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Remarks on the comparison principle for quasilinear PDE with no zeroth order terms
Convergence of equilibria for incompressible elastic plates in the von Kármán regime
1. | University of Pittsburgh, Department of Mathematics, 301 Thackeray Hall, Pittsburgh, PA 15260 |
2. | 2030 Mary Ellen Lane, State College, PA 16803, United States |
References:
[1] |
J. M. Ball, Minimizers and the Euler-Lagrange equations, In Proc. ISIMM conference, Paris, Springer, 1983.
doi: 10.1007/3-540-12916-2_47. |
[2] |
J. M. Ball, Some open problems in elasticity, Geometry, mechanics, and dynamics, 3-59, Springer, New York, 2002.
doi: 10.1007/0-387-21791-6_1. |
[3] |
P. G. Ciarlet, Mathematical Elasticity, North-Holland, Amsterdam, 2000. |
[4] |
P. G. Ciarlet and P. Rabier, Les Equations de von Karman, Lecture Notes in Mathematics, 826, Berlin-Heidelberg-New York, Springer-Verlag 1980. |
[5] |
S. Conti and G. Dolzmann, Derivation of a plate theory for incompressible materials, C.R. Math. Acad. Sci. Paris, 344 (2007), 541-544.
doi: 10.1016/j.crma.2007.03.013. |
[6] |
S. Conti and G. Dolzmann, Gamma-convergence for incompressible elastic plates, Calc. Var. PDE, 34 (2009), 531-551.
doi: 10.1007/s00526-008-0194-1. |
[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, C. R. Math. Acad. Sci. Paris, 336 (2003), 697-702.
doi: 10.1016/S1631-073X(03)00028-1. |
[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, Comm. Pure Appl. Math., 55 (2002), 1461-1506.
doi: 10.1002/cpa.10048. |
[9] |
G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.
doi: 10.1007/s00205-005-0400-7. |
[10] |
M. Lewicka, A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011) 493-505.
doi: 10.1051/cocv/2010002. |
[11] |
M. Lewicka, L. Mahadevan and M. R. Pakzad, The Foppl-von Karman equations for plates with incompatible strains, Proceedings of the Royal Society A, 467 (2011), 402-426.
doi: 10.1098/rspa.2010.0138. |
[12] |
M. Lewicka, L. Mahadevan and M. R. Pakzad, Models for elastic shells with incompatible strains, Proceedings of the Royal Society A, 470 (2014), 1471-2946.
doi: 10.1098/rspa.2013.0604. |
[13] |
M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. IX (2010), 1-43. |
[14] |
M. Lewicka, M. G. Mora and M. R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Rational Mech. Anal., 200 (2011), 1023-1050.
doi: 10.1007/s00205-010-0387-6. |
[15] |
M. Lewicka and M. R. Pakzad, The infinite hierarchy of elastic shell models; some recent results and a conjecture, Infinite Dimensional Dynamical Systems, Fields Institute Communications, 64 (2013), 407-420.
doi: 10.1007/978-1-4614-4523-4_16. |
[16] |
M. Lewicka and M. R. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 1158-1173.
doi: 10.1051/cocv/2010039. |
[17] |
H. Li, Topics in the Mathematical Theory of Nonlinear Elasticity, Ph. D thesis, University of Minnesota, 2012. |
[18] |
H. Li and M. Chermisi, The von Karman theory for incompressible elastic shells, Calculus of Variations and PDE, 48 (2013), 185-209.
doi: 10.1007/s00526-012-0549-5. |
[19] |
M.G. Mora, S. Muller and M. G. Schultz, Convergence of equilibria of planar thin elastic beams, Indiana Univ. Math. J., 56 (2007), 2413-2438.
doi: 10.1512/iumj.2007.56.3023. |
[20] |
M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differential Equations, 252 (2012) 35-55.
doi: 10.1016/j.jde.2011.09.009. |
[21] |
S. Müller and M. R. Pakzad, Convergence of Equilibria of Thin Elastic Plates-The Von Karman Case, Comm. Partial Differential Equations, 33 (2008), 1018-1032.
