Analysis of Bloch’s Method in Structures With Energy Dissipation

Bloch analysis was originally developed to solve Schro¨dinger’s equation for the electron wave function in a periodic potential field, such as found in a pristine crystalline solid. In the context of Schro¨dinger’s equation, damping is absent and energy is conserved. More recently, Bloch analysis has found application in periodic macroscale materials, such as photonic and phononic crystals. In the vibration analysis of phononic crystals, structural damping is present together with energy dissipation. As a result, application of Bloch analysis is not straight-forward and requires additional considerations in order to obtain valid results. It is the intent of this paper to propose a general framework for applying Bloch analysis in such systems. Results are presented in which the approach is applied to example phononic crystals. These results reveal the manner in which damping affects dispersion and the presence of band gaps in periodic systems.