Available online at www.sciencedirect.com
ScienceDirect
Comput. Methods Appl. Mech. Engrg. 368 (2020) 113183
www.elsevier.com/locate/cma
A micro-mechanical model for unsaturated soils based on DEM
Xin Liua , Annan Zhoua ,∗, Shui-long Shena,b , Jie Lia , Daichao Shengc
a Discipline of Civil and Infrastructure Engineering, School of Engineering, Royal Melbourne Institute of Technology (RMIT
University), Melbourne, VIC 3001, Australia
b College of Engineering, Shantou University, No. 243, Da Xue Road, Shantou, Guangdong 515063, China
c School of Civil and Environmental Engineering, University of Technology Sydney (UTS), Broadway, Sydney, NSW 2007, Australia
Received 23 October 2019; received in revised form 15 April 2020; accepted 23 May 2020
Available online xxxx
Graphical Abstract
Abstract
A micro-mechanical model to study the microscopic and macroscopic behavior of unsaturated soils under different suctions
is proposed in this study. In the model, a novel pore-scale numerical method for simulating the liquid–solid interfaces is
proposed first. A discretization of the particle surface using Fibonacci-Lattice is then introduced to calculate the capillary
forces from the complex liquid–solid interfaces. The joint influence of capillary forces and the interparticle contact forces
on the motion of the particles are handled by the discrete element method (DEM). The effective stress parameter estimated
by the model is compared with the experimental results for unsaturated soils, which confirms the validity of the proposed
micro-mechanical model. The microscopic responses (liquid–solid interfaces, capillary forces, contact forces and coordination
numbers) and macroscopic responses (strength, stress–strain relationship and volume change) of unsaturated soils in desaturation
tests and triaxial tests are studied by the proposed model.
c 2020 Elsevier B.V. All rights reserved.
⃝
Keywords: Unsaturated soils; Micro-mechanical model; Liquid–solid interfaces; Capillary forces; Discrete element method
∗ Corresponding author.
E-mail address: annan.zhou@rmit.edu.au (A. Zhou).
https://doi.org/10.1016/j.cma.2020.113183
c 2020 Elsevier B.V. All rights reserved.
0045-7825/⃝
2
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
1. Introduction
Unsaturated soil, composed of solid (soil particles), liquid (water) and gas (air), is one of the most common
materials in nature, which covers significant areas of earth’s surface and embraces a wide range of soil types. The
liquid pressures and gas–liquid interfacial tension introduce additional forces to the particles [1,2], which make
the mechanical behavior (like strength and deformation) of unsaturated soil outstandingly complicated, compared
with saturated or dry soils [3,4]. Therefore, the contents and distributions of gas and liquid play very fundamental
roles in the hydro-mechanical behavior of unsaturated soils. In the past decades, considerable efforts have been
made by many researchers for the development of a consistent continuum-based framework for unsaturated soils by
using laboratory tests and theoretical and numerical analyses, encompassing unsaturated soil mechanics problems
such as shear strength and volume change [5–7]. However, continuum-based framework for unsaturated soils still
need to face many challenges and limitations because the continuum-based models rely on the phenomenological
relations based on macroscopic experimental observations. Furthermore, by using the continuum-based framework,
it is difficult to obtain insights into some microscopic aspects that affects the macroscopic responses inherently.
The discrete element method (DEM) is an alternative approach to overcome these limitations of macro- and
continuum-based models. In a DEM model, the materials are modeled as discrete particles interacting with
neighboring ones. The interparticle force increments are calculated at all contacts from the relative velocities of
the contacting particles, and the particle motion is governed by Newton’s second law [8]. Thus, DEM provides
both direct and indirect accesses to evaluate the contact stress, displacement and other detailed internal information
such as average coordination number and fabric tensor of microstructures [9]. Recently, DEM has been extended
to model the mechanical behavior of partially saturated porous media like unsaturated soils.
DEM simulations of unsaturated soils are majorly conducted in the pendular regime (the degree of saturation is
generally less than 15%), in which liquid occurs only as isolated liquid bridges in between independent particles.
In this case, the geometry of the isolated liquid bridge, which is defined by the Young–Laplace equation, can be
determined either by using numerical approaches (e.g. [10–15]) or by using the toroidal approximation proposed
by Fisher [1] (e.g. [16–19]). Based on the obtained geometry, then the capillary force can be conveniently evaluated,
either in the neck of the liquid bridge (a circle area) or in the liquid–solid interface (a spherical cap), because
both geometric shapes mentioned above are regular. According to the Fisher’s approximation, the external radius
of curvature (r1 ) of the liquid bridge is constant and the minimum value (r2 ) at the bridge neck is donated to the
internal radius of curvature. Because of the simplification on the geometry, the methods using Fisher’s approximation
is much more efficient compared to the numerical approaches. According to Lian et al. [2], if only the isolated liquid
bridges are taken into consideration, the errors in the capillary force calculated by the methods based on Fisher’s
approximation are less than 10% of those obtained by the numerical approach.
For the scenarios of higher degree of saturation, it is difficult to reproduce the liquid morphologies since the
liquid can be formed as complex clusters. Therefore, it is very challenging to evaluate the capillary force from
such complicated geometries of liquid–solid interfaces. A few attempts have been made to extend DEM to the
high saturation scenarios. Researchers such as Liu and Sun [20], Kim et al. [21] and Li et al. [22] have adopted
an empirical method by directly installing adhesive forces at contacts to represent capillary forces to simulate
high saturation cases. These studies can only simulate the capillary force qualitatively, since the effect of spatial
distributions of liquid is omitted. Jiang et al. [23] proposed a method to consider the disappearance of menisci and
air bubbles. This method, however, is restricted in two-dimensional scenario under dense conditions. Besides, in this
method, the capillary forces were approximately calculated based on isolated capillary bridge model in the contact
area and did not consider the real liquid geometry. Melnikov et al. [24] proposed a numerical approach which can
resolve and combine capillary bridges, menisci, and fully saturated pores to form local liquid clusters of different
shapes. It is therefore capable of simulating the liquid redistribution at various saturation levels beyond the pendular
regime. Based on this approach, Melnikov et al. [25] further developed a fluid–particle coupling model to study
the shearing behavior of granular soils under different degrees of saturation. However, the major problem of this
method is that it can only be used for granular media composed of equal size particles. Yuan and Chareyre [26]
and Yuan et al. [27] proposed a 2PFV-DEM scheme (two-phase pore-scale finite volumes coupled with the DEM)
to solve quasi-static hydro-mechanical problems. In the 2PFV-DEM scheme, the 2PFV is used for calculating the
quasi-static pore water distribution, the pore space is decomposed by regular triangulations, from which a set of
pores connected by throats can be identified. A local entry capillary pressure is evaluated for each throat, based
on the balance of capillary pressure and surface tension at equilibrium. However, as no capillary bridges can be
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
3
Table 1
DEM models for unsaturated soils.
Models
Consider liquid
phase?
Limited to
particle
contact point?
Threedimensional?
Polydisperse
particles?
Limited to
dense
packing?
Liu and Sun [20]
Gröger et al. [17]
Jiang et al. [23]
Soulié et al. [14]
Richefeu et al. [11]
Scholtès et al. [12]
Kim et al. [21]
Than et al. [19]
Shen et al. [18]
Melnikov et al. [25]
Yuan and Chareyre [26]
Wang et al. [15]
Duriez and Wan [10]
Li et al. [22]
Duriez et al. [28]
Asadi and Mirghasemi [16]
No
Isolated bridges
Liquid bridges
Isolated bridges
Isolated bridges
Isolated bridges
No
Isolated bridges
Isolated bridges
Yes
Liquid clusters
Isolated bridges
Isolated bridges
No
Isolated bridges
Isolated bridges
Yes
No
Yes
No
No
No
Yes
Yes
Yes
No
No
No
No
Yes
No
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Yes
Yes
No
No
Yes
No
No
No
No
No
No
Yes
Yes
No
No
No
No
No
modeled, this method cannot be applied to the low saturation regime where capillary bridge water exists. In addition,
the application of this method is restricted for dense packing media only. The advantages and limitations of the
current DEM models for unsaturated soils are summarized in Table 1.
It is clear that, in terms of DEM simulations of unsaturated soil, it is important to calculate the liquid induced
force on particles, namely the capillary force. The capillary force (F c ) includes two parts: (1) F s , the force
determined by the fluid pressure ( pl ) when the gas pressure is the atmosphere pressure ( pg = 0) and (2) F γ , the
force determined by the interfacial tension (γ ). F s is determined by pl and the corresponding area of liquid–solid
interfaces on the particles (Θ, the blue areas in Fig. 1). F γ is determined by γ and the corresponding gas–liquid–
solid contact lines (Γ is the contour of the liquid–solid interface Θ, also known as the tri-phase contact line, the
red line in Fig. 1). Thus, an accurate simulation of liquid–solid interface is of pivotal importance in the modeling
of unsaturated soil by using microscopic methods like DEM. Moreover, the liquid–solid interface is not a constant
but keeps changing with the deformation process of the solid structure, which makes the simulation of liquid–solid
interface more challenging but more demanding. Unfortunately, both determining the liquid–solid interfaces and
further evaluating the capillary forces from the interface are very difficult due to the complex morphologies of the
liquid–solid interfaces which limit the application of the DEM to unsaturated media. As summarized in Table 1,
when the liquid phase is considered, there is no three-dimensional DEM model so far that can model the unsaturated
soils containing polydisperse particles with various densities in entire saturation scope.
