An unconstrained third-order plate theory

Abstract

The Mindlin plate theory is an unconstrained first-order theory because no constraint is assumed for the three displacement functions (one transverse displacement and two rotations). There are three sets of natural boundary conditions. The classical theory constrains the rotations as partial derivations of the transverse displacement. There are only two sets of natural boundary conditions and the effective shear must be introduced to satisfy the variational principles. The Mindlin plate requires an experimentally determined shear correction factor. Various third-order theories have been proposed to eliminate the shear correction factor. A general third-order plate theory involves five displacement functions: one transverse displacement, two linear variations of the inplane displacements (rotations) and two cubic variations of the inplane displacements. The Reddy theory is variationally consistent among the third-order theories. However, the cubic variations are constrained by the transverse displacement and the rotations. Effective generalized forces of higher dimensions must be introduced to achieve the variational consistency along the boundaries. It is shown here that if all the five displacement functions are unconstrained, consistent natural boundary conditions are satisfied without the introduction of effective forces.