Free vibration analysis of pre-stretched plates with electromechanical coupling

With the development of soft electro-active materials, their electromechanical coupling behaviors have become a focus of intense research interest in fields of mechanics and related engineering disciplines. However, the complexity associated with multi-field coupling is further convoluted with the nonlinearity caused by finite deformation, making the problems very difficult to be solved. In this study, the free vibration of a simply-supported pre-stretched rectangular plate with electromechanical coupling is considered. An exact three-dimensional solution is derived based on the linearized biasing theory proposed by Dorfmann and Ogden, using the new state-space formulations. In contrast to the conventional state-space formulations, two displacement functions and two stress functions are introduced to express the two in-plane displacements and two transverse shear stresses respectively. Thus, two decoupled state equations of lower order are derived, which correspond to different deformation modes. Numerical examples are considered to illustrate the effects of various parameters on the natural frequencies of the plate. Particular phenomena observed from the numerical results, e.g. the electromechanical coupling, the size-increasing induced softening and the stretch induced stiffening, are discussed.