INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
Published online 22 August 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.728
DEM analysis of bonded granular geomaterials
S. Utili1, ∗, †, ‡ and R. Nova2
1 Civil
Engineering Department, Strathclyde University, Glasgow, U.K.
Engineering Department, Politecnico di Milano, Italy
2 Structural
SUMMARY
In this paper, the application of the distinct element method (DEM) to frictional cohesive (c, ) geomaterials is described. A new contact bond model based on the Mohr–Coulomb failure criterion has been
implemented in PFC2D. According to this model, the bond strength can be clearly divided into two distinct
micromechanical contributions: an intergranular friction angle and a cohesive bond force. A parametric
analysis, based on several biaxial tests, has been run to validate the proposed model and to calibrate
the micromechanical parameters. Simple relationships between the macromechanical strength parameters
(c, ) and the corresponding micromechanical quantities have been obtained so that they can be used to
model boundary value problems with the DEM without need of further calibration.
As an example application, the evolution of natural cliffs subject to weathering has been studied.
Different weathering scenarios have been considered for an initially vertical cliff. Firstly, the case of uniform
weathering has been studied. Although unrealistic, this case has been considered in order to validate the
DEM approach by comparison against analytical predictions available from limit analysis. Secondly, nonuniform weathering has been studied. The results obtained clearly show that with the DEM it is possible
to realistically model boundary value problems of bonded geomaterials, which would be overwhelmingly
difficult to do with other numerical techniques. Copyright q 2008 John Wiley & Sons, Ltd.
Received 18 October 2007; Revised 20 May 2008; Accepted 22 May 2008
KEY WORDS:
distinct element method; bonded geomaterials; cohesive frictional materials; slope stability
1. INTRODUCTION
Many aspects of the behaviour of granular unbonded geomaterials have been successfully
modelled by distinct element method (DEM) in the past, for instance, shear banding [1], induced
anisotropy [2], elastic deformability [3], particle crushing [4] and plastic deformability [5]. In this
paper, DEM has been applied to bonded granular geomaterials. A new contact bond model for
∗ Correspondence
to: S. Utili, Civil Engineering Department, Strathclyde University, Glasgow, U.K.
stefano.utili@strath.ac.uk
Formerly at Structural Engineering Department, Politecnico di Milano, Italy.
†
E-mail:
‡
Contract/grant sponsor: Italian Ministry of Education
Copyright q
2008 John Wiley & Sons, Ltd.
1998
S. UTILI AND R. NOVA
2D analyses and implemented in PFC2D [6] will be presented to reproduce the global behaviour
of cohesive frictional geomaterials, which could be both clayey soils and soft rocks.
In recent years, some authors used DEM to reproduce the behaviour of bonded grains in 2D in
order to model rocks. Potyondy and Cundall [7] used the parallel bond model offered in PFC to
study the brittle behaviour of granites under compressive loading. Jiang et al. [8, 9] and Toboada
et al. [10] introduced cohesive bonding and rolling friction to study the behaviour of bonded
granulates. Delenne et al. [11] performed experimental tests on pairs of aluminium rods glued by
an epoxy resin to deduce micromechanical relationships between relative displacements and forces
at the contact between rod pairs. In all these cases, the bond between two particles possesses a
finite width and therefore the particles exchange not only a normal and tangential force but also
a moment at their contact point. As a consequence, the number of micromechanical parameters
requiring to be calibrated becomes larger than the case of unbonded particles. In the present paper,
a model for cohesive frictional geomaterials is proposed, in which particles only exchange forces
at the contact points. Therefore, only one additional micromechanical parameter is required to pass
from the unbonded case to the bonded case. From a physical point of view, bonded granulates and
cemented soils, in general, are all characterized by bonds having a finite width and therefore able
to transfer not only forces but also bending moments. However, for the problem investigated in this
paper, the DEM has been used as a numerical tool to investigate cohesive frictional geomaterials as
continua rather than as a means of investigation of the real micromechanical behaviour of bonded
grains. In other words, the micromechanical bond implemented is good to reproduce a cohesive
frictional continuum characterized by an assigned couple of c, values so that the boundary value
problem of interest may be studied by the use of DEM. Therefore, the type of bond adopted has
been sought to be as simple as possible to minimize the micromechanical parameters needed in
the calibration process rather than attempting at reproducing the real grain-to-grain interaction.
A calibration procedure based on biaxial tests was established to reproduce the global behaviour
of c, geomaterials [12]. Simple relationships between the macromechanical strength parameters
(c, ) and the corresponding micromechanical quantities have been obtained so that they can be
used to model boundary value problems with DEM without need of further calibration. In the
following, as it is customary in the soil mechanics of DEM literature (see e.g. [3, 7]), the equivalent
continuum stresses acting at the boundaries of a selected material element will be expressed in kPa.
The stiffness coefficients of the contact springs have been calibrated, in fact, in such a way that
the overall 2D specimen behaviour mimics the behaviour of an actual 3D soil specimen in plane
strain conditions, assuming a unit thickness of the discs in the direction orthogonal to the plane
of deformation. The springs at the contacts should not therefore be considered to have a direct
physical meaning, as a 2D world of discs has no physical meaning. However, both the springs and
the discs are used as a means to reproduce the behaviour of a 3D soil element, in biaxial strain
conditions.
As an example of an application of the implemented model to boundary value problems, the
evolution of natural cliffs subject to weathering has been studied. Weathering can be due to a
variety of both physical and chemical actions of degradation because of environment agents. This
degradation causes a progressive deterioration of the mechanical properties of many hard soils
and rocks present on the earth’s crust. Various types of geomaterials are affected, e.g. clay shales,
sandtones, chalk, pyroclastic deposits. This phenomenon may become relevant in particular for the
stability of slopes exposed to aggressive environments such as cliffs in coastal areas. The progressive
reduction of soil strength causes subsequent slope failures and the progressive retreat of the cliff
crest. Examples of such a phenomenon are described for instance by Hutchinson [13, 14]. The
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
1999
speed at which slope retreat occurs is usually rather low, but, in clayey soils and volcanic or
carbonatic rocks, it is fast enough to jeopardize the safety of buildings or infrastructure located
nearby. In this study, the soil strength of the considered cliffs has been characterized by cohesion
and internal friction according to the Mohr–Coulomb criterion so that a procedure has been
established to determine the speed of cliff retreat as a function of weathering-induced soil strength
decrease. In the same spirit as above, to model the effect of volume forces on the cliff collapse,
the weight of the discs simulating the cliff material was calibrated in such a way that the stresses
generated by the rock weight are those one could calculate in a 3D continuum of unit thickness
in the direction orthogonal to the plane of deformation. The overall unit weight of the material is
therefore expressed in kN/m3 .
The paper is divided into four parts. In Section 2, the procedure used to execute the numerical
biaxial tests run will be illustrated. In Section 3 the calibration of PFC parameters to reproduce
the behaviour of frictional cohesionless geomaterials will be shown, whereas in Section 4 the
calibration for frictional cohesive geomaterials will be shown. Finally in Section 5, the study
performed about the evolution of natural cliffs subject to weathering will be illustrated.
2. CALIBRATION PROCEDURE
The numerical specimens used in the biaxial tests are featured by a uniform random distribution
of particle radii varying from 1.5 to 4.5 mm contained by four rigid frictionless walls making a
square 0.2 m long (see Figure 1). It is known in the literature (e.g. [15, 16]) that a number of
particles larger than 1000 elements is enough to obtain meaningful biaxial tests. In all the tests
run, the number of particles used has been around 1220. A total of 130 tests were run.
Since the walls containing the particles were frictionless, the principal directions of stresses and
strains coincide with the coordinate axes x and y (see Figure 1). Therefore, principal strains were
calculated directly from wall displacements, whereas principal stresses were obtained as the sum
y
x
Figure 1. Example of the specimen used in the calibration process.
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2000
S. UTILI AND R. NOVA
of the normal forces exchanged between discs and the bordering wall divided by the wall length
and disc thickness.
In the description of the state of stress, the following 2D invariant variables have been used:
s = ( y +x )/2 (mean effective pressure) and t = ( y −x )/2 (deviatoric stress). In the description
of the state of strain, the associate strain invariant variables have been used: v = x + y (volumetric strain) and d = y −x (deviatoric strain). This particular choice of invariants guarantees
the energetic equivalence relative to the first-order work: dW = x ˙ x + y ˙ y = s˙v +t ˙d .
