Infinite penetration of a projectile into a granular medium

An object falling in a fluid reaches a terminal velocity when the drag force and its weight are balanced. Contrastingly, an object impacting into a granular medium rapidly dissipates all its energy and comes to rest always at a shallow depth. Here we study, experimentally and theoretically, the penetration dynamics of a projectile in a very long silo filled with expanded polystyrene particles. We discovered that, above a critical mass, the projectile reaches a terminal velocity and, therefore, an endless penetration.     

The scaling and dynamics of a projectile obliquely impacting a granular medium

In this paper, the dynamics of a spherical projectile obliquely impacting into a two-dimensional granular bed is numerically investigated using the discrete element method. The influences of projectile’s initial velocities and impacting angles are mainly considered. Numerical results show that the relationship between the final penetration depth and the initial impact velocity is very similar to that in the vertical-impact case. However, the dependence of the stopping time on the impact velocity of the projectile exhibits critical characteristics at different impact angles: the stopping time approximately increases linearly with the impact velocity for small impact angles but decreases in an exponential form for larger impact angles, which demonstrates the existence of two different regimes at low and high impact angles. When the impact angle is regarded as a parametric variable, a phenomenological force model at large impact angles is eventually proposed based on the simulation results, which can accurately describe the nature of the resistance force exerted on the projectile by the granular medium at different impact angels during the whole oblique-impact process. The degenerate model agrees well with the existing experimental results in the vertical-impact cases.     

Dynamics of drag and force distributions for projectile impact in a granular medium

Our experiments and molecular dynamics simulations on a projectile penetrating a two-dimensional granular medium reveal that the mean deceleration of the projectile is constant and proportional to the impact velocity. Thus, the time taken for a projectile to decelerate to a stop is independent of its impact velocity. The simulations show that the probability distribution function of forces on grains is time independent during a projectile's deceleration in the medium. At all times the force distribution function decreases exponentially for large forces.     

Propagation of an ultimately short electromagnetic pulse in a nonlinear medium described by the fifth-order Duffing model

We discuss propagation of an ultimately short (single-cycle) pulse of an electromagnetic field in a medium whose dispersion and nonlinear properties can be described by the cubic-quintic Duffing model, i.e., by an oscillator with third-and fifth-order anharmonicity. A system of equations governing the evolution of a unidirectional electromagnetic wave is analyzed without using the approximation of slowly varying envelopes. Three types of solutions of this system describing stationary propagation of a pulse in such a medium are found. When the signs of the anharmonicity constants are different, then the amplitude of a steady-state pulse is limited, but its energy may grow on account of an increase in its duration. The characteristics of such a pulse, referred to as an electromagnetic domain, are discussed.     

Two dimensional fractional projectile motion in a resisting medium

In this paper we propose a fractional differential equation describing the behavior of a two dimensional projectile in a resisting medium. In order to maintain the dimensionality of the physical quantities in the system, an auxiliary parameter k was introduced in the derivative operator. This parameter has a dimension of inverse of seconds (sec)^?1 and characterizes the existence of fractional time components in the given system. It will be shown that the trajectories of the projectile at different values of γ and different fixed values of velocity v ₀ and angle θ, in the fractional approach, are always less than the classical one, unlike the results obtained in other studies. All the results obtained in the ordinary case may be obtained from the fractional case when γ = 1.     

Stochastic dynamics of a sheared granular medium

We experimentally investigate the response of a sheared granular medium in a Couette geometry. The apparatus exhibits the expected stick-slip motion and we probe it in the very intermittent regime resulting from low driving. Statistical analysis of the dynamic fluctuations reveals notable regularities. We observe a possible stability property for the torque distribution, reminiscent of the stability of Gaussian independent variables. In this case, however, the variables are correlated and the distribution is skewed. Moreover, the whole dynamical intermittent regime can be described with a simple stochastic model, finding good quantitative agreement with the experimental data. Interestingly, a similar model has been previously introduced in the study of magnetic domain wall motion, a source of Barkhausen noise. Our study suggests interesting connections between different complex phenomena and reveals some unexpected features that remain to be explained.     