The main objective of this study is to build up an advanced micro-mechanical model for unsaturated soil. In this
new model, we firstly propose a robust approach to directly model the liquid–solid interfaces on particles, which is
validated against the results by the pore morphology (PM) method [29]. Then the Fibonacci-Lattice method [30] to
produce the discretization of the particle surface is introduced to evaluate the capillary force from the complicated
liquid–solid interfaces. In the proposed model, the liquid–solid interfaces are able to be updated synchronously
with the deformation of the solid skeleton. The capillary forces and the contact forces exerted on the solid particles
and the corresponding deformation of soil skeleton are then handled by a DEM solver (ESyS-Particle, [31]). The
proposed model is then employed to simulate a desaturation test under a constant confining pressure (100 kPa)
for a dense sample (e = 0.66 at 10 kPa) and a series of triaxial tests under various confining pressures (10, 50,
100, 200, 400 kPa) and suctions (0, 20, 40, 60, 80 and 100 kPa) for both dense and loose samples (e = 0.66 and
0.73, respectively). For desaturation tests, the evolution of volumetric strain, the liquid–solid interfaces and capillary
forces with suction increase are presented and discussed. The average contact stress and effective stress parameter are
calculated, and the effective stress parameter estimated by the model is compared with the experimental results for
unsaturated soils. Furthermore, the simulation results of desaturation tests by the proposed model are also compared
with the computed results by the 2PFV-DEM scheme [26]. For unsaturated triaxial tests, the effect of the suction
4
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 1. The force determined by the fluid pressure (F s ) and the force determined by the interfacial tension (F γ ). The blue areas represent
liquid–solid interface (Θ), and the red lines represent gas–liquid–solid contact lines (Γ ).
and shearing strain on the deviator stress, volume change, liquid–solid interfaces, capillary forces, contact force
networks and coordination numbers are illustrated and discussed.
2. Methodology
2.1. Determination of the liquid–solid interface
Liquid inside the porous materials like sandy or silty soils can be formed as isolated bridges (pendular
regime), funicular bridges (funicular regime) or more complicated clusters (capillary regime) depending on various
saturations, suctions and pore structures [32,33]. To simulate such complex morphologies of liquid, a hemisphere
approximation (name after [34]) was adopted in many studies to replace the general form of the Young–Laplace
equation [29,35]. The hemisphere approximation assumes that the gas–liquid interface can be represented by a
hemisphere where the radius of that sphere, r , is related to the capillary pressure, s = pg − pl , as below.
2γ
(1)
r
In this case, any two particles with the separate distance (ds ) smaller than 2r are connected with a liquid bridge.
The hemisphere assumption is also adopted in this paper to model the liquid–solid interfaces. Eq. (1) is applied
to every particle pair with a separate distance ds < 2r . Two spherical caps are then predicted on each particle
and the caps are parts of the liquid–solid interfaces. The proposed algorithm to calculate the polar angle, θ , and
the normal, n, of each cap is listed in Appendix A. Since every particle in the packing is surrounded by multiple
particles, multiple caps are generated on the same particle and form the liquid–solid interfaces. These caps can be
isolated (corresponding to the liquid–solid interface of pendular liquid bridges) or overlapped (corresponding to the
liquid–solid interfaces of funicular liquid bridges or more complex liquid clusters) depending on the size (θ ) and the
orientation (n) of the caps (see Fig. 2). For example, as shown in Fig. 2, there are three spherical caps on the surface
of a particle, each is defined by a normal, n and a polar angle, θ . Caps 1 and 2 are overlapped (α < θ1 + θ2 , α is the
angle between n1 and n2 ) and form a complex liquid–solid interface existed in either funicular or capillary regimes.
Cap 3 is isolated thus represent the interface between particle and an isolated liquid bridge (pendular regime).
However, for the isolated liquid bridges, the previous study [29] suggested the general form of the Young–Laplace
equation rather than the hemisphere assumption. The general form of the Young–Laplace equation can be written
as
1
1
s = γ( + )
(2)
r1 r2
where r1 and r2 are two principal radii of curvature of the gas–liquid interface. The isolated caps need be detected
first and recalculated by using Eq. (2) (rather than by using Eq. (1)). To detect the isolated caps, every two caps
s = pg − pl =
5
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 2. Liquid–solid interfaces on particle. Each spherical cap is defined by a normal, n and a polar angle, θ . Caps 1 and 2 are overlapped
(α < θ1 + θ2 , α is the angle between n1 and n2 ) and form a complex liquid–solid interface existed in either funicular or capillary regimes.
Cap 3 is isolated thus represent the interface between particle and an isolated liquid bridge (pendular regime).
calculated by Eq. (1) on a particle pair are labeled with a same number, for example, caps C iA and C Bi represent
such paired caps on particle A and B, i is the labeled number. On particle A, an isolated cap C iA is detected when
αi j > θi + θ j
(3)
j
j
where αi j is the angle between n iA and n A , which are the normal of cap C iA and any other cap on particle A, C A ,
j
respectively; θi and θ j are the polar angles of caps C iA and C A . If C Bi is also an isolated cap on particle B, then caps
i
i
C A and C B are recalculated by using Eq. (2). A semi-analytical approach is developed to determine the values of r1
and r2 (see Appendix A). The steps to calculate the liquid–solid interfaces proposed in this paper are summarized
in Fig. 3 as a flow chart. It should be noted that there is hysteresis in the existence of menisci. For instance, if
particles separate more than 2r, the liquid bridge will break, but if the particles approach again to each other, a new
liquid bridge is reappeared only when the particles coming to contact again. However, similar to the existing work
[11,15], the menisci hysteresis is not taken into account in the proposed method since it may significantly increase
computational cost, while no obvious difference has been found in our several trial simulations if the menisci
hysteresis is considered. In addition, like the most existing DEM models for unsaturated system [10–21,26], the
proposed method is suction controlled, therefore it cannot be directly applied for constant water content tests.
2.2. Validation on the calculated liquid–solid interface
The pore morphology (PM) method [29] is used in this study to validate the proposed calculation method for the
liquid–solid interface. The PM method reproduces the gas and liquid distribution in a porous material for a given
capillary pressure in static conditions. The gas–liquid interfacial area can be calculated by using erosion and dilation
algorithms, which are widely used in image analysis procedures. For example, Turner et al. [36] have demonstrated
that the 3D morphology of the fluid phases at the pore-scale predicted by PM method for granular media match well
with the experimental images. Recently, the PM method has been widely used in the literature to predict the soil
water characteristic curves (SWCC) of porous media which produce a good agreement with the experimental results
[29,34,37,38]. By adopting the assumption that liquid in the larger pores is always excavated before the liquid in the
smaller pores that is widely accepted in soil water research [39–43], we compare the liquid–solid interfaces predicted
by the proposed method and that by the PM method (see Fig. 4). Since the computing power of the PM method is
significantly higher than the proposed method, for a clear comparison, only a small sample containing 22 particles
are employed. Fig. 4a shows the liquid–solid interfaces on particles under suctions of 100 kPa, 50 kPa, 33 kPa
and 20 kPa. Fig. 4b illustrates the comparisons of the liquid–solid interfaces computed by two different methods
(particles have been removed). In Fig. 4b, the white voxels represent the liquid predicted by the PM method, and
blue surfaces represent the liquid–solid interfaces predicted by the proposed method. The comparisons indicate that
the proposed method can well simulate the distribution of liquid–solid interfaces for unsaturated soils at different
suctions.
Fig. 5 shows the quantitative comparison of the area of liquid–solid interface computed by the PM method
and the proposed method. With the increase of the suction, the interface area sharply decreases within a suction
range from 0 to 20 kPa, then gently decreases within a suction range from 20 kPa to 100 kPa. The coefficient of
determination R2 of the proposed method is 0.99. The liquid–solid interface area obtained from the proposed method
6
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 3. The flow chart for calculating the liquid–solid interfaces.
agrees reasonably well with that from the PM method. It should be noted that the PM method is a comprehensive
method for liquid/gas analysis in pore structures. It can be used to produce the liquid/gas distributions, liquid–solid
interfaces and water retention behavior but correspondingly requires a large amount of computing power especially
when considering thousands of polydisperse spherical particles. While the proposed method has clear objective to
calculate the liquid–solid interface, which is an essential part for the DEM analysis to calculate the liquid induced
force for the unsaturated soils. Table 2 compares the calculation time required for applying the above two different
methods to predict the liquid–solid interfaces on 22 particles (see Fig. 4a) over varying suctions. The running time
by the PM method ranges from 14.77 to 221.163 sec, while that by the proposed method ranges from 0.071 to
0.352 sec It is observed that, in contrast to the PM method, the proposed method with respect to computing time
is 200∼600 times faster. The proposed method is fast and robust, which makes it suitable to be integrated with the
DEM analysis.
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
7
Fig. 4. (a) Evolution of the liquid–solid interfaces on particles predicted by the proposed method with suction decreasing from 100 kPa to
20 kPa. The Blue surfaces represent the liquid–solid interfaces predicted by the proposed method. (b) Comparisons between the liquid–solid
interfaces (without particles) predicted by pore morphology (PM) and the proposed method under different suctions. The white voxels
represent the liquid predicted by the PM method, and blue surfaces represent the liquid–solid interfaces predicted by the proposed method.
2.3. Determination of capillary forces
The liquid–solid interfaces become highly complicated in the funicular and capillary regime (see Fig. 4), which
is extremely difficult to measure the area analytically [44]. Therefore, a discretization of the particle surface is
introduced here to evaluate capillary force from the complex liquid–solid interfaces. The Fibonacci-Lattice method
[30,45], which was initially used for measuring the Earth coverage of satellite constellations, allows an almost
8
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 5. Comparisons of computed liquid–solid interface areas between the PM method and the proposed method. The liquid–solid interface
is calculate based on the same sample in Fig. 4. The coefficient of determination R2 of the proposed method is 0.99.
Table 2
Comparisons of running time between PM method and the proposed method in
simulating the liquid–solid interfaces on particles over varying suctions.
Suction (kPa)
Running times (sec)
PM method
The proposed method
10
20
33
50
100
221.16
77.61
30.81
19.12
14.77
0.352
0.203
0.084
0.062
0.071
uniform allocating of the center of sub-divided surfaces on the particle surface. It has important applications
in photon manipulation and nanophotonic devices [46] and in the evaluation and optimization of map projections [47].
To elaborate the lattice, it is based on the golden ratio Φ −1 = 0.618 and can have any odd number of points
[45]. Let N be any natural number. The number of points is
P = 2N + 1
(4)
and the longitudinal coordinate (lon i ) and latitudinal coordinate (lati ) of the ith point (i is an integer from −N to
N ) are
2i
lati = ar csin(
)
(5)
2N + 1
lon i = 2πiΦ −1
(6)
Fig. 6a shows an example of particle surface discretized with 1001 Fibonacci-Lattice points. Once the liquid–
solid interfaces (marked in blue) on a particle are determined, as shown in Fig. 6b, the Fibonacci-Lattice points are
used to evenly sub-divide the surface and the Fibonacci-Lattice points within the liquid–solid interfaces are used to
calculate the capillary force (F c ). This force includes two parts, F s , the force by fluid pressure pl (the gas pressure
is the atmosphere pressure, pg = 0) and F γ , the force by interfacial surface tension γ . Namely, F c = F s + F γ .
