An analytical framework for Kirchhoff plate-type locally resonant piezoelectric metastructures

We present a modeling framework for thin piezoelectric bimorph plates with segmented electrodes acting as electromechanical metastructures (i.e. finite metamaterial structures). Using Hamilton's extended principle and the assumptions of classical plate theory, the governing equations and boundary conditions for the fully coupled electromechanical system are obtained. The surfaces of the piezoelectric material are segmented into opposing pairs of electrodes of arbitrary shape, and each pair of electrodes is shunted to an external circuit. 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 damping, among other effects, as discussed in our previous work. Numerical validations are performed using commercially available finite element software (COMSOL Multiphysics).