Computers and Geotechnics 38 (2011) 14–29
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Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
Modeling shear behavior and strain localization in cemented sands
by two-dimensional distinct element method analyses
M.J. Jiang a,⇑, H.B. Yan a, H.H. Zhu a, S. Utili b,1
a
b
Dept. of Geotechnical Engineering, Tongji University, Shanghai 200092, China
Dept. of Engineering Science, Oxford University, Oxford OX1 3PJ, UK
a r t i c l e
i n f o
Article history:
Received 25 November 2009
Received in revised form 7 September 2010
Accepted 8 September 2010
Available online 12 October 2010
Keywords:
Cemented sand
Bond breakage
Strain localization
Numerical analyses
Distinct element method
a b s t r a c t
This paper presents a numerical investigation of shear behavior and strain localization in cemented sands
using the distinct element method (DEM), employing two different failure criteria for grain bonding. The
first criterion is characterized by a Mohr–Coulomb failure line with two distinctive contributions, cohesive and frictional, which sum to give the total bond resistance; the second features a constant, pressureindependent strength at low compressive forces and purely frictional resistance at high forces, which is
the standard bond model implemented in the Particle Flow Code (PFC2D). Dilatancy, material friction
angle and cohesion, strain and stress fields, the distribution of bond breakages, the void ratio and the
averaged pure rotation rate (APR) were examined to elucidate the relations between micromechanical
variables and macromechanical responses in DEM specimens subjected to biaxial compression tests.
A good agreement was found between the predictions of the numerical analyses and the available
experimental results in terms of macromechanical responses. In addition, with the onset of shear banding, inhomogeneous fields of void ratio, bond breakage and APR emerged in the numerical specimens.
Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Most natural soils are characterized by a bonded structure arising from various processes, for example, the solution and deposition of silica at particle contacts [1]. Soils are also sometimes
artificially cemented by chemical agents in ground-treatment processes. Experimental results in the literature have shown that cemented soils have peculiar behaviors different from uncemented
ones, such as stiffening at low pressure followed by yielding in a
manner similar to overconsolidated soils [2,3], enhanced strength
[4–8], a relatively brittle stress–strain response and a more dilative
volumetric response [6,9–11]. These findings have encouraged
extensive research into constitutive models of cemented soils.
Several continuum constitutive models have been suggested to
describe some important features of structured soils by previous
researchers [12–21]. Although the models differ in mathematical
details, they are all based on the principle that the size of the
state-boundary surface increases with interparticle bonding (for
instance, see [20]). However, concerning cemented soils, only the
macroscopic response can be observed in the laboratory, while
the mechanisms taking place at the micromechanical level remain
mostly unknown. One of the motivations for the present study was
⇑ Corresponding author. Tel./fax: +86 21 65980238.
1
E-mail address: mingjing.jiang@tongji.edu.cn (M.J. Jiang).
Formerly at Strathclyde University, Glasgow, UK.
0266-352X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compgeo.2010.09.001
to bridge the gap between macro- and micromechanics for these
types of soils.
In the past decades, a number of theoretical, experimental and
numerical works have been carried out on strain localization, particularly in granulates [22–26]. Regarding experimental works, the
1c2e apparatus developed in Grenoble in the 1990s was the first
intended to relate micromechanics to macromechanics by running
biaxial tests on Schneebeli wooden rods [27,28]. However, only a
few reports can be found on strain-localization analysis in clays
due to the theoretical, numerical and technical difficulties related
to this research field [29–33]. For example, it is very difficult to
gather sufficient microscopic data from specimens in the laboratory even with advanced technologies such as X-ray techniques
[34–36], stereophotogrammetric techniques [37], or particle-image velocimetry [38]. Such an unsatisfactory situation extends to
the analysis of strain localization in cemented sands, which constituted another strong motivation for this study.
It is authors’ opinion that the distinct element method (DEM)
presents an effective method to investigate the global mechanical
behavior, strain localization, and associated micromechanisms
occurring in cemented sands. The DEM was first proposed by
Cundall and Strack [39], in which a detailed description of the
method can be found. The main objective of this study was to
provide insight into the shear behavior and strain localization taking place in cemented sands by DEM analyses. For this purpose, a
series of biaxial compression tests was run on assemblies of 2D
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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
bonded disks. Two different bond models were employed, which
will be introduced in the following section. All the simulations
were run using PFC2D [40].
2. DEM numerical modeling
Two bond models were used in this study. The first, hereafter referred to as Jiang’s model, was initially proposed by Jiang et al.
[41,42] to investigate the yielding of microstructured sands,
whereas the second model is a standard model offered by PFC2D,
the so-called ‘‘contact-bond” model (see [40]). The common physical features of both models are illustrated in Fig. 1. Both bond models are made by a combination of a spring, a dashpot and a divider in
the normal direction, whereas a spring, a dashpot and a slider are
present in the tangential direction. However, in all the simulations
performed in this study, damping was set to either zero or to a very
small value; this was verified not to influence the results obtained.
Hence, no dashpot was actually present in the bonds employed in
the performed simulations. The global behavior of a geomaterial depends on the type of breakage it undergoes, which can be either
fragile or ductile (see [43,44]). It is generally accepted that the
breakage process is mainly fragile in cemented sands. This is also
underpinned by tests recently performed on glued steel rods, as reported in [45]. Therefore, the bonds used in the presented simulations break in a purely fragile fashion, as illustrated in Fig. 2.
Here, the main mechanical features of the employed bond models
are shown in terms of the normal and tangential relative displacements versus the respective contact forces, Fn and Fs. Fig. 2a2 and b2
show the relationships between force and displacement achieved
by experimental tests carried out on pairs of Schneebeli steel rods
glued together. These experimental tests were meant to mimic
the mechanics of two-dimensional bonded disks. According to the
results of these experiments, the relationship between force and
displacement is linear until the bond breaks. For uncemented granular geomaterials, several researchers have proposed nonlinear
relationships to take into account the dependency of the material
stiffness on the magnitude of normal forces exchanged at the contacts. However, in the case of bonded materials, the force–displacement relationship can be satisfactorily approximated by a simpler
linear one, which requires the calibration of fewer parameters, as
the bonding material is responsible for most of the compliancy
(i.e., compressive stresses remain sufficiently low to avoid significant compression of the grains), as also indicated by the available
experimental evidence [45,46]. Therefore, the parameters determining bond behavior (see Fig. 2) are Rn, the normal bond strength,
Rt, the tangential bond strength, Kn, the normal contact stiffness, Ks,
the tangential contact stiffness, and /l, the interparticle friction angle. Fig. 2a shows that the two models display the same perfor-
Fig. 1. Physical analogue of the employed bond models.
