Shear behaviors of granular mixtures of gravel-shaped

coarse and spherical fine particles investigated via discrete

element method

Jian Gong a,b

Ph.D student

E-mail: gjdlut@mail.dlut.edu.cna State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China.

b School of Civil Engineering, Central South University, Changsha 410075, China

Jun Liu a (corresponding author) *

Professor

E-mail: junliu@dlut.edu.cn

Tel: +86 411 84708894a State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China.

Liang Cui c

LecturerEmail: l.cui@surrey.ac.ukTel: 01483 68 6214c Department of Civil and Environmental Engineering, University of Surrey, UK

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171819202122

Shear behaviors of granular mixtures of gravel-shaped

coarse and spherical fine particles investigated via discrete

element method

Abstract: The shear behaviors of granular mixtures are studied using the discrete

element method. These granular materials contain real gravel-shaped coarse particles

and spherical fine particles. Dense samples have been created by the isotropic

compression method. The samples are then sheared under drained triaxial

compression to a large strain to determine the peak and residual shear strengths. The

emphasis of this study is placed on assessing the evolutions of contributions of the

coarse-coarse (CC) contacts, coarse-fine (CF) contacts and fine-fine (FF) contacts to

the peak and critical deviator stresses. The results are used to classify the structure of

granular mixtures. Specifically, the granular mixtures are fine-dominated or coarse-

dominated materials when the coarse particle content is less than 30%-40% or greater

than 65%-70%, respectively. A comparison with previous findings suggests that the

spherical binary mixtures will become coarse-dominated materials at a relatively

larger coarse particle content (i.e., 75%-80%) than this study (i.e., 65%-70%), which

is attributed to the particle shape effect of coarse particles. A microscopic analysis of

CC, CF and FF contacts at the peak and critical states, including normal contact forces

and proportions of strong and weak contacts of each contact type to total contacts,

reveals why the contributions of CC, CF and FF contacts to the peak and residual

shear strengths are varied. Finally, a detailed analysis of the anisotropies indicates that

the increases of peak and residual shear strengths are primarily related to the gradual

increases in geometrical anisotropy ac and tangential contact force anisotropy at to

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compensate for the continuous decrease in normal contact force anisotropy an.

Furthermore, it is interesting to note that the branch vector frame provides a better

linear relationship between the stress ratio and the geometric anisotropy of the strong

and nonsliding subnetwork than the contact frame for the coarse-dominated materials.

Keys words: Granular mixtures; Contact type; Classification; Coarse particle content;

Particle shape; Drained triaxial tests; Anisotropy.

1. Introduction

Granular mixtures due to multiple mechanisms are common media in civil

engineering applications. For example, rockfill-sand mixtures due to weathering and

deposition are widespread in many natural slopes and rockfill structures [1]. Waste

rock-tailing mixtures are often encountered in tailing dams to address impoundment

stability and acid rock drainage concerns for tailing waste management [2]. Ballast-

fouling mixtures exist in railway transport structures widely due to ballast degradation

and infiltration of external fine particles such as coal and clay [3, 4]. These granular

mixtures constitute a distinct structure that consists of coarse and fine particles with

different shapes and sizes. The mechanical behaviors of granular mixtures are

complex, mainly due to their discrete and heterogeneous attributes. In fact, these

attributes lead to specific evolutions of internal texture under loading. To obtain a

better understanding of the mechanical behaviors of granular mixtures, it is necessary

to probe the evolutions of these internal textures.

A binary mixture is the simplest case among mixture packings, and consists of

two materials with particles of diameter Dc (coarse particles) and Df (fine particles).

The discrete element method (DEM) is widely known to be a powerful and

computationally intensive approach to model granular materials. Recently, the effect

of coarse particle content (or fines content) on the mechanical behaviors of binary

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mixtures has been studied using the DEM at both macroscopic and microscopic

levels. For example, Minh et al. [5, 6] have explored the force transmission through

binary mixtures of sand- and silt-sized spheres under one-dimensional compression.

Langroudi et al. [7, 8] and Shire et al. [9-11] have investigated the micromechanical

behavior of internally unstable/stable gap-graded soils under isotropic compression.

Furthermore, Ueda et al. [12], Zhou et al. [13], and Gong and Liu [14, 15] have

studied the shear strength of binary mixtures by means of biaxial and direct shear

tests. These paradigms commonly use sphere or disc packing to model particle

interactions, because of the expensive computational costs incurred when simulating a

binary mixture system [16]. Obviously, the spherical and discoid particles are overly

simplified when compared with real granular materials, especially for extremely

irregular coarse particles such as gravels, waste rocks and railway ballasts. To date,

few studies have attempted to explore the mechanical behaviors of binary mixtures

with different particle shapes. For example, Zhou et al. [17] have used spheres and a

rolling resistance model to study the undrained behavior of binary granular mixtures

with different fines contents. Yang et al. [18], Azema et al. [19], Lu et al. [20] and

Gong et al. [21] have conducted two-dimensional DEM simulations to investigate the

shear behaviors of binary mixtures composed of irregular (i.e., elongate and

polygonal) coarse particles or discoidal fines. Furthermore, Ng et al. [22-24] have

adopted ellipsoidal coarse and fine particles in three-dimensional DEM simulations to

probe the effects of particle shape and fines content on the packing density and shear

strength of binary mixtures. In general, investigations on the binary mixtures of

irregular particles are rare, especially for cases considering actual coarse particles.

The interparticle contacts in granular materials can be considered as a complex

network forming a highly inhomogeneous structure. Previous DEM studies have

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defined some meso-structures to explore the mechanism of the load resistance within

granular systems. The descriptor of mesoscopic structures includes two major

methods, i.e., using the contact loops [25, 26] or the contact types [5, 14, 18, 19, 24,

27, 28]. The contact types are easier to determine than contact loops; thus, researchers

prefer to employ different contact types to explore the microscale statistical

information of granular mixtures. Contact types in binary mixtures can be classified as

coarse particle–coarse particle (CC) contacts, coarse particle–fine particle (CF)

contacts and fine particle–fine particle (FF) contacts. The developments of CC, CF

and FF contacts with the associated force chains under loading, sliding as well as their

contributions to deviator stress are interesting and can enhance our understanding of

granular mixtures to a deeper level. Based on the discrete expression of the stress

tensor under triaxial loading, networks of different contact types (CC, CF and FF)

contributing to the overall stress tensor σ ijand the deviator stress σ d can be evaluated

with [5, 14, 18, 19, 24, 28]:

σ ij=σ ijCC+σ ij

CF+σ ijFF (1)

σ d=σ11−σ 22+σ33

2=σ

d

CC

+σdCF+σd

FF (2)

Based on DEM triaxial simulations of spherical binary mixtures with different fines

content, De Frias Lopez et al. [28] have quantified σ dCC, σ d

CF and σ dFF of all samples at

σ d = 100 kPa. All binary mixtures then can be classified into four structures: an

overfilled structure when σ dFF > σ d

CF > σ dCC; an interactive-overfilled structure when σ d

CF

> σ dFF > σ d

CC; an interactive-uderfilled structure when σ dCF > σ d

CC > σ dFF; and an

underfilled structure when σ dCC > σ d

CF > σ dFF. Such a classification system enables

engineers to predict the behaviors of granular mixtures by grouping them into similar

response categories. Gong and Liu [14] have quantified the σ dCC, σ d

CF and σ dFF values of

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binary mixtures of spheres at the peak state. The classification results of granular

mixtures as reported in De Frias Lopez et al. [28] and Gong and Liu [14] are

comparable to the experimental results. Clearly, the classification result is directly

related to the relatively values ofσ dCC, σ d

CFandσ dFF. As the strain is developed, the

contacts will separate and recontact under an external loading, indicating that the σ dCC,

σ dCFandσ d

FF values may change. Therefore, the effectiveness of the classified method

proposed by De Frias Lopez et al. [28] needs further verification at relatively large

strain level. This requirement is the motivation of the current work, wherein we

conduct DEM simulations on binary mixtures using gravel-shaped coarse particles

and quantify the σ dCC, σ d

CF and σ dFF values during the entire strain development process.

The main aim of this paper is to systematically explore the mechanical behaviors

of binary mixtures with different coarse particle contents at both the peak state and the

critical state using the DEM. The binary mixtures consist of real gravel-shaped coarse

particles and spherical fine particles. To highlight the effect of particle shape,

supplemental tests with binary mixtures consisting of spherical coarse particles and

spherical fine particles were also conducted. In particular, the variations of

contributions of CC, CF and FF to the deviator stress during the entire strain

development process are addressed, and used to classify the structure of granular

mixtures. This paper is organized as follows. First, an introduction to DEM modeling

is provided. Then, several macroscale simulation results are presented. Then, the

variations of CC, CF and FF contacts contributing to the peak and residual deviator

stress are quantified, and then a classification of binary mixtures is performed. In

addition, the effects of coarse particle shape on the contributions of CC, CF and FF

contacts to the peak deviator stress are investigated. Afterwards, the microscopic

mechanisms underlying the varying contributions of CC, CF and FF contacts to the

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deviator stress are revealed. Ultimately, the stress-force-fabric is analyzed, and any

fabric anisotropies affected by the coarse particle content are evaluated at the peak

and critical states. Finally, the main conclusions of this study are presented.