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
9
Fig. 6. (a) Particle surface discretized with 1001 Fibonacci-Lattice points. Each Fibonacci-Lattice point is described by longitudinal and
latitudinal angles. (b) The capillary force due to liquid–solid interface (marked as blue). Point i locates inside the liquid–solid interface, the
j
forces by fluid pressure f si = Apl n i . Point j locates on the tri-phase line, the force by interfacial surface tension f γ = lγ τ i , the forces by
j
fluid pressure f s = 0.5Apl n j .
F s is calculated as follows.
∫
pl nds
Fs =
(7)
∂Θ
where ∂Θ represents the liquid–solid interface (i.e., the contact surface between the liquid and the particle).
By discretizing the particle surface by Fibonacci-Lattice points, Eq. (7) can be replaced by
F s = Apl
Nw
∑
n i + 0.5Apl
i=0
Ne
∑
nj
(8)
j=0
2
where A = 4πPR is the area of a sub-divided surface (full area is considered for the points inside and half area
for points on the tri-phase contact lines, by assuming taking half area for points on the tri-phase contact lines, for
a Fibonacci-Lattice with 4001 points for example, the relative error between the theoretical and predicted area of
randomly placed cap is generally less than 0.06%); Nw is the number of the Fibonacci-Lattice points inside the
interface (the algorithm to find these points is presented in Appendix B); Ne is the number of the Fibonacci-Lattice
points located on the tri-phase contact lines (the algorithm to find these points is presented in Appendix B); n i and
n j are unit vectors of points i and j from the center of the particle, respectively. n i can be written as follows (n j
is similar to n i ).
⎛
⎞
cos(lon i )cos(lati )
n i = ⎝ sin(lon i )cos(lati ) ⎠
(9)
sin(lati )
The forces by fluid pressure pl of point i inside the liquid–solid interface ( f si ) and of point j located on the tri-phase
line ( f sj ) are shown in Fig. 6b.
The force by interfacial surface tension (F γ ) can be calculated as below.
∫
Fγ =
γ τ dl
(10)
∂Γ
where ∂Γ represents the gas–liquid–solid contact lines (also known as the tri-phase contact line). By discretizing
the particle surface with Fibonacci-Lattice points, Eq. (10) can be rewritten as
F γ = lγ
Ne
∑
i=0
τi
(11)
10
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 7. Forces on a particle. The liquid–solid interface i is consisted of two overlapped spherical caps which exist in funicular and capillary
regimes. There are two interparticle contact points inside liquid–solid interface i, hence, two pairs of contact forces act on liquid–solid
interface i. The liquid–solid interface j is a single spherical cap corresponding to an isolated liquid bridge, only one interparticle contact
point located on liquid–solid interface j, hence, only one pair of contact forces acted, and the orientation of the normal contact force F nc
j
j
is opposite to the orientation of F s and F γ .
√
where l = A is the nominal distance between two neighboring points [45]; τ i is the direction of the interfacial
surface tension induced force of point i, which is defined and calculated by Eq. (B.3) (see Appendix B). The force
by interfacial surface tension γ of point j located on the tri-phase line ( f γj ) is shown in Fig. 6b.
It should be noted that since the Fibonacci-Lattice points are dispersedly distributed on the surface of particle,
the tri-phase line cannot be perfectly reproduced by these points, the resulted tri-phase line are slightly discretized.
Points located on tri-phase line are defined by the condition that the distance between the point and the outline of
the cap is smaller than 0.5l (see Appendix B). Under this assumption, for a Fibonacci-Lattice with 4001 points as
an example, the relative error between the theoretical and predicted perimeter of randomly placed cap is generally
less than 0.25%.
The computational efficiency is highly influenced by the size of the Fibonacci-Lattice. As reported by González
[30], when the number of Fibonacci-Lattice points is over 4001, the maximum error measured for randomly located
spherical caps of any size is less than 0.05%. Thus, a Fibonacci-Lattice with 4001 points is employed in this
study.
2.4. DEM modeling of unsaturated soils
The total force exerted on a particle, F T , consists of the forces induced by the fluid, F s (see Eq. (8)) and F γ
(see Eq. (11)), and the contact forces between particles, F n and F τ (see Fig. 7). In general, F T can be written as
follows.
F T = F s + Fγ + Fn + Fτ
(12)
The interparticle contact force between particles can be calculated by a linear force–displacement contact law. The
reason for choosing the simple contact law instead of a more sophisticated Hertzian type contact law is that it
was found that the different contact formulations yield qualitatively very similar results that regarding the shear
strength of the particle sample and its related characteristics, while the Hertzian contact formulation required more
computational power [48,49].
The normal contact force is linearly proportional to the relative normal displacement
Fn = K nU n
(13)
where U n is the relative displacement in the normal direction; K n is the normal contact stiffness, we assume
A RB
K n /R = 100 MPa. R = R2RA +R
is the equivalent radius between two contacting particles with radius R A and
B
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
11
Table 3
Parameters used in the DEM simulations.
Parameters
Values
Particle number in the sample
Particle density, kg m−3
K n /R, Pa
K s /K n
Interparticle frictional coefficient, µ
9977
2710 × 106
1.5 × 108
2/3
0.5
R B . The shear force is calculated in an incremental pattern as follows:
F τt = F τt−1 + K s dU s
(14)
where F τt and F τt−1 are the shear forces at current and previous time steps, respectively; dU s is the incremental
displacement in tangential direction; K s is the shear contact stiffness, it is suggested by Goldenberg and Goldhirsc
[50] that the value of K s /K n for realistic granular materials should be around 2/3 < K s /K n < 1. Hence, we
assume K s /K n = 2/3. The interparticle sliding is assumed to be governed by Coulomb’s friction law, which adopts
a sliding frictional coefficient µ = 0.5. The parameters used in the DEM simulations are listed in Table 3. Following
a common particle rotation assumption [51–53], each particle in a granular system is considered freely rotatable.
The rolling of a particle is directly governed by the inter-particle friction F τ and the force determined by the
interfacial tension F γ , and the angular velocity of particle is updated by the moments calculated from the F τ and
F γ applied to the particle surface. Considering rolling resistance between particles may help produce more realistic
behaviors, which can be employed to enhance the model in the future work.
The combined effects of capillary forces and the interparticle contact forces on the motion of the particles are
handled by the DEM solver, ESyS-particle [31], where the particle motion is governed by Newton’s second law.
The flow chart for modeling the unsaturated soil by using micro-mechanical methodology is shown in Fig. 8. In
Fig. 8, Nt is the current step of the simulation, N is the total steps of the simulation.
3. Numerical setup
A cubic packing of polydisperse spherical particles is employed here. A total of 9977 particles are randomly
generated inside a cubic box confined by six rigid frictionless walls. Fig. 9 presents the particle size distribution
(PSD) of the hypothetical soil, the particle sizes range from 11.7 µm to 34.5 µm. The hypothetical soil is a silt
according to the ISO 14688-1 [54] which grades silt particle sizes between 2 µm and 63 µm. After the desired
number of particles is generated in the cubic box, the packing is then compressed isotopically by two stages. In the
first stage, the confining pressure is gradually increased to the aimed confining pressure. To generate packings with
different initial densities, different interparticle frictional coefficients µ are employed. In specific, a high frictional
coefficient (µ = 1.0) and a low frictional coefficient (µ = 0.001) are employed to generate loose and dense samples,
respectively. When finishing the first stage, µ is set to be 0.5. The consolidation process is then continued a sufficient
number of iteration steps to reach quasi-static conditions. After the isotropic compression, the packings obtained
with different void ratios are ready for the next-step testing. The void ratios for the dense and loose samples are
0.66 and 0.73, respectively, at a confining stress of 10 kPa. A dense packing confined by six rigid frictionless walls
is shown in Fig. 9, in which the color of each particle represents its size. During the simulation, for achieving
quasi-static condition within a reasonable computational time in DEM modeling, the particle density was scaled up
by a factor of 106 as suggested by several researchers ([55]; [53,56]). The parameters used in the DEM simulations
are listed in Table 3.
The procedure to calculate the liquid–solid interfaces and especially the capillary force is very time-consuming
compared to the regular DEM calculation procedure. However, the skeleton deformation within a small amount of
time steps remains limited and do not cause tangible changes on liquid–solid interfaces and the capillary forces.
Thus, in this study, the liquid–solid interfaces are updated every 10,000-time steps, within this period, the increment
of axial strain is 0.2% during the triaxial shearing, the difference of capillary forces calculated between two adjacent
updates is negligible. In the desaturation test, the skeleton deformation and evolution of capillary forces are more
negligible within this period. Some other criteria like inertia number, alternatively, can be also selected to determine
the period for updating the liquid–solid interfaces.
12
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 8. Flow chart for modeling of unsaturated soil. The simulation begins with time step Nt = 0, and end up with Nt = N, N is the
specified total time steps.
A desaturation process (i.e., drying tests) is simulated on the dense sample. The initial suction is equal to zero
(s = pl − pg = 0), which stands for a fully saturated scenario (Sr = 1). The desaturation is controlled by increasing
suction progressively (specific steps can be found in Table 4). In addition to desaturation tests, a series of suctioncontrolled triaxial tests with different confining pressures (from 10 kPa to 400 kPa) and various suctions (from 0 kPa
to 100 kPa) are simulated on both dense and loose samples (see Table 3). The simulation results for desaturation
and triaxial tests are presented in Sections 4 and 5.
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
13
Fig. 9. Particle size distribution of the hypothetical soil. The particle sizes range from 11.7 µm to 34.5 µm lies within the range of silt
particle (2 µm to 63 µm.). The small illustration in Fig. 9 represents a dense packing confined by six rigid frictionless walls, in which the
color of each particle represents its size.
Table 4
Numerical tests on the dense and loose samples.
