Solitary waves in a general class of granular dimer chains

We report on a countable infinity of traveling solitary waves in a class of highly heterogeneous ordered one-dimensional granular media, in particular, granular dimers composed of an infinite number of periodic sets of “heavy” elastic spherical beads in contact with N “light” ones; these media are denoted as 1:N granular dimers. Perfectly elastic Hertzian interaction between beads is assumed and no dissipative forces are taken into account in our study; moreover, zero pre-compression is assumed, rendering the dynamics strongly nonlinear through complete elimination of linear acoustics from the problem. After developing a general asymptotic methodology for the 1:N granular dimer, we focus on the case N=2 and prove numerically and asymptotically the existence of a countable infinity of traveling solitary waves in the 1:2 dimer chain. These solitary waves, which can be regarded as anti-resonances in these strongly nonlinear media, are found to be qualitatively different than those previously studied in homogeneous and 1:1 dimer chains (i.e., composed of alternating heavy and light beads) which possess symmetric velocity waveforms. In contrast, for traveling solitary waves in 1:2 dimers, the velocity waveforms of the responses of the heavy beads are symmetric, whereas those of the light beads are non-symmetric. Interestingly, we show that no such solitary waves can be realized in general 1:N granular dimers with N>2, although near-solitary waves can exist in these systems based on slow-fast frequency approximations.