2. DEM modeling

The well recognized DEM program PFC3D [29] was used to perform the

numerical simulation in this study. Bidisperse samples were created by mixing two

component materials—one with realistic gravel geometries and the other consisting of

finer spherical sand-sized particles. Spherical fines are used in this study because the

particle shapes of fines are relatively regular, and the sizes of fines are much smaller

compared with coarse particles. Previous numerical studies [18-20] have also

simulated binary mixtures consisting of irregularly shaped coarse particles and

spherical fine particles. The gravel-shaped coarse particles were modeled using a

clump multisphere approach. The particle-forming method described by Taghavi [30]

was a built-in function of PFC3D and was used to determine the positions and radii of

the constitutive spheres. The gravel geometries were obtained from industrial

computerized tomography (CT), which has been shown to be a reliable way to obtain

the 3D microstructure of materials to a satisfactory resolution [31, 32]. A CT scan of

gravel can be converted into a STL-file, which can then be directly imported into

PFC3D [29]. Note that the STL-file stores a triangular surface mesh used by the rapid

prototyping industry as a standard file format. The triangular surfaces were then filled

with subspheres. Figs. 1(a)-(c) show examples of a real gravel particle, a triangular

surface stored in a STL-file and a corresponding sphere-filled gravel-shaped particle,

respectively. A comparison between Fig. 1(a) and Fig. 1(c) indicates that the geometry

of the gravel-shaped particle is similar to that of real gravel. A total of 33 different

shaped gravels were scanned, and the corresponding sphere-filled gravel-shaped

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particles are illustrated in Fig. 2. The particle information are provided in each

subbox. The NS value denotes the corresponding number of subspheres to fill the

particle, which is controlled by two parameters of particle shape description as

defined by Taghavi [30], i.e., distance and ratio. The distance corresponds to an

angular measure of particle smoothness in degrees in the range of 0 to 180; the greater

the distance, the smoother the subspheres distribution. The ratio is the smallest to

largest subspheres kept in the particle with 0 < ratio < 1. Generally, the greater the

distance and the lower the ratio in our multisphere approach, the greater the number

of subspheres that are needed to generate the particle. Taking into account

computational costs, the distance and ratio of all coarse particles are set to 150 and

0.3 respectively, the same values used in our previous study [33]. Sphericity (S) and

roundness (R) are two important scales characterizing particle shape. Sphericity can

be quantified as the diameter of the largest inscribed sphere relative to the diameter of

the smallest circumscribed sphere. Roundness is quantified as the average radius of

curvature of surface features relative to the maximum sphere that can be inscribed in

the particle. Based on the sphere-fitting algorithms that can be directly utilized for

STL-files proposed by Nie et al. [34], both the S and R values of all STL-files are

calculated. In addition, previous experimental studies [35-37] have reported that

regularity (Re = (S+R)/2) is a good index to quantify the particle shape effect on the

shear strength of granular materials. The S, R and Re values of corresponding

particles are included in each subbox. The R values are close in this study. According

to Simon’s classification of particle roundness [38], all coarse particles can be

classified as subangular (when R values range from 0.13 to 0.25) except when

individual particle (i.e., No. 9). When generating the sample, the particles as shown in

Fig. 2 are randomly selected as coarse particles. All coarse particles have the same

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volume, equal to the volume of an equivalent sphere with a diameter Dc at 6.71 mm,

which is the average diameter of the scanned gravels. The diameters of fine particles,

Df, are set to 1.51 mm. Accordingly, the particle size ratio = Dc/Df = 4.44, which is

the same as that used in De Frias Lopez et al. [28, 39], who explored the resilient

properties, force transmission and soil fabric of binary granular mixtures. Bidisperse

cubic samples with coarse contents ranging from 0% to 100% (by weight) in steps of

approximately 10% were generated. For convenience, each test is identified by the

percentage of the coarse particles by weight, W. For example, W0 indicates pure fine

particles (i.e., W=0%), and W100 indicates pure coarse particles (i.e., W=100%). Note

that the same DEM modeling was also prepared for binary mixtures with spherical

coarse and fine particles with = 4.44 to evaluate the effect of particle shapes of

coarse particle son the mechanical behavior of binary mixtures.

To generate a sample, the number of coarse particles, Ncp, is first determined, and

then the number of fine particles, Nfp, can be calculated by the specific W and the

known particle density. The Ncp of each bidisperse sample is almost the same as that

used in Minh et al. [5, 6]. Greater Ncp may be preferred, but the simulation requires

more computational resources. Particles with random orientations were initially

generated within a cube, with zero contacts. The gravitational acceleration and friction

coefficients between particle–particle and particle–wall were temporarily set to zero

to avoid force gradients and obtain isotropically dense samples. A zero friction

coefficient during specimen generation can produce the densest specimens for a given

generation procedure [40, 41]. Frictionless conditions were also used in previous

studies dealing with DEM bidisperse samples (e.g., [5, 6, 12, 14, 15, 19, 28, 39]).

Particles were subjected to an isotropic compression with a low strain rate over a large

number of small time steps. A servo-controlled mechanism was introduced to achieve

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and maintain the desired confining pressure c = 200 kPa during the isotropic

compression (Itasca [29]). The sample was considered to be in equilibrium when the

ratio of the mean static unbalanced force to the mean contact force was less than 10-5,

and the difference between the stress obtained from the walls and c was smaller than

a tolerance of 0.5%. After the isotropic compression, the specimen dimensions and

initial porosity were measured, and the particle–particle friction coefficients were

set to 0.5 for shear. Table 1 lists the details of the experimental program, including

coarse particle content, number of particles, specimen dimensions and initial

porosities for the generated samples. It must be note that, in general, the obtained W

slightly deviates from the originally specified Wo (e.g., for Wo = 30%, the obtained

value was 30.04% for gravel-shaped coarse particles and 29.99% for spherical coarse

particles). This result was due to some particles escaping from the material vessel

during the generation procedure because of large contact forces. Ncp, Ncp-sub and Nfp

denote the number of coarse particles, number of subspheres constituting coarse

particles and number of fine particles, respectively. A mass of fine particles results in

high computational expenses. A workstation with an IntelXeon CPU E5-

2690v4(2) was used in this study, and the average calculation time of each DEM

numerical test was approximately 25 days. Jamilkowski et al. [42] suggested that the

ratio of the sample size to the maximum particle size should be greater than 5, with an

ideal ratio of 8, to eliminate the effect of the specimen size and minimize stress

nonuniformities inside a sample. The l0, w0 and h0 represent length, width and height

of the initial specimen before shear, respectively. The minimum ratio of l0/Dc

approaches 8.77 at W70 for gravel-shaped coarse particles and 9.17 at W70 for

spherical coarse particles, as listed in Tab. 1. Note that the dimensions of W0 are

different from the other samples because a smaller number of fines is used in view of

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the high computational expenses. Nevertheless, the minimum ratio of w0/Df of W0

approaches 20.1, which is greater than the other samples because of the smaller size

of fines. Huang et al. [43] discussed the potential effect of the sample size on the

DEM simulation results. They found that the stiffness and the peak shear strength of

the sample only slightly reduced with an increase in the sample size. Therefore,

modeling a smaller sample here would not affect the observed trends and conclusions.

Based on the initial porosity n0 in Tab. 1, Fig. 3 shows the effect of the coarse particle

content on the porosity on completion of the generation procedure. The numerical

results from Minh et al. [5, 6] using balls with 10.0, De Frias Lopez et al. [28, 39]

using balls with 4.44, and Ng et al. [23, 24] using different aspect ratios (AR for

short) of ellipsoids with 5.0 are also plotted for comparison. It can be observed in

Fig. 3 that the porosity gradually decreases with increasing W until the porosity

reaches the minimum value at W 70%, and then the trend revers with a further

increase in W. The porosity versus W thus has a ‘V’ shape, which compares well with

other numerical results [5, 6, 23, 24, 28, 39]. This implies that the particle shape and

particle size ratio have insignificant effects on the evolution of initial porosity with W

when 4.44 10.0.

