有摩擦的颗粒链脉冲动力学

ar X iv :c on d- m at /0 30 70 80 v1 [ co nd -m at. sta t-m ec h] 3 Ju l 2 00 3 Pulse Dynamics in a Chain of Granules With Friction Alexandre Rosas and Katja Lindenberg Department of Chemistry and Biochemistry, and Institute for Nonlinear Science University of California San Diego La Jolla, CA 92093-0340 (Dated: February 2, 2008) We study the dynamics of a pulse in a chain of granules with friction. We present theories for chains of cylindrical granules (Hertz potential with exponent n = 2) and of granules with other geometries (n > 2). Our results are supported via numerical simulations for cylindrical and for spherical granules (n = 5/2). PACS numbers: 45.70.-n,05.45.-a,45.05.+x I. INTRODUCTION The propagation of pulses in granular materials has been a subject of vigorous recent interest. Seminal work on this subject was carried out by Nesterenko [1] two decades ago in his analysis of the propagation of nonlinear compression pulses along a line of particles. 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]. The preponderance of the work has been numerical, but important analytic advances have also been made, as well as experimental verifications of some of the theoretical predictions [6, 16]. The standard generic model potential between monodisperse elastic granules that repel upon overlap according to the Hertz law is given by [19, 20] V (δk,k+1) = a|δ| n k,k+1, δ ≤ 0, V (δk,k+1) = 0, δ > 0. (1) Here δk,k+1 ≡ 2R− [(zk+1 + yk+1)− (zk + yk)] , (2) a = 2 5D(Y, σ) ( R 2 )1/2 , D(Y, σ) = 3 2 ( (1− σ2) Y ) , (3) Y and σ denote Young’s modulus and Poisson’s ratio, zk denotes the initial equilibrium position of granule k in the chain, and yk is the displacement of granule k from this equilibrium position. The geometric parameter R is the radius if the particles are elastic spheres. More generally, R is determined by the principal radii of curvature of the surfaces at the point of contact [20]. The exponent n is 5/2 for spheres, it is 2 for cylinders, and in general depends on geometry. The displacement of the k-th granule (k = 1, 2, . . . , L) in the chain from its equilibrium position in a frictional medium is governed by the equation of motion m d2yk dτ2 = −γ˜ dyk dτ − a(yk − yk+1) n−1θ(yk − yk+1) + a(yk−1 − yk) n−1θ(yk−1 − yk). (4) Here θ(y) is the Heavyside function, θ(y) = 1 for y > 0, θ(y) = 0 for y < 0, and θ(0) = 1/2. It ensures that the particles interact only when in contact. Note that for open boundary conditions the second term on the right hand 2 side of this equation is absent for the last granule and the third term is absent for the first, while for periodic boundary conditions yk+L = yk. In terms of the rescaled variables and parameters yk = ( mv20 a )1/n xk, τ = 1 v0 ( mv20 a )1/n t, γ = γ˜ mv0 ( mv20 a )1/n , (5) Eq. (4) can be rewritten as x¨k = −γx˙k − (xk − xk+1) n−1θ(xk − xk+1) + (xk−1 − xk) n−1θ(xk−1 − xk), (6) where a dot denotes a derivative with respect to t. Initially the granules are placed along a line so that they just touch their neighbors in their equilibrium positions (no precompression), and all but one granule, granule i, are at rest. The velocity of granule i is v0 (the impulse). In the rescaled variables the initial conditions become xk(0) = x˙k(0) = 0, ∀k 6= i, xi(0) = 0, and x˙i(0) = 1. In our work we set i = 1. In the absence of the frictional contribution −γ˜dyk/dτ or −γx˙k, when n > 2 an initial impulse settles into a pulse that becomes increasingly narrow with increasing n, and propagates at a velocity that is essentially constant and determined by n and by the amplitude of the pulse. For n = 2 the pulse spreads in time and travels at a constant velocity independent of pulse amplitude. In the latter case there is considerable backscattering that leads to backward motion of all the granules behind the pulse, whereas the backscattering is minimal for n > 2 [9]. The pulse is a completely conservative solitary wave in the limit n→∞. Our interest lies in ascertaining the effects of the frictional contributions on these results. Although friction is on occasion mentioned in theoretical and numerical work, it is usually mentioned as an element that is neglected or omitted. However, its presence and importance in experiments is recognized as inevitable [16]. Note that frictional effects may arise not only from the forces between the granular chain and the surrounding medium but also from the conversion of translational motion to other degrees of freedom (e.g. rotation) [16]. In an earlier paper we analyzed in detail the effects of frictional forces on the dynamics