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A DEM Study of the Evolution of Fabric of Coarse-Grained Materials during

Oedometric and Isotropic Compression

Mandeep Singh Basson1 and Alejandro Martinez2

1Ph.D. Student, Department of Civil and Environmental Engineering, University of California,

Davis; e-mail: mbasson@ucdavis.edu

2Assistant Professor, Department of Civil and Environmental Engineering, University of

California, Davis; e-mail:amart@ucdavis.edu

ABSTRACT

Granular materials like sandy and gravely soils are composed of an assembly of individual

particles. A given soil deposit or specimen has a specific arrangement of individual particles,

clumps of particles, and void spaces, which along with the orientation of inter-particle contacts,

encompasses the basis of the soil fabric concept. Experimentally soil, fabric can be quantified by

non-destructive methods such as image microscopy and X-ray computed tomography. While these

state-of-the-art methods provide accurate and useful information, they require specialized

equipment and analysis methods. This has ushered a rapid increase in the use of particle-based

numerical tools, such as the Discrete Element Method (DEM), which enable the study of behaviors

at the micromechanical- and element-scales. In this paper, we investigate the behavior of

specimens in oedometric (1D) and isotropic compression using 2D DEM simulations. To quantify

the fabric of the specimens in DEM, scalar descriptors (void ratio, coordination number, and

number of rattlers) and vector fabric descriptors (contact normal direction, contact normal and

shear forces) were monitored at different stages of oedometric and isotropic compression. The

results presented herein provide a comparison of the trends captured by the different fabric

descriptors and highlight the effect of fabric anisotropy on the macroscopic response.

INTRODUCTION

Experimental and numerical investigations on stress transmission in granular systems show that

the particle level interactions control the element-scale behavior (Drescher and De Jong 1972, Oda

1972, Peters et al. 2005, Khalili et al. 2017). Under the application of external stress, the

interparticle contacts align in a specific direction to transmit the stress in the form of force chains.

These force chains and the constraint of equilibrium on each particle leads to strong inhomogeneity

in the transmission of forces. This causes the system to have a directional characteristic, which is

often reflected in the anisotropy of the internal fabric (Rothenburg and Bathurst 1989, Kuhn et al.

2015, Fu and Dafalias 2015). Destructive and non-destructive experimental techniques, such as X-

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Ray computed tomography, optical microscopy and acoustic wave velocity, have been used to

study the evolution of fabric during conventional testing (Ando et al. 2013, Wiebicke et al. 2017).

Yet, Discrete Element Method (DEM) remains a popular choice due to its ability to track

micromechanical interactions between the particles in terms of forces and displacements.

In this study, we present the results of oedometric and isotropic compression two-

dimensional DEM simulations. Extending previous studies on oedometric and isotropic

compression, the effect of a single cycle of loading and unloading in the evolution of fabric is

explored, both on the scalar and macroscopic response (void ratio, coordination number, rattlers

fraction, K0) and the micro-scale fabric anisotropy (contact normal, contact normal and shear

forces). Specimens with various initial states were created, and the effect of differences in void

ratio, coordination number, rattlers fraction and stress state on the macroscopic response during

compression is investigated. The force chains formed during compression were divided into strong

and weak force networks and the evolution of their anisotropies is analyzed. Most importantly, the

equivalence of micro-scale anisotropy and the macro-scale response is investigated to explore

micromechanical aspects of anisotropic behavior of granular systems.

MODEL AND SIMULATION DETAILS

YADE, an open source DEM code, was used in this study (Šmilauer et al. 2015). 2-D simulations

are used here as they facilitate fabric analysis as shown in experimental (Drescher and De Jong

1972) and numerical studies (Rothenburg and Bathurst 1989, Radjai et al. 1998, Peters et al. 2005,

Fu and Dafalias 2015). Specimens with dimensions of 63.5 mm by 63.5 mm were prepared and

confined by periodic cells. Circular particles were represented as two-dimensional disks with a

mean particle diameter (D50) of 3.15 mm and uniformity coefficient (Cu) of 2.6. The non-linear

Hertz Mindlin contact law was used, with Elastic Modulus of 77 GPa and Poisson’s ratio of 0.3.

Although these material properties are close to the properties of glass, once normalized, these

elastic properties can be suitably applied to a variety of materials (Ahmed et al. 2019). The density

of the particles was taken as 2650 kg/m3 and the global damping coefficient as 0.05. Different

sample preparation procedures were employed to create different initial stress states, void ratios,

and fabric anisotropies. Samples were either prepared under oedometric or isotropic compression

to a target mean stress of 25 kPa. Finally, four different initial states were obtained by taking the

interparticle friction coefficient of 0.1 and 0.4. The samples prepared with an interparticle friction

angle of 0.1 resulted in a void ratio of 0.237 when prepared isotropically and 0.246 when prepared

oedomterically, which are denoted as dense samples. On the other hand, the samples prepared with

an interparticle friction angle of 0.4 resulted in a void ratio of 0.249 when prepared isotropically

and 0.256 when prepared oedomterically, which are denoted as loose samples.