The particle–particle interactions and the particle–wall interactions obey simple

linear force–displacement contact laws. The basic parameters in the simulations are

listed in Tab. 2. The normal contact stiffness of particle kn varies according to kn =

Ecr/(ra+rb), where Ec denotes the contact effective modulus, ra and rb denote the radii

of particles in contact, and r represents the smaller value of ra and rb. The Ec values for

particle–particle (=108 Pa) and particle–wall (=109 Pa) in this study are the same as

our previous study [33]. The value of kn/ks (ks represents the shear contact stiffness of

particles) suggested by Goldenberg and Goldhirsch [44] for realistic granular

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materials falls within the range of 1.0 < kn/ks < 1.5, which correlates well with the

Cattaneo–Mindlin model [45] for elastic sphere contacts; kn/ks = 4/3 was used in this

study. In the contact law, the tangential component of the contact force between two

particles, ft, is capped as ft ≤ fn, where fn is the normal contact force and = 0.5 is the

sliding friction coefficient. Note that = 0.5 was also adopted in previous DEM

simulations on probing the properties and behaviors of binary mixtures (e.g., [5, 6, 8,

18, 19, 28, 39]).

3. Macroscopic behaviors

3.1 Macroscopic variable definitions

Conventional drained triaxial compressions were simulated in this study. The

isotropic samples were subjected to a vertical compression by the downward

displacement of the top wall at a constant velocity, while a constant confining

pressure acted on the lateral walls via servo control. The effective mean (σ m) and

deviator (σ d) stresses are defined as:

σ m=(σ1+σ 2+σ3) /3 (3)

σ d=σ1−(σ2+σ 3)/2 (4)

where σ 1 denotes the axial stress; σ 2 and σ 3 (σ 2=σ3) denote the lateral stresses. The

axial strain ❑1 and volumetric strain ❑v are derived from the movements of the rigid

walls:

❑1=(h0−h)/h0 (5)

❑v=(v0−v )/v0 (6)

where h0 is the initial height of the sample; h is the height at a given deformation; v0 is

the initial volume of the sample; and v is the volume of the sample at the same given

deformation. The volumetric compression is considered to be positive in this study.

The internal angle of friction, , which represents the shear strength of the granular

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material, can be defined from the stress ratio in a drained triaxial loadings based on

the Mohr-Coulomb criterion:

sin ϕ=σ 1−σ3

σ1+σ3=

3σd /σm

σ d/σm+6 (7)

The dilatancy angle in the triaxial loadings is defined as follows:

sin ψ=−d εv /d ε1

2−d ε v /d ε1 (8)

To ensure quasistatic deformation, the shear rate should be slow enough such that the

kinetic energy supplied by shearing is negligible compared with the static pressure.

This can be formulated in terms of an inertia parameter I defined by [46]:

I= ε̇ D√σm/ ρ (9)

where ε̇ is the axial compression strain rate, and D is the ensemble average of the

particle diameters. Quasistatic shear was ensured by the condition with I≪1. The

constant velocity applied on the top wall was set to 0.5 m/s in the simulation, which

ensures that I will be below 10-5 throughout the test.

3.2 Shear strength and dilatancy

To explore the macroscopic and microscopic behaviors at the critical state, all

samples are sheared to approximately 50% of 1. At such large deformations, the

characteristic critical state conditions (i.e., a constant d/m and porosity) are

approximately satisfied. Figs. 4(a)-(b) present the stress ratio, d/m, versus the axial

strain, 1, for gravel-shaped and spherical coarse particles, respectively. Due to

initially dense isotropic packing, all samples exhibit stiff responses at the beginning of

shear, and the peak stress ratios are reached at a small axial strain. All d/m values

pass peak and then gradually decrease until eventually approaching the residual shear

strength. For gravel-shaped coarse particles, Fig. 4(a) shows that W affects both the

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peak and residual shear strengths. However, for spherical coarse particles, Fig. 4(b)

indicates that W has only a small influence on the peak and residual shear strengths.

The peak friction angle, p, and the residual friction angle, c, (average value for 1

ranges from 40% to 50%) for samples with different W are plotted in Fig. 5. For

comparison, the p values in using true ellipsoids with different aspect ratios (AR for

short) and 5.0 are also plotted. Fig. 5 indicates that the p values of spherical

binary mixtures are similar to those of ellipsoidal binary mixtures in Ng et al. [23].

This is an unexpected result since the p values of ellipsoids are generally greater than

those of spheres, reported by Gong and Liu [33]. In effect, the unexpected result is

attributed to the fact that the initial condition of samples in this study is different from

that in Ng et al. [23]. Specifically, an interparticle friction coefficient of 0.1 was used

in Ng et al. [23] to generate initial samples, which are looser than the initial samples

produced by the frictionless condition in this study. Alternatively, for gravel-shaped

coarse particles, the increase of W could not enhance the peak and residual shear

strengths when W is less than 60%, although both p and c exhibit small fluctuations.

However, for W greater than 60%, both p and c increase significantly. The

evolutions of p and c with W are roughly consistent. Similar evolutions of peak

shear strengths can also be observed in Lu et al. [20], who investigated biaxial

compressions of binary mixtures of polygonal coarse and discoidal fine particles. Fig.

5 also indicates that for gravel-shaped coarse particles, both p and c increase

significantly as W is greater than 60%. This can be mainly attributed to three potential

factors, including the particle shape effect of coarse particles (i.e., angularity and

nonconvexity), the combination of different coarse particle shapes and the change of

particle size distribution (i.e., W increases). Both p and c values of the spherical

coarse particles in this study and p values in Ng et al. [20] remain nearly unchanged

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as W increases, despite some fluctuations. This finding implies that the change of

particle size distribution has a limited effect on the shear strengths of binary mixtures.

In addition, the combination of different coarse particle shapes is also demonstrated to

have an insignificant effect on the shear strength of binary mixtures in this study,

which will be discussed in the following paragraph. Therefore, the increases in the

peak and critical shear strengths are thought to be underlied by the particle shape

effect of the coarse particles. Specifically, when W is greater than 60%, the coarse

particles in a binary mixture form a skeleton structure. A further increase in coarse

particle content will enhance the interlocking effect among the coarse particles, and

thus both the p and c values are gradually increased.

Following Scholtes et al. [47], the degree of interlocking of a particle p is

estimated through the assessment of its mean internal moment mp, defined as

mp=tr (M p). Here, M p is the internal moment tensor given by Mp=∑

α∈ pf i

α d jα [48],

where d⃗ denotes the vector connecting the center of particle p to the contact point

associated with the contact force f⃗ involved in the contact . Thus, the average degree

of interlocking of a particle in the sample, DI, can be quantified as: DI= 1N p

∑p∈ N p

mp,

where Np represents the number of particles. To quantify the interlocking effect of

samples with gravel-shaped particle, the evolution of DI with respect to W is

illustrated in Fig. 6. It can be seen that the evolutions of DI at the peak and critical

states with W are roughly consistent. Specifically, the DI value remains nearly

unchanged when W is less than 60% but increases significantly when W is greater

than 60%. The shape of the curve in Fig. 6 is quite similar to that shown by the

relationship between the friction angles and W for gravel-shaped coarse particles, as

shown in Fig. 5. This finding also confirms that when W is greater than 60%, the

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increase of p and c values are mainly caused by the interaction among the coarse

particles. The peak and residual shear strengths of granular materials are mainly

influenced by the contact friction and particle shape. As W increases when W ≤ 60%,

the binary mixture gradually changes from a fine-dominated material to a coarse-fine

interaction material, but the mixture has not formed a skeleton structure of coarse

particles yet. Namely, the shear strength of a binary mixture is mainly controlled by

the contact friction of FF contacts or CF contacts but is less affected by the

interlocking effect between the coarse particles. The contact friction coefficients of

FF, CF and CC contacts are identical in this study (i.e., = 0.5). This could be the

reason that the peak and residual shear strengths are almost invariant when W is less

than 60%. This observation suggests that W = 60%-70% is a watershed where the

binary mixture transitions from a coarse-fine-interaction material to a coarse-

dominated material. The watershed is consistent with the critical coarse particle

content when binary mixtures reach their minimum porosity (i.e., W 70%, as

observed in Fig. 3). In addition, the corresponding watershed of the fines content (i.e.,

30%-40%) is usually regarded as a critical turning point of liquefaction resistance of a

silt-sand-mixture, termed as threshold fines content by Thevanayagam et al. [49],

transitional fines content by Yang et al. [50] or limiting fines content by Polito and

Martin [51].

Although the coarse particles are selected randomly, the current results cannot

separate the effects between particle shape and coarse fraction. The results in Fig. 5

may be different with different combinations of coarse particles because of the

variation in particle shape. To evaluate the potential effects of coarse particle shapes,

extra triaxial tests were conducted with W80. Previous experimental studies (e.g., [35-

37]) found that there is a strong correlation between the Re (or R) value and the shear

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strength of granular materials. Using a single type of coarse particle can avoid the

combination effect of multiple particle shapes. Therefore, a single coarse particle type

with minimum (i.e., No. 9), mean (i.e., No. 23) or maximum (i.e., No. 17) Re value as

shown in Fig. 2 was used to generate the W80 samples and conduct the triaxial tests.

