Abstract
The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin shell is considered, as the thickness h of the shell tends to zero. Given the appropriate scalings of the applied force and of the initial data in terms of h, it’s verified that three-dimesional solutions of the nonlinear elastodynamic equations converge to solutions of the time-dependent von KMorármMorán equations or dynamic linear equations for shell of arbitrary geometry.
Similar content being viewed by others
References
Friesecke G, James R D, and Müller S, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimesional elasticity, Comm. Pure Appl. Math., 2002, 55: 1461–1506.
Friesecke G, James R D, and Müller S, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 2006, 180: 183–236.
Lewicka M, Mora M G, and Pakzad M R, A nonlinear theory for shells with slowly varying thickness, C. R. Acda. Sci. Paris, Sér. I, 2009, 347: 211–216.
Lewicka M, Mora M G, and Pakzad M R, Shell theory arising as low energy Γ-limit of 3d nonlinear elasticity, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2010, IX: 253–295.
Lewicka M, Mora M G, and Pakzad M R, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal., 2011, 200: 1023–1050.
Lewicka M and Pakzad M R, The infinite hierarchy of elastic shell models: Some recent results and a conjecture, Infinite Dimensional Dynamical Systems, Fields Institute Communications, Springer, New York, 2013, 64: 407–420.
Yao P F, Linear strain tensors on hyperbolic surfaces and asymptotic theories for thin shells, SIAM J. Math. Anal., 2019, 51: 1387–1435.
Müller S and Pakzad M R, Convergence of equilibria of thin elastic plates-the von Kármán case, Comm. Part. Differ. Equ., 2008, 33: 1018–1032.
Mora M G and Müller S, Convergence of equilibria of three-dimensional thin elastic beams, Proc. Roy. Soc. Edinburgh Sect. A. Math., 2008, 138: 873–896.
Mora M G, Müller S, and Schultz M G, Convergence of equilibria of planar thin elastic beams, Indiana Univ. Math. J., 2007, 56: 2413–2438.
Lewicka M, A note on convergence of low energy critical points of nonlinear elasticity fünctionals, for thin shells of arbitrary geometry, ESAIM: COCV, 2011, 17: 493–505.
Mora M G and Scardia L, Convergence of eqüilibria of thin elastic plates ünder physical growth conditions for the energy density, J. Diff. Equ., 2012, 252: 35–55.
Müller S, Mathematical problems in thin elastic sheets: Scaling limits, packing, crumpling and singularities, Vector-Valued Partial Differential Equations and Applications, 125–193, LNM 2179, Springer, Cham., 2017.
Abels H, Mora M G, and Müller S, Large time existence for thin vibrating plates, Comm. Part. Diff. Equ., 2011, 36: 2062–2102.
Abels H, Mora M G, and Müller S, The time-dependent von Karmán plate eqüation as a limit of 3d nonlinear elasticity, Calc. Var., 2011, 41: 241–259.
Lecümberry M and Müller S, Stability of slender bodies ünder compression and validity of the von Karmán theory, Arch. Ration. Mech. Anal., 2009, 193: 255–310.
Simon J, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 1987, 146: 65–96.
Spivak M, A Comprehensive Introduction to Differential Geometry, vol V. Second Edition. Püblish or Perish Inc. Aüstralia, 1979.
Taylor M, Partial Differential Equations I: Basic Theory, Second Edition, Springer, 2011.
Author information
Authors and Affiliations
Corresponding authors
Additional information
This research was supported by the National Science Foundation of China under Grant Nos. 61473126 and 61573342, and Key Research Program of Frontier Sciences, CAS, under Grant No. QYZDJ-SSW-SYS011.
This paper was recommended for publication by Editor HU Xiaoming.
Rights and permissions
About this article
Cite this article
Qin, Y., Yao, PF. The Time-Dependent Von Kármán Shell Equation as a Limit of Three-Dimensional Nonlinear Elasticity. J Syst Sci Complex 34, 465–482 (2021). https://doi.org/10.1007/s11424-020-9146-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-020-9146-4