Nonlinear Dynamics in Granular Columns

Interactions of solitary waves in integrable and nonintegrable lattices

We study how the dynamics of solitary wave (SW) interactions in integrable systems is different from that in nonintegrable systems in the context of crossing of two identical SWs in the (integrable) Toda and the (non-integrable) Hertz systems. We show that the collision process in the Toda system is perfectly symmetric about the collision point, whereas in the Hertz system, the collision process involves more complex dynamics. The symmetry in the Toda system forbids the formation of secondary SWs (SSWs), while the absence of symmetry in the Hertz system allows the generation of SSWs. We next show why the experimentally observed by-products of SW-SW interactions, the SSWs, must form in the Hertz system. We present quantitative estimations of the amount of energy that transfers from the SW after collision to the SSWs using (i) dynamical simulations, (ii) a phenomenological approach using energy and momentum conservation, and (iii) using an analytical solution introduced earlier to describe the SW in the Hertz system. We show that all three approaches lead to reliable estimations of the energy in the SSWs.

Solitary waves in a granular chain of elastic spheres: Multiple solitary solutions and their stabilities

A granular chain of elastic spheres via Hertzian contact incorporates a classical nonlinear force model describing dynamical elastic solitary wave propagation. In this paper, the multiple solitary waves and their dynamic behaviors and stability in such a system are considered. An approximate KdV equation with the standard form is derived under the long-wavelength approximation and small deformation. The closed-form analytical single- and multiple-soliton solutions are obtained. The construction of the multiple-soliton solutions is analyzed by using the functional analysis. It is found that the multiple-soliton solution can be excited by the single-soliton solutions. This result is confirmed by the numerical analysis. Based on the soliton solutions of the KdV equation, the analytic solutions of multiple dark solitary waves are obtained from the original dynamic equation of the granular chain in the long-wavelength approximation. The stability of the single and multiple dark solitary wave solutions are numerically analyzed by using both split-step Fourier transform method and Runge-Kutta method. The results show that the single dark solitary wave solution is stable, and the multiple dark solitary waves are unstable.

Possibility of the existence of the rogue wave and the super rogue wave in granular matter

By using the traditional perturbation technique, a focusing nonlinear Schrödinger equation (NLSE) for the one-dimensional bead chain with the initial prestress is first obtained. The Peregrine soliton, called the rogue wave in the present paper, and the super rogue wave are investigated both numerically and analytically. It is noted that both the rogue wave and the super rogue wave do exist in the one-dimensional bead chain. The solutions from the NLSE can correctly describe the real rogue wave as well as the real super rogue wave in the limiting case of small amplitude. Both the rogue wave and the super rogue wave propagate in the granular bead chain as if they are solitary waves.

Fluctuations in Hertz chains at equilibrium

We examine the long-term behavior of nonintegrable, energy-conserved, one-dimensional systems of macroscopic grains interacting via a contact-only generalized Hertz potential and held between stationary walls. Such systems can be set up to have no phononic background excitation and represent examples of a sonic vacuum. Existing dynamical studies showed the absence of energy equipartitioning in such systems, hence their long-term dynamics was described as quasiequilibrium. Here we show that these systems do in fact reach thermal equilibrium at sufficiently long times, as indicated by the calculated heat capacity. As a by-product, we show how fluctuations of system quantities, and thus the distribution functions, are influenced by the Hertz potential. In particular, the variance of the system's kinetic energy probability density function is reduced by a factor related to the contact potential.

Shock wave in a one-dimensional granular chain under Hertz contact

The shock wave in one-dimensional bead chain is studied numerically. When the shock wave arrives, the bead velocity oscillates around the piston velocity. It is found that the shock front is composed of several solitary waves and the limitation of the maximum bead velocity is 2 times the piston velocity in the limiting case where the initial overlap is zero. If the initial overlap is not zero, then the maximum bead velocity is less than 2 times the piston velocity but larger than the piston velocity. As the initial overlap increases from zero to the finite value, the shock velocity depends on not only the piston velocity but also the initial overlap. The crossover of the dependence of the shock velocity on the piston velocity from the zero initial prestress to the finite value is obtained in the present manuscript. It is an improvement of the results presented in Phys. Rev. Lett. 108, 058001 (2012)10.1103/PhysRevLett.108.058001. In other words, the dependence of the shock velocity on the parameters of the granular materials is given.