In addition, another three different W80 samples with randomly selected coarse

particles were modeled to conduct the triaxial tests. Fig. 7 presents the information

(including Re of single type of coarse particle and n0 of each sample) and stress-strain

relationships of these extra tests. For the results of a single type of coarse particle,

both the n0 and the shear strength slightly increase as the Re value decreases. This is

consistent with the observations in previous experimental studies (e.g., [35-37]). For

the results of four different random samplings and a single type of coarse particle (No.

23), the n0 and stress-strain relationships are generally coincident, indicating that

different combinations of coarse particles have an insignificant effect on the initial

porosity and shear strength of W80 in this study. The possible reason could be the fact

that the Re values of particles are close in this study, except for an individual particle

type (i.e., No. 9) and a random mixture of various shapes averages the effect of Re.

Alternatively, previous experimental studies have reported that the c value of sands

linearly decreases with an increase in R, e.g., c = 42-17R as reported in Cho et al.

[35], c = 41.20-21.21R as reported in Yang and Luo [36] and c = 25.02(1-R)+20 as

reported in Suh et al. [37]. In this study, the relationship between c and R for pure

fines (W0) and pure coarse particles (W100) can be fitted by c = 38.01-19.72R,

which is closer to the correlation developed by c = 41.20-21.21R as reported by Yang

and Luo [36]. Future work may include studying the effects of particle shapes by

running simulations of coarse particles that gradually vary from well-rounded to

angular and establishing the relationship between the macroscopic and microscopic

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mechanical parameters and the shape parameters of coarse particles.

The evolution of volumetric strain, v, with axial strain, 1, is illustrated in Fig.

8(a). First, all samples undergo a slight contraction before expanding gradually to

nearly plateau values, corresponding to a state of isochoric deformation (i.e., critical

state). Volume expansion can be expressed in terms of dilatancy angle, . Fig. 8(b)

illustrates the evolution of as a function of 1. In all cases, starts from a negative

value, consistent with the variation in v, before increasing gradually to a positive

peak value p, finally decreasing and approaching zero (i.e., critical state). The

relationship between p and W is inserted in Fig. 8(b), together with the results of

spherical coarse particles for comparison. A widely used empirical stress-dilatancy

relationship focused on the strength parameters of uniform sands developed by Bolton

[52] yields:

ϕ p=ϕc+a ψ p (10)

where the dilatancy coefficient, a, implies the contribution of dilatancy to the peak-

state strength, and a varies based on soil type. Bolton [52] suggested that a = 0.48 in

the triaxial compression condition for clean sands. Fig. 9 illustrates the evolution of a

¿) with W, aiming at exploring the stress-dilatancy relationship of binary mixtures. It

is observed that for gravel-shaped and spherical coarse particle, the variations of a

with respect to W are similar. When W < 60%-70%, a is observed to slowly increase

with increasing W, implying that the contribution of the dilatancy to the peak shear

strength slowly increases as W is increased. However, a gradually decreases with

increasing W when W ≥ 60%-70%, which suggests that the contribution of the

dilatancy to the peak shear strength rapidly decreases as W is increased.

4. Mesoscopic behaviors

4.1 Role of contact type and classification results

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The emphasis of the present study is placed on quantifying the effects of

deformation and coarse particle content on the contribution of each contact type (i.e.,

CC, CF and FF) to the deviator stress, which cannot be observed in physical

experiments. The average stress tensor in the volume v of the granular assembly is

given by the following expression [53]:

σ ij=1v∑c=1

Nc

f ic d j

c (11)

where Nc indicates the contact number in a granular assembly; c represents a specific

contact; f cdenotes the corresponding contact force; and dc denotes the corresponding

branch vector joining the centers of the two particles in contact. The contribution of

CC, CF and FF contacts to the deviator stress can be quantified as follows [5, 14, 28,

39]:

CCC=σ d

CC

σ d×100 %=

(σ1CC−(σ 2

CC+σ 3CC)/2)

σd×100 % (12)

CCF=σd

cF

σ d×100 %=

(σ1CF−(σ2

CF+σ3CF )/2)

σd× 100 % (13)

CFF=σd

FF

σ d×100 %=

(σ 1FF−(σ2

FF +σ3FF)/2 )

σd×100 % (14)

where σ kCC, σ k

CF and σ kFF are obtained based on Eq. (11) by restricting the summation to

CC, CF and FF contacts, respectively, with k = 1, 2 and 3, representing the axial and

two lateral directions, respectively. Figs. 10(a)-(c) display the evolution of

contribution of each contact type to the deviator stresses, CCC, CCF and CFF, with axial

strain, 1. The black point and dotted box in the figures represent the peak state and

critical state, respectively. The CCC, CCF and CFF values at the peak and critical states,

as well as the axial strain corresponding to the peak (p) and critical (c) states, are

also listed in Tab. 3. Obviously, Figs. 10(a)-(c) and Tab. 3 show that W affects the CCC,

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CCF and CFF values at the peak and critical states. The relationships between the CCC,

CCF and CFF values at the peak and critical states and W are illustrated in Fig. 11.

Clearly, it is observed that CCC continually increases and CFF continually decreases

with increasing W, which is evident for both the peak state and the critical state. The

CCF value first increases with increasing W, reaches a maximum when W = 60% and

then decreases with a further increase in W. Comparing CCC, CCF and CFF between the

peak state and the critical state for a specified W, it is found that CCF at the critical

state is always greater, whereas CFF at the critical state is always lower. Moreover,

when W ≤ 50%, CCC at the critical state is nearly the same as that at the peak state,

whereas when W > 50%, CCC at the critical state is relatively lower. In addition, Fig. 11

indicates that the magnitudes of CCC, CCF and CFF are dependent on both coarse

particle content and axial strain. For W ≤ 40% at the peak state and W ≤ 30% at the

critical state, it is clear that the FF contacts play a primary role in providing the

deviator stress and the CF contacts provide a secondary effect, whereas the

contribution of the CC contacts is insignificant. For 40% < W ≤ 55% at the peak state

and 30% < W ≤ 48% at the critical state, the CF contacts start to dominate the deviator

stress and the FF contacts provide a secondary contribution, while the contribution of

the CC contacts is still the smallest. For 55% < W ≤ 65% at the peak state and 48% <

W ≤ 70% at the critical state, the CF contacts still dominate the deviator stress,

whereas the CC contacts begin to play a more important role than the FF contacts. For

W > 65% at the peak state and W > 70% at the critical state, the contribution of CC

contacts is the largest, while the CF contacts provide a secondary contribution and the

contribution of the FF contacts is insignificant.

Following De Frias Lopez et al. [28], Tab. 4 establishes the criteria to determine

the limits of the mechanical transitional behaviors of binary mixtures based on the

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relative contribution of each contact type to the deviator stress. For comparison, Tab.

4 also includes the limits of numerical spherical binary mixtures with = 4.44 after

De Frias Lopez et al. [28], experimental results for sandy gravel after Kuenza et al.

[55], experimental results on glass beads mixtures and Ottawa sand-clay mixtures

from Vallejo [1, 56] and limits of soil-rock mixtures from Xu et al. [57]. The fabric

structure of the binary mixtures can be classified into four groups as used by dam

engineers [58]: an overfilled structure (G1); an interactive-overfilled structure (G2);

an interactive-underfilled structure (G3); and an underfilled structure (G4). Referring

to the fabric structures in detail following De Frias Lopez et al. [28], an overfilled

structure indicates that most coarse particles float in a matrix of fines. An interactive-

overfilled structure indicates that the coarse particles begin to contact and interact

with each other. An interactive-underfilled structure indicates that coarse particles

start to form the main load-bearing skeleton, whereas the fines optimally fill their

voids. An underfilled structure indicates that most of the fines fill the voids between

the coarse particles with little contribution to the load-bearing skeleton. As shown in

Tab. 4, in this study, the ranges of W ≤ 40%, 40% < W ≤ 55%, 55% < W ≤ 65% and W

> 65% can be used to classify groups G1-G4 at the peak state, respectively. In the

same way, the ranges of W ≤ 30%, 30% < W ≤ 48%, 48% < W ≤ 70% and W > 70%

can be used to classify groups G1-G4, respectively, at the critical state. This finding

indicates that the classification results of granular mixtures are dependent on the

development of strain. Gong and Liu [14] conclude that the boundaries between G1

and G2 (named Wf) and between G3 and G4 (named Wc) are two thresholds of

granular mixtures. In this study, the two thresholds signify that the shear strengtsh of

the granular mixtures are dominated by the fine particles or the coarse particles at W ≤

Wf or W ≥ Wc, respectively. It should be note that the two thresholds are important to

