A justification of nonlinear properly invariant plate theories |
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Authors: | D. D. Fox A. Raoult J. C. Simo |
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Affiliation: | (1) Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street, 02860 Pawtucket, Rhode Island;(2) Laboratoire de Modelisation et Calcul, Université J. Fourier, B.P. 53x-38041, Grenoble cedex;(3) Division of Applied Mechanics, Stanford University, 94305 Stanford, California |
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Abstract: | A single asymptotic derivation of three classical nonlinear plate theories is presented in a setting which preserves the frame-invariance properties of three-dimensional finite elasticity. By a successive scaling of the external loading on the three-dimensional body, the nonlinear membrane theory, the nonlinear inextensional theory and the von Kármán equations are derived as the leading-order terms in the asymptotic expansion of finite elasticity. The governing equations of the nonlinear inextensional theory are of particular interest where 1) plane-strain kinematics and plane-stress constitutive equations are derived simultaneously from the asymptotic analysis, 2) the theory can be phrased as a minimization problem over the space of isometric deformations of a surface, and 3) the local equilibrium equations are identical to those arising in the one-director Cosserat shell model. Furthermore, it can be concluded that with a regular, single-scale asymptotic expansion it is not possible to obtain a system of plate equations in which finite membrane strain and finite bending strain occur simultaneously in the leading-order term of an asymptotic analysis. |
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