Power-Laws of Nonlinear Granular Chain under Gravity

The soliton propagating in a horizontal granular chain with Hertzian contact damps due to graity in a vertical chain. We find that the type of damping is power-law in depth and the exponent of power-law for the amplitude approaches −1/12 for weak impulse. We also show that other quantities such as grain velocity, kinetic energy, and number of grains participating to signal as well as phase and group velocity follow power-laws. We explain these powerlaw behaviors by solving the equation of motion analytically in the small oscillation and continuum limit in which the nonlinear equation of motion is mapped into the linearized equation of motion with varying force constant. 45.70.-n, 46.40.Cd, 02.70.Ns, 43.25.+y Typeset using REVTEX 1 Recently, physics of granular materials attracts great interest, since the materials are ubiquitous around us and their properties are unique and also useful in many applications [1]. The propagation of sound or elastic wave in granular medium is also one of interesting subjects related to the properties of granular matter[2]. A rather simple system, the granular chain with nonlinear contact force such as Hertzian contact [3], has been revived by finding a soliton in transmitting elastic impulse. This soliton existing in a highly nonlinear regime of horizontal Hertzian chain was first predicted by Nesterenko [4] and its experimental verification was performed by Lazaridi and Nesterenko [5] and recently by Coste et al. [6]. Even though three-dimensional granular systems may not follow simple Hertzian contact force law due to geometrical effect [7], the simple nonlinear contact force is still interesting because of the existence of soliton which may provide a possibility to get information inside granular matter. In addition, one-dimensional system is usually the starting point for studying higher dimensional systems. It is well-known that the velocity of elastic impulse scales as P 1/6 or h for a vertical Hertzian chain [7], where P is the pressure which is linearly proportional to the depth h of vertical chain. Sinkovits and Sen [8] extended it to arbitrary nonlinear contact force F ∝ δ, where δ means overlapped distance. They showed that the velocity scales as h 1 p ) 1 2 for p ≥ 1 at large h. As far as we know, no power-law dependence on depth has been found in a gravitationally compacted chain except propagating speed. In this Letter, we study the characteristics of shock propagation in the gravitationally compacted granular chain and show many power-law behaviors depending on depth. We treat here rather weak impulse which gives the motion of grains oscillatory and can be treated analytically. Initial impulse can be used as a parameter which controls solitariness of signal. The power-law behaviors for a wide range of impulse will be discussed in a separated work [9]. We first obtain various exponents of power-laws, such as amplitude, characteristic time (period) and length (wavelength), grain velocity, and kinetic energy through molecular dynamics simulations for the gravitationally compacted granular chain with Hertzian contact. These power-law behaviors have never been studied before except signal velocity. Our sec2 ond work is to derive these power-laws analytically in the small oscillation or weak impulse regime in which the equation of motion of a grain can be mapped into the equation of the linear chain with varying force constants at each contact. This equation of motion can be solved analytically in the continuum or long wavelength limit. The asymptotic behavior of the solution gives rise to all sorts of power-laws mentioned above. The solution is quite general and gives power-laws for arbitrary exponent p of contact force. It is usually impossible to treat general nonlinear problem analytically, so we use molecular dynamics simulations to study the dynamics of grains in the gravitationally compacted chain with nonlinear contact force. We solve numerically the equation of motion of a grain at zn, mz̈n = η[{∆0 − (zn − zn−1)} p − {∆0 − (zn+1 − zn)} ] + mg, (1) where zn is the distance from the top of chain to the center of i-th spherical grain, m is the mass of grain, ∆0 is the distance between adjacent centers of the spherical grain, and η = (p+1)b where b is the constant defined in Eq. (2) below. Therefore, the overlap between adjacent grains at nth contact is δn = ∆0 − (zn+1 − zn). We perform simulations for p = 3/2 which is the Hertzian contact for this study. This equation of motion comes from the Hertzian interaction energy between neighboring granular spheres which is given by [3] V (δn) = 2 5D ( RnRn+1 Rn +Rn+1 )1/2 δ n ≡ bδ 5/2 n , (2) where Rn is the radius of the spherical grain and D = 3 4 ( 1− σ n En + 1− σ n+1 En+1 ) , (3) where σn, σn+1 and En, En+1 are Poisson’s ratios and Young’s moduli of the bodies at neighboring positions, respectively [3]. To perform numerical simulation for Eq. (1), we choose a vertical chain of N = 2× 10 grains and neglect plastic deformation. As a calculational tool, we use the third-order Gear 3 predictor-corrector algorithm [10]. We choose 10m, 2.36 × 10kg, and 1.0102 × 10s as the units of distance, mass, and time, respectively. These units gives the gravitational acceleration g = 1. We set the grain diameter 100, mass 1, and the constant b of Eq. (1) 5657 for molecular dynamics simulation. The equilibrium condition