1. Introduction
Ternary mixtures, after binary ones, are the simplest polydisperse granular packings, which exhibit interesting and unexplained behavior. An examination of their properties is very important and necessary because the particulate systems composed of three particle size fractions are commonly used in many branches of industry (e.g., chemistry, pharmacy, and metallurgy).
In the past, extensive efforts have been made to investigate the structural properties of granular packings that are composed of non-uniformly sized particles by using experimental [
1,
2,
3,
4] and numerical methods [
4,
5,
6,
7,
8]. A number of these studies has been devoted to the analysis of properties of ternary granular mixtures, however, they mainly included an analysis of the packing density and the coordination number, the most important parameters in describing the geometrical arrangement of particles in grain bedding [
1,
9,
10,
11,
12]
An experimental study on ternary mixtures of approximately spherical particles, which was conducted by Westman and Hugill [
9], showed that the apparent volumes of mixtures containing unit real volume of solid fall between limiting values that can be calculated from simple assumptions, and that their deviation from these limits depends in a definite manner upon the diameter ratios of the component particles. In 1961, experimental and theoretical studies on density of ternary packings of nearly spherical steel shots, as conducted by McGeary [
1], extended the study of Westman and Hugill [
9]. A number of attempts have been made for the next few years to model mathematically the packing density of multi-sized granular assemblies [
10,
11,
12]. In 1986, Stovall et al. [
11] proposed a linear density model of multicomponent systems, which was applicable inter alia for binary and ternary mixtures. It assumed that the specific volume of the mixture was a linear function of the volumetric fractions of the individual size classes. Deviations from this theory at certain volumetric fractions, which lead to decrease in packing density, and result in a non-linear particle packing model, as developed by Kwan et al. [
13] and Wong and Kwan [
10] for binary and ternary mixtures, respectively. In 2018, Yan et al. [
12] proposed a mathematical model for calculating the primary porosity of unconsolidated sands based on the grain size distribution and packing texture. The authors proved that the new method is applicable for binary and ternary mixtures of sand.
Two-dimensional image modelling of binary and ternary granular packings, as performed by Mota et al. [
14], have shown a strong relationship between the ratio of the diameter of the largest and the smallest particle in system and the volume fraction on the porosity and tortuosity of granular system. Numerical study on the structural properties of granular assemblies with particle size distributions uniform by number of particles, divided into three, five and seven particle size classes, as conducted by Wiącek et al. [
15], by using a discrete element method (DEM), revealed that the packing density of ternary samples increased when the particle size ratio increased. The opposite tendency was observed for mean overall coordination number, which decreased with increasing particle size ratio. An experimental study on the effect of the volume fraction of the components on the coordination number in ternary mixtures of spheres, which were carried out in 2003 by Zou et al. [
3], indicated that the overall coordination number remained independent of volume fraction of particle size classes. However, the authors observed that the partial coordination numbers varied with volume fraction of components. These findings were later confirmed by the numerical results that were obtained by Yi et al. [
16]. Based on the DEM simulated results, the authors proposed a mathematical model to describe the distributions of partial coordination number in ternary granular packings.
The composition of multi-sized mixtures, determining packing density and distribution of contacts in granular assemblies, also has great influence on their mechanical properties. Micromechanical properties (e.g., distribution of normal contact forces, stress transmission, mobilisation of friction at contact points) and macromechanical properties (e.g., flowability, permeability, elasticity and strength) of granular packings were both found to be affected by degree of particle size heterogeneity and volume fraction of particle size classes [
17,
18,
19,
20,
21].
Martin and Bouvard [
17] observed that the stiffness of binary packings subjected to isostatic compaction increased with an increase in the number of large particles and the particle size ratio. Numerical simulations of compression of polydisperse packings, as performed by Gu and Yang [
22], have shown a decrease in the modulus of elasticity with an increase in the particle size heterogeneity that was confirmed by Wiącek and Molenda [
23] for polydisperse mixtures with normal particle size distribution (PSD). Later studies conducted by Wiącek et al. [
21] for binary mixtures of spheres have shown a significant effect of both, the ratio of the diameters of the smallest and the largest particles and the number fraction of particle size classes on the mechanical properties of packings. The distributions of normal contact forces and the partial and global stresses varied with an increase in particle size ratio and content of small particles in mixture. A decrease in the energy dissipated in granular bedding with an increase in the contribution of small spheres in system and with a decrease in ratio of the diameters of the smallest and the largest particles was observed. Numerical simulations of the compression of polydisperse packings, as prepared by Göncü et al. [
19] and Wiącek et al. [
21], revealed higher pressure in the samples with greater particle size heterogeneity. Wiącek et al. [
21] found that the global stress and packing density followed the same path with increasing contribution of small spheres in binary mixtures. Iddir et al. also reported a strong relationship between the particle size ratio and the contribution of particle size classes and the stresses in binary and ternary particulate systems in simple shear flow [
18].
