In Neohookean elasticity one minimizes functionals which depend on the L2 norm of the deformation gradient, plus a nonlinear function of the determinant, with some notion of invertibility to represent non-interpenetrability of matter. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector, however only for the case where some Lp-norm of the gradient with p \ensuremath> 2 is controlled (in three dimensions). We first characterize their class of functions in terms of properties of the associated rectifiable current. Then we address the physically relevant p = 2 case, and show how their notion of invertibility can be extended to p = 2. The class of functions so obtained is however not closed. We prove this by giving an explicit construction, which has interesting consequences even in other frameworks.

Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 521–549.