Numerical tests
Sample
Confining pressure (kPa)
Suction (kPa)
Desaturation tests
Dense
100
0, 10, 20, 30, 40, 50, 60, 70,
80, 90, 100, 200, 300, 400,
500, 600, 700, 800, 900
Triaxial tests
Dense
Loose
10, 50, 100, 200, 400
10, 50, 100, 200, 400
0, 20, 40, 60, 80, 100
0, 20, 40, 60, 80, 100
4. Computed behavior of soil in a desaturation process
4.1. Volume change
Fig. 10a and b present the volumetric strain and degree of saturation when suction increases by the proposed
method and the 2PFV-DEM scheme proposed by Yuan and Chareyre [26], respectively. In the proposed method,
water bridges (or liquid–solid interface) are generated between every two particles with the separate distance smaller
than 2r, regardless the connectedness of the liquid phase. Considering connectedness of the generated liquid–solid
interfaces may be helpful to model the hysteresis loops of the complete SWCC but will not be pursued here to
avoid excessive complication.
Since the proposed method (see Section 2) can simulate the liquid–solid interface only, the degrees of saturation
in Fig. 10a are calculated by the PM method [29,38], and further fitted by the van Genuchten (VG) model [57].
[
](1− n1 )
1
(15)
Sr =
1 + (αs)n
where α and n are fitting parameters. On the other hand, the 2PFV-DEM scheme can explicitly and directly model
the transport of two-phase fluids in the pore space, as well as the interactions between the solids and fluids. The
SWCC curve shown in Fig. 10a is different from that in Fig. 10b. The air entry value is 23 kPa in Fig. 10a
however 18 kPa in Fig. 10b, while the residual suction is 58 kPa in Fig. 10a however 22 kPa in Fig. 10b. This
difference between Fig. 10a and b is due to the less agreement prediction of the 2PFV-DEM scheme [26]. It has
been addressed by Yuan et al. [58] that the SWCC predicted by the 2PFV-DEM scheme was significantly biased
with the experiment results. The capillary forces are determined by the liquid–solid interfaces (see Eqs. (7) and
(10)). In the 2PFV-DEM scheme, the liquid–solid interfaces are indirectly predicted by modeling the distributions
of the two-phase fluids. Considering the SWCC predicted by the 2PFV-DEM scheme was significantly biased with
14
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 10. Volumetric strain and degree of saturation during desaturation process predicted by: (a) the proposed model; (b) the 2PFV-DEM
scheme [26]. The degrees of saturation in (a) are calculated by the PM method [29,38], and is further fitted by the van Genuchten (VG)
model [57].
the experiment results, the produced liquid–solid interfaces by the 2PFV-DEM scheme can be also different from the
real conditions, which may result in inaccurate capillary forces and some computational error. However, in this study,
the liquid–solid interfaces are directly predicted by the proposed method (thus, the SWCC is not compulsory for the
micro-mechanical analysis) and agree well with the PM method (see Figs. 4 and 5). Moreover, the corresponding
SWCC calculated by the PM method match well with the experimental results [29,34,36–38].
Three different saturation regimes namely pendular, funicular and capillary regimes are determined based on the
SWCC curve [59]. As illustrated in Fig. 10a, within the capillary regime (s < 23 kPa and Sr > 91%), the volumetric
strain computed by the proposed method increases with suction increase. Within the funicular regime (23 kPa <
s < 58 kPa and 5% < Sr < 91%), the volumetric strain firstly increases with suction increase, then maximizes at
suction of 40 kPa around, and decreases with suction increase afterwards. Within the pendular regime (s > 58 kPa
and Sr < 5%), the volumetric strain firstly decreases with suction increase, and finally reaches a residual value
at s = 100 kPa. As shown in Fig. 10b, when s < 30 kPa, the volumetric strain predicted by the 2PFV-DEM
scheme is firstly increased and then decreased with the increase of suction similar to Fig. 10a. However, when s >
30 kPa, the volumetric strain further increases with suction increase which is unreasonable and also disagrees with
the general experimental observations [60]. The volumetric evolution of the sample during desaturation test reflects
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
15
the evolution of capillary cohesion effect with the change of suction. Experimental observations by Schubert [61]
and Lu et al. [62] indicate that, with the increase of suction, the capillary cohesion sharply increases in the capillary
regime, and maximizes in the capillary or funicular regime, decreases in the pendular regime. The tendency of the
evolution of the volumetric strain obtained by the micro-mechanical model proposed by this study (see Fig. 10a)
agrees with the experimental observations.
4.2. Average contact stress and effective stress parameter
Christoffersen et al. [63] have proposed the definition of stress tensor based on micro-mechanical quantities for
fully saturated and dry systems as follows.
1 ∑ c c
σicj =
f d
(16)
V c∈N i j
c
where V is the total volume of the assembly; Nc is the total number of contacts; f ic is the contact force at a contact
and d cj is the branch vector joining the centers of two contacting particles. Term σicj in Eq. (16) represents the
average contact stress within the solid skeleton network in a statistical manner, which is similar to the average
skeleton stress [64]. For fully saturated or dry systems, it has been validated to be equal to Terzaghi’s effective
stress ([51,52,65,66]).
For unsaturated soils, Bishop and Donald [67] proposed an effective stress equation, in which the suction was
modified by a factor χ as below.
σ ′ = (σ − pg ) + χ( pg − pl )
(17)
where χ is an empirical parameter (i.e., effective stress parameter) with values between 0 and 1. The terms (σ − pg )
and ( pg − pl ) are also known as the net stress (σ net ) and suction (s), respectively. The expression of χ is still
questionable. Many attempts have been made to quantify χ both theoretically and experimentally [68–76]. It is
generally acknowledged that the effective stress parameter χ to be a function of the degree of saturation Sr , and this
relationship has been widely observed in experimental tests [68,73,75–78]. One of the widely accepted relationships
between χ and Sr is χ = Sr [71]. The χ − Sr relationship is generally independent on specific soil properties because
the influence of soil types, soil structures, stress conditions and soil-water characteristics are generally included in
the variation of Sr . This form has been widely used in constitutive models for many types of unsaturated soil
[79–84] and in engineer practices [85,86], and was the basis of many other equations to quantify χ [68,75,77,87].
Considering most of the experimental data for χ − Sr relationship is based on non-plastic soils like various silts
[67,68,88–91]. It is reasonable to compare the hypothetical soil, which is a silt according to the ISO 14688-1 [54],
with two different silts tested in the laboratory.
Following in Scholtès et al. [13] and Yuan and Chareyre [26], the average contact stress (σicj ) is assumed to
be equivalent to the effective stress for unsaturated systems. Therefore, we define the effective stress parameter by
combining Eqs. (16) and (17) as follows.
1 c
σ − σ net
p ′ − ( p − pg )
= 3 ii
(18)
s
s
The comparisons of χ calculated by Eq. (18) with that obtained from experimental results are presented in Fig. 11a.
σ net is the confining pressure (i.e. 100 kPa, see Table 4); σiic is calculated by Eq. (16), in which f ic and dic at every
contact are resulting from the joint effects of capillary force and external pressure (see Eq. (12)). Sr are estimated
from the SWCC shown in Fig. 10a. The typical experimental results are collected from Bishop and Donald [67] on
Braehead silt and Bishop and Blight [88] on Vaich moraine. As shown in Fig. 11a, the calculated effective stress
parameter results match very well with the experiments at various degrees of saturation. The computed results by
using the 2PFV-DEM scheme developed by Yuan and Chareyre [26] are presented in Fig. 11b. The relation between
χ and Sr is almost linear. The predicted χ by the 2PFV-DEM scheme lies close to the 1:1 line (χ = Sr ) and is
slightly below the 1:1 line when Sr > 62% and Sr < 17% and slightly above the 1:1 line when 17% < Sr < 62%.
However, the χ predicted by Yuan and Chareyre [26] are much lower than that of experimental results at high
level of saturation, and no prediction is reported when the degree of saturation is lower about 15%. The reason
for the less agreement in the 2PFV-DEM scheme may be the liquid–solid interfaces were not reasonably predicted,
χ=
16
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 11. Comparisons between the experimental results with effective stress parameter χ predicted by: (a) the proposed model; and (b)
2PFV-DEM scheme [26]. The typical experimental results are collected from Bishop and Donald [67] on Braehead silt and Bishop and
Blight [88] on Vaich moraine.
which is result from both the less agreement prediction of SWCC (see Fig. 10b) and the shape of the liquid–solid
interfaces. In the 2PFV-DEM scheme [26], water only exists in the defined regular triangulations, plane gas–liquid
interfaces are generated, which cannot exist in real thermal-dynamic system, thus the capillary force cannot be
evaluated accurately. Furthermore, as no capillary bridges can be modeled, this method cannot be applied to the
low saturation regime where only capillary bridge water exists.
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
17
Fig. 12. Evolution of liquid–solid interfaces (a, b and c) and configurations (d, e, and f) and angular distributions (g, h and i) of capillary
forces under different suctions during desaturation. The arrows in (d, e and f) demonstrate the total capillary forces acting on particles and
are added on the centers of particles. The white meshes in (g, h and i) are fitted by Eq. (19). Evolutions are for (a, d and g) s = 20 kPa,
(b, e and h) s = 50 kPa, (c, f and i) s = 100 kPa.
4.3. Variation of liquid–solid interfaces and capillary forces
Fig. 12 presents the evolution of liquid–solid interfaces and capillary forces during the desaturation process.
Fig. 12a, b and c present the diagrams of the liquid–solid interfaces under suctions of 20 kPa, 50 kPa and 100 kPa,
respectively. The liquid–solid interfaces in Fig. 12a almost cover all the surfaces of the solid particles as the suction
is about the air entry value (see Fig. 10). While in Fig. 12b and c, as the degree of saturation decreases with suction
increase, the area of the liquid–solid interfaces also decreases correspondingly. Fig. 12d, e and f show the capillary
forces acting on particles under suctions of 20 kPa, 50 kPa and 100 kPa, respectively. Capillary force arrows with
different sizes and directions are employed to demonstrate the capillary forces acting on particles. And the arrows are
added on the centers of particles. As the liquid–solid interfaces cover most of the solid particle surfaces (Fig. 12a),
only 1980 capillary force arrows are observed when s = 20 kPa (as shown in Fig. 12d). While 9977 capillary force
arrows are observed when s = 50 or 100 kPa, as shown in Fig. 12e and f, respectively. The capillary force arrows
shown in Fig. 12e (s = 50 kPa, the funicular regime) are thickest and longest among three suctions (see Fig. 12d,
e and f). Fig. 12g, h and i illustrate angular distribution E (θ, β) of the orientations of the capillary forces under
three different suctions. θ and β are zenith angle and azimuth angle in the spherical coordinate system. E (θ, β)
is defined as the portion of capillary forces falling within an angular interval ∆θ and ∆β. Following Chang and
18
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 13. Frequency distributions of the magnitude of the capillary forces during desaturation predicted by: (a) the proposed model; and (b)
2PFV-DEM scheme [26]. The frequency distributions of the same degrees of saturation (Sr = 20%, 60% and 90%) are plotted in (a) and
(b). The frequencies are calculated with an interval of 1 µN.
