Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators

Abstract Beam lattice materials are characterized by a periodic microstructure realizing geometrically regular pattern of elementary cells. The dispersion properties governing the free dynamic propagation elastic waves can be studied formulating parametric discrete models cellular and applying Floquet-Bloch theory. Within this framework, wave means spectral design techniques and/or energy dissipation mechanisms is major issue theoretical applied interest. Specifically, inhibited purposely designing microstructural parameters to open stop bands in material spectrum at target center frequencies. Based on these motivations, general formulation presented for determining mechanical metamaterials, modeled as locally resonant beam lattices with generic coordination number. mechanism local resonance realized tuning auxiliary oscillators, viscoelastically coupled microstructure. As peculiar aspect, viscoelastic coupling derived based Boltzmann superposition integral, whose kernel approximated Prony series. Consequently, damped governed linear homogeneous system integro-differential equations motion. Therefore, differential motion frequency-dependent coefficients obtained in-space Z -transform in-time bilateral Laplace transform. complex-valued branches characterizing determined parametrically analyzed quadrilateral cell. may exceed model dimension, due occurrence pure-damping components. Particularly, spectra corresponding Laurent series approximations investigated solution admissibility convergence increasing order analyzed. standard viscous damping recovered first-order approximation. Low-order found underestimate real imaginary parts spectrum, well low-frequency bandwidth. Finally, forced response harmonic mono-frequent external point excitation investigated. metamaterial responses non-resonant, quasi-resonant forces compared discussed from qualitative quantitative viewpoint.