Fractured membranes under determinant constraints: the case of cohesive surface energies

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 orientation-preserving regime, in which the deformation is required to preserve orientation locally ($\det \nabla u > 0$), 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.