Computers and Geotechnics 157 (2023) 105268
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Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
Study on soil-rock slope instability at mesoscopic scale using discrete
element method
Yangyu Hu a, Ye Lu b, *
a
b
Graduate Student, Department of Civil Engineering, Shanghai University, 99 Shangda Road, Shanghai 200444, China
Assistance Professor, Department of Civil Engineering, Shanghai University, 99 Shangda Road, Shanghai 200444, China
A R T I C L E I N F O
A B S T R A C T
Keywords:
Soil-rock mixtures (S-RM) slope
Bearing capacity
Rock-wrapping movement
Micro-scale mechanism
Discrete element method
Soil-rock mixtures (S-RM) are pervasive in nature and are heterogeneous media with continuous and discrete
properties. The S-RM slope has become one of the fundamental geological environments for engineering activ­
ities due to the constant growth of engineering construction. In this study, a slope system was constructed based
on the discrete element method (DEM) to analyze the micro-scale phenomena of rock-wrapping in the slope shear
band and the micro-scale mechanisms, including force chains and anisotropy, under the ultimate load-bearing
state. The results indicate that the slope ultimate bearing capacity increased with rock content and angularity
but decreased with aspect ratio and was also affected by the spatial distribution of rocks. During the slope
destabilization process, there were at least four modes of rock-wrapping movement in the shear band, and the
variability of soil and rock movement caused the irregularity of shear surface in the S-RM slope. The disturbance
range of slope shear dilatation was heavily influenced by rock content and angularity, and the aspect ratio was an
important factor in determining the morphological characteristics of shear dilatation. The content, distribution,
and shape of rocks influenced the regularity of the distribution of force chains and the anisotropy of a slope.
1. Introduction
Soil-rock mixtures (S-RM), which are primarily composed of rock
and soil, are macroscopically a mixed medium between soil and rock
(Lindquist, 1994; Medley, 2001; Medley and Rehermann, 2004; Xu
et al., 2007). Gravity deposits, water accumulation, and weathering are
the geological causes of S-RM (Cen et al., 2017; Coli et al., 2010; Gao
et al., 2018). S-RM is prevalent in nature, and many slopes are composed
of S-RM, which has mechanical properties between soil and rock and the
characteristics of both continuous and discrete media (Xu et al., 2011).
As large-scale engineering construction and geotechnical engineering
continue to grow, S-RM slopes are being used in more and more projects,
such as the maintenance of water conservancy and the building of tailing
dams, among other things.
S-RM slopes must therefore be evaluated for stability. The limit
equilibrium method (LEM) is a traditional method for analyzing slope
stability. The LEM computes the slope safety factor based on the rela­
tionship between shear resistance and shear force along the potential
damage surface of the shear direction in the slope (Ahmed et al., 2012;
Hazari et al., 2020). This typically requires simplifying the geological
model by treating the S-RM as a continuous media material and
substituting the macroscopic soil parameters (Li et al., 2011). However,
the LEM disregards the properties of discrete media in S-RM, so there
may be significant discrepancies between the assumed slip surface and
the actual slip surface. With the development of computer technology,
researchers began to study slopes by using numerical models. Since its
introduction, the finite element method (FEM) has been widely utilized
for the study of slope stability (Abusharar and Han, 2011; Dawson et al.,
1999; Naylor, 1982; Subramanian et al., 2017; Zhao et al., 2021). The
FEM does not require the assumption of a slip surface and considers the
internal stress field conditions of the slope. The FEM can simulate the
nonlinear estimation of soils and the progressive deterioration of slopes
with strain-softening characteristics. The problem with the FEM, how­
ever, is that it treats all media as a continuum, which is perhaps less
reasonable when a micro-grained matrix of lower relative strength is
embedded in a rock mass of higher relative strength. On the other hand,
the discrete element method (DEM) (Cundall and Strack, 1979), which
treats materials as assemblies of individual particles that can interact
with each other at their contact points, takes into account the different
properties of bulk materials and has become a new way for many re­
searchers (Deluzarche and Cambou, 2006; Gao et al., 2022; Gao et al.,
2021; Huang et al., 2021; Yang et al., 2022) to solve problems. Other
* Corresponding author.
E-mail address: ye.lu@shu.edu.cn (Y. Lu).
https://doi.org/10.1016/j.compgeo.2023.105268
Received 8 October 2022; Received in revised form 8 December 2022; Accepted 11 January 2023
Available online 24 February 2023
0266-352X/© 2023 Elsevier Ltd. All rights reserved.
Y. Hu and Y. Lu
Computers and Geotechnics 157 (2023) 105268
rock-wrapping (Montoya-Araque and Suarez-Burgoa, 2019; Wang and
Zhang, 2019), a phenomenon in which the shear zone of an S-RM slope
assumes the shape of avoiding rock or the slip surface becomes distorted
around the rock. However, the study of rock-wrapping movement in
shear zones is insufficient, and this phenomenon is only mentioned in
the current work. Further research is required to determine the reason,
nature, and effects of rock-wrapping movement, which may be related to
the cause of S-RM slope failure. The formation of shear zones is inti­
mately related to slope deformation, and the study of the rock-wrapping
form of shear zones is useful for analyzing the damage mode of S-RM
slopes and for preventing slope-related natural disasters.
In this study, a series of laboratory experiments were performed to
determine soil strength and slope bearing capacity, and DIC was utilized
to determine information about the landslide surface and local mesoscale properties such as rotation and porosity in order to calibrate the
DEM model. Then, the corresponding DEM slope model was constructed,
and the effect of different rock shapes was approximated using a random
construction of blocks. After confirming the accuracy of the DEM model
by comparing it to laboratory test results, the micro-scale rock-wrapping
phenomenon of the slope shear band and the micro-scale mechanism
changes at the ultimate bearing state of the slope were investigated
under the influence of various factors.
