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Cohesive Membranes under determinant constraints

Nicola Pio Melillo, Dario Reggiani

Abstract

This paper is devoted to the variational derivation of reduced models for elastic membranes with fracture under constraints on the determinant of the deformation gradient. We consider two physically relevant settings: the non-interpenetration regime, in which the deformation is required to be orientation-preserving ($\det \nabla u > 0$), and the incompressible regime, in which the deformation preserves volume ($\det \nabla u = 1$). In both cases, the surface energy density is allowed to depend on the jump amplitude, thus encompassing cohesive fracture models with activation threshold. The main technical contribution is the construction of recovery sequences that simultaneously satisfy the determinant constraint and optimize the surface energy. This is achieved through a combination of $C^\infty$ diffeomorphisms converging to the identity (which rotate the normal to the jump set so as to minimize the reduced surface energy), and a new smooth approximation result for $GSBV^p$ functions.

Cohesive Membranes under determinant constraints

Abstract

This paper is devoted to the variational derivation of reduced models for elastic membranes with fracture under constraints on the determinant of the deformation gradient. We consider two physically relevant settings: the non-interpenetration regime, in which the deformation is required to be orientation-preserving (), and the incompressible regime, in which the deformation preserves volume (). In both cases, the surface energy density is allowed to depend on the jump amplitude, thus encompassing cohesive fracture models with activation threshold. The main technical contribution is the construction of recovery sequences that simultaneously satisfy the determinant constraint and optimize the surface energy. This is achieved through a combination of diffeomorphisms converging to the identity (which rotate the normal to the jump set so as to minimize the reduced surface energy), and a new smooth approximation result for functions.
Paper Structure (11 sections, 29 theorems, 282 equations, 4 figures)

This paper contains 11 sections, 29 theorems, 282 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Basic notation
  4. Functions of bounded variation
  5. Approximation results
  6. Approximation results with a maximal rank condition:
  7. Dimension Reduction under non-interpenetration
  8. Properties of $W_0$ and $\psi_0$, and lower bound
  9. Upper bound
  10. Main result
  11. Dimension Reduction under Incompressibility

Key Result

Proposition 2.11

Let $\Omega\subseteq {R}^N$ be an open set, let $u \in SBV^{p}(\Omega;{R}^{m})$, and let $(u_n)_{n\ge1} \subseteq SBV^{p}(\Omega;{R}^{m})$ be such that, as $n\to\infty$, Then, up to a subsequence (not relabeled), there exist a set $A \subseteq {R}^N$ with $\mathcal{H}^{N-1}(A)<+\infty$ and a non-negative function $g \in L^1(\mathcal{H}^{N-1}\!\llcorner\, A)$ such that:

Figures (4)

  • Figure 3.1: Construction in Lemma \ref{['lemma01']}: The shaded part represents $\Delta$.
  • Figure 4.1: The plane $\Pi_{x_0,\kappa}$ and its normal vector before and after the transformation through $O_\rho$.
  • Figure 4.2: Construction of the segments $\alpha_i'$ and their neighborhoods $U^i \Subset V^i$ along a connected component $\gamma_l$ of the jump set.
  • Figure A.1: The curve $S$, its tubular neighborhood $(S)_{2\sigma}$, and the balls $B_\delta(g(a))$, $B_\delta(g(b))$ around the endpoints.

Theorems & Definitions (63)

  • Definition 2.1: Quasiconvexity
  • Definition 2.2: Quasiconvex envelope
  • Remark 2.3
  • Definition 2.4: Rank-one convexity
  • Definition 2.5: Piecewise affine functions
  • Definition 2.6: Triangulation
  • Definition 2.7: Clarke subdifferential
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 53 more