mance in compression and tension, but with different shear
strengths. Fig. 2b shows that the shear strength in the Jiang model
is pressure dependent and increases linearly with the normal force
according to the Mohr–Coulomb criterion. The bond strength can be
thought of as the sum of frictional and cohesive contributions. The
shear strength of the standard bond model in the PFC2D code (PFC
model) is instead pressure independent at low stresses, while it increases linearly with the normal pressure at high stresses. Therefore, this bond strength can be assumed to be provided by a
purely cohesive contribution when the normal contact force acting
on the bond is low (Fn < Rt/tan /l), and a purely frictional one when
the force is high (Fn > Rt/tan /l). With the current experimental
apparatus, an accurate determination of the failure envelope for
cementitious bonds is a difficult task. Moreover, there exists no
consensus among researchers on what bond model best represents
reality. Therefore, both models were used in this work. In summary,
the failure envelope according to the Jiang model is:
F n;f ¼ Rn
F s;f ¼ Rt þ F n tan /l
ð1Þ
and for the PFC model is:
(
F n;f ¼ Rn
F s;f ¼
Rt
if F n 6 Rt = tan /l
F n tan /l
if F n > Rt = tan /l
ð2Þ
It may be argued that the sharp transition between tensile and
shear resistance employed in both bond models is not realistic because there is a kink point along the strength envelope (point B in
Fig. 2c), whereas it would be reasonable to expect a smooth
transition, for instance, given by a curve starting from point A and
tangentially reaching the inclined straight line Fs = Fntan /l (see
Fig 2c). However, the shape of such a curve is uncertain and the
adoption of a nonlinear failure envelope rather than a linear one
is likely to have a small influence on the global behavior of the granular material. Therefore, it can be concluded that the adoption of a
nonlinear failure envelope for bonds is not justified given the current state of knowledge of bonded granulates. For the same reason,
Rn was taken as equal to Rt. Concerning the contact-stiffness parameters, the values used in all the performed simulations were independent of the amount of cement. Although the small strainelastic stiffness of cemented soils depends on the cement content
(see [47]), the values of the contact stiffnesses, Kn and Ks, employed
in the simulations were constant, as the soil elastic response at
small strains was not the focus of this study. The same choice of
constant contact stiffnesses was utilized by Wang and Leung (see
[54]).
The particle-size distribution shown in Fig. 3, featuring ten different particle radii, was used to generate the DEM specimens. The
numerical specimens were 800 mm wide and 1680 mm high. The
planar void ratio at the beginning of the biaxial tests was 0.27,
which is representative of a loose sample. The total number of sand
particles employed in each numerical specimen was around
24,000. In some preliminary simulations, fewer particles (6000)
were used, with the macromechanical response showing no significant differences. However, if accurate information on shear-band
width and strain and rotation fields within the shear bands is
sought, such a small number of particles would be insufficient.
With the adoption of 24,000 particles per specimen, the resulting
ratios of shear-band width to average particle diameter were
approximately 14 at an axial strain of 6% and approximately 17
at an axial strain of 13%, which are large enough to obtain reliable
information on the shear bands. This can be regarded as a
outstanding feature of the simulations performed here, as in previous DEM studies on other geomaterials the number of particles lying within the shear band appeared to be too small to draw reliable
conclusions about the kinematic mechanisms taking place,
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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
10
Normal force (kN)
8
6
3
(a2)
Jiang model (R=1500N)
PFC model (R=1500N)
Jiang model (R=5000N)
PFC model (R=5000N)
2
Normal force (kN)
(a1)
4
2
0
1
Compression
Jiang et al, 2007
0
Delenne et al, 2004
-1
-2
Tension
-2
-4
-6
-0.06 -0.04 -0.02
0.00
0.02
0.04
0.06
0.08
-3
-0.20 -0.15 -0.10 -0.05
0.10
(b2)
10
Shear force (kN)
Shear force (kN)
5
0
-10
-0.10
Jiang model (R=1500N)
PFC model (R=1500N)
Jiang model (R=5000N)
PFC model (R=5000N)
-0.05
0.00
0.15
0.20
Normal force: 4kN
0.05
Jiang et al, 2007
Normal force: 2kN
2.0
Normal force: 0kN
Delenne et al, 2004
1.5
Normal force: 0kN
1.0
0.5
0.0
0.0
0.10
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Shear displacement (mm)
Shear displacement (mm)
(c)
0.10
3.0
2.5
Normal force: 0 kN
-5
0.05
Normal displacement (mm)
Normal displacement (mm)
(b1)
0.00
2.0
Fn=-0.5
Fs= 0.5
3 Fs= Fntanφμ
1
Shear strength (kN)
2
1.5
1
1.0
2
Jiang model
PFC model
Fn=-0.5
Fs= Fntanφμ+0.5
2
3
0.5
2
B
1
0.0
-1.0
A
Residual strength
of both models
1
-0.5
ϕu =° 26.6
0.0
0.5
1.0
1.5
2.0
Normal force (kN)
Fig. 2. Mechanical responses of the contact-bond models used in the DEM analyses. (The data in (a2) and (b2) come from [45,56]). (a1) Model prediction in normal direction.
(a2) Experimental data in normal direction. (b1) Model prediction in tangential direction. (b2) Experimental data in tangential direction. (c) Relationships between shear
strength and normal force. (The solid line represents peak strength while dashed line residual strength).
rotation rates, bond-breakage rates and strain field inside and outside the shear bands.
To simulate the soft rubber membrane used to confine soil samples in geotechnical tests, flexible side boundaries consisting of
bonded particles were employed, as originally proposed in [48].
The stress-controlled flexible boundary implemented in the code
is the same as that used by Wang and Leung [49], to which the reader is referred to for details of the boundary implementation in DEM
code. As already shown in [50,51,58], the use of flexible membrane
boundaries allows the model to capture the deformation characteristics of developing shear bands with a good degree of accuracy. The
input parameters for sand grains and membrane particles are summarized in Table 1. The top and bottom boundaries were simulated
by rigid walls having the same normal and tangential contact stiffnesses as the sand particles. The coefficient of friction between
walls and particles was set to zero to reproduce ideal experimental
conditions.
The multilayer undercompaction method proposed by Jiang
et al. [52] was used to generate the packing of particles so as to obtain loose and homogeneous specimens. Details on the generation
procedure can be found in [52]. Five horizontal layers were used
during specimen generation, with each layer containing 4800 particles randomly distributed in a rectangular area 800 mm wide and
437 mm high. Particles were compacted to the target planar void
ratio, ep = 0.27, by moving the top rigid wall downward at a constant speed of 5.0 m/s with the lateral and bottom walls fixed.
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
17
Percentage of finer (%)
100
80
60
40
20
0
5
6
7
8
9
10
Grain diameter (mm)
Fig. 4. DEM specimen after generation.
Fig. 3. Particle size distribution used in the DEM analyses.
Table 1
Sample size and material parameters used in the DEM simulations.
