Effect of Particle Shape and Fine Content on the ...

Effect of Particle Shape and Fine Content on the Behavior of Binary Mixture

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Tang-Tat Ng, F.ASCE 1; Wei Zhou 2; and Xiao-Lin Chang 3

Abstract: The behavior of binary granular mixture is studied by the discrete element method. This granular material contains large and small particles that are similar ellipsoids of different sizes. The size ratio of the major-axes of the large and small ellipsoids is 5. Samples of ellipsoids of different particle shapes have been created by pluviation method and the particle growth technique. Segregation is observed in samples prepared by pluviation method but not particle growth technique. Thus, nine very dense samples (≈ the lowest void ratio) with various fine contents (volume fraction of the small ellipsoids) are generated using the particle growth technique. The result indicates that the packing density is a function of particle shape and fine content. These very dense samples are then sheared under drained triaxial compression to very large strain to determine peak shear strength, critical state void ratio, and ultimate shear strength. Peak shear strength is affected more by particle shape than by fine content. The authors found that the effect of particle shape on the material behavior at critical state is not significant. Fine content affects the critical state void ratio but not the shear strength at very large strain. DOI: 10.1061/(ASCE) EM.1943-7889.0001070. © 2016 American Society of Civil Engineers.

Introduction Granular materials like sands and gravels contain a wide range of particle sizes and particle shapes that affect the bulk properties. The frictions of angular granular soils are higher than those of round granular soils. The shear strength of well graded sands is greater than those of poorly graded (uniform) sands. Particle shape and grain size distribution are very important to the mechanical behavior (Mitchell and Soga 2005). Particle shape can be described by form, roundness, and surface texture. Fig. 1 shows the definition of form and roundness. Form can be described using elongation and sphericity. Elongation is defined as the ratio of smallest diameter to the diameter perpendicular to the smallest diameter. For ellipsoids, elongation is identical to aspect ratio, AR . It ranges from 1 for a sphere to approaching 0 for rod-like particles. Sphericity denotes to the degree to which the shape of a particle approaches a sphere. Wadell’s sphericity is the ratio of particle’s surface area and the surface area of a sphere having the same volume (1932). One of many alternative definitions of sphericity is the ratio of particle volume to the volume of smallest circumscribing sphere. Roundness is the measure of the sharpness of a particle’s edges and corners. Wadell defined roundness (1932) as the ratio of the average radius of curvature of the corners to the radius of the largest inscribed sphere. Another definition is the ratio between the radius of curvature of the most convex part and the mean radius. Texture reflects the local smoothness that can be determined by either the Fourier method (Bowman et al. 2001) or fractal analysis 1 Professor, Dept. Civil Engineering, Univ. of New Mexico, Albuquerque, NM 87131 (corresponding author). E-mail: [email protected] 2 Professor, State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan Univ., Wuhan 430072, China. 3 Professor, State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan Univ., Wuhan 430072, China. Note. This manuscript was submitted on April 20, 2015; approved on December 9, 2015; published online on February 23, 2016. Discussion period open until July 23, 2016; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399.

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(Santamarina et al. 2001). Textural feature can be obtained by the higher harmonics (>8th) of the Fourier method (Bowman et al. 2001). The surface texture affects the coating process as well as the bulk materials packing and shear behavior. Smooth-surfaced particles may be easier to coat with a binder, and have a lower void ratio, but rough-surfaced particles may form stronger mechanical bonds. The particle shape description of a prolate ellipsoid (Ra > Rb ¼ Rc ) is relatively simple (a single parameter). Ra , Rb , and Rc are the major axis, and the two minor axes of an ellipsoid, respectively. The surface texture of an ellipsoid is a constant because the highest harmonics of the Fourier analysis is second order. Table 1 shows the particle radius ratio (PRR ¼ Ra ∶Rc ), elongation, Wadell’s sphericity, and roundness for the three particle shapes considered in this paper. PRR is the reciprocal of elongation. Roundness is estimated as the smallest radius of curvature to the mean radius of curvature that is determined at a number of points on the ellipsoid’s surface. Sphericity various only from 0.956 to 0.994. Sphericity of most natural sands also varies within a narrow range (Edil et al. 1975). Thus, sphericity will not be used to represent the particle shape. Natural gravel and sand grains have been described by the length, width, and height. PRR will be used to denote the particle shape because it is similar to the length and width (or height) ratio of a particle. Effect of particle shape on the behavior of monodisperse samples of ellipsoids have been studied using the discrete element method (DEM) at both macroscopic and microscopic levels (Ng 2001, 2009). Identical sample preparation procedures are used to create samples of different PRR . The initial void ratios of these samples are similar in general. The peak and ultimate friction angles decrease slightly with the increase of PRR . The relationship between the critical state void ratio and PRR is nonmonotonic. When the particle shape deviates from a sphere (increasing PRR ), the critical state void ratio decreases and reaches a lowest value. Then critical state void ratio increases with further increase of PRR . The effect of particle shape on interface behavior has been studied using different clumps of various angularities (Jensen 2001). The clumps are formed by overlapping of multiple discs. The result confirms that shear strength increases with particle angularity.