Misra [92], the angular distributions are approximated by
]
1 [
a20
E (θ, β) =
1+
(19)
(3cos2θ + 1) + 3 sin2 θ (a22 cos2β + b22 sin2β)
4π
4
where a20 and a22 are the anisotropy coefficients; b22 = 0 for orthotropic material. The angular distributions of
the capillary forces for the desaturation process are almost isotropic (see Fig. 12g, h and i), since the values of
orientation anisotropy coefficient a20 are very small (a20 = −0.38, −0.28 and −0.026 when s = 20, 50, 100 kPa,
respectively).
Fig. 13a and b show the frequency distribution of the magnitude of the capillary forces (F c = F s + F γ ) under
different suctions in desaturation process by using the proposed model and the 2PFV-DEM scheme, respectively.
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
19
Fig. 14. Evolution of deviator stress (a and c) and void ratio (b and d) with axial strain during triaxial test with confining pressure of
10 kPa under various suctions. The evolutions are for (a and b) dense sample, (c and d) loose sample.
The frequencies are calculated with an interval of 1 µN (=1 × 10−6 N) in Fig. 13a and b. When using the proposed
method (Fig. 13a), the mean capillary force is 0.11 µN at s = 20 kPa (Sr = 90%), 1.67 µN at s = 30 kPa (Sr =
60%) and 8.94 µN at s = 50 kPa (Sr = 20%). When modeling by the 2PFV-DEM (Fig. 13b), however, the mean
capillary force is 1.65 µN at s = 18 kPa (Sr = 90%), 5.19 µN at s = 19 kPa (Sr = 60%) and 6.46 µN at s = 23
kPa (Sr = 20%).
5. Computed behavior of saturated and unsaturated soil in triaxial shearing
5.1. Stress–strain behavior for different packings under shearing
Fig. 14 presents the evolution of the deviator stress and void ratio with the development of the axial strain during
triaxial tests under confining pressure of 10 kPa for both dense and loose samples. As shown in Fig. 14a, postpeak strain softening behavior is observed in general for the dense sample, while the loose sample (see Fig. 14c)
shows strain hardening behavior with no clear peak strength. Dense samples (see Fig. 14b) in general show more
distinct shear dilation than the loose samples (see Fig. 14d) under constant suctions. For both dense and loose
samples, when suction increases from 0 kPa to 40 kPa, the peak and residual strengths increase as well as the strain
softening, and shear dilation become more distinct. When suction increases from 40 kPa to 80 kPa, the peak and
residual strengths decrease. The strain softening and shear dilation become less distinct with increase of suction.
The stress–strain curves are almost identical when suction is equal to 80 kPa and 100 kPa (pendular regime). It
is clear from the numerical simulation presented here that the capillary cohesion effect reflected by the average
contact stress is maximized in the funicular regime, which lead to significant peak strength, strain softening and
shear dilation [61,62].
Fig. 15 shows the calculated stress–strain relationships for dense/loose saturated/unsaturated samples obtained
from triaxial tests with confining pressures (σ3 ) of 50 kPa, 100 kPa, 200 kPa and 400 kPa. As shown in Fig. 15, at
various confining pressures, the dense samples show distinct post-peak strain softening while loose samples only
show strain hardening. Along with the increase of confining pressure, the axial strain at the peak deviator stress
is increasing for dense sample. The axial strain at the peak is about 7% when σ3 is 400 kPa which is greater
than 2% when σ3 is 10 kPa (see Fig. 14). More importantly, the effect of suction becomes less distinct along with
20
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 15. Evolution of deviator stress versus axial strain during triaxial tests under various suctions. Evolutions for dense samples are for
(a) σ3 = 50 kPa, (c) σ3 = 100 kPa, (e) σ3 = 200 kPa, (g) σ3 = 400 kPa. Evolutions for loose samples are for (b) σ3 = 50 kPa, (d)
σ3 = 100 kPa, (f) σ3 = 200 kPa, (h) σ3 = 400 kPa.
the increase of confining pressures, for dense and loose samples. As shown in Fig. 15, all the stress–strain curves
with different suctions merge together when confining pressure increases. However, the tendencies of the change
of deviator stress with suctions are still similar under different confining pressures.
5.2. Shear strength
As shown in Fig. 16, the peak and residual deviator stresses collected from saturated and unsaturated triaxial
tests (see Figs. 14 and 15) are plotted against the average contact stress calculated by Eq. (16). The peak stresses
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
21
Fig. 16. Strength envelopes illustrated in the average contact stress space. The residual deviator stresses include both the dense and loose
cases.
are chosen as the largest stress from the curves in Figs. 14 and 15. An excellent linear relationship between the
deviator stress and the average contact stress can be regressed with the values of R2 greater than 0.99. Therefore,
in the space of deviator stress and average contact stress, the strength behavior for unsaturated soils can be well
described by using the Mohr–Coulomb strength criterion [93]. As shown in Fig. 16, the peak strength envelopes
for dense sample lie above that for loose sample with a greater slope (M = 1.243 for dense sample and M = 0.882
for loose sample). While the residual envelops for both samples are almost the same (M = 0.736).
5.3. Variation of liquid–solid interfaces and capillary forces
Fig. 17 presents the change of the liquid–solid interfaces and capillary forces for a dense sample at triaxial strains
of 0 (beginning of the shearing), 2% (corresponding to peak stress) and 20% (corresponding to residual stress), when
s = 40 kPa and σ = 10 kPa. The change of solid structure leads to the redistribution of the liquid–solid interfaces
(see Fig. 17a, b and c) and further to the evolution of the capillary forces (see Fig. 17d, e and f). As shown in
Fig. 17d, e and f, the capillary arrows become thicker and longer with the increase of axial strain, which means the
capillary forces are enhanced during the shearing process. Fig. 18 shows the frequency distribution of the magnitude
of the capillary forces during shearing. The frequencies are calculated with an interval of 1 µN. As shown in Fig. 18,
the magnitude of the capillary force generally increased with shearing process, with the mean force increases from
4.16 µN at the beginning of the shearing to 6.14 µN at the end of the shearing. During the shearing, due to the
dilation of the samples, the degree of saturation decreases from 40% at εa = 0% to 25% at εa = 20% and to 15%
at εa = 20%. Therefore, it is safe to say that the capillary forces are enhanced with the decreasing of degree of
saturation.
The orientation of the capillary forces also changes with the increase of axial strain (see Fig. 17g, h and i). At the
beginning of the shearing, an approximately homogeneous orientation distribution of capillary forces is observed (the
anisotropic coefficient a20 = −0.27 in Fig. 17g). Along the shearing, however, the orientation distribution becomes
more anisotropic with increasing axial strain, there are more capillary forces oriented in horizontal direction than
that in vertical direction, with the anisotropic coefficient a20 decreases to −0.41 at εa = 2% (Fig. 17h) and to −0.52
at εa = 20% (Fig. 17i). This phenomenon that structure deformation can induce anisotropic distribution of capillary
force is also reported by Scholtès et al. [13]. This microscopic observation challenges a widely used assumption in
unsaturated soil mechanics that suction can be treated as an isotropic stress variable.
22
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 17. Evolution of liquid–solid interfaces (a, b and c) and configurations of capillary forces (d, e and f) and angular distributions of
capillary forces in three-dimensions (g, h and i) during the shearing under suction of 40 kPa and confining pressure of 10 kPa. The arrows
in (d, e and f) demonstrate the total capillary forces acting on particles and are added on the centers of particles. The white meshes in
(g, h and i) are fitted by Eq. (19). Evolutions are for (a, d and g) εa = 0% (beginning), (b, e and h) εa = 2% (peak stage), (c, f and i)
εa = 20% (residual stage).
The evolutions of both the magnitude and the distribution of capillary forces can be analyzed by using the
volumetric deformation and the associated particle rearrangement. During the shearing, in the load direction (y),
the number of contacts increases and the distance between particles decrease, leading to the increase of the area of
liquid–solid interfaces in the load direction, and further leading to the decrease of total capillary forces in the load
direction. While in the lateral directions (x and z), the distance between particles increases, leading to the decrease of
the area and increase of the number of liquid–solid interfaces in the lateral directions, and further to the significantly
increase of total capillary forces in the lateral directions. The magnitude of the increase in total capillary forces in
the lateral directions (x and z) is larger than the magnitude of the decrease in the load direction (y). Hence, with the
increase of axial strain, the magnitudes of capillary forces are enhanced, and the orientation distribution becomes
more anisotropic with most of the capillary forces orient in lateral directions. The same phenomenon regarding to
the evolutions of the magnitude and the distribution of capillary forces may be also observed in constant water
content test, since a similar volumetric deformation and particle rearrangement pattern can be also observed in
constant water content test.
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
23
Fig. 18. Frequency distributions of the magnitude of the capillary forces at εa = 0% (beginning), εa = 2% (peak stage) and εa = 20%
(residual stage) under suction of 40 kPa and confining pressure of 10 kPa. The frequencies are calculated with an interval of 1 µN.
5.4. The variation of contact force network and coordination number
Fig. 19 shows the change of contact force networks under different suctions (s = 0 and 40 kPa) at different
strains (0%, 2% and 20%) of dense sample under a confining pressure of 10 kPa. Fig. 19a–c represent the contact
force networks for s = 0 and Fig. 19d–f represent those for s = 40 kPa at different shearing stages (initial, peak and
residual states). Each colorful column in figures connects the centroids of two contacted particles, and its thickness
denotes the magnitude of the contact force. Due to the capillary cohesion effect, stronger contact force networks
can be observed in Fig. 19d–f compared to Fig. 19a–c at given strains. Fig. 19g–l illustrate the angular distributions
of the contact forces, and Eq. (19) is used to fit the distribution. At the beginning of the shearing (see Fig. 19a, d, g
and j), contact forces are randomly oriented and the overall force networks are generally isotropic (the anisotropic
coefficient a20 = 0.18 for s = 0 kPa and a20 = 0.19 for s = 40 kPa). When the axial strain increases from 0 to 2%,
the initially randomly oriented forces gradually change their preferential direction to align with the direction of the
vertical force (see Fig. 19b, e, h and k), with a20 increasing to 0.68 for s = 0 kPa and to 0.71 for s = 40 kPa. As
shown in Fig. 19c, f, i and l, the contact force networks become a bit less anisotropic when the axial strain increases
from peak stage (2%) to residual stage (20%), with a20 slightly decreasing from 0.68 to 0.45 for s = 0 kPa and
from 0.71 to 0.47 for s = 40 kPa. It is observed in Fig. 19g–l that capillary force generally enhances the anisotropic
nature of the contact force network (for example, at peak strength, a20 = 0.68 when s = 0 and a20 = 0.71 when
s = 40 kPa), but this effect is not significant, which is also observed in [22].
