Packing Of Identical Spheres Research Articles

A micromechanical model for estimating the shear modulus and damping ratio of loose sands under low stresses. Application to a Mars regolith simulant

The dynamic properties of loose sands under low stresses have been poorly investigated because of the higher order of magnitude of stress levels in terrestrial geotechnical structures. However, low densities and low stresses prevail in the sandy surface deposits of some other rocky planets, making low stress conditions relevant for extra-terrestrial soil mechanics. This is the case of Mars, on the surface of which a seismometer has been placed during the InSight mission. In this context, a dynamic shear rheometer was used to measure the shear modulus and damping ratio of a Martian regolith simulant under very low stresses to improve the interpretation of the InSight dataset on surface materials. This paper also revisits the grain contact stiffness and the overall modulus of a random packing of identical spheres, based on the Hertz-Mindlin contact theory. A micromechanical model accounting for the effects of both grain roughness and slipping in the soil degradation curve is proposed. The results of the model show a good agreement with experimental data, capturing the non-linear transition from low to high-shear strains. The model hence provides a new framework for a better understanding of the behaviour of granular materials in low gravity (extra-terrestrial) conditions.

Discrete element simulation of surface mechanical attrition treatment with rough-surface sonotrode

Surface mechanical attrition treatment (SMAT) is a specialized cold working method that is used to induce compressive residual stresses and refine crystalline grains at the surface of metal components. This technique is increasingly employed in different industries, and the control and optimization of the method require fundamental understanding and accurate process modelling, in particularly detailed information regarding the shot-target interaction. This paper presents a Discrete Element Method (DEM) study of the shot dynamics with rough sonotrodes. These rough surfaces are created by grouping in a hexagonal packing of identical spheres. Different roughness values are obtained by varying the sphere diameter and spacing. The coefficient of restitution (CoR) between shot and target used in our DEM simulations was obtained from Finite Element Method (FEM) simulations of multiple shot-target impacts. The DEM results are analyzed after the SMAT process has reached the steady state. The relationships between surface roughness and the resulting impact angle and velocity distributions are investigated. Furthermore, based on a parametric study, we conducted correlation analyses between processing parameters and shot dynamics, and identified key conditions, using the mechanistic models. The proposed numerical method and findings of this study could be used to tailor the SMAT processes.

Perspective: Basic understanding of condensed phases of matter via packing models.

Packing problems have been a source of fascination for millennia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals, and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable, and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.

Genesis of Hexagonal Close Packed (HCP) Structures from Packing of Identical Spheres (Atoms)

Genesis of Hexagonal Close Packed (HCP) Structures from Packing of Identical Spheres (Atoms)

A simple method for determining sphere packed bed radial porosity

Abstract A new and simple method is established to examine the radial porosity profile of mono-sized spheres in packed bed cylinders based on the sphere center coordinates. The method is derived from geometrical and analytical techniques and uses arc lengths to calculate the radial porosity profile at any given axial position, a given interval axial position, or the total axial position, thus evaluating the local, interval, or axially averaged radial porosity profile, respectively. The new method and analytical functions are simple and straightforward and provide an accurate representation of the radial porosity profile given the sphere center coordinates. The simple method is used to calculate the radial porosity profile for the fixed packing of identical spheres in cylindrical containers with D/d ≥ 1.0. The evaluated results for the radial porosity profile are benchmarked with existing analytical equations and available experimental and numerical data, respectively, for mono-sized spheres in cylindrical containers. A concise FORTRAN program for the new arc length-based technique is presented for this simple method for calculating radial porosity profiles.

Etudes in computational rock physics: Alterations and benchmarking

We tested computational benchmarking data for the absolute permeability, electrical formation factor, and elastic moduli based on the Finney pack, a physical dense random pack of identical spheres digitally rendered into a 3D rectangular coordinate system, as the starting digital object. It is altered by (a) changing the radius of each sphere and (b) geometrically inverting these new packs by swapping grains and pores. Porosity, the absolute permeability, electrical formation factor, and elastic moduli are computed for all these alterations. The direct (grain-based) objects are relevant to clastic rock, and the inverse objects are proxies for carbonates with moldic pores. To corroborate these computational results, we matched the permeability versus porosity, formation factor versus porosity, and elastic moduli versus porosity trends they form by established theoretical rock physics models. These trends persisted when we reduced the scale of investigation by subsampling some of the digital objects under examination.

