A Geometrically Nonlinear Theory of Elastic Plates

Abstract

A set of kinematical and intrinsic equilibrium equations are derived for plates undergoing large deflection and rotation but with small strain. The large rotation is handled by means of the general finite rotation of a frame in which the material points that are originally along a normal line in the undeformedplate undergo only small displacements. The unit vector fixed in this frame, which coincides with the normal when the plate is undeformed, is not in general normal to the deformed plate average surface because of transverse shear. The arbitrarily large displacement and rotation of this frame, which vary over the surface of the plate, are termed global deformation; the small relative displacement is termed warping. It is shown that rotation of the frame about the normal is not zero and that it can be expressed in terms of other global deformation variables. Exact intrinsic virtual strain-displacement relations are derived; based on a reduced two-dimensional strain energy function from which the warping has been systematically eliminated, a set of intrinsic equilibrium equations follow. It is shown that only five equilibrium equations can be derived in this manner, because the component of virtual rotation about the normal is not independent. These equilibrium equations contain terms which cannot be obtained without the use of a finite rotation vector which contains three nonzero components. These extra terms correspond to the difference of in-plane shear stress resultants in other theories; this difference is a reactive quantity in the present theory.