Nonlinear dispersion properties of one-dimensional mechanical metamaterials with inertia amplification

Abstract

Abstract Architected metamaterials offering superior dynamic performances can be conceived by inducing local mechanisms of inertia amplification in the periodic microstructure. A one-dimensional cellular lattice characterized by a pantograph mechanism in the tetra-atomic cell is proposed as minimal physical realization of inertially amplified metamaterial. A discrete model is formulated to describe the undamped free dynamics of the cell microstructure. The ordinary differential equations of motion feature quadratic and cubic inertial nonlinearities, induced by the indeformability of the pantograph arms connecting the principal atoms with the secondary atoms, serving as inertial amplifiers. An asymptotic approach is employed to analytically determine the dispersion properties governing the free propagation of harmonic waves in the pantographic metamaterial. First, the linear wavefrequencies and waveforms are obtained by solving the eigenproblem governing the lowest asymptotic order. An invariant parametric form is achieved for the pass and stop band structure, corresponding to propagation and attenuation branches of the dispersion spectrum in the plane of complex wavenumbers. The major effects due to the mass ratio of the inertial amplifiers are discussed. Particularly, the existence conditions, amplitude and centerfrequency of the band gap separating the acoustic and optical pass bands are determined analytically. Second, the nonlinear wavefrequencies and waveforms are obtained by solving the hierarchical linear problems governing the higher asymptotic orders. Analytical, although asymptotically approximate, functions are achieved for the nonlinear wavefrequencies and waveforms, which show quadratic dependence on the oscillation amplitudes. The mechanical conditions for the softening/hardening behaviour of the nonlinear wavefrequencies and the different topological properties of the invariant manifolds associated to the nonlinear waveforms are discussed. Finally, numerical simulations are provided to validate the analytical results.