Wave propagation in viscoelastic metamaterials via added-state formulation

Mechanical metamaterials composed of a non dissipative periodic microstructure of flexible ligaments and stiff rings, viscoelastically coupled with local resonators, are considered. By following a variational approach, the linear damped dynamics are described according to a beam lattice formulation, valid for a generic coordination number. A consistent theoretical description of the viscoelastic ring-resonator coupling is introduced, leading to an integral–differential form of the governing equations. The enlargement of the model dimension by addition of viscoelastic states allows to conveniently describe the metamaterial dynamics through a system of ordinary differential equations. Therefore, the application of the Z-transform and a suitable mapping in the wavevector space allows to obtain the system of ordinary differential equations governing the free and forced propagation of damped waves. Subsequently, the polynomial eigenproblem ruling the harmonic free wave propagation in the complex frequency domain is stated by applying the bilateral Laplace transform, whereas an analytical procedure is outlined to solve the forced wave propagation problem in the time domain. The triangular beam lattice metamaterial is considered as significant benchmark for application of the theoretical and methodological framework. Considering first the free wave propagation, the properties of the dynamic stiffness matrix are discussed and the dispersion frequency spectrum is parametrically investigated, with focus on the spectral effects of the viscoelastic coupling. The dependence of the stop bandwidth amplitude between the low and high-frequency spectral branches on the relaxation time characterizing the viscous function is also highlighted and discussed. Moreover, the forced response to a harmonic external point source is numerically investigated in the time domain. The results are qualitatively compared and quantitatively discussed by distinguishing the fundamental cases of non-resonant, resonant and quasi-resonant excitation frequency.