Remarks on the theory of elasticity
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, p. 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
@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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