Magnetizations of a nanoparticle described by the transverse Ising model

The characteristic influences of size S, exchange interaction and transverse field on the longitudinal and transverse magnetizations of a ferroelectric small particle described by the transverse Ising model are investigated by the use of the standard mean-field theory. In particular, the longitudinal magnetization of a nanoparticle is strongly affected by the surface situations. The effective exponent β_eff of the longitudinal magnetization is also studied. We find some characteristic phenomena of β_eff, depending on the values of S and the ratios of the physical parameters. In relation of recent investigations, the thermal variations of longitudinal and transverse magnetizations in the nanoparticle, consisting of a ferromagnetic core with size S=3 surrounded by a ferromagnetic surface shell with an antiferromagnetic inter-shell coupling, are examined and some typical ferrimagnetic behaviors are found in them. In relation to these phenomena, the effects of surface dilution on the magnetizations are investigated and some novel features are found in the system with size S=3 surrounded by such a ferromagnetic diluted surface shell.     

Starting to move through a granular medium

We explore the process of initiating motion through a granular medium by measuring the force required to push a flat circular plate upward from underneath the medium. In contrast with previous measurements of the drag and penetration forces, which were conducted during steady state motion, the initiation force has a robust dependence on the diameter of the grains in the medium. We attribute this dependence to the requirement for local dilation of the grains around the circumference of the plate, as evidenced by an observed linear dependence of the initiation force on the plate diameter.     

Upward penetration of grains through a granular medium

We study experimentally the creeping penetration of guest (percolating) grains through densely packed granular media in two dimensions. The evolution of the system of the guest grains during the penetration is studied by image analysis. To quantify the changes in the internal structure of the packing, we use Voronoï tessellation and a certain shape factor which is a clear indicator of the presence of different underlying substructures (domains). We first consider the impact of the effective gravitational acceleration on upward penetration of grains. It is found that the higher effective gravity increases the resistance to upward penetration and enhances structural organization in the system of the percolating grains. We also focus our attention on the dependence of the structural rearrangements of percolating grains on some parameters like polydispersity and the initial packing fraction of the host granular system. It is found that the anisotropy of penetration is larger in the monodisperse case than in the bidisperse one, for the same value of the packing fraction of the host medium. Compaction of initial host granular packing also increases anisotropy of penetration of guest grains. When a binary mixture of large and small guest grains is penetrated into the host granular medium, we observe size segregation patterns.     

Self-preservation of large-scale structures in a nonlinear viscous medium described by the Burgers equation

We use the asymptotic solution of the one-dimensional Burgers equation to study the self-preservation of large-scale random structures. We show that in the process of their evolution, large-scale structures remain stable against small-scale perturbations for the case of a continuous initial spectrum with a spectral index smaller than unity. We study both analytically and numerically the correlation coefficient of a large-scale structure and of the same structure with a high-frequency perturbation and show that with the passage of time the coefficient tends to unity. Using the asymptotic formulas of the theory of random excursion of stochastic processes, we study the statistical properties of the perturbing field and find that the effect of high-frequency perturbations is equivalent to the introduction of effective viscosity. Zh. éksp. Teor. Fiz. 115, 564–583 (February 1999)     

A nonlinear granular medium with particle rotation: A one-dimensional model

Equations of nonlinear acoustics are derived from the micromechanical representation of a granular medium as a system of elastically interacting particles possessing translational and rotational degrees of freedom. The structure of the equations is invariant with respect to the shape and size of the particles. The changes in the latter affect only the coefficients in the equations. The inclusion of microrotations and moment interactions of particles leads to the formation of a new type of waves in the medium—microrotational waves. Their dispersion properties are similar to those of spin waves propagating in a magnetoelastic medium. In the low-frequency approximation, the microrotational waves disappear, and the equation describing the transverse waves acquires a term with quadratic nonlinearity. The latter provides an explanation for the generation of the second shear harmonic that is observed in real solids contrary to the predictions of the nonlinear theory of elasticity, which prohibits such phenomena.     

Mixing and condensation in a wet granular medium

We have studied the effect of small amounts of added liquid on the dynamic behavior of a granular system consisting of a mixture of glass beads of two different sizes. Segregation of the large beads to the top of the sample is found to depend in a nontrivial way on the liquid content. A transition to viscoplastic behavior occurs at a critical liquid content, which depends upon the bead size. We show that this transition can be interpreted as a condensation due to the hysteretic liquid bridge forces connecting the beads, and we provide the corresponding phase diagram.     

