Power-Laws in Nonlinear Granular Chain under Gravity

The signal generated by a weak impulse propagates in an oscillatory way and dispersively in a gravitationally compacted granular chain. For the power-law type contact force, we show analytically that the type of dispersion follows power-laws in depth. The power-law for grain displacement signal is given by h − 1 4 (1− 1 p ) where h and p denote depth and the exponent of contact force, and the power-law for the grain velocity is h − 1 4 ( 1 3 + 1 p ) . Other depth-dependent power-laws for oscillation frequency, wavelength, and period are given by combining above two and the phase velocity power-law h 1 2 (1− 1 p ) . We verify above power-laws by comparing with the data obtained by numerical simulations. 45.70.-n, 46.40.Cd, 02.70.Ns, 43.25.+y Typeset using REVTEX 1 Physics of granular materials attracts great interest recently [1], since these materials are ubiquitous around us and their properties are unique and also useful in many applications [2,3]. The propagation of a sound or a weak elastic wave in a granular medium is also one of interesting subjects related to the properties of granular matter [4]. A rather simple system, the granular chain with Hertzian contact [5], has been revived by finding a soliton in transmitting elastic impulse. This soliton, existing in a highly nonlinear regime of a horizontal Hertzian chain was first predicted by Nesterenko [6] and its experimental verification was performed by Lazaridi and Nesterenko [7] and recently by Coste et al. [8]. Even though three-dimensional granular systems may not follow simple Hertzian contact force law due to geometrical effect [9], the one-dimensional granular chain with nonlinear contact force is still interesting. It may describe a fundamental feature of the dynamics of nonlinear granular chain which appears in many areas of nature. In addition, one-dimensional system is usually the starting point of studying higher dimensional systems. It is well-known that the velocity of an elastic impulse scales as P 1/6 or h for the Hertzian chain [9], where P is the pressure, linearly proportional to the depth h for vertical chain. Sinkovits and Sen [10] extended this to arbitrary nonlinear contact force of power-law type F ∝ δ, where δ denotes overlapped distance between adjacent grains. They showed that the signal velocity vph scales as h (1− 1 p ) 1 2 for p ≥ 1 at large h. This has been simply obtained by considering the well-known relation vph ∝ √ μ, where μ is the elastic constant which is given by μ ∼ h for the above power-law type contact force. As far as we know, however, no power-law dependences on depth of the signal characteristics, such as oscillation frequency, period, and wavelength have been found in the gravitationally compacted chain. In this work, we study the propagation of acoustic or weak impulses in the gravitationally compacted granular chain. We derive analytically the power-law behaviors of signal characteristics which depend on depth or time. We treat here a rather weak impulse which makes grain motion oscillatory and can be treated analytically even though it contains nonlinearity. The other extreme which is a highly nonlinear regime has been studied by Nesterenko [6]. Initial impulse may be used as a parameter which controls the solitariness of signal. The 2 power-law behaviors for a wide range of impulse will be discussed in a separate work [11]. We would like to obtain analytically the exponents of various power-laws, such as grain displacement, grain velocity, and oscillation period, frequency, and wavelength. We first solve the equation of motion of a grain displacement under gravity in the small oscillation or weak impulse regime in which the equation of motion under gravity can be mapped into the equation for the horizontal linear chain with varying force constant at each contact. The normal mode solution of the equation of motion can be obtained analytically in the continuum or long wavelength limit. The asymptotic behavior of the normal mode gives rise to the correct power-law behavior in depth, since the equation of motion has been changed into a linear form. Once we get the information on the grain velocity, all sorts of power-laws mentioned above can be obtained. Since the equation of motion for grain velocity is not linear, the normal mode solution may not work to obtain power-law in depth. Therefore, we construct fully nonlinear forms describing displacement and velocity signal and obtain their depth-dependence behaviors. Our solution is quite general and gives rise to generic power-laws for arbitrary exponent p of the contact force in the oscillating regime. The equation of motion of n-th grain at zn is given by mz̈n = η[{∆0 − (zn − zn−1)} − {∆0 − (zn+1 − zn)}] + mg, (1) where zn is the distance from the top of chain to the center of the n-th spherical grain, m is the mass of grain, ∆0 is the distance between adjacent centers of the spherical grain, and η is the elastic constant of grain. Therefore, the overlap between adjacent grains at nth contact is δn = ∆0 − (zn+1 − zn). It is usually impossible to solve general nonlinear problems in an analytical way. Therefore we may not solve the nonlinear differential equation of Eq. (1) exactly. But we may treat it analytically in a small oscillation regime which can be achieved by applying a weak impulse. For this purpose, we introduce a new variable