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models for minimum void ratio and critical state void ratio for various fine contents. The models have found to be useful to represent the experimental observation of six sand-silt mixtures. This paper presents the study of binary mixtures of similar ellipsoids. The particle size ratio between the large and small ellipsoids is 5. Samples of three different particle shapes (PRR ¼ 1.2, 1.5, and 1.7) and three different fine contents (Fc ¼ 10, 30, and 50%) are created and sheared under triaxial compression loading. The numerical result of void and the material behavior is presented.


DEM Input Parameters

Fig. 1. Form and roundness of a grain

Table 1. Shape Parameters: PRR , Elongation, Sphericity, and Roundness of an Ellipsoid PRR Ra=Rc 1.2 1.5 1.7

Elongation or AR

Sphericity

Roundness

0.833 0.667 0.588

0.994 0.973 0.956

0.736 0.467 0.351

Effect of particle shape of large particles in binary mixtures of 50% fine have also been investigated using DEM (Ng 2004a). The large and small ellipsoids are not similar. The particle shape of the large particles affects the peak friction angle. Two-dimensional DEM simulations of binary mixtures of various sizes of discs have been carried out to investigate the contributions of large and small particles on the mechanical behavior (Ueda et al. 2011). The result shows two particular fine contents (Fc1 and Fc2 , Fc1 < Fc2 ) such that the mechanical contribution of the small particles is negligible when Fc ≤ Fc1 and the mechanical contribution of the large particles is negligible when Fc ≥ Fc2 . Effect of Fc has been studied experimentally on samples of binary mixtures such as gravel-sand, sand-silt, sand-clay, or gravel-clay. The clay content of sand-clay mixtures affects packing and shear strength (Kenney 1967; Pakbaz and Moqaddam 2012). Liquefaction potential is affected by the clay content of natural soils. The shear strength of gravel-sand mixtures increases with the gravel content (Holtz and Gibbs 1956; Simoni and Houlsby 2006). The presence of silt of sand-silt mixtures affects the volume-change behavior significantly but not the overall trends of the stress-strain behavior. (Shapiro and Yamamuro 2003). Result of direct shear tests on binary mixtures of fine (clay and silt) and coarse material (sand, glass beads, river sand, and gravel) indicates that the friction angle at constant volume (ϕcv ) is related to particle shape and content of the coarse particles (Li 2013). ϕcv decreases with the elongation of coarse fraction and with the increase of coarse portion. The effect of particle size distribution on liquefaction potential has been examined through physical testing of glass balls and Hostun sand, as well as DEM simulations of spheres (Liu et al. 2014). The static liquefaction potential increases with the increase of uniformity. Theoretical and empirical models have been developed for the packing density of binary mixture (Furnas 1931; Westman 1936; McGeary 1936; Yu et al. 1995; Lade et al. 1998; Rassouly 1999; Elliott et al. 2002). Recently, Yin et al. (2014) proposed empirical © ASCE

Triaxial simulations were carried out using the DEM program ELLIPSE3 H (Ng 2002). The Hertzian and simplified Mindlin’s solutions are employed as the normal and tangential contact laws, respectively (Lin and Ng 1997). The boundaries are rigid. The properties of the elastic ellipsoids are shear modulus of 29 GPa and the Poisson’s ratio of 0.15. The friction coefficient between particles is 0.5. The material properties are consistent with those of quartz sands. Mass scaling is used to speed up the simulations such that the density of the ellipsoid (2; 650 Mg=m3 ) is 1,000 times the real value. Previous studies on binary mixtures indicates that the damping ratio between 0.2 and 2% yield consistent result (Ng 2006; Ng et al. 2015). Therefore 0.2% is used in the study.

