Wave propagation in elastic metamaterials with nonlinear local resonators

In this work, wave propagation in nonlinear elastic metamaterials is investigated. The nonlinear elastic metamaterial is modeled as two nonlinear mass-in-mass chains: one is the nonlinear chain with linear local resonators, and the other is the nonlinear chain with nonlinear local resonators. The amplitude-dependent dispersion relations are obtained by the perturbation method. Some interesting bandgap characteristics such as the emergence of inflection point, branch overtaking and branch cut-off in the dispersion curve can be observed in two softening nonlinear mass-in-mass chains. Accordingly, these characteristics are studied, and special attention is given to the bandgap bounding frequencies. The functional relationship between the bounding frequency and amplitude is expressed based on a semi-analytical hybrid method combining perturbation method and function fitting. Especially, we found several critical amplitudes to play important roles in forming the inflection points. The numerical results show that the bandgaps disappear as the amplitude increases in two nonlinear chains. The inflection points of the acoustical branch do not appear before the bandgap disappears for the nonlinear chain with linear local resonators. However, they have appeared before the bandgap disappears for the nonlinear chain with nonlinear local resonators. Also, effects of the nonlinear local resonators on the bandgap are analyzed.