2.1. Specimen generation
PFC adopts a procedure of random generation for the system of particles. The generated particles
cannot overlap one another. The method used to generate samples with a prescribed porosity
was the so-called expansion method: first, a specimen is generated with particle radii reduced by
a constant factor; second, all particle radii are multiplied by the factor; third, some cycles are
performed until the system reaches an equilibrium state. Some cycles are needed by the system to
reach equilibrium since after radii expansion particles overlap greatly and therefore strong repulsive
forces arise. Since all particles are expanded by the same factor, the contact forces throughout
the assembly are similar, resulting in an isotropic uniform network of contact forces. This is an
effective technique of sample generation since it allows the generation of a homogeneous network
of contact forces and particle distribution within the created samples; it is computationally efficient
and avoids the onset of high lock-in contact forces, the equivalent of self-stresses in terms of
continuum mechanics (see [6, 7, 17]).
An important factor affecting the initial stress state 0 of the assembly is porosity. A limit
value of specimen porosity, n = 0.17, was found such that if a higher value of porosity is taken,
particles do not exchange contact forces so that the stress is nil. This limit value depends on the
particle size distribution adopted: Rmax /Rmin = 3. In fact, if a smaller range of radii were adopted, a
higher limit value of porosity would have been obtained and if the range reduces to a unique value
σ0
k
0.16
0.17
n
Figure 2. Qualitative relationship between porosity and initial stress for various contact stiffnesses. n = 0.16
is the porosity value adopted for all the calibration simulations.
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2001
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
(all particles with the same radius), the highest limit value of porosity is achieved. The initial stress
depends also on the contact stiffness adopted. This dependence is illustrated in Figure 2 where
the relationship between porosity and initial stress is qualitatively drawn. It is impossible, except
in case of regular packing, to create simultaneously a specified porosity and a specified isotropic
stress. Therefore, an initial porosity value was chosen for all the tests and loading/unloading was
applied to bring the specimens to the prescribed isotropic confining pressure. The chosen initial
porosity value used in all the biaxial tests was n = 0.16. Lower porosity values led to too high
initial stresses for the range of confining pressure that needs to be investigated (see Section 3):
i.e. a very large unloading would be necessary to bring the specimen to the prescribed confining
pressures and the state of stress at failure would be of the same order of magnitude as the initial
stress.
2.2. Biaxial test
The test can be divided into two phases: isotropic compression and biaxial loading. During the
first phase y = x , whereas during the second phase x = p, where p is the confining pressure
reached at the end of the first phase. No gravity was applied to the specimens.
After the specimens were generated, they were brought to the prescribed confining pressure p.
This was done by moving the boundary walls in stress control. In order to avoid meaningless
results, loading has to be imposed maintaining quasi-static conditions. To fulfil this requirement,
only one wall per direction was moved. Doing so, the difference of forces exerted by particles
on the opposing walls at any instant t during the test is entirely due to inertial forces induced by
the movement of the walls. If this difference is small in comparison with the forces exerted on
the walls, the inertial forces can be considered negligible. A numerical servo-control prescribed
wall velocities in such a way that the target confining pressure was reached smoothly without
introducing oscillations of stresses around the target value. In Figure 3 the stresses exerted on
walls during the isotropic compression phase are shown.
250
stresses on walls [kPa]
200
150
upper wall (servo-controlled velocity)
lower wall (fixed)
100
right wall (servo-controlled velocity)
50
left wall (fixed)
0
0.0E+00
5.0E-05
1.0E-04
1.5E-04
2.0E-04
2.5E-04
3.0E-04
eps_y
Figure 3. Stresses acting on the boundary walls during isotropic compression up to 200 kPa.
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2002
S. UTILI AND R. NOVA
Figure 4. Fields of displacement at the end of the biaxial test: (a) field obtained by PFC2D and (b) field
obtained by LMGC software (contact dynamics method), after Lanier and Jean [18].
After isotropic compression, friction was assigned to the particles and the specimens were loaded
in biaxial compression. In this case, a prescribed velocity was applied only to the upper boundary
wall, which moves downwards. A numerical servo-control prescribed the motion of the right
boundary wall in such a way that the lateral stress remains constant during the test. In addition, the
velocity of the upper wall was calibrated in order to maintain quasi-static conditions during the test.
In Figure 4(a), the typical displacement field obtained at the end of a biaxial test for the case
of a purely frictional material is shown. This is in agreement with experimental tests on wooden
rods [16] and numerical tests run by other authors. Lanier and Jean [18], for instance, performed
biaxial tests on an assembly of rigid non-overlapping disks according to the contact dynamics
method using the LMGC software. In Figure 4(b) it is shown that the displacement field obtained
from their test is in very good agreement with that obtained by PFC.
In all simulations, a small numerical damping was used (5%). This damping consists in
decreasing the particle accelerations by a factor proportional to the unbalanced forces acting on
each particle. It was only used to shorten the computation time and therefore it has no particular
physical meaning but its use is justified in so far as it does not affect the global response of the
assembly to the mechanical actions imposed. Some tests with no damping were run and the fact
that the results obtained were not affected by the numerical damping introduced was verified.
3. CALIBRATION OF MICROMECHANICAL PARAMETERS FOR A FRICTIONAL
COHESIONLESS MATERIAL
PFC offers the possibility of inhibiting particle rotation. As observed elsewhere [19], if particle
rotations are allowed, no matter how large the interparticle friction angle, a limit value of the
macroscopic friction angle is achieved. This value depends on the particle size distribution adopted
and in our case is no larger than 25◦ , far lower than that observed in actual geomaterials. Therefore,
rotations were inhibited in all the simulations run. The imposed constraint is equivalent to adopting
an infinite rotational stiffness for the particles while no moment can be exchanged between two
particles at a contact point since this would require the definition of a finite bond width that
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2003
is instead infinitely small in the proposed model. This assumption does not correspond to a physical
reality since particles do both rotate and slide at the micromechanical level, whereas according
to our assumption, they can only slide. It would be more realistic to assume a certain rolling
resistance at contacts as done by [10], but as already observed in the Introduction; here, the
objective is to achieve as simple micromechanical relationships as possible in order to apply the
DEM to geotechnical problems with cohesive frictional continua rather than to reproduce the ‘true’
micromechanical interactions among grains.
It was decided to run biaxial tests for a range of confining pressures varying between 0 and
500 kPa. This range covers the existing confining pressures for the problem tackled in Section 5
where a 40 m high cliff subject to weathering was studied.
The interparticle friction angle represents the slope of the yield line in the F N –F S plane,
where F N is the normal force and F S the shear force acting at a contact, respectively. Considering
the loading phase, each time |F S | exceeds F N tan at a contact, a new value of F S is assigned
to the contact: F S = F N tan . This bond model is the default model in PFC and it gives rise to
a rather ductile global mechanical response for a system of particles subject to biaxial loading as
it can be clearly seen in Figure 5(a).
Linear elastic relationships between relative displacements between two particles at a contact
and the forces exchanged have been adopted (this is the default option in PFC, whereas another
possibility is to use the so-called Hertz–Mindlin non-linear relationships that present the disadvantage of requiring more micromechanical parameters to be introduced). The normal contact stiffness
adopted controls the amount of particle overlapping (see Table I). Given a compressive force,
particle overlapping increases with decreasing normal contact stiffness. But there is a physical
limit that must be taken into account. In fact, particle overlapping has been introduced in PFC as a
fictitious numerical way to model the dependence of soil compliance on pressure in a simple way.
In Table I, the maximum overlapping between particles recorded in correspondence to the highest
loading conditions ( p = 500 kPa, end of biaxial test) are reported for various normal contact stiffnesses. Therefore, K N must be at least 5×104 kN/m; otherwise an excessive overlapping would
occur leading to meaningless results. In all the simulations run, tangential and normal stiffnesses
have been assumed equal: K N = k S = k. Finally, in Section 5, the predictions obtained by DEM
will be compared with results obtained by the limit analysis upper bound method. According to
this method, soil is assumed to be characterized by a rigid perfectly plastic behaviour; hence, it
can be concluded that the higher the k, the better. However, the value of k greatly influences the
time needed to run a simulation since the stability of the integration scheme used to solve the
equations of motion limits the time step to a value that is inversely proportional to the square
root of k. Therefore, there are two conflicting objectives that can be summarized as realistically
deformable material and fast computations.
Moreover, the contact stiffness usually has an influence on the global strength. In Figure 5(a)
the stress–strain relationship obtained for different values of contact stiffness (from k = 5×103 to
5×106 kN/m) at p = 500 kPa are shown. Similar trends were recorded for other values of p. In
the figure, the existence of a limit value klim is shown, such that if k>klim is assumed, the material
strength is unaffected by contact stiffness. In Figure 5(b), the influence of k on is summarized
and it emerges that klim = 5×105 kN/m is the limit value at p = 500 kPa. Therefore, a compromise
was found by selecting a value of k, which guarantees an acceptable computational time and a
slight dependence of the strength on the contact stiffness: k = 5×104 kN/m was selected as the
default value for all the subsequent biaxial tests.