Samples
Width of sample (mm)
Height of sample (mm)
Initial void ratio
Sand particles
Total number in sample
Density (kg/m3)
Diameter (mm)
Normal contact stiffness for test (N/m)
Tangential contact stiffness for test (N/m)
Interparticle coefficient of friction for test
Coefficient of friction between wall and particle
Normal contact stiffness for specimen generation (N/m)
Tangential contact stiffness for specimen generation (N/m)
Interparticle coefficient of friction for specimen generation
Normal contact stiffness between sand and membrane
particles (N/m)
Local damping coefficient
Viscous damping coefficient
In normal and tangential directions
Membrane particles
Diameter (mm)
Density (kg/m3)
Normal bond strength in normal direction (N)
Bond strength in tangential direction (N)
Normal contact stiffness (N/m)
Tangential contact stiffness (N/m)
800
1680
0.27
24,000
2600
Gradation in
Fig. 3
7.5 107
5.0 107
0.5
0.0
7.5 109
5.0 109
1.0
3.75 106
0.5
0.0
2.0
1000
1.0 10100
1.0 10100
3.75 106
2.5 106
In order to generate a sufficiently loose sample, a coefficient of
interparticle friction, tan /l = 1.0, was used during the generation
process. The numerical specimen obtained is shown in Fig. 4.
After specimen generation, the coefficient of interparticle friction was set to tan /l = 0.5. The specimens were then consolidated
in 1D under a small vertical pressure of 12.5 kPa with the side
walls kept fixed. In this stage, contacts between particles were
found to be distributed uniformly within the specimens. Then
bonds were formed at all particle contacts. The specimens were
subsequently subjected to an isotropic constant confining pressure.
Finally, the specimens were vertically compressed, with the top
wall moving downward and the bottom wall moving upward, both
at a constant speed, while the lateral pressure acting on the particle membranes was kept unchanged. The strain rate adopted in all
the simulations was 6.0% axial strain per minute. This low strain
rate ensured that quasistatic conditions were always present during testing. Moreover, this low rate also ensured that the pressures
measured at the top and bottom walls remained similar throughout the test. These conditions imply that the stress field within
the specimen can be considered uniform on average until the
occurrence of localization. Note that, although the numbers of contacts was the same when bonds were assigned to particles, the
number of bonds at the beginning of the biaxial test differed
depending on the bond model employed. This is due to the difference in implementation of the force-relative-displacement relationship in the normal direction between the Jiang and PFC
models. In the Jiang bond model (see [41,42]), particles always
overlap at the bonded contacts. In the PFC bond model (see [40]),
bonds can exist with particles separated by a certain distance proportional to the exchanged tension force Fn. When the contact normal force is zero, the overlap is nil. However, such a difference does
not affect the main conclusions in this study.
A series of biaxial compression tests was performed with different bond strengths, i.e., 0 N, 1.5 kN, and 5 kN, and under different
confining pressures, i.e., 50 kPa, 100 kPa, 200 kPa, 400 kPa, and
800 kPa. To describe the macroscopic mechanical responses of
the specimens, two-dimensional stress invariants were employed:
the mean effective stress, s = (ry + rx)/2, and the deviatoric stress,
t = (ry rx)/2.
3. Mechanical behavior of bonded DEM specimens
3.1. Stress–strain and volumetric responses
Figs. 5 and 6 present the numerical results obtained from biaxial compression tests on the Jiang and PFC models under confining
pressures of 50 kPa, 200 kPa and 800 kPa. For reference, the responses of uncemented specimens having the same initial void
ratio are also plotted in the figures. Figs. 5 and 6 show that with
both the Jiang and PFC bond models specimens were characterized
by strain softening and shear dilatancy, unlike the uncemented
case. Moreover, peak deviatoric stress and dilatancy increased with
bond strength. These trends are in agreement with the available
experimental data, shown in Fig. 7, reported by [49]. From these
data emerges evidence that grain cementation significantly alters
the stress–strain response of loose sands, which here changed from
strain hardening to strain softening with an increasing degree of
cementation. Analogously, the volumetric response switched from
contractive to dilative. At a micromechanical level, this type of response is due to the formation of particle arches within the shear
band. This in turn occurs because of the breakage of some bonds,
causing the formation of clusters of bonded particles which are free
to rotate, thus contributing to volumetric dilation. A detailed
description of the kinematics of cluster formation in loose bonded
granulates and the consequent dilative behavior can be found in
Wang and Leung [54].
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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
(b)
350
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
Deviatoric stress (kPa)
300
250
Deviatoric stress (kPa)
(a)
200
150
100
500
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
400
300
200
100
50
0
0
0
2
4
6
8
10
12
14
0
2
4
(c)
1000
Deviatoric stress (kPa)
Axial strain (%)
800
6
8
10
12
14
Axial strain (%)
600
400
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
200
0
0
2
4
6
8
10
12
14
Axial strain (%)
Fig. 5. Stress–strain responses of cemented sands for various bond strengths under different confining pressures from DEM simulations: (a) r3 = 50 kPa; (b) r3 = 200 kPa;
r3 = 800 kPa.
Comparing the stress–strain response obtained from the Jiang
model with that from the PFC model, it was observed that, at
low confining stresses, the specimen in the PFC bond model exhibited a higher peak stress and a more pronounced strain softening
than the specimen in Jiang’s model. For instance, the response obtained at R = 5 kN and r3 = 50 kPa was characterized by a peak
deviatoric stress of 325 kPa in the Jiang model and 175 kPa in the
PFC model. However, at high confinement, the specimen in the
Jiang model exhibited the higher peak deviatoric stress and more
pronounced strain softening. This can be explained by considering
the strength envelopes of the two bonds, which are shown in Fig. 2.
The tensile strength of the two bonds is identical but the shear
strength is different. Therefore, although bonds may fail in either
tension or shear, the different macromechanical behavior obtained
was mainly due to the different bond breakage under shear. In the
tension zone (e.g., Point B in Fig. 2c), the shear strength of the specimen in the Jiang model was lower than that in the PFC model,
whereas for bonds in compression it was the opposite. This explains why the peak deviatoric stress obtained from the PFC model
with R = 5 kN was higher than that in the Jiang model at r3 =
50 kPa but was the same at r3 = 200 kPa and lower at
r3 = 800 kPa. These results, especially those obtained from the
Jiang model, show a good agreement with the available experimental data from a qualitative viewpoint (see Fig. 7). In particular,
examining the experimental curves corresponding to various cement contents, the marked increase of peak deviatoric stress and
softening taking place when the cement content changed from
2% to 3% was captured well by the Jiang model. In addition, Fig. 6
shows that specimens with Jiang-model bonds demonstrated more
dilation than those with PFC-model bonds.
Finally, it can be noted that in case of the PFC model the resulting stress–strain curve became progressively closer to the curve of
the unbonded specimen with increasing confinement; these ultimately coincide in the case of R = 1.5 kN and r3 = 800 kPa (see
Fig. 5c). To explain the observed trend, it is necessary to consider
that, according to the PFC model, the peak and the residual bond
strengths coincide for high normal contact forces whereas they
are distinctively different when the normal forces are low. This
means that a higher confining stress leads to a higher number of
bonds in the residual state and thus a smaller difference between
bonded and unbonded specimens. Therefore, the fact that the
bonded and unbonded curves coincide means that the majority
of the PFC bonds reached their residual states.
