Universal relations for solitary waves in granular crystals under shocks with finite rise and decay times

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 than, equal to, 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 or mitigation. We have supported the analytical results by direct numerical simulations.

Figure 1

Schematic showing the correction plot to estimate $F_{{\mathrm{SW}}_{max}}.$

Figure 2

Comparison of a time series of the contact force profile between the exact numerical solution (numerical) and the Padé approximation (theory).

Figure 3

Representation of the time-parameter $t^P$ when $t^P<n{t}_0.$

Figure 4

(a) Triangular input force profile with fixed total pulse duration but variable rise time and (b) the first granule's velocity profile; here $\mathrm{Min}\left(V_{1\mathrm{st}}\right)$ stands for the end condition of the first granule; negative velocity shows the granule will be ejected.