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2004
S. UTILI AND R. NOVA
k=5e3 [kN/m]
k=5e4 [kN/m]
k=5e5 [kN/m]
k=5e6 [kN/m]
700
600
t [kPa]
500
400
300
200
100
0
0.00
0.02
0.04
0.06
0.08
(a)
0.10
0.12
0.14
0.16
0.18
0.20
eps_dev
40
35
φ [degrees]
30
34.4
34.3
33.0
29.0
25
20
15
10
5
0
1E3
5E3
1E4
(b)
5E4
5E5
1E5
5E6
1E6
1E7
contact stiffness [kN/m]
Figure 5. (a) Biaxial test at p = 500 kPa with tan = 0.2 for different contact stiffness values and
(b) contact stiffness vs internal friction angle in semi-logarithmic scale ( p = 500 kPa and tan = 0.2).
Table I. Normalized overlapping (U = particle overlapping, D = particle diameter)
for various contact stiffnesses.
K N = k S (kN/m)
U/D
5e3
5e4
5e5
5e6
1/1.2
1/35
1/188
1/3075
U refers to the maximum overlapping existing at the end of the biaxial test ( p = 500 kPa).
Up to here, it has been shown that both parameters p and k influence the global behaviour
obtained. But they do not act independently as already reported in [20]. In fact, if the same ratio
K / p is taken, the stress–strain relationships obtained tend to coincide as shown in Figure 6(a)
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2005
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
k=5e5, p=500
k=5e4, p=50
1.4
1.2
1.0
t/p
0.8
0.6
0.4
0.2
0.0
0.00
0.02
0.04
(a)
0.06
0.08
0.10
0.12
eps_dev
k=5e5, p=500
k=5e4, p=50
k=5e4, p=500
k=5e3, p=50
eps_dev
-0.040
-0.035
-0.030
eps_vol
-0.025
-0.020
K/p=10^3
-0.015
-0.010
K/p=10^2
-0.005
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.000
0.005
(b)
Figure 6. Curves obtained keeping the K / p ratio constant: (a) deviatoric stress vs deviatoric strain in the
case of K / p = 103 and (b) volumetric strain vs deviatoric strain in the case of K / p = 103 and 102 .
where the deviatoric stress has been normalized by the confining pressure p. Hence now also,
the v −d relationship achieved can be conveniently presented in terms of K / p rather than by
varying the parameter p alone (see Figure 6(b)). A physically sound trend has been achieved. The
compacting phase is larger for lower K / p ratios, which therefore can be thought of as a measure
of the rigidity level adopted. Concerning dilatancy, the same final value (slope of the curves in the
right side of the graph) independent of the K / p ratio adopted was obtained.
Another aspect that deserves attention concerns the observed dependence of the internal friction
angle on the confining pressure. In Figure 7, the variation of the internal friction angle against the
confining pressure is shown for two values of contact stiffness: k = 5×104 and 5×105 kN/m. This
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2006
S. UTILI AND R. NOVA
k=5e5 [kN/m]
40
35
φ [degrees]
30
k=5e4 [kN/m]
25
20
15
10
5
0
0
50
100
150
200
250
300
350
400
450
500
confining pressure [kPa]
Figure 7. Angle of internal friction vs confining pressure for two values of contact stiffness (tan = 0.2).
variation is featured by a few points of percentage in both cases analysed. The results achieved
are in accordance with experimental [21] and numerical [22] data reported in the literature relative
to biaxial tests on granular materials where the friction angle is reported to slightly decrease
with increasing confining pressure. Tests run at higher confining pressures (1 and 5 MPa) put in
evidence a stronger decrease in internal friction angle. This is due to the fact that the studied
contact stiffness values (5×104 and 5×105 kN/m) are too low for such confining pressures. Values
adequate to reproduce the global behaviour in a middle-low range of pressures are not adequate
in a middle-high range. This fact highlights the importance of having defined a precise range of
confining pressures to investigate.
3.1. Determination of – relationship
In order to determine a – relationship, failure envelopes have been obtained by linearly
interpolating failure values in the s–t plane for different values of (see Figure 8(a)). The slope
of the lines in the figure gives the internal friction angle of the material. In Figure 8(b), the values
of internal friction angle against intergranular friction are shown. The relationship achieved is
linear in a wide range of such that all the values of engineering interest (from 15 to 45◦ ) are
covered:
= K 1 + K 2
(1)
where K 1 and K 2 are two material constants. It is therefore possible to conclude that a simple
relationship between micro- and macromechanical strength parameters has been obtained and a
frictional cohesionless material has been satisfactorily reproduced. If the particles were left free
to rotate, the obtained relationship in terms of – would have been non-linear (see e.g. [23]).
Hence the achieved linearity is due to the imposed kinematic constraint according to which particle
rotations are fully prevented so that relative movement along the tangential direction between two
particles in contact is entirely due to slip.
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2007
1100
1000
900
tanφ µ =0.3
800
t [kPa]
700
600
tanφ µ =0.2
500
400
300
tanφ µ =0.1
200
tanφ µ =0.04
100
0
0
100
200
300
400
500
600
700
800
900 1000 1100 1200 1300 1400 1500 1600
s [kPa]
(a)
50
45
40
φ [degrees]
35
30
25
φ = 1.90φ µ + 11.2
20
15
10
5
0
0
2
(b)
4
6
8
10
12
14
16
18
φ µ [degrees]
Figure 8. (a) Failure envelopes (for various tan ) achieved by linear interpolation of failure values in
the s–t plane (k = 5×104 kN/m) and (b) – relationship inferred from the envelopes.
4. CALIBRATION OF MICROMECHANICAL PARAMETERS FOR
A FRICTIONAL COHESIVE MATERIAL
In order to simulate a frictional cohesive material, PFC offers the possibility of assigning a
shear and normal contact bond: SB and NB, respectively, both positive values. If these bonds are
assigned to contacts, the force–displacement laws along both the tangential and normal directions
are linearly elastic so long as the contact forces remain lower than the strength values SB and NB
(see Figure 9(a)). Biaxial tests showed that with this type of bond it is not possible to get a global
strength higher than the case of unbonded particles retaining a ductile behaviour at the macroscale.
This is due to the fact that the bonds considered are fragile: as soon as a contact force reaches
the limit values, either NB or SB, the contact breaks. Moreover, from a conceptual point of view,
this bond model is far from the macromechanical behaviour that we want to reproduce since the
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2008
S. UTILI AND R. NOVA
FS
Strength of intact
contacts
B2
A2
A1
B1
SB
Strength of broken or
non-bonded contacts
A3
φµ
(a)
FN
B3
NB
A2
FS
Strength of intact
contacts
A3D
A1
Strength of broken or
non-bonded contacts
A3F
cµ
B1
B2 B3D
φµ
tµ
B3F
FN
(b)
Figure 9. (a) PFC contact bond model. A1 and B1 represent the forces relative to a compressive and
non-compressive contact, respectively, just before the imposition of the force–displacement law. A2 and B2
represent the contact forces after the elastic trial. A3 and B3 represent the final contact forces after contact
breakage. (b) Implementation of the proposed model in PFC. When the contact strength is exceeded two
possibilities are given: fragile behaviour (F, grey), the contact breaks and the failure surface reduces
to the failure surface of non-bonded contact, and ductile behaviour (D, black), the contact remains
intact and the strength unaltered.
global strength of a c − material is characterized by a shear strength increasing with the normal
pressure, whereas the shear resistance offered in the PFC contact bond model, SB, is independent
of the amount of normal pressure exerted on the contact.
Therefore, a new contact model was implemented via a routine executed every time step. The
proposed contact model is characterized by the adoption of the Mohr–Coulomb failure criterion
into the F N –F S plane (see Figure 17(a)) and linear elastic laws until the contact forces remain
within the M–C failure line (see Figure 10). According to this criterion, two distinct strength
contributions are present: a cohesive one ruled by the parameter c and a frictional one ruled by
the parameter . The tensile strength is simply given by t = c / tan .
In Figure 9(b), the way the model works is shown. Two types of behaviour could be used: fragile
and ductile. In the former case, when the contact shear force F S reaches the contact strength, the
contact breaks and the shear strength is reduced to F N tan . Conversely, in the latter case, when
the shear strength is exceeded, the contact does not break, the shear strength remains unaltered
and the shear force is brought to F N tan +c . Concerning the implementation of the bond
model into the calculation cycle, executed by the code at every time step, the routine gets into
action just after the contact forces have been calculated. The first check is on F N . In the case
of F N <−t , if the contact is fragile, it breaks (contact declared broken and contact forces set to
Copyright q
2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2009
FN
k
1
UN
tµ
A
(a)
FS
B
S
Fmax
k
1
US
(b)
Figure 10. Contact laws in terms of relative particle displacement vs contact force: (a) normal direction
(compression assumed as positive) and (b) tangential direction. In point A if the bond is fragile, it
breaks; otherwise, if ductile the normal force and gap between particles remain unchanged. Looking at
the tangential contact law: if the bond is fragile, the graph is ‘linear elastic’ (when the force reaches
point B, the contact breaks), whereas if it is ductile the graph is ‘linear elastic–perfectly plastic’.
zero F N = F S = 0); otherwise, if ductile, it remains intact (F N = −t and F S = 0). In Figure 10(a),
the contact law along the normal direction is shown. The second check is on F S . In the case
of F S >F N tan +c , if the contact is fragile, it breaks (contact declared broken, F N remains
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Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2010
S. UTILI AND R. NOVA
unaltered whereas F S = F N tan ); otherwise, if ductile, it remains intact (F N remains unaltered
whereas F S = F N tan +c ). In Figure 10(b), the contact law along the tangential direction is
shown.