3.2. Failure envelopes
Fig. 8 shows the envelopes of the peak and residual strengths
obtained by the numerical simulations in the Jiang and PFC bond
models and in the uncemented case. It can be seen here that both
the peak and residual strengths of the cemented specimens are larger than the strengths exhibited by the uncemented ones. This is
due to bonding effects. The apparent cohesion increases with bond
strength in both bond models. However, the obtained friction angles varied in different ways. The peak friction angle, /peak, obtained from the Jiang specimens increased with bond strength;
conversely, /peak obtained from the PFC specimens decreased with
bond strength. This difference can be linked to the different peakstrength envelopes present in the two adopted bond models. The
obtained residual friction angle, /res, increased slightly with bond
strength in both bond models, the increase being higher in the case
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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
(b)
-8
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
-6
-4
-2.5
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
-2.0
Volumetric strain (%)
Volumetric strain (%)
(a)
-2
0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2
0
2
4
6
8
10
12
0
14
2
4
Axial strain (%)
(c)
8
10
12
14
0.0
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
0.2
Volumetric strain (%)
6
Axial strain (%)
0.4
0.6
0.8
1.0
1.2
0
2
4
6
8
10
12
14
Axial strain (%)
Fig. 6. Volumetric vs. axial strain of cemented sands for various bond strengths under different confining pressures from DEM simulations: (a) r3 = 50 kPa; (b) r3 = 200 kPa;
r3 = 800 kPa.
(a)
(b)
Fig. 7. Experimental data obtained from laboratory tests on sands with various degrees of cementation at 50 kPa confining pressure (after Wang and Leung [49]): (a) stress–
strain relationship, and (b) void ratio-strain relationship.
of the Jiang bond model. Fig. 9 shows the peak and residual
strength envelopes observed experimentally by Wang and Leung
[49] for cemented sands; here, apparent cohesion, peak and residual friction angles all increased with cement content. Comparing
our numerical results with these experimental data, the Jiang model proved to be better than the PFC model in capturing the material
behavior. The degree of agreement with the experimental data is
quite remarkable considering that the bond models employed in
these analyses were fairly simple, as they assumed linear strength
envelopes and did not account for any exchange of moments.
4. Strain localization in bonded DEM specimens
In this section, both the Jiang and PFC bond models were investigated with regard to strain localization. For brevity’s sake, in the
following, the results from only two specimens were selected to
present and discuss: one each with bonds in the Jiang model and
PFC models, both specimens being confined at a pressure of
r3 = 100 kPa and with a bond strength of R = 5 kN. However, the
choice of restricting the description of the results obtained to just
these two samples was justified by the fact that the behaviors
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M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
(a) 600
(b)
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
400
300
200
400
300
200
100
100
0
Jiang sample (R=5000N)
Jiang sample (R=1500N)
PFC sample (R=5000N)
PFC sample (R=1500N)
Uncemented sample
500
Shear strength (kPa)
Shear strength (kPa)
500
600
0
0
200
400
600
800
1000
1200
1400
0
Mean normal stress (kPa)
200
400
600
800
1000
1200
1400
Mean normal stress (kPa)
Fig. 8. Achieved strength envelopes for various bond strengths: (a) peak strength envelopes, and (b) residual strength envelopes.
(a)
(b)
Fig. 9. Strength envelopes of cemented sand observed experimentally at different cement contents (after Wang and Leung [49]): (a) peak strength envelopes, and (b) residual
strength envelopes.
observed in tests run for different bond strengths and confining
pressures were similar (data not shown).
In Fig. 10, the obtained stress–strain and volumetric responses
for the aforementioned specimens are shown. The test can be divided into five stages, with starting and ending points as indicated
by points O, A–E. Point O indicates the initial state before any vertical load is applied. Point A can be defined as a ‘‘yielding point”, as
this is where some bonds begin to break. This point also marks the
transition between dilative and contractive behaviors. Point B
marks the peak of the deviatoric stress and point C the occurrence
of maximum dilatancy. At point D, the volumetric strain becomes
nearly constant. Finally, point E signals the end of the test. These
stages will be used in the next subsections to describe the features
of the observed strain localization in the two specimens.
4.1. Deformed specimens
The deformation patterns of the two numerical specimens observed at different axial strains are shown in Fig. 11. Square grids
of particles with different colors were employed to illustrate the
characteristics of the strain field in the specimens [53]. Fig. 11a
and f shows that the strain field at the yielding (point A in Fig. 9)
was homogeneous in both specimens. Fig. 11b and g shows that
the development of a shear band when the peak of the deviatoric
stress is reached, i.e., point B in Fig. 10, was more pronounced in
the specimen in the Jiang model than in the specimen in the PFC
model. Fig. 11c–e and h–j shows that the evolution of the strain
fields in the two numerical specimens was similar in the remaining
stages, with shear bands developing from the top-left to the bottom-right corner of the specimens and showing a similar inclination of about 52° to the horizontal.
4.2. Contact-force chains and stress fields
In Fig. 12, the formation of contact force chains can be observed
in the two bonded specimens at different stages. The thickness of
the lines in the figure is proportional to the magnitude of the contact forces. A webbed pattern of force chains, with thick chains distributed mainly in the vertical direction, can be observed until the
peak of the deviatoric stress is reached. This is in agreement with
previously published works. Then, some thicker, columnar chains
progressively developed with increasing axial strain, as illustrated
in Fig. 12c–e and h–j. Moreover, it was found that these columnar
force chains, which were initially oriented along the vertical direction, gradually rotated away from the shear band inclination but
not so much as to become perpendicular to it.
One way of describing the state of stress within the specimens at the onset of shear banding is to calculate the average
stresses. This can be done by considering a suitable number of
particles within a representative area and calculating the equivalent stress tensor from contact orientations and forces [39]. In
this study, a simplified method included in the PFC2D code
(see [40]) was used to obtain a stress tensor by averaging stresses over a circular area inside the specimen. Preliminary analyses
21
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
(b) 400
400
Deviatoric stress (kPa)
Volumetric strain:
P: 0.38% Q: -0.74%
B
350
A
300
B
250
V o lu m e tric s tra in :
P : 0 .1 9 % Q : -1 .4 %
C
A
200
S tre s s -s tra in c u rv e
D
150
E
100
V o lu m e tric re s p o n s e
Q
50
0
O
0
2
4
6
8
10
12
A
300
Stress-strain curve
250
C
14
Volumetric response
200
Bonding breakage
D
150
E
Q
100
50
O
0
B o n d in g b re a k a g e
P
Volumetric strain:
P: 0.38% Q: -0.74%
B
350
Deviatoric stress (kPa)
(a)
P
0
2
4
6
8
10
12
14
Axial strain (%)
Axial strain (%)
Fig. 10. Mechanical responses for the cemented samples used to describe strain localization (R = 5 kN, r3 = 100 kPa): (a) Jiang sample, and (b) PFC sample.
