Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories

This paper is concerned with the dynamic stability response of an embedded piezoelectric nanoplate made of polyvinylidene fluoride (PVDF). In order to present a realistic model, the material properties of nanoplate are assumed viscoelastic using Kelvin-Voigt model. The visco-nanoplate is surrounded by viscoelastic medium which is simulated by orthotropic visco-Pasternak foundation. The PVDF visco-nanoplate is subjected to an applied voltage in the thickness direction. In order to satisfy the Maxwell equation, electric potential distribution is assumed as a combination of a half-cosine and linear variation. Adopting the nonlocal Mindlin plate theory, the governing equations are derived based on the energy method and Hamilton's principle. A novel numerical method namely as differential cubature method (DCM) in conjunction with the Bolotin's method is applied to obtain the dynamic instability region (DIR) and parametric resonance responses of the visco-nanoplate. The effects of different parameters such as nonlocal parameter, external electric voltage, structural damping, boundary condition, dimension of nanoplate and viscoelastic medium are shown on DIR and parametric resonance frequency of structure. The accuracy of the proposed method is verified by comparing its numerical prediction with other theoretical and experimental published works as well as solution of system with differential quadrature method (DQM). Results indicate that the external electric voltage increases the frequency of the system especially in thick nanoplates.