Assembly Of Disks Research Articles

Scale Separation in Granular Packings: Stress Plateaus and Fluctuations

It is demonstrated, by numerical simulations of a 2D assembly of polydisperse disks, that there exists a range (plateau) of coarse-graining scales for which the stress tensor field in a granular solid is nearly resolution independent, thereby enabling an "objective" definition of this field. Expectedly, it is not the mere size of the system but the (related) magnitudes of the gradients that determine the widths of the plateaus. Ensemble averaging (even over "small" ensembles) extends the widths of the plateaus to subparticle scales. The fluctuations within the ensemble are studied as well. Both the response to homogeneous forcing and to an external compressive localized load (and gravity) are studied. Implications to small solid systems and constitutive relations are briefly discussed.

Grain-boundary free energy via thermodynamic integration

In a previous publication by Lusk and Beale [Phys. Rev. E 69, 026117 (2004)], fluctuating cell (FC) theory was used to estimate the free energy of symmetric tilt grain boundaries in an assembly of nearly hard disks. The FC method is much faster than the more traditional thermodynamic integration, but the accuracy of the algorithm has not been assessed in association with persistent defect structures. This motivated the present work wherein the FC free energies are compared directly with the data obtained via thermodynamic integration from an Einstein crystal to an assembly of hard disks. This comparison is made over the range of possible misorientations for symmetric tilt boundaries and indicates that the FC method gives quantitatively accurate estimates for grain-boundary free energy. We also demsonstrate that the FC approximation is quantitatively accurate at determining the free-energy contribution of each particle whether in the bulk or the grain boundary. The FC calculation is about two orders of magnitude faster than a full thermodynamic integration. This approach may offer a numerically efficient means of estimating the free energy of persistent defect structures to greater accuracy than is afforded by the quasiharmonic and local harmonic approximations.

Rheophysics of dense granular materials: Discrete simulation of plane shear flows

We study the plane shear flow of a dense assembly of dissipative disks using discrete simulation and prescribing the pressure and the shear rate. Those shear states are steady and uniform, and become intermittent in the quasistatic regime. In the limit of rigid grains, the shear state is determined by a single dimensionless number, called the inertial number I , which describes the ratio of inertial to pressure forces. Small values of I correspond to the quasistatic critical state of soil mechanics, while large values of I correspond to the fully collisional regime of kinetic theory. When I increases in the intermediate dense flow regime, we measure an approximately linear decrease of the solid fraction from the maximum packing value, and an approximately linear increase of the effective friction coefficient from the static internal friction value. From those dilatancy and friction laws, we deduce the constitutive law for dense granular flows, with a plastic Coulomb term and a viscous Bagnold term. The mechanical characteristics of the grains (restitution, friction, and elasticity) have a small influence in the dense flow regime. Finally, we show that the evolution of the relative velocity fluctuations and of the contact force anisotropy as a function of I provides a simple explanation of the friction law.

Multiscaling at PointJ: Jamming is a Critical Phenomenon

We analyze the jamming transition that occurs as a function of increasing packing density in a disordered two-dimensional assembly of disks at zero temperature for "Point J" of the recently proposed jamming phase diagram. We measure the total number of moving disks and the transverse length of the moving region, and find a power law divergence as the packing density increases toward a critical jamming density. This provides evidence that the T=0 jamming transition as a function of packing density is a second order phase transition. Additionally, we find evidence for multiscaling, indicating the importance of long tails in the velocity fluctuations.

Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.

Modelling of cutting tool – soil interaction – part II: macromechanical model and upscaling

In the foregoing paper we have proposed a strategy for soil modelling based on the discrete approach. By considering the soil as an assembly of rigid disks we have developed a local model. The model concentrates at the contact level between the disks the real mechanical behaviour of the soil. For this purpose suitable contact models have been developed, where specific elasto-plastic laws have been implemented in the node-to-segment contact formulation.

Are granular materials simple? An experimental study of strain gradient effects and localization

Experiments test the dependence of shearing stress on the first two gradients of shear strain. The tests were conducted by direct numerical simulation using the Discrete Element Method (DEM) on a large two-dimensional (2D) assembly of circular disks. The assembly was coerced into non-uniformly deformed shapes by applying body forces to the material region. The tests show that shearing stress is affected by both the first and second gradients of shear strain, and the measured responses to strain and its gradients are all incrementally non-linear. The dilation rate is unaffected by strain gradients. Particle rotations, although highly erratic, are, on average, consistent with the mean-field rotation and unaffected by strain gradients. In independent unconstrained tests, the material was sheared without body forces so that localization could freely occur. Three localization patterns were observed: microbands at very small strains; non-persistent shear bands at moderate strains; and persistent bands at large strains. The observed features of microbands and shear bands are consistent with the measured influences of shearing strain and its first two derivatives.

