On a Consistent Dynamic Finite-Strain Plate Theory and Its Linearization

Abstract

This paper develops a dynamic finite-strain plate theory consistent with three-dimensional Hamilton’s principle under general loadings with a fourth-order error. Starting from the three-dimensional field equations for a compressible hyperelastic material and by a series expansion about the bottom surface, we deduce a vector dynamic plate equation with three unknowns, which exhibits the local momentum-balance structure. Associated weak formulations are considered, in connection with various boundary conditions. Then, by linearization, we provide a novel linear plate theory for orthotropic materials, which takes into account both stretching and bending effects. For isotropic materials, it is further modified to a refined linear plate theory, leading to higher-order results under certain circumstances. To verify the present plate theory, an exhaustive study of the free vibration and static bending problems is carried out. Comparing with the available exact three-dimensional solutions for these problems, it is shown that the plate theory indeed provides correct asymptotic results for displacements and stresses distributions. The advantages of this linear plate theory include (i) its simplicity (as simple as the Kirchhoff-Love theory), (ii) high accuracy for frequencies and deflections, as well as capability of capturing the distributions of all concerned quantities, and (iii) applicability for general loadings.