Impulse propagation in granular systems

Impulse propagation in granular systems Surajit Sen,* Soumya Chakravarti,+ Donald P. Visco, Jr.,¶ Masami Nakagawa,† David T. Wu‡ and Juan Agui, Jr.♦ *Department of Physics State University of New York at Buffalo, Buffalo, NY 14260-1500, USA +Department of Physics, California State Polytechnic University, Pomona, CA 91768, USA ¶ Department of Chemical Engineering, Tennessee Technological University, Cookeville, TN 38305, USA † Department of Mining Engineering, Colorado School of Mines, Golden. CO 80401, USA ‡ Department of Chemistry and Department of Chemical Engineering, Colorado School of Mines, Golden. CO 80401, USA ♦ NASA - Glenn Research Center, MS 500-102, 21000 Brookpark Road, Cleveland, OH 44135, USA Abstract: We review recent developments on the topic of impulse propagation in assemblies of elastic beads, where the beads interact upon contact via the (nonlinear) Hertz potential. The role of solitary waves in granular systems is emphasized. The effects of impurities, disorder and restitution are discussed. The possible applications of this research in the development of new technologies are mentioned.

INTRODUCTION The problem of sound propagation in granular beds such as in soil has a long history. It has been known, for instance that low frequency sound, typically in the range of some 100-300 Hz, penetrates well through shallow soil [1-5]. It has also been known that large impulses can be used to penetrate deep into soil and can be used to ascertain the presence of oil deposits and rock formations and the like [5]. Systematic analyses of the propagation of mechanical shocks into shallow depths of granular beds had remained unexplored until relatively recently [6-12]. In this work, we first present a study of impulse propagation in a chain of grains and show that any impulse propagates as a rather tight energy bundle or more precisely as a solitary wave in such a system. The discussion goes on to explore the properties of the solitary waves. Our fundamental studies lend themselves to several technologically important applications. We discuss two such applications. One of these concerns the possibility of designing novel shock absorption systems with the possibility of partially recovering the energy of the impulse. A second concerns the possibility of using acoustic images to image buried objects in complex media such as soil.

EQUATION OF MOTION AND SOLITARY WAVES We consider a monodisperse chain of elastic spheres of radius R in which the spheres barely touch one another, i.e., there is “zero loading” between the spheres. The spheres repel upon overlap by an amount δi,i+1 ≡ 2R-((zi+1+ui+1) - (zi+ui)), where zi describes the initial equilibrium position of some grain i in the chain, and ui is the displacement of the same grain from the equilibrium position. Then, according to the Hertz law [13] the repulsive potential between two adjacent spheres is given by, V(δi,i+1) = a|δi,i+1 | 5/2, δ > 0, V(δi,i+1) ≡ 0, δ2. The value of n depends upon the geometry of the contact region between two adjacent elastic bodies in contact and in general, these bodies do not have to be spheres. When one considers spheres, n = 5/2 [13-14]. Given the magnitude of n, it may be noted that the repulsive potential V(δi,i+1), as stated in Eq.(1), is intrinsically nonlinear in the sense that one cannot write the force between the compressed spheres as having any linear component. The absence of a linear component in the force law (i.e., absence of Hooke’s law) implies that sound propagation is not possible in a chain of elastic beads at zero precompression, a phenomenon that has been referred to in the literature as "sonic vacuum" [6]. To initiate sound propagation, one must introduce some precompression in the system. The simplest case is uniform precompression, say by an amount ∆, effected on every grain. The equation of motion of a bead in the chain (not at the boundaries) then becomes, m d2ui(t)/dt2 = na{[∆ + ui-1(t) - ui(t)]n-1 - {[∆ + ui(t) - ui+1(t)]n-1},

(2)

where the right hand side (RHS) of Eq.(2) can be expanded as a perturbation against ∆ when ∆>0. Nesterenko [6] showed that if ui varies slowly in space, i.e., if the longwavelength limit is invoked, and if ∆→0 and 2