Fig. 1. Grading curves of particles.
than theoretical and numerical methods, laboratory tests are a crucial
method for solving geotechnical engineering problems (Bate and Zhang,
2013; Cheng et al., 2021; Deng et al., 2020), particularly at an early
stage when the theoretical basis is not yet complete and the numerical
simulation technology is not yet well-established. The digital image
correlation (DIC) method (Bickel et al., 2018) is a highly accurate noncontact measurement technique. Small-scale tests in the lab are a good
way to use DIC method to figure out how S-RM slopes get worse over
time. In addition, the local damage features of the slope may be captured
by image analysis technology, and new insights into the damage
mechanism of the S-RM slope can be gained by calibrating the DEM
model with DIC technology and conducting joint analysis on macro and
micro perspectives.
In general, the studies of S-RM slopes can be divided into two aspects:
analysis of sliding processes (Gao et al., 2022) and analysis of mecha­
nisms (Lu et al., 2018; Napoli et al., 2018). Sliding process analysis is
concerned with dynamic processes, and research points include sliding
velocity, impact forces, etc.; mechanism analysis is concerned with
stability, deformation, and damage characteristics. The majority of
current research findings on mechanistic analysis are macro-focused.
Facors including rock content, gradation, the shape, and spatial loca­
tion of rock have significant effects on the stability of slopes. However,
few research has been conducted on the effects of these factors on the
evolution of internal force chains and the anisotropy of slopes which can
result in changes in mechanical properties. At present, most researches
on the micro-scale mechanism of S-RM focuses on triaxial tests and
direct shear tests (Li et al., 2022; Xu et al., 2019). In triaxial tests, the
boundary conditions are symmetric, and in direct shear tests, the shear
surface is artificially defined. Nevertheless, these boundary conditions
do not apply to the actual S-RM slope environment. To better understand
the physical and mechanical properties of S-RM slopes, a micro-scale
mechanistic investigation of S-RM slopes is required. When an S-RM
slope becomes unstable, the shear zone is frequently accompanied by
2. DEM model for lab tests
2.1. Test materials and setup
For the test, S-RM samples with rock contents (RC) of 0%, 10%, 50%,
and 80% were prepared. The soil consisted of sand with a particle size of
less than 2 mm and greater than 0.075 mm, while the rock consisted of
gravel with a particle size of 2–10 mm. The grading curves obtained by
sieve analysis method (Germaine and Germaine, 2009) are presented in
Fig. 1.
The S-RM slopes with different rock content was subjected to a static
overload test, and the deformation process of slope instability was
recorded by DIC (see Fig. 2). This test loading apparatus is a universal
material testing machine. This universal material testing machine has a
maximum loading capacity of 500 kN, a loading range of 700 mm, and a
maximum loading speed of 500 mm/min. During the test, the slope top
settlement and load data were automatically recorded, and the machine
was loaded with 8 N/s of incremental axial force on a (100 mm × 290
mm × 50 mm) loading plate.
2.2. Construction of DEM block templates
In order to simulate the complexity and diversity of natural rock
forms, many researchers (Cui et al., 2020; Lu et al., 2017; Wang et al.,
1999; Xu et al., 2015; Zhao et al., 2021) have analyzed the methods for
Fig. 2. Schematic of lab test.
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Computers and Geotechnics 157 (2023) 105268
Fig. 3. Random block generation.
constructing stochastic fine structure models for blocks. In this study,
the FISH programming language is used to generate random block
shapes. Two factors determine the randomness of the block: the
randomness of the control point angle and the randomness of the control
point itself. There are two randomness of control points: one is the
randomness of the total number of control points, and the other is the
randomness of the distance between individual control points and the
origin. As shown in Fig. 3a and Eqs. (1), (2), and (3). The control points
of the block shape are the points A-F, whose number Q follows a uniform
random distribution. The distance l of each control point to the origin O
and the angle α of the line connecting adjacent control points to the
origin obey a normal random distribution, respectively:
Q ∼ U(Qmin , Qmax )
(1)
(
)
l ∼ N l0 , σ2l
(2)
(
)
◦
/
α ∼ N α0 , σ2α α0 = 360 Q
Fig. 4. Comparison of images and templates (partial).
(3)
where Qmin, Qmax are the minimum and maximum number of control
points respectively, l0 is the expected value of the distance from the
control point to the origin, σl is the variance of the distance, α0 is the
expected value of the angle of the control point, σ α is the variance of the
angle.
Indicators of the shape of geotechnical materials (Blott and Pye,
2008; Zhou et al., 2020) include angularity, roundness, aspect ratio,
concavity, etc. The physical and mechanical properties of S-RM are
similarly affected by the shape parameter indices of the blocks (Lu et al.,
Fig. 5. DEM model of the slope with 50% rock content.
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Computers and Geotechnics 157 (2023) 105268
Fig. 6. Comparison of test and simulation of direct shear test.
2017), so they should be controlled. This study focuses on the two fac­
tors of angularity and aspect ratio because the indicators are coupled
and correlated with each other along with the geometry. The angularity
can be approximated as decreasing as the number of polygon edges in­
creases (Mirghasemi et al., 2002), so the angularity can be controlled by
the number of edges. It has been shown that the aspect ratio can be
defined as the ratio of the long axis to the short axis of the external
rectangle with the smallest width among those that coincide with the
polygon sides (Preparata and Shamos, 2012), as presented in Fig. 3b.
Fig. 3c shows the block templates generated by the aforementioned al­
gorithm, which were compared to the realistic gravel images in Fig. 4,
and eight groups of block templates were chosen for subsequent simu­
lations. According to geotechnical test standards, it is acceptable to use
the minimum width of the rock as the gravel particle size, and the rock is
generated according to the grading curve in Fig. 1.
Table 1
Microparameters.