Fig. 20 presents the frequency distributions of the magnitude of the contact forces during different shearing
stages under different suctions. The frequencies are calculated with an interval of 1 µN. The suction significantly
increases the magnitudes of the contact forces. When s = 0 kPa, the mean contact force is 7.49 µN at εa = 0%,
20.81 µN at εa = 2% and 23.04 µN at εa = 20%. When s = 40 kPa, the mean contact force is 24.52 µN at εa =
0%, 48.71 µN at εa = 2% and 45.63 µN at εa = 20%. The contact forces for s = 40 kPa are much greater than
those for s = 0 kPa at given shearing strains. Noted that when s = 40 kPa, the contact forces at εa = 20% are
less than that at εa = 2% in contrast to the case of s = 0 kPa. This is due to the distinct strain softening effect for
s = 40 kPa, while there is no strain softening for s = 0 kPa (see Fig. 14a). It is difficult to separate the mechanical
effects from capillary effects regarding the contribution to the contact forces especially in funicular and capillary
regimes. In funicular and capillary regimes, complex liquid clusters generate among more than two particles, in
this case, several interparticle contact points can coexist in one liquid–solid interface. As shown in Fig. 7 (see the
liquid–solid interface i), there may be several contact forces and only one capillary force acting on the liquid–solid
interface.
24
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. 19. Evolution of networks (a–f) and angular distributions (g–l) of the contact forces of dense sample under a confining pressure of
10 kPa and different suctions. Each colorful column in (a–f) connects the centroids of two contacted particles, and its thickness denotes the
magnitude of the contact force. The white meshes in (g–l) are fitted by Eq. (19). The evolutions in (a–c and g–i) correspond to s = 0 kPa.
The evolutions in (d–f and j–l) correspond to s = 40 kPa. Evolutions are for (a, d, g and j) εa = 0% (beginning), (b, e, h and k) εa = 2%
(peak stage), (c, f, i and l) εa = 20% (residual stage).
c
Fig. 21 shows the evolution in coordination number Z (= 2N
, where Nc is the total number of contacts and N p is
Np
the particle number) with suctions under various confining pressures at the residual state (εa = 20%) of the triaxial
tests. Generally, the coordination numbers for the dense samples and for the loose samples are very similar, which
implies the density does not affect coordination number significantly at the residual state. As shown in Fig. 21, the
coordination number increases with the confining pressure. Under the same confining pressure, the coordination
number increases when suction is less than 60 kPa, and then decreases. The variation magnitude of coordination
number with suction change becomes weak when the confining pressure increases. The coordination number Z
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
25
Fig. 20. Frequency distributions of the magnitude of the contact forces at εa = 0% (beginning), εa = 2% (peak stage) and εa = 20%
(residual stage) under suction of 0 and 40 kPa and confining pressure of 10 kPa. The frequencies are calculated with an interval of 1
µN. The solid lines denote the frequency distributions of s = 0 kPa, corresponding to Fig. 19a–c. The dash lines denote the frequency
distributions of s = 40 kPa, corresponding to Fig. 19d–f.
Fig. 21. Evolution in coordination number with suctions under various confining pressures at the residual state of the numerical triaxial
tests. The coordination number is calculated as Z = 2Nc /N p , where Nc is the total number of contacts and N p is the particle number.
increases rapidly during shear until becoming approximately constant, and the rapid increase in Z is associated
with volumetric contraction, then Z becomes essentially constant during volumetric expansion [22]. As shown in
Fig. 14b and d, for both dense and loose samples, when suction increases from 0 kPa to 40 kPa, the volumetric
contraction increases. When suction increases from 40 kPa to 100 kPa, the volumetric contraction decreases. The
variation of Z with suction is generally associated with the variation of volumetric contraction. The variation of
coordination number with suction reflects another aspect of micro-mechanism of suction effect on unsaturated soil
26
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
behavior, in addition to magnitudes of contact forces. In other words, the suction not only increases the interparticle
contact force (see Figs. 19 and 20) but also increase the coordination number of the specimen in the shearing process,
both of which enhance the shear resistance of soils [23].
6. Conclusions
This study develops a novel three-dimensional numerical model to simulate the macro- and micro-mechanical
behavior of unsaturated soils. A pore-scale numerical method is proposed to directly simulate the liquid–solid
interfaces. A discretization of the particle surface using Fibonacci-Lattice is introduced to calculate the capillary
forces from the complicated liquid–solid interfaces. The joint effects of capillary forces and interparticle contact
forces on the motion of the particles are handled by the DEM solver. This micro-mechanical model is then employed
to reproduce the micro- and macroscopic responses of the unsaturated soil in desaturation and triaxial tests. Some
major conclusions can be drawn as follows.
(1) The proposed model can reproduce the micro- and macro-mechanical responses of both dense and loose
unsaturated assemblies under full range of saturation in desaturation test and in triaxial compression tests.
Compared to the 2PFV-DEM scheme [26], the proposed model predicts more reasonable results. The effective
stress parameter predicted by the proposed micro-mechanical model agrees well with experimental results.
(2) It is observed that in both simulations of desaturation test and triaxial tests, the capillary forces firstly increase
with suction at higher levels of saturation (in capillary regime), then maximizes in funicular regime, finally
decreases in lower levels of saturation (pendular regime).
(3) Structure deformation can increase the magnitude of the capillary forces and induce and/or enhance
anisotropic distribution of capillary forces, since anisotropy can naturally appear depending on the structure
of the solid skeleton.
(4) During the shearing, the capillary forces will not only increase the interparticle contact force but also increase
the coordination number of the specimen. Both enhance the shear resistance of unsaturated soils.
Finally, it is noted that the method proposed in this paper is applicable to granular soils, for which interparticle
interactions are purely mechanical, and liquid/solid interactions are limited to capillary effects. In addition, like the
most existing DEM models for unsaturated system [10–21,26], the proposed method is suction controlled, therefore
it cannot be directly applied for constant water content tests. The method presented in the paper is still in its early
development stage; however, the proposed method demonstrates its capacity in simulating the desaturation and
triaxial tests of unsaturated soils. The enhancement to the proposed model is expected in the future work.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could
have appeared to influence the work reported in this paper.
Acknowledgments
The financial supports from Australian Research Council (LP160100649; DP200100549; IH180100010) are
greatly appreciated.
Appendix A. Calculate the liquid–solid interfaces
In the funicular and capillary regimes (Fig. A.1a), the principle curvature radius is assumed to be r = r1 = r2 .
According to Eq. (1), r is calculated by
r=
2s
γ
(A.1)
27
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
Fig. A.1. Morphology of spherical caps forming the liquid–solid interface between particle A and B (a) based on the hemisphere assumption
and (b) based on Laplace equation.
Thus, the polar angles of the spherical caps forming the liquid–solid interface on the particles are calculated by
(
)
(r + R B )2 + d 2 − (r + R A )2
θ A = acos
(A.2)
2d (r + R B )
(
)
(r + R A )2 + d 2 − (r + R B )2
θ B = acos
(A.3)
2d (r + R A )
where d is the distance between the centroid of particle A and B. And the orientations of the liquid–solid interfaces
are determined by
⎛ xB − xA ⎞
d
⎜
⎟
⎜y − y ⎟
⎜ B
A⎟
nA = ⎜
⎟
⎟
⎜
d
⎠
⎝
zB − zA
d
n B = −n A
(A.4)
(A.5)
Within the framework of Young–Laplace theory (Eq. (2)), in the pendular regime (Fig. A.1b), the geometry of
the liquid bridge is related to the interparticle separation distance, suction, and the ratio of particle radii. In this
paper, we propose a semi-analytical framework where the toroidal approximation is implemented, and the geometry
prediction achieved through an iterative process. Based on the toroidal approximation [1], the relationship between
r1 and r2 is given by,
( √
)
1
r2 = −
(A.6)
(2r1 + R A + R B + d) (R B + d − R A ) (R A + d − R B ) (2r1 + R A + R B − d) − r1
2d
Since the suction s > 0, according to Eqs. (2) and (A.6), we obtain the minimum and maximum value of the r1
⎡
)(
))
((
R
RB
RB
A
2
2
⎣
)
r1max = ((
2d
−
1
−
d
+
1
−
+
)2
RA
RA
RB
2
2 1 − R A + 3d

⎤
)(
((
)(
)
)2 ((
)2
)
2

R
R
R
R
B
B
B
B
√ 1−
d2 + 1 −
− 2d 2 −
1−
+ 3d 2
d2 + 1 −
− 4d 2 ⎦
RA
RA
RA
RA
(A.7)
28
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
⎡
r1min =
2
((
RA
)2
1 − RR BA
) ⎣2d 2 −
+ 3d 2
((
)(
))
RB
RB
2
1−
d +1−
−
RA
RA

⎤
)(
((
)(
)
)2 ((
)2
)
2

RB
RB
RB
√ 1 − RB
d2 + 1 −
− 2d 2 −
1−
+ 3d 2
d2 + 1 −
− 4d 2 ⎦
RA
RA
RA
RA
(A.8)
For a given configuration of the particle pair and suction, the main steps of the iterative approach for the calculation
of the r1 and r2 are summarized below:
The determined r1 is applied in Eqs. (A.2) and (A.3) to calculate the polar angles of the spherical caps forming
the liquid–solid interface of the isolated liquid bridges.
