An analytical framework for locally resonant piezoelectric metamaterial plates

Abstract

We present an analytical modeling framework and its analysis for thin piezoelectric metamaterial plates to enable and predict low-frequency bandgap formation in finite structural configurations with specified boundary conditions. Using Hamilton’s extended principle and the assumptions of classical (Kirchhoff) plate theory, the governing equations and boundary conditions for the fully coupled two-dimensional electromechanical system are obtained. The two surfaces of the piezoelectric bimorph are segmented into non-overlapping opposing pairs of electrodes of arbitrary shape, and each pair of electrodes is shunted to an external circuit. This formulation can be used to study the effect of electrode shape on plate response for topology optimization and other vibration control applications. Using modal analysis, we show that for a sufficient number of electrodes distributed across the surface of the plate, the effective dynamic stiffness of the plate is determined by the shunt circuit admittance applied to each pair of electrodes and the system-level electromechanical coupling. This enables the creation of locally resonant bandgaps and broadband attenuation, among other effects, in analogy with our previous work for one-dimensional piezoelectric structures with synthetic impedance shunt circuits. It is also demonstrated that piezoelectric bimorph plates display significantly improved performance (i.e. electromechanical coupling) over bimorph beams, but require additional electrode segmentation to achieve metamaterial-type performance. The governing equations are also used for dispersion analysis using the plane wave expansion method, enabling the analytical dispersion analysis of unit cells with arbitrary electrode shapes. The modeling framework and approximate closed-form bandgap expressions are numerically validated using finite element analysis.