Mechanical impulse propagation in a three-dimensional packing of spheres confined at constant pressure

Mechanical impulse propagation in granular media depends strongly on the imposed confinement conditions. In this work, the propagation of sound in a granular packing contained by flexible walls that enable confinement under hydrostatic pressure conditions is investigated. This configuration also allows the form of the input impulse to be controlled by means of an instrumented impact pendulum. The main characteristics of mechanical wave propagation are analyzed, and it is found that the wave speed as function of the wave amplitude of the propagating pulse obeys the predictions of the Hertz contact law. Upon increasing the confinement pressure, a continuous transition from nonlinear to linear propagation is observed. Our results show that in the low-confinement regime, the attenuation increases with an increasing impulse amplitude for nonlinear pulses, whereas it is a weak function of the confinement pressure for linear waves.

Pulse reflection and transmission due to impurities in a granular chain

Reflection and transmission due to the incident wave in one-dimensional bead chains when there are impurities have been studied. The impurities can be any kind of material, any size, and their numbers are arbitrary. The dependence of the transmission and the reflection on the numbers and the material parameters of the impurities are given. The analytical results are given by using the inverse scattering method. Substantial reflection is observed if there is only one steel bead. However, the reflection is negligible if there are two steel beads. The reflection monotonously increases as the numbers of the steel beads increase. The reflection remains a constant when the numbers of the steel beads are so many that the length composed by the steel beads is larger than the width of the solitary wave. It can be used to detect the impurities in the beads' chain by measuring the reflection of a pulse.

Granular chains with soft boundaries: Slowing the transition to quasiequilibrium

We present here a detailed numerical study of the dynamical behavior of "soft" uncompressed grains in a granular chain where the grains interact via the intrinsically nonlinear Hertz force. It is well known that such a chain supports the formation of solitary waves (SWs). Here, however, the system response to the material properties of the grains and boundaries is explored further. In particular, we examine the details of the transition of the system from a SW phase to an equilibrium-like (or quasiequilibrium) phase, and for this reason we ignore the effects of dissipation in this study. We find that the soft walls slow the reflection of SWs at the boundaries of the system, which in turn slows the journey to quasiequilibrium. Moreover, the increased grain-wall compression as the boundaries are softened results in fewer average grain-grain contacts at any given time in the quasiequilibrium phase. These effects lead to increased kinetic energy fluctuations in the short term in softer systems. We conclude with a toy model that exploits the results of soft-wall systems. This toy model supports the formation of breather-like entities and may therefore be useful for localizing energy in desired places in the granular chain.

Characterizing traveling-wave collisions in granular chains starting from integrable limits: the case of the Korteweg-de Vries equation and the Toda lattice

Our aim in the present work is to develop approximations for the collisional dynamics of traveling waves in the context of granular chains in the presence of precompression. To that effect, we aim to quantify approximations of the relevant Hertzian FPU-type lattice through both the Korteweg-de Vries (KdV) equation and the Toda lattice. Using the availability in such settings of both one-soliton and two-soliton solutions in explicit analytical form, we initialize such coherent structures in the granular chain and observe the proximity of the resulting evolution to the underlying integrable (KdV or Toda) model. While the KdV offers the possibility to accurately capture collisions of solitary waves propagating in the same direction, the Toda lattice enables capturing both copropagating and counterpropagating soliton collisions. The error in the approximation is quantified numerically and connections to bounds established in the mathematical literature are also given.

