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January  2015, 14(1): 143-166. doi: 10.3934/cpaa.2015.14.143

Convergence of equilibria for incompressible elastic plates in the von Kármán regime

Marta Lewicka 1, and  Hui Li 2,

1. 

University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260
2. 

2030 Mary Ellen Lane, State College, PA 16803, United States

Received  March 2014 Revised  April 2014 Published  September 2014

We prove convergence of critical points to the nonlinear elastic energies $J^h$ of 3d thin incompressible plates, to critical points of the 2d energy obtained as the $\Gamma$-limit of $J^h$ in the von Kármán scaling regime. The presence of incompressibility constraint requires to restrict the class of admissible test functions to bounded divergence-free variations on the 3d deformations. This poses new technical obstacles, which we resolve by means of introducing 3d extensions and truncations of the 2d limiting deformations, specific to the problem at hand.
Keywords: calculus of variations., plate theories, Gamma convergence, nonlinear elasticity, Convergence of equilibria.
Mathematics Subject Classification: Primary: 74K20; Secondary: 74B2.
Citation: Marta Lewicka, Hui Li. Convergence of equilibria for incompressible elastic plates in the von Kármán regime. Communications on Pure & Applied Analysis, 2015, 14 (1) : 143-166. doi: 10.3934/cpaa.2015.14.143
References:
[1]

J. M. Ball, Minimizers and the Euler-Lagrange equations,, In \emph{Proc. ISIMM conference}, (1983).  doi: 10.1007/3-540-12916-2_47.  Google Scholar

[2]

J. M. Ball, Some open problems in elasticity,, \emph{Geometry, (2002), 3.  doi: 10.1007/0-387-21791-6_1.  Google Scholar

[3]

P. G. Ciarlet, Mathematical Elasticity,, North-Holland, (2000).   Google Scholar

[4]

P. G. Ciarlet and P. Rabier, Les Equations de von Karman,, \emph{Lecture Notes in Mathematics}, 826 (1980).   Google Scholar

[5]

S. Conti and G. Dolzmann, Derivation of a plate theory for incompressible materials,, \emph{C.R. Math. Acad. Sci. Paris}, 344 (2007), 541.  doi: 10.1016/j.crma.2007.03.013.  Google Scholar

[6]

S. Conti and G. Dolzmann, Gamma-convergence for incompressible elastic plates,, \emph{Calc. Var. PDE}, 34 (2009), 531.  doi: 10.1007/s00526-008-0194-1.  Google Scholar

[7]

G. Friesecke, R. D. James, M. G. Mora and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence,, \emph{C. R. Math. Acad. Sci. Paris}, 336 (2003), 697.  doi: 10.1016/S1631-073X(03)00028-1.  Google Scholar

[8]

G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity,, \emph{Comm. Pure Appl. Math.}, 55 (2002), 1461.  doi: 10.1002/cpa.10048.  Google Scholar

[9]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, \emph{Arch. Ration. Mech. Anal.}, 180 (2006), 183.  doi: 10.1007/s00205-005-0400-7.  Google Scholar

[10]

M. Lewicka, A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry,, \emph{ESAIM: Control, 17 (2011), 493.  doi: 10.1051/cocv/2010002.  Google Scholar

[11]

M. Lewicka, L. Mahadevan and M. R. Pakzad, The Foppl-von Karman equations for plates with incompatible strains,, \emph{Proceedings of the Royal Society A}, 467 (2011), 402.  doi: 10.1098/rspa.2010.0138.  Google Scholar

[12]

M. Lewicka, L. Mahadevan and M. R. Pakzad, Models for elastic shells with incompatible strains,, \emph{Proceedings of the Royal Society A}, 470 (2014), 1471.  doi: 10.1098/rspa.2013.0604.  Google Scholar

[13]

M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, Vol. IX (2010), 1.   Google Scholar

[14]

M. Lewicka, M. G. Mora and M. R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells,, \emph{Arch. Rational Mech. Anal.}, 200 (2011), 1023.  doi: 10.1007/s00205-010-0387-6.  Google Scholar

[15]

M. Lewicka and M. R. Pakzad, The infinite hierarchy of elastic shell models; some recent results and a conjecture,, \emph{Infinite Dimensional Dynamical Systems, 64 (2013), 407.  doi: 10.1007/978-1-4614-4523-4_16.  Google Scholar

[16]

M. Lewicka and M. R. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics,, \emph{ESAIM: Control, 17 (2011), 1158.  doi: 10.1051/cocv/2010039.  Google Scholar

[17]

H. Li, Topics in the Mathematical Theory of Nonlinear Elasticity,, Ph. D thesis, (2012).   Google Scholar

[18]

H. Li and M. Chermisi, The von Karman theory for incompressible elastic shells,, \emph{Calculus of Variations and PDE}, 48 (2013), 185.  doi: 10.1007/s00526-012-0549-5.  Google Scholar

[19]

