Homogenization of piezo-flexoelectric phononic structures with embedded electric circuits

We consider the problem of wave propagation in periodic microstructures composed of flexoelectric and piezoelectric (PZFX) materials and involving electrodes which enable to control the response due to the electromechanical transmission and tunable external electric circuits (EEC). The asymptotic homogenization is employed to derive models describing the effective properties of such acoustic metamaterials. It is known that strongly heterogeneous elastic structures with large contrast in the elasticity provide the metamaterial property. The band gap structures are retained in the homogenization limit (w.r.t the scale 'epsilon' going to zero) due to the elasticity scaling by 'epsilon' square in the ``soft'' parts. In analogy, we treat the PZFX heterogeneous continuum extended by the EEC. To this aim, some of the material parameters of the piezo- and flexoelectric constitutive laws must be related to the powers of 'epsilon', to capture the following: a) the large contrasts in the elastic stiffness (soft flexoelectric inclusions, stiffer piezoelectric matrix, b) finite electric potential inducing very strong electric field in the microstructure (the controlling electric voltage is prescribed, depending on the EEC properties), c) the 2nd order gradient boundedness associated with the weak flexoelectric effect. We derive the homogenized model of the piezo-flexo metamaterial being coupled with a homogenized lattice of the EEC. It is shown, how an artificial effective piezoelectric medium can be generated by purely flexoelectric microstructure. Moreover, the dispersion analysis based on the Bloch wave decomposition is modified by the EEC. Due to the soft parts and the EEC, band gaps can be tuned.