(a) 1.2%
(b) 2%
(f) 1.2%
(g) 1.7%
(c) 6%
(d) 8%
(h) 6%
(i) 8%
(e) 12%
(j) 12%
Fig. 11. The deformed DEM specimens controlled by two bond models at different axial strain that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample:
(a–e), PFC sample: (f–j)).
showed that the accuracy of the obtained stresses depended on
the ratio between the radius of the chosen circular area and
the radii of the soil particles. If the radius of the circle was sufficiently large, e.g., 10–12 times the average particle radius, the
calculated stress values were practically unaffected by the size
of the radius of the circle. In Fig. 13, vectors of the principal
stresses obtained in this way are shown at various stages of
loading for both numerical specimens. Note here that the stress
fields remained homogeneous until the onset of shear banding,
with the direction of the major principal stress almost vertical
everywhere until the peak of the deviatoric stress was reached
(point B in Fig. 10). Once the deviatoric stress exceeded the peak,
the direction of the major principal stress inside the shear band
started to rotate gradually away from the shear-band direction.
In contrast, outside the shear band the principal stresses
remained unchanged.
22
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
(a) 1.2%
(b) 2%
(f) 1.2%
(g) 1.7%
(c) 6%
(h) 6%
(d) 8%
(i) 8%
(e) 12%
(j) 12%
Fig. 12. Contact force chains obtained in two different bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample:
(a–e), PFC sample: (f–j)).
4.3. Bond-breakage fields
It is generally accepted, although not yet confirmed either
experimentally or numerically, that the formation of shear bands
in cemented sands is associated with bond breakage. To investigate
this aspect, the relationship between bond-breakage rate and axial
strain obtained for the two analyzed numerical specimens is
shown in Fig. 14. The bond-breakage rate g was defined as:
g¼
ðN1 N2 Þ=N
e2 e1
ð3Þ
where N is the total number of bonds at the beginning of the compression test and N1 and N2 represent the number of intact bonds at
strains e1 and e2, respectively. Note that N amounted to 25,365 for
the specimen in the Jiang model and 33,543 for the specimen in
the PFC model. This is due to the fact that bonds were assigned to
particle contacts in different ways, as described in Section 2.
As shown in Fig. 14, most bonds remained intact until the two
specimens reached point A, previously defined as the ‘‘yielding
point”, where the volumetric response changed from contractive
to dilative. Then, bond breakages occurred differently in the two
specimens. In the case of the Jiang model, the peak value of
the bond-breakage rate was quite large and occurred soon after
the peak of the deviatoric stress was reached. After this peak, the
bond-breakage rate was significantly reduced; from this much
lower rate, it then decreased gradually until the end of the test
(see CDE in the figure). Conversely, the specimen in the PFC model
showed a lower peak value but a higher rate in the remaining part
of the test, with a non-negligible number of bonds still breaking at
constant volumetric strain at the end of the test. Considering the
results shown for both specimens, it was concluded that bond
breakage started at the material yielding point and reached its
maximum value during strain softening.
Fig. 15 illustrates the distribution of bond breakages observed in
the two bonded specimens at different values of axial strain. Bond
breakage was concentrated within a narrow region, tending to
coincide with the shear bands forming in the two specimens. In
the specimen in the Jiang model, only one shear band developed
continuously during the test. Conversely, several shear bands
formed along different conjugate lines in the specimen in the PFC
model, with an inclination of about 52° to the horizontal. Among
the several shear bands taking place, only one became significantly
thick, with all the other bands remaining relatively thin. Measuring
the bond breakages occurring in five circular regions, it was observed that only a few bonds broke outside the shear bands, as
can also be seen in the images of Fig. 15. This indicates a strong link
between bond breakage and the formation of either permanent or
transient shear bands.
To assess the cumulative effect of bond breakage over time, the
evolution of the number of bonds within each circular region is
shown in Fig. 16 for both bond models. In the case of circles lying
entirely outside a shear band, the number of bonds remained prac-
23
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
Fig. 13. Stress fields in the two bonded samples measured numerically at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample:
(a–e), PFC sample: (f–j).
Volumetric strain:
P: 0.38% Q: -0.74%
300
6
Stress-strain curve
250
C
5
Volumetric response
200
Bonding breakage
D
150
4
3
E
Q
100
50
2
1
O
0
0
P
0
2
4
6
8
10
12
3
B
250
7
A
14
Axial strain (%)
Deviatoric stress (kPa)
B
350
Deviatoric stress (kPa)
(b) 300
8
Volumetric strain:
P: 0.19% Q: -1.4%
C
A
200
Stress-strain curve
2
D
150
E
Volumetric response
100
1
Q
50
0
O
Bonding breakage
P
0
2
4
6
8
10
12
Bonding breakage rate
400
Bonding breakage rate
(a)
0
14
Axial strain (%)
Fig. 14. Bonds breakage rate against axial strain observed in the DEM cemented samples (R = 5 kN, r3 = 100 kPa): (a) Jiang sample, and (b) PFC sample.
tically unchanged, as expected. On the contrary, in the case of circles lying inside a shear band, it is worth noting that the evolution
of the bond breakages was different for the two specimens (see
curve 5 in Fig. 16a and b). In fact, in the case of the Jiang bond model, a roughly linear decrease of intact bonds over time was observed, whereas in the case of the PFC bond model the decrease
was parabolic, with more bonds breaking at small strains than at
large strains. Moreover, the final number of broken and intact
bonds at the end of the test (ea = 12%) also differed: in the case of
the Jiang bond model, 79% of bonds were still intact, whereas in
the case of the PFC bond model the percentage of intact bonds
was only 63%.
These results are different from those shown in [49], where a
significant number of bond breakages occurred outside the shear
bands as well (see Fig. 17). Such a difference may be attributed
to the features of the bond model employed by Wang and Leung,
which make use of several small particles one order of magnitude
smaller than the sand grains to reproduce the bonding cement. At
present, it would be premature to conclude which bond model is
more realistic, as simplifying assumptions were introduced in both
24
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
Fig. 15. Distributions of bond breakage in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa). Jiang sample:
(a–e); PFC sample: (f–j).
(a)
Number of bonds
Number of bonds
(b)
Axial strain (%)
Axial strain (%)
Fig. 16. Number of bonds within and outside shear bands (cycles 1–5 in Fig. 14) in the two bonded samples at different axial strains (R = 5 kN, r3 = 100 kPa). Jiang sample (a);
PFC sample (b).
[49] and the present models. In [49], the PFC parallel bond model
was employed at the contacts between the small particles representing the bonding cement and the large particles representing
the sand grains. In the present study, a simpler numerical modeling
of the cementing material was used to make an investigation of the
influence of bond strength and microscopic failure criteria on both
macroscopic mechanical response and the onset of shear banding
more feasible.