Granular chain between asymmetric boundaries and the quasiequilibrium state

Some 30 years have passed since we learned that any velocity perturbation develops into a propagating solitary wave in a granular chain, and over a decade has passed since we learned that these solitary waves break and reform upon collision, leaving behind small secondary solitary waves. The production of the latter eventually precipitates the quasiequilibrium state characterized by large energy fluctuations in dissipation-free granular systems. Here we present dynamical simulations on the effects of soft boundaries on solitary wave interaction in granular chains held between fixed walls. We show that at short time scales, a gradient in the distribution of kinetic energy between the boundaries is indeed sustained. At long times, however, such a gradient gets obliterated and there is no measurable difference between the average kinetic energies of the particles adjacent to walls. Our findings suggest that (i) the quasiequilibrium state can effectively erase small gradients of the average kinetic energies of the particles adjacent to walls in a system, (ii) Boltzmann distribution of grain speeds is realized in the system of interest, and (iii) time and space averages yield the same result, thus suggesting that the system is ergodic.

Wave propagation in square granular crystals with spherical interstitial intruders

We investigate the propagation and scattering of highly nonlinear waves in granular systems composed of spheres in contact arranged in a square packing, and study how the presence of small and light spherical interstitial defects, also referred to as intruders, affects the wave propagation. The effects of a single defect are investigated experimentally and compared to numerical simulations, showing very good quantitative agreement. Transmitted and scattered waves are formed, whose characteristics depend on the material properties of the defect in relation to the properties of the particles in the lattice. Experiments and numerical simulations reveal that stiffer defects are more efficient at redistributing energy outside the impacted chain and soft defects induce a localization of the energy at the defect. Finally, the effects of the presence of two defects, placed diagonally or aligned in the square packing are also investigated, as well as how their interaction depends on their relative positions.

Uniform shock waves in disordered granular matter

The confining pressure P is perhaps the most important parameter controlling the properties of granular matter. Strongly compressed granular media are, in many respects, simple solids in which elastic perturbations travel as ordinary phonons. However, the speed of sound in granular aggregates continuously decreases as the confining pressure decreases, completely vanishing at the jamming-unjamming transition. This anomalous behavior suggests that the transport of energy at low pressures should not be dominated by phonons. In this work we use simulations and theory to show how the response of granular systems becomes increasingly nonlinear as pressure decreases. In the low-pressure regime the elastic energy is found to be mainly transported through nonlinear waves and shocks. We numerically characterize the propagation speed, shape, and stability of these shocks and model the dependence of the shock speed on pressure and impact intensity by a simple analytical approach.

Amplitude-dependent phononic processes in a diatomic granular chain in the weakly nonlinear regime

Nonlinear acoustic processes of second harmonic generation and nonlinear resonances in a diatomic granular chain (a granular phononic crystal) with static precompression are reported. The observed nonlinear self-action process which manifests itself as shifts in resonance frequencies of the chain leads to amplitude-dependent band edges: the properties of the phononic crystal change as a function of wave amplitude. Observed nonlinear effects at the band edges are exceptionally strong (self-induced attenuation and self-induced transparency) due to the peculiar frequency dependence of the attenuation in these frequency regions. The reported effects open the way for applications in wave tailoring by nonlinear phononic crystals, using amplitude-dependent processes, such as passive amplitude-dependent attenuators or amplifiers and various logical elements.

Sustained strong fluctuations in a nonlinear chain at acoustic vacuum: beyond equilibrium

