Energy Relaxation in Damped Two-Dimensional Graded Elastic Lattices

Abstract

We have studied the relaxation rate of vibrational modes in damped two-dimensional graded mass lattices. The relaxation rate spectrum in the weak damping limit can be obtained analytically through a perturbation theory based on the semiclassical quantum analogue envelope function. We found dip or peak structures on the relaxation rate spectrum. The dip or peak structures can be described quantitatively by the asymptotic behavior of relaxation rate at the transition frequencies. The frequency dependence of the relaxation rate is qualitatively explained by the mode patterns of gradon modes. The validity of the analytic results is confirmed by numerical solution with weak damping. In the strong damping case, we need to retain higher-order perturbations. These results can be applied to the energy relaxation in analogous systems.