

Preprint 22/2002
Some remarks on the theory of elasticity for compressible Neohookean materials
Sergio Conti and Camillo De Lellis
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Submission date: 04. Mar. 2002
Pages: 30
published in: Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2 (2003) 3, p. 521-549
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Abstract:
In compressible Neohookean elasticity one minimizes functionals
which are composed by the sum of the 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
-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.