Here we consider dynamical problems as in linear response theory but for purely nonlinear systems where acoustic propagation is prohibited by the potential, e.g., the case of an alignment of elastic grains confined between walls. Our simulations suggest that in the absence of acoustic propagation, the system relaxes using only solitary waves and the eventual state does not resemble an equilibrium state. Further, the studies reveal that multiple perturbations could give rise to hot and cold spots in these systems. We first use particle dynamics based simulations to understand how one of the two unequal colliding solitary waves in the chain can gain energy. Specifically, we find that for head-on collisions the smaller wave gains energy, whereas when a more energetic wave overtakes a less energetic wave, the latter gains energy. The balance between the rate at which the solitary waves break down and the rate at which they grow eventually makes it possible for the system to reach a peculiar equilibriumlike phase that is characteristic of these purely nonlinear systems. The study of the features and the robustness of the fluctuations in time has been addressed next. A particular characteristic of this equilibriumlike or quasiequilibrium phase is that very large energy fluctuations are possible--and by very large, we mean that the energy can vary between zero and several times the average energy per grain. We argue that the magnitude of the fluctuations depend on the nature of the nonlinearity in the potential energy function and the feature that any energy must eventually travel as a compact solitary wave in these systems where the solitary wave energies may vary widely. In closing we address whether these fluctuations are peculiar to one dimension or can exist in higher dimensions. The study hence raises the following intriguing possibility. Are there physical or biological systems where these kinds of nonlinear forces exist, and if so, can such large fluctuations actually be seen? Implications of the study are briefly discussed.

Nonlinear repulsive force between two solids with axial symmetry

We modify the formulation of Hertz contact theory between two elastic half-solids with axial symmetry and show that these modifications to Hertz's original framework allow the development of force laws of the form F [Please see symbol] z(n), 1 < n < ∞, where F is the force and z is the distance between the centers of the two solids. The study suggests that it may be possible to design physical systems that can realize such force laws. We let the half-solids be characterized by radii of curvatures R(1) and R(2) and invoke a factor m>0 to describe any aspect ratio in the two bodies, all being valid near the contact surface. We let the x-y plane be the contact surface with an averaged pressure across the same as opposed to a pressure profile that depends on the contact area of a nonconformal contact as originally used by Hertz. We let the z axis connect the centers of the masses and define z1,2 = x(α) / R(1,2)(α-1) + y(α)/(mR(1,2))(α-1), where z(1,2) ≥ 0 refers to the compression of bodies 1, 2, α > 1, m > 0, x, y ≥ 0. The full cross section can be generated by appropriate reflections using the first quadrant part of the area. We show that the nonlinear repulsive force is F = az(n), where n ≡ 1 + 1/α, and z ≡ z(1) + z(2) is the overlap and we present an expression for a = f(E, σ, m, α, R(1), R(2)) with E and σ as Young's modulus and the Poisson ratio, respectively. For α = 2, ∞, to similar geometry-dependent constants, we recover Hertz's law and the linear law, describing the repulsion between compressed spheres and disks, respectively. The work provides a connection between the contact geometry and the nonlinear repulsive law via α and m.

Discrete breathers at the interface between a diatomic and a monoatomic granular chain

In the present work, we develop a systematic examination of the existence, stability, and dynamical properties of a discrete breather at the interface between a diatomic and a monoatomic granular chain. We remarkably find that such an "interface breather" is more robust than its bulk diatomic counterpart throughout the gap of the linear spectrum. The latter linear spectral gap needs to exist for the breather state to arise and the relevant spectral conditions are discussed. We illustrate the minimal excitation conditions under which such an interface breather can be "nucleated" and analyze its apparently weak interaction with regular highly nonlinear solitary waveforms.

Pulse propagation in a chain of o-rings with and without precompression

We implement a binary collision approximation to study pulse propagation in a chain of o-rings. In particular, we arrive at analytic results from which the pulse velocity is obtained by simple quadrature. The predicted pulse velocity is compared to the velocity obtained from the far more resource-intensive numerical integration of the equations of motion. We study chains without precompression, chains precompressed by a constant force at the chain ends (constant precompression), and chains precompressed by gravity (variable precompression). The application of the binary collision approximation to precompressed chains provides an important generalization of a successful theory that had up to this point only been implemented to chains without precompression, that is, to chains in a sonic vacuum.

