Otis R. Walton

Viscosity, granular‐temperature, and stress calculations for shearing assemblies of inelastic, frictional disks

Employing nonequilibrium molecular‐dynamics methods the effects of two energy loss mechanisms on viscosity, stress, and granular‐temperature in assemblies of nearly rigid, inelastic frictional disks undergoing steady‐state shearing are calculated. Energy introduced into the system through forced shearing is dissipated by inelastic normal forces or through frictional sliding during collisions resulting in a natural steady‐state kinetic energy density (granular‐temperature) that depends on the density and shear rate of the assembly and on the friction and inelasticity properties of the disks. The calculations show that both the mean deviatoric particle velocity and the effective viscosity of a system of particles with fixed friction and restitution coefficients increase almost linearly with strain rate. Particles with a velocity‐dependent coefficient of restitution show a less rapid increase in both deviatoric velocity and viscosity as strain rate increases. Particles with highly dissipative interactions result in anisotropicpressure and velocity distributions in the assembly, particularly at low densities. At very high densities the pressure also becomes anisotropic due to high contact forces perpendicular to the shearing direction. The mean rotational velocity of the frictional disks is nearly equal to one‐half the shear rate. The calculated ratio of shear stress to normal stress varies significantly with density while the ratio of shear stress to total pressure shows much less variation. The inclusion of surface friction (and thus particle rotation) decreases shear stress at low density but increases shear stress under steady shearing at higher densities.

Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres

Abstract Molecular-dynamics-like simulations are utilized to map regions of flow parameter space where steady flows occur for monodisperse assemblies of inelastic, frictional spheres, flowing down frictional, inclined planar surfaces. The trajectory-following technique utilizes nearly-rigid particles interacting via contact forces and gravity. Energy losses in the simulations occur only via displacement-dependent hysteretic loading/unloading paths and sliding friction. Initial scoping calculations are examining flows on an incline tilted 17° from the horizontal, determining the boundaries of various flow regimes. Assemblies of inelastic spheres with interparticle friction coefficients less than the tangent of the inclination angle accelerate unboundedly. Those with friction coefficients somewhat greater than the tangent of the inclination angle develop steady velocity and density profiles for a variety of flow depths but result in arrested flow for coefficients of friction greatly exceeding the inclination angle tangent (e.g., by more than a factor of 2). Conversely, a significant range of inclination angles that will result in steady flows for a given set of assembly properties, is expected to exist.

Simulations and physical measurements of glass spheres flowing down a bumpy incline

Abstract An inclined chute facility and its associated diagnostics has been developed and utilized to study the flow of granular materials. A variety of flow regimes and flow phenomena were observed. Fully developed flows were observed over a bumpy base for a range of slopes. Under some conditions, these flows were dominated by friction and under other conditions, collisions played a dominant role. A variety of unsteady flows were also observed. These include decelerating flows, accelerating flows, and wavy (periodic) flows. The characteristics of the base strongly influenced the flow regime and flow dynamics. Discrete particle simulation model parameters were determined from individual particle tests and particle impact experiments. Simulations of nominally steady flows at two fixed angles showed relatively good agreement with experimental values for particle velocities near the side-walls and on the top surface. The mass flow rate and the flow depth were also consistent with the experiments; however, both experiments and simulations exhibited significant fluctuations about the nominal mean values. The simulations were utilized to interpret flow parameters interior to the flow (i.e., in regions that cannot easily be measured non-intrusively). Far from the side-walls, the granular temperature was found to have a maximum near the bumpy base and to decrease toward the top surface — consistent with granular kinetic theory predictions for flows on bumpy inclines, without side-walls. Near the side-walls the behavior was substantially different with granular temperature decreasing from the top to a minimum at the lower `corners of the chute. This behavior is consistent with experimental measurements of fluctuation velocities near the side-walls. The simulations confirm that the previous discrepancy in the variation of the granular temperature with depth between kinetic theory and near-side-wall measurements was a result of the side-walls, which cause strong three-dimensional structure in the flow.

Particle-Dynamics Calculations of Shear Flow*

Abstract Two-dimensional discrete particle computer models similar to those of P. Cundall (ref. 1, 2) are described. The “soft-particle” approach used in these models allows them to be applied over a wide range of conditions from static situations through rapid shear conditions. Surface friction between particles and with boundary walls is explicitly modeled. Particular attention is paid to the modeling of dynamic situations wherein the energy losses and momentum transfer during inter-particulate collisions play important roles. Comparisons with analytic solutions have verified the numerical techniques and direct comparison with physical tests involving several particles have verified the models ability to calculate the motion of real materials. Direct shear tests on oil shale rubble and corresponding calculations indicate qualitatively similar circulation phenomena and both showed large fluctuations in the magnitude of the shearing force. Incline chute flow calculations are providing detailed descriptions of individual particle paths in which shearing and size segregation phenomena can be observed. Initial comparisons with experiments indicate somewhat slower segregation in two-dimensional calculations than in experiments with spherical particles.

