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.

@article{ASNSP_2003_5_2_3_521_0, author = {Conti, Sergio and de Lellis, Camillo}, title = {Remarks on the theory of elasticity}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {3}, year = {2003}, pages = {521-549}, zbl = {1114.74004}, mrnumber = {2020859}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0} }

Conti, Sergio; de Lellis, Camillo. Remarks on the theory of elasticity. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 521-549. http://www.numdam.org/item/ASNSP_2003_5_2_3_521_0/

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