The samples were then subjected to a single cycle of quasi-static oedometric or isotropic

compression to 1000 kPa. The friction coefficient during compression was the same as that used

during preparation. The applied strain rate was kept low enough to keep the Inertial Number (I)

less than 10-4 during sample preparation and loading, and 10-5 during unloading. Along with

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inertial number, the samples were equilibrated to an unbalanced force ratio less than 10-5 and a

stress tolerance ratio less than 10-2. In total, eight simulations were performed, which are

summarized in Table 1. The first letter in the name denotes the preparation technique, the second

letter denotes the compression technique, and the number denotes the inter particle friction

coefficient. By comparing, for example, the samples IO0.4 and OO0.4, the effect of preparation

technique can be explored. Comparison of, for example, samples II0.4 and IO0.4 allows for the

effect of compression technique to be explored. Finally, comparison of samples II0.1 and II0.4

allows for investigation of the effect of inter particle coefficient.

Table 1. DEM simulation details

Name

Preparation

Compression

Interparticle

Friction

Coefficient

Initial

void ratio

Final

void ratio

Initial

CN

Final

CN

II0.1

Isotropic

Isotropic

0.1

0.237

0.194

3.51

4.25

II0.4

Isotropic

Isotropic

0.4

0.249

0.201

2.98

3.96

IO0.1

Isotropic

Oedometric

0.1

0.239

0.198

3.43

4.21

IO0.4

Isotropic

Oedometric

0.4

0.252

0.212

2.93

3.72

OI0.1

Oedometric

Isotropic

0.1

0.245

0.200

3.45

4.23

OI0.4

Oedometric

Isotropic

0.4

0.256

0.213

2.90

3.91

OO0.1

Oedometric

Oedometric

0.1

0.246

0.205

3.53

4.21

OO0.4

Oedometric

Oedometric

0.4

0.254

0.219

2.89

3.73

EVOLUTION OF SCALAR DESCRIPTORS

Void Ratio. The void ratio of the 2-D sample was calculated using the cross-sectional areas of the

disks and the overall sample. Due to particle movements near the periodic boundaries, the void

ratio computations were performed in a square box with dimensions that were reduced by a

distance of 2.5 times the mean particle diameter (D50) (Reboul et al. 2008, Basson et al. 2019).

The evolution of void ratio for different samples with increasing vertical stresses is shown in Fig.

1 (a). A comparison of oedometric compression and isotropic compression reveals that the changes

in the void ratio are smaller in oedometric compression than isotropic compression. Additionally,

the compression slope is steeper for isotropic compression than for oedometric compression. These

observations are similar to those by other authors (Khalili et al. 2013, McDowell and Bono 2013).

It is interesting to note that the compression slope is dependent only on the compression technique,

while the effect of sample preparation, which controls fabric anisotropy, does not have a significant

effect. This may indicate that after a specific stress state is applied, the fabric of the sample evolves

according to the on the magnitude and direction of the applied boundary stresses. This ‘swept out

of memory’ phenomena is unique to granular materials and explains their asymptotic behavior

(Mašín 2012). Apart from the rearrangement of particles, the void ratio change is minimal as no

plastic deformation occurs in the simulation.

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Coordination Number. Two different coordination numbers were monitored during the

simulation. Coordination Number (CN) was computed for all the particles contacts, and

Mechanical Coordination Number (MCN) was computed for particle contacts which participate in

the transmission of forces. The definition of CN and MCN is same as described in Thornton (2000).

Fig 1 (b) shows the evolution of coordination number (CN) and mechanical coordination number

(MCN) for samples II0.4, IO0.1, OI0.4, and OO0.4. Unsurprisingly, all the samples show an

increase in coordination number as the vertical stress increases. This is because when the stress is

applied and the void ratio decreases, the radial distance between the particles decreases, causing

more contacts to form. Interestingly, the compression technique affects the formation of contacts

which is highlighted in the difference of coordination number. The oedometrically compressed

samples reach a lower coordination number than the isotropically compressed particles. This is

due to the fact that the oedometrically compressed samples are compressed in one direction causing

the force network to become highly anisotropic in the direction of the major principal stress. Due

to this anisotropy, more particles in the minor principal stress direction remain as rattlers. When

the particles are isotropically compressed, there is minimal anisotropy and more particles

participate in the load bearing structure. Similar observations are recorded in Khalili et al. (2017).

