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

TL;DR

This paper characterizes the weak limits of axisymmetric maps with bounded neo-Hookean energy, revealing a dipole structure and providing explicit relaxation, thus advancing understanding of singularities in nonlinear elasticity models.

Contribution

It offers a detailed description of weak limits and explicit relaxation of the neo-Hookean energy for axisymmetric maps, highlighting the generic dipole structure.

Findings

Weak limits have a dipole structure.

Explicit relaxation of the neo-Hookean energy is provided.

Connections with Cartesian currents and harmonic maps are established.

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.