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Some remarks on the theory of elasticity for compressible Neohookean materials


Conti, S; De Lellis, C (2003). Some remarks on the theory of elasticity for compressible Neohookean materials. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V, 2(3):521-549.

Abstract

In Neohookean elasticity one minimizes functionals which depend on the L2 norm of the deformation gradient, plus a nonlinear function of the determinant, with some notion of invertibility to represent non-interpenetrability of matter. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector, however only for the case where some Lp-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, which has interesting consequences even in other frameworks.

In Neohookean elasticity one minimizes functionals which depend on the L2 norm of the deformation gradient, plus a nonlinear function of the determinant, with some notion of invertibility to represent non-interpenetrability of matter. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector, however only for the case where some Lp-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, which has interesting consequences even in other frameworks.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2003
Deposited On:17 Sep 2010 09:08
Last Modified:05 Apr 2016 13:25
Publisher:Scuola Normale Superiore di Pisa
ISSN:0391-173X
Additional Information:© 2003 Annali della Scuola Normale Superiore
Official URL:http://annaliscienze.sns.it
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2020859
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1114.74004
http://www.numdam.org/item?id=ASNSP_2003_5_2_3_521_0
Permanent URL: https://doi.org/10.5167/uzh-21869

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