Numerical Study of Bending Test On Compacted Clay by Discrete Element Method: Tensile Strength Determination
Numerical Study of Bending Test On Compacted Clay by Discrete Element Method: Tensile Strength Determination
Numerical Study of Bending Test On Compacted Clay by Discrete Element Method: Tensile Strength Determination
Fourier of Grenoble, France; (d) Laboratoire THE, UMR 5564, University Joseph Fourier of Grenoble, France
Abstract: The study of the tensile behaviour of clay is one of the topics which requires a specific
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lighting especially when one pays a close attention to the pathology of work built with or on clays
subjected to significant tensile forces. Therefore, failure or damage of clay, under specific conditions, can
be related to tensile stress limit and not to shear stress limit. It is the case of compacted clay liners in
landfill cap cover subjected to differential settlements within the waste. In order to study the tensile
compacted clay behaviour, a series of laboratory bending tests were carried out. The analysis of the
results and the determination of relevant parameters are difficult. Extrapolation to mechanical clay
behaviour could not be directly obtained. On the basis of the test results, different numerical simulations
using Discrete Element Method were performed and compared with an analytical model, which seems to
be the most suitable to interpret bending tests results for the clay soils. As main result, the Discrete
Element Method confirms that the compression stress remains linear function of the strain during the
bending test until the failure. However, the tensile stresses increase as a non-linear function. For the
tensile strength determination, the analytical differential model remains efficient only when the tensile
Corresponding authors: E-mail: mehrez.jamei@enit.rnu.tn and olivier.ple@ujf-grenoble.fr
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1 INTRODUCTION
Implementation of clay liners in landfill cap cover poses many problems. Because of the real presence of
compressible waste as foundation, this operation requires to carry out a compaction of weak energy.
However, potential differential settlements within the waste should be considered. This phenomenon can
induce bending strains in the clay layer and leads to damage once the cracks developed. As the clay
barrier is a major element of site safety, the mechanical behaviour of the clay must be investigated both
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before and after cracking. Despite the fact of several constitutive laws for compression or for shear were
proposed in the literature, some aspects of clay behaviour were poorly investigated such as the tensile
strength. As experiments are very difficult to conduct under tensile effort, the challenge to set up an
efficient protocol which leads to the tensile strength is not completely resolved. For rock or concrete
materials, according to many authors (Cai and Kaiser, 2004; Claesson and Bohloli, 2002), it is often
necessary to resort to indirect tensile tests (such flat Brazilian test or beam test). Indeed, Kaklis et al.,
(2005) reported indirect tensile tests are simple to perform.. However, many problems are associated
with indirect tensile tests especially when conducted on clay soils, such as the flatten effect in the
Brazilian test, in which shear stresses are induced at the vicinity of the loading plate preventing the
tensile stresses to develop. Furthermore, preparing a beam of clayey soil with an appropriate twinge,
necessary to well predict the bending behaviour, is also a real constraint. Hence, for clayey soils, the
theoretical framework classically used to interpret the indirect tensile tests for rock and concrete
materials seems to be questioned. Then, in case of brittle materials, the difficulties are essentially related
to indirect tests interpretation. Therefore, for materials with ductile behaviour that are used in
compact materials. Unfortunately, the results deduced from this test depend on the experimental protocol
and on the interpretation method (Satyanarayana and Satyanarayana, 1972). The tensile strength deduced
from bending tests depends on the validity of models assumptions. In the literature, three methods were
proposed. The first model is the classical elastic beam theory, based on the assumption that the Youngs
modulus has the same value in tension as in compression (Ajaz and Parry, 1975). The tensile strength is
directly obtained using the measurement of applied force during the test. The second model, called the
direct method or bi-modular method, keeps the assumption that the stress is proportional to strain, but the
value of Youngs modulus in tension may differ from that in compression (Ajaz and Parry, 1975). In this
method the measurement of the strains in the extreme fibres is required in addition. The constraint related
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to the using of this method is that the strains should be recorded using special measurement techniques as
conduct.
The differential method does not assume a preferred stress-strain relationship. The behaviour law is so
unknown (Prentis, 1951). Besides, this method has been appeared as an alternative analytical model to
A body of research works on clayey soils show that tensile strength deduced from bending test may be
higher than that measured by a direct tensile test, the ratio varies between 0.66 and 3 (Berenbaum and
Brodie, 1959; Narain and Rawat, 1970; Addanki et al., 1974). For clay Bricks (Hughes et al., 2000) the
corresponding beams were 65x65x215 mm and the above ratio was 2.3. For a silty clay (Plasticity Index
= 15%, Optimum water content = 22.55%, Maximum dry density = 16.4 KN/m3), Jack et al. recorded
0.66 as a ratio.
Because of the experimental and theoretical analysis difficulties, alternative numerical technique was
developed. The traditional step consists in characterizing the behaviour with experimental data for a
phenomenological modelling and then to extend this behaviour to real geotechnical structure thanks to a
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numerical simulation (FEM). Unfortunately, the numerical simulations usually used have difficulties in
taking into account discontinuities for these materials (especially damage). To get round this difficulty,
one can describe the behaviour of the clay by the distinct elements method (DEM). Already developed to
model the total behaviour of grains (Cundall and Strack, 1979), this method is not of current use to take
into account the cohesive material behaviour. As the soil tested in the study is a silty clay (not a soft soil),
the distinct elements method was used. Firstly it was calibrated with the result of Brazilian and
unconfined compression tests and then applied to bending beam tests. This paper discusses the validity of
Experimental tests were carried out at LIRIGM and ENIT laboratories on a silty clay soil coming
from a site that is selected for a landfill project (near the city of Nabeul in Tunisia). The physical
characteristics of the clayey soil are given in Table 1. Four-points bending tests were selected (Fig. 1)
specifically because of constant bending moment in the middle zone of the specimen. Beam tests have
between the supporting rods must be large enough to limit the influence of the compression zone and to
initiate the failure mode under tensile strength (Fig. 1 and Fig. 2). Due to the low twinges of the tested
beams, we note that the essential assumptions of beams theory are not valid. Consequently, numerical
and analytical models which are not based on the assumptions of beams theory were used to interpret
Series of experimental quasi-static tests were carried out on unsaturated silty clay. In order to keep
the same initial conditions, like the water content and the dry density, the initial water content is fixed to
a standard value equal to 2 % higher than the optimum water content (the average of the water content is
5
17 % 0.5 %, the degree of saturation is 95%). The dry density is almost the Proctor optimum value and
Four displacement transducers were used; two were placed on the support rods, and the others in the
middle part of the specimen (Fig. 1). The load was measured by means of a load cell mounted on the
upper axial support rods of the press. A constant displacement rate (0.2 mm/mn) was applied through the
lower plate of the press. Further, the middle part of the specimen was covered with artificial roughcast
made of black and white painting. This part was filmed continuously with a camera (5 million pixel with
1 pixel = 0.11 mm) to monitor the initiation and the progressive propagation of cracks from the bottom to
The Discrete Element Method is one of the emerging numerical techniques used to investigate the
behaviour of several materials like granular materials. The method consists of a discrete modelling of the
material seen like as structure. Each particle (inclusion, aggregate, grain) interacts with its neighbours by
the effects of the interactions at different scales. Moreover, by this method it is possible to describe the
The complexity of the behaviour is taken into account by methods of assembly. Already developed to