Dynamic behaviour of piezoelectric nanoribbons with wavy configurations on an elastomeric substrate

To meet the great needs for high-performance of piezoelectric nano-scaled structures, the study of their dynamic behaviour is essential in the design of stable and controllable nano-devices, such as energy harvesters, actuators and so on. Brittle wrinkled thin nanoribbons of lead zirconate titanate (Pb(Zr1−xTix)O3, abbreviated as PZT) bound to an elastomeric substrate form a wavy configuration, whose amplitude and wavelength can be designed to accommodate different deformation ranges that would suit real operating environments, including vibration. In this paper, the PZT nanoribbon is modelled as an elastic nonlinear beam with von Karman approximation, and the elastomeric substrate is modelled as a semi-infinite linear elastic medium. Electrical-mechanical coupling is considered in the constitutive relation of the PZT nanoribbon. By utilizing the extended Lagrangian principle, the equation of motion of the PZT nanoribbon-substrate structure with damping, which can more accurately characterise the electromechanical dynamic behaviour of the structure, is derived. To evaluate the dynamic performance of the partly buckled structure, the symplectic Runge–Kutta method (SRK), which has been developed to solve linear damped ordinary differential equations, is adopted to solve the corresponding equations. Bifurcation diagrams are depicted, and the effects of the applied voltage and pre-strain to the elastomeric substrate on the static bifurcation are analysed, indicating that the pre-strain can easily induce surface wrinkling and the applied voltage can be used to more precisely control surface wrinkling. Moreover, the effect of damping on the dynamic behaviour is discussed through the results of time histories and phase portraits for different values of the applied voltage and pre-strain, confirming that as time progresses, the motion gradually vanishes around the static buckling amplitude. The conclusions of this study are useful for the wavy-design strategy of the PZT nanostructure-based stretchable electronics devices, and for the prediction and passive control of the dynamic behaviour of these devices.