A JUSTIFICATION OF NONLINEAR PROPERLY INVARIANT PLATE THEORIES

A single asymptotic derivation of three classical nonlinear plate theo ries is presented in a setting which preserves the frame-invariance pr operties of three-dimensional finite elasticity. By a successive scali ng of the external loading on the three-dimensional body, the nonlinea r membrane theory, the nonlinear inextensional theory and the von Karm an equations are derived as the leading-order terms in the asymptotic expansion of finite elasticity. The governing equations of the nonline ar inextensional theory are of particular interest where 1) plane-stra in kinematics and plane-stress constitutive equations are derived simu ltaneously 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 tho se arising in the one-director Cosserat shell model. Furthermore, it c an be concluded that with a regular, single-scale asymptotic expansion it is not possible to obtain a system of plate equations in which fin ite membrane strain and finite bending strain occur simultaneously in the leading-order term of an asymptotic analysis.