How solitary waves collide in discrete granular alignments

Solitary waves in continuum media pass through each other with only a slight phase change. However, in an intrinsically nonlinear many-body system such solitary waves could behave differently. It was predicted and experimentally confirmed that head-on solitary wave collisions in granular alignments are followed by the formation of tiny secondary solitary waves in the vicinity of the collision point. While it remains a challenge to provide an analytical treatment of the local time evolution, we present arguments and associated simulations to address a crucial unknown, namely, why the secondary solitary waves must form. Extensive numerical investigations on solitary wave collisions at a grain center and at an edge show marked differences. The effects of softening the grain repulsion are discussed to validate the arguments.

Dynamics of metastable breathers in nonlinear chains in acoustic vacuum

The study of the dynamics of one-dimensional chains with both harmonic and nonlinear interactions, as in the Fermi-Pasta-Ulam and related problems, has played a central role in efforts to identify the broad consequences of nonlinearity in these systems. Nevertheless, little is known about the dynamical behavior of purely nonlinear chains where there is a complete absence of the harmonic term, and hence sound propagation is not admissible, i.e., under conditions of "acoustic vacuum." Here we study the dynamics of highly localized excitations, or breathers, which are known to be initiated by the quasistatic stretching of the bonds between adjacent particles. We show via detailed particle-dynamics-based studies that many low-energy pulses also form in the vicinity of the perturbation, and the breathers that form are "fragile" in the sense that they can be easily delocalized by scattering events in the system. We show that the localized excitations eventually disperse, allowing the system to attain an equilibrium-like state that is realizable in acoustic vacuum. We conclude with a discussion of how the dynamics is affected by the presence of acoustic oscillations.

Compactons and chaos in strongly nonlinear lattices

We study localized traveling waves and chaotic states in strongly nonlinear one-dimensional Hamiltonian lattices. We show that the solitary waves are superexponentially localized and present an accurate numerical method allowing one to find them for an arbitrary nonlinearity index. Compactons evolve from rather general initially localized perturbations and collide nearly elastically. Nevertheless, on a long time scale for finite lattices an extensive chaotic state is generally observed. Because of the system's scaling, these dynamical properties are valid for any energy.

Nonlinear waves in dry and wet Hertzian granular chains

A one-dimensional dry granular medium, a chain of beads which interact via the nonlinear Hertz potential, exhibits strongly nonlinear behaviors. When such an alignment further contains some fluid in the interstices between grains, it may exhibit new interesting features. We report some recent experiments, analysis and numerical simulations concerning nonlinear wave propagation in dry and wet chains of spheres. We consider first a monodisperse chain as a reference case. We then analyze how the pulse characteristics are modified in the presence of an interstitial viscous fluid. The fluid not only induces dissipation but also strongly affect the intergrain stiffness: in a wet chain, wave speed is enhanced and pulses are shorter. Simple experiments performed with a single sphere colliding a wall covered by a thin film of fluid confirm these observations. We demonstrate that even a very small amount of fluid can overcome the Hertzian potential and is responsible for a large increase of contact stiffness. Possible mechanisms for wet contact hardening are related to large fluid shear rate during fast elastohydrodynamic collision between grains.

Characterization of reflection intermittency in a composite granular chain

The physical factors controlling the power-law behavior of impact energy in a composite granular chain remain elusive. Based on event-driven simulations and the on-off intermittency of wave reflections, we obtain the probability distribution functions of the waiting time tau and the energy leakage DeltaE . They exhibit lognormal distributions, which together with the relationship between DeltaE and tau allow one to explain directly the power-law behavior of the confined energy. This work may be extended to higher dimensions and help us understand the complex dynamics in granular materials.

Crossover in the power-law behavior of confined energy in a composite granular chain

We present a numerical study of the impact energy decay in a composite granular chain containing two heavy and one light sections. We observe a marked crossover in the power-law behavior of the impact-energy decay. The average reflection frequency first increases with a decreasing acceleration, and arrives at its maximum at "crossing" time then decays almost exponentially. The analysis demonstrates that this phenomenon is related to the structural transition from compression to dilation state in both heavy-particle sections. The further calculations suggest the dependence relation of the power-law exponent (gammacb) in compression state on the mass ratio (m2/m1) and the Hertz law exponent (n) of the composite granular chain gammacb approximately (m2/m1)1/(n+1).

