A note on convergence of low energy critical points of nonlinear elasticity functionals, for thin shells of arbitrary geometry

Lewicka, Marta

ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 493-505.

We prove that the critical points of the 3d nonlinear elasticity functional on shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0 to the critical points of the von Kármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)]. This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ. 33 (2008) 1018-1032], derived for the case of plates when $S\subset {ℝ}^2$. The convergence holds provided the elastic energies of the 3d deformations scale like h⁴ and the external body forces scale like h³.

[1] J.M. Ball, Some open problems in elasticity, in Geometry, mechanics, and dynamics, Springer, New York, USA (2002) 3-59. | MR | Zbl

[2] P.G. Ciarlet, Mathematical Elasticity, Vol. 3: Theory of Shells. North-Holland, Amsterdam (2000). | MR | Zbl

[3] G. Dal Maso, An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser, USA (1993). | MR | Zbl

[4] G. Friesecke, R. 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. | MR | Zbl

[5] G. Friesecke, R. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183-236. | MR | Zbl

[6] H. Ledret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 73 (1995) 549-578. | MR | Zbl

[7] M. Lewicka and M. Pakzad, The infinite hierarchy of elastic shell models: some recent results and a conjecture. Preprint (2009) http://arxiv.org/abs/0907.1585. | MR | Zbl

[8] M. Lewicka, M.G. Mora and M.R. Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells. Preprint (2008) http://arxiv.org/abs/0811.2238. | MR | Zbl

[9] M. Lewicka, M.G. Mora and M.R. Pakzad, A nonlinear theory for shells with slowly varying thickness. C. R. Acad. Sci. Paris, Sér. I 347 (2009) 211-216. | MR | Zbl

[10] M. Lewicka, M.G. Mora and M.R. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear). | Numdam | MR

[11] A.E.H. Love, A treatise on the mathematical theory of elasticity. 4th Edn., Cambridge University Press, Cambridge, UK (1927). | JFM | Zbl

[12] M.G. Mora and S. Müller, Convergence of equilibria of three-dimensional thin elastic beams. Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 873-896. | MR | Zbl

[13] M.G. Mora and L. Scardia, Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density. Preprint (2008). | MR | Zbl

[14] M.G. Mora, S. Müller and M.G. Schultz, Convergence of equilibria of planar thin elastic beams. Indiana Univ. Math. J. 56 (2007) 2413-2438. | MR | Zbl

[15] S. Müller and M.R. Pakzad, Convergence of equilibria of thin elastic plates - the von Kármán case. Comm. Part. Differ. Equ. 33 (2008) 1018-1032. | Zbl

[16] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. V. Second Edn., Publish or Perish Inc., Australia (1979). | MR | Zbl

[17] T. Von Kármán, Festigkeitsprobleme im Maschinenbau, in Encyclopädie der Mathematischen Wissenschaften IV. B.G. Teubner, Leipzig, Germany (1910) 311-385. | JFM