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A Lavrentiev phenomenon in the neo-Hookean model

Marco Barchiesi, Duvan Henao, Carlos Mora-Corral, Rémy Rodiac

Abstract

We exhibit a Lavrentiev gap phenomenon for the neo-Hookean energy in three-dimensional nonlinear elasticity. More precisely, we construct boundary data for which the infimum of the neo-Hookean energy over deformations satisfying a natural regularity and invertibility condition is strictly larger than the infimum over the weak $H^1$-closure of that class. The mechanism underlying the gap is a deformation with a dipole-type singularity.

A Lavrentiev phenomenon in the neo-Hookean model

Abstract

We exhibit a Lavrentiev gap phenomenon for the neo-Hookean energy in three-dimensional nonlinear elasticity. More precisely, we construct boundary data for which the infimum of the neo-Hookean energy over deformations satisfying a natural regularity and invertibility condition is strictly larger than the infimum over the weak -closure of that class. The mechanism underlying the gap is a deformation with a dipole-type singularity.
Paper Structure (14 sections, 7 theorems, 116 equations, 5 figures)

This paper contains 14 sections, 7 theorems, 116 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of the problem and statement of the result
  3. Lavrentiev gap phenomena in elasticity
  4. Strategy of the proof and organisation of the paper
  5. The relaxed energy and the Conti--De Lellis map
  6. The relaxed energy
  7. The Conti--De Lellis map
  8. Proof of Theorem \ref{['th:main1']}
  9. Variants of Theorem \ref{['th:main1']}
  10. The axisymmetric setting
  11. Working with the weak closure of Sobolev homeomorphisms
  12. The dipole map
  13. The approximating sequence
  14. Patch of the dipole

Key Result

Theorem 1.1

Let $\Omega=B(\bm0,4)$, and assume that $H$ is convex and satisfies eq:explosiveH and for some $\alpha<\tfrac{1}{3}$, $\beta < \tfrac{3}{2}$, and $c>0$. Then, there exist a subdomain $\widetilde{\Omega} \subset \subset \Omega$ and a bi-Lipschitz homeomorphism $\bm b : \Omega \rightarrow \mathbb{R}^3$ such that where $\overline{\mathcal{A}^r}$ is the closure of $\mathcal{A}^r$ with respect to the

Figures (5)

  • Figure 1: Reference (left) and deformed (right) configurations ($2D$ sections) of the map $\bm v$ by Conti and De Lellis
  • Figure 2: Reference (left) and deformed (right) configurations ($2D$ sections) of the approximating map $\bm u_\varepsilon$
  • Figure 3: In blue the $2D$ section of the region $U_\delta$
  • Figure 4: The transformation $\bm g = s(r,x_3)\bm e_r + z(r,x_3)\bm e_3$ from the (planar representative slice of) region $d$ in the reference configuration (left) onto the intermediate $(s,z)$ configuration (right).
  • Figure 5: The red arrows illustrate the effect of the function $\omega_\varepsilon(z)$, which extrudes horizontally the image by $\bm v$ of the level curve $\{\bm x\in\Omega: s=\mathop{\mathrm{dist}}\nolimits(\bm x, U)=\tfrac{\delta}{2}\}$. This gives rise to the blue curve, which is the image of the same level set by the approximating map $\bm u$. Our construction $\bm b_\delta$ interpolates linearly between that blue curve and the curve $\bm v(\{s=\delta\})$ on the right-end of the transition layer $T_\delta$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 5 more