Impulse penetration into idealized granular beds: behavior of cumulative surface kinetic energy

We report a particle dynamics based simulational study of the propagation of delta function mechanical impulses in idealized three-dimensional hexagonal close packed lattices of monosized Hertz spheres. This paper presents five key results on the kinetic energy of grains at the surface of a granular bed after the generation of a normal impulse into the bed. (i) We find that the time integrated or cumulative average kinetic energy per surface grain, kappa, drops as an impulse penetrates into the bed. The minimum value of kappa, say kappa(0), is reached at some time t=tau after the impulse has been generated. (ii) This value, kappa(0), depends upon the restitutional losses at the grain contacts and kappa(0) increases as restitutional losses at granular contacts increase in magnitude. (iii) The asymptotic value of kappa is denoted by kappa(final) . Our data show that increasing the area across which an impulse is generated, A, leads to kappa(final) proportional to A(-1/2) . (iv) If we assign random masses to our monosized grains, kappa(final) grows quadratically as a function of the range of mass variation about a mean mass. We find that at large times, i.e., t>>tau , kappa proportional to (1-exp [k (1-t/tau)]) , where the constant k is roughly independent of restitution for the typical values of restitution encountered. (v) Our data suggest that at early times, the backscattering process carries signatures of ballistic propagation of the mechanical energy while at late times, the backscattering process is reminiscent of vibrations of an essentially ergodic system. Given the ballisticlike propagation of mechanical energy into granular beds, we conclude that a wave equation based description of mechanical energy propagation into granular beds may not always be appropriate.