Contact type
7
2
Effective modulus (10 N/m )
Stiffness ratio
Friction coefficient
Rolling friction coefficient
Maximum attractive force (N)
Attraction range (mm)
Sand-sand
Sand-rock
Rock-rock
6.5
1.5
0.4
0.05
3
7.5 × 10-2
80
2.7
0.4
—
—
—
80
3.0
1.5
—
—
—
3. Validation of dem models
3.1. Calibration of microparameters
In this study, a numerical simulation model of a direct shear test with
the same dimensions as the laboratory test was developed, and samples
with 0% and 40% rock content were subjected to direct shear with 100
kPa, 200 kPa, 400 kPa, and 800 kPa normal stresses, respectively. To
calibrate the parameters, the microparameters are changed so that the
physical and mechanical property indexes derived from numerical tests
match the results of laboratory tests.
Fig. 6 depicts a comparison of simulation and test results. The graph
demonstrates that the shear strength-shear displacement curve derived
from the simulation closely resembles the curve derived from the labo­
ratory direct shear test. As shear displacement increases, the simulation
curve begins to fluctuate within a range, which is caused by the
continuous breakage and reorganization of the contact force chain
within the S-RM during the shear process and the redistribution of the
rock and soil particles forming the spatial structure. At high vertical
stress, there is a substantial difference between the experimental and
simulated results in terms of soil shear swell ability (positive vertical
displacement represents shear dilation). There are numerous explana­
tions for this. First, the simulation was set up in 2-D model while real test
was 3-D. Second, there are some differences in the size and shape of
particles between the simulation and lab test. These factors make it
challenging to simulate the soil density with 100% accuracy in DEM.
Third, the simulation particles cannot be broken into smaller particles to
fit the pores of the soil, and they can only respond to the high stress
boundary conditions by misalignment and rearrangement, resulting in a
2.3. Setup of S-RM slope DEM models
Fig. 5 depicts the numerical model of the S-RM slope. Using the slope
with a 50% rock content as an example, the slope height is 350 mm, the
slope angle is 50◦ , and the dimensions correspond to the model test.
According to the grading curve in Fig. 1, the particle size was doubled to
improve the calculation efficiency. Previous research indicates that the
particle size effect can be disregarded when the loading plate width is
greater than 40d50 (Craig, 1983). Rocks are randomly distributed on the
slope. The surrounding walls are fixed boundaries, and the foundation
model is generated utilizing the grid method (Duan and Cheng, 2016)
and fully compacted under gravity to simulate the initial ground stress
field, followed by slope cutting and calculation until equilibrium
convergence. At the top of the slope, a square wall was installed as a
loading plate, and a quasi-static constant loading rate method of 0.01
mm/s was used until the slope bearing capacity reached its maximum
and large area sliding occurred (Chen et al., 2018; Garakani et al., 2020).
The local damping mechanism permits the model to achieve static
equilibrium as quickly as possible, and the local damping coefficient is
set to be 0.7 (Hassan and El Shamy, 2019).
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Computers and Geotechnics 157 (2023) 105268
Fig. 7. Comparison of test and simulation results.
simulation that is more likely to emphasize the effects of shear expan­
sion. The maximum shear dilation gradually increases as the stone
content rises, but the maximum shear contraction continues to decrease;
hence, the higher the rock content, the more pronounced the shear
dilation characteristics. This rule applies to both experiments and sim­
ulations. For the sand-sand contact, the adhesive rolling resistance linear
model (Gilabert et al., 2007) was used, while the linear model was used
for the sand-rock and rock-rock contacts. The deformation parameters of
S-RM are controlled by the effective modulus and stiffness ratio, and the
contact stiffness between particles is determined by Eqs. (4) and (5).
Table 1 displays the microparameters obtained from numerical tests.
Sand had a density of 2650 kg/m3, while rock had a density of 2700 kg/
m3.
kn = AE* /L
(4)
ks = kn /κ*
(5)
where kn, ks are the contact stiffnesses in the normal and shear di­
rections respectively, E* is the effective modulus, κ* is the stiffness ratio,
A is inter-particle contact area (the diameter of smaller particles in 2D),
L is the distance to the center of contact particles.
3.2. Validation of slope models
Fig. 7a depicts the displacement vector when the slope with 50%
rock content reaches its ultimate bearing capacity. The arrow direction
and length indicate the direction and magnitude of soil displacement,
respectively. The soil displacement on the left side of the middle line of
the loading plate is negligible, while the vertical displacement is pre­
dominant. On the right side of the center line of the loading plate, the
soil near the slope moves horizontally most of the time, and the trends of
movement in the simulation and experiment are similar.
Fig. 7b illustrates the load (P) and displacement (s) curves of slopes
with different RC. In contrast, the P-s curves derived from the simulation
Fig. 8. Setup of model under different condition.
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comparisons, with edge numbers of 3–4, 7–8, and aspect ratios of
1.0–1.5, 2.0–2.5. In order to study the effect of the relative initial po­
sition of particles and rocks, the rock content was held constant at 50%
and only the initial position of rocks was altered; two cases were set up
to discuss the rocks concentrated on the slope top and surface, respec­
tively. Fig. 8 depicts the case-specific model diagrams (for the change in
the rock shape, see Fig. 3c). The calculated bearing capacity for all cases
is shown in Table 2.
Table 2
Bearing capacity under different conditions.
Influence of rock content (RC)
RC
0%
10 %
P (kPa)
32
39
Influence of initial position of rock blocks
Rock
Randomly
Densely located
distribution
located
at top
P (kPa)
97
137
Influence of angularity of rock blocks
Edge number
3–4
5–6
P (kPa)
242
157
Influence of aspect ratio (AR) of rock blocks
AR
1.0–1.5
1.5–2.0
P (kPa)
205
157
50 %
80
%
97
157
Densely located at
surface
84
4. Analyses on dem simulation results
7–8
118
4.1. Analysis of deformation mechanism
2.0–2.5
143
It is widely known that particles within the shear band undergo
significant rotation. The investigation of the rotational regularity of soil
and rock particles within the S-RM slope can lead to an analysis of the
shear band formation process and damage.