of two granules, specifically the way in which forward and backward motion of the granules is affected by these forces [21]. Herein we extend that work to a chain of granules. Since our approach will in general follow established methods, it is useful to lay out at the outset an overview of the principal approaches that have been implemented in the study of pulse dynamics in frictionless chains, and the various features that determine these dynamics. Theoretical studies of pulse dynamics in frictionless chains have mainly relied on three approaches: (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]. From these studies it is clear that there three features determine the dynamics in these chains: (1) The power n in the potential; (2) The absence of a restoring force; and (3) The discreteness of the system. While each of these leads to essential aspects of the dynamics, the literature is not always entirely clear on which feature is determinant in a particular behavior, nor is it always clear which of these features is being approximated or even ignored. One example is the equivocal connection between the dynamics of granular systems and systems with power law interaction potentials that include a restoring force. For instance, there is an extensive literature on the Fermi-Pasta-Ulam (FPU) chain with purely nonlinear interactions of the form appearing in Eq. (6) with n > 2 but without the Heavyside theta functions [23]. Highly localized pulses propagate in these systems [14, 24], but their relation to the localized solutions in granular systems is by no means clear. This ambiguity is exacerbated by the fact that discrete systems are frequently approximated by continuum equations. While the pulse-like (soliton or solitary wave) solutions that emerge from these approximations are assumed (and even shown numerically) to describe certain aspects of granular system, the continuum approximations never explicitly respect the absence of a restoring force, i.e., they are more clearly justifiable for FPU-like systems. And so with these solutions in hand, it is not clear what consequences of the absence of a restoring force and of the discreteness of the system have been lost. In some sense opposite to the continuum approximations are two-body or three-body phenomenologies. These provide considerable qualitative insight into some consequences of discreteness, but are quantitatively not sufficiently accurate because the true dynamics extend over more than two or three granules. In this paper, even while focusing on the effects of friction, we make an attempt to provide some clarity on these issues. Numerical simulations are of course extremely helpful and ultimately an accurate test of theory. Note that the rescaled system Eq. (6) has no free parameters in the absence of friction and only a single parameter when there is lenovo 高亮 3 friction, and therefore numerical characterization is particularly straightforward in this particular system. However, this universality is lost with any of a number of modifications that might be interesting and have been considered recently such as, for example, tapering of the masses in the chain [3, 15, 16] and mass [7, 11] or frictional [21] disorder. Consequently, it is desirable to have a strong analytic backdrop. The paper is organized as follows. Section II deals with a chain of cylindrical granules. First we review existing results for frictionless granules, discussing them in the context of the issues and uncertainties noted earlier. We then extend the theory to granules subject to friction, and present numerical simulations in support of these results. In Sec. III we follow the same presentation sequence for granules with n > 2, with special attention to spherical granules in our numerical simulations. A summary of results and of future directions is presented in Sec. IV. II. CYLINDERS (n = 2) Cylindrical granules have provided a theoretical testbed for dynamics in granular chains because the exponent n = 2 leads to analytic manageability not available for other potentials. Although sometimes called the “harmonic” case, it should be remembered that the Hertz potential is quite different because there is no restoring force, that is, the cylindrical Hertz potential is half of a harmonic potential. The derivative of the force law therefore changes discontinuously between extension and compression. This leads to considerable differences in the chain dynamics. Nevertheless, some aspects of the cylindrical chain dynamics can be inferred from those of a harmonic chain, and this can be