Motivated by the tensile experiments on titanium alloys of Petrinic et al
(2006), which show the formation of cracks through the formation and
coalescence of voids in ductile fracture, we consider the problem of
formulating a variational model in nonlinear elasticity compatible both
with cavitation and with the appearance of discontinuities across
two-dimensional surfaces. As in the model for cavitation of Müller and
Spector (1995) we address this problem, which is connected to the
sequential weak continuity of the determinant of the deformation gradient
in spaces of functions having low regularity, by means of adding an
appropriate surface energy term to the elastic energy. Based upon
considerations of invertibility we are led to an expression for the
surface energy that admits a physical and a geometrical interpretation,
and that allows for the formulation of a model with better analytical
properties. We obtain, in particular, important regularity properites of
the inverses of deformations, as well as the weak continuity of the
determinants and the existence of minimizers. We show further that the
creation of surface can be modelled by carefully analyzing the jump set of
the inverses, and we point out some connections between the analysis of
cavitation and fracture, the theory of SBV functions, and the theory of
cartesian currents of Giaquinta, Modica and Soucek. (Joint work with
Carlos Mora-Corral, Basque Center for Applied Mathematics).