# 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 (2011)

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

## Access Full Article

top## Abstract

top## How to cite

topLewicka, 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

top- J.M. Ball, Some open problems in elasticity, in Geometry, mechanics, and dynamics, Springer, New York, USA (2002) 3–59. Zbl1054.74008
- P.G. Ciarlet, Mathematical Elasticity, Vol. 3: Theory of Shells. North-Holland, Amsterdam (2000). Zbl0953.74004
- G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications8. Birkhäuser, USA (1993).
- G. Friesecke, R. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity. Comm. Pure. Appl. Math.55 (2002) 1461–1506. Zbl1021.74024
- G. Friesecke, R. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal.180 (2006) 183–236. Zbl1100.74039
- H. LeDret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl.73 (1995) 549–578. Zbl0847.73025
- M. Lewicka and M. Pakzad, The infinite hierarchy of elastic shell models: some recent results and a conjecture. Preprint (2009) . Zbl1263.74035URIhttp://arxiv.org/abs/0907.1585
- M. Lewicka, M.G. Mora and M.R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Preprint (2008) . Zbl1291.74130URIhttp://arxiv.org/abs/0811.2238
- M. Lewicka, M.G. Mora and M.R. Pakzad, A nonlinear theory for shells with slowly varying thickness. C. R. Acad. Sci. Paris, Sér. I347 (2009) 211–216. Zbl1168.74036
- M. Lewicka, M.G. Mora and M.R. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear). Zbl05791996
- A.E.H. Love, A treatise on the mathematical theory of elasticity. 4th Edn., Cambridge University Press, Cambridge, UK (1927). Zbl53.0752.01
- M.G. Mora and S. Müller, Convergence of equilibria of three-dimensional thin elastic beams. Proc. Roy. Soc. Edinburgh Sect. A138 (2008) 873–896. Zbl1142.74022
- M.G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. Preprint (2008). Zbl1291.74128
- M.G. Mora, S. Müller and M.G. Schultz, Convergence of equilibria of planar thin elastic beams. Indiana Univ. Math. J.56 (2007) 2413–2438. Zbl1125.74026
- S. Müller and M.R. Pakzad, Convergence of equilibria of thin elastic plates – the von Kármán case. Comm. Part. Differ. Equ.33 (2008) 1018–1032. Zbl1141.74034
- M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. V. Second Edn., Publish or Perish Inc., Australia (1979). Zbl0439.53001
- T. von Kármán, Festigkeitsprobleme im Maschinenbau, in Encyclopädie der Mathematischen WissenschaftenIV. B.G. Teubner, Leipzig, Germany (1910) 311–385.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.