doi: 10.1080/03605300701629443. |
show all references
References:
[1] |
J. M. Ball, Minimizers and the Euler-Lagrange equations, In Proc. ISIMM conference, Paris, Springer, 1983.
doi: 10.1007/3-540-12916-2_47. |
[2] |
J. M. Ball, Some open problems in elasticity, Geometry, mechanics, and dynamics, 3-59, Springer, New York, 2002.
doi: 10.1007/0-387-21791-6_1. |
[3] |
P. G. Ciarlet, Mathematical Elasticity, North-Holland, Amsterdam, 2000. |
[4] |
P. G. Ciarlet and P. Rabier, Les Equations de von Karman, Lecture Notes in Mathematics, 826, Berlin-Heidelberg-New York, Springer-Verlag 1980. |
[5] |
S. Conti and G. Dolzmann, Derivation of a plate theory for incompressible materials, C.R. Math. Acad. Sci. Paris, 344 (2007), 541-544.
doi: 10.1016/j.crma.2007.03.013. |
[6] |
S. Conti and G. Dolzmann, Gamma-convergence for incompressible elastic plates, Calc. Var. PDE, 34 (2009), 531-551.
doi: 10.1007/s00526-008-0194-1. |
[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, C. R. Math. Acad. Sci. Paris, 336 (2003), 697-702.
doi: 10.1016/S1631-073X(03)00028-1. |
[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, Comm. Pure Appl. Math., 55 (2002), 1461-1506.
doi: 10.1002/cpa.10048. |
[9] |
G. Friesecke, R. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.
doi: 10.1007/s00205-005-0400-7. |
[10] |
M. Lewicka, A note on the convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011) 493-505.
doi: 10.1051/cocv/2010002. |
[11] |
M. Lewicka, L. Mahadevan and M. R. Pakzad, The Foppl-von Karman equations for plates with incompatible strains, Proceedings of the Royal Society A, 467 (2011), 402-426.
doi: 10.1098/rspa.2010.0138. |
[12] |
M. Lewicka, L. Mahadevan and M. R. Pakzad, Models for elastic shells with incompatible strains, Proceedings of the Royal Society A, 470 (2014), 1471-2946.
doi: 10.1098/rspa.2013.0604. |
[13] |
M. Lewicka, M. G. Mora and M. R. Pakzad, Shell theories arising as low energy $\Gamma$-limit of 3d nonlinear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. IX (2010), 1-43. |
[14] |
M. Lewicka, M. G. Mora and M. R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Rational Mech. Anal., 200 (2011), 1023-1050.
doi: 10.1007/s00205-010-0387-6. |
[15] |
M. Lewicka and M. R. Pakzad, The infinite hierarchy of elastic shell models; some recent results and a conjecture, Infinite Dimensional Dynamical Systems, Fields Institute Communications, 64 (2013), 407-420.
doi: 10.1007/978-1-4614-4523-4_16. |
[16] |
M. Lewicka and M. R. Pakzad, Scaling laws for non-Euclidean plates and the $W^{2,2}$ isometric immersions of Riemannian metrics, ESAIM: Control, Optimisation and Calculus of Variations, 17 (2011), 1158-1173.
doi: 10.1051/cocv/2010039. |
[17] |
H. Li, Topics in the Mathematical Theory of Nonlinear Elasticity, Ph. D thesis, University of Minnesota, 2012. |
[18] |
H. Li and M. Chermisi, The von Karman theory for incompressible elastic shells, Calculus of Variations and PDE, 48 (2013), 185-209.
doi: 10.1007/s00526-012-0549-5. |
[19] |
M.G. Mora, S. Muller and M. G. Schultz, Convergence of equilibria of planar thin elastic beams, Indiana Univ. Math. J., 56 (2007), 2413-2438.
doi: 10.1512/iumj.2007.56.3023. |
[20] |
M. G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density, J. Differential Equations, 252 (2012) 35-55.
doi: 10.1016/j.jde.2011.09.009. |
[21] |
S. Müller and M. R. Pakzad, Convergence of Equilibria of Thin Elastic Plates-The Von Karman Case, Comm. Partial Differential Equations, 33 (2008), 1018-1032.
doi: 10.1080/03605300701629443. |
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