Segregation of Binary Mixtures A binary mixture should have enough large particles for the possibility of the development of packing among large particles. The authors prefer the sample’s dimensions to be large enough to accommodate two large particles along each dimension at the minimum. Larger sample size (more particles) may be better but the simulation requires more computational resources. The sample for this study contains 25 large particle (>8) and 3,125 small particles for Fc ¼ 50%. The total number of particles (Np) is 3,150. Pluviation method is commonly used to create samples to mimic the in-situ soil condition. Particles are generated randomly without initial contacts and settled by gravity. A smaller particle-to-particle friction coefficient (μ ¼ 0.1) is used for easy particle adjustment and to achieve greater density. Fig. 2 shows the snapshots of a sample (PRR ¼ 1.5, Fc ¼ 50%) during deposition phase. After equilibrium is reached, μ is increased to 0.5. The system is allowed to reach another state of equilibrium. Finally, the sample is compressed with an isotropic confining pressure (100 kPa) and the final configuration is shown in Fig. 3. The void ratio and average coordination number of this sample are 0.478 and 4.684, respectively. The average coordination number is similar to those of the binary mixtures with smaller RPS (1.25) (Ng 2004a). The void ratio is much lower than those of the binary mixtures (0.609 ∼ 0.639) as expected. Given the same Fc, the void ratio should decrease or remain constant with the increase of RPS . Two more samples (Fc ¼ 30% and 10%) are prepared using the pluviation method. To keep similar sample size as the sample of Fc ¼ 50% (Np ¼ 3,150), the sample (Fc ¼ 30%) contains of 56 large particles and 3,000 small particles (Np ¼ 3,056). Another sample (Fc ¼ 10%) is created with 150 large particles and 3,125 small particles (Np ¼ 3,275). Figs. 4 and 5 show the snapshots during particle deposition. Particle segregation is very clear. They show the Brazil nut effect (the large particles end up on the surface), which is not observed in previous systems of smaller RPS.

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Fig. 2. Snap shots of a sample (Fc ¼ 50%) during deposition: (a) initial stage; (b) intermediate stage; (c) final stage

the particles’ size increases gradually to their final sizes. Internal stress may build up depending on the desired final void ratio. If the internal stress is greater than 10 kPa, the dimensions of the cube will be adjusted to reduce the internal stress. 10 kPa (1=10 of the final confining pressure) is arbitrary selected. The authors will use particle growth technique to create the very dense samples presented in this paper.

Numerical Samples and Results

Fig. 3. Final configuration of a sample (PRR ¼ 1.5, Fc ¼ 50%)

Nonuniform samples have been created with alternating smallparticle dominated layers and large-particle dominated layers. Another issue on this sample size is found. The dimension of the sample (Fc ¼ 50%) are 81:81:79 mm. In order to maintain the possibility of the development of a contact between two large ellipsoids in all directions, the dimension of a sample at critical state should be greater than 60 mm (possibility of contact at the major axis between two large ellipsoids). A 15% vertical strain will reduce the vertical dimension from 79 to 60 mm. The authors past experience indicates that critical state requires a strain no less than 30%. This sample size is good for peak shear strength but not large enough to study the critical state, therefore a larger sample size (Np > 3,275) is needed. The authors have used other sample preparation methods such as isotropic compression method and particle growth technique. Homogeneous samples have been created using the isotropic compression method (Ng 2001). In this method, particles are created randomly in a cube without initial contact. The dimensions of the cube are reduced gradually with the desired confining pressure. However, a desired final void ratio may not be obtained easily. Particle growth technique is similar to the radius expansion method used in PFC3D for spheres (Itasca 2008). The dimensions of a cube are determined by the desired final void ratio. Particles of reduced sizes are created randomly in the cube without initial contact. Then © ASCE