Grain-boundary free energy in an assembly of elastic disks.

Grain-boundary free energy is estimated as a function of misoriention for symmetric tilt boundaries in an assembly of nearly hard disks. Fluctuating cell theory is used to accomplish this since the most common techniques for calculating interfacial free energy cannot be applied to such assemblies. The results are analogous to those obtained using a Leonard-Jones potential, but in this case the interfacial energy is dominated by an entropic contribution. Disk assemblies colorized with free and specific volume elucidate differences between these two characteristics of boundary structure. Profiles are also provided of the Helmholtz and Gibbs free energies as a function of distance from the grain boundaries. Low angle grain boundaries are shown to follow the classical relationship between dislocation orientation/spacing and misorientation angle.

Structural studies on Δ 3-Δ 2-enoyl-CoA isomerase: the variable mode of assembly of the trimeric disks of the crotonase superfamily

Subunits of the enzymes in the crotonase superfamily form tight trimeric disks. In most members of this protein superfamily these disks assemble further into hexamers. Here we report on the 2.1 Å structure of a tight hexameric crystal form of the yeast peroxisomal Δ 3-Δ 2-enoyl-CoA isomerase (Eci1p). A comparison of this structure to a previously solved crystal form of Eci1p and other structures of this superfamily shows that there is much variability with respect to the relative distance between the disks and their relative orientations. In particular helices H2 and H9 are involved in the inter-trimer contacts and there are considerable structural differences in these helices in this superfamily. Helices H2 and H9 are near the catalytic cavity and it is postulated that the observed structural variability of these helices, stabilized by the different modes of assembly, has allowed the evolution of the wide range of substrate and catalytic specificity within this enzyme superfamily.

A boundary integral for gradient averaging in two dimensions: application to polygonal regions in granular materials

AbstractA boundary integral is derived for averaging the gradients of a function within a two‐dimensional (2D) region. The double integral uses the boundary derivative of the function to compute the average gradients, and it accounts for possible discontinuities in the function along the boundary. The integral reduces to a simple matrix expression for a polygonal region. When applied to a 2D granular material, the matrix expression can be used to compute the averaged local strains within polygonal void regions (particle clusters). In this situation, a realistic calculation of strain must account for the discontinuous movements among rigid particles along the polygon sides, as might occur if the particles are rotating as well as translating. The matrix expression provides a simple and efficient means of correcting the average strain to account for the discontinuous movements. For a cluster of circular disks, the correction is a consequence of the rolling and sliding among particles. The significance of the correction is illustrated with an example simulation of a large dense assembly of circular disks. Although the paper applies the boundary integral to the kinematics of granular regions, the integral will likely find other applications in 2D situations that involve discontinuous fields. Copyright © 2003 John Wiley & Sons, Ltd.

Simulations of Galaxy Formation in a Λ Cold Dark Matter Universe. II. The Fine Structure of Simulated Galactic Disks

We present a detailed analysis of the disk component of a simulated galaxy formed in the ΛCDM cosmogony. At redshift z = 0, two distinct dynamical components are easily identified solely on the basis of the orbital parameters of stars in the galaxy: a slowly rotating, centrally concentrated spheroid and a disklike component largely supported by rotation. The disk may be further decomposed into a thin, dynamically cold component with stars on nearly circular orbits and a hotter, thicker component with orbital parameters transitional between the thin disk and the spheroid. Supporting evidence for the presence of distinct thick- and thin-disk components is provided, as in the Milky Way, by the double-exponential vertical structure of the disk and in abrupt changes in the vertical velocity distribution as a function of stellar age. The dynamical origin of these components offers intriguing clues to the assembly of spheroids and disks in the Milky Way and other spiral galaxies. The spheroid is old and has essentially no stars younger than the time elapsed since the last major accretion event, ~8 Gyr ago for the system we consider here. The majority of thin-disk stars, on the other hand, form after the merging activity is over, although a significant fraction (~15%) of thin-disk stars are old enough to predate the last major merger event. This unexpected population of old-disk stars consists mainly of the tidal debris of satellites whose orbital plane was coincident with the disk and whose orbits were circularized by dynamical friction prior to full disruption. More than half of the stars in the thick disk share this origin, part of a trend that becomes more pronounced with age: 9 out of 10 stars presently in the old (age of ≳10 Gyr) disk component were actually brought into the disk by satellites. By contrast, only one in two stars belonging to the old spheroid are tidal debris; the rest may be traced to a major merger event that dispersed the luminous progenitor at z ~ 1.5 and seeded the formation of the spheroid. Our results highlight the role of satellite accretion events in shaping the disk, as well as the spheroidal, component and reveal some of the clues to the assembly process of a galaxy preserved in the detailed dynamics of old stellar populations.

