Cohesive Membranes under determinant constraints

TL;DR

This paper derives reduced models for elastic membranes with fracture under determinant constraints, incorporating cohesive fracture models with activation thresholds, through advanced variational techniques and recovery sequence constructions.

Contribution

It introduces a novel method for constructing recovery sequences that satisfy determinant constraints while optimizing surface energy in fracture models.

Findings

Successfully constructed recovery sequences satisfying determinant constraints.

Extended the variational framework to include cohesive fracture with activation thresholds.

Provided new smooth approximation results for $GSBV^p$ functions.

Abstract

This paper is devoted to the variational derivation of reduced models for elastic membranes with fracture under constraints on the determinant of the deformation gradient. We consider two physically relevant settings: the non-interpenetration regime, in which the deformation is required to be orientation-preserving ($\det \nabla u > 0$), and the incompressible regime, in which the deformation preserves volume ($\det \nabla u = 1$). In both cases, the surface energy density is allowed to depend on the jump amplitude, thus encompassing cohesive fracture models with activation threshold. The main technical contribution is the construction of recovery sequences that simultaneously satisfy the determinant constraint and optimize the surface energy. This is achieved through a combination of $C^\infty$ diffeomorphisms converging to the identity (which rotate the normal to the jump set so as to minimize the reduced surface energy), and a new smooth approximation result for $GSBV^p$ functions.