Nine very dense samples (3 PRR and 3 Fc) have been created. For each PRR and Fc, the authors first select a desired void ratio. If the internal stress is less than 10 kPa when all particles are of the correct sizes, the authors will reduce the desired void ratio and repeat the process (particle growth). The final assemblage should be very dense because the particles are not able to fit in the initial cube without increasing the dimensions of the cube. The process is repeated with another assembly of different initial particle locations. If the final void ratios of these two assemblies differ more than 0.01, a third assembly will be generated. The final sample is the assembly with the lowest void ratio. Fig. 6 shows the snap shots during the growth of large ellipsoids. Small ellipsoids of final size (no particle growth is needed) and large ellipsoids of size smaller than the final size are created randomly. A very small μ is used in the process for easy adjustment of particle locations because of the small maximum tangential force at the contact. After every particle size is at the final size, μ is increased to 0.5 and isotropic compression (100 kPa) is applied. Figs. 7–9 show the final configurations of all particles and of the large particles only. Similar configurations of large particles for various PRR are observed when Fc ¼ 10% (Fig. 7). The degree of variation of the configurations of large particles increases as the Fc increases (Figs. 8 and 9). Table 2 shows the characteristics of the nine samples. The dimensions of these samples are large enough for a minimum of 35% vertical strain. The CN is twice the number of contacts divided by the total number of particles. CN increases with the decrease of void ratio and the increase of Fc for all PRR . Low CN is observed because these samples contain a large number of rattlers, particles have fewer than two contacts. The void ratios (em ) of different PRR is plotted in Fig. 10. Although em is very close to the minimum void ratio, em is different

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Fig. 4. Snap shots of deposition of a sample (Fc ¼ 30%): (a) initial stage; (b) intermediate stage; (c) final stage

Fig. 5. Snap shots of deposition of a sample (Fc ¼ 10%): (a) initial stage; (b) intermediate stage; (c) final stage

Fig. 6. Snap shots during growing of particles (Sample R123): (a) RPS ¼ 0.83; (b) RPS ¼ 2.9; (c) RPS ¼ 5 © ASCE

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Fig. 7. Configurations of samples with Fc ¼ 10%: (a) R121, PRR ¼ 1.2; (b) R151, PRR ¼ 1.5; (c) R171, PRR ¼ 1.7


from the minimum void ratio determined in laboratory because of different procedures and stress condition. The general variation of em with Fc is unique for these three PRR . The dashed lines are based on the Furnas model (Furnas 1931), which assumes the small particles are fine enough to fit inside the voidage formed by the large particles. Result of the data (RPS ¼ 5) should plot above the theoretical lines. The concave upward shape is similar to the experimental observation. It is not prudent to fit the data (three data points) with the recent model by Yin et al. (2014), which requires five material constants. © ASCE

In a binary mixture, small particles can be considered to occupy the voids defined by the large particles. Let eL be the void ratio calculated based on large particles only. Fig. 11 shows eL for different PRR and Fc. eL increases with the increase of Fc to accommodate more small particles. Void is then reduced when portion of the space is occupied by the small particles. Reduction of void ratio is represented as the ratio between the void reduction and eL and the result is plotted in Fig. 12. The reduction ratio increases with Fc and is independent of PRR . Samples of ellipsoids with larger PRR (rod like particles) are needed to confirm this finding.

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Table 2. Characteristics of Binary Mixtures of Similar Ellipsoids Sample R121 R123 R125 R151 R153 R155 R171 R173 R175

PRR (Ra=Rc)

Fc (%)

Sample size (Np)

Void ratio (e)

Coordination number (CN)

1.2 1.2 1.2 1.5 1.5 1.5 1.7 1.7 1.7

10 30 50 10 30 50 10 30 50

6,750 6,570 7,056 6,750 6,570 7,056 6,750 6,570 7,056

0.537 0.355 0.416 0.479 0.322 0.375 0.447 0.342 0.377

0.355 1.682 3.662 0.817 5.594 5.428 0.674 4.223 4.863

Fig. 11. Variations of void ratio calculated based on large particles only

Fig. 10. Variation of void ratio of very dense samples

Table 3 lists the coordination number of large particles (CN L ) and coordination number of small particles (CN f ). They are plotted against Fc as shown in Fig. 13. CN L is twice the number of contacts between large particles divided by the number of large © ASCE