Radial porosity in packed beds of spheres

A functional approach has been developed to investigate the radial porosity of mono-sized spheres in cylinders. Analytical and semi-analytical equations have been developed to calculate the local radial porosity and the radial porosity distribution, respectively, within a cylindrical packing structure. The analytical equations are based upon fundamental principles and are simple, straightforward and provide highly accurate results for the radial porosity with minimal computational prerequisites. The analytical equations have been developed for the fixed packing of identical spheres in cylindrical containers with D/d ≥ 2.0. The predicted results for the local radial porosity and the radial porosity distributions are benchmarked with an existing analytical equation and available experimental data, respectively, for mono-sized spheres in cylindrical containers.

Using Mindlin theory to model friction-dependent shear modulus in granular media

An explicit expression for the effective shear modulus of a random packing of identical spheres is derived as a function of Mindlin’s tangential stiffness with interparticle contact friction. The motivation behind the approach is to incorporate the effect of intergrain friction to predict velocities in unconsolidated sands. The Mindlin friction term, allowing partial slip across the contact area between pairs of spheres, can be viewed as a parameter accounting for the growing macroscopic intergrain friction in sands as burial progresses. Hence, both moduli and velocities will gradually increase as the compressional- to shear-wave velocity ratio [Formula: see text] or Poisson’s ratio [Formula: see text] decreases. An estimate of effective elastic constants in particular shear modulus can be obtained for a spherical grain pack with an arbitrary frictional behavior ranging between two special contact boundary conditions representing infinite friction and zero friction. The proposedmodel predicts a nonlinear transition between the two special grain-contact conditions when compared to previously published linear relationships. Comparison of elastic properties, i.e., dynamic shear-modulus predictions assuming zero contact friction with experimental data on loose glass bead and sand samples undergoing hydrostatic compression, appears to match reasonably well at low confining stress (less than [Formula: see text]) but deviates gradually as stress increases. It is advocated that the increasing effective internal frictional resistance of the experimental core samples control both the frictional attenuation mechanism in loose grain packs under low confining stress for strain amplitudes typical of seismic waves (less than [Formula: see text]) and the higher stress-velocity sensitivity. Circumstantial evidence of this is found in publications describing both laboratory attenuation analysis and consolidation experiments on granular materials with different degrees of competence or static shear strength.

Microstructures and macroscopic conductivity of randomly packed and uniaxially pressed sphere compacts

The present paper deals with simulation of microstructures and macroscopic conductivity of randomly packed, uniaxially pressed and sintered particles. Random packings of identical spheres are constructed by using a sequential deposition method and their microgeometry after the compaction in sintering is geometrically created by proportional reduction in distances between the sphere centers only in the vertical direction and by mass addition around overlapped necks. Some of microgeometrical characteristics of the created compacts are statistically examined. Using the data on the packings and geometrical models, macroscopic thermal conductivities of the compacts are estimated. It is found that the conductivities are greatly different from those of simple cubic packings, although both the packings have almost the same coordination number, and that anisotropy in the conductivity is induced by the compaction in addition to gravity. The conductivities are expressed as a function of the compaction and sintering degrees for practical purposes.

On the definition of an ideal amorphous solid of uniform hard spheres

Perfection of structure is defined firstly by the definition of imperfections that may occur in that structure, and secondly by the strict requirement of absence of those imperfections. An ideal amorphous solid is a geometrical structure with perfectly random (as distinct from disordered) packing of spheres/atoms. This is achieved by requiring all spheres to be in fixed positions (no rattlers) and the packing to obey certain statistical rules (without exceptions). The random configurations of local clusters are described by the mathematics of self-avoiding random walks, and the distribution of mutual contacts (coordination numbers) is described by combinatorics developed in connection with an earlier work on the structure of liquids. Flaws in the structure are defined. An ideal amorphous solid, based on packing of identical spheres and without any flaws, appears to have packing density close to approximately 0.61. Flaws which form clusters with close packing configurations (fcc and hcp) have the effect of increasing the packing density, whereas other type of flaws, i.e., loose spheres or vacancies will inevitably decrease the packing density. This relationship is revealed by analysis of recently published experimental packings and computer simulations. In that sense, the ideal amorphous solid described here is entirely new and original.

