Calculus of Variations and Geometric Measure Theory

S. Conti - C. De Lellis

Some remarks on the theory of elasticity for compressible Neohookean materials

created on 23 Nov 2001
modified by delellis on 03 May 2011


Published Paper

Inserted: 23 nov 2001
Last Updated: 3 may 2011

Journal: Ann. Scuola Norm. Sup. Cl. Sci. (5)
Volume: II (2003)
Pages: 521-549
Year: 2001


In Neohookean elasticity one minimizes functionals which depend on the $L^2$ 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 $L^p$-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, which has interesting consequences even in other frameworks.

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