Appendix B. Determine the Fibonacci-Lattice points inside the liquid–solid interfaces and the points
located on the gas–liquid–solid lines
As described in Section 2.1, the liquid–solid interfaces are consisting of multiple isolated or overlapped spherical
caps, in which the isolated caps representing the liquid–solid interfaces of isolated liquid bridges which often
formed in pendular regime, and the overlapped caps representing the complex interfaces which formed in funicular
regime. Accurately measuring the liquid–solid interfaces and the gas–liquid–solid lines is fundamental important
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
29
Fig. B.1. Fibonacci-Lattice points inside the liquid–solid interface (marked as cyan)
√ and located on the tri-phase line (marked as read). The
point inside the liquid–solid interface is determined by αk j < θkc and dk j > 0.5 A. The point located on the tri-phase line is determined
√
by dki ≤ 0.5 A.
to calculate the capillary forces on particles. By using the Fibonacci-Lattice points, this can be easily achieved.
This starts by finding which points of the lattice are inside the liquid–solid interfaces and lied on the gas–liquid–
solid line, see Fig. B.1. For a cap k with a normal n ck and a polar angle θkc , and a point i with a normal n i , if
n c ·n
the angle between n ck and n i , αki = cos−1 ( |nck||ni | ) is smaller than θkc , and the distance between the point and
i
k
√
A is the nominal distance of two neighboring points),
the outline of the cap, dki , is bigger than 0.5l (l =
i.e.
√
(
)
dki = R θkc − αki > 0.5 A
(B.1)
2
where R is the radius of the particle, and A = 4πPR is the area of a sub-divided surface. Then point i is inside the
cap. If the distance between the point and the outline of the cap, dki , is smaller than 0.5l, i.e.
√
⏐
⏐
dki = R ⏐θkc − αki ⏐ ≤ 0.5 A
(B.2)
Then the point i is assumed to be lied on the outline of the cap k. The direction of the surface tension induced
force, τ i is tangent to n i and perpendicular to the tri-phase line, thus
|n i ||n ck | c
nk − ni
n i ·n ck
⏐
τi = ⏐
⏐ |n i ||nck | c
⏐
⏐ n i ·nc n k − n i ⏐
(B.3)
k
For a complex liquid–solid interface consisting of multiple overlapping caps, a lattice point may be covered by
several caps; and only the parts of outline of the caps outside the overlapping area are the segment of the tri-phase
line. To find all the points inside the liquid–solid interfaces and those located on the tri-phase lines, a procedure
is introduced in this appendix. The caps are firstly determined during the simulation. To illustrate the procedure,
assuming there are total N caps, 1, 2, ···, N , on particle m. For point i in the Fibonacci-Lattice, Eq. (B.1) is firstly
applied to caps 1, 2, ···, N . If Eq. (B.1) is satisfied to certain cap, for example cap k, then point i is inside the
liquid–solid interfaces. If Eq. (B.1) is not satisfied to any cap, then check out whether Eq. (B.2) is satisfied. If Eq.
(B.2) is satisfied, point i is located on the tri-phase line. Otherwise, point i is outside the liquid–solid interfaces.
This procedure is also summarized as pseudocode below:
30
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
References
[1] R.A. Fisher, On the capillary forces in an ideal soil, J. Agric. Sci. 16 (1926) 13.
[2] G. Lian, C. Thornton, M.J. Adams, A theoretical study of the liquid bridge forces between two rigid spherical bodies, J. Colloid
Interface Sci. 161 (1993) 138–147.
[3] K.H. Roscoe, A.N. Schofield, A. Thurairajah, Yielding of clay in states wetter than critical, Geotechnique 13 (1963) 211–240.
[4] Y.P. Yao, W. Hou, A.N. Zhou, UH model: three-dimensional unified hardening model for overconsolidated clays, Geotechnique 59
(2009) 451–469.
[5] A. Gens, Soil–environment interactions in geotechnical engineering, Géotechnique 60 (2010) 3–74.
[6] D. Sheng, Review of fundamental principles in modelling unsaturated soil behaviour, Comput. Geotech. 38 (2011) 757–776.
[7] S.K. Vanapalli, D.G. Fredlund, D.E. Pufahl, A.W. Clifton, Model for the prediction of shear strength with respect to soil suction, Can.
Geotech. J. 33 (1996) 379–392.
[8] P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies, Géotechnique 29 (1979) 47–65.
[9] J. Lin, W. Wu, A comparative study between DEM and micropolar hypoplasticity, Powder Technol. 293 (2016) 121–129.
[10] J. Duriez, R. Wan, Contact angle mechanical influence in wet granular soils, Acta Geotech. 12 (2017) 67–83.
[11] V. Richefeu, M.S. El Youssoufi, F. Radjai, Shear strength properties of wet granular materials, Phys. Rev. E 73 (2006) 051304.
[12] L. Scholtès, B. Chareyre, F. Nicot, F. Darve, Micromechanics of granular materials with capillary effects, Internat. J. Engrg. Sci. 47
(2009) 64–75.
[13] L. Scholtès, P.Y. Hicher, F. Nicot, B. Chareyre, F. Darve, On the capillary stress tensor in wet granular materials, Int. J. Numer. Anal.
Methods Geomech. 33 (2009) 1289–1313.
[14] F. Soulié, F. Cherblanc, M.S. El Youssoufi, C. Saix, Influence of liquid bridges on the mechanical behaviour of polydisperse granular
materials, Int. J. Numer. Anal. Methods Geomech. 30 (2006) 213–228.
[15] J.-P. Wang, X. Li, H.-S. Yu, A micro–macro investigation of the capillary strengthening effect in wet granular materials, Acta Geotech.
13 (2017) 513–533.
[16] R. Asadi, A.A. Mirghasemi, Numerical investigation of particle shape on mechanical behaviour of unsaturated granular soils using
elliptical particles, Adv. Powder Technol. (2018).
[17] T. Gröger, U. Tüzün, D.M. Heyes, Modelling and measuring of cohesion in wet granular materials, Powder Technol. 133 (2003)
203–215.
[18] Z. Shen, M.J. Jiang, C. Thornton, Shear strength of unsaturated granular soils: three-dimensional discrete element analyses, Granul.
Matter 18 (2016).
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
31
[19] V.D. Than, S. Khamseh, A.M. Tang, J.M. Pereira, F. Chevoir, J.N. Roux, Basic mechanical properties of wet granular materials: a
DEM study, J. Eng. Mech. 143 (1) (2016) C4016001.
[20] S.H. Liu, D.A. Sun, Simulating the collapse of unsaturated soil by DEM, Int. J. Numer. Anal. Methods Geomech. 26 (2002) 633–646.
[21] B.S. Kim, S.W. Park, S. Kato, DEM simulation of collapse behaviours of unsaturated granular materials under general stress states,
Comput. Geotech. 42 (2012) 52–61.
[22] T. Li, M.J. Jiang, C. Thornton, Three-dimensional discrete element analysis of triaxial tests and wetting tests on unsaturated compacted
silt, Comput. Geotech. 97 (2018) 90–102.
[23] M.J. Jiang, S. Leroueil, J.M. Konrad, Insight into shear strength functions of unsaturated granulates by DEM analyses, Comput.
Geotech. 31 (2004) 473–489.
[24] K. Melnikov, R. Mani, F.K. Wittel, M. Thielmann, H.J. Herrmann, Grain-scale modeling of arbitrary fluid saturation in random packings,
Phys. Rev. E 92 (2015) 022206.
[25] K. Melnikov, F.K. Wittel, H.J. Herrmann, Micro-mechanical failure analysis of wet granular matter, Acta Geotech. 11 (2016) 539–548.
[26] C. Yuan, B. Chareyre, A pore-scale method for hydromechanical coupling in deformable granular media, Comput. Methods Appl.
Mech. Engrg. 318 (2017) 1066–1079.
[27] C. Yuan, B. Chareyre, F. Darve, Deformation and stresses upon drainage of an idealized granular material, Acta Geotech. 13 (2017)
961–972.
[28] J. Duriez, R. Wan, M. Pouragha, F. Darve, Revisiting the existence of an effective stress for wet granular soils with micromechanics,
Int. J. Numer. Anal. Methods Geomech 42 (2018) 959–978.
[29] M. Hilpert, C.T. Miller, Pore-morphology-based simulation of drainage in totally wetting porous media, Adv. Water Resour. 24 (2001)
243–255.
[30] Á. González, Measurement of areas on a sphere using fibonacci and latitude–longitude lattices, Math. Geosci. 42 (2009) 49–64.
[31] Y. Wang, F. Alonso-Marroquin, A finite deformation method for discrete modeling: particle rotation and parameter calibration, Granul.
Matter 11 (2009) 331–343.
[32] S.M. Iveson, J.D. Litster, K. Hapgood, B.J. Ennis, Nucleation, growth and breakage phenomena in agitated wet granulation processes:
a review, Powder Technol. 117 (2001) 3–39.
[33] N. Mitarai, F. Nori, Wet granular materials, Adv. Phys. 55 (2006) 1–45.
[34] T. Sweijen, H. Aslannejad, S.M. Hassanizadeh, Capillary pressure–saturation relationships for porous granular materials: Pore
morphology method vs. pore unit assembly method, Adv. Water Resour. 107 (2017) 22–31.
[35] V.P. Schulz, J.r. Becker, A. Wiegmann, P.P. Mukherjee, C.-Y. Wang, Modeling of two-phase behavior in the gas diffusion medium of
PEFCs via full morphology approach, J. Electrochem. Soc. 154 (2007) B419.
[36] M.L. Turner, L. Knüfing, C.H. Arns, A. Sakellariou, T.J. Senden, A.P. Sheppard, R.M. Sok, A. Limaye, W.V. Pinczewski, M.A.
Knackstedt, Three-dimensional imaging of multiphase flow in porous media, Physica A 339 (2004) 166–172.
[37] B. Ahrenholz, J. Tölke, P. Lehmann, A. Peters, A. Kaestner, M. Krafczyk, W. Durner, Prediction of capillary hysteresis in a porous
material using lattice-Boltzmann methods and comparison to experimental data and a morphological pore network model, Adv. Water
Resour. 31 (2008) 1151–1173.
[38] X. Liu, A.N. Zhou, J. Li, S.J. Feng, Reproducing micro X-ray computed tomography (microXCT) observations of air–water distribution
in porous media using revised pore-morphology method, Can. Geotech. J. 57 (2020) 149–156.
[39] D.G. Fredlund, A. Xing, Equations for the Soil- Water Characteristic Curve, Can. Geotech. J. 31 (1994) 533–546.