Numerically packing spheres in cylinders

A novel approach has been developed to numerically pack spheres in cylinders. The packing algorithm uses a simple sequential technique that is based on a dimensionless packing parameter. The dimensionless packing parameter includes both axial and radial variables in order to determine a sphere's sequential placement within a cylindrical packing structure. The numerical simulation has been applied to the loose packing of identical spheres in cylindrical containers with D/ d ≥ 2.0. The predicted results for the mean porosity and the radial porosity distributions are validated against the available experimental data for spheres in cylindrical containers. The simulated results are highly accurate and the simple packing algorithm requires minimal computational prerequisites.

Residue packing in proteins: Uniform distribution on a coarse-grained scale

The high packing density of residues in proteins ought to be manifested in some order; to date this packing order has not been thoroughly characterized. The packing regularity in proteins is important because the internal organization of proteins can have a dominant effect on functional dynamics, and it can aid in the design, simulation and evaluation of structures. Packing metrics could also inform us about normal sequence variability, an issue that, with the accumulating genome data, becomes increasingly important. Other studies, indicating a possible correlation between packing density, sequence conservation, and folding nucleation [O. B. Ptitsyn, J. Mol. Biol. 278, 655 (1998)], have emphasized the importance of packing. Here, residue clusters from protein databank structures, each comprised of a central residue and all neighbors located within the first coordination shell, have been rigidly re-oriented and superimposed in a self-consistent optimization. About two-thirds of residues are found to follow approximately the relative orientation preferences of face-centered-cubic packing, when examined on a coarse-grained scale (one site per residue), while the remaining one-third occupy random positions. The observed regularity, which becomes more pronounced after optimal superimposition of core residues, appears to be the result of uniform sampling of the coordination space around each residue on a coarse-grained scale with hydrophobic clustering and volume exclusion, to achieve packing densities close to that of the universal closest packing of identical spheres.

Computational rock physics at the pore scale: Transport properties and diagenesis in realistic pore geometries

Transport properties, such as permeability, are important in many geophysical and petroleum applications. However, complex pore geometry often makes modeling and simulation of transport properties in porous media very difficult. Conventional methods are usually based on partial differential equations, but the implementation of these techniques becomes very complicated when the geometry is extremely complex. As a result, simplified geometry is often used; thus the implementation depends heavily on the model. Solving the same problem in a different geometry often means that many parts of the implementation have to be changed. Therefore, a more robust and simple tool, which can handle complex pore space without oversimplification or modification of the model, is needed. The Lattice-Boltzmann method (LBM), based on statistical description of microscopic phenomena, is one alternative. LBM describes fluid motion as collisions of imaginary particles, which are much bigger than the real fluid molecules. These particles have nearly nothing in common with real fluids, but they show almost the same behavior at a macroscopic scale. This simple collision rule is exactly equivalent to the Navier-Stokes equation within certain appropriate limits. Figure 1 shows a synthetic pore geometry (a random dense pack of identical spheres) and the corresponding digital 3-D structure. Uniform spheres in a random dense pack are a reasonable first approximation to real sandstones. Another advantage for this model is that changing porosity is relatively easy, and laboratory measurements can be mimicked using sintered glass beads. Figure 2 shows electric flux (current) calculated by the finite-element method (FEM) and hydraulic flux calculated by LBM. Both are normalized by each maximum value. The hydraulic flux shows much more sensitivity to the grain boundary than the electric current; the hydraulic flux decreases rapidly near the boundary. Also, the high flux regions are different in Figure 2, even though pore geometry …

Intergranular squirt flow in sand: grains with viscous cement

We offer an exact solution to the problem of deformation of two elastic spherical particles with viscous cement at their contact. This model is intended to mimic a granular geomaterial whose grains are covered with viscous fluid of geologic or biogenic nature (e.g., marine sediment, heavy-oil sand, asphalt concrete). The solution for the normal stiffness of a two-grain combination is reduced to an ordinary integro-differential equation that has to be solved numerically. We find two approximations for the rigorous numerical solution. The simplest one, based on the Maxwell viscoelastic model, cannot accurately reproduce the relaxational behavior of the system. The second one is based on the Cole–Cole model that allows one to introduce a spectrum of relaxation times. This expression is very accurate and can be used instead of the numerical solution. The complex effective elastic moduli of the aggregate are calculated from statistical averaging for a dense random pack of identical spheres. The theoretical results match well experimental data obtained on a glass bead pack with viscous epoxy cement.