ΔS represents the static load settlement (s) when the slope ultimate
bearing capacity is reached. The rotation field of the S-RM slope with a
50% rock content is depicted in Fig. 9. The rotation of particles is pos­
itive in the anticlockwise direction. When s = 1/4 ΔS, the particles on
the left side of the loading plate are predominantly rotated in a clock­
wise direction, while those on the right side are predominantly rotated
in a counterclockwise direction. When s = 1/2 ΔS, the shear bands on
both sides of the loading plate cross and combine into a single shear
band beneath the loading plate. The merged shear band rotates pre­
dominantly counterclockwise and develops and expands toward the
surface of the slope. The performance demonstrates a process of pro­
gressive destruction.
To validate the correctness of the simulation, we chose the upper and
lower local fields in the slope with a 50% rock content and calculated
dimensions of 84 mm × 48 mm and 44 mm × 28 mm, respectively. As
shown in Fig. 10, the shear strains of the two regions were computed
using DIC, and nine representative regions (S1-S9) were chosen based on
the thickness of the shear zone (Nübel, 2002). Fig. 11 illustrates the rock
rotation-time curves for the two local fields. Within the two local fields,
the rotation of the rocks in the shear zone is significantly greater than
that of the rocks in the other regions, and the moment at which the rocks
began to violently rotate is extremely nearby. The onset of considerable
rotation of the rocks in the higher local field occurs earlier than in the
lower local field. These resemble the simulated occurrences and
demonstrate the validity of the simulations, after which the remaining
scenarios are investigated.
In Fig. 12a, the shear band is circular and smooth when RC = 0% and
10%. When RC = 50%, the shear band becomes more curved and a rockwrapping phenomenon becomes evident. And when RC = 80%, the
shear band demonstrates a large, complex, multi-forked form of damage.
Table 2 demonstrates that the slope bearing capacity increases signifi­
cantly when the rocks are concentrated on the slope crest and decreases
when the rocks are concentrated on the sloping surface; the slope
Fig. 9. Contour maps of the particle rotation with RC = 50 % (unit: deg).
are more congruent with those from the experiment. The evolution of
the P-s curve consists of two stages. First, as the load increases, the
settlement at the top of the slope approaches a linear increase. The S-RM
at the incline crest is in the compacting stage. In the second stage, the
settlement continues to increase, the vertical load reaches its maximum
and begins to decline, and the slope has sustained instability damage.
When RC = 0% and 10%, the slope peak ultimate bearing capacity is
relatively similar. When RC = 50%, the slope bearing capacity rises
sharply, whereas when RC = 50% to 80%, the rise in ultimate bearing
capacity begins to slow. This indicates that when RC is low, the prop­
erties of S-RM are controlled primarily by sand, whereas when RC rea­
ches a certain level, the properties of S-RM are controlled jointly by sand
and rock. Since the angle of friction between rock and sand is higher for
rock than for sand, adding rock to an S-RM slope will make it much
stronger.
The preceding results demonstrate that the DEM slope model is
reasonable. In this set of simulations, the number of edges and the aspect
ratio (AR) of the block templates are primarily positioned between 5 and
6 and 1.5–2.0, respectively, to improve computational efficiency. Ac­
cording to the previous study, when RC = 80%, the presence of rock has
a significant impact on the slope bearing capacity. To investigate the
effect of angularity and aspect ratio, the rock content of 80% was chosen
as a constant variable, and four simulation groups were set up to serve as
Fig. 10. Schematic of local field.
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Computers and Geotechnics 157 (2023) 105268
Fig. 11. Rock rotation-time curves.
Fig. 12. Contour maps of the particle rotation (unit: deg).
bearing capacity decreases as the number of rock edges and aspect ratio
increase. In Fig. 12b, when the rocks are concentrated at the top of the
slope, the top of the slope will rotate violently, like the case of a high
rock content, whereas when the rocks are concentrated at the slope face,
the disturbance of the slope is smaller, and the shear zone in the middle
of the slope does not appear to extend to the sloping surface. Fig. 12c and
d demonstrate that slopes with varying angularity and aspect ratio
exhibit a wide range of irregular damage forms; however, the vertical
distance from the merging point of the shear bands to the bottom of the
loading plate increases as the edge number and aspect ratio increase.
This phenomenon is explicable by the following: the greater the angu­
larity and aspect ratio, the closer the gravel is to the angular rock, and
the angular rock is more likely than the subrounded rock to produce an
interlocking effect; the interlocking force is an important factor in
maintaining the slope stability, and the greater the aspect ratio of a rock,
the greater its propensity to rotate under the influence of contact force.
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Computers and Geotechnics 157 (2023) 105268
Fig. 13. Trajectories of the particles in three areas.
To further investigate the influence of rocks on the shear zone, three
representative areas near the loading plate, in the middle of the slope,
and near the slope surface where violent rotations occurred were
selected in the 50% rock-bearing S-RM slope (A1, A2, and A3), and the
trajectories of rocks and 12 surrounding sand grains were traced, as
shown in Fig. 13a. Fig. 13b depicts sand particles S1-S8 on the upper left
side of the rock R1, and S9-S12 on the lower right side of R1 in area A1.
When s = 2/3 ΔS, the relative positions of sand and gravel particles are
significantly altered. Most sand particles move along the rock left side.
The sand particles S6-S8 are close to the right edge of the gravel and have
a small relative displacement. The sand predominantly surrounds the
rock on the left side. According to Fig. 13c, the particle distribution in
the region A2 is comparable to that in the region A1. When s = 2/3 ΔS,
sand particles S1-S4 move along the lower left side of the rock R1,
whereas S5-S8 move in the direction of the upper right side of R1. The
relative positions of S9-S12, which are situated on the lower left side of
R1, and R1 remain unaffected. The sand apparent slip trajectory
appeared to split along both sides of the rock. Fig. 13d depicts the mo­
tion trajectory in the A3 region. Sand grains S1-S9 are dispersed outside
the skeleton of the rock, whereas S10-S12 are located within the skeleton.
