Power-Law Behaviors in Nonlinearly Coupled Granular Chain under Gravity

We find power-law behaviors of grain velocity in both propagation and backscattering in a gravitationally compacted granular chain with nonlinear contact force. We focus on the leading peak of the velocity signal which decreases in a power-law d−α, where d is the location of the peak, as the signal goes down. The ratio of backscattered to incident leading velocity also follows a power-law d −β i , where di is the depth of impurity. The up-going backscattered signal is nearly solitary. Therefore, the overall change of the leading velocity peak is given by the power-law d −(α+β) i . We find α = 0.250 and β = 0.167 for the Hertzian contact force. 46.10.+z, 03.40.Kf, 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 including Hertzian contact [3], has been revived by finding a soliton in transmitting elastic impulse. This soliton found in a loaded 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 law is still interesting because the existence of solitary wave may provide a possibility to get information inside granular matter and the one-dimensional system usually provides the starting point for studying more complex systems. In addition to studying horizontal chain, the gravitationally compacted chain and column of grains have been studied by Sinkovits and Sen [8–10] in terms of molecular dynamics simulations. The two cases are not the same in the following sense that the former shows robust solitary wave within certain amount of loading, while the latter shows solitonlike wave whose waveform disperses little by little as it propagates down. Sen at el. [10] studied backscattering of the solitonlike wave by impurity in a gravitationally compacted chain to get the information on buried impurity such metal-poor land mines. They obtained interesting relations between the phase of grain-velocity signal and the mass of impurity. The result reflects the well-known phase-density relation in the connected string with different densities. In this Letter, we focus on the propagating and backscattering characteristics of the leading part of grain-velocity signal in the gravitationally compacted granular chain with Hertzian contact force especially. We also treat other types of nonlinear forces. We report interesting power-law behaviors of the velocity signal. This work is attractive because it supplies new laws of signal propagation in the gravitationally compacted granular chain with nonlinear contact force and a possibility of identifying the location of buried impurity by observing the peak-velocity of the leading part of returned signal at surface. 2 To study the dynamics of grains in the gravitationally compacted Hertzian chain, we solve numerically the equation of motion of a grain at zi, mz̈i = 5 2 a[{∆0 − (zi − zi−1)} 3/2 − {∆0 − (zi+1 − zi)} ] − mg, (1) where m is the mass of grain, ∆0 = Ri + Ri+1, and a is the constant defined in Eq. (2) below. We neglect plastic deformation in Eq. (1). This equation of motion comes from the Hertzian interaction energy between neighboring granular spheres which is given by [3]