Consistent nonlinear plate equations to arbitrary order for anisotropic, electroelastic crystals

This paper derives nonlinear plate equations for electroelastic crystals using both power series and trigonometric expansions of the three-dimensional equations. Unlike existing theories, material nonlinearities are included to cubic order in the gradients of the field variables, which allows Duffing behavior to be properly modeled. Moreover, inconsistencies in existing nonlinear power series expansions are revealed, and a consistent expansion is given. Next, a Galerkin truncation is applied to the variational formulation of the plate equations to give a very general reduced-order model of its dynamics near primary resonance. By comparison with the Galerkin discretization of the exact equations, nonlinear correction factors are derived for both power series and trigonometric expansions. Numerical continuation of the resulting nonlinear ODEs demonstrates the effect of lateral eigenmodes on the Duffing behavior of the frequency response. Both power series and trigonometric expansions produce results in close agreement. In the limit of purely thickness vibrations, the nonlinear plate equations reduce to the Galerkin truncation of the exact equations.