The study of the relationship between the particle size distribution and the structural and mechanical properties of grain assemblies is of great importance to the university research and the industry because both natural granular materials and the ones involved in industrial processes are nonuniform in size. Over the last few decades, a number of studies have been conducted to investigate the relationship between the structural properties and the mechanical behavior of particulate assemblies with the particle size heterogeneity. Although few investigations were devoted to ternary mixtures with various degree of polydispersity and various volume fractions of particle size classes, a review of the literature shows that these studies mainly focused on the analysis of packing density and coordination number of mixtures. A detailed studies on the micromechanical properties of ternary granular packings with PSD uniform and nonuniform by number of particles, including a contribution of the mechanically stable particles or particles of various sizes into the force transmission through the system, the relationship between the partial coordination number and particle size ratio or distribution of contact forces in the samples with various particle size ratios and number fractions of particle size classes, are very scarce and insufficient. Knowledge in that field seems to be the most demanded in the branches of industry, wherein polydisperse granular materials are subjected to external loads during such processes as tableting, pressing, agglomeration et al. These are, inter alia, pharmaceutical, metallurgical, ceramic, food, and energy industries. Granular materials are subjected to various types of loading conditions during technological processes that frequently result in products of properties different than the properties of components. A production of the stable and uniform products of high quality and the establishment of the optimal handling, storage, and conveying conditions for these materials requires good knowledge of their structural and micromechanical properties. Many of these solids exhibit difficult handling behaviors, which gives rise to considerable challenges in the design and operation of the handling and processing.
An increase in the demand for the knowledge of the micromechanical properties of polydisperse granular packings, which results from increasing interest in granular materials and their application in many branches of industry makes that issue very important and worthy of further investigation.
Therefore, this study focuses on the micro-scale analysis of the effect of particle size ratio and the contribution of particle size fraction on structural and micromechanical properties of mixtures composed of three grain size classes. A discrete element method DEM [
24] was applied to model a confined uniaxial compression test of granular packings, which is one of the methods most frequently recommended for measuring mechanical parameters of grain beddings. A study on ternary granular materials with PSDs uniform and nonuniform by number of particles was conducted, providing a significant knowledge on these idealized sets of spheres. Knowledge gained from presented project may lead to better understanding of real granular mixtures with more complex PSDs and might find application in industries dealing with granular materials.
2. Materials and Methods
The three-dimensional simulations were conducted using the EDEM software (version 2.7) [
24], based on the Discrete Element Method [
25]. A simplified viscous-elastic non-linear Hertz–Mindlin contact model was used [
26], wherein the normal contact force (
), which results from the contact of particle
i with particle
j, is expressed as:
The tangential force is obtained from:
where
In Equations (1)–(3),
kn and
kt are the normal and tangential stiffness coefficients,
δn and
δt are the normal and cumulative shear displacements,
m* is an equivalent mass, and
and
are the normal and tangential components of the relative velocity. In Equation (2), Δ
δ is the relative tangential displacement of the contacting surfaces occurring in the
n-th timestep. The stiffness coefficients may be expressed as:
where
is an equivalent Young’s modulus;
is an effective radius of contacting particles; and
is an equivalent shear modulus. The elastic constants, Young’s modulus
Y, and shear modulus
G, are related to each other, as follows:
where
ν is a Poisson’s ratio.
The coefficient of restitution (
e) dependent coefficient
β in Equations (1) and (3) is given by:
The time step is set to small to allow for the assumption of constant translational and rotational accelerations. The motion of each particle is given by:
where
mi,
Ri,
Ii,
Vi, and
ωi are, respectively, the mass, radius, moment of inertia, linear velocity, and angular velocity of particle
i;
g is acceleration due to gravity, which is assumed to be constant. The normal and tangential forces result from the contact of particle
i with particle
j. Integrating Equations (8) and (9) gives the velocity and position of the particles. The torque of particle, related to rolling friction
µr, is expressed, as follows:
with
li being a distance of the contact point from the center of mass of particle
i.
Tangential contact force was limited by the Coulomb friction law assuming that particles slide over each other when the tangential force is at limiting friction. The Coulomb friction law is expressed as:
where
μs is the coefficient of static friction.
The rigid particles were allowed to locally overlap at contact points while using a soft contact approach. A conservation of volume of bedding was assumed since the overlaps between particles did not exceed 0.01% of particle diameters.
Table 1 lists the input parameters for DEM simulations, corresponding to the mechanical parameters of steel rods and steel walls.
