Strongly nonlinear traveling waves in granular dimer chains

Abstract

This study is concerned with the dynamics of spatially periodic travelling waves supported by strongly nonlinear granular dimer chains with no pre-compression. In particular, we demonstrate numerically the formation of special families of travelling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains. These traveling waves depend on a single system parameter defined as the mass ratio of the two granules forming each dimer pair of the chain. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. In the present work we demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter and the energy of the motion vary. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.