Study on Large Deformation of Laminated Piezoelectric Rectangular Plate

Abstract

For the laminated piezoelectric rectangular plate with large deflection and large rotation, the nonlinear equilibrium differential equations are derived and solved. Firstly, the global Cartesian coordinate system to describe the undeformed geometry and the local orthogonal curvilinear coordinate system to describe the deformed geometry are established respectively on the mid-plane of the plate before and after the deformation, and the relationship between the two coordinates is expressed by transformation matrix. For the convenience of calculation, the expressions of the nonlinear curvatures and inplane strains are obtained by Taylor series expansion. Considering the piezoelectric effect, three equilibrium partial differential equations describing nonlinear bending problems are obtained by the principle of virtual work. Furthermore, in order to simplify the solution process, the stress function is introduced to automatically satisfy the first two equations for the large deformation of the cantilever plate, and the relationship between stress function, the mid-plane internal force and shear force is also given for the first time. Therefore, the stress function and the transversal displacement are the main unknowns of the governing equation and compatibility equation. Additionally, the approximate deflection function and stress function are given which can satisfy all the displacement boundary conditions and only part of the force boundary conditions. Thereby, the generalized Galerkin method is used to obtain the approximate solution of the nonlinear bending problem. Finally, the results in the study are verified by comparison with the results obtained from the finite element method. It also provides a theoretical basis for the engineering application of the large deformation of the piezoelectric cantilever plate.