A Simple Numerical Framework for Finite Deflection of Piezoelectric Beams

Abstract

Piezoelectric materials can develop mechanical strain upon applying electric voltage and vice-versa. A piezoelectric bimorph, widely used in various sensors and actuator applications, essentially behaves like a beam and consists of a nonpiezoelectric material substrate layer glued between two piezoelectric layers. The application of an electric field alone can induce the bending of such a beam. Studies on the modeling of the piezoelectric bimorph are mostly restricted to the small deflection regime. In the present work, a simple numerical method is proposed to obtain the large deflection response of any piezoelectric bimorph. To begin with, the governing equation of a cantilever bimorph under electric field and end load is obtained. The nonlinear governing equation is then linearized with respect to the current time step. Subsequently, the linearized equation is solved using the RK4 method. From the numerical results, it is found that the response of the key design parameter, namely free displacement is considerably different from that predicted from small deflection analysis. Also, as the entities involved are suitably nondimensionalized, the results are directly relatable to all classes of piezoelectric materials. The nondimensionalization has also paved the way for better insight into the physical problem by rendering a simple mathematical representation.