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Some remarks on the theory of elasticity for compressible Neohookean materials
Sergio Conti and Camillo De Lellis
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the $L^2$ norm of the deformation gradient and a nonlinear function of the determinant. Non-interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some $L^p$-norm of the gradient with $p>2$ 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.