An investigation of electroelastic bandgap formation in locally resonant piezoelectric metastructures

Locally resonant electromechanical metastructures made from flexible substrates with piezoelectric layers connected to resonant shunt circuits exhibit vibration attenuation properties similar to those of purely mechanical metastructures. Thus, in analogy, these locally resonant electromechanical metastructures can exhibit electroelastic bandgaps at wavelengths much larger than the lattice size. In order to effectively design such metastructures, the modal behavior of the finite structure with given boundary conditions must be reconciled with the electromechanical behavior of the piezoelectric layers and shunt circuits. To this end, we develop the theory for a piezoelectric bimorph beam with segmented electrodes under transverse vibrations, and extract analytical results for bandgap estimation using modal analysis. Under the assumption of an infinite number of segmented electrodes, the locally resonant bandgap is estimated in closed form and shown to depend only on the target frequency and the system-level electromechanical coupling. It is shown that bandgap formation in piezoelectric metastructures is associated with a frequency-dependent modal stiffness, unlike the frequency-dependent modal mass in mechanical metastructures. The relevant electromechanical coupling term and the normalized bandgap size are calculated for a representative structure and a selection of piezoelectric ceramics and single crystals, revealing that single crystals (e.g. PMN-PT) result in significantly wider bandgap than ceramics (e.g. PZT-5A). Numerical studies are performed to demonstrate that the closed-form bandgap expression derived in this work holds for a finite number of electrode segments. It is shown that the number of electrodes required to create the bandgap increases as the target frequency is increased.