On the Ordinary, Generalized, and Pseudo-Variational Equations of Motion in Nonlinear Elasticity, Piezoelectricity, and Classical Plate Theories

Abstract

Ordinary, generalized, and pseudo-variational equations of motion in three-dimensional theories of nonlinear elasticity and piezoelectricity are presented systematically. These are applied to the derivations of plate equations of the classical type. In contrast to the derivations of plate equations that include thickness and higher-order effects, it is shown that the volume and surface integrals in a three-dimensional ordinary variational equation of motion must now be used jointly in a coupled manner. Details are demonstrated by first treating a classical linear plate. Equations of the classical type for large deflections of laminated composite and piezoelectric plates are then derived, with the famous von Kármán equations of an isotropic homogeneous plate deducible as a special case. Interrelationship among various plate equations is emphasized.