Variational models with Eulerian-Lagrangian formulation allowing for material failure

TL;DR

This paper develops a variational framework with Eulerian-Lagrangian formulation to model material failure, including cavitation and fracture, extending previous elastic models to account for discontinuities and complex material behaviors.

Contribution

It introduces a new existence theory for minimizers in models allowing for free discontinuities in both deformation and Eulerian fields, applicable to complex materials like liquid crystals and ferromagnetic elastomers.

Findings

Existence of minimizers in models with material failure.

Application to liquid crystals, phase transitions, and elastomers.

Illustration of theory's effectiveness and limitations.

Abstract

We investigate the existence of minimizers of variational models with Eulerian-Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do so, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.