Three possible micromechanical models, summarized in Table II, were tested: normal and
shear contact behaviours fragile, normal and shear contact behaviours ductile, and normal contact
behaviour fragile and shear contact behaviour ductile. Bonds were assigned to contacts at the end
of the isotropic loading/unloading phase. In Figure 11, the network of bonds is shown. It can be
seen that bonds are approximately uniformly distributed within the specimen, although the network
web is irregular. This is due to the wide particle distribution adopted. This network appears suitable
to reproduce real bonds in soil. Moreover, an irregular network is a necessary condition in order to
reproduce an homogeneous cohesive material as much as to reproduce an homogeneous frictional
material.
In Figure 12, the stress–strain relationship achieved from all the three models is shown. In order
to make a comparison among the models possible, the same values of bonds have been assigned in
all the tests: c = 0.1 kN, where c is the contact shear strength when F N = 0. Since t = c / tan ,
Table II. Micromechanical models tested.
Model
Normal
Shear
Global behaviour
Adopted
DD
DF
FF
Ductile
Fragile
Fragile
Ductile
Ductile
Fragile
Infinite strength
Ductile
Not modified
No
Yes
No
Figure 11. Network of bonds ( p = 500 kPa).
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DOI: 10.1002/nag
2011
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
500
450
400
DF≡DD
350
t [kPa]
300
FF
250
200
UN
150
100
50
0
0.0E+00
5.0E-04
1.0E-03
1.5E-03
(a)
2.0E-03
2.5E-03
3.0E-03
3.5E-03
4.0E-03
eps_dev
3500
3000
2500
t [kPa]
DD
2000
1500
DF
1000
FF≡UN
500
0
0.00
0.01
(b)
0.02
0.03
0.04
0.05
eps_dev
Figure 12. Stress–strain relationship achieved for models DD, DF and FF, all with c = 0.1 kN, and
unbonded particles (curve named UN) with k = 5×104 kN/m, p = 500 kPa and tan = 0.2: (a) range of
small strains: it can be seen that the curves DD and DF coincide, whereas the curve FF initially coincides
with DF and then it moves towards the UN and (b) range of large strains: the difference between the
curve DF and DD can be seen, whereas the curves FF and UN coincide.
the parameter c alone is needed to fully characterize bonds. Hence, c together with fully
characterize the material strength at the micromechanical level.
The specimens that adopted the FF model gave rise to a mechanical response, at low strains,
identical to the DF and DD cases and stiffer than the case of unbonded material (see Figure 12(a)),
but, at larger strains, identical to the unbonded case with a global peak strength substantially
unaffected by the presence of bonds (see the curves UN and FF in Figure 12(b)). This behaviour
could appear odd, at first, since bonds of the FF model are characterized by a peak strength given
by a frictional and a cohesive contribution clearly larger than the unbonded case characterized
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Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2012
S. UTILI AND R. NOVA
by the frictional contribution alone. It happens that the FF curve starts to deviate from the DF
curve when about the first 10 bonds have broken in shear (d = 7×10−4 ); then an increasingly
larger number of bonds progressively break so that the number of bonds still unbroken at the peak
of the FF curve is negligible. This explains why the model is unable to add extra strength to the
system of particles in comparison with the unbonded case.
The specimens that adopted the DF model gave rise to a satisfactory mechanical response
since the global behaviour achieved is ductile and the resulting mechanical strength shows a finite
increment in comparison with the unbonded case (see Figure 12(b)). The global strength obtained
is finite since the bonds are allowed to break only when the normal forces exchanged at contacts
exceed the tensile limit value t . In this case bonds are also subject to progressive breakage under
an increasing load, but the model is able to add a significant extra strength in comparison with
the FF case because of the ‘elastic–perfectly plastic’ law adopted for the shear action. In order to
understand what happens at micromechanical level, it is useful to look at the normal and tangential
forces exchanged at contacts. These forces are plotted in Figure 13 where each graph corresponds
to a different situation during the biaxial loading. The contact forces first reach the M–C yield
line in the compression zone (see Figure 13(a)); then they move along the M–C line towards the
tensile limit t (see Figure 13(b)), and when t is eventually reached, bonds break. If we consider
0.50
0.50
yield line
0.10
-0.5
0.0
-0.10
0.5
1.5
1.0
0.10
-0.5
-0.50
0.0
-0.10
0.5
1.0
1.5
-0.30
-0.30
(a)
yield line
0.30
FS [kN]
FS [kN]
0.30
FN [kN]
-0.50
(b)
FN [kN]
0.50
line
yield line
FS [kN]
0.30
0.10
-0.5
0.0
-0.10
0.5
1.0
1.5
-0.30
-0.50
(c)
FN [kN]
Figure 13. Forces exchanged at contacts during biaxial loading for the DF bond model. Each force is
represented by a dot in the graphs: (a) d = 7×10−4 corresponding to the fork point between the FF and
DF curves in Figure 11(a). The forces are compressive and only seven dots lie along the yield lines (dashed
grey); (b) d = 7×10−3 , corresponding to the fork point between the DF and DD curve in Figure 11(b).
The highest density of the dots are along the yield lines. Eight bonds have already reached the tensile
limit and broken off; and (c) d = 4.5×10−2 , corresponding to the peak of the DF curve in Figure 11(b).
The highest dot density is still along the yield lines but the presence of dots, relative to broken bonds,
lying along the inner F N tan lines (solid grey) is also apparent.
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DOI: 10.1002/nag
2013
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
the instant when the DF curve reaches the peak in the t −d plane, the broken bonds amount to
11.5% of the total number of bonds (see Figure 13(c)). From a macromechanical point of view,
two phases can be identified: a purely elastic one and a subsequent elasto-plastic one. The former
takes place from d = 0 to 7×10−4 , corresponding to the fork point between the DF curve and
the FF curve (see Figure 12(a)), whereas the latter takes place from d = 7×10−4 onwards and
it is characterized by increasingly higher irrecoverable relative tangential displacements among
particles.
The DD model gave rise to a response characterized by infinite strength, as it is shown in
Figure 12(b): the deviatoric stress continued to increase at d = 0.4 (in Figure 12(b) d only up to
0.05 is shown for space reason). In fact, according to this model, bonds are not allowed to break
either when the shear limit or when the tensile limit is reached by the contact forces. Hence, it is
clear that this model is not suitable to reproduce a geomaterial characterized by a finite strength.
In conclusion, the only model suitable to reproduce a c, soil is the DF, which has been adopted
in all the successive biaxial tests.
4.1. Determination of c –c relationships
In this section, the calibration of c will be illustrated in detail. In Figure 14 the stress–strain
relationships achieved for some values of contact shear strength (c = 0.1, 0.05, 0.025 kN) are
shown. In the figure, it can be easily seen that the global behaviour obtained by the DF model is
ductile. In order to evaluate the influence of c on the increase in global strength, the values of s
and t at failure are shown in Figure 15(a) for various contact shear strengths for the same confining
pressures previously investigated (see Section 3). As it can be seen in the figure, these values are
well interpolated by linear functions. All the linear functions achieved have the same slope. This
implies that the global strength obtained for different c is characterized by the same internal
friction angle. Moreover, two distinct contributions can be clearly identified: one expressed by a
frictional term, the slope of the lines equal to sin , and the other one by a cohesive term, the
intercept of the lines with the t-axis equal to c cos . From what has been shown up to here, it is
possible to conclude that cohesion can be related to the c parameter alone.
In Figure 15(b), the values of s and t at failure can be seen for the same tests but with a
much lower intergranular friction angle, tan = 0.04 instead of tan = 0.2. These values are
not fitted any more by linear interpolation and the lines obtained by linear interpolations are no
longer parallel. Instead, if lower values of shear contact strength (c = 0.01, 0.005, 0.025 kN) are
1200
c =0.1 kN
c =0.05 kN
800
t [kPa]
t [kPa]
1000
c =0.025 kN
600
c =0 kN
400
200
0
0.00
(a)
0.02
0.04
0.06
0.08
eps_dev
0.10
0.12
500
450
400
350
300
250
200
150
100
50
0
c =0.1 kN
c =0.05 kN
c =0.025 kN
c =0 kN
0.00
(b)
0.02
0.04
0.06
0.08
0.10
0.12
eps_dev
Figure 14. Deviatoric stress vs deviatoric strain for k = 5×104 kN/m and tan = 0.2: (a) confining
pressure p = 500 kPa and (b) confining pressure p = 50 kPa.
