Transients in sheared granular matter

As dense granular materials are sheared, a shear band and an anisotropic force network form. The approach to steady-state behavior depends on the history of the packing and the existing force and contact network. We present experiments on shearing of dense granular matter in a 2D Couette geometry in which we probe the history and evolution of shear bands by measuring particle trajectories and stresses during transients. We find that when shearing is stopped and restarted in the same direction, steady-state behavior is immediately reached, in agreement with the typical assumption that the system is quasistatic. Although some relaxation of the force network is observed when shearing is stopped, quasistatic behavior is maintained because the contact network remains essentially unchanged. When the direction of shear is reversed, a transient occurs in which stresses initially decrease, changes in the force network reach further into the bulk, and particles far from the wheel become more mobile. This occurs because the force network is fragile to changes transverse to the force network established under previous shear; particles must rearrange before becoming jammed again, thereby providing resistance to shear in the reversed direction. The strong force network is re-established after displacing the shearing surface , where d is the mean grain diameter. Steady-state velocity profiles are reached after a shear of . Particles immediately outside of the shear band move on average less than 1 diameter before becoming jammed again. We also examine particle rotation during this transient and find that mean particle spin decreases during the transient, which is related to the fact that grains are not interlocked as strongly.Received: 5 March 2004, Published online: 24 August 2004PACS: 45.70.-n Granular systems - 83.80.Fg Granular solids     

Particle dynamics in sheared granular matter

The particle dynamics and shear forces of granular matter in a Couette geometry are determined experimentally. The normalized tangential velocity V(y) declines strongly with distance y from the moving wall, independent of the shear rate and of the shear dynamics. Local rms velocity fluctuations deltaV(y) scale with the local velocity gradient to the power 0.4+/-0.05. These results agree with a locally Newtonian, continuum model, where the granular medium is assumed to behave as a liquid with a local temperature [deltaV(y)](2) and density dependent viscosity.     

Kinetic theory for sheared granular flows

Rapid granular flows are far-from-equilibrium-driven dissipative systems where the interaction between the particles dissipates energy, and so a continuous supply of energy is required to agitate the particles and facilitate the rearrangement required for the flow. This is in contrast to flows of molecular fluids, which are usually close to equilibrium, where the molecules are agitated by thermal fluctuations. Sheared granular flows form a class of flows where the energy required for agitating the particles in the flowing state is provided by the mean shear. These flows have been studied using the methods of kinetic theory of gases, where the particles are treated in a manner similar to molecules in a molecular gas, and the interactions between particles are treated as instantaneous energy-dissipating binary collisions. The validity of the assumptions underlying kinetic theory, and their applicability to the idealistic case of dilute sheared granular flows are first discussed. The successes and challenges for applying kinetic theory for realistic dense sheared granular flows are then summarised.     

Rheology of sheared monodisperse granular suspensions

Sheared granular suspensions can either flow or be jammed. They show as well a ‘melting’ transition: partially ordered flowing states are found which can be melted into fully disordered arrangements of grains by sufficient shear. While these are well documented phenomena, the underlying mechanisms and their control parameters are still far from clear. Via Molecular Dynamics simulations, we study the rheology of a model system of sheared frictional monodisperse granular materials [7, 8]. In particular, we aim to understand the nature of a critical line separating crystallised and melted states and the “jammed” region in the phase diagram. We outline as well connections and differences with thermal glass formers and colloidal suspensions.     

Subdiffusion and cage effect in a sheared granular material

We investigate experimentally the diffusion properties of a bidimensional bidisperse dry granular material under quasistatic cyclic shear. The comparison of these properties with results obtained both in computer simulations of hard spheres systems and Lennard-Jones liquids and experiments on colloidal systems near the glass transition demonstrates a strong analogy between the statistical behavior of granular matter and these systems, despite their intrinsic microscopic differences (thermal vs athermal). More specifically, we study in detail the cage dynamics responsible for the subdiffusion in the slow relaxation regime, and obtain the values of relevant time and length scales.     

Velocity autocorrelations and viscosity renormalization in sheared granular flows

The decay of the velocity autocorrelation function in a sheared granular flow is analyzed in the limit where the wavelength of fluctuations is larger than the "conduction length," so that energy is a nonconserved variable. The decay of the velocity autocorrelation function is much faster than that in a fluid at equilibrium for which energy is a conserved variable. Specifically, the autocorrelation function in a sheared granular flow decays proportional to t-3 in 2D and t-9/2 in 3D, in contrast with the decay proportional to t-1 in 2D and t-3/2 in 3D for a fluid at equilibrium. The renormalization of the viscosity due to mode coupling is evaluated using this form of the decay of the autocorrelation function. It is found that the logarithmic divergence in the viscosity in 2D, and the divergence of the Burnett coefficients in 3D, which is characteristic of a fluid of elastic particles at equilibrium, is absent in a sheared granular flow.     

