Shock propagation in a granular chain

Abstract

We numerically solve the propagation of a shock wave in a chain of elastic beads with no restoring forces under traction (no-tension elasticity). We find a sequence of peaks of decreasing amplitude and velocity. Analyzing the main peak at different times we confirm a recently proposed scaling law for its decay.

How Hertzian Solitary Waves Interact with Boundaries in a 1D Granular Medium

“…It was recently reported [10] that identical and opposite propagating solitons do not preserve themselves upon collision and hence these are solitary waves rather than solitons. Several detailed numerical studies have been devoted to understand the interactions of solitary waves with a perfectly reflecting wall [10,11,12,13,14] and show that tiny secondary solitary waves are generated as a solitary wave is reflected off a wall [10,12]. However, due to experimental difficulties, no close comparison between experiments and simulations has so far been established.…”

Elastic–Plastic Wave Propagation in Uniform and Periodic Granular Chains

“…frequency components of the initial disturbance move through the chain [39][40][41]. Because the different frequency components move at different velocities, a steady front is not formed in linear materials and the pulse widens as it travels down the chain.…”

“…6(c)) are visibly different than the behavior of the single material chains. Heterogeneous chains of Hertzian materials have been shown to have wave speed dependence on the mass ratio of the constituent spheres, whereas wave speeds in the harmonic linear chains only depend on the average density of the constituents [11][12][13][39][40][41]. For chains of spheres with identical radii but two alternating materials, the leading wave velocity in the dimer elastic-plastic materials shown in Fig.…”

Pulse dynamics in a chain of granules with friction

“…This early work firmly established the nonlinear flavor of the problem: Nesterenko showed that under appropriate assumptions, among them the slow spatial variation of the displacements of the particles, the equations of motion for granular particles could in most cases be approximated by a continuous nonlinear partial differential equation that admits a soliton solution (later shown to actually be a solitary wave solution [2,3]) for a propagating perturbation in the chain. The recent revival of interest in the subject [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16] has been triggered in part by a concern with important technological applications such as the design of efficient shock absorbers [14], the detection of buried objects [5,6,7,8], and the fragmentation of granular chains [9]. The revival has involved advances in the modeling, simulation, and solution of the equations associated with important features of granular materials such as their discreteness [2,3,10,14], dimensionality [14], disorder [7,11,14], and loading provided by gravitational forces [4,7,13,14,17,18].…”

“…(1) Numerical solution of the equations of motion [4,7,9,10,11,14,18]; (2) Continuum approximations to the equations of motion followed by exact or approximate solutions of these approximate equations [1,9,18]; and (3) Phenomenology about properties of pairwise (or at times three-body) collisions together with the assumption that pulses are sufficiently narrow to be principally determined by these properties [15,22].…”