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DOI: 10.1002/nag
2014
S. UTILI AND R. NOVA
c =0.1 kN
c =0.1 kN
1000
c =0.05 kN
800
c =0.025 kN
t [kPa]
t [kPa]
1200
1100
1000
900
800
700
600
500
400
300
200
100
0
c =0 kN
600
c =0.05 kN
400
c =0.025 kN
c =0.01 kN
c =0.005 kN
c =0 kN
200
0
0
200 400 600 800 1000 1200 1400 1600 1800
(a)
0
200 400 600 800 1000 1200 1400 1600 1800
(b)
s [kPa]
s [kPa]
Figure 15. Failure envelopes achieved by linear interpolation of failure values in the s–t plane for various
c with k = 5×104 kN/m: (a) case of tan = 0.2 and (b) case of tan = 0.04.
250
6000
slope of c -cµ curve
200
jf=0.2
c [kPa]
jf=0.04
150
jf=0.1
jf=0.3
100
50
0
(a)
4000
y = 775*x
2000
0
0
0.025
0.05
0.075
0.1
0
0.125
(b)
c µ [kN]
0.1
0.2
0.3
0.4
tanφ µ
Figure 16. (a) c vs c for various values of intergranular friction angle (k = 5×104 kN/m). The dashed
line indicates the upper limit of validity of the achieved c–c relationships and (b) slope of the c–c curve
vs tan . Points interpolated by a power relationship.
considered, the values at failure are again fitted by straight lines having the same slope. Therefore,
it is still possible to consider the global strength as given by a frictional part, depending on ,
clearly distinct from the cohesive one, ruled by c , but only within a certain range. The limit of
validity of this range is given by a limit value of global cohesion (see Figure 16).
The c–c relationships achieved by running tests at different values of intergranular friction
angle are shown in Figure 16(a). According to the figure, c and c can be linearly related within
a limit value of cohesion conservatively taken as clim = 220 kPa. Higher cohesion values cannot be
linearly related to c because tests run with linearly inferred values of c , showed failure points
on the s–t plane not lying on parallel lines any more. It is reasonable to suppose that clim depends
on the characteristics of the geometry of the specimens analysed, namely, particle size distribution
and porosity.
The relationships between c and c depend on the intergranular friction angle . Such a
dependence is well illustrated in Figure 16(b) where the slope of the c–c relationships is plotted
against the internal friction angle . The dots, corresponding to the different values of analysed,
are well interpolated by a power relationship expressed by
c = K ·c =
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K3
c
(tan ) K 4
(2)
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2015
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
K 3 = 775 m−2 and K 4 = 0.6 being two material constants whose value depends on the porosity
and particle size distribution adopted. From Equation (2) the presence of a vertical asymptote for
= 0 can be inferred. When = 0, the failure line in the F N –F S plane becomes a horizontal
line according to the model adopted (see Figure 9(b)); therefore, if any value of c is assigned, t
becomes infinite and the specimen assumes infinite strength.
It can be concluded that simple relationships between the strength parameters , c and , c
have been obtained for a range of values of engineering interest and the mechanical strength of
a frictional cohesive geomaterial can be satisfactorily reproduced by DEM. Hence, it is possible
to derive the micromechanical parameters needed by DEM from the known macromechanical
strength parameters (an example is given in Table III). If a different particle size distribution
were chosen, the obtained values could be different. However, the distribution adopted is large
(Rmax /Rmin = 3) when compared with other numerical biaxial simulations reported in the literature
(e.g. [17, 24]); therefore, the relationships obtained can be considered to be of general validity and
the values obtained can be used for simulation of other boundary value problems involving c −
soils, without the need of recalibration.
Table III. Determination of , c from known values of , c.
c (kPa)
20
50
100
200
= 15◦
= 22◦
= 33◦
= 42◦
= 2.3◦
= 5.7◦
= 11◦
= 17◦
c = 3.5e−3 kN
c = 6.6e−3 kN
c = 9.7e−3 kN
c = 12e−3 kN
= 2.3◦
= 5.7◦
= 11◦
= 17◦
c = 8.7e−3 kN
c = 17e−3 kN
c = 24e−3 kN
c = 30e−3 kN
= 2.3◦
= 5.7◦
= 11◦
= 17◦
c = 17e−3 kN
c = 33e−3 kN
c = 49e−3 kN
c = 60e−3 kN
= 2.3◦
= 5.7◦
= 11◦
= 17◦
c = 35e−3 kN
c = 66e−3 kN
c = 97e−3 kN
c = 0.12 kN
FS
time increasing
fresh material
cµ
partially weathered
material
uncemented soil
φµ
FN
tµ
Figure 17. New model implemented in PFC: the contact strength is given by the summation of two distinct
contributions: and c . The strength can be diminished as a function of weathering action.
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DOI: 10.1002/nag
2016
S. UTILI AND R. NOVA
In the next section, a study on the evolution of cliffs subject to weathering will be shown.
To simulate the effect of weathering, the macromechanical cohesion needs to be progressively
diminished, whereas the friction angle can be considered constant, as a first approximation at least.
This can be done simply by reducing c , as shown in Figure 17. When the material becomes fully
weathered, hence c = 0, the contact bond reduces to being a slip contact. According to this, the
DF model implemented in PFC2D can be considered as a natural extension of the slip model,
already present in the code, to include cohesion in a consistent way.
5. APPLICATION: A STUDY ON THE EVOLUTION OF CLIFFS
SUBJECT TO WEATHERING
5.1. Introduction
In this section, a study on the evolution of natural cliffs subject to weathering will be illustrated.
The strength of the cliffs considered is expressed by cohesion and friction according to the Mohr–
Coulomb strength criterion. Weathering was assumed to make cohesion progressively decrease
over time, whereas the friction angle remained constant. Kimmance [25], for instance, reported
experimental data relative to weathering of granites, which underpin this assumption (see Figure 18).
First, the case of uniform weathering will be dealt with. The assumption of uniform weathering
is clearly unrealistic, since weathering affects soil zones close to the exposed surfaces of the cliffs
more rather than zones well within the soil mass. But this case is worth considering since the results
obtained will be compared against analytical predictions available from limit analysis [26]. In that
work, the evolution of homogeneous cliffs subject to a uniform decrease in cohesion was studied
by repeatedly applying the limit analysis upper bound method on the different forefront profiles
Figure 18. Shear strength envelope for different degrees of weathering of granites (after Kimmance [25]).
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DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2017
Figure 19. Evolution of a cliff initially vertical ( = 34◦ ) according to the limit analysis upper bound
method [26]. Grey lines indicate the successive failure mechanisms taking place at different times ti
whereas black lines indicate the final cliff profile.
assumed by the cliffs subject to subsequent retrogressive failures (see Figure 19). The critical
cohesion value responsible for each failure was found by imposing the balance of the external
work done by the soil mass slipping away and the energy dissipated along the corresponding
failure surface. The upper bound solution obtained assuming a logarithmic spiral single wedge
mechanism, which is slightly higher than multiple-wedge logarithmic spiral mechanisms (see [27]),
is particularly significant since it differs by less than 1.6% (1.6 in the case of = 0◦ , less in the
case of >0◦ ) from the best lower bound solutions available today by numerical limit analysis
(see [28]). Therefore, it is possible to say that the value given by the upper bound single mechanism
is a good approximation of the collapse value. It can be argued that natural cliffs are made of
cohesive soils and soft rocks that usually do not follow an associative flow rule whose validity is
assumed by limit analysis. Nevertheless, Manzari and Nour [29] showed that from an engineering
viewpoint, for homogeneous slopes the assumption of a non-associative flow rule would make little
difference in terms of critical collapse values. Therefore, the results obtained by limit analysis can
be deemed meaningful and retained to validate the DEM approach.
Next, the case of non-uniform weathering will be illustrated. Different cases were analysed
depending on different hypotheses about the way weathering propagates within cliffs and the
initial cohesion assumed. From a qualitative point of view, three different mechanisms of cliff
retrogression took place depending on the choice of the parameters adopted, which will be described
in the following.
5.2. Features of the slope analysed
A vertical uniform slope 40 m high with unit weight = 20 kN/m3 was analysed. A quite typical
value of friction angle ( = 34◦ ) for natural cliffs was adopted. According to the slope height
chosen, the relationships between c, and c , achieved for a middle-low range of pressures
(from 0 to 500 kPa) can be used. After some trials, a suitable geometric domain (see Figure 20) was
determined so that H and V were linearly distributed along the lateral left and right boundary
walls. A large length (3H) behind the cliff front was chosen in order to follow the progressive
retreat of the cliffs.