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directly assess the engineering properties of granular mixtures, which can result in

improved designs, small-scale tests and knowledge-based design decisions. The

present simulations indicate that Wf = 30%-40% and Wc = 65%-70%, depending on

the strain level. Some differences in the two thresholds are observed compared with

the results of De Frias Lopez et al. [28]; for example, Wf is 30%-40% in this study,

while 45% in De Frias Lopez et al. [28], and Wc is 65%-70% in this study, while 75%

in De Frias Lopez et al. [28]. Both Wf and Wc in this study are relatively lower than

those in De Frias Lopez et al. [28], which can be attributed to two causes. One is that

De Frias Lopez et al. [28] determined CCC, CCF and CFF before the peak state (i.e., d =

100 kPa). The other cause is related to the effect of coarse particle shapes, which will

be discussed in the next section. It is remarkable that Wf = 30%-40% and Wc = 65%-

70% in this study agree well with many experimental results. For example, Kuenza et

al. [55] found that the peak shear strengths of gravelly soils were controlled by the

sand matrix when W ≤ 40%, and Vallejo [1, 56] arrived at same conclusion in testing

glass bead mixtures and Ottawa sand-clay mixtures. In addition, Xu et al. [57]

concluded that the residual shear strength of soil-rock mixtures were dominated by

fines at W ≤ 30% and coarse particles at W ≥ 70%. Nevertheless, diverse views also

exist, such as Wc = 75%-80% for glass bead mixtures and Ottawa sand-clay mixtures

in experimental studies [1, 56] and spherical and discoidal coarse particles in

numerical studies [14, 28]. This indicates that rounded coarse particles lead to a

greater Wc (i.e., Wc = 75%-80%) when compared with the results in this study (i.e., Wc

= 65%-70%). Namely, a binary mixture with spherical coarse particles will form a

skeleton of coarse particles at W = 75%-80%, which is greater than W = 60%-70% for

binary mixtures with gravel-shaped coarse particles. The explanation for this

phenomenon will be given in the following.

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4.2 Effect of coarse particle shape

A matrix model concerning the thresholds of sandy gravel was described by

Fragaszey et al. [59]. Specifically, Fragaszey et al. [59] concluded that the mechanical

behavior of sandy gravel was related to the state of the coarse–coarse contacts. In fact,

the state of coarse–coarse contacts can be quantified by the partial coordination

number of coarse particles, Zc, which is defined as twice the total number of contacts

between coarse particles, NCcc, (not between subspheres) divided by the total number

of coarse particles Ncp, i.e., Zc =2NCcc/Ncp. Fig. 12 displays Zc with respect to the

coarse particle content, W. The binary mixtures data with spherical coarse particles in

other studies are also included for comparison, i.e., steel balls with = 2 and = 4 in

Pinson et al. [60], numerical ball mixtures with = 2 and = 4 in Biazzo et al. [61],

numerical ball mixtures with = 2 and = 5 in Meng et al. [62], and numerical ball

mixtures with = 5 in Rodriguez et al. [63]. As shown in Fig. 12, all Zc values

gradually increase with the increase in W, as expected. Furthermore, Zc values in this

study are nearly the same as those in other studies when W ≤ 40%. This is attributed to

the fact that the coarse particles are almost floating in the matrix of fine particles at

this moment; thus, the particle shapes of coarse particles have an insignificant effect

on the Zc value. However, the Zc values of gravel-shaped coarse particles is clearly

greater than those of spherical coarse particles in this study or previous physical or

numerical tests with spheres when W > 40%. Because of these larger Zc values, it is

easy to understand that the skeleton structure of coarse particles will be formed at a

lower W for gravel-shaped coarse particles. Therefore, the lower Wc for gravel-shaped

coarse particles is primarily due to the particle shape effect of coarse particles.

5 Microscopic analyses

5.1 Normal contact forces and proportion of each contact type

Tab. 4 shows that the classification results of binary mixtures at the peak state are

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different from those at the critical state. This finding is related to the fact that the

magnitudes of CCC, CCF and CFF are changed as the strain develops from the peak state

to the critical state, as shown in Fig. 11. In this section, the underlying microscopic

mechanism for the variations of CCC, CCF and CFF from the peak state to the critical

state for gravel-shaped coarse particle will be investigated. The CCC, CCF and CFF

values are affected by the contact forces and proportions of the CC, CF and FF

contacts, respectively. Therefore, the normal contact forces and proportions of each

contact type at the peak and critical states are examined.

The mean normal contact force f n is defined as the average of normal contact

forces over all contacts. Tab. 5 lists the f n of each contact type of various samples at

the peak and critical states. There is a significant variation in f n for various samples,

indicating that f n is coarse particle content-dependent [5, 14]. Among the contact

forces of each contact type for a specific sample, the CC contact is the largest on

average, CF contact comes second, and FF contact is the smallest, which is consistent

with the previous numerical observations (e.g., [5, 6, 14, 54]). Under external loads,

the coarse particles are relatively stable because of the greater contact numbers per

particle when compared with fine particles. Therefore, the CC contacts are prone to

behave like a backbone being able to transmit forces stably. This could be the reason

why the mean contact force of CC contacts is the largest, followed by CF and FF

contacts. The values of f n of each contact type at the peak and critical states are

significantly different. Specifically for the CC and CF contacts, f n at the critical state

is larger than that at the peak state, while f n for the FF contacts at the critical state is

smaller than that at the peak state.

It is well known that force transmission within sheared granular materials occurs

via coexisting strong and weak contacts, which form the corresponding strong and

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weak contact force networks [64]. The strong contacts carry forces larger than the

average contact force, whereas the weak contacts carry forces smaller than the

average contact force. In fact, the strong and weak contacts play different roles in the

sheared granular materials. Strong contacts form a solid-like backbone that can

transmit forces, whereas weak contacts behave like an interstitial liquid providing

stability against forces propagating through strong contacts [64]. The proportions of

the strong and weak contacts of each contact type, PS mn and PW

mn (mn denotes CC, CF or

FF contact), are defined as the number of the strong and weak contacts of each contact

type over all contacts, respectively. The superscripts S and W represent strong and

weak contacts, respectively. Figs. 13(a)-(b) show the PS mn and PW

mn values, respectively.

Clearly, the evolutions of PS mn and PW

mn values with W are similar. Specifically, PS CC and P

W CC continually increase with W, while PS

FF and PW FF continually decrease with W. In

addition, the PS CF and PW

CF values first increase with an increase of W, reach a maximum

when W = 70%-80% and then decrease with a further increase in W. These trends are

related to the fact that as W increass, the number of CC and FF contacts will

continually increase and continually decrease, respectively; the number of CF contacts

will start from zero (i.e., W0), reaching a peak at the intermediate W, and then

gradually decrease to zero (i.e., W100). Alternatively, for strong contacts as shown in

Fig. 13(a), the PS FF and PS

CF values at the critical state are smaller and larger than that at

the peak state, respectively. Previous studies [65] have reported that the strong

contacts are preferentially aligned with 1 and bear most of the deviator stress.

Therefore, when strain develops from the peak state to the critical state, the decreased

strong FF contacts lead to a decrease of CFF, while the increased strong CF contacts

result in an increase of CCF, as shown in Fig. 11. Alternatively, Fig. 13(a) indicates that

the PS CC values at the peak and critical states are generally consistent. However, Fig.

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13(b) makes clear that when W > 60%, the PW CC values at the critical state are larger

than those at the peak state. Previous studies [65] have indicated that the weak

contacts tend to align themselves in the 3 direction and are negatively correlated with

the deviator stress. Therefore, when compared to the peak state, the increased weak

CC contacts at the critical state lead to a decreased CCC at the critical state, as

observed in Fig. 11.

5.2 Sliding contacts percentage

In a DEM simulation, the sliding of a contact is governed by the Coulomb

friction law. In this study, a contact sliding is assumed to occur when |ft|/(fn) >

0.9999. For a specific binary mixture, the sliding contact percentage (i.e., SCP) is

obtained as follows:

SCP=NC s

N c×100 % (15)

where NCs denotes the number of sliding contacts in the sample. Fig. 14 illustrates the

variation of SCP with axial strain 1. It is observed that SCP increases gradually to a

peak value, then decrease continuously to a steady state, as also observed by Gong

and Liu [33] and Gu et al. [66]. The SCP value at the steady state slowly increases

from 15% to 28% as W increases from 0% to 60%, and then rapidly increases from

28% to 52% as W increases from 60% to 100%. It is clear that the SCP of W100 is

larger than that of W0. This is attributed to the fact that the rotation resistance

between the nonspherical particles pushes more contacts to slide to accommodate the

imposed deformations [67]. It has been previously concluded that the skeleton

structure of coarse particles is formed when W = 60%-70%. A further increase in W

will enhance the strong interlocking effect between coarse particles as show in Fig. 6,

and thus result in a rapid increase of SCP when W ≥ 60% as observed in Fig. 14.