exploited to great advantage in the analysis. If one were to ignore the absence of a restoring force and implement the simplest continuum approximation to Eq. (6) in the absence of friction, the result would be a simple wave equation with diffusive coupling whose solutions do not represent the observed behavior of the n = 2 chain. In reality, an initial impulse in the chain described by Eq. (6) in the absence of friction moves as a spreading pulse. Although the pulse spreads and sheds some energy, the waveform can nevertheless be clearly identified as a pulse [9]. Its maximum kmax(t) travels forward with a constant unit velocity and a displacement amplitude that increases as t1/6. The pulse spreads in time as t1/3, more slowly than it would in a system with diffusive coupling. It is interesting to understand which of these features are due to the discrete nature of the chain, and which are due to the absence of a restoring force. Further, once these questions have been answered, it is interesting to explore what happens to these features in the presence of friction. Our presentation of the chain of cylindrical granules consists of three parts. First we review the results of Hinch and Saint-Jean (HSJ) [9] for a frictionless chain, and we recast the problem in a way that clarifies some of the approximations made in that work and pinpoints the sources of differences between the n = 2 chain and a purely diffusive coupling. We supplement this review with our analytic results that reproduce some of their purely numerical ones. Then we modify this analysis to include the effects of weak hydrodynamic friction on the granules. Finally, we complement this analytic (and necessarily approximate) treatment with a comparison with numerical simulation results for the frictional chain of cylindrical granules. A. Frictionless Granules - Theory The HSJ theory is based on the following approximations implemented consecutively and independently. 1. The solution is assumed to be described by a traveling pulse of constant form which propagates at constant unit speed and has an amplitude b and width λ that vary slowly with time: x(k, t) = b(t)f ( k − t λ(t) ) . (7) 2. The pulse is assumed to retain almost all of its initial energy. HSJ show that assumption 1. leads to equipartition of this energy between potential and kinetic forms, as should be the case for a harmonic potential. These two assumptions are sufficient to lead to the conclusion that λ ∝ b2. 3. A continuum approximation is implemented that takes into account some discreteness effects. In the lowest order diffusive coupling approximation one would set the difference xk+1 + xk−1 − 2xk ≈ ∂ 2x(k)/∂k2. Retention of the next term in a Taylor series expansion of xk±1 about xk leads to the continuum approximation that incorporates some of the effects of discreteness: ∂2x(k, t) ∂t2 = ∂2x(k, t) ∂k2 + 1 12 ∂4x(k, t) ∂k4 . (8) lenovo 高亮 lenovo 高亮 lenovo 高亮 lenovo 高亮 4 Note that this expansion includes a restoring force. A transformation to a moving frame with unit propagation velocity, i.e., a change of variables from k and t to ν = k − t and t, transforms this equation to ∂2x ∂t2 − ∂2x ∂t∂ν = 1 12 ∂4x ∂ν4 . (9) Regardless of the form of b(t) or of the function f , this is sufficient to establish that the solution Eq. (7) is consistent with this equation only if λ ∼ t1/3, and also that the next term in the Taylor series expansion is unimportant for this result. The width of the pulse is therefore governed by the first manifestation of discreteness. One assumes that the absence of a restoring force does not affect this result because the granules within the pulse are in fact overlapping most of the time and hence subject mainly to the repulsive portion of the potential. Note that the continuum approximation together with the conservation of energy are then sufficient to arrive at the conclusion that b(t) ∼ t1/6, i.e., that the pulse amplitude actually grows with time. 4. It is not yet clear that (7) is actually compatible with (8) until one determines the function f . Substitution of (7) into (8) and retention of leading terms in t leads to the equation for f(ξ), f ′′′′ − 8ξf ′′ − 4f ′ = 0. (10) This equation has four solutions. The one consistent with the requirement that f(ξ) decays for large ξ (i.e., ahead of the wave) and consistent with the assumption of energy conservation by the pulse is [26] f(ξ) = N ∫ ∞ ξ Ai2(21/3y)dy (11) where Ai(z) is an Airy function and N = [ Ep∫∞ ξ0 Ai4(21/3y)dy ]1/2 = 3.533 √ Ep. (12) Here Ep is the pulse energy and 2 1/3ξ0 is the first zero of Ai(z). These