Large force fluctuations in a flowing granular medium

We study fluctuations in the force at the boundary of a 2D granular flow. The forces are mainly impulsive at all flow rates. The probability distribution of impulses decays exponentially at large impulses, as do the forces in a static granular medium. At small impulses, the distribution evolves continuously with flow rate with no indication of the transition from collisional flow to intermittently jamming flows. However, the distribution of the time interval between collisions tends to a power law, P(tau) - tau(-3/2), showing a clear dynamical signature of the approach to jamming.     

Potential energy in a three-dimensional vibrated granular medium measured by NMR imaging

We use molecular dynamics simulations to study the swelling of randomly end-cross-linked polymer networks in good solvent conditions. We find that the equilibrium degree of swelling saturates at Q(eq) approximately N(3/5)(e) for mean strand lengths &Nmacr;(s) exceeding the melt entanglement length N(e). The internal structure of the network strands in the swollen state is characterized by a new exponent nu = 0.72+/-0.02. Our findings can be rationalized by a Flory argument for a self-similar structure of mutually interpenetrating network strands, agree partially with the classical Flory-Rehner theory, and are in contradiction to de Gennes' c(*)-theorem.     

Numerical studies of a granular gas in a host medium

The nonlinear dissipative Boltzmann equation for an granular gas diffusing in a elastically scattering host medium is investigated and numerically solved by means of direct stochastic simulation. The procedure requires an appropriate treatment of the two collision integrals involved, and of their different features. The algorithm is first tested versus exact results for macroscopic moments worked out in the Maxwellian-pseudo-molecules approximation, and then it is applied to the more realistic collision model of hard spheres for both elastic and inelastic encounters. When collisions with background are dominant, reliability of some hydrodynamic closures is then discussed by comparison of their outputs to the kinetic results achieved by the present DSMC approach.     

Nonlinear electrical conductivity in a 1D granular medium

We report on observations of the electrical transport within a chain of metallic beads (slightly oxidized) under an applied stress. A transition from an insulating to a conductive state is observed as the applied current is increased. The voltage-current (U-I) characteristics are nonlinear and hysteretic, and saturate to a low voltage per contact (0.4 V). Our 1D experiment allows us to understand phenomena (such as the Branly effect) related to this conduction transition by focusing on the nature of the contacts instead of the structure of the granular network. We show that this transition comes from an electro-thermal coupling in the vicinity of the microcontacts between each bead - the current flowing through these contact points generates their local heating which leads to an increase of their contact areas, and thus enhances their conduction. This current-induced temperature rise (up to 1050 C) results in the microsoldering of the contact points (even for voltages as low as 0.4 V). Based on this self-regulated temperature mechanism, an analytical expression for the nonlinear U-I back trajectory is derived, and is found to be in very good agreement with the experiments. In addition, we can determine the microcontact temperature with no adjustable parameters. Finally, the stress dependence of the resistance is found to be strongly non-hertzian due to the presence of the surface films. This dependence cannot be usually distinguished from the one due to the disorder of the granular contact network in 2D or 3D experiments.Received: 7 November 2003, Published online: 28 May 2004PACS: 45.70.-n Granular systems - 72.80.-r Conductivity of specific materials     

Experimental study of nonlinear acoustic effects in a granular medium

Results of a series of experimental studies of nonlinear acoustic effects in a granular medium are presented. Different effects observed in the experiments simultaneously testify that the nonlinearity of granular media is governed by the weakest intergrain contacts. The behavior of the observed dependences suggests that the distribution function of contact forces strongly increases in the range of forces much smaller than the mean force value, which is inaccessible for conventional experimental measuring techniques. For shear waves in a granular medium, the effects of demodulation and second harmonic generation with conversion to longitudinal waves are studied. These effects are caused by the nonlinear dilatancy of the medium, i.e., by the nonlinear law of its volume variation in the shear stress field. With the use of shear waves of different polarizations, the anisotropy of the nonlinearity of the medium is demonstrated. The observation of the cross-modulation effect shows that the nonlinearity-induced modulation components of the probe wave are much more sensitive to weak nonstationary perturbations of the medium, as compared to the linearly propagating fundamental harmonic. The nonlinear effects under study offer promise for diagnostic applications in laboratory measurements and in seismic monitoring systems.