When s = 2/3 ΔS, the rock R1 rotates significantly, and the two rocks
begin to gradually separate. When s = ΔS, most sand particles outside
the skeleton shift to both sides of the skeleton, resulting in bifurcation.
S8 and S9 squeeze between the rock skeletons, causing the relative po­
sitions of the rocks to shift noticeably.
Table 3 lists the displacement and rotation of the rock and sand
particles in three regions. The displacement is positive downwards and
to the right, and the rotation is positive anticlockwise. According to the
table, along the shear direction, the displacement of sand particles on
the upper right side of the rock and outside the skeleton was greater than
that of the rock, whereas the displacement of sand particles on the lower
left side of the rock and within the skeleton was not significantly
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Table 3
Particle displacement and rotation in three areas.
Particles
Area A1
S1
S2
S3
S4
S5
S6
R1
Area A2
S1
S2
S3
S4
S5
S6
R1
Area A3
S1
S2
S3
S4
S5
S6
R1
Horizontal displacement
(mm)
Vertical displacement
(mm)
Rotation
(◦ )
Particles
Horizontal displacement
(mm)
Vertical displacement
(mm)
Rotation
(◦ )
9.14
10.59
9.61
10.79
10.23
9.9
11.05
13.38
14.31
14.25
12.29
13.12
9.81
7.8
19
− 50
23
− 31
–23
27
− 14
S7
S8
S9
S10
S11
S12
8.8
9.71
10.84
9.35
9.45
9.66
9.97
13.17
11.41
11.31
11.83
12.4
− 37
− 21
− 35
40
50
25
11.99
11.69
13.3
12.3
13.52
12.1
10.04
7.44
7.5
7.54
7.1
7.4
6.87
3.94
− 37
− 21
–23
− 21
–33
16
− 16
S7
S8
S9
S10
S11
S12
12.12
12.87
13.11
14.54
13.78
13.45
7.16
7.05
7.4
6.79
6.4
6.83
− 42
77
37
28
38
− 2
7.88
8.9
10.19
9.41
9.25
8.39
3.87
8.54
7.3
5.68
6.94
5.98
4.31
5.17
52
60
50
74
40
32
6
S7
S8
S9
S10
S11
S12
R2
5.74
4.26
7.23
2.75
1.78
2.84
1.75
7.67
4.51
1.71
8.25
5.93
7.4
3.01
22
110
10
− 36
− 20
− 25
− 11
Fig. 14. Schematic of rock-wrapping modes.
Fig. 15. Contour maps of the porosity with RC = 50 %.
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Fig. 17. In Fig. 17a, the upper local field porosity initially exhibits a
declining trend, and the slope is in the compression process. Then, the
porosity in the shear zone and slip zone increases noticeably, with the
greatest increase in the shear zone, indicating the creation of the shear
band. Compared with Fig. 17b, the shear band develops earlier in the
upper local field than in the lower local field, and the overall amplitude
of shear expansion is also greater in the upper local field than in the
lower local field. The pattern of local porosity development in the test is
remarkably close to the results of the simulation, further validating the
validity of the simulation. By comparing Figs. 17 and 11, we can
determine that the time when the rock begins to violently rotate and the
shear zone initiates is the same, while the displacement of sand and rock
has been continuous with the increase of load, and from Fig. 13 and
Table 3, we can again determine that the rotation of sand is earlier and
larger than that of rock, indicating that the rotation of rock on the shear
zone is the primary cause of the instability of the S-RM slopes.
Fig. 18a demonstrates that the greater the rock content, the greater
the shear expansion phenomenon when the slope is unstable. When
displacement and rotation occur, the soil internal structure will be
drastically altered. Because rocks are significantly larger than sand.
Fig. 18b demonstrates that when rocks are concentrated at the top of the
slope, a significant shear dilatation occurs in the upper portion of the
slope due to the load. When the rocks are concentrated on the sloping
surface, the connection of the shear zone is less coherent in the middle of
the slope, and the porosity at the foot of the slope is greater, even though
the sloping surface is not in direct contact with the loading plate. This
indicates that the distribution of the rocks has a significant effect on the
shear dilatation of the slope. Fig. 18c depicts a nephogram of porosity
with different angularities of S-RM slopes to the ultimate bearing state.
As shown in the diagram, the greater the angularity of the rock, the
greater the shear swelling area when the slope is unstable. When the
edge number of the rock is between 3 and 4, slope instability manifests
as shear swelling from the top of the slope to the bottom. As the edge
number increases to 5–6 and 7–8, the local shear swelling area decreases
from the foot of the slope to the middle of the slope. As for the aspect
ratio (AR), it can be seen in Fig. 18d that when AR = 1.0–1.5, the contour
of the shear swelling region is relatively smooth, whereas as AR in­
creases, the contour becomes increasingly curved and rough. This is
because the skeleton of angular rocks can more easily be reconstructed
into a spatial structure with larger internal pores. From a single shape
factor, angularity primarily affects the extent of shear swelling gener­
ated during slope instability, whereas aspect ratio dominates the shear
swelling area morphology.
Fig. 16. Schematic of morphological processing.
different from that of the rock. Typically, the rocks within the shear band
rotate more slowly than the sand particles. Four rock-wrapping modes of
rock influence on the shear band were summarized by combining Fig. 13
and Table 3 – unilateral rock bypass, bifurcation, bifurcation + crossing
rock, and unilateral rock bypassing + bifurcation + crossing rock. As
shown in Fig. 14, the blue arrows indicate the direction of shear zone
expansion, and the black and red arrows indicate the direction of rock
displacement and rotation, respectively. When the S-RM slope un­
dergoes local damage deformation, the soil moves along the edge of the
rock or the contact surface between the rocks, causing differential
rotation and movement between the soil and rock. The uncoordinated
differential rotation and rock-wrapping motion gradually develop from
local to global, resulting in the rock-wrapping phenomenon of the shear
band. Consequently, the slope destabilization process culminates in an
irregular and gradual damage process.
