Wave propagation in fractionally damped nonlinear phononic crystals

Studying wave propagation in phononic crystals (PCs) in the presence of energy dissipation is a crucial step toward the precise dynamic modeling of periodic structures. Fractional calculus is an appropriate tool to reach a more perceptive idea of energy dissipation compared to other damping models. Therefore, in this work, we aim to provide a semi-analytical model for wave propagation in fractionally damped nonlinear PCs. For this purpose, the method of multiple scales is used to solve the governing equations of PCs, and the nonlinear dispersion relations of fractionally damped monoatomic chains and lattices are obtained. The Caputo definition of fractional derivatives is used to model damping. Besides providing new insight into the energy dissipation in PCs, the results of this research emphasize the importance of considering nonlinearities in modeling periodic materials, especially because the propagation frequency in nonlinear crystals is amplitude-dependent. The obtained results are validated with numerical modeling of fractionally damped PCs.