Remarks on the theory of elasticity
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 521-549.

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 of the gradient. 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.

Classification : 74B20, 35D05, 46E35, 49J45
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Conti, Sergio; de Lellis, Camillo. Remarks on the theory of elasticity. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 521-549. http://archive.numdam.org/item/ASNSP_2003_5_2_3_521_0/

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