We discuss the limiting behavior (using the notion of -limit) of the 3d nonlinear elasticity for thin shells around an arbitrary smooth 2d surface. In particular, under the assumption that the elastic energy of deformations scales like , being the thickness of a shell, we derive a limiting theory which is a generalization of the von Kármán theory for plates.
Lewicka, Marta 1 ; Mora, Maria Giovanna 2 ; Pakzad, Mohammad Reza 3
@article{ASNSP_2010_5_9_2_253_0,
author = {Lewicka, Marta and Mora, Maria Giovanna and Pakzad, Mohammad Reza},
title = {Shell theories arising as low energy $\mathbf{\Gamma }$-limit of 3d nonlinear elasticity},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
pages = {253--295},
year = {2010},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
mrnumber = {2731157},
zbl = {05791996},
language = {en},
url = {https://www.numdam.org/item/ASNSP_2010_5_9_2_253_0/}
}
TY - JOUR
AU - Lewicka, Marta
AU - Mora, Maria Giovanna
AU - Pakzad, Mohammad Reza
TI - Shell theories arising as low energy $\mathbf{\Gamma }$-limit of 3d nonlinear elasticity
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2010
SP - 253
EP - 295
VL - 9
IS - 2
PB - Scuola Normale Superiore, Pisa
UR - https://www.numdam.org/item/ASNSP_2010_5_9_2_253_0/
LA - en
ID - ASNSP_2010_5_9_2_253_0
ER -
%0 Journal Article
%A Lewicka, Marta
%A Mora, Maria Giovanna
%A Pakzad, Mohammad Reza
%T Shell theories arising as low energy $\mathbf{\Gamma }$-limit of 3d nonlinear elasticity
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2010
%P 253-295
%V 9
%N 2
%I Scuola Normale Superiore, Pisa
%U https://www.numdam.org/item/ASNSP_2010_5_9_2_253_0/
%G en
%F ASNSP_2010_5_9_2_253_0
Lewicka, Marta; Mora, Maria Giovanna; Pakzad, Mohammad Reza. Shell theories arising as low energy $\mathbf{\Gamma }$-limit of 3d nonlinear elasticity. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 253-295. https://www.numdam.org/item/ASNSP_2010_5_9_2_253_0/
[1] , “Mathematical Elasticity”, Vol. 3, “Theory of Shells”, North-Holland, Amsterdam, 2000. | MR | Zbl
[2] , “Habilitation Thesis”, University of Leipzig, 2003.
[3] and , Derivation of elastic theories for thin sheets and the constraint of incompressibility, In: “Analysis, Modeling and Simulation of Multiscale Problems”, Springer, Berlin, 2006, 225–247. | MR
[4] and , Derivation of a plate theory for incompressible materials, C. R. Math. Acad. Sci. Paris 344 (2007), 541–544. | MR | Zbl
[5] and , Confining thin sheets and folding paper, Arch. Ration. Mech. Anal. 187 (2008), 1–48. | MR | Zbl
[6] , “An Introduction to -Convergence”, Progress in Nonlinear Differential Equations and their Applications, Vol. 8, Birkhäuser, MA, 1993. | MR | Zbl
[7] , “Partial Differential Equations”, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. | MR
[8] , , and , Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris 336 (2003), 697–702. | Zbl
[9] , and , A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity, Comm. Pure. Appl. Math. 55 (2002), 1461–1506. | MR | Zbl
[10] , and , A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence, Arch. Ration. Mech. Anal. 180 (2006), 183–236. | MR | Zbl
[11] and , On the rigidity of certain surfaces with folds and applications to shell theory, Arch. Ration. Mech. Anal. 129 (1995), 11–45. | MR | Zbl
[12] and , “Isometric Embedding of Riemannian Manifolds in Euclidean Spaces”, Mathematical Surveys and Monographs, Vol. 130, American Mathematical Society, Providence, RI, 2006. | MR
[13] , Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21–64. | MR | Zbl
[14] , Festigkeitsprobleme im Maschinenbau, In: “Encyclopädie der Mathematischen Wissenschaften”, Vol. IV/4, 311–385, Leipzig, 1910. | JFM
[15] and , The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 73 (1995), 549–578. | MR | Zbl
[16] and , The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci. 6 (1996), 59–84. | MR | Zbl
[17] and , The uniform Korn-Poincaré inequality in thin domains, arXiv: 0803.0355. | MR | Zbl | Numdam
[18] , A Lusin property of Sobolev functions, Indiana Univ. Math. J. 26 (1977), 645–651. | MR | Zbl
[19] , “A Treatise on the Mathematical Theory of Elasticity”, 4th edition, Cambridge University Press, Cambridge, 1927. | JFM
[20] , Justification of nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method, Arch. Rational Mech. Anal. 169 (2003), 1–34. | MR | Zbl
[21] , Unpublished note.
[22] and , Convergence of equilibria of thin elastic plates – the von Kármán case, Comm. Partial Differential Equations 33 (2008), 1018–1032. | MR | Zbl
[23] , The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. | MR | Zbl
[24] , On the justification of the nonlinear inextensional plate model, Arch. Ration. Mech. Anal. 167 (2003), 179–209. | MR | Zbl
[25] , Statique et dynamique des coques minces. II. Cas de flexion pure inhibée. Approximation membranaire, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 531–537. | MR | Zbl
[26] , Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl. 88 (2007), 107–122. | MR | Zbl
[27] , “A Comprehensive Introduction to Differential Geometry”, Vol V, 2nd edition, Publish or Perish Inc., 1979. | MR | Zbl
[28] , Modeling of a membrane for nonlinearly elastic incompressible materials via gamma-convergence, Anal. Appl. (Singap.) 4 (2006), 31–60. | MR | Zbl
