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Space of Infinitesimal Isometries and Bending of Shells
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We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on the Gaussian curvature of the middle surfaces: Normal components of infinitesimal isometries satisfy an elliptic problem, or a parabolic one, or a hyperbolic one according to the middle surface being elliptic, or parabolic, or hyperbolic, respectively. In those cases, a problem of determining an infinitesimal isometry is changed into that of 1-dimension. Then we apply those results to the energy functionals of bending of shells which has been obtained as two-dimensional problems by the limit theory of Gamma-convergence from the three-dimensional nonlinear elasticity. Therefore the limit theory of Gamma-convergence reduces to be a one-dimensional problem in the those cases.
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Cited by 1 Pith paper
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Lower Bounds of Optimal Exponentials of Thickness in Geometry Rigidity Inequality for Shells
The optimal thickness exponent in the nonlinear geometric rigidity inequality is at least 4/3 (hyperbolic), 1 (elliptic), and 3/2 (parabolic) shells.
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