Experimental evidence of shock mitigation in a Hertzian tapered chain

We present an experimental study of the mechanical impulse propagation through a horizontal alignment of elastic spheres of progressively decreasing diameter phi(n): namely, a tapered chain. Experimentally, the diameters of spheres which interact via the Hertz potential are selected to keep as close as possible to an exponential decrease, phi(n+1) = (1-q)phi(n), where the experimental tapering factor is either q(1) approximately equal to 5.60% or q(2) approximately equal to 8.27%. In agreement with recent numerical results, an impulse initiated in a monodisperse chain (a chain of identical beads) propagates without shape changes and progressively transfers its energy and momentum to a propagating tail when it further travels in a tapered chain. As a result, the front pulse of this wave decreases in amplitude and accelerates. Both effects are satisfactorily described by the hard-sphere approximation, and basically, the shock mitigation is due to partial transmissions, from one bead to the next, of momentum and energy of the front pulse. In addition when small dissipation is included, better agreement with experiments is found. A close analysis of the loading part of the experimental pulses demonstrates that the front wave adopts a self-similar solution as it propagates in the tapered chain. Finally, our results corroborate the capability of these chains to thermalize propagating impulses and thereby act as shock absorbing devices.

Impulse absorption by tapered horizontal alignments of elastic spheres

We present an analytical and numerical study of the problem of mechanical impulse propagation through a horizontal alignment of progressively shrinking (tapered) elastic spheres that are placed between two rigid end walls. The studies are confined to cases where initial loading between the spheres is zero (i.e., in the "sonic vacuum" region). The spheres are assumed to interact via the Hertz potential. Force and energy as a function of time for selected grains that comprise the solitary wave are provided and shed light on the system's behavior. Propagation of energy is analytically studied in the hard-sphere approximation and phase diagrams plotting normalized kinetic energy of the smallest grain at the tapered end are developed for various chain lengths and tapering factors. These details are then compared to kinetic energy phase diagrams obtained via extensive dynamical simulations. Our figures indicate that the ratios of the kinetic energies of the smallest to largest grains possess a Gaussian dependence on tapering and an exponential decay when the number of grains increases. The conclusions are independent of system size, thus being applicable to tapered alignments of micron-sized spheres as well as those that are macroscopic and more easily realizable in the laboratory. Results demonstrate the capabililty of these chains to thermalize propagating impulses and thereby act as potential shock absorbing devices.

Universal power-law decay of the impulse energy in granular protectors

Protecting a big impulse from outside is one of the important issues of our everyday life. A granular medium is often used as a protecting material. The impulse inside a granular medium is a solitary wave which may be confined temporarily to a particular region of the medium, which we call the granular container that plays the role of the protector. We find a universal power-law behavior in time for the leakage of the impulse energy confined inside various granular containers.

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.

Pulse velocity in a granular chain

We discuss the applicability of two very different analytic approaches to the study of pulse propagation in a chain of particles interacting via a Hertz potential, namely, a continuum model and a binary collision approximation. While both methods capture some qualitative features equally well, the first is quantitatively good for softer potentials and the latter is better for harder potentials.

Pulse dynamics in a chain of granules with friction

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).

Dynamics of two granules

We study the dynamics of two particles that interact only when in contact. In this sense, although not in every particular, the interactions mimic those in granular materials. The detailed solution of the dynamics allows an analysis of the backscattering behavior of the first particle and of the energy dissipation in the system as a function of various parameters.

