Harmonic and superharmonic wave propagation in 2D mechanical metamaterials with inertia amplification

An original parametric lattice model is proposed to investigate harmonic and superharmonic planar waves propagating in a two-dimensional mechanical metamaterial, whose periodic microstructure is characterized by local linkage mechanisms for pantographic inertia amplification. The free undamped dynamics in the metamaterial plane is governed by differential difference equations of motion, featuring geometric nonlinearities of both elastic and inertial nature. Within the weakly nonlinear oscillation regime, multi-harmonic wave solutions are achieved analytically, although asymptotically, by means of a suited perturbation method. At the lowest perturbation order, the linear dispersion properties (wavefrequencies and waveforms) of freely propagating monoharmonic waves are determined analytically as functions of the mechanical parameters. At higher perturbation orders, the amplitudes of the superharmonic wave components generated by quadratic and cubic nonlinearities are determined analytically, in the absence of internal resonances. Furthermore, the nonlinear corrections of the linear wavefrequencies are obtained. Smooth transitions from hardening to softening behaviors (or viceversa) are found to occur along particular propagation directions, depending on the wavelength. Physically, a pair of unexplored and interesting dynamic phenomena are disclosed. First, the free propagation of transversal waves along particular directions is characterized - independently of the wavenumber - by essentially nonlinear waveforms ( floppy modes), featuring evanescent amplitude-dependent wavefrequency. Second, the generation of superharmonic components oscillating with double and triple frequency multiples - caused by quadratic and cubic nonlinearities - can determine a loss of polarization ( superharmonic depolarization) in waves propagating with perfectly polarized waveforms in the linear field.