features describe the traveling displacement pulse of increasing width and amplitude quite accurately, as shown by the numerical simulations in [9]. We note that the total energy of the system with the initial unit velocity impulse at one granule is 1/2. The numerical results of HSJ lead to an asymptotic pulse energy of Ep = 0.48, that is, 96.2% of the energy resides in the pulse. Below we calculate the pulse energy analytically. In addition to the forward traveling pulse, HSJ also investigate the momentum of the particles ejected backward as the pulse goes by. That particles must be ejected is a consequence of the conservation of momentum: the traveling pulse of constant energy and increasing width and displacement amplitude carries increasing forward momentum, which must be balanced by the backward momentum of the ejected particles. One then arrives at the next item on the list of assumptions: 5. The absence of a restoring force is explicitly recognized in the calculation of the momentum of the ejected particles, which simply keep traveling backward with the momentum they acquire at the moment of separation from the pulse. Equating the rate of change of the momentum of the forward pulse at time t to that of the particle ejected at that time leads to the conclusion that the backward momentum of the nth particle is x˙n = −ct −5/6 = −cn−5/6, the latter equality arising from the unit speed of pulse propagation. The numerical simulations of HSJ lead to the value c = 0.158, a value that we obtain analytically. To obtain analytic results for the pulse energy Ep and the constant of proportionality c, we note that the forward momentum of the propagating pulse is P = ∑ x˙(k,t)>0 x˙(k, t) = ∑ x˙(k,t)>0 b(t) λ(t) f ′(ξ) = b(t) ∫ ∞ ξ0 f ′(ξ)dξ = N˜b(t) (13) where N˜ = N ∫ ∞ ξ0 Ai2(21/3ξ)dξ = 1.379 √ Ep. (14) lenovo 高亮 lenovo 高亮 lenovo 高亮 lenovo 高亮 lenovo 高亮 lenovo 高亮 lenovo 高亮 比例常数 5 The rate of change of the forward momentum then is P˙ = N˜ b˙(t). Since the pulse velocity is unity, this is the momentum transferred to the last particle as it is ejected: vb = −P˙ = −N˜ b˙(t) = − N˜ 6 t−5/6 = − N˜ 6 n−5/6. (15) The backscattered energy then is Eb = 1 2 ( N˜ 6 )2 ∞∑ k=1 k−5/3 = 0.056Ep (16) and for the total energy we obtain E = Ep + Eb = 1.056Ep, (17) from which it immediately follows that asymptotically the energy in the pulse is Ep = 0.947E. (18) Thus, 94.7% of the energy resides in the pulse (to be compared to the HSJ value of 96.2% obtained from simulations) and the remainder is backscattered. With the initial condition E = 1/2 used in all simulations we thus have for the constants defined earlier N = 2.431 and N˜ = 0.949. The constant c in HSJ is c = N˜/6 = 0.158, exactly as they obtained from numerical simulations. This essentially completes the solution. The summary description is then that an initial velocity impulse propagates forward at unit speed, with a width that grows as t1/3 and a displacement amplitude that grows as t1/6. This pulse carries almost all of the initial energy and its momentum increases. This increase in momentum is compensated by granules that are ejected backward as the pulse passes. The speed of the ejected granules decreases, the nth granule being ejected with a speed proportional to n−5/6. Further insights can be gained by viewing this problem a bit differently. Suppose that we do not implement a continuum approximation at all, but instead focus on the full discrete equation x¨k = xk+1 + xk−1 − 2xk. (19) Like the continuum equation, this of course includes restoring forces. This equation can be solved exactly [24], xk(t) = ∫ t 0 J2k−2(2t ′)dt′, (20) where Jn(z) is the Bessel function of the first kind [26]. The solution is a moving spreading pulse along with oscillatory displacements that are left in its wake (precisely because there are restoring forces). The peak and width of the pulse can be obtained from the properties of the Bessel functions, in particular from knowledge of their first two zeroes and the first maximum [27]. The maximum of the pulse occurs at kmax(t) = t+O(t 1/3), so the pulse velocity is unity. Its width increases as t1/3. In these features the solution is similar to the one assumed in Eq. (7). However, the amplitude of the displacement pulse does not grow but is instead constant in time. (If we write the Bessel function solution in the form (7) but with b(t) = const independent of t, we find from substitution into Eq. (8) that f satisfies the equation f ′′′′ − 8zf ′′ − 8f ′ = 0.) The pulse energy according to this description is not constant but instead decreases as t−