Abstract
We rigorously derive the von Kármán shell theory for incompressible materials, starting from the 3D nonlinear elasticity. In case of thin plates, the Euler-Lagrange equations of the limiting energy functional reduce to the incompressible version of the classical von Kármán equations, obtained formally in the limit of Poisson’s ratio ν → 1/2. More generally, the midsurface of the shell to which our analysis applies, is only assumed to have the following approximation property: \({\mathcal C^3}\) first order infinitesimal isometries are dense in the space of all W 2,2 infinitesimal isometries. The class of surfaces with this property includes: subsets of \({\mathbb R^2}\), convex surfaces, developable surfaces and rotationally invariant surfaces. Our analysis relies on the methods and extends the results of Conti and Dolzmann (Calc Var PDE 34:531–551, 2009, Lewicka et al. (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IX:253–295, 2010, Friesecke et al. (Comm. Pure. Appl. Math. 55, no. 2, 1461–1506, 2002).
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Communicated by J. Ball.
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Li, H., Chermisi, M. The von Kármán theory for incompressible elastic shells. Calc. Var. 48, 185–209 (2013). https://doi.org/10.1007/s00526-012-0549-5
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DOI: https://doi.org/10.1007/s00526-012-0549-5