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
4.4. Void-ratio distribution
In Fig. 18, maps of the distribution of void ratios in the two
cemented specimens obtained at different axial strains, corresponding to points A–E in Fig. 10, are shown. The results obtained
from both specimens show that the void ratio remained homogeneous until the peak of the deviatoric stress was reached. After this
(a)
25
point, the void ratio became increasingly larger (i.e., dilation) within the shear bands as the axial strain increased. In contrast, the
void ratio outside the shear bands changed only slightly during
the test. This indicates that the volumetric dilation of the whole
specimen was almost entirely due to the dilation occurring within
the shear bands. This observation is in good agreement with the
available experimental evidence on structured soils [30].
(b)
Fig. 17. Numerical results obtained from biaxial test on cemented sand with different bond model (after [49]): (a) bonding network and location of shear band, and (b)
distribution of bond breakage.
Fig. 18. Distributions of void ratio in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC
sample: (f–j)).
26
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
In addition, Fig. 18 shows that the void ratio within the shear band
at large axial strain was larger in case of the Jiang bond model than
in the case of the PFC bond model. This is also in agreement with
the results presented in Section 3.1, namely, that the specimen in
the Jiang bond model demonstrated a larger overall dilation than
the specimen in the PFC bond model (see Fig. 6).
It is interesting to compare the maps of void-ratio distribution
in Fig. 18 with the results reported in [55] relative to biaxial tests
run on uncemented sand specimens using flexible boundaries. In
[55], Bardet and Proubet found that the determining the distribution of the void ratio was not helpful in identifying shear bands
and therefore concluded that ‘‘in two-dimensional materials, the
(the use of the) volumetric strain calculated from the void ratio
is inappropriate to detect shear bands.” However, they acknowledged that ‘‘this observation is in disagreement with the radiographic measurements of shear bands on real sands”, referring to
those reported in [56,57]. In our other simulations (data not
shown), we found that strain localization did not occur in the case
of an uncemented specimen with the same initial void ratio as the
cemented specimens studied in this work. On the contrary, shear
banding did occur in the case of a dense packing of uncemented
particles, as reported in [58]. Therefore, we concluded that the
0.1
0.0
-0.1
-0.2
1
2
3
4
5
-0.3
-0.4
-0.5
A B
0
2
C
D
4
6
8
Axial strain (%)
10
12
14
(b)
0.2
Averaged pure rotation rate
(a)
Averaged pure rotation rate
Fig. 19. Distributions of APR in the two bonded samples at different axial strains that correspond to points A–E in Fig. 9 (R = 5 kN, r3 = 100 kPa, Jiang sample: (a–e), PFC
sample: (f–j)).
0.1
0.0
-0.1
-0.2
-0.3
1
2
3
4
5
-0.4
-0.5
-0.6
A B
0
2
C
D
4
6
8
10
12
14
Axial strain (%)
Fig. 20. Relationships between APR and axial strain observed in the two cemented samples at measurement circles 1–5 shown in Fig. 14 (R = 5 kN, r3 = 100 kPa): (a) Jiang
sample, and (b) PFC sample.
27
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
presence of bonds substantially changes the characteristics of
strain localization because persistent shear bands were detected
in the void-ratio distribution maps at axial strains as low as 1.8%
in the Jiang specimen and 2.8% in the PFC2D specimen.
4.5. Distributions of averaged pure rotation rate (APR)
The first author [59] has recently proven that the energy dissipation between sand particles is related to their relative sliding
displacement and can be further expressed in terms of their sliding
rotation rate. The sliding rotation rate consists of two parts: one related to particle translation and the other to particle rotation and
radius. The second part, hereafter termed the pure rotation rate,
h_ p (see [59]), can be expressed by:
1
h_ p ¼ ðr 1 h_ 1 þ r 2 h_ 2 Þ
r
ð4Þ
shear band developed in the specimens. Fig. 21 displays the evolution of both average void ratio and APR for each band at the different stages of the test. In Fig. 21a, the adopted local coordinate
system and measurement bands are shown. It can be observed that
at yielding (point A) both the average void ratios and APRs in each
measurement band were very similar, with the APR values being
almost zero. After point B was reached, the average void ratio
and APR inside the persistent shear band drifted away from the
values measured outside the shear band. A comparison between
Fig. 21b and c shows that: (1) the thickness of the shear band in
the Jiang specimen was larger than in the PFC specimen; and (2)
the thickness of the shear band depended on which variable was
used to identify it. For instance, if the void ratio was used, this
(a)
where r1 and r2 are the radii and h_ 1 and h_ 2 are the rotation rates of
the two particles in contact; here, r is the equivalent radius of the
two particles in contact, defined as:
r¼
2r 1 r 2
r1 þ r2
ð5Þ
Therefore, the averaged pure rotation rate x (APR) can be expressed by [59–61]:
ð6Þ
4.6. Thickness of shear bands
To investigate the thickness of the shear bands, the specimens
were divided into 25 measurement bands, all parallel to the main
0.3
0.35
0.2
0.30
0.25
0.1
Shear band
0.20
0.0
APR
A
B
C
D
E
-0.1
-0.2
-0.3
0.0
0.1
0.2
Void ratio
A
B
C
D
E
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.15
Void ratio
(b)
0.10
0.05
0.00
1.0
Distance from origin (m)
(c)
0.3
0.35
0.2
0.30
0.25
0.1
Shear band
0.20
0.0
APR
A
B
C
D
E
-0.1
-0.2
-0.3
0.0
0.1
0.2
Void ratio
A
B
C
D
E
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.15
Void ratio
where N is the total number of contacts and rk is the equivalent radius for the two particles at the kth contact, which can be calculated
from Eq. (5). APR is thus a local variable linking the macro- and
micromechanics of sand motion which does not exist in classical
continuum mechanics; it was first introduced in [59–61].
In Fig. 19 are shown maps of the APR distributions in the two
cemented specimens obtained at different axial strains, corresponding to points A–E in Fig. 10. Measurement circles with a radius equal to 10–12 times the average particle radius were
chosen to carry out APR measures. From Fig. 19 it is evident that
the APR was almost zero in the whole specimen until the peak of
the deviatoric stress was reached (point B in Fig. 10); after point
B, it grew continuously within the shear bands. However, the
APR was nearly zero and changed only slightly outside the shear
bands during the test. These observations are consistent with the
maps of void ratios shown in Fig. 18. In general, at large strains
the PFC specimen manifested a larger APR within the shear bands
than did the Jiang specimen.
Fig. 20 displays the APR values measured at different strain levels from five measurement circles (see Fig. 15). Here, the APR in all
the circles was almost zero until yielding (point A) and increased
later. The APR in circle 5 was the largest for both specimens. These
observations are consistent with the results shown in Figs. 15 and
19. The APRs calculated for circles 1–4 in the case of the PFC specimen became relatively small after the specimen had experienced
maximum dilatancy (point C in Fig. 10), while the APRs calculated
for circles 3 and 4 in the case of the Jiang specimen assumed significantly larger values than their initial ones after the peak of the
deviatoric stress (point B in Fig. 10) was reached. This is because
circles 3–4 are quite close to the shear band in the Jiang specimen
and the enhanced APR zone was wider in the Jiang specimen than
in the PFC specimen.
