Nonlinear mixed solitary - Shear waves and pulse equi-partition in a granular network

We study primary pulse transmission in a two-dimensional granular network composed of two ordered chains that are nonlinearly coupled through Hertzian interactions. Impulsive excitation is applied to one of the chains (designated as 'excited chain'), and the resulting transmitted primary pulses in both chains are considered, especially in the non-directly excited chain (the 'absorbing chain'). A new type of mixed nonlinear solitary pulses-shear waves is predicted for this system, leading to primary pulse equi-partition between chains. An analytical reduced model for primary pulse transmission is derived to study the strongly nonlinear acoustics in the small-amplitude approximation. The model is re-scalable with energy and parameter-free, and is asymptotically solved by extending the one-dimensional nonlinear mapping technique of Starosvetsky (2012). The resulting nonlinear maps governing the amplitudes of the mixed-type waves accurately capture the primary pulse propagation in this system and predict the first occurrence of energy equipartition in the network. To confirm, in part, the theoretical results we experimentally test a series of two-dimensional granular networks, and prove the occurrence of strong energy exchanges leading to eventual pulse equi-partition between the excited and absorbing chains, provided that the number of beads is sufficiently large.