SUMMARY

The effect of finite-amplitude wave propagation in nonlinear periodic structures is analyzed within the framework of dispersion band structures. Recent research shows that certain periodic arrangements of materials with high acoustic impedance contrasts can be employed to control wave propagation. These systems are often termed phononic crystals or metamaterials. Design of these systems usually relies on computation and analysis of dispersion band structures, which contain information about wave propagation speed and direction. The location and influence of complete (and partial) band gaps is a particularly interesting characteristic. Wave propagation is prohibited for frequencies that correspond to band gaps; thus, periodic systems behave as filters, wave guides, and lenses at certain frequencies. Most phononic crystals or metamaterials are designed with the intent of controlling small-amplitude waves within a limited frequency range. Most methods proposed for adding tunability to these systems utilize external stimuli such as pre-compression in the form of physical contact or electromagnetic forces to change the physical topology or material properties. Our focus on finite-amplitude wave propagation has similar effects without the need for external stimuli or modification to the periodic system.
We analyze wave propagation in nonlinear metamaterials using a Bloch wave-based perturbation analysis. The analysis leads to an open set of finite-difference equations which are solved for first-order corrections to the dispersion band structure. We will investigate the influence of material and geometric nonlinearities on the dispersion relationship. A novel aspect of the proposed work is the ease with which discretized systems may be analyzed. Geometrically complex structures may be analyzed using system matrices that result from a finite-element discretization. This connection enables the design and optimization of unit cells with nonlinear elements. The multilayer material is explored as a first demonstration of the nonlinear analysis. Higher-dimensional systems with more complex geometry and nonlinearities will follow. The nonlinear interaction of two waves and its effect on the dispersion structure is further considered as an option for wave propagation management and control.