Nonlinear Elasticity of Granular Media

The finite and incremental elasticity of a random packing of identical spheres is derived using energy methods. We consider different models for the contact forces between spheres, all of which are based upon or related to the fundamental Hertz theory; we consider only the special cases of perfect friction (no tangential slip) or no tangential friction. The existence of a strain energy function for the medium depends critically upon the type of contact. If the tangential contact stiffness is independent of the normal force, then the energy is well defined for all values of the macroscopic strain. Otherwise, the strain energy of the system is path dependent, in general. However, the concept of a quadratic strain energy function is always well defined for incremental motion superimposed on large confining stress and strain. For all models considered, we derive the changes in wave speeds due to incremental strains. For the models based upon an energy function we derive expressions for the third-order elastic constants as a function of confining pressure.

Plastic compaction of cemented granular materials

Abstract We analytically relate hydrostatic stress to strain in a random dense pack of identical spheres cemented at their contacts. The spheres are elastic and the cement is perfectly plastic. This solution for the sphere pack is based on a solution for the normal interaction of two cemented spheres. Initially, the two spheres touch each other at a point. We show that, as loading increases and cement becomes plastic, a finite (Hertzian) direct-contact area between the spheres necessarily has to develop and progress. The stress-strain behavior of the pack depends on the cement's yield limit and on the amount of cement. At the same hydrostatic stress, the deformation of the cemented aggregate is smaller than that of the uncemented one. This difference becomes large as the yield limit increases. We calculate the bulk modulus of an aggregate from the stress-strain curve. In the plasticity domain, the bulk modulus of the cemented aggregate is smaller than that of the uncemented one. The difference between the two may easily reach 50%. Of course, as the cement's yield limit decreases, the aggregate's stress-strain curve and the bulk modulus approach those of the uncemented sphere pack. This theoretical conclusion is qualitatively supported by experiments on epoxy-cemented glass beads. The maximum contact stress in the cemented aggregate may be less than a half of that in the uncemented one. This result explains an experiment where an uncemented glass bead sample failed at a hydrostatic stress of 50 MPa, whereas an epoxy-cemented sample stayed intact.

Effective properties of cemented granular materials

An analytical model is developed to describe the effective elastic properties of a cemented granular material that is modeled as a random packing of identical spheres. The elastic moduli of grains may differ from those of cement. The effective bulk and shear moduli of the packing are calculated from geometrical parameters (the average number of contacts per sphere and porosity), and from the normal and tangential stiffnesses of a two-grain combination. The latter are found by solving the problems of normal and tangential deformation of two elastic spherical grains cemented at their contact. A thin cement layer is approximated by an elastic foundation, and the grain-cement interaction problems are reduced to linear integral equations. The solution reveals a peculiar distribution pattern of normal and shear stresses at the cemented grain contacts: the stresses are maximum at the center of the contact region when the cement is soft relative to the grain, and are maximum at the periphery of the contact region when the cement is stiff. Stress distribution shape gradually varies between these two extremes as the cement's stiffness increases. The solution shows that it is mainly the amount of cement that influences the effective elastic properties of cemented granular materials. The radius of the cement layer affects the stiffness of a granular assembly much more strongly than the stiffness of the cement does. This theoretical model is supported by experimental results.

Random packings of spheres built with sequential models

Several models have been introduced to construct random packings of identical spheres. The packings are built on a horizontal basal layer using iterative sequential algorithms in which a new added sphere is always in contact with three other spheres in the packing. In the Bennet model the position with the lowest vertical coordinate is selected. In the 'anti-Bennet' model, only positions that are stable under gravity are considered, and that with the highest vertical coordinate is selected. In the Eden model the new sphere is selected at random among all the possibilities and, in the 'stable Eden' model, the random choice is limited to positions that are stable under gravity. In all cases, the density (packing fraction) is determined within an uncertainty of +or-10-4. The histogram for the number of contacts between spheres, the bond angle distribution and the distance distribution were also determined. The results are compared with corresponding ballistic model results.

Close packing of identical spheres

Ordinary marbles can be placed in a wooden rack on the stage of an overhead projector to illustrate the close-pack coordination of 12, the arrangement with 12 nearest neighbor atoms.

Simulation of small particle penetration in a random medium

Random packings of identical spheres of unit diameter were built according to a procedure in which spheres are released one after another along randomly positionned vertical trajectories and then follow the path of steepest descent on the others until they reach a stable position under gravity. Once a packing has been built, smaller spheres, of diameter d does not diverge but saturates to c 11 and the histogram N reaches an exponential shape whose large ΔZ tail is well fitted by N α exp(−0.103 ΔZ). It is shown that such behavior is due to a non zero proportion of equilateral triangles of tangent spheres in the random packing