6 Anisotropic analysis

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The macroscopic characteristics and contact-scale characteristics described

above provide a basic picture of the effect of particle shapes and content of coarse

particles on shear behaviors of binary mixtures. To relate the microscopic phenomena

to the macroscale behavior, anisotropy is analyzed in this section. Anisotropy is one of

the most important characteristics of granular materials and can be categorized as

geometrical anisotropy and mechanical anisotropy [33, 64, 66, 69]. Geometrical

anisotropy is defined by the local orientations of contact planes, which produce a

global anisotropic phenomenon. Mechanical anisotropy is caused by external forces,

and depends on contact forces induced between particles with respect to the local

orientations of the contact planes. Both geometrical and mechanical anisotropies

affect each other, contributing to the stress tensor [33, 64, 66, 69].

6.1 Quantification of anisotropy

Satake [70] proposed a quantitative measure of fabric anisotropy using the fabric

tensor as follows:

ϕij=∫Ω

❑

E(Ω)ni n jd Ω= 1N c

∑1

N c

n in j (16)

where n is the unit contact vector; i, j = 1, 2, 3 represent the axial and two lateral

directions, respectively; note that unless indicated otherwise an Einstein summation

convention is adopted for repeated subscripts; Ω denotes the orientation of n relative

to the global coordination system; E(Ω) indicates the probability density function of

contact normal at a unit spherical surface, which can be expressed as:

E(Ω)= 14 π

(1+aijc n in j) (17)

where a ijc is the second order anisotropy tensor, and used to characterize the fabric

anisotropy. Substituting Eq. (17) into Eq. (16) and integrating it, Eq. (18) can be

obtained.

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a ijc=15

2ϕij

' (18)

where ϕ ij' is the deviator part of the fabric tensor, ϕij. In a similar fashion, the

anisotropy of normal and tangential contact forces can be derived with respect to the

corresponding fabric tensor. Specifically, the average normal and tangential contact

force tensors, as described in Guo and Zhao [69], can be expressed as:

F ijn= 1

4 π∫Ω❑

f n(Ω)ni n j d Ω=∑1

Nc f n nin j

N c(1+aklc nk nl)

(19)

f n(Ω)=f 0(1+aijn ni n j) (19a)

F ijt = 1

4 π∫Ω❑

f t(Ω) ti n j d Ω=∑1

N c f t ti n j

N c(1+aklc nk nl)

(20)

f t(Ω)=f 0 ¿ (20a)

Eqs. (19a) and (20a) indicate the probability distributions of F ijn and F ij

t , respectively.

a ijn and a ij

t are the second-order anisotropy tensors and used to characterize the normal

and tangential contact force anisotropy, respectively. a ijn and a ij

t are given as:

a ijn=

15 Fijn'

2 f 0aij

t =15 F ij

t '

3 f 0

(21)

where F ijn' and F ij

t ' are the deviatoric parts of F ijn and F ij

t , respectively; f 0=F iin. Similar

to the derivation of anisotropy of normal and tangential contact forces, the anisotropy

of normal and tangential parts of branch vectors, a ijbn and a ij

bt, are derived with respect

to the fabric tensors Bijn and Bij

t as:

a ijbn=

15 Bijn '

2 b0aij

bt=15 Bij

t '

3 b0

(22)

where b0=Biin. Here, the normal and tangential parts of branch vectors indicate that

branch vectors project to the normal and tangential directions of corresponding

contacts, respectively. Because the fabric tensor is deviatoric in nature, it is

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convenient to use a scalar a¿, obtained from the invariant of each anisotropy tensor a ij¿

(i.e., a ijc, a ij

n, a ijt , a ij

bn and a ijbt), which can reflect the degree of fabric anisotropy as

follows:

a¿=Sign(aij¿ σ ij

' )√ 32

aij¿ a ij

¿ (23)

A similar definition of a¿ can be found in the literature [69]. Guo and Zhao derived a

relationship between the stress ratio and various anisotropies, namely, the stress-force-

fabric relationship as follows [69]:

σd

σm=0.4 ¿¿) (24)

where the cross products of the two anisotropic tensors are neglected in the right-hand

term. Furthermore, it should be noted that Eq. (24) is obtained under the assumption

that the contact forces and the branch vectors are uncorrelated in the granular system.

However, in binary mixtures, the CC contacts often capture the largest contact force,

the contact force of the CF contacts come second, and the contact force of the FF

contacts is the smallest. In other words, the magnitudes of contact forces are

dependent on the length of branch vectors. The uncorrelated assumption between the

contact forces and the branch vectors may not be fulfilled for binary mixtures.

Therefore, the effectiveness of Eq. (24) needs verification.

To validate the stress-force-fabric relationship given in Eq. (24), the stress ratios

σd

σm

obtained from DEM data for various coarse particle contents based on Eq. (11) is

compared with those derived from the parameters of anisotropy, as shown in Figs.

15(a)-(b). To be concise, Fig. 15(a) illustrates the contribution of each item of

anisotropy to the stress ratio σd

σm for a representative case with W50, while Fig. 15(b)

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displays the variations in σd

σm determined by Eq. (24) at the peak and critical states,

with varying coarse particle content. From Fig. 15(a), it can be seen that Eq. (24)

provides an excellent fit at 1 ≥ 30%, whereas Eq. (24) underestimates the stress ratio

at 1 < 30%. Similar observations can also be found in Fig. 15(b), where Eq. (24)

provides an excellent fit at the critical state for various samples, whereas Eq. (24)

underestimates the shear strength at the peak state of various samples. This

underestimation could be attributed to the fact that Eq. (24) neglects the cross-

products of the two anisotropic tensors and assumes uncorrelated contact forces and

branch vectors in granular systems. Although Eq. (24) underestimates the peak stress

ratio, Fig. 15(b) demonstrates that the evolutions of σd

σm at the peak and critical states

obtained by Eq. (24) are generally consistent with those obtained from DEM data.

That is, the parameters a* for the degree of anisotropy defined in Eq. (23) are

generally reasonable. It is, thus of interest to understand what role these quantities

play in the stress ratio (or shear strength). For example, as indicated in Fig. 15(a), an

underpins the shear strength, ac and at respectively make a secondary and tertiary

contribution to the shear strength, and the contributions of abn and abt are trivial.

6.2 Anisotropies and origins of shear strength

The variations of geometrical anisotropies (i.e., ac, abn and abt) and mechanical

anisotropies (i.e., an and at) with respect to coarse particle content are illustrated in

Figs. 16(a)-(b). It is observed that abn and abt are slight positives at the peak state and

the critical state when W < 50%-60%. However, the abn and abt values at the critical

state become negative when W ≥ 50%-60%. Based on Eq. (23), negative abn and abt

values are attributed to that the fact the branch vectors between the coarse particles

are nearly perpendicular to the axial stress direction because of particle rotations

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occurring during shear [33]. In general, the geometrical anisotropies, abn and abt, at the

peak and critical states are relatively small. The evolutions of ac at the peak and

critical states exhibit a similar trend. That is, the ac values are nearly constant with

small fluctuations when W < 60%. However, ac rapidly increases with an increase of

W when W ≥ 60%, which could be related to the fact that a skeleton of coarse particles

has formed at this moment. The larger ac value generally indicates that the probability

density of contact normally becomes greater along the axial direction and lower along

the two lateral directions. As clearly shown in Fig. 16(a), the ac of W100 is much

larger than that of W0 for both the peak and critical states. This is in accordance with

the notion that when compared with the packing of spherical particles, the packing of

larger sized of nonspherical particles needs fewer contacts along the lateral directions

to maintain stability and more contacts along the axial direction to transmit strong

contact forces. For the mechanical anisotropies shown in Fig. 16(b), when W < 60%,

an slowly decreases while at slowly increases with increasing W at the peak state. In

addition, when W < 60%, an and at at the critical state are considered to be a constant

despite slight fluctuations. However, when W ≥ 60%, an and at at both the peak and

critical states decrease and increase, respectively, with increasing W. Azema et al. [27]

concluded that at has a positive correlation with the mean friction mobilization of

granular systems, under the assumption that friction coefficient and normal contact

force fn are weakly correlated. As a result, when W ≥ 60%, the increase of at with W

implies that at a higher coarse particle content, the force balance is secured by a

strong activation of frictional forces. This effect can also be evidenced by the

observation in Fig. 14 that the sliding contact percentage at the steady state rapidly

increases with respect to W when W ≥ 60%. The smaller an value generally represents

that the average normal contact force becomes decreases along the axial direction and

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increases along the lateral direction. As previously stated, an increase of ac indicates a

lower probability density of contact normal along the two lateral directions. It is

remarkable that the lateral confining pressure is maintained as a constant value during

shear (c = 200 kPa). Thus, along the two lateral directions, a lower probability

density of contact normal is accompanied by a larger average normal contact force to

maintain the steady lateral confining pressure. This is the reason why the ac value

increases, whereas the an value decreases with respect to W when W ≥ 60%.