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DOI: 10.1002/nag
2018
S. UTILI AND R. NOVA
3
4
3H
H
5
2
H
6
H
1
Figure 20. Domain adopted in the numerical simulations with the numbered
boundary walls used during the generation phase.
Concerning grain size, it was obviously not possible to assume the particle sizes adopted in the
biaxial tests, since the number of elements required to simulate the problem would be too large
for the capabilities of the computers available today. Hence, particle sizes were suitably upscaled
retaining the same uniform distribution adopted in the biaxial tests (Rmax /Rmin = 3). The particle
radii were multiplied by a common factor = 66.7 in a first simulation case and = 200 in a second
simulation case, the total number of particles being 74 405 in the first case and 21 385 in the second
case. All the other parameters, such as contact stiffness, porosity, etc., remained unchanged. In
order to assign the chosen internal friction angle = 34◦ to the slope, was determined from
Equation (1). Hence the value of tan = 0.214 was assigned to the discs and their rotations were
inhibited.
Particles were generated by the same procedure used during biaxial tests with the only difference that six instead of four rigid frictionless boundary walls were used to contain them. After
the completion of the generation procedure, gravity was applied. Once the system reached static
equilibrium, bonds were assigned to contacts. The network of contact bonds obtained was characterized by the same irregular fabric as the networks obtained in the biaxial tests. An initial very
high value of cohesion, about 10 times the cohesion value recorded at the occurrence of the first
failure, was assigned to particles. Successively, the 4th and 5th walls (see Figure 20) were moved
away from the soil domain with a speed slow enough to maintain quasi-static conditions up to
complete detachment of the walls from the soil. At the end of this operation, a vertical cohesive
slope under geostatic conditions was reproduced. Considering the region behind the slope forefront,
compressive vertical chains of contact forces were obtained, whereas tensile contact forces were
present in the upper cliff region along the horizontal direction.
5.3. Uniform degradation
Cohesion within the cliff was decreased by finite steps starting from the value assigned at the end of
the cliff generation procedure. After each decrease in cohesion, some cycles were executed. Since
the loading conditions did not change, unbalanced forces did not immediately arise. Therefore,
some cycles were needed in order to make the strength reduction affecting the static equilibrium
of the system. In order to identify the onset of a failure, the kinetic energy of the system was
monitored. The values of c corresponding to the occurrence of each failure were recorded. Since
these values correspond to values of cohesion lower than clim , the c–c relationship (Equation (2))
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DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2019
achieved in Section 4 is valid and it was used to calculate the corresponding values of c. Since the
particle radii have been upscaled by a factor ( = 66.7 in the first case and = 200 in the second
case) and the unit weight assigned to particles was kept the same, the magnitude of contact forces
and therefore the magnitude of the contact shear strength will result in being upscaled by the same
factor because contact forces are directly proportional to particle radii. Therefore, the value of the
adhesive force actually assigned to contacts is given by
ccont = ·c = (tan ) K 4
K3
c
(3)
In these simulations, time had no real physical meaning until the onset of a failure line and the
arousal of significant dynamic forces. In fact, the weathering action occurs in quasi-static conditions
and therefore the time needed to bring cohesion from the initial value cini to the critical value
corresponding to the occurrence of the first failure c1 depends on the number of cycles run after
the imposition of each cohesion decrement c. The time steps used for each cohesion decrement
were high enough to let the system reach a new equilibrium under a different value of cohesion
for the bonds, but as few as possible to save computational time. At the onset of dynamic forces in
the system, the features of the simulation change radically since the computational elapsing time
assumes a real meaning for the soil slipping away. In terms of computational effort, the monitoring
of debris flow is the most onerous task.
In all the simulations, no numerical damping was used. Since the variable used to monitor the
onset of a failure mechanism was kinetic energy, the presence of numerical damping acting to
diminish particle accelerations would delay the detection of the failure, which therefore would be
attributed to a lower cohesion value leading to an underestimation of the critical value of cohesion
at failure. Moreover, the use of numerical damping to follow the soil movement when a failure
takes place would alter the results at the very least in terms of the final position taken by the debris
material.
Figure 21. Fields of velocities just before the occurrence of the first failure: (a) according to the
distinct element method (PFC-2D) and (b) according to limit analysis (rigid rotation mechanism). The
scale is the same for both figures.
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Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
2020
S. UTILI AND R. NOVA
Figure 22. Cliff just before the occurrence of the first failure. Representation of regions where bonds are
broken. Light colour = disks with intact bonds; dark colour = disks with at least one bond broken.
5.3.1. First failure occurrence. The field of velocities of the particles recorded just before the
occurrence of the first failure is shown in Figure 21(a). From the figure, a log-spiral contour line
delimiting the soil region involved in the failure mechanism can be recognized. This line is very
similar to the logarithmic spiral assumed by limit analysis and it passes through the slope toe as
in the limit analysis solution (see Figure 21(b)). In Figure 22, particles are plotted with different
colours depending on whether their contacts have broken or are still intact. From this representation
it can be seen that contact breakage occurs in a limited soil region enclosing the failure line and
that the soil slipping away is formed by a wedge of particles with intact bonds. Moreover, no
contact breakage occurs below the slope toe.
At this point of the numerical analysis, it becomes necessary to introduce hypotheses concerning
erosion. Two extreme conditions of erosion were assumed in this first set of simulations: either
strong or absent erosion. In the former case, all the material detached from the slope front is eroded
before the occurrence of any successive landslide. This case was analysed in order to compare the
predictions obtained by DEM with those achieved by the limit analysis upper bound method in
[26] since the presence of the debris material at the base of the cliff cannot be taken into account
by the limit analysis solution. Although an extreme condition, there are real cases reported in the
literature where this condition is met. For instance, Hutchinson [14] reported experimental data
about the retreat of a coastal cliff in Kent (U.K.) subject to such a condition of strong erosion.
The case of no erosion was then studied in order to investigate how the soil detached from the
cliff front impacts at the cliff base and how much its weight on the cliff toe stabilizes the slope
forefront, delaying the occurrence of successive failures.
5.3.2. Case of strong erosion conditions. The condition of strong erosion was simulated by deleting
all the particles detached from the slope forefront before they impacted the ground level at the
cliff base. In Figure 23(a), the field of velocities recorded just before the occurrence of the second
failure is shown. The shape of the soil wedge slipping away is well fitted by a logarithmic spiral and
therefore is in good agreement with the failure line predicted by limit analysis (see Figure 23(b)).
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2021
Figure 23. Fields of velocities: (a) at the onset of the second failure according to the DEM; (b)
at the onset of the second failure according to limit analysis; (c) at the onset of the third failure
according to the DEM; and (d) at the onset of the third failure according to limit analysis. The
scale is the same for all figures.
The successive failures are characterized by mechanisms that involve only a part (the upper one)
of the slope front (see Figure 23(c)). This result also agrees with the limit analysis solution (see
Figure 23(d)). Unlike the first failure mechanism, bond breakage was not limited to the soil region
close to the failure line but it was smeared over a larger region behind the slope forefront.
In Figure 24, the evolution of the studied slope is shown in terms of normalized cohesion,
c/(H ) the inverse of the stability number, against dimensionless crest retreat. The values obtained
by limit analysis and by the numerical simulations (two cases are shown: slope with = 66.7 and
slope with = 200) are compared. The agreement of the solutions achieved by the two methods is
very good from a qualitative viewpoint and quite good from a quantitative viewpoint. In fact, the
trend of the crest retreat as a function of the cohesion decrease is the same. The limit values of
cohesion predicted by the numerical method (case = 66.7) differ, from the limit analysis solution,
by no more than 20% and the values of crest retreat by no more than 14%.
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0.16
0.14
L.A.
0.12
λ=66.7
c/(γ*H)
0.1
λ=200
0.08
0.06
0.04
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
L/H
Figure 24. Dimensionless crest retreat vs normalized cohesion ( = 34). Circles represent values achieved
by limit analysis; triangles are values relative to the case of = 66.7; squares are relative to = 200. Lines
indicate crest retreat and cohesion values at failure for increasing time.
If the two numerical simulations run are compared, convergence of the numerical solution to the
limit analysis solution is observed since the values of cohesion and crest retreat determined when
the number of particles is higher ( = 66.7) are closer to the limit analysis solution in comparison
with the other case ( = 200). Therefore, increasing the accuracy of the numerical analysis leads
to a better fitting between the two methods.
After some failures occurred, a drift between the crest retreat predicted by the numerical
simulations and limit analysis was observed (right side of Figure 24). This is probably due to the
fact that the number of particles involved in the landsliding soil mass becomes too few and the
crest retreat results affected by the particle size adopted. For this reason, simulations were stopped
in correspondence to the observed drift.