[40] D. Or, M. Tuller, Liquid retention and interfacial area in variably saturated porous media: Upscaling from single-pore to sample-scale
model, Water Resour. Res. 35 (1999) 3591–3605.
[41] D. Or, M. Tuller, Cavitation during desaturation of porous media under tension, Water Resour. Res. 38 (2002) 19-1–19-14.
[42] V. Richefeu, M.S. El Youssoufi, F. Radjai, Shear strength properties of wet granular materials, Phys. Rev. E Stat. Nonlin. Soft Matter
Phys. 73 (2006) 051304.
[43] S.W. Tyler, S.W. Wheatcraft, Fractal processes in soil water retention, Water Resour. Res. 26 (1990) 1047–1054.
[44] B. Kantsiper, S. Weiss, An analytic approach to calculating earth coverage, Adv Astronaut Sci 97 (1998) 313–332.
[45] R. Swinbank, R.J. Purser, Fibonacci grids: A novel approach to global modelling, Q. J. R. Meteorol. Soc. 132 (2006) 1769–1793.
[46] X. Hui, C. Yu, Photonic bandgap structure and long-range periodicity of a cumulative fibonacci lattice, Photonics Res. 5 (2016) 11–14.
[47] S. Baselga, Fibonacci lattices for the evaluation and optimization of map projections, Comput. Geosci. 117 (2018) 1–8.
[48] A. Di renzo, F.P. Di maio, Comparison of contact-force models for the simulation of collisions in DEM-based granular flow codes,
Chem. Eng. Sci. 59 (2004) 525–541.
[49] C. Wellmann, C. Lillie, P. Wriggers, Comparison of the macroscopic behavior of granular materials modeled by different constitutive
equations on the microscale, Finite Elem. Anal. Des. 44 (2008) 259–271.
[50] C. Goldenberg, I. Goldhirsc, Friction enhances elasticity in granular solids, Nature 435 (2005) 188–91.
[51] N. Guo, J. Zhao, The signature of shear-induced anisotropy in granular media, Comput. Geotech. 47 (2013) 1–15.
[52] N. Guo, J. Zhao, A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media, Int. J. Numer. Methods
Eng. 99 (2013) 789–818.
[53] S.-J. Feng, X. Liu, H.-X. Chen, T. Zhao, Micro-mechanical analysis of geomembrane-sand interactions using DEM, Comput. Geotech.
94 (2018) 58–71.
[54] E. ISO, 14688-1: 2017: Geotechnical Investigation and Testing–Identification and Classification of Soil–Part 1: Identification and
Description, British Standards Institution, Switzerland, 2017.
[55] C. Thornton, Numerical simulations of deviatoric shear deformation of granular media, Géotechnique 50 (2000) 43–53.
[56] C. Shen, S.H. Liu, L. Wang, Y. Wang, Micromechanical modeling of particle breakage of granular materials in the framework of
thermomechanics, Acta Geotech. 14 (4) (2019) 939–954.
32
X. Liu, A. Zhou, S.-l. Shen et al. / Computer Methods in Applied Mechanics and Engineering 368 (2020) 113183
[57] M.T. van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils1, Soil Sci. Am. J. 44 (1980).
[58] C. Yuan, B. Chareyre, F. Darve, Pore-scale simulations of drainage in granular materials: Finite size effects and the representative
elementary volume, Adv. Water Resour. 95 (2016) 109–124.
[59] K.N. Manahiloh, B. Muhunthan, Characterizing Liquid Phase Fabric of Unsaturated Specimens from X-Ray Computed Tomography
Images, Springer, Berlin, Heidelberg, 2012.
[60] M.R. Cunningham, A.M. Ridley, K. Dineen, J.B. Burland, The mechanical behaviour of a reconstituted unsaturated silty clay,
Géotechnique 53 (2003) 183–194.
[61] H. Schubert, Capillary forces - modeling and application in particulate technology, Powder Technol. 37 (1984) 105–116.
[62] N. Lu, T.-H. Kim, S. Sture, W.J. Likos, Tensile strength of unsaturated sand, J. Eng. Mech. 135 (2009) 1410–1419.
[63] J. Christoffersen, M.M. Mehrabadi, S. Nemat-Nasser, A micromechanical description of granular material behavior, J. Appl. Mech. 48
(1981).
[64] C. Jommi, Remarks on the constitutive modelling of unsaturated soils, in: T. Mancuso (Ed.), Experimental Evidence and Theoretical
Approaches in Unsaturated Soils, 2000, pp. 139–153.
[65] E. Catalano, B. Chareyre, E. Barthélémy, Pore-scale modeling of fluid-particles interaction and emerging poromechanical effects, Int.
J. Numer. Anal. Methods Geomech. 38 (2014) 51–71.
[66] N. Guo, J. Zhao, A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media, Int. J. Numer. Meth. Eng.
99 (2014) 789–818.
[67] A.W.T. Bishop, I.B. Donald, Experimental study of partly saturated soil in the triaxial apparatus, in: Proceedings of the 5th International
Conference on Soil Mechanics and Foundation Engineering, Dunod, Paris, France, 1961.
[68] D.G. Fredlund, A. Xing, M.D. Fredlund, S.L. Barbour, The relationship of the unsaturated soil shear to the soil-water characteristic
curve, Can. Geotech. J. 33 (1996) 440–448.
[69] Y. Kohgo, M. Nakano, T. Miyazaki, Theoretical aspects of constitutive modelling for unsaturated soils, Soils Found. 33 (1993) 49–63.
[70] A.L. Öberg, G. Sällfors, Determination of shear strength parameters of unsaturated silts and sands based on the water retention curve,
Geotech. Test. J. 20 (1997).
[71] B.A. Schrefler, The Finite Element Method in Soil Consolidation, University College of Swansea, 1984.
[72] D. Sheng, D.G. Fredlund, A. Gens, A new modelling approach for unsaturated soils using independent stress variables, Can. Geotech.
J. 45 (2008) 511–534.
[73] D. Sheng, A.N. Zhou, D.G. Fredlund, Shear Strength Criteria for Unsaturated Soils, Geotech. Geol. Engi. 29 (2011) 145–159.
[74] D.A. Sun, H. Matsuoka, Y. Yao, W. Ichihara, An elasto-plastic model for unsaturated soil in three-dimensional stresses, Soils Found.
40 (2000) 17–28.
[75] S.K. Vanapalli, D.G. Fredlund, D.E. Pufahl, A.W. Clifton, Model for the prediction of shear strength with respect to soil suction, Can.
Geotech. J. 33 (1996) 379–392.
[76] A.N. Zhou, R.-Q. Huang, D. Sheng, Capillary water retention curve and shear strength of unsaturated soils, Can. Geotech. J. 53 (2016)
974–987.
[77] D.G. Toll, B.H. Ong, Critical-state parameters for an unsaturated residual sandy clay, Géotechnique 53 (2003) 93–103.
[78] Y. Gao, D.A. Sun, Z.C. Zhu, X.F. Xu, Hydromechanical behavior of unsaturated soil with different initial densities over a wide suction
range, Acta Geotech. 14 (2019) 417–428.
[79] B. François, L. Laloui, ACMEG-TS: A constitutive model for unsaturated soils under non-isothermal conditions, Int. J. Numer. Anal.
Methods Geomech. 32 (2008) 1955–1988.
[80] L. Laloui, M. Nuth, On the use of the generalised effective stress in the constitutive modelling of unsaturated soils, Comput. Geotech.
36 (2009) 20–23.
[81] D.A. Sun, D.C. Sheng, H.B. Cui, S.W. Sloan, A density-dependent elastoplastic hydro-mechanical model for unsaturated compacted
soils, Int. J. Numer. Anal. Methods Geomech. 31 (2007) 1257–1279.
[82] A.N. Zhou, D. Sheng, S.W. Sloan, A. Gens, Interpretation of unsaturated soil behaviour in the stress-saturation space, I: Volume change
and water retention behaviours, Comput. Geotech. 43 (2012) 178–187.
[83] A.N. Zhou, D. Sheng, S.W. Sloan, A. Gens, Interpretation of unsaturated soil behaviour in the stress-saturation space, II: Constitutive
relationships and validations, Comput. Geotech. 43 (2012) 111–123.
[84] A.N. Zhou, D. Sheng, An advanced hydro-mechanical constitutive model for unsaturated soils with different initial densities, Comput.
Geotech. 63 (2015) 46–66.
[85] B. François, L. Tacher, C. Bonnard, L. Laloui, V. Triguero, Numerical modelling of the hydrogeological and geomechanical behaviour
of a large slope movement: the Triesenberg landslide (Liechtenstein), Can. Geotech. J. 44 (2007) 840–857.
[86] G. Klubertanz, L. Laloui, L. Vulliet, Identification of mechanisms for landslide type initiation of debris flows, Eng. Geol. 109 (2009)
114–123.
[87] A.N. Zhou, S. Wu, J. Li, D. Sheng, Including degree of capillary saturation into constitutive modelling of unsaturated soils, Comput.
Geotech. 95 (2018) 82–98.
[88] A.W. Bishop, G.E. Blight, Some aspects of effective stress in saturated and partly saturated soils, Géotechnique 13 (1963) 177–197.
[89] S. Nam, M. Gutierrez, P. Diplas, J. Petrie, Determination of the shear strength of unsaturated soils using the multistage direct shear
test, Eng. Geol. 122 (2011) 272–280.
[90] Y.S. Song, W.K. Hwang, S.J. Jung, T.H. Kim, A comparative study of suction stress between sand and silt under unsaturated conditions,
Eng. Geol. 124 (2012) 90–97.
[91] S. Oh, N. Lu, Y.K. Kim, S.J. Lee, S.R. Lee, Relationship between the Soil-Water Characteristic Curve and the Suction Stress
Characteristic Curve: Experimental Evidence from Residual Soils, J. Geotech. Geoenviron. 138 (2012) 47–57.
[92] C.S. Chang, A. Misra, Application of uniform strain theory to heterogeneous granular solids, J. Eng. Mech. 116 (1990) 2310–2328.
[93] S.S. Wu, A.N. Zhou, J. Li, J. Kodikara, W.C. Cheng, Hydromechanical behaviour of overconsolidated unsaturated soil in undrained
conditions, Can. Geotech. J. 56 (2019) 1609–1621.