Fig. 12. Void reduction after filling of small particles

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Table 3. Coordination Numbers of the Samples Sample

PRR

Fc (%)

CN L

CN f

R121 R123 R125 R151 R153 R155 R171 R173 R175

1.2 1.2 1.2 1.5 1.5 1.5 1.7 1.7 1.7

10 30 50 10 30 50 10 30 50

4.596 3.233 1.929 5.916 3.733 1.750 6.264 4.000 1.714

0.008 1.118 3.129 0.768 3.725 4.438 0.051 2.881 3.955

Table 4. Eigenvalues of Fabrics of Particle Orientation, Branch Vector, and Unit Normal Sample R121 R123 R125 R151 R153 R155 R171 R173 R175


a

Particle orientation 0.342 0.337 0.341 0.338 0.337 0.341 0.346 0.340 0.338

0.325 0.335 0.328 0.333 0.333 0.333 0.329 0.333 0.332

a

0.323 0.328 0.312 0.329 0.330 0.326 0.326 0.326 0.330

Branch vector 0.346 0.343 0.345 0.340 0.347 0.341 0.346 0.348 0.344

0.325 0.331 0.332 0.333 0.331 0.336 0.332 0.335 0.332

0.321 0.326 0.323 0.328 0.322 0.324 0.322 0.317 0.324

Unit normal 0.344 0.414 0.379 0.348 0.396 0.349 0.344 0.403 0.436

0.336 0.311 0.315 0.331 0.310 0.342 0.333 0.321 0.302

0.321 0.277 0.306 0.322 0.295 0.309 0.323 0.276 0.262

τ 1 ¼ 0.342, τ 2 ¼ 0.323, τ 3 ¼ 0.325.

Fig. 14. β (degree of anisotropy) of unit normal for all samples

centers of contacting ellipsoids. In a loose definition, the unit normal is the direction cosines of the contact vector normal to the assumed planar contact surface. A second-order tensor is commonly used to describe the distribution of these vectors. Let Fi ¼ hxi ; yi ; zi i be the ith vector. The fabric tensor is defined as M ¼ ΣFi ⊗ Fi

Fig. 13. CN L and CN f versus fine content for various PRR

particles. CN f is the ratio of twice the number of contacts between small particles and the number of small particles. The general trend between Fc and CN L (or CN f ) is similar to the finding of binary mixtures of circles (Ueda et al. 2011). As Fc increases, CN L decreases while CN f decreases. When Fc ¼ 50%, similar CN L is observed for various PRR . Large particles are mostly coated by small particles because there are less than two contacts between large particles. When Fc ¼ 10% (very few small particles), CN f is so small such that particle shape is not matter as these small particles are floating in the voidage. The load is carried by the network of large particles. It is conjectured that the effect of small particle shape is not significant for small Fc and the effect of large particle shape is not significant at high fine content. Particle orientation, branch vector, and unit contact normal vector (unit normal) are examined for these nine isotropically compressed samples. Particle orientation is the direction of the major axis of an ellipsoid. The branch vector is the vector jointing the © ASCE

ð1Þ

Three eigenvalues (τ 1 , τ 2 , τ 3 , and τ 1 ≥ τ 2 ≥ τ 3 ), can be obtained from M. Table 4 lists the eigenvalues for these three fabrics. Particle orientation and branch vector are very close to isotropic since τ 1 ≈ τ 3 for all samples. The ratio βð¼ τ 1 =τ 3 Þ has been used to quantify the degree of fabric anisotropy (Ng 2004b). β ¼ 1 implies isotropic fabric. Greater deviation from 1 associates with greater degree of anisotropy. Fig. 14 shows the β of unit normal for all samples. The unit normal is almost isotropic (β ≤ 1.08) for Fc ¼ 10% regardless of particle shape. The shape of the function between β and Fc is concave down for PRR ≤ 1.5 while a monotonic increase function is found for PRR ¼ 1.7. Particle shape affects the distribution of unit normal but not the particle orientation and branch vector for these dense samples. Mechanical behavior of these samples has been obtained by drained compression simulations. Result of samples (PRR ¼ 1.5) is shown in Fig. 15. Similar results are found for samples of other particle shapes (PRR ¼ 1.2 and 1.7). Similar shear strength at very large strain is observed for all nine samples regardless of particle shape or fine content that is similar to that in the DEM simulations of spheres (Yan and Dong 2011). However, the result is conflict with the experimental finding of direct shear tests (Li 2013). The contrary finding may be attributable to the limitations of direct shear tests such as the weakest plane may not be coincide with the

