An asymptotic theory of thin hyperelastic plates—I. General theory

Abstract

A large deflection theory of thin hyperelastic plates is developed using an asymptotic analysis of three-dimensional equations of nonlinear elasticity in the reference configuration. The domain occupied by a thin plate is first transformed into a domain of comparable dimensions and then the displacement and stress components are scaled as in the von Kármán theory. The displacement vector and the stress tensor are expanded in terms of powers of an appropriate thickness parameter and a hierarchy of equilibrium equations and boundary conditions are derived following the usual procedure. In a similar fashion the hierarchy of constitutive equations is found for an arbitrary form of strain energy function. The theories corresponding to the lowest two order members in this hierarchy are studied in detail and the zeroth order theory is shown to correspond to the celebrated von Kármán theory. Moreover, it is demonstrated that the effect of physical nonlinearity becomes significant in the first order approximation.