5.3.3. Case of no erosion conditions. In this set of simulations, the particles involved in the
detaching soil mass have not been deleted in order to study more realistically the evolution of the
cliffs. Not only the occurrence of a failure mechanism but also the movement of the detaching
soil until the end of the flow of any debris has been modelled. In order to detect the end of the
debris movement the kinetic energy of the system has been monitored. Once the soil flow had
ended, cohesion was further reduced up to the detection of the subsequent landslide. Doing so,
the stabilizing effect of the debris material lying on the toe of the cliff was not neglected unlike
the previously studied case of strong erosion conditions. In Figure 25, the slope profile is shown
after the first landslide occurrence: the presence of three distinct wedges of bonded particles can
be observed, lying above an inclined plane of debonded particles.
During the occurrence of the second failure, unlike the case of strong erosion (see Figure 23(a)),
only a part of the slope front is involved in the failure mechanism. This is due to the stabilizing
effect exerted by the debris material that fell during the first failure. This effect is also responsible
for values of cohesion at failure lower than those recorded in the case of strong erosion (second
failure occurs at c/(H ) = 0.036 instead of 0.049). In Figure 25(b), the slope profile achieved at
the end of the debris flow is shown. The debris particles lie above the intact soil that is made by
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Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2023
Figure 25. Profile and network of bonds at the end of the soil flow ( = 66.7): (a) after the first landslide
and (b) after the second landslide.
bonded particles making a plane whose inclination was equal to the global friction angle of the
material, = 34◦ , which is also the natural angle of repose for the debonded particles.
Further weathering did not change the situation significantly since it led to subsequent small
detachments of lumps of soil accumulating at the base of the cliff according to the angle of repose.
5.4. Non-uniform degradation
With the model set-up, it is now possible to simulate more and more realistic weathering scenarios.
In fact, weathering is due to an ensemble of chemical and physical actions on hard soils and soft
rocks that are certainly non-uniform in space and change over time. A second set of simulations
was run assuming that weathering propagates within the cliff along two fronts of degradation
parallel to the exposed surfaces of the cliff. Therefore, a horizontal and a vertical front were
assumed. Each front was assumed to progress inwards at constant speed (see Figure 26):
xfront (t) = 0+vx ∗t,
yfront (t) = H −v y ∗t
(4)
where xfront and yfront are the x coordinate of the vertical front and the y coordinate of the horizontal
front, respectively, whereas vx and v y , both positive values, are the x velocity component of the
vertical front and the y velocity component of the horizontal front, respectively. The assumption
of fronts of weathering propagating perpendicularly to the exposed surfaces and at a constant rate
of weathering penetration over time is underpinned for instance by the data reported by Yokota
and Iwamatsu [30], from penetrometer tests on soft pyroclastic rocks.
The degradation of the cliff region subject to weathering (grey zone in Figure 26) was assumed
to be
Dx (t) = (xfront (t)− x)/x
with 0<Dx <1
(5a)
D y (t) = (y − yfront (t))/ y
with 0<D y <1
(5b)
where Dx and D y are damage indices for the generic contact bond located at (x, y), whereas x
and y represent the distance over which damage linearly varies from 0 to 1. Considering a generic
instant of time t, damage will be zero within the inner intact zone of the slope; then it will linearly
vary from 0 to 1 and it will be uniformly 1 in the zone of fully weathered material (see Figure 26).
There is a region within the slope where the horizontal and vertical propagating weathering fronts
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2008 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
DOI: 10.1002/nag
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S. UTILI AND R. NOVA
Figure 26. Weathering within the slope as time elapses.
0.5<D<0.75
0.25<D<0.5
0<D<0.25
D=0
Figure 27. Example of spatial distribution of weathering during the degradation process. The degradation
is plotted in bands. Initial cohesion c = 150 kPa. Arrows represent the velocity field.
overlap (upper left in Figure 26). In this region, soil is subject to a double exposure to weathering
coming from the top and the forefront of the cliff. Here the damage is given by a combination
of the exposure to two different weathering sources.
It seemed reasonable to combine the two
2
damage indices by the following formula: D = Dx + D 2y . This simple formula ensures that the
total damage is higher than the damage due to each single weathering action and it guarantees
a smooth transition between the horizontal and vertical lines at equal damage (see Figure 27).
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DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
2025
On the contrary, the mere sum of the two contributions would imply the presence of an unrealistic
angled point between the lines.
Since the spatial distribution of weathering changes over time, it is necessary to explicitly assume
an initial condition to simulate the progression of degradation over time unlike the case of uniform
degradation. The initial condition assumed at t0 was that of intact cliff: D = 0 all throughout the
slope. This assumption is debatable since it supposes the existence of a time when the cliff was
characterized by a uniform strength, which could be thought as the time of formation of the cliff:
for instance, the settling of a pyroclastic flow. However, the main idea of this study was to model
the problem by steps with increasing complexity: first considering the rather academic case of
uniform weathering and comparing it with the results obtained by limit analysis to validate the
DEM approach, then introducing more and more realistic hypotheses. Hence, assuming cohesion
linearly related to the damage index D, the parameters c and t are given by
c (x, y) = cini ·(1− D(x, y)),
t (x, y) = c (x, y)/ tan (6)
where cini is the uniform initial value of cohesion for the cliff.
Concerning the degradation rate, a constant rate of degradation v D was assumed in all the points
subject to weathering. This parameter represents the damage rate, that is, the increment of D over
time. Therefore, the distances x and y in Equation (5) were determined accordingly:
x = vx ·
1
,
vD
y = vy ·
1
vD
(7)
The duration of weathering is usually quite slow: from some decades to several centuries
depending on the type of soil. Therefore, the speed of weathering needs to be scaled to a much
higher value to run simulations in a feasible computational time. The velocities of the two fronts
of propagation, vx and v y , were scaled by a factor so that if tsim is the fictitious time in the
simulation and treal the real one, tsim = C ×treal . At first, it could be thought that any scaling factor
is fine since a cliff subject to weathering remains under quasi-static conditions. However, if the
scaling factor adopted is too high the bonds do not have time to break off and then the soil failure
is detected with delay. This leads to an overestimation of the times when soil failure takes place
leading, in turn, to an overestimation of the cliff resistance. In conclusion, weathering is induced so
quickly that the increment of weathering per time step is too high to simulate the process correctly.
To choose a proper factor, some sensitivity analyses were performed and C = 1010 proved to be
a suitable value. In order to identify the onset of a failure, the kinetic energy of the system was
monitored as in the case of uniform weathering. When dynamic forces within the system became
significant, weathering was stopped and the elapsed time was recorded. Then cycles were run to
follow the evolution of the moving particles until the system reached again a condition of static
equilibrium. At this point, weathering was resumed and the process of degradation continued until
the subsequent soil movement in the cliff took place.
5.4.1. Results: qualitative description. Two different failure mechanisms took place depending on
the assumed spatial distribution of weathering. If the transition between the intact and the fully
damaged zone was sharp ( y = 1 m), particles started to detach from the vertical slope, just at the
top or along the whole cliff, depending on the ratio between the two velocities of the fronts of
propagation vx /v y (see Figure 28). On the contrary, when the transition was smooth, i.e. there was
a large distance between the intact and the fully damaged zone ( y = 80 m), a full block of soil
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L2
L1
L1
A
t1
t1
v
A
ζ
A
vx
L3
L2
t2
φ
t3
vy
t2
(a)
(b)
Figure 28. Scratch failure. The arrows indicate the velocities of the detaching particles. The dashed lines
indicate the position of the full damage line, D = 1, at different times: (a) crest retreat governed by
horizontal front (case of vx /v y = 0.1) and (b) crest retreat governed by vertical front (case of vx /v y = 0.5).
Figure 29. Cliff at the onset of the first failure. Different colours for different levels of degradation. The
black arrows represent the velocity field. Initial cohesion c = 150 kPa.
detached from the cliff (see Figure 29). The two mechanisms could be defined as ‘scratch’ and
wedge failures, respectively. The first term refers to the fact that material fell off the cliff as if
it were the result of a scratch action; the second term recalls the typical landslide involving the
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2027
detachment of a wedge of soil material. In the first case, particles started falling off the cliff when
a layer of fully weathered material with fully debonded particles was formed. The particles fell
off since they could not stand in vertical position without cohesion. In the second case, when
the wedge failure took place along a log-spiral surface, all the particles were still bonded (see
Figure 29).
