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.