The detailed analyses of the corresponding geometrical and mechanical

anisotropies allow us to highlight the microscopic mechanisms that underlie the

dependence of shear strength with respect to W. Based on the analysis in Figs. 16(a)-

(b), the peak and residual shear strengths are a joint effect of ac, an and at. The roughly

constant peak shear strength when W < 60% is attributed to the fact that the slow

decrease of an is compensated by the slow increase of at, while ac is nearly unchanged.

In addition, when W < 60%, the nearly constant residual shear strength is caused by

the almost constant anisotropies ac, an and at. However, when W ≥ 60%, a skeleton

structure of coarse particles is gradually formed. The mechanical behaviors of the

binary mixtures start to be controlled by the coarse–coarse contacts. The great

interlocking effect between coarse particles results in an increase of ac and at with

respect to W. Nevertheless, the requirement of steady lateral confining pressures of

the sheared samples result in a decrease of an with respect to W. Therefore, when W ≥

60%, the increases of the peak and residual shear strengths are primarily related to a

gradual increase in ac and at to compensate for the continuous decrease in an. In

general, abn and abt offer small contributions to the peak and residual shear strengths.

In DEM simulations, a contact frame or a branch vector frame joining the centers

of two touching particles (see Fig. 17) are two common approaches used to describe

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the contact plane. Note that these two frames coincide for spherical particles. Both

frames are helpful for understanding the microscopic origins of the mechanical

behavior of granular materials [27]. The above anisotropic analyses are based on the

contact frame (i.e., n-t frame as shown in Fig. 17). As discussed by several previous

researchers ([17, 71]), in spherical particle systems, the geometrical anisotropy within

the strong and nonsliding contacts acsn based on the contact frame provide a linear

relationship with the stress ratio d/m (i.e., ❑d

❑m=k ac

sn). Note that the value of acsn is

obtained based on Eq. (23) by restricting to the strong and nonsliding contacts. Figs.

18(a)-(b) present the relationships between d/m and the geometrical anisotropy acsn

for various samples based on the contact frame and branch vector frame, respectively.

The dotted lines in figures denote the linear fitting for these relationships. In both

frames, it can be seen that the fitting lines pass through the origin when W < 60% but

slightly deviate from the origin when W ≥ 60%. Alternatively, based on the branch

vector frame as shown in Fig. 18(b), the d/m values linear increase with increasing

acsn for various samples. However, based on the contact frame as shown in Fig. 18(a),

the good linear relationships between d/m and acsn are applicable for W < 60% only.

The relationships between d/m and acsn are not strictly linear when W ≥ 60%. These

findings can also be reflected by the R2 values of fitting lines, as shown in Fig. 19.

Clearly, the R2 values remain high when W < 60% and sequentially maintain high

when W ≥ 60% for the branch vector frame but change to a relatively low value when

W ≥ 60% for the contact frame. As previously stated, a skeleton structure of coarse

particle has been formed when W = 60%-70%. Therefore, when W < 60% for both

frames, the good linear relationships between d/m and acsn are related to the fact that

the spherical fines dominant the mechanical behaviors of binary mixtures. In addition,

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when comparing with the contact frame as W ≥ 60%, the R2 values of fitting lines for

the branch vector frame are obviously greater. That is, when W ≥ 60%, the branch

vector frame provides a better linear relationship between d/m and acsn than the

contact frame for the coarse-dominated materials in this study. This result may be

related to fact that the geometrical anisotropy expressed in the branch vector frame for

nonspherical particles are more accurate than in the contact frame, as reported by

Azema et al [27]. Fig. 19 also illustrates the evolutions of slope of fitting lines with W.

Clearly, the slope of fitting lines exhibits unimodal characteristics with W in both

frames. Namely, the slope first increases, reaching a peak value at W 70%, and then

decreases with an increase in W. Note that the similar unimodal characteristics (peak

W = 75%) are also reported in Zhou et al. [17] through a series of numerically

undrained triaxial tests on spherical binary mixtures.

7. Conclusion remarks

The shear behaviors of binary mixtures were studied numerically under triaxial

loading. The binary mixtures consist of gravel-shaped coarse particles and spherical

fine particles, with particle size ratio = 4.44. For comparison, spherical binary

mixtures with = 4.44 were also conducted. Dense samples were created by isotropic

compression method. After the confining pressure applied on the walls was stable, all

samples were then sheared in a quasistatic way. Macroscopic, mesoscopic and

microscopic characteristics, anisotropic properties were obtained from the present

simulations.

In terms of macroscopic shear characteristics, the peak and residual shear

strengths and the stress-dilatancy relationship affected by W were investigated. The

increase of coarse particle content, W, could not enhance the peak and residual shear

strengths when W < 60%. However, for W ≥ 60%, both the peak and residual shear

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strengths increased significantly, which caused by the interlocking effect between the

coarse particles. This finding indicated that a skeleton structure of coarse particle has

been formed when W = 60%-70%. The stress-dilatancy relationship indicated that the

contribution of dilatancy to the peak shear strength was slowly increased when W <

60%-70%, then rapidly decreased when W ≥ 60%-70%.

In terms of mesoscopic shear characteristics, the effects of deformation and

coarse particle content on the contributions of the CC, CF and FF contacts to the

deviator stress were quantified. For a given binary mixture, the CFF, CCF and CCC

values are varied when the axial strain is increased. Results of CCC, CCF and CFF at the

peak and critical states were used to classify the binary mixtures. Specifically, the

granular mixtures were fine-dominated or coarse-dominated materials when the

coarse particle content was less than 30%-40% or greater than 65%-70%, respectively.

Comparison with the results of spherical binary mixtures indicated that the spherical

binary mixtures will become coarse-dominated materials at relatively larger coarse

particle content (i.e., 75%-80%) than that of gravel-shaped coarse particle (i.e., 65%-

70%), which was attributed to the particle shape effect of the coarse particles.

In terms of microscopic shear characteristics, the proportion of strong and weak

contacts of each contact type were investigated to explore why the CCC, CCF and CFF

values were varied. From the peak state to the critical state, the decreased proportion

of FF contacts was mainly strong contacts, whereas the increased proportion of CF

contacts was mainly strong contacts. Therefore, at the critical state, the CFF and CCF

values are smaller and larger than that at the peak state, respectively. The increase

proportion of CC contacts at the critical state when W > 60% is mainly weak contacts.

Therefore, CCC at the critical state is lower than that at the peak state for W ≥ 50%.

In terms of anisotropic properties, the variations of geometrical and mechanical

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anisotropies with respect to W were studied. This enabled us to understand the

microscopic mechanism that underlies the dependency of the peak and residual shear

strengths on the coarse particle content. The roughly constant peak shear strength

when W < 60% is attributed to the fact that the slow decrease of an is compensated by

the slow increase of at, while ac is nearly unchanged. In addition, when W < 60%, the

nearly constant residual shear strength is caused by the almost constant anisotropies

ac, an and at. When W ≥ 60%, the increases of peak and residual shear strengths were

primarily related to gradual increases in ac and at to compensate the continuous

decrease in an. Furthermore, it was interesting to note that the branch vector frame

provides a better linear relationship between d/m and acsn than the contact frame for

the gravel-shaped coarse particles in this study.

8. Acknowledgments

This research was supported by the National Natural Science Foundation of China

(51890915, 51479027 and 51809292). This support is gratefully acknowledged.