M.G. Mora, S. Muller and M. G. Schultz, Convergence of equilibria of planar thin elastic beams,, \emph{Indiana Univ. Math. J.}, 56 (2007), 2413.  doi: 10.1512/iumj.2007.56.3023.  Google Scholar

[20]

M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density,, \emph{J. Differential Equations}, 252 (2012), 35.  doi: 10.1016/j.jde.2011.09.009.  Google Scholar

[21]

S. Müller and M. R. Pakzad, Convergence of Equilibria of Thin Elastic Plates-The Von Karman Case,, \emph{Comm. Partial Differential Equations}, 33 (2008), 1018.  doi: 10.1080/03605300701629443.  Google Scholar

show all references

References:
[1]

J. M. Ball, Minimizers and the Euler-Lagrange equations,, In \emph{Proc. ISIMM conference}, (1983).  doi: 10.1007/3-540-12916-2_47.  Google Scholar

[2]

J. M. Ball, Some open problems in elasticity,, \emph{Geometry, (2002), 3.  doi: 10.1007/0-387-21791-6_1.  Google Scholar

[3]

P. G. Ciarlet, Mathematical Elasticity,, North-Holland, (2000).   Google Scholar

[4]

P. G. Ciarlet and P. Rabier, Les Equations de von Karman,, \emph{Lecture Notes in Mathematics}, 826 (1980).   Google Scholar

[5]

S. Conti and G. Dolzmann, Derivation of a plate theory for incompressible materials,, \emph{C.R. Math. Acad. Sci. Paris}, 344 (2007), 541.  doi: 10.1016/j.crma.2007.03.013.  Google Scholar

[6]

S. Conti and G. Dolzmann, Gamma-convergence for incompressible elastic plates,, \emph{Calc. Var. PDE}, 34 (2009), 531.  doi: 10.1007/s00526-008-0194-1.  Google Scholar

[7]

G. Friesecke, R. D. James, M. G. Mora and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence,, \emph{C. R. Math. Acad. Sci. Paris}, 336 (2003), 697.  doi: 10.1016/S1631-073X(03)00028-1.  Google Scholar

[8]

G. Friesecke, R. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity,, \emph{Comm. Pure Appl. Math.}, 55 (2002), 1461.  doi: 10.1002/cpa.10048.  Google Scholar

[9]

G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, \emph{Arch. Ration. Mech. Anal.}, 180 (2006), 183.  doi: 10.1007/s00205-005-0400-7.  Google Scholar

[10]

M. Lewicka, A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry,, \emph{ESAIM: Control, 17 (2011), 493.  doi: 10.1051/cocv/2010002.  Google Scholar

[11]

M. Lewicka, L. Mahadevan and M. R. Pakzad, The Foppl-von Karman equations for plates with incompatible strains,, \emph{Proceedings of the Royal Society A}, 467 (2011), 402.  doi: 10.1098/rspa.2010.0138.  Google Scholar

[12]

M. Lewicka, L. Mahadevan and M. R. Pakzad, Models for elastic shells with incompatible strains,, \emph{Proceedings of the Royal Society A}, 470 (2014), 1471.  doi: 10.1098/rspa.2013.0604.  Google Scholar

[13]

M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity,, \emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, Vol. IX (2010), 1.   Google Scholar

[14]

M. Lewicka, M. G. Mora and M. R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells,, \emph{Arch. Rational Mech. Anal.}, 200 (2011), 1023.  doi: 10.1007/s00205-010-0387-6.  Google Scholar

[15]

M. Lewicka and M. R. Pakzad, The infinite hierarchy of elastic shell models; some recent results and a conjecture,, \emph{Infinite Dimensional Dynamical Systems, 64 (2013), 407.  doi: 10.1007/978-1-4614-4523-4_16.  Google Scholar

[16]

M. Lewicka and M. R. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics,, \emph{ESAIM: Control, 17 (2011), 1158.  doi: 10.1051/cocv/2010039.  Google Scholar

[17]

H. Li, Topics in the Mathematical Theory of Nonlinear Elasticity,, Ph. D thesis, (2012).   Google Scholar

[18]

H. Li and M. Chermisi, The von Karman theory for incompressible elastic shells,, \emph{Calculus of Variations and PDE}, 48 (2013), 185.  doi: 10.1007/s00526-012-0549-5.  Google Scholar

[19]

M.G. Mora, S. Muller and M. G. Schultz, Convergence of equilibria of planar thin elastic beams,, \emph{Indiana Univ. Math. J.}, 56 (2007), 2413.  doi: 10.1512/iumj.2007.56.3023.  Google Scholar

[20]

M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density,, \emph{J. Differential Equations}, 252 (2012), 35.  doi: 10.1016/j.jde.2011.09.009.  Google Scholar

[21]

S. Müller and M. R. Pakzad, Convergence of Equilibria of Thin Elastic Plates-The Von Karman Case,, \emph{Comm. Partial Differential Equations}, 33 (2008), 1018.  doi: 10.1080/03605300701629443.  Google Scholar

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