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Thin vibrating plates: long time existence and convergence to the von Kármán plate equations. (English) Zbl 1278.74071

Summary: The asymptotic behavior of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness \(h\) of the plate tends to zero. We discuss the long time existence and convergence to solutions of the time-dependent von-Kármán and linear-plate equation under appropriate scalings of the applied force and initial values in terms of \(h\).

MSC:

74H40 Long-time behavior of solutions for dynamical problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
74K20 Plates
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References:

[1] H. Abels M. G. Mora S. Müller Large time existence for thin vibrating plates. Preprint , arXiv:0909.2796v1, 2009.
[2] H. Abels M. G. Mora S. Müller The time-dependent von Kármán plate equation as a limit of 3d nonlinear elasticity. Preprint , arXiv:0912.4135v1, 2009.
[3] G. Friesecke R. D. James S. Müller A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. , 180(2):183-236, 2006. · Zbl 1100.74039
[4] R. Monneau Justification of the nonlinear Kirchhoff-Love theory of plates as the application of a new singular inverse method. Arch. Ration. Mech. Anal. , 169(1):1-34, 2003. · Zbl 1030.74030
[5] S. Müller M.R. Pakzad Convergence of equilibria of thin elastic plates - the von Kármán case. Comm. Partial Differential Equations , 33:1018-1032, 2008. · Zbl 1141.74034
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