Averaged pure rotation rate (APR)
N
N 1X
1 X
1 _k k _k k
h_ p ¼
h
r
þ
h
r
2 2
N k¼1
N k¼1 r k 1 1
Averaged pure rotation rate (APR)
x¼
0.10
0.05
0.00
1.0
Distance from origin (m)
Fig. 21. Averaged APR and void ratio obtained from different bands for two bond
models (R = 5000 N, r3 = 100 kPa): (a) definition of local coordinates and measurement bands, (b) Jiang sample, and (c) PFC sample.
28
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
resulted in the thickness of the detected shear band being larger
than the thickness measured employing the APR.
5. Conclusions
This study presented an insight into the mechanical behavior
and strain localization of cemented sands by means of DEM analyses. Two types of specimens characterized by two different bond
models were investigated by performing numerical compressive
biaxial tests. One bond model was originally proposed by Jiang
et al. [41,42] (Jiang’s model), while the other is the standard contact-bond model available in PFC2D (PFC model). The multilayer
undercompaction technique was employed to generate loose
bonded specimens. Stress–strain relationships, overall material
strength, stress fields, bond-breakage fields, and APR fields, as well
as microscopic responses were analyzed. The main conclusions of
the study are summarized as follows:
(1) In both the Jiang and PFC bond models, numerical specimens
exhibited strain softening and shear dilatancy. The opposite
behavior was exhibited by an unbonded specimen with the
same initial void ratio. The peak of the deviatoric stress
and the angle of dilatancy increased with increasing bond
strength. In the case of the PFC bond, a larger strain softening and a higher deviatoric stress peak were observed at low
confinement. The opposite was observed at high confinement. In addition, the obtained global friction angle in the
case of the PFC bond model decreased with increasing bond
strength. The opposite was true in the case of the Jiang bond
model. In general, the obtained numerical results were in
good agreement with the experimental data reported in
[49] relative to Portland-cemented sands, especially in case
of the Jiang bond model.
(2) Although the mechanical responses of the two investigated
bonded granulates were different, the main features of the
observed strain-localization processes were the same. Both
granulates developed shear bands with an inclination of
approximately 52° to the horizontal during biaxial compression. Bond breakage, void ratio and averaged pure rotation
rate exhibited higher values inside the developed shear
bands than outside. Finally, it was shown that during biaxial
compression contact-force directions and principal stresses
rotate within the shear bands.
(3) There were also some differences observed between the two
bonded granulates concerning the mechanisms taking place
during shear banding. The maximum rate of bond breakage
and the void ratio inside the shear bands were larger for the
Jiang bond model than for the PFC bond model; the opposite
is true for the APR. The thickness exhibited by the main
shear band in the case of the Jiang bond model was wider
than the one developed by specimens with the PFC bond
model. Such differences are likely due to the different failure
criteria adopted in the two bond models.
Acknowledgments
This research was financially supported by the National Science
Foundation, China, with Grant No. 10972158, China National Funds
for Distinguished Young Scientists with Grant No. 521025932, the
Fund for Chinese Researchers Returning from Overseas, Ministry of
Education, China, Grant No. 2007-1108, and a Travel Grant from
the Royal Society, Grant No. 2008/R2 for the fourth author. The
support of the Itasca Consulting Group is also gratefully
acknowledged.
References
[1] Mitchell JK, Solymar ZV. Time-dependent strength gain in freshly deposited or
densified sand. J Geotech Div, ASCE 1984;110:1559–76.
[2] Leroueil S, Vaughan PR. The general and congruent effects of structure in
natural soils and weak rocks. Geotechnique 1990;40(3):467–88.
[3] Airey DW. Triaxial testing of naturally cemented carbonate soil. J Geotech Eng,
ASCE 1993;119:1379–98.
[4] Dupas J, Pecker A. Static and dynamic properties of sand-cement. J Geotech
Eng, ASCE 1979;105:419–36.
[5] Acar Y B, El-Tahir AE. Low strain dynamic properties of artificially cemented
sand. J Geotech Eng, ASCE 1986;112:1001–15.
[6] Clough GW, Sitar N, Bachus RC, Rad NS. Cemented sands under static loading. J
Geotech Eng, ASCE 1981;107:799–817.
[7] Clough GW, Iwabuchi J, Rad NS, Kuppusamy T. Influence of cementation on
liquefaction of sands. J Geotech Eng, ASCE 1989;115:1102–17.
[8] Huang JT, Airey DW. Properties of artificially cemented carbonate sand. J
Geotech Geoenviron Eng 1998;124(6):492–9.
[9] Lade PV, Overton DD. Cementation effects in frictional materials. J Geotech
Eng, ASCE 1989;115:1373–87.
[10] Abdulla AA, Kiousis PD. Behavior of cemented sands – I. Testing. Int J Numer
Anal Methods Geomech 1997;21(8):533–47.
[11] Schnaid F, Prietto PDM, Consoli NC. Characterization of cemented sand in
triaxial compression. J Geotech Geoenviron Eng 2001;127(10):857–68.
[12] Kavvadas J, Amorosi A. A constitutive model for structured soils. Geotechnique
2000;50(3):263–73.
[13] Rouainia M, Muir Wood D. A kinematic hardening constitutive model for
natural clays with loss of structure. Geotechnique 2000;50(2):152–64.
[14] Vatsala A, Nova R, Murthy BRS. Elastoplastic model for cemented soils. J
Geotech Geoenviron Eng, ASCE 2001;127(8):679–87.
[15] Rocchi G, Fontana M, Prat MD. Modelling of natural soft clay
destruction
processes
using
viscoplasticity
theory.
Geotechnique
2003;53(8):729–45.
[16] Baudet B, Stallebrass S. A constitutive model for structured soils. Geotechnique
2004;54(4):269–78.
[17] Shen ZJ. A masonry constitutive model for structured clays. Rock Soil Mech
2000;21(1):1–4 [in Chinese].
[18] Jiang MJ, Shen ZJ. A structural suction model for structured clays. In:
Proceedings of 2nd international conference on soft soil engineering, 1996.
Nanjing (China); 1996. p. 221–40.
[19] Nova R, Castellanza R, Tamagnini C. A constitutive model for bonded
geomaterials subject to mechanical and/or chemical degradation. Int J
Numer Anal Methods Geomech 2003;27:705–32.
[20] Cotecchia F, Chandler J. A general framework for the mechanical behaviour of
clays. Géotechnique 2000;50(4):431–47.
[21] Lagioia R, Nova R. An experimental and theoretical study of the behaviour of a
calcarenite in triaxial compression. Géotechnique 1995;45(4):633–48.
