Nonlinear wave propagation in a two-dimensional lattice model of textile metamaterials

An original parametric lattice model is formulated to describe the propagation of harmonic elastic waves in two-dimensional textile metamaterials. Within a weak nonlinear regime, the free undamped motion of the textile metamaterial, starting from a spatially periodic pretensioned configuration, is governed by nonlinear differential difference equations. Quadratic and cubic nonlinearities arise from the elastic contact between plain woven yarns. By applying the asymptotic method of multiple scales, the nonlinear dynamics of the periodic cell are governed by an ordered hierarchy of linear perturbation equations. Therefore, by virtue of the linearity and spatial periodicity, the Floquet-Bloch theory is recursively applied at each order of the perturbation equations to study the linear and nonlinear dispersion properties. Specifically, the lowest order solutions return the linear dispersion diagram characterizing the free undamped propagation of small-amplitude harmonic waves. Within the technical range of the parameters, the dispersion diagram shows the coexistence of two passbands, separated by a large mid-frequency stopband. By virtue of an energy-based classification criterion, the different polarizations of the waves propagating in the low-frequency and high-frequency bands are disclosed. The higher orders allow to determine analytically the combined effects of the nonlinearities on the dispersion properties, in the absence of internal resonances. In particular, the wavefrequencies exhibit a quadratic dependence on the wave oscillation amplitude, characterized by a systematic softening behavior. Moreover, the amplitudes of the damped nonlinear response induced by the external excitation due to a harmonically oscillating pretension are analyzed in the frequency domain and the instability regions of the primary resonance are obtained in the whole range of feasible mechanical parameters. Finally, analytical results are successfully validated by numerical simulations in the time domain.