16. ...and 9 more sections

Key Result

Theorem 2.3

Let $\varepsilon_n\to 0^+$ and let $((A_n,\boldsymbol{y}_n))_n$ be a sequence with $(A_n,\boldsymbol{y}_n)\in\mathcal{Q}_{\varepsilon_n}$ for all $n\in \mathbb{N}$ satisfying Then, there exists $(A,\boldsymbol{y})\in \mathcal{Q}$ such that, up to subsequences, we have and

Figures (4)

  • Figure 1: The deformation in Example \ref{['ex:radial']} for $b=1/2$.
  • Figure 2: The deformation in Example \ref{['ex:cor']} for $b=1/2$.
  • Figure 3: The deformation in Example \ref{['ex:superpos']}.
  • Figure 4: The deformation in Example \ref{['ex:counterexample']}.

Theorems & Definitions (64)

  • Definition 2.1: Deformations
  • Definition 2.2: Approximating deformations
  • Theorem 2.3: Compactness
  • Theorem 2.4: Lower bound
  • Theorem 2.5: Optimality of the lower bound
  • Definition 3.1: Geometric domain and image for maps on domains
  • Proposition 3.2: Area formula for maps on domains
  • Definition 3.3: Geometric domain and image for maps on hypersurfaces
  • Proposition 3.4: Area formula for maps on hypersurfaces
  • Definition 3.5: Topological degree and topological image
  • ...and 54 more