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Fig. 17. Peak friction angle and degree of anisotropy

Fig. 15. Stress-strain curves for Samples R151, R153, and R155

predetermined shear plane and the nonuniform stress on the shear plane. The result of peak friction angle and critical state void ratio is plotted in Fig. 16. Particle shape clearly affects the peak friction angle. Fc is not important as the peak friction angles of various Fc only differs by 2°. A significant decrease of initial void ratio is found between samples of Fc ¼ 10% and 30% but the friction angle only differs by 1°. The initial void ratio alone is not sufficient to predict peak shear strength. Because these samples are very dense that can be considered to be at similar relative density (≈100%). Relative density may correlate better to peak shear strength than void ratio does for samples having similar fabric. PRR does not affect the critical state void ratio [Fig. 16(b)]. The curve shape between critical state void ratio and Fc is concave upward that is similar to that of em and Fc (Fig. 10). The result confirms that the empirical model for minimum void ratio and the empirical model for critical state void ratio have similar forms (Yin et al. 2014). The variation of peak friction angle and the initial anisotropy of unit normal are presented in Fig. 17. The result indicates that initial anisotropy of unit normal has no effect on peak friction angle.

Conclusions

Fig. 16. Peak friction angles and critical void ratios of samples of different fine contents © ASCE

Effect of particle shape on void ratio and mechanical behavior of binary mixture of similar ellipsoids has been examined. Because of the selected particle size ratio (RPS ¼ 5) between the large and small ellipsoids, segregation occurs when particles are settled by gravity. Thus, the authors used particle growth technique to prepare nine very dense samples without segregation. The micromechanics of these samples indicate isotropic distributions of particle orientation and branch vector. Both fabrics are not affected by PRR and Fc. The distribution of unit normal shows anisotropy when Fc ≥ 30%. Initial void ratios are functions of PRR and Fc. The general function shape between void ratio and Fc is concave upward with the lowest void ratio when Fc ¼ 30%. There are fewer selfcontacts between large particles as Fc increases. Large particles are surrounded by the small particles for samples of higher fine content (e.g., ≥30%). Particle shape affects the peak shear strength with the greatest peak shear strength when PRR ¼ 1.5. There is no relationship between the initial anisotropic fabric of unit normal and the strength of material. Critical state void ratio is a function of Fc but not particle shape. Shear strength at very large strain is similar for samples of different particle shape and Fc. The effect of particle shape has clearly been observed in this study.

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Acknowledgments This work was partially supported by the National Natural Science Foundation of China (No. 51322905 and No. 51379161) and the National Basic Research Program of China (Project 2013CB035901).


Notation The following symbols are used in this paper: AR = aspect ratio, the ratio between the minor axis and major axis of an ellipsoid; CN = number of contacts per particle of all ellipsoids; CN f = number of contacts per particle among small ellipsoids; CN L = number of contacts per particle among large ellipsoids; ecr = critical state void ratio; eL = void ratio calculated with the large ellipsoids only; em = minimum void ratio of a sample prepared by particle growth method; Fc = fine content, the weight of small ellipsoids of the total weights of all ellipsoids; Np = number of total ellipsoids (Sample size); PRR = the ratio between major axis and minor axis of an ellipsoid (Ra=Rc); Ra = major axis of an ellipsoid; Rb ≥ = minor axis of an ellipsoid; Rc RPS = the ratio of the major axis of the large ellipsoid and major axis of the small ellipsoid; β = ratio of the major and minor eigenvalues of the secondorder fabric tensor; and μ = friction at the contact between two ellipsoids.

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