Refined theory for linearly elastic plates: Laminae and laminates

It is well known that bending and stretching modes of deformation in linearly-elastic plates decouple at leading order in thickness provided that the mid-plane is a plane of reflection symmetry of the three-dimensional material properties. This situation being the exception rather than the norm, it is important to have a model that accommodates coupling in a manner consistent with the threedimensional nature of the problem while retaining the analytical tractability of conventional engineering plate theory. Here, this is achieved through an expansion of the potential energy of the plate in powers of its thickness to derive an optimal approximate two-dimensional model for equilibria. The variables are the displacement field of a surface parallel to the lateral surfaces of the plate and associated director fields that emerge naturally in the expansion procedure. To achieve a model that is as accurate as possible, the directors are constrained in accordance with necessary conditions arising in the exact three-dimensional theory. These are incorporated at the level of the energy functional, yielding a model accurate to third order in the small thickness.