9. Nomenclature

Symbolsa dilatancy coefficienta ij

¿anisotropy tensor a ij

c , a ijn, a ij

t , a ijbn or a ij

bt

abn, abt anisotropies of normal branch vector and tangential branch vector, respectivelyac anisotropies of the contact normalan, at anisotropies of normal contact force and tangential contact force, respectivelya* anisotropy ac, an, at, abn or abt

b0 average branch vector calculated for the entire range of

Bijn,Bij

n ' normal branch vector tensor and its deviatoric part

Bijt ,Bij

t ' tangential branch vector tensor and its deviatoric part

CCC contribution of coarse–coarse (CC for short) contacts to deviator stressCCF contribution of coarse–fine (CF for short) contacts to deviator stressCFF contribution of fine–fine (FF for short) contacts to deviator stressD the ensemble average of the particle diameterdc branch vector joining the centers of the two particles in contactDc diameter of coarse particlesDf diameter of fine particlesDI the average degree of interlocking of a particle in the sample

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871

Ec contact effective modulusE() probability density function of contact normal at a unit spherical surfacefc contact force at a given contactf 0 average normal contact force calculated for the entire range of fn normal contact force at a specific contactft tangential contact force at a specific contactF ij

n,F ijn' normal contact force tensor and its deviatoric part

F ijt ,F ij

t ' tangential contact force tensor and its deviatoric part

f n,f t probability distributions of F ijn and F ij

t , respectivelyh height of the sample at a given deformation stateh0 initial height of the sample after isotropic compressionkn, ks normal and shear contact stiffness of the particlesl0 initial length of the sample after isotropic compressionn, ni, nj unit contact normal vector and its component in the i, j directionn0 initial porosity of sample after compressionNc number of contacts in particle systemNcp number of coarse particles in particle systemNcp_sub number of subspheres constituent coarse particlesNfp number of fine particles in particle systemNCCC number of contacts between coarse particlesNCCF number of contacts between coarse particle-fine particleNCFF number of contacts between fine particle-fine particleNCs number of sliding contacts in the sampler the smaller value of ra and rb

ra, rb the radius of particles in contactSCP sliding contact percentage in the particle systemti, tj component in the i, j direction for unit contact tangential vectorv volume of the sample at a given deformation statev0 initial volume of the sample after isotropic compressionW actual coarse particle content in the samplew0 initial width of the sample after isotropic compressionWf, Wc thresholds of granular mixturesWo objective coarse particle content for binary mixtureZc partial coordination number of coarse particle

Greek symbols particle size ratio of binary mixture, i.e., Dc/Df

1, v axial strain and volumetric strain, respectivelyε̇ axial strain and volumetric strain, respectively interparticle friction coefficient internal angle of frictionϕij, ϕij

' fabric tensor and its deviatoric part

p, c peak and residual friction angle, respectively the orientation of the contact normal in spherical coordinates density of the particles1 axial stress2, 3 lateral stressesc confining pressured deviator stressσ d

CC,σ dCF, CC, CF and FF contacts contribute to deviator stress, respectively

872873

σ dFF

σ ij, σ ij' stress tensor and its deviatoric part

σ ijCC,σ ij

CF,

σ ijFF

CC, CF and FF contacts contribute to overall stress tensor, respectively

m mean stress dilatancy anglep peak dilatancy angle

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Five tables in this studyTab.1 Samples dimensions, particle numbers and initial porosities after isotropic compressionTab. 2 Input parameters in the DEM simulationsTab. 3 The CCC, CCF and CFF values at peak and critical states and axial strain corresponds to peak (i.e., p) and critical (i.e., c) statesTab. 4 Criteria for determination of the mechanical transitional behavior limits and fabric classification of binary mixturesTab. 5 The mean normal contact force f nof each contact type of various assemblies at peak stress and critical state

983984985986987988989990991992993994995996997998999

100010011002100310041005100610071008100910101011101210131014101510161017101810191020102110221023102410251026102710281029

Tab.1 Samples dimensions, particle numbers and initial porosities after isotropic compression

Samples **Expressionis faulty **

Wo

(%)W

(%)Dimensions

h0w0l0 (mm) min(h0, w0, l0)/Dc Ncp Ncp-sub Nfp n0

W0 0 0 30.7630.3330.67 20.09* 0 0 9589 0.370

W10 10 9.92 63.1963.7763.69 9.42 104 19136 82974 0.352

W20 20 19.80 63.0563.5663.25 9.40 210 39250 74458 0.339

W30 30 30.04 62.7863.1962.92 9.37 320 59976 65586 0.324

W40 40 40.05 63.1663.0462.00 9.25 427 80206 56421 0.314

W50 50 50.00 62.9662.6961.74 9.34 535 101219 47127 0.304

W60 60 60.01 62.7061.9161.57 9.18 642 121559 37695 0.290

W70 70 69.98 61.7560.8358.85 8.77 749 141614 28300 0.233

W80 80 79.99 61.6059.7061.33 8.90 855 162032 18837 0.249

W90 90 90.00 60.3263.7064.18 8.99 971 185064 9506 0.307

W100 100 100 60.3263.7064.18 8.99 2000 381126 0 0.347

Samples **Expressionis faulty **

Wo

(%)W

(%)Dimensions

h0w0l0 (mm) min(h0, w0, l0)/Dc Ncp Ncp-sub Nfp n0

W0 0 0 30.7630.3330.67 20.09* 0 - 9589 0.370

W10 10 104 - 0.350

W20 20 20.02 64.4364.6464.01 9.54 210 - 79410 0.329

W30 30 29.99 64.1364.4063.89 9.52 319 - 70491 0.312

W40 40 40.04 63.8963.5863.63 9.48 427 - 60544 0.296

W50 50 50.06 63.4962.9763.26 9.39 535 - 50536 0.279

W60 60 59.98 62.9262.6462.66 9.34 640 - 40420 0.263

W70 70 70.04 61.8361.5463.79 9.17 748 - 30294 0.249

W80 80 80.03 63.9161.8563.09 9.22 854 - 20170 0.270

W90 90 90.01 65.0865.1265.04 9.69 967 - 10159 0.335

W100 100 100 85.3877.7383.48 11.58 1969 - 0 0.393

Samples I represents the samples with gravel-shaped coarse particles, while Samples II denotes the samples with spherical coarse particles; *min(h0, w0, l0)/Df=20.09 for W0, Df=1.51 mm denotes

the particle size of fine particles

10301031

103210331034

Tab. 2 Input parameters in the DEM simulationsParameter Value

Particle density, 2600 kg/m3

Inter-particle friction, 0.5Wall-particle friction, w 0.0

Contact modulus of particle-wall 1109 Pa

Contact modulus of particle-particle 1108 Pa

kn/ks 4/3Damping constant 0.7

1035

10361037

Tab. 3 The CCC, CCF and CFF values at peak and critical states and axial strain corresponds to peak (i.e., p) and critical (i.e., c) states

Samples p cPeak state Critical state

CFF CCF CCC CFF CCF CCC

W0 3.0% 40%-50% 100% 0% 0% 100% 0% 0%

W10 3.0% 40%-50% 87% 13% 0% 84% 16% 0%W20 3.0% 40%-50% 73% 25% 2% 67% 31% 2%W30 3.0% 40%-50% 59% 27% 4% 50% 45% 5%W40 3.4% 40%-50% 44% 44% 12% 30% 55% 15%W50 3.8% 40%-50% 32% 50% 18% 15% 61% 24%W60 3.8% 40%-50% 17% 53% 30% 6% 64% 30%W70 5.0% 40%-50% 4% 35% 61% 1% 45% 54%W80 6.6% 40%-50% 1% 24% 75% 0% 32% 68%W90 6.8% 40%-50% 0% 15% 85% 0% 19% 81%

W100 7.8% 40%-50% 0% 0% 100% 0% 0% 100%

10381039

1040

Tab. 4 Criteria for determination of the mechanical transitional behavior limits and fabric classification of binary mixtures

Group G1 G2 G3 G4

Fabric structure Overfilled structure

Interactive-overfilled structure

Interactive-underfilled structure

Uderfilled structure

Contribution of each contact type FF>CF>CCCC~0% CF>FF>CC CF>CC>FF CC>CF>FF

FF~0%

Numerical results

limits in this study (peak state) W≤40% 40%<W≤55% 55%<W≤65% W>65%

limits in this study (critical state) W≤30% 30%<W≤48% 48%<W≤70% W>70%

limits in De Frias Lopez et al. [25] (at d=100

kPa)W<45% 45%<W<60% 60%<W<75% W>75%

Experimental results

limits in Vallejo [1] (peak state) W≤40% 40%<W≤70% 70%<W≤80% W>80%

limits in Vallejo [45] (peak state) W≤40% - - W>75%

limits in Kuenza et al. [44] (peak state) W≤40% - - -

limits in Xu et al. [46](critical state) W≤30% - - W≥70%

10411042

1043

Tab. 5 The mean normal contact force f n of each contact type of various assemblies at peak stress and critical state

Assemblyf n of CC contacts (N) f nof CF contacts (N) f nof FF contacts (N)

Peak state Critical state Peak state Critical state Peak state Critical state

W0 0.00 0.00 0.00 0.00 0.84 0.74

W10 2.40 4.01 1.23 1.50 0.98 0.90W20 2.34 4.11 1.25 1.58 1.01 0.96W30 2.50 3.14 1.28 1.66 1.02 0.77W40 3.33 3.50 1.41 1.77 0.90 0.69W50 2.81 3.55 1.40 2.03 1.02 0.77W60 3.06 3.59 1.60 2.03 1.01 0.63W70 3.50 3.66 1.60 2.94 0.82 0.45W80 3.91 3.92 2.25 3.67 0.70 0.56W90 4.02 4.04 3.28 4.45 0.99 0.93W100 6.99 8.21 0.00 0.00 0.00 0.00

10441045

10461047