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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.
REFERENCES
Ahmed S.S., M. S. Basson, and A. Martinez. (2019) "Particle-Scale Contact Response of 3D
Printed Particle Analogs." Engineering Mechanics Institute Conference. Pasadena, CA, USA.
Andò, E., G. Viggiani, S. A. Hall, and J. Desrues. (2013). "Experimental Micro-mechanics of
Granular Media Studied by X-ray Tomography: Recent Results and Challenges."
Géotechnique Letters 3, no. 3 : 142-146.
– 10 –
Basson, M. S., R. Cudmani, and G. V. Ramana. (2019) “Evaluation of Macroscopic Soil Model
Parameters Using the Discrete Element Method,” International IACMAG Symposium. IIT
Gandhinagar, Gujarat, India, Paper 112.
Drescher, A., and G. J. de Jong. (1972) “Photoelastic Verification of a Mechanical Model for the
Flow of a Granular Material.” Journal of the Mechanics and Physics of Solids, 20, 337-340.
Fu, P., and Y. F. Dafalias. (2015). "Relationship between Void- and Contact Normal-based Fabric
Tensors for 2D Idealized Granular Materials." International Journal of S. and S. 63, 68-81.
Gao, Y., and Y. H. Wang. (2014) “Experimental and DEM Examinations of K0 in Sand under
Different Loading Conditions.” Journal of G. and G. Engineering, 140-5.
Gu, X., J. Hu, and M. Huang. (2015). “K0 of Granular Soils: A Particulate Approach.” Granular
Matter, 17-6, 703-715.
Imole, O. I.. (2014). “Discrete Element Simulations and Experiments: Toward Applications for
Cohesive Powders”. PhD Thesis, Universiteit Twente.
Khalili, M. H., Roux, J.-N., Pereira, J.-M., Brisard, S., and Bornert, M. (2017). “Numerical study
of one-dimensional compression of granular materials. I. Stress-strain behavior,
microstructure, and irreversibility.” Physical Review E, 95(3), 032907.
Kuhn, M. R., W. Sun, and Q. Wang. "Stress-induced Anisotropy in Granular Materials: Fabric,
Stiffness, and Permeability." Acta Geotechnica 10, no. 4 (2015): 399-419.
Lopera Perez, J. C., C. Y. Kwok, C. O’Sullivan, X. Huang, and K. J. Hanley. (2015) “Numerical
Study of One-Dimensional Compression in Granular Materials.” Géotechnique Letters, 5, 96-
103.
Mašín, D. "Asymptotic Behaviour of Granular Materials." Granular Matter 14, no. 6 (2012): 759-
74
Mcdowell, G. R., and J. P. De Bono. (2013). “On the Micro Mechanics of One-dimensional
Normal Compression." Géotechnique 63, no. 11, 895-908.
Mital, U., R. Kawamoto, and J. E. Andrade. (2019) “Effect of Fabric on Shear Wave Velocity in
Granular Soils.” Acta Geotechnica, 1-15.
Oda, M.. "The Mechanism of Fabric Changes During Compressional Deformation Of Sand." Soils
and Foundations 12, no. 2 (1972): 1-18.
Peters, J. F., M. Muthuswamy, J. Wibowo, and A. Tordesillas. (2005) “Characterization of Force
Chains in Granular Material.” Physical Review E - Statistical, Nonlinear, and Soft Matter
Physics, E 72, No. 4.
Radjai, F., D. E. Wolf, M. Jean, and J.-J. Moreau. (1998) "Bimodal Character of Stress
Transmission in Granular Packings." Physical Review Letters 80, no. 1 : 61-64.
Reboul, N., E. Vincens, and B. Cambou. (2008) "A Statistical Analysis of Void Size Distribution
in a Simulated Narrowly Graded Packing of Spheres." Granular Matter 10, no. 6: 457-68.
Rothenburg, L, and R. J. Bathurst. (1989) “Analytical Study of Induced Anisotropy Granular
Materials in Idealized.” Geotechnique, 39-4, 601-614.
Šmilauer et al. (2015), Yade Documentation 2nd ed. The Yade Project.
Satake, M., (1982). Fabric tensor in granular materials. P. A. Vermeer and H. J. Luger, eds. In:
Deformation and Failure of Granular Materials. Delft, 63–68.
Thornton, C. (2000) “Numerical Simulations of Deviatoric Shear Deformation of Granular
Media.” Géotechnique, 50-1, 43-53.
Wiebicke, M., E. Andò, I. Herle, and G. Viggiani. (2017) "On the Metrology of Interparticle
Contacts in Sand from X-ray Tomography Images." Measurement Science and Technology
28, no. 12 : 124007.