4.2. Analysis of local porosity
Damage to the S-RM slope will lead to a change in porosity as a result
of the altered soil structure. Shear band formation is accompanied by a
strong shear expansion phenomenon, and the evolution of a shear band
can be reflected by porosity (Li et al., 2015).
As illustrated in Fig. 15a, the local porosity can be determined by
scattering and overlapping 492 measuring spheres with a diameter of 35
mm within the slope. Fig. 15b depicts the nephogram of porosity for the
slope with a 50% rock content. When s = 1/3 ΔS, the porosity in the
vicinity of the loading plate edges begins to increase. When s = 2/3 ΔS,
the porosity of the lower portion of the loading plate, i.e., the middle and
upper portions of the slope body, grows and tends to extend to the
adjacent slope surface. When s = ΔS, the region with increased porosity
forms a zone of through connectivity. There is a strong relationship
between porosity and the shear strength of the soil, which decreases
with increasing porosity (Tang et al., 2021), and the slope will be
damaged along the shear strength weak side (Zhou et al., 2009).
Similarly, the porosity was calculated by morphologically processing
the high-resolution pictures of the two local areas in Fig. 10, and the
results refer to Fig. 16. The local fields can be separated into shear, slip,
and stability zones based on the thickness of the shear zone (Lu et al.,
2022). The porosity-time curves of the two local fields are depicted in
4.3. Analysis of force chain
In discrete media, the structure formed by the transfer of forces be­
tween particles is referred to as a force chain (Cates et al., 1998), and the
evolution of this structure can reflect the macroscopic mechanical
Fig. 17. Local porosity-time curves.
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Computers and Geotechnics 157 (2023) 105268
Fig. 18. Contour maps of the porosity.
behavior of the material.
In this study, four conditions (Otto et al., 2003; Peters et al., 2005)
define the force chain on the slope: (I) There are more than three par­
ticles (including sand and rock) in the force chain; (II) As shown in the
Eq. (6), the absolute value of the major principal stress of each particle
in the force chain must be greater than the average value of the major
principal stress of all particles in the entire slope. The value and direc­
tion of the major principal stress in the particle are determined by Eqs.
(7), (8) and (9); (III) The angle between the line of the center of
neighboring particles in the force chain and the direction of the major
principal stress is less than a predetermined value θc , as depicted in
Fig. 19a and Eqs. (10) and (11); (IV) The direction of the force chain is
defined as the direction of the line connecting the center of the first and
last particle (Fu et al., 2019), as shown in Fig. 19b.
σ1 >
N ⃒ ⃒
1 ∑
⃒σ i ⃒
N i=1 1
σ (ϕ)
ij =
σ1 =
11
(6)
)
1 ∑( (c)
xi − x(ϕ)
Fj(c,ϕ)
i
V nc
σ 11 + σ 33
2
+
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
(σ − σ )2
11
33
+ (σ13 )2
2
(7)
(8)
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Computers and Geotechnics 157 (2023) 105268
becomes significantly larger and more irregular. This phenomenon in­
dicates a strong correlation between the strain localization in the slope
at the macroscopic level and the force chain transfer path among the
particles at the microscopic level.
To investigate the composition of the force chain and its develop­
ment in the slope, the aggregate amount ratio is evaluated according to
the following equations:
Fig. 19. Schematic of the force chain.
2σ13
σ 11 − σ 33
◦
θc =
180
〈Z〉
cosθc <
jσnext
1
≤1
|j||σnext
1 |
Mfc
× 100%
M
(12)
Rr =
Mr
× 100%
Mfc
(13)
where Rfc is the percentage of force chain content in the slope, Mfc is
the total mass of force chain, M is the mass of the slope, Rr is the per­
centage of rock content in force chain, and Mr is the total mass of high
stress rock.
The variation of Rfc and Rr, as well as the P-s curves for each scenario,
are depicted in Fig. 21. The variation curve of Rfc can be separated into
two stages, the first of which corresponds to the particle compression
and force chain expansion stages. In the second step, compaction of the
slope is complete, and the force chain network has developed and is
transferring stress. This resembles the pattern of the P-s curve; however,
the initial stage of Rfc occurs somewhat sooner than that of the P-s curve,
implying that the structure of the stable force chain is established before
the slope reaches its ultimate state. The change regularity of Rr is
comparable to Rfc. Table 4 lists the values of Rfc and Rr under various
conditions. From Fig. 21a and Table 4, Rfc fluctuates in a range once the
force chain reaches a stable state, and this fluctuation is more pro­
nounced as the rock content increases. During the process of static
loading, the force chain network is reorganized in response to the
modification of the particle skeleton structure, and the force chain
transfer path is modified accordingly. The Rfc falls as rock content in­
creases, which is interestingly the exact reverse of the tendency of the P-s
curve, whereas Rr approaches the rock content in the slope. The results
indicate that sand particles bear less load in slopes with a high rock
content, and that rock becomes the primary bearing medium, with rock
controlling the mechanical properties of S-RM. It can be seen from
Fig. 21b that when the rocks are concentrated at the top of the slope, i.e.,
the closer the location of the rocks is to the loaded place, the higher the
ratio of Rr is, and it has even surpassed the rock content in the slope, but
it has almost no effect on Rfc. Fig. 21c, d and Table 4 demonstrate that
the variation of Rfc and Rr under various rock shapes is similar. Conse­
quently, the composition ratio of rock in the force chain skeleton is
influenced by rock content and rock distribution but is unrelated to rock
shape.