Rattles Fraction. The fraction of rattlers is the ratio of the particles which have one or no contact

(no force transmission) to the total number of the particles in the simulation. Fig 1(c) shows the

evolution of rattler fraction with an increase in vertical stress. As the sample is compressed, a

greater number of particles participate in force transmission, decreasing the rattlers percentage

(from 6% to 1.5%). There are 50% more rattlers in sample IO0.4 than in OI0.4 at the end of

compression. This supports the results indicating lower CN and MCN in the case of oedometrically

compressed samples in Fig. 1(b). For samples subjected to isotropic compression, the decrease in

the rattlers fraction is nearly identical, showing that the rattlers evolution of these samples is not

affected by the initial preparation technique.

Stress Ratio during 1D compression. K0 is a response of a granular system under oedometric

loading. It is defined as the ratio of horizontal effective stress to vertical effective stress. Fig. 1 (d)

shows the evolution of K0 for the different tests. For II samples, the values of K0 remain close to

1. Surprisingly, the IO and OO samples tend to reach a similar value of K0 as the compression

progresses, indicating that the final K0 values are independent of the initial stress state. A sharp

decrease is observed in the IO samples, as the K0 drops from the isotropic value of 1, and converges

to an asymptotic value. For the OI samples, the K0 values increase towards 1 and reach a stable

value asymptotically. The dense samples attain a lower K0 value, which is in line with previous

numerical and experimental studies (Gao and Wang 2014, Perez et al. 2015). A reversal of K0 is

observed during the unloading phase, leading to values that are greater than 1. The

micromechanical explanation for this behavior is presented in the next sections.

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Figure 1. a) Evolution of void ratio, (b) CN and MCN, (c) rattlers fraction, and (d) K0 for

“loose” samples.

EVOLUTION OF ANISOTROPY

Strong and weak force networks. As normal forces is transmitted through the particle contacts,

there are contacts which transmit a force greater than the average contact normal force. These

particles form the strong force (SF) network and the remaining contacts form the weak force (WF)

network. These two subsets of the force network influence the anisotropy, stress, and dissipation.

The bimodal characteristic of force transmission network has been investigated in literature (Peters

et al. 2005, Radjai et al. 1998) and their definition of strong and weak force networks is used in

this study. For all the simulations, the force network was divided into strong and weak force

networks, and their anisotropies were studied.

Fabric anisotropy. Vector analysis of contact normal unit vectors, contact normal force (CNF)

and contact shear forces (CSF) reveals the directional details of fabric and its evolution as a

response to applied loading. During compression, the spatial distribution of contact normal unit

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vectors and CNFs orients itself towards the major and minor principal stress directions. In this

section, the fabric is quantified using the Fabric Tensor (for the contact normal unit vectors) and

angular distributions of contact normal unit vectors for CNF and CSF. A fabric tensor is a metric

that quantifies microstructural orientation-related characteristics of the material in a tensorial form

(Oda 1972, Satake 1982). The fabric tensor (Fij) is a two by two matrix for 2D analysis, and the

contact normal anisotropy (denoted as ‘a’ hereafter) is calculated as the difference between the

diagonal items (F11 – F22).

Although the fabric tensor provides mathematically descriptive information of anisotropy

and direction of that anisotropy at a given stress state, it fails to provide the graphical representation

of contacts in different directions. Graphical representation of contact orientation for different

directions uses angular distributions of vectors, as presented in Rothenburg and Bathurst (1989).

These angular distributions are represented as:

Contact Normal distribution =1

2π {1+a cos 2(θ-θa)}

(1)

Contact Normal Force (CNF) distribution =fo{1+ancos 2(θ-θn)}

(2)

Contact Shear Force (CSF) distribution = fo{a 2(θ-θ)}

(3)

where a, an and as are the contact normal, CNF and CSF anisotropies respectively, θa, θn, and θs are

the direction of the respective anisotropies and f0 is the average normal or shear force over all the

contacts in the assembly. Although there are various ways of working with the contact data (Imole

2014), angular binning was used here. In angular binning, the data extracted from the simulation

in the form of contact vectors are binned and represented in the form of polar plots. These polar

plots were then curve fitted with the equation 1, 2 and 3, to quantify anisotropy. Fig 2 (a) and (b)

shows two such polar plots for CNF for isotropic (circular) and oedometric (peanut shaped)

compression for samples II0.1 and IO0.1, respectively.

