Modeling Bloch Waves in Prestressed Phononic Crystal Plates

The study aims at investigating the effect of a generic state of prestress on the passbands and bandgaps of a phononic crystal plate. To this end, an Updated Lagrangian scheme is developed, consisting in a two-step procedure: first, a static geometrically nonlinear analysis of a representative unit cell undergoing the action of an applied external load is conducted and then the Floquet-Bloch decomposition is applied to the linearized equations of the acousto-elasticity for the unit cell in the deformed configuration. In addition, a formula for the calculation of the energy velocity is proposed. In the case of an epoxy plate with cylindrical steel inclusions, it is shown that, even in the presence of prestress inducing full reversible deformation state, the bandgap experiences a shift towards higher frequencies when the cell is subjected to a compressive prestress, whereas a frequency downshift is registered when the cell is subjected to traction. In particular, it is demonstrated that the frequency downshift of the bandgap for the phononic plate undergoing a tensile prestress is approximately 3.5% with respect to the case of the phononic plate under compression. The results presented herein provide insights in the behavior of phononic crystal plates with tunable dispersive properties, and suggest new leverages for wave manipulation valuable in many application fields such as wave filters, waveguiding and beam splitting, sensing devices, and vibration shielding.