Particle Motion Tracking Utilizing a High-Resolution Digital CCD Camera

Abstract Digital imaging acquisition and processing is applied for the tracking of a two-dimensional assembly of disks. The disks are used to assemble models for studying granular material behavior for which the kinematics of the individual particles, and hence the disks' trajectory during motion, are of cardinal importance. Large particle size is dictated by photoelastic interparticle interaction analysis. Therefore, a sufficiently large representative number of particles results in a large physical model. The system requirements of a sizable physical model comprised of a large number of particles, and simultaneous precise tracking of all particles, are achieved through the use of high-resolution digital CCD camera and semi-automatic image processing and analysis algorithms. The physical system setup, image acquisition procedure, calibration, accuracy, and algorithms of analysis are described. The system's performance is demonstrated through test results describing the displacement field along with the motion and interaction of individual particles. The system's advantages in providing detailed tracking of a large number of particles with minimal labor are unmatched by any other technique. The data obtained by such systems allow a better understanding of particulate media behavior with the ability to examine and calibrate numerical models on an individual grain basis.

Structured deformation in granular materials

Microscale deformations are investigated in numerical DEM experiments of a large two dimensional assembly of disks. The assembly was subjected to quasi-static biaxial loading at small to moderate strains. Deformations within individual voids were computed from the relative motions of surrounding particles. Evolution of the local fabric was measured in terms of void-based parameters, including effective void ratio, void cell valence, and shape-elongation of the voids, all of which increased monotonically during loading. A direct correlation was measured between local void shape and dilation, which accounts for the transition from compressive to dilatant behavior. Deformation was very nonuniform at the microscale of individual voids. The predominant deformation structures were thin obliquely trending bands of void cells within which slip deformation was most intense. These “microbands” appeared spontaneously throughout the test, even at the start of loading. The microbands ranged in thickness between one and four particle diameters. Unlike shear bands, the microbands were neither static nor persistent: they would emerge, move, and disappear. Their orientation angle increased as deformation proceeded. Dilation was slightly larger within the microbands than in the surrounding material. Void shapes within the microbands were somewhat elongated, with an elongation direction that was related to the orientations of the microbands. Energy dissipation was concentrated within the microbands, even at small strains. In a small cycle of loading and unloading, local fluctuations in the elastic and plastic slips occurred in opposite directions. No spatial relation was found between the deformation microbands and chains of the most heavily loaded particles. Particle rotations were structured, with the most rapid rotations occurring within and near microbands. The rotations tended to relieve sliding between most particles, but transferred the sliding to a few contacts at which frictional slipping was most intense.

Flow and Heat Transfer in Rotating Cavities.

One of the promising trends to increase the thermal efficiency of gas turbines is to raise the gas temperature at the turbine inlet. In modern gas turbines, turbine section operates under severe thermal conditions. Cooling of the turbine section is one of the key technologies for further improvement of gas turbines to keep their safe working. The objective of this study is to investigate the flow and heat transfer in a low-pressure turbine which has cavities formed by an assembly of turbine disks. This study employs two dimensionless parameters which govern flow and heat transfer in rotating cavities. These are throughflow Reynolds number and rotational Reynolds number. Flow visualization and heat transfer experiments in rotating situation were carried out in wide range of experimental conditions. This paper focuses on the effect of forced convection by the throughflow on flow and heat transfer characteristics in rotating cavities.

Yield and deformation of an assembly of disks subjected to a deviatoric stress loading

The deformation of an assembly of particles is examined using a discrete element method (DEM) numerical model. The particles are modeled as random-sized, rough, inelastic, circular two-dimensional disks. The simulations keep track of the displacements, velocities and contact forces of each particle in order to examine the local rearrangements of the particles during the deformation and to determine the bulk stress states. The behavior of cohesive materials can be examined by introducing tensile forces between particles. The tests are done by applying constant confining pressures on two parallel flexible boundaries and a constant displacement rate on the two other flat frictionless walls. During the loading, the axial stress on the moving walls reaches a peak value and afterwards remains essentially constant for large strains. Stress–strain curves are obtained for a large range of confining pressures. They show that the yield envelopes follow the linear Mohr–Coulomb criterion. The global angle of friction is determined for a large range of particle–particle friction angles and particle size distributions. It was found that the global friction angle φ cv increased with interparticle friction angle φ i for φ i<6°; for larger values of φ i, the global friction angle is essentially constant. As the spread in the particle size distribution increased, the magnitude of the internal angle of friction was found to increase for a given interparticle friction angle.

Correlated light scattering by a dense distribution of condensation droplets on a window pane

An analytical model of the scattering structure factor for an assembly of noninteracting hard disks has recently appeared in the literature [Phys. Rev. A 42, 5978-5989 (1990)]. We employ this model to calculate correlated light scattering by monodispersions and binary mixtures of condensation droplets on a window pane. We find that an area fraction of ƒ ≥ 0.6 is required for producing the near-forward direction scattering suppression and that a moderately wide polydispersion of droplet sizes is capable of producing the experimentally observed bright ring of colored light.

