A justification of nonlinear properly invariant plate theories

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.