Doubly nonlinear waveguides with self-switching functionality selection capabilities

In this article, we investigate the effects of the interplay between quadratic and cubic nonlinearities on the propagation of elastic waves in periodic waveguides. Through this framework, we unveil an array of wave control strategies that are intrinsically available in the response of doubly nonlinear systems and we infer some basic design principles for tunable elastic metamaterials. The objective is to simultaneously account for two sources of nonlinearity that are responsible for distinct and complementary phenomena and whose effects are therefore typically discussed separately in the literature. Our study explicitly targets the intertwined effects that the two types of nonlinearity exert on each other, which modify the way in which their respective signatures are observed in the dynamic response. Through two illustrative examples we show how the dispersion correction caused by cubic nonlinearity can be used as an internal switch, or mode selector, capable of tuning on or off certain high-frequency response features that are generated through quadratic mechanisms. To this end, a multiple scale analysis is employed to obtain a full analytical solution for the nonlinear response that includes a complete description of the dual frequency-wave number dispersion correction shifts induced on all the branches, and elucidates the conditions necessary for the establishment of phase matching conditions.