Exact theory of dispersion for nonlinear wave propagation and extension to phononic materials
| Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this work, we consider strongly nonlinear wave propagation in elastic solids, although the theory presented is in principle applicable to other types of waves such as waves in fluids, gases, and plasma.
We investigate a thick elastic rod admitting longitudinal motion. In the linear limit, this rod is dispersive due to the effect of lateral inertia. The nonlinearity is introduced through either the stress-strain relation and/or the strain-displacement gradient relation. Using a theory we have developed earlier and demonstrated on thin rods and beams , we derive an exact nonlinear dispersion relation for the thick rod.
The derived relation is validated by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations). The study is then extended to a continuous thin rod with a periodic arrangement of resonators (nonlinear elastic metamaterial)  or material properties (nonlinear phononic crystal) . For this problem we introduce a new method that is based on a standard transfer matrix augmented with a nonlinear enrichment at the constitutive material level. This method yields an approximate band structure that accounts for the finite wave amplitude. Finally, we present an analysis on the condition required for the existence of spatial invariance in the wave profile.