The length of the force chain (FCL) is defined as the number of
particles in the force chain, while the strength of the force chain (FCS) is
defined as the average large principal stress in the force chain. By
integrating all the FCS, the average force chain strength on the slope can
be determined. An FCS of less than is referred to as a weak force chain,
while an FCS of greater than is referred to as a strong force chain. To
further analyze the distribution regularity of FCL and FCS, the proba­
bility density of FCL and FCS is normalized in this study, and the fitting
Eqs. (14) and (15) are presented. Where Eq. (14) represents the FCL
probability density formula and Eq. (15) represents the FCS probability
density formula:
Fig. 20. Distribution of force chains on slopes with different rock content
(unit: ΔSmax ).
tan(2θ) =
Rfc =
(9)
(10)
(11)
where N is the total number of particles, σ ij is the stress of the particle
in different directions, V is the volume of the particle (the area in 2D), θ
is the direction of σ 1 , Z is the average coordination number inside the
model, j is the branch vector connecting adjacent particles, and σ next
is
1
the major principal stress vector of the neighboring particles.
Fig. 20 depicts the distribution of force chains on slopes of varying
rock content. To accurately describe the distribution characteristics of
the slope sliding surface, the particles were colored after being divided
into 10 parts based on the slope maximum displacement ΔSmax (Zhang
et al., 2022). The black disks in the diagram represent high stress par­
ticles, while the gray polygons represent high stress rocks. The illus­
tration demonstrates that the force chain begins to develop beneath the
loading plate and extends primarily to the stable zone on the lower left
side of the boundary. Many continuous force chains are distributed
throughout the slope stable zone, which bears the bulk of the load,
although the slope slip surface has a discontinuous force chain. The slip
surface is the shear strength weakest facet. Fracture occurs when the
shear stress of the force chain on the slip surface exceeds the shear
strength of the particles. On a slope with a greater proportion of rock,
where the force chain distribution is relatively loose, the slip surface
y = y0 + Aexp( − x/t)
(14)
[
(
)]
y = A 1 − Bexp − Cx2 exp( − Dx)
(15)
Where A, B, C and D are all calculated coefficients.
Fig. 22 illustrates the probability density (Pl) curve of FCL for slopes
with varying rock contents. The curve fit reaches a value of 0.996. As
derived from the papers (Fu et al., 2019; Pöschel and Schwager, 2005),
the probability density curves of FCL are exponentially distributed, and
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Computers and Geotechnics 157 (2023) 105268
Fig. 21. Force chains constitution aggregate amount ratio under different condition.
the probability density decreases as FCL increases. The probability
density of the short force chains will increase as the slope rock content
increases. After reorganization of force chains, shorter force chains are
more likely to form on slopes with high rock content.
Fig. 23 depicts the normalized probability density (Ps) curves of FCS,
and Table 5 lists the peak probability and its corresponding FCS. Based
on Fig. 23 and Table 5, the probability density curve of the force chain
strength exhibits an ascending and then descending trend, with the
probability density peaking at 0.58–0.67 as the force chain strength
increases. Currently, FCS belongs to the interval of weak force chains,
indicating that most of the slope system force chain network is
composed of weak force chains. The interweaving of strong and weak
force chains constitutes a complete network of force chains that main­
tains the system stability.
According to Liu et al. (2022), the lower probability density of weak
force chains denotes a greater proportion of strong force chains in the
system. The higher FCS indicates that the system has a higher loadbearing capacity and facilitates the transfer of external loads. Fig. 23a
and Table 5 demonstrate that the slope has the greatest capacity to
transfer and diffuse external loads when RC = 80%, followed by RC =
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Computers and Geotechnics 157 (2023) 105268
deflection vector within the slope, with the direction of the vector arrow
representing the principal stress deflection. The major principal stress
field and the vector deflection area resemble the force chain distribu­
tion, as shown in the diagram. The stress is greatest beneath the
deflected loading plate, and the stress vector is nearly vertical. This area
is designated as the stress concentration zone and assigned with the
number 1. The stress is greater in the region adjacent to the left
boundary, the stress vector is deflected to the left with a greater
magnitude, and the area of deflection is elliptical. This area is designated
as the stress transfer area and is designated by the number 2. The area
below the center of the loading plate to the right of the slope where the
stress is lower and the stress vector is deflected to the right is known as
the stress diffusion area, designated by the number 3.
During stress redistribution, the state of contact between particles
adapts to the shift in the stress field. Using the contact distribution
function proposed by Rothenburg and Bathurst (1989):
Table 4
Rfc and Rr under different conditions.
Influence of rock content (RC)
RC
0%
10 %
Rfc (%)
25–26
23–25
Rr (%)
—
15.9
Influence of initial position of rock blocks
Rock
Randomly
Densely located
distribution
located
at top
Rfc (%)
21–24
21–25
Rr (%)
56.7
61
Influence of angularity of rock blocks
Edge number
3–4
5–6
19–24
19–23
Rfc (%)
Rr (%)
81
80.5
Influence of aspect ratio (AR) of rock blocks
AR
1.0–1.5
1.5–2.0
Rfc (%)
19–25
19–23
82.4
80.5
Rr (%)
50 %
80 %
21–24
56.7
19–23
80.5
Densely located
at surface
21–24
48.9
7–8
20–24
78.7
2.0–2.5
19–24
79.5
E(θ) =
1
[1 + an cos2(θ − θn ) ]
2π
(16)
where E(θ) is the density function of the contact normal distribution;
θn is the main direction of the contact normal anisotropy (angle with the
horizontal line); an is the Fourier fit coefficient, describing the
complexity of the anisotropy.