Nucleation and crystal growth in sheared granular sphere packings

We investigate the nucleation of ordered phases, their symmetries, and distributions in dense frictional hard sphere packings as a function of particle volume fraction ?, by imposing cyclic shear and constant applied pressure conditions. We show, with internal imaging, that the nucleating crystallites in the bulk consist of 10-60 spheres with hexagonal close packed (hcp) order and nonspherical shape, that are oriented preferentially along the shear axis. Above ?=0.62±0.005, crystallites with face centered cubic (fcc) order are observed with increasing probability, and ordered domains grow rapidly. A polycrystalline phase with domains of fcc and hcp order is observed after hundreds of thousands of shear cycles.     

Thermodynamics and effective temperatures in sheared granular matter and emulsions

Recent theories postulate that the non-equilibrium behavior of systems experiencing jamming or structural arrest could be described by equilibrium thermodynamic concepts. If a thermodynamic framework can describe the behavior of systems far from equilibrium, then an effective temperature with a true thermodynamic meaning exists as a key parameter in characterizing the material's properties. In order to examine the validity of the thermodynamics for jammed systems, we perform a numerical experiment with a realistic granular matter model specially conceived to be reproducible in the laboratory. The results strongly support the thermodynamic picture.     

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.     

Numerical simulation of two-dimensional granular shearing flows and the friction force of a moving slab on the granular media

This paper studies some interesting features of two-dimensional granular shearing flow by using molecular dynamic approach for a specific granular system.The obtained results show that the probability distribution function of velocities of particles is Gaussian at the central part,but diverts from Gaussian distribution nearby the wall.The macroscopic stress along the vertical direction has large fluctuation around a constant value,the non-zero average velocity occurs mainly near the moving wall,which forms a shearing zone.In the shearing movement,the volume of the granular material behaves in a random manner.The equivalent friction coefficient between moving slab and granular material correlates with the moving speed at low velocity,and approaches constant as the velocity is large enough.     

Flow, ordering, and jamming of sheared granular suspensions

We study the rheological properties of a granular suspension subject to constant shear stress by constant volume molecular dynamics simulations. We derive the system "flow diagram" in the volume fraction or stress plane (phi, F): at low phi the flow is disordered, with the viscosity obeying a Bagnold-like scaling only at small F and diverging as the jamming point is approached; if the shear stress is strong enough, at higher phi an ordered flow regime is found, the order-disorder transition being marked by a sharp drop of the viscosity. A broad jamming region is also observed where, in analogy with the glassy region of thermal systems, slow dynamics followed by kinetic arrest occurs when the ordering transition is prevented.     

Fluctuations in granular media

Dense slowly evolving or static granular materials exhibit strong force fluctuations even though the spatial disorder of the grains is relatively weak. Typically, forces are carried preferentially along a network of "force chains." These consist of linearly aligned grains with larger-than-average force. A growing body of work has explored the nature of these fluctuations. We first briefly review recent work concerning stress fluctuations. We then focus on a series of experiments in both two- and three-dimension [(2D) and (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/force; the length scales of these fluctuations extend up to 10(2) grains. In 2D, we use photoelastic disks that permit visualization of the internal force structure. From this we can make comparisons to recent models and calculations that predict the distributions of forces. Typically, these models indicate that the distributions should fall off exponentially at large force. We find in the experiments that the force distributions change systematically as we change the mean packing fraction, gamma. For gamma's typical of dense packings of nondeformable grains, we see distributions that are consistent with an exponential decrease at large forces. For both lower and higher gamma, the observed force distributions appear to differ from this prediction, with a more Gaussian distribution at larger gamma and perhaps a power law at lower gamma. For high gamma, the distributions differ from this prediction because the grains begin to deform, allowing more grains to carry the applied force, and causing the distributions to have a local maximum at nonzero force. It is less clear why the distributions differ from the models at lower gamma. An exploration in gamma has led to the discovery of an interesting continuous or "critical" transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, gamma, must be adjusted to a value gamma(c) to reach the "critical point." We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities and particle rotations or spin. Distributions for the azimuthal velocity, V(theta), and spin, S, of the particles are nearly rate invariant, which is consistent with conventional wisdom. Near gamma(c), the grain motion becomes intermittent causing the mean velocity of grains to slow down. Also, the length of stress chains grows as gamma-->gamma(c). The 3D experiments show statistical rate invariance for the stress in the sense that when the power spectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Omega, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectra can be characterized by two different power laws, P proportional, variant omega(-alpha), in the high and low frequency regimes: alpha approximately 2 at high omega; alpha<2 at low omega. The force distributions computed from the 3D stress time series are at least qualitatively consistent with exponential fall-off at large stresses. (c) 1999 American Institute of Physics.     