Application of molecular dynamics to macroscopic particles

Abstract The development of molecular dynamics research is briefly reviewed with emphasis on current non-equilibrium computational techniques used to calculate transport coefficients in systems of polyatomic molecules. Many of the numerical methods and numerical- experiment evaluation techniques of that field could be adapted to macroscopic granular material studies. There are, however, significant differences between macroscopic and molecular interactions — differences that considerably modify some of the computational methods and also the interpretation of the calculational results. Macroscopic particles interact with non-conservative interaction forces. Macroscopically available energy is lost from the system during most dynamic interactions — through plastic deformation, friction and breakage. Cundall and Strack are using simplified models of the interaction forces acting between essentially rigid macroscopic particles to study quasi-static deformations in assemblies of two-dimensional circular particles. We are using somewhat similar twodimensional models in a study of non-equilibrium shearing flow of granular materials. Several verification calculations and tests have confirmed the ability of these computer models to predict the dynamic interactions of macroscopic particles. Preliminary results from shearing flow calculations and corresponding laboratory tests are also in qualitative agreement. Plans for a quantitative study of shear stress dependence on shear rate and other parameters are discussed.

Particle-Dynamics Calculations of Gravity Flow of Inelastic, Frictional Spheres

Summary Three-dimensional discrete-particle computer models that calculate the motion of each individual grain in assemblies of hundreds of particles in steady shearing flows with either periodic or real boundaries have been modified to simulate gravity flow of particles through arrays of cylindrical horizontal rods and down inclined chutes. The particle interaction models reproduce experimentally measured recoil trajectories for colliding frictional particles, including rotation effects. Laboratory measurements of the flow of glass beads cascading down through an array of horizontal cylindrical rods correlate well with gravity flow calculations of inelastic, frictional spheres falling through a similar rod array. Less elastic particles are found to cascade through the array faster than nearly elastic particles. Likewise, smaller particles are found to flow faster than large ones. Model simulations of nearly two-dimensional inclined chute flow tests of 6 mm diameter cellulose-acetate spheres flowing over a rough surface between parallel vertical glass plates, result in particle velocities that are considerably higher than values measured in similar laboratory tests at UCLA; however, inclusion of approximate air drag effects in the calculational model eliminates most of the discrepancy producing both density and velocity profiles that are close to the measured values.

Concentration non-uniformity in simple shear flow of cohesive powders

Abstract A method is developed to quantify the particle concentration non-uniformity of cohesive powders in a simple shear flow for both dilute and dense conditions. The ratio of the variance of the particle number concentration, n, to the square of mean particle number concentration, r=〈n′2〉/〈n〉2, is first obtained using a series of sub-cells of different volumes based on a discrete element simulation of particle dynamics in a simple shear flow. The deviation, Δr, of r from the rapid shear rate limit at the same bulk concentration is taken as the measure of the particle concentration non-uniformity. The effects of shear rates on this concentration non-uniformity are quantitatively examined. The quantitative value of this non-uniformity is used to identify a change in the micro-scale cluster structure of dense powder flows under various shear rates and to help in understanding the mechanism for the observed non-monotonic stress–strain rate behavior.

Analysis of explosions in hard rocks: The power of discrete element modeling

Publisher Summary nThis chapter presents an analysis of explosions in hard rocks—the power of discrete element modeling. The salient characteristic of hard rock masses is that they are seldom massive monolithic formations but rather are penetrated by numerous geological discontinuities such as joint sets, faults, shears, and contacts. These discontinuities control the propagation of ground shock and the kinematics of the resulting motions. A realistic analysis of explosions in hard rocks must include the effect of the discrete discontinuities. The fundamental interaction of blocks is through their colliding and rebounding. Except for rare geologies, the block motion has a three-dimensional character. For a complete simulation of explosion effects, the discrete element analysis can be interfaced with the results of analyses in the very near source region of extreme pressures, which can be modeled with continuum-based hydrocodes. Simulation of discrete particle motion has its origins in the field of molecular dynamics, where the bulk properties of systems of particles on a molecular scale are obtained by space and time averaging the velocities and forces acting on the individual particles. However, such molecular-scale models do not handle properties specific to rock masses such as inelastic normal forces, irregular shapes, and tangential friction at contacts.

Potential discrete element simulation applications ranging from airborne fines to pellet beds

Under micro-gravity, lack of sedimentation allows all scales of airborne particulates to participate in the formation of clusters and aggregates. As observed in the International Space Station (ISS), the resulting verylow-density dust aggregates can collect on ventilation inlet screens and duct walls. Discrete Element Method (DEM) simulations, utilizing cohesive interparticle forces and bending-moment interactions, are a tool that can assist in understanding the build-up, compaction, and removal of such agglomerate beds. At a different length-scale, high pellet-pellet contact stresses can be developed in the thermally cycled packed granular beds of air revitalization equipment (possibly fracturing pellets and/or producing unwanted fines). The limits and capabilities of DEM models to simulate these and other particulate systems is discussed.

SIMULATION OF FILLING AND EMPTYING IN A HEXAGONAL-SHAPE SOLAR GRAIN SILO

ABSTRACT A new silo design for grain-storage is examined using a numerical procedure to model its 3D granular flows during the filling and emptying processes. The authors in Hemández-Cordero, et al. Korea-Australia Rheol. J. 12(1)269-281(2000)] have previously presented the design of the new silo and its observed flow behavior. Its main characteristic is the almost complete elimination of excessive dynamics stresses. Since the required computational resources to model the transient phenomena in these experiments are enormous, here, we present basic numerical results related to packing and dynamics of grains considering this complex design. Especial emphasis is given to simulate in great detail collisions of spherical grains with the walls, between themselves, as well as the complex geometry of the new silo, such as filling and unloading openings, inclined walls, etc. The interactions include compressive normal forces between grains as well as tangential forces involved in sliding and rolling between two kernels. Virtual contact mechanics valid in the vicinity of the symmetry plane of the silo are also prescribed, permitting predictions closer to experimentally observed behavior. In spite of the complexity of interactions, steady flow patterns results obtained with reasonable computational times are presented.