Considering now the case of scratch failure, the evolution of the slope can be described in the
following way: once particles started falling off, they progressively accumulated at the base of the
slope and the process of detachment continued until the debonded particles at the top of the cliff
got sustained by the debris accumulated at the cliff base. In this case, single or small groups of
particles detached from the cliff as weathering progressed; therefore, it was not possible to identify
discrete events of failure and the crest retreat can be considered continuous over time. This type
of failure mechanism is very far from the case of uniform weathering where only wedge-type
failure mechanisms took place. The crest retreat–time relationship achieved in this case is linear
unlike the case of uniform weathering where it is step-like. Linearity is due to the assumed front
of degradation moving at a constant velocity. Hence, if we consider the line of full degradation,
characterized by D = 1, it propagated inwards at constant velocity too. Obviously, as soon as
particles were reached by this line, they started to move if not sustained. Depending on the ratio
of the velocities of the two fronts, either the vertical or the horizontal one resulted to govern the
crest retreat. If the horizontal front was fast, vx /v y = 0.5, particles started falling from the vertical
forefront and the crest retreat rate was governed by the speed of horizontal weathering. As it can
be seen in the scheme depicted in Figure 28(a), the line of particle detachment moves constantly
inwards with speed vx . If the horizontal front was slow, vx /v y = 0.1, particles started falling from
the corner at the top of the cliff (see Figure 27). Then as degradation proceeded, particles started
falling from the vertical forefront as well, but the crest retreat rate was governed by v y , the
velocity of propagation of the horizontal front. This can be explained looking at Figure 28(b).
The progressive crest retreat L i is governed by the soil detaching from the horizontal top layer of
debonded particles. This detachment, in turn, occurs roughly according to the angle of friction of
the material: particles start moving only if >, being the horizontal slope of the line joining each
debonded particle with the corner where soil is still bonded: point A in Figure 28(b). Therefore,
the crest retreat progressed at a velocity close to v y .
Considering the case of wedge failure, a sequence of discrete landslides was recorded over time.
The retrogression of the cliff was characterized by a discrete number of landslide events taking
place similarly to the previously analysed case of uniform weathering. In Figure 29, the weathering
distribution within the cliff at the onset of the first failure can be seen. In Figure 30 instead, the
velocity field at impending second failure is plotted. The movement involved only the upper part
of the cliff forefront since the lower part was sustained by the debris accumulated at the cliff base.
The relationship between time and crest retreat is step-like since crest retreat occurs only at the
occurrence of discrete failure events.
Considering the geological literature [31], the ‘scratch’ mechanism is quite typical and it can
take place for a wide variety of soft rocks and hard soils. The wedge mechanism, instead, is
typical of many coastal areas made of chalks or other soft rocks. For example, Hutchinson [13, 14]
described some cliffs in Kent (U.K.) subject to a series of discrete crest retreats.
5.4.2. Results: quantitative description. Many parameters were involved in the analyses run but
they can all be derived from four independent parameters: v y , vx , v D and cini . The speed of the
horizontal front of weathering propagation, v y , varies greatly depending on the type of material
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S. UTILI AND R. NOVA
Y
X
Figure 30. Cliff at the onset of the second landslide. Light grey, original material; dark grey, debris particles.
Arrows represent the velocity field. Initial cohesion cini = 150 kPa.
considered. For instance, a typical value for a carbonatic rock would be v y1 = 0.007 m/year, whereas
a possible value for clay soils could be 100 times higher, v y2 = 0.7 m/year. A reasonable value of
v D for a carbonatic rock would be v D1 = 0.0061 year−1 , whereas for clay soils we can assume
v D2 = 0.61 year−1 . The speed of crest retreat, as already observed, depends on the speed ratio
of the propagation fronts of weathering; hence, two cases have been considered: vx1 = 0.1v y1 ,
vx2 = 0.1v y2 and vx1 = 0.5v y1 , vx2 = 0.5v y2 . Finally, a value of cini = 500 kPa was chosen for both
cases. In Figure 31(a), the relationship achieved in terms of time–crest retreat for the two cases
analysed is shown: vx = 0.1v y and vx = 0.5v y for both types of the geomaterials considered. Since
v y2 /v y1 = v D2 /v D1 = 100, the length of the transition zones, x and y , is the same for both
geomaterials and therefore the results of one simulation can be used for both geomaterials. In the
case of clay soils, the complete retrogression of the cliff occurs in about 80 years, whereas, in the
case of a carbonatic rock, the retrogression of the cliff occurs in about 8000 years. In Figure 31(a),
the time of crest retreat is scaled on being = 1 for clay soils and = 100 for the carbonatic rock.
A second set of simulations was run keeping v y1 = 0.007 m/year, v y2 = 100×v y1 , vx1 = 0.5v y1
and vx2 = 0.5v y2 but changing both cini and v D in order to find when a wedge-type failure
mechanism took place. To this end, cini has to be as small as possible and therefore it was assumed
that is the value at which the first contacts started to break for the case of uniform weathering equal
to 150 kPa. Then, smaller values of v D were assumed until the sought mechanism was found (see
Figure 29). In Figure 31(b), the obtained relationships in terms of time–crest retreat are shown.
When y = 20 m, hence v D1 = 3.5×10−4 and v D2 = 100×v D1 , a scratch-type mechanism still
takes place, whereas for y = 80 m, hence v D1 = 8.8×10−5 and v D2 = 100×v D1 , the relationship
becomes step-like since the cliff evolution is governed by a succession of subsequent landslides.
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DOI: 10.1002/nag
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DEM ANALYSIS OF BONDED GRANULAR GEOMATERIALS
0.8
0.8
Final L/H
0.6
0.6
0.5
scratch failure
0.4
L/H
L/H
0.5
scratch failure
0.3
0.2
0.2
0.1
0.1
0
20ρ
40ρ
60ρ
time [years]
wedge failure
0.4
0.3
0
(a)
Final L/H
0.7
0.7
scratch failure
0
80ρ
0
(b)
20ρ
40ρ
60ρ
80ρ
100ρ
120ρ
time [years]
Figure 31. Relationships achieved in terms of crest retreat–time. = 1 for clay soils; = 100
for carbonatic rocks. (a) Initial cohesion cini = 500 kPa and v D1 = 0.0061 year−1 and (b) initial
cohesion cini = 150 kPa and vx /v y = 0.5.
In this set of simulations, the times needed for a complete retrogression of the cliff are of the same
order of magnitude as those of the other set of simulations (see Figure 31(b)).
6. CONCLUSIONS
A contact bond based on the Mohr–Coulomb strength criterion, assumed to govern the normal
and tangential forces exchanged at the contact points among soil particles, was used to extend the
DEM to the study of bonded granular materials. Unlike other models proposed in the literature,
only one micromechanical parameter is required to pass from an unbonded contact to a bonded
contact. Although simple, the bond model resulted successfully in reproducing the mechanical
behaviour of frictional cohesive geomaterials. The micromechanical parameters involved have been
calibrated for a wide range of c and values so that it is now possible to analyse 2D boundary
value problems of bonded granular materials by DEM adopting the proposed bond model without
the need of recalibration. Simple relationships and a table have been provided for the choice of
the suitable micromechanical parameters from c and .
As an example application, the evolution of natural cliffs subject to weathering has been studied.
Considering homogeneous cliffs, the height of a cliff is directly proportional to the value of cohesion
and weathering causes a progressive reduction of cohesion leading to successive soil failures and
a progressive retreat of the cliff crest. Different weathering scenarios have been analysed for an
initially vertical cliff. Firstly, the case of uniform weathering has been studied. Although unrealistic,
this case was useful to validate the DEM approach against analytical predictions available from
limit analysis [26]. Good agreement between the predictions from limit analysis and the DEM was
achieved. Secondly, a more realistic non-uniform weathering has been modelled. Weathering has
been assumed to propagate from the exposed surfaces of the cliffs inwards and increase linearly
over time. Two kinds of failure mechanisms have been observed, which could be labelled as
‘scratch’ and ‘wedge’ failures. The crest retreat–time relationships obtained are linear in the first
case and step-like in the second case. Both cases are present in nature. The results obtained clearly
show that with the DEM it is possible to realistically model the retrogression of cliffs subject to
weathering and to study the stability of cohesive slopes.
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Int. J. Numer. Anal. Meth. Geomech. 2008; 32:1997–2031
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S. UTILI AND R. NOVA
In this work, an important characteristic of DEM was fully exploited: this method can follow
the evolution of mechanical systems under both static and dynamic conditions without requiring
any particular arrangement or modification to the code used, since equilibrium is always imposed
by cardinal equations of dynamics. It was possible to follow the evolution of a mechanical system
under quasi-static conditions, as in the case of the biaxial tests run or for the simulation of cliff
weathering prior to the onset of soil movements, within a reasonable computational time. Another
aspect deserving some comment is that the geometry of the cliff analysed changed several times.
This would not be easily modelled using FE codes.
ACKNOWLEDGEMENTS
This work was developed during the Ph.D. of the first author at Politecnico di Milano. The financial
support of the Italian Ministry of Education, and the support offered by Ce.A.S. and Itasca Consulting
Group are gratefully acknowledged.
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DOI: 10.1002/nag