Three-Dimensional Closed Form Solutions and ∞3Theories for Orthotropic Plates

A Navier-type method for finding the exact three-dimensional solution for orthotropic rectangular plates is presented. The present procedure is an extension of a previous author's work (which was valid for isotropic plates) and uses the Mixed Form of Hooke's Law (MFHL) which leads to the formulation of the boundary conditions on the top and bottom surfaces of the plate directly in terms of transverse stresses. The paper also extends the Generalized Unified Formulation (GUF) to the case of orthotropic materials. ∞3 theories are presented. The unknown displacements are expanded along the thickness of the plate by using a powerful compact formulation which leads to the writing, with a single formal theory, of a class of ∞3 theories. This formal technique allows the user to treat each unknown independently and, therefore, different numerical approaches can be used in the FEM codes based on this Generalized Unified Formulation. This technique can be very useful in multifield problems such as thermoelastic applications and multilayered plates embedding piezo-layers. The paper analyzes the effects of the different orders of expansion for the displacement variables for various material properties. It is shown that when the anisotropy of the mechanical properties is strong the results are more sensitive to the order used to approximate the displacements. Thirty different theories are assessed for both thick and thin plates and the results are compared against the present exact solution.