Numerical tests of the confined uniaxial compression of granular packings was performed, which is a standard laboratory test procedure to measure the mechanical properties of granular materials [
27]. The simulations contained three stages: sample generation, compression test, and relaxation of granular packing. In the sample generation stage, spheres with random initial coordinates were generated inside the box with rectangular cross-section of 0.12 m × 0.1 m and 0.12 m thick, which were placed above a chamber of uniaxial compression apparatus of the same size (see
Figure 1). The size of the sample, regarded by authors as a representative elementary volume [
28], allowed for neglecting the wall effects. The spheres settled down onto the bottom of the test chamber under gravity in a dispersed stream, so-called rain filling. In the second stage of simulation, spheres were compressed through the top cover of the chamber that moved vertically downwards at a constant velocity of 3 m/min until a maximum vertical pressure on the uppermost particles reached 100 kPa. In the last stage of simulations, a relaxation of compressed granular packings was conducted until a total kinetic energy of particles reached 10
−6 J. The rigid and frictional walls of apparatus were modelled, which did not deform under the applied load. The test procedure followed the recommendations of the Eurocode 1 [
27]. Three replicate tests were run for each sample to verify the repeatability.
The simulations were carried out while using samples composed of three particle size fractions. The samples were described by particle size ratio
g, defined as a ratio between the diameter of the largest and the smallest spheres. Mixtures with particle size ratio varying from 1.25 to 5 were generated in simulations. Each sample consisted granulometric fraction composed of spheres with diameter of 6 mm and two fractions with smaller particles. The differences between the diameters of particles in neighboring size fractions were equal. The samples had particle size distribution uniform by number of particles (i.e., an equal number of particles of different sizes) or nonuniform by number of particles (i.e., a different number of particles of different sizes). Mixtures with PSD nonuniform by number of particles were described by number fraction of particle size classes
f, which defined a percentage content of spheres representing different particle size classes to total number of particles in mixture.
Table 2 presents the total number (
N) and diameters of particles (
Dk) in mixtures with uniform PSD, while number fraction of particle size classes (
fk), number of particles in fractions (
Nk), and total number of particles (
N) in mixtures with uniform PSD and
g = 1.25 and 3.75 are shown in
Table 3.
4. Conclusions
Wiącek et al.previously conducted the detailed studies on the effect of the particle size heterogeneity and the content of particle size classes on the structural and micromechanical properties of binary granular packings [
4,
21]. Because ternary particulate systems are commonly used in many branches of industry and the knowledge of some of their properties remains scarce, the previous study has been extended to ternary mixtures. Although, in some cases, the binary and ternary granular mixtures exhibit similar behaviors, the latter ones exhibit certain characteristics, typical of these materials. Therefore, in this study, the relationship between the particle size ratio and the number fraction of particle size classes in ternary granular packings, and their structural and mechanical properties were investigated in the micro-scale. A confined uniaxial compression of mixtures that were composed of three particle size fractions was simulated with a discrete element method.
In the last few decades, various efforts have been made to study the properties of ternary packings, however, they were mainly devoted to beddings with different volume fractions of the components. In this study, packings with PSD uniform by number of particles (with equal number of spheres of each diameter), or nonuniform by number of particles (with content of spheres of each diameter equaled to 20% or 40% of total number of spheres) were generated. These idealized ternary mixtures represent interesting sets of spheres whose properties are scarcely known.
The porosity of samples with uniform PSD was found to decrease with an increase in the particle size ratio up to the certain critical particle size ratio above which the porosity remained unchanged. In spite of a decrease in the porosity of ternary mixtures, a few percentage decrease in the average coordination number was observed in the samples. Simultaneously, the corrected coordination number, including mechanically stable particles with more than three contacts, substantially increased with an increase in value of g. A study on the effect of the particle size ratio on the contribution of partial contact numbers into the global contact number and on the transmission of pressures through the contact network in ternary mixtures with PSD being uniform by number of particles has shown a presence of a critical g value above which the properties of packings remained unchanged. Analogous results have been found by authors for binary mixtures composed of spheres with different particle size ratios.
A study of the relationship between the fraction of particle size classes and the microstructure of ternary mixtures revealed a small effect of the content of particle fractions on the porosity and coordination numbers in packings, regardless of the g value. A strong influence of the number fraction of particle size classes on the partial stress in ternary mixtures was observed, which increased with an increase in the value of g. It was found that in ternary mixtures with small g value, the contribution of the partial stress into the global stress is determined mainly by number fraction of particle size classes, while it is determined by particle size ratio to a greater extent in packings with larger g values.
The findings of this study indicate some similarities, but also some differences between binary and ternary granular materials, which confirms a necessity for the investigation of the latter ones as thoroughly as mixtures comprising two components. A study on ternary materials with PSD uniform and nonuniform by the number of particles may lead to better understanding of real granular mixtures with more complex PSDs and more accurate prediction of effects that were observed in particulate assemblies.