Invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity

In this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea to unify the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of creation of new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies the sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one a.e. deformations is also one-to-one a.e. We then use these results to obtain existence of minimizers of variational models that incorporate the elastic energy and this created surface energy, taking into account the orientation-preserving and the non-interpenetration conditions.