Nonlinear wave propagation in locally dissipative metamaterials via Hamiltonian perturbation approach

The cellular microstructure of periodic architected materials can be enriched by local intracellular mechanisms providing innovative distributed functionalities. Specifically, high-performing mechanical metamaterials can be realized by coupling the low-dissipative cellular microstructure with a periodic distribution of tunable damped oscillators, or resonators, vibrating at relatively high amplitudes. The benefit is the actual possibility of combining the design of wave-stopping bands with enhanced energy dissipation properties. This paper investigates the nonlinear dispersion properties of an archetypal mechanical metamaterial, represented by a one-dimensional lattice model characterized by a diatomic periodic cell. The intracellular interatomic interactions feature geometric and constitutive nonlinearities, which determine cubic coupling between the lattice and the resonators. The non-dissipative part of the coupling can be designed to exhibit a softening or a hardening behavior, by independently tuning the geometric and elastic stiffnesses. The nonlinear wavefrequencies and waveforms away from internal resonances are analytically determined by adopting a perturbation technique. The employed approach makes use of tools borrowed from Hamiltonian perturbation theory, together with techniques often used in the context of nearly-integrable Hamiltonian systems.The dispersion spectra are determined in closed, asymptotically approximate, form as a nonlinear function of the time-dependent decreasing amplitude decrement. The invariant manifolds defined by the harmonic periodic motions are also analytically determined. The asymptotic results are further validated numerically.