Weakly nonlinear wave interactions in multi-degree of freedom periodic structures

•We analyze elastic wave interactions in nonlinear, periodic materials.•We consider multi-degree of freedom and multidimensional systems.•Wave interactions allow for novel control of wave direction and group velocity.•Wave interactions suggest superprism, focusing, and multiplexing applications.

This work presents a multiple time scales perturbation analysis for analyzing weakly nonlinear wave interactions in multi-degree of freedom periodic structures. The perturbation analysis is broadly applicable to (discretized) periodic systems in any dimensional space and with a wide range of constitutive nonlinearities. Specific emphasis is placed on cubic nonlinearity, as dispersion shifts typically arise from the cubic components in nonlinear restoring forces. The procedure is first presented in general. Then, application to the diatomic chain and monoatomic two-dimensional lattice demonstrates, individually, the treatment of multiple degree of freedom systems and higher dimensional spaces. The dispersion relations are modified by weakly nonlinear wave interactions and lead to additional opportunities to control wave propagation direction, band gap size, and group velocity. Numerical simulations validate the expected dispersion shifts. An amplitude-tunable focus device demonstrates the viability of utilizing dynamically-introduced dispersion to produce beam steering that may, ultimately, lead to a phononic superprism effect as well as multiplexing/demultiplexing behavior.