Fig. 25 depicts a comparison of the direction of the contact normal
when the slope reaches its maximum bearing capacity. Within a statis­
tical interval of 10◦ , the main direction of contact normal is counted, and
the rose slices show the ratio of the number of contacts in that direction
to the average number of contacts along the whole direction. Fig. 25a
depicts the distribution of the contact normal direction in three areas of
the slope containing 50% rock. The degree of contact anisotropy (an) is
depicted in the figure as area1 > area3 > area2. As the slope primary
bearing area, the contact in area 1 will be concentrated to resist external
loads. In area 2, the contact distribution will be more uniform under the
influence of both boundary and transferred loads, exhibiting the same
isotropic consolidation property. Since the area1 directly resists the
external load, it is primarily discussed in later sections. Fig. 25b, c, d,
and e show that the principal direction of the contact normal in area 1
remains essentially unchanged under the influence of various factors,
but an increases with increasing rock content, rock concentration in the
loaded region, rock angularity, and decreasing rock aspect ratio. Due to
its irregular shape and greater impedance effect, when rock is used as
the primary bearing medium of a slope, the misalignment produced in
space is significantly smaller than when sand particles are used. The
shear strength of granular material increases with angularity, and it has
been demonstrated that shear strength is also related to anisotropy
within the structure (Azema et al., 2013). Although the smaller aspect
ratio of the rock is not conducive to improved interlocking properties,
the anisotropy is compensated for by the rock reduced tendency to rotate
under contact forces. Higher anisotropy indicates that the contact forces
resisting external loads are more concentrated and that the intergranular
contact is tighter. The slope squeezing effect intensifies, causing the
bearing capacity to increase.
Fig. 22. Probability density curve of FCL with different rock content.
50%, 0%, and 10%. The slope ability to transfer and diffuse external
loads tends to decrease and then increase as the rock content rises,
different to the rule in Table 2 that the bearing capacity rises as the rock
content rises. This can be explained as follows: when the rock content is
low, the sand envelops the rocks, preventing direct contact between the
sands, and the rocks may interrupt the force chain transmission within
the slope, resulting in a reduction in force transmission capacity. How­
ever, because the resistance of rock is significantly greater than that of
sand, the slope bearing capacity is increased to some degree. When RC
= 50% and 80%, the probability density curves of FCS are similar,
indicating that when the rock content reaches a certain level, the slope
force transmission capacity tends to stabilize. In Fig. 23b, the slope with
a random distribution of rocks has the greatest force transfer capability,
whereas the slope with a more concentrated distribution of rocks away
from the loading location has a lower force transfer capability. It sug­
gests that the inhomogeneous media distribution may also easily break
the force chain within the slope, which is not conducive to force trans­
mission within the slope. Based on Fig. 23c and Table 5, it can be
determined that the slope force transfer capability increases as the an­
gularity of the rock increases. The effect of the rocks interlocking con­
tributes to the load transfer between the skeletons. In comparison, when
the aspect ratio is altered, the curve shifts slightly (Fig. 23d), but the
overall trend is similar, indicating that the effect of aspect ratio on the
force transfer capacity is minor.
5. Conclusions
In this study, DEM was utilized to investigate the micro-scale
mechanism when the S-RM slope reaches the ultimate bearing state.
Furthermore, the rock-wrapping motion in the shear zone was analyzed
in detail, and the effects of rock content, initial rock position, and rock
shape were considered, leading to the following conclusions:
(1) The slope ultimate bearing capacity increases with increasing
rock content and angularity but decreases with increasing aspect
ratio. Increasing the concentration of rock at the slope pressurebearing portion is conducive to enhancing the slope bearing
capacity.
4.4. Analysis of stress field and anisotropy
The process of static loading is accompanied by a redistribution of
stresses within the slope soil. Fig. 24 depicts the principal stress field and
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Y. Hu and Y. Lu
Computers and Geotechnics 157 (2023) 105268
Fig. 23. Probability density curves of FCS under different condition.
Table 5
Peak probability (Ps) of FCS under different conditions.
Influence of rock content (RC)
RC
0%
10 %
FCS (σ1 )
0.66
0.59
62.9
70.8
Ps (%)
Influence of initial position of rock blocks
Rock
Randomly
Densely located
distribution
located
at top
FCS (σ1 )
0.62
0.6
Ps (%)
58.4
65.8
Influence of angularity of rock blocks
Edge number
3–4
5–6
FCS (σ1 )
0.59
0.62
62.3
57.3
Ps (%)
Influence of aspect ratio (AR) of rock blocks
AR
1.0–1.5
1.5–2.0
FCS (σ1 )
0.61
0.62
Ps (%)
58.7
57.3
50 %
80 %
0.62
58.4
0.62
57.3
Densely located
at surface
0.58
73.2
7–8
0.67
53
Fig. 24. Major principal stress field and deflection vector.
2.0–2.5
0.63
56
rocks in the shear zone. The phenomenon of soil shear swelling
has a broader range as angularity increases. The rock-wrapping
phenomenon in the shear band becomes more obvious as the
aspect ratio increases. The angularity primarily affects the shear
swelling range, whereas the aspect ratio has a significant impact
on the shear swelling morphology.
(4) A strong correlation exists between the transmission path of the
force chain and the slip fracture deformation caused by slope
instability. When a certain percentage of rock is present, rock
becomes the primary load-bearing medium. The slope load
transfer capacity decreases and then increases with increasing
rock content, increases with increasing angularity, and decreases
as aspect ratio increases. Among them, aspect ratio has the least
impact. The uniformly distributed S-RM medium facilitates the
transmission of forces within the slope.
(5) The rock content of the slope, the initial position of the rock, and
the rock shape all influence the degree of anisotropy of the
(2) When the rock content reaches a certain threshold, the rock in­
fluence on the shear band becomes evident, and the shear band
becomes more curved and exhibits an obvious rock-wrapping
phenomenon. In the shear band, there are four types of rock
winding modes – unilateral rock bypass, bifurcation, bifurcation
+ crossing rock, and unilateral rock bypass + bifurcation +
crossing rock. The rotation amplitude of rock particles in the
shear band is less than that of sand particles. This differential
rotation and uncoordinated rock-wrapping movement are
responsible for the irregular destabilization damage to the S-RM
slope.
(3) The shear band will be accompanied by an obvious shear swelling
phenomenon when the S-RM slope is destabilized. The rapid
destabilization of the slope is caused primarily by the rotation of
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Fig. 25. Distribution of the contact normal direction.
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Computers and Geotechnics 157 (2023) 105268
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CRediT authorship contribution statement
Yangyu Hu: Methodology, Investigation, Writing – original draft. Ye
Lu: Investigation, Writing – review & editing.
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.
Data availability
Data will be made available on request.
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