[22] Labuz
J,
Drescher
A.
Bifurcations
and
instabilities
in
geomechanics. Swets: Zeitlinger; 2003.
[23] Rudnicki JW, Rice JR. Conditions for localization of deformation in pressuresensitive dilatant materials. J Mech Phys Solids 1975; 23: 371-394.
[24] Vardoulakis I. Shear band inclination and shear modulus of sand in biaxial
tests. Int J Numer Anal Methods Geomech 1980;4:103–19.
[25] Papamichos E, Vardoulakis I. Shear band formation in sand according to noncoaxial plasticity model. Géotechnique 1995;45:649–61.
[26] Vardoulakis I, Sulem J. Bifurcation analysis in geomechanics. London: Blackie;
1995.
[27] Lanier J, Jean M. Pow Technol 2000;109:206–21.
[28] Calvetti F, Combe G, Lanier J. Experimental micromechanical analysis of a 2D
granular material: relation between structure evolution and loading path.
Mech Cohes Frict Mater 1997;2:121–63.
[29] Hicher PY, Wahyudi H, Tessied D. Micro-structural analysis of strain
localization in clay. Comput Geotech 1994;16:205–22.
[30] Jiang MJ, Shen ZJ. Microscopic analysis of shear band in structured clay. China J
Geotech Eng 1998;20(2):102–8.
[31] Jiang MJ, Hongo T, Fukuda M. Pre-failure behaviour of deep-situated Osaka
clay. China Ocean Eng 1998;12(4):453–65.
[32] Jiang MJ, Peng LC, Zhu HH, Lin YX, Huang LJ. Macro- and micro properties
of two natural marine clays in China. China Ocean Eng 2009;23(2):
329–44.
[33] Yatomi C, Yashima A, Iizuka A, Sano I. General theory of shear bands formation
by a noncoaxial Cam-clay model. Soils Found 1989;29(3):41–53.
[34] Otani J, Mukunoki T, Obara Y. In: Characterization of failure and density
distribution in soils using X-ray CT scanner, China–Japan joint symposium on
resent development of theory & practice in geotechnology. Shanghai; 1997. p.
45–50.
[35] Nemat-Nasser S, Okada N. Radiographic and microscopic observation of shear
bands in granular materials. Geotechnique 2001;51(9):753–65.
[36] Viggiani G, Lenoir N, Besuelle P, Di Michiel M, Desrues J, Kretzschmer M. X-ray
microtomography for studying localized deformation in fine-grained
geomaterials under triaxial compression. Comptes Rendues Mécanique
2004;332:816–26.
[37] Harris WW, Viggiani G, Mooney MA, Finno RJ. Use of stereophotogrammetry to
analyze the development of shear bands in sand. Geotech Test J (ASTM)
1995;18(4):405–20.
M.J. Jiang et al. / Computers and Geotechnics 38 (2011) 14–29
[38] White DJ, Take WA, Bolton MD. Soil deformation measurement using particle
image
velocimetry
(PIV)
and
photogrammetry.
Geotechnique
2003;53(7):619–31.
[39] Cundall PA, Strack ODL. Discrete numerical model for granular assemblies.
Géotechnique 1979;29:47–65.
[40] Itasca Consulting Group Inc. Particle flow code in 2 dimensions, version 3.1.
Minnesota, USA; 2004
[41] Jiang MJ, Lerouil S, Konrad JM. Yielding of microstructured geomaterial by
DEM analysis. J Eng Mech, ASCE 2005;131(11):1209–13.
[42] Jiang MJ, Yu HS, Leroueil S. A simple and efficient approach to capturing
bonding effect in naturally-microstructured sands by discrete element
method. Int J Numer Methods Eng 2007;69:1158–93.
[43] Utili S, Nova R. DEM analysis of bonded granular geomaterials. Int J Numer
Anal Methods Geomech 2008;32(17):1997–2031.
[44] Utili S, Crosta GB. Modelling the evolution of cliffs subject to weathering: II.
Discrete elements approach. J Geophys Res – Earth Surface; accepted for
publication.
[45] Delenne JY, El Youssoufi MS, Cherblanc F, Beneet JC. Mechanical behaviour and
failure of cohesive granular materials. Int J Numer Anal Methods Geomech
2004;28:1577–94.
[46] Jiang MJ, Yan HB. Micro-contact laws of bonded granular materials for DEM
numerical analyses. EPMESC XI (APCOM07), Kyoto, Japan; 2007.
[47] Yun TS, Santamarina JC. Decementation, softening and collapse: changes in
small-strain shear stiffness in K0 loading. J Geotech Geoenviron Eng, ASCE
2005;131(3):350–8.
[48] Kuhn MR. A flexible boundary for three-dimensional DEM particle assemblies.
Eng Comput 1995;12:175–83.
[49] Wang YH, Leung SC. Characterization of cemented sand by experimental and
numerical investigations. J Geotech Geoenviron Eng, ASCE 2008;134(7):
992–1004.
29
[50] Corriveau D, Savage SB, Oger L. Internal friction angles: characterisation using
biaxial test simulations. In IUTAM symposium; 1997.
[51] Iwashita K, Oda M. Rolling resistance at contacts in simulation of shear band
development by DEM. J Eng Mech 1998;124(3):285–92.
[52] Jiang MJ, Konrad JM, Leroueil S. An efficient technique for generating
homogeneous specimens for DEM studies. Comput Geotechnol
2003;30(7):579–97.
[53] Jiang MJ, Yu HS, Harris D. Discrete element modelling of deep penetration in
granular soils. Int J Numer Anal Methods Geomech 2006;30(4):335–61.
[54] Wang YH, Leung SC. A particulate scale investigation of cemented sand
behaviour. Can Geotech J 2008;45:29–44.
[55] Bardet JP, Proubet J. A numerical investigation of the structure of persistent
shear bands in granular media. Géotechnique 1991;41:599–613.
[56] Desrues J. Localisation de la déformation plastique dans le materiaux
granulaires. Thèse de doctorat, Grenoble University; 1984. p. 185.
[57] Vardoulakis I, Graf B. Calibration of constitutive models for granular materials
using data from biaxial experiments. Géotechnique 1985;35:299–317.
[58] Jiang MJ, Zhu HH, Li XM. Strain localisation analyses of idealised sands in
biaxial tests by distinct element method. Frontiers of Architecture and Civil
Engineering in China 2010;4(2):208–22.
[59] Jiang MJ, Yu HS, Harris D. Kinematic variables bridging discrete and continuum
granular mechanics. Mech Res Commun 2006;33:651–66.
[60] Jiang MJ, Harris D, Yu HS. Kinematic models for non-coaxial granular
materials: Part II: evaluation. Int J Numer Anal Methods Geomech
2005;29(7):663–89.
[61] Jiang MJ, Harris D, Yu HS. Kinematic models for non-coaxial granular
materials: part I: theories. Int J Numer Anal Methods Geomech
2005;29(7):643–61.