A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

Marta Lewicka

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 2, page 493-505
  • ISSN: 1292-8119

Abstract

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We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ.33 (2008) 1018–1032], derived for the case of plates when S 2 . The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

How to cite

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Lewicka, Marta. "A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 493-505. <http://eudml.org/doc/276328>.

@article{Lewicka2011,
abstract = { We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ.33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb\{R\}^2$. The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3. },
author = {Lewicka, Marta},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Shell theories; nonlinear elasticity; Gamma convergence; calculus of variations; shell theories; gamma convergence; calculus of variations},
language = {eng},
month = {5},
number = {2},
pages = {493-505},
publisher = {EDP Sciences},
title = {A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry},
url = {http://eudml.org/doc/276328},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Lewicka, Marta
TI - A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/5//
PB - EDP Sciences
VL - 17
IS - 2
SP - 493
EP - 505
AB - We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ.33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb{R}^2$. The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.
LA - eng
KW - Shell theories; nonlinear elasticity; Gamma convergence; calculus of variations; shell theories; gamma convergence; calculus of variations
UR - http://eudml.org/doc/276328
ER -

References

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