Figure 2. (a) CNF polar plot for isotropic (sample II0.1, an = 0.01) and (b) oedometric

compression (sample IO0.1, an = 0.72) (blue line is curve with fitted parameters), and (c)

comparison of contact normal anisotropy as obtained from fabric tensor and polar plot curve

fitting for test IO0.1

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As the fabric tensor provides the anisotropy of contact normal, a successful curve fitting of

contact normal polar plots should provide a similar anisotropy value. To confirm the accuracy of

the curve fitting, the fabric tensor anisotropy and anisotropy from fitted curves for the IO0.1 test

are plotted in Fig 2(c). As the values from both methods are close to identical, this curve fitting

method is deemed appropriate to fit the contact normal, CNF, and CSF for all the simulations.

Evolution of different anisotropies during oedometric compression. Figs. 3(a), 3(b), and 3(c)

shows the evolution of a, an, and as for sample IO0.1. As this sample was prepared by isotropic

compression, the initial anisotropies are close to zero. During compression, as the vertical stress

increases, the magnitude of all the anisotropies increase. Along with this change in magnitude,

Fig. 3 reveals the directional aspect of SF and WF networks. The SF network aligns vertically with

the principal stress direction, which is shown in the positive sign of the anisotropy. On the other

hand, the WF network aligns horizontally, in the direction of the minor principal stress

(perpendicular to direction of load application). This directional anisotropy shows that the SF

network carries most of the compression load by forming columns of force network. These

columns percolate downwards from the location of load application and need stability against

buckling. Therefore, the WF network aligns in the perpendicular direction, providing lateral

support against buckling to SF networks. Radjai et al. (1998) noted that this lateral support causes

the WF network to behave similarly to an interstitial liquid. This observation is essential when the

stiffness of the granular system is measured using shear wave velocities (Mital et al. 2019). During

the unloading phase, a decrease in the magnitude of the anisotropies is observed. For instance, at

a vertical stress value 150 kPa, the sample reaches an isotropic state with near zero anisotropy.

Further unloading to a vertical stress of 25 kPa results in an increase in the anisotropy magnitude,

but the direction is perpendicular to the direction observed during the loading phase.

Stress Ratio (K0) and Anisotropy. Rothenburg and Bathurst (1989) studied biaxial compression

of 2D disks using DEM and concluded that the deviatoric stress could be related to anisotropy

parameters as:

σz-σx

σz+σx=1

2 ( a+an+as)

(4)

The authors also note that K0 can be expressed as:

K0=σx

σz=2- ( a+an+as)

2+( a+an+as)

(5)

Using the anisotropy parameters from all force (AF) chains, K0 values are calculated and

compared to those obtained from the stresses measured at the boundaries of the specimens. It is

apparent that the K0 values obtained from the anisotropy parameters are almost identical to the K0

values obtained from the boundary stresses. This equation not only applies to the loading cycle but

also the unloading cycle, as the changes in the principal direction during unloading are reflected

in the signs of the anisotropy parameter. This change in the direction of anisotropies during

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unloading explains the phenomena of the increase of K0 with overconsolidation. Fig 4 shows the

contact polar plots for sample IO0.1 at 75kPa during loading and unloading. As the sample

becomes overconsolidated, some particles experience force interlocking and get jammed in the

minor principal direction. The SF networks align themselves in the horizontal direction during

unloading, which is reflected in the contact polar plots. Higher forces in the horizontal direction

cause the K0 reach values greater than 1. This interlocking of forces is observed experimentally

and numerically by Gu et al. (2015) and Gao and Wang (2014).

Figure 3. Evolution of (a) contact normal anisotropy (a), (b) CNF anisotropy (an), (c) CSF

anisotropy (as), and (d) comparison of K0 as obtained from stresses and anisotropy for sample

IO0.1.

CONCLUSION

This paper presents the results of two dimensional compression tests performed using DEM on

samples with different initial fabric and compressed by different compression techniques. Four

samples were prepared with different initial void ratios, coordination numbers and number of

rattlers. These samples were compressed isotropically and oedometrically, and the evolution of

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void ratio, coordination numbers, numbers of rattlers, K0 and anisotropy parameters was tracked

for a total of eight simulations. Although the compression response under increasing vertical stress

seems nearly reversible (as shown in the void ratio, coordination number and number of rattlers)

and independent of the initial state, the internal state of the material shows irreversible changes (as

shown in anisotropy parameters and K0). Also, a direct correlation between the fabric anisotropy

and K0 was observed, and the value of K0 is correctly predicted in both loading and unloading

curves by the anisotropy parameters. The difference in anisotropy parameters of strong and weak

force chains shows that the weak force chains align to provide stability to buckling of the strong

force chains. All these observations highlight the effect of particle-scale fabric on the macro-scale

behavior of granular materials.

Fig 4. Comparison of (a) strong force network, (b) weak force network and (c) average force

network polar plots for sample IO0.1 at 75 kPa both during loading and unloading.

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