Universal Relations for Solitary Waves in Granular Crystals under Finite Rise-decay Duration Shocks

We focus on solitary waves generated in arrays of lightly contacting spherical elastic granules by shock forces of steep rise and slow decay durations, and establish a priori: (i) whether the peak value of the resulting solitary wave would be greater, equal, or less than the peak value of the input shock force; (ii) the magnitude of the peak value of the solitary waves; (iii) the magnitude of the linear momentum in each solitary wave; (iv) the magnitude of the linear momentum added to the remaining granules, if the first granule is ejected; and (v) a quantitative estimate of the effect of the granules' radius, density and stiffness on force amplification/mitigation. We have supported the analytical results by direct numerical simulations.