Static and dynamic evanescent wave light scattering studies of diblock copolymers adsorbed at the air/water interface

We report the results of static and dynamic evanescent wave light scattering studies of a monolayer of a diblock copolymer, polystyrene-b-polymethylmethacrylate (PS-b-PMMA) with weight averaged molecular weights (Mw) of 880 000:290 000 supported at the air/water interface. Our studies probe the interfacial structural and dynamic properties of the monolayer on a length scale which is a fraction of the wavelength of light. The static light scattering studies were carried out as a function of polymer surface coverage and temperature; we also report some preliminary data for the dependence of the static structure function on the relative molecular weights of the PS and PMMA blocks. The complementary dynamic light scattering studies were carried out only as a function of surface coverage. Our data suggest that, upon spreading in the air/water interface, PS-b-PMMA (880:290 K) copolymers form thin disklike aggregates containing about 240 molecules. These data are consistent with a model in which each such aggregate is a ‘‘furry disk’’ with a dense core consisting of a layer of collapsed PS blocks atop a thin layer of extended PMMA blocks on the water surface and a brushlike boundary of extended PMMA blocks. The data show that the furry disks diffuse freely when the surface coverage is small, but when the surface coverage is large, they are immobile. Our data also suggest that the furry disks can aggregate to form even larger ‘‘islands’’ of disks with an extension greater than 20 μm. The static structure function of the assembly of furry disks is well described, over a wide range of surface coverage, by the structure factor of a two-dimensional hard disk fluid modulated by a two-dimensional hard disk form factor.

Influence of loading pulse duration on dynamic load transfer in a simulated granular medium

An experimental and numerical investigation was conducted to study the dynamic response of granular media when subjected to impact loadings with different periods or wavelengths. The granular medium was simulated by a one-dimensional assembly of circular disks arranged in a straight single chain. In the experimental study, the dynamic loading was produced using projectile impact from a gas gun onto one end of the granular assembly, and the measured wave signal was collected using strain gages. The numerical simulations were conducted using the distinct element method. It was found from the experiments and numerical simulations that input waves with a short period (τ ≈ 90 μs) will propagate in this granular medium with little waveform change under steady amplitude attenuation ; whereas longer waves (τ $ ̆ 200 μs) will propagate with significant waveform dispersion. For these longer wavelength signals, the smooth waveform will undergo separation into a series of short oscillatory signals, and this rearrangement of energy allows a portion of the transmitted signal to increase in amplitude during the initial phases of propagation. Thus the granular medium acts as a nonlinear wave guide, and local microstructure and contact nonlinearity will allow input signals of sufficiently long wavelength to excite resonant sub-units of the medium to produce this observed ringing separation. Following a modeling scheme originally proposed by Nesterenko [J. Appl. Mech. Tech. Phys. 5,733 (1983)], a nonlinear wave equation model was developed which is related to soliton dynamics and leads to travelling wave solutions of specific wavelength found in our experimental and numerical studies.

Incremental constitutive relations for granular materials based on micromechanics

Micromechanically based constitutive relations for two-dimensional flow of granular materials are presented. First, overall stresses are related to the interparticle forces and microstructural parameters. Then, the overall velocity gradient is related to measures of relative sliding and rotation of granules. The notion of the class of granules with continuously evolving distribution of contact normals, is introduced. Simple local constitutive relations are considered for the rate of change of the contact forces, the evolution of the contact normals, the mechanism of local failure, and the density of contacts in a particular class. This leads to macroscopic rate constitutive equations through a Taylor averaging method. Due to nonlinearity of the rate constitutive equations, the response is computed by an incremental procedure. As an illustration, the overall response of a two-dimensional assembly of discs subjected to an overall shearing deformation is determined. In addition, explicit results are presented for the evolution of fabric, contact forces, and the history of active and inactive classes of contacts. The stress-strain relations and the evolution of fabric and contact forces are in qualitative agreement with the observed behaviour of granular materials. In light of these results, the mechanisms of failure and inelastic deformation of dense as well as loose granular materials are discussed.

Evaluation of Static and Dynamic Contact Stresses in Simulated Granular Particles Using Strain Gages

Abstract The application of strain gages for the determination of static and dynamic contact loads in granular perticles is demonstrated. For experimental convenience, the grandar particles are simulated by circular disks fabricated from Homalite-100, a brittle polyester material. Stress field equations in the vicinity of the contact points are carefully evaluated to optimize the relative position of strain gages. The results obtained from strain gages were compared with those obtained using the optical technique of photoelasticity for both static and dynamic problems. Finally, as an example, strain gages are used to study wave propagation in a single chain assembly of disks.