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On the lack of compactness in the axisymmetric neo-Hookean model

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

TL;DR

The study of the relaxation of the neoHookean energy defined originally in an axisymmetric setting and in a functional space where no cavitation is allowed is pursued, and it is shown that the lower bound on the relaxation obtained presents strong similarities with the relaxed energy in the context of S-valued harmonic maps.

Abstract

We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti-- De Lellis is generic in some sense. On this map we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of \(\mathbb{S}^2\)-valued harmonic maps.

On the lack of compactness in the axisymmetric neo-Hookean model

TL;DR

The study of the relaxation of the neoHookean energy defined originally in an axisymmetric setting and in a functional space where no cavitation is allowed is pursued, and it is shown that the lower bound on the relaxation obtained presents strong similarities with the relaxed energy in the context of S-valued harmonic maps.

Abstract

We provide a fine description of the weak limit of sequences of regular axisymmetric maps with equibounded neo-Hookean energy, under the assumption that they have finite surface energy. We prove that these weak limits have a dipole structure, showing that the singular map described by Conti-- De Lellis is generic in some sense. On this map we provide the explicit relaxation of the neo-Hookean energy. We also make a link with Cartesian currents showing that the candidate for the relaxation we obtained presents strong similarities with the relaxed energy in the context of -valued harmonic maps.

Paper Structure

This paper contains 38 sections, 39 theorems, 424 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notations and definitions
  3. Different coordinates systems and axisymmetry
  4. Topological images
  5. The surface energy
  6. Geometric image and area formula
  7. Change of variables formula for surfaces
  8. Towards an upper bound for the relaxed energy
  9. Definition of the limit Conti--De Lellis map
  10. Definition of the new recovery sequence $\text{\boldmath $u$}_\varepsilon$
  11. Motivation for our choice of recovery sequence
  12. Region $c_\varepsilon$
  13. Region $a_\varepsilon'$
  14. Region $a_\varepsilon$
  15. Region $b$
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $(\text{\boldmath $u$}_n)_n$ be a sequence in $\mathcal{B}$ such that $(F(\text{\boldmath $u$}_n))_n$ is equibounded. Then there exists $\text{\boldmath $u$}\in\mathcal{B}$ such that, up to a subsequence, $\text{\boldmath $u$}_n \rightharpoonup \text{\boldmath $u$}$ in $H^1(\Omega,\mathbb{R}^3)$ In particular, the energy $F$ has a minimiser in $\mathcal{B}$.

Figures (8)

  • Figure 1: The $2D$ section of (a possible realization of) the Conti--De Lellis map CoDeLe03. The purple circle $\{y_1^2 + y_2^2+ (y_3-\tfrac{1}{2})^2=\tfrac{1}{2}^2\}$ on the right is not attained as the image of any set of material points $\text{\boldmath $x$}$ in $\Omega=B(\text{\boldmath $0$},3)$. It is, instead, new surface created by the map, that is, part of the boundary of the image of $\Omega \setminus \{\text{\boldmath $0$},\text{\boldmath $0$}'\}$ by $\text{\boldmath $u$}$, where $\text{\boldmath $0$}=(0,0,0)$ and $\text{\boldmath $0$}'=(0,0,1)$ are the only points where $\text{\boldmath $u$}$ is singular
  • Figure 2: The Conti--De Lellis map CoDeLe03 takes a portion of a given region $O$ and sends it outside itself. The two closed curves in the right figure play a prominent role. One, on top, $\Gamma$, represented with a dashed circle, is a bubble created from two cavitation-like singularities. The other, $\text{\boldmath $u$} (\partial O)$, with self-intersections, enclosing three connected components, is represented with a dash-dotted line. Part of the coloured region on the right figure lies outside the dash-dotted loop, even though it consists of material points that were inside the dash-dotted curve in the reference configuration. Regions $a$--$f$ are defined in Section 3; see also Figure \ref{['fig:regions']}
  • Figure 3: An extension of the Conti--De Lellis map CoDeLe03 that satisfies the Dirichlet condition $\text{\boldmath $u$}(\text{\boldmath $x$})=\text{\boldmath $x$}$ on the boundary
  • Figure 4: The map by Conti--De Lellis is defined differently in regions $a$ to $f$. The reference and deformed configurations appear, respectively, on the left and on the right
  • Figure 5: Reference and deformed configurations for the map $\text{\boldmath $u$}_\varepsilon$
  • ...and 3 more figures

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • ...and 83 more