NeoHookean energies, cavitation, and relaxation in nonlinear elasticity

Abstract: The neoHookean model is one of the most commonly used approaches to study the mechanical response of elastic bodies undergoing large deformations. However, the neoHookean energy is expected to possess no minimizers in the Sobolev class naturally associated to its quadratic coercivity. This is connected to the formation and sudden expansion of voids observed in confined elastomers. There is analytical evidence for the conjecture that the nonexistence is due to the opening of an ever larger number of cavities. Regularizations of the neoHookean model either impose a length-scale for the cavities created (with a second gradient, or taking into account the energy required to stretch the created surface) or impose a bound in the number of cavities that the body is allowed to open. In the second approach, the first existence results are due to Henao & Rodiac (2018) in the axisymmetric class for hollow domains, and to Doležalová, Hencl & Molchanova (2024) in the weak closure of homeomorphisms in 3D. In more general classes where harmonic dipoles are admitted, a relaxation approach has been proposed by Barchiesi, Henao, Mora-Corral & Rodiac (2023, 2024), where the mass of the singular part of the derivative of the inverse is found to accurately give the cost of creating dipole singularities.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

Function spaces