The propagation and backscattering of soliton-like pulses in a chain of quartz beads and related problems. (I). Propagation

Abstract

We confirm that for vanishingly small loading and large impact condition, it may be possible to generate solitons in a chain of grains that are characterized by Hertzian contacts. For uniform or progressive loading conditions throughout the chain, one generates soft-solitons which are weakly dispersive in space and time. Under conditions of weak impact, one generates acoustic pulses through the chain. We describe the displacements, velocities and accelerations suffered by the individual grains when subjected to solitons, soft-solitons and acoustic pulses and describe the effects of restitution on the propagating pulse.

Introduction

Solitons are among the most fascinating objects known to the scientist [1], [2]. From a physical perspective, one can describe the dynamical soliton as a “tight bundle” of energy that can travel through a medium without any dispersion in space or in time. Solitons are encountered in a handful of physical systems and usually, under special initial conditions. As shown first theoretically, simulationally and experimentally by Nesterenko [3], [4], [5], granular systems belong to the select category of systems that support solitons. Somehow, the discovery of solitons in granular, or more specifically, Hertzian systems received limited attention during the decade that followed Nesterenko's initial work [3], [4], [5]. Recently, Sinkovits and Sen [6], [7] reported studies on impulse propagation in granular media in one and two spatial dimensions. Later, Coste et al. [8] reported experiments that confirmed the theoretical and experimental work of Nesterenko [3], [4], [5] and was consistent with the studies of Sinkovits and Sen [6], [7]. More recently, MacKay [9] has published a proof (as opposed to a solution to the dynamics problem) that Hertzian chains support solitons and Sen and Manciu have proposed a simple formula to describe the grain displacements during the passage of a soliton [10].

The key property of a “Hertzian system” [11] is that two elastic grains in intimate contact would exert mutual repulsion that would be proportional to their overlap raised to some power. Thus, if 2R denotes the diameter of a spherical grain, then the overlap δ⩾0 would be $\text{δ≡2R−r}{}_{\mathrm{<mtext>i,</mtext><mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>i+1</mtext>}}$, where $\text{r}{}_{\mathrm{<mtext>i,</mtext><mspace\; sp="0.16"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>i+1</mtext>}}$ is the actual distance between the centers of the adjacent grains. The potential energy gained by the compressed two-grain system turns out to be V(δ)=aδⁿ where a is some constant [11]. In Hertzian systems, one can calculate a which depends upon the elastic properties of the media that the grains are constituted of and n which is sensitive to the contact geometry of the grains [12]. Typically, Hertzian systems are anharmonic, i.e., n>2 and hence, they exhibit nonlinear dynamics. One expects interesting dynamical behavior in these systems. As we shall see, the problem of acoustic transport through granular systems reveals a variety of interesting phenomena. Under special conditions, one can support solitons in these systems as has been claimed by Nesterenko and others.

A few years ago, Sinkovits and Sen [6], [7] carried out particle-dynamics-based analyses of the propagation of sound and shock impulses through one and two-dimensional granular systems that are subjected to gravity. The key objective of these analyses was to confirm that particle-dynamics-based studies could readily recover the predictions regarding acoustic propagation in Hertzian systems subjected to gravity which are based upon elasticity theory [13] and probe the possible non-linear behavior at the shallow reaches of the bed. Among the results presented in [6], [7] was a finding that the kinetic energy of the granular bed, which has been subjected to an impulse at its surface, when plotted as a function of time and space, appeared to travel down the bed as a pulse that suffered very little dispersion. In summary, the gravitationally loaded granular beds that were probed, supported soliton-like excitations when they were subjected to appropriate impulses. A soliton-like object can be envisioned as a softer bundle of energy than a true soliton. It travels like a soliton but experiences some interaction with the medium as it travels. The softness of the energy bundle, or the dispersion suffered by the pulse, is significant even for pulses with relatively small amplitudes.

The presence of soliton-like pulses in perturbed Hertzian systems presents interesting possibilities. One such possibility is to exploit these pulses, which can travel through a granular medium with little dispersion, to probe for buried objects which may be indetectable by electromagnetic, ultrasonic and purely acoustic probes [14], [15]. Metal-poor landmines, unexploded ordnance, and buried structures in granular beds are examples of underground objects that may be detectable using backscattering of soliton-like pulses. We shall discuss our studies on the backscattering of soliton-like pulses from buried objects in the following article [16].

We discuss the following problems in this article: (i) the conditions under which solitons can form; (ii) the conditions under which soliton-like pulses can be sustained and (iii) the conditions under which acoustic pulses can form.

Details of our models and of the particle dynamics simulations are presented in Section 2. The results of our study are presented in Section 3. We close in Section 4 with a summary of the work and open questions.