General Framework for Modeling Multifunctional Metamaterial Beam Based on a Derived One-Dimensional Piezoelectric Composite Finite Element

Phononic crystals and metamaterials have been widely investigated over the last decade. In recent years, by integration with piezoelectric transducers, phononic/metamaterial-based piezoelectric energy harvesters (PEHs) have gained increasing research interest for achieving multifunctionalities. This paper proposes a general framework for modelling phononic/metamaterial beams bonded with piezoelectric transducers based on a one-dimensional piezoelectric composite finite element derived using the generalized Hamilton's principle. A method for calculating band structures of infinitely long models of phononic/metamaterial beams that can carry piezoelectric transducers is then developed. This method is demonstrated via two case studies. The first case study investigates a metamaterial beam without piezoelectric coverage, and the proposed method is verified by the transfer matrix method (TMM). Compared with the TMM, the proposed method provides a dispersion relationship in a simpler form and thus demonstrates higher computational efficiency. The second case study investigates a metamaterial beam with periodic piezoelectric coverage. The proposed method takes into consideration the piezoelectric effect. Band structures of such a piezoelectric metamaterial beam under short-circuit and open-circuit conditions are evaluated. Subsequently, corresponding finitely long models of the two case studies are analyzed. The transmittances and open-circuit voltage responses of the piezoelectric transducers are then calculated. The predicted band gaps from transmittances match well with those from band structures. In addition, the transmittances and open-circuit voltage responses of piezoelectric transducers predicted based on the proposed model are verified against the finite-element solution produced by the ANSYS FE program.