Effective temperature and jamming transition in dense, gently sheared granular assemblies

We present extensive computational results for the effective temperature, defined by the fluctuation-dissipation relation between the mean square displacement and the average displacement of grains, under the action of a weak, external perturbation, of a sheared, bi-disperse granular packing of compressible spheres. We study the dependence of this parameter on the shear rate and volume fractions, the type of particle and the observable in the fluctuation-dissipation relation. We find the same temperature for different tracer particles in the system. The temperature becomes independent on the shear rate for slow enough shear suggesting that it is the effective temperature of the jammed packing. However, we also show that the agreement of the effective temperature for different observables is only approximate, for very long times, suggesting that this defintion may not capture the full thermodynamics of the system. On the other hand, we find good agreement between the dynamical effective temperature and a compactivity calculated assuming that all jammed states are equiprobable. Therefore, this definition of temperature may capture an instance of the ergodic hypothesis for granular materials as proposed by theoretical formalisms for jamming. Finally, our simulations indicate that the average shear stress and apparent shear viscosity follow the usual relation with the shear rate for complex fluids. Our results show that the application of shear induces jamming in packings whose particles interact by tangential forces.     

Hysteresis and competition between disorder and crystallization in sheared and vibrated granular flow

Experiments on spherical particles in a 3D annular shear cell vibrated from below and sheared from above show a hysteretic freezing or melting transition. Under sufficient vibration a crystallized state is observed, which can be melted by sufficient shear. The critical line for this transition coincides with equal kinetic energies for vibration and shear. The force distribution is double peaked in the crystalline state and single peaked with an approximately exponential tail in the disordered state. Continuous relations between pressure and volume (with dP/dV>0) exist for a continuum of partially and/or intermittently melted states over a range of parameters.     

Strain localisation in granular media

This paper discusses strain localisation in granular media by presenting experimental, full-field analysis of mechanical tests on sand, both at a continuum level, as well as at the grain scale. At the continuum level, the development of structures of localised strain can be studied. Even at this scale, the characteristic size of the phenomena observed is in the order of a few grains. In the second part of this paper, therefore, the development of shear bands within specimen of different sands is studied at the level of the individual grains, measuring grains kinematics with x-ray tomography. The link between grain angularity and grain rotation within shear bands is shown, allowing a grain-scale explanation of the difference in macroscopic residual stresses for materials with different grain shapes. Finally, rarely described precursors of localisation, emerging well before the stress peak are observed and commented.     

Kinetic model for sheared granular flows in the high Knudsen number limit

The sheared granular flow of rough inelastic granular disks is analyzed in the high Knudsen number limit, where the frequency of particle-wall collisions is large compared with particle-particle collisions, using a kinetic theory approach. An asymptotic expansion is used in the small parameter epsilon =(nsigmaL), which is the ratio of the frequencies of particle-particle and particle-wall collisions, where n is the number of disks per unit area, sigma is the disk diameter, and L is the channel width. The collisions are specified using a normal coefficient of restitution e(n) and a tangential coefficient of restitution e(t). The analysis identifies two regions in the e(t) - e(n) parameter space, one where the final steady state is a static one in which the translational velocities of all particles decrease to zero, and the second where the final steady state is a dynamic one in which the mean square velocities scale as a power of epsilon in the limit epsilon --> 0. Both of these predictions are shown to be in quantitative agreement with computer simulations.     

Dynamical heterogeneity close to the jamming transition in a sheared granular material

The dynamics of a bidimensional dense granular packing under cyclic shear is experimentally investigated close to the jamming transition. Measurement of multipoint correlation functions are produced. The self-intermediate scattering function, displaying slower than exponential relaxation, suggests dynamic heterogeneity. Further analysis of four point correlation functions reveal that the grain relaxations are strongly correlated and spatially heterogeneous, especially at the time scale of the collective rearrangements. Finally, a dynamical correlation length is extracted from a spatiotemporal pattern of mobility. Our experimental results open the way to a systematic study of dynamic correlation functions in granular materials.     

Structure and cluster formation in granular media

The two most important phenomena at the basis of granular media are excluded volume and dissipation. The former is captured by the hard sphere model and is responsible for, e.g., crystallization, the latter leads to interesting structures like clusters in non-equilibrium dynamical, freely cooling states. The freely cooling system is examined concerning the energy decay and the cluster evolution in time. Corrections for crystallization and multi-particle contacts are provided, which become more and more important with increasing density.     

Structures and non-equilibrium dynamics in granular media

In granular media, dissipation leads to interesting phenomena like cluster formation in non-equilibrium dynamical states. As an example, the freely cooling system is examined concerning the energy decay and the cluster evolution with time. Furthermore, the probability distribution of the collision frequency is discussed. Uncorrelated events lead to a Poisson distribution for the collision frequencies in the homogeneous system, whereas cooperative phenomena can be related to a power-law decay of the collision probability per unit time. To cite this article: S. Luding, C. R. Physique 3 (2002) 153–161.