Impulse backscattering in granular beds: introducing a toy model

Impulses efficiently propagate into nominally dry granular beds and backscatter from buried inclusions in such beds may be potentially exploited to image shallow buried objects (SBOs). However, reliable imaging of SBOs requires "cleaning up" of surface vibrations, and, in addition to three-dimensional (3D) particle dynamics simulations, a phenomenological model to parametrize the bed surface may be useful for field applications. We introduce a 1D mean-field-like toy model with two parameters, which allows one to model surface vibrations, is consistent with experiments in a granular bed, and can help estimate the approximate signal transmission properties of the bed.

Secondary solitary wave formation in systems with generalized Hertz interactions

We consider a chain of monodisperse elastic grains of radius R where the grains are barely in contact. The grains repel upon contact via the Hertz-type potential, V proportional to delta(n), n > 2, where delta > or = 0, is the grain-grain overlap, delta identical with 2R-(u(i+1)-u(i)), where u(i) denotes the displacement of grain i from its original equilibrium position. This being a computational study, we consider n to be arbitrary. Our dynamical simulations build on several earlier studies by Nesterenko, Coste, and Sen and co-workers that have shown that an impulse propagates as a solitary wave of fixed spatial extent, infinity < L(n) < 1, through a chain of grains. Here, we develop on a recent study by Manciu, Sen, and Hurd [Phys. Rev. E 63, 016614 (2001)] that shows that colliding solitary waves in the chains of interest spawn a well-defined hierarchy of multiple secondary solitary waves (SSWs) that carry approximately 0.5% or less of the energy of the original solitary waves. We show that the emergence of SSWs is a complex process where nonlinear forces and the discreteness of the grains lead to the partitioning of the available energy into hierarchies of SSWs. The process of formation of SSWs involves length scales and time scales that are controlled by the strength of the nonlinearity in the system. To the best of our knowledge, there is no formal theory that describes the dynamics associated with the formation of SSWs. Calculations for cases where the Hertz-type potential can be symmetric in the overlap parameter delta, i.e., where delta can be both positive and negative, suggest that the formation of secondary solitary waves may be a fundamental property of certain discrete, nonlinear systems.

Linearized impulse wave propagating down a vertical column of heavy particles

A granular column is subjected to a small amplitude impact on its top. For a generalized power-law contact force between neighboring grains, numerical simulations show that the propagation of the impulse wave is controlled by dispersion. This leads quantitatively to a power-law decrease of the amplitude of the wave with depth. We find numerically the dependence of this power-law exponent on the force-law exponent. An analytic expression for the decrease is then derived from a long-wave approximation.

Solitary wave dynamics in generalized Hertz chains: an improved solution of the equation of motion

The equation of motion for a bead in a chain of uncompressed elastic beads in contact that interact via the potential V(delta) approximately delta( n), n>2, delta being overlap, supports solitary waves and does not accommodate sound propagation [V. Nesterenko, J. Appl. Mech. Tech. Phys. 5, 733 (1983)]. We present an iteratively exact solution to describe the solitary wave as a function of material parameters and a universal, infinite set of coefficients, which depend only on n. We compute any arbitrary number of coefficients to desired accuracy and show that only the first few coefficients of our solution significantly improves upon Nesterenko's solution. The improved solution is a necessary step to develop a theoretical understanding of the formation of secondary solitary waves [M. Manciu, et al., Phys. Rev. E 63, 011614 (2001)].

Effects of gravity and nonlinearity on the waves in the granular chain

The solitary signal observed in a horizontal granular chain changes its speed and form due to gravity in a vertical chain. We find that all the propagating signals in a vertical chain follow power laws in depth for propagating speed, grain velocity, amplitude, and width. This stems from the power-law type changing of elastic properties in a medium under gravity. The propagation may be separated into two types according to the behavior of the power-law exponents, depending on the strength of the nonlinearity. We show that the power-law exponents are constants in the strength of the impulse in the weakly nonlinear regime, while they depend on the strength of the impulse in the strongly nonlinear regime. We derive power-law exponents for the weakly nonlinear regime analytically and try to understand the behaviors of the strongly nonlinear regime through analytical treatment.

