Remarks on the theory of elasticity

Sergio Conti; Camillo de Lellis

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 3, page 521-549
  • ISSN: 0391-173X

Abstract

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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.

How to cite

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Conti, Sergio, and de Lellis, Camillo. "Remarks on the theory of elasticity." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.3 (2003): 521-549. <http://eudml.org/doc/84511>.

@article{Conti2003,
abstract = {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 $p&gt;2$ 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.},
author = {Conti, Sergio, de Lellis, Camillo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {521-549},
publisher = {Scuola normale superiore},
title = {Remarks on the theory of elasticity},
url = {http://eudml.org/doc/84511},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Conti, Sergio
AU - de Lellis, Camillo
TI - Remarks on the theory of elasticity
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 3
SP - 521
EP - 549
AB - 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 $p&gt;2$ 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.
LA - eng
UR - http://eudml.org/doc/84511
ER -

References

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