Crossing of identical solitary waves in a chain of elastic beads

We consider a chain of elastic beads subjected to vanishingly weak loading conditions, i.e., the beads are barely in contact. The grains repel upon contact via the Hertz-type potential, Vinfinitydelta(n), n>2, where delta> or =0, delta being the grain-grain overlap. Our dynamical simulations build on several earlier studies by Nesterenko, Coste, and Sen and co-workers that have shown that an impulse propagates as a solitary wave of fixed spatial extent (dependent only upon n) through a chain of Hertzian beads and demonstrate, to our knowledge for the first time, that colliding solitary waves in the chain spawn a well-defined hierarchy of multiple secondary solitary waves, which is approximately 0.5% of the energy of the original solitary waves. Our findings have interesting parallels with earlier observations by Rosenau and colleagues [P. Rosenau and J. M. Hyman, Phys. Rev. Lett. 70, 564 (1993); P. Rosenau, ibid. 73, 1737 (1994); Phys. Lett. A 211, 265 (1996)] regarding colliding compactons. To the best of our knowledge, there is no formal theory that describes the dynamics associated with the formation of secondary solitary waves. Calculations suggest that the formation of secondary solitary waves may be a fundamental property of certain discrete systems.

Dynamics of a gravitationally loaded chain of elastic beads

Elastic beads repel in a highly nonlinear fashion, as described by Hertz law, when they are compressed against one another. Vertical stacking results in significant compressions of beads at finite distances from the surface of the stack due to gravity. Analytic studies that have been reported in the literature assume acoustic excitations upon weak perturbation [J. Hong et al., Phys. Rev. Lett. 82, 3058 (1999)] and soliton-like excitations upon strong perturbation [V. Nesterenko, J. Appl. Mech. Tech. Phys. 5, 733 (1983); S. Sen and M. Manciu, Physica A 268, 644 (1999)]. The present study probes the position, velocity and acceleration and selected two-point temporal correlations and their power spectra for individual beads for cases in which the system has been (i) weakly, (ii) strongly, and (iii) moderately perturbed at the surface in the sense specified in the text. Our studies reveal the existence of distinctly different dynamical behavior of the tagged beads, in contrast to conventional acoustic response, as the strength of the perturbation is varied at fixed gravitational loading. We also comment on the effects of polydispersity on system dynamics and probe the relaxation of isolated light and heavy beads in the chain. (c) 2000 American Institute of Physics.

Generalized langevin equation and recurrence relations

The generalized Langevin equation (GLE) is a reformulation of the Heisenberg equation of motion, and hence, an exact equation. It is the basis of the memory function approach, a very widely used method for studying dynamics of classical and quantum fluids. The GLE was first derived by Mori in a very formal way. A much simpler and more physically motivated derivation was given by us some years later. In this work we provide perhaps the simplest possible derivation of the GLE. The simplicity of the derivation helps to bring out the subtleties present in this important dynamical relationship.

Heisenberg, langevin, and current equations via the recurrence relations approach

Some years ago the Heisenberg equation of motion was formally solved by the recurrence relations approach. It is shown here that the Langevin equation represents a structural property of the recurrence relations. The Langevin equation is useful for studying the time evolution of the current. The resulting current-current correlation function is compared with Luttinger's phenomenological theory. Geometric interpretations are made for the conductivity and the dielectric function.

Characterization of soliton damping in the granular chain under gravity

A soliton created in the horizontal granular chain damps due to gravity in the vertical chain. We show that there are two types of propagating modes, quasisolitary and oscillatory, in the vertical chain, depending on the strength of impulse. We find that the type of damping is a power law in depth or time. We also find that the absolute value of the exponent of the power law decreases as the strength of the initial impulse increases in the quasisolitary regime. In the oscillatory regime, however, in which the initial impulse is weak, the power-law exponent is independent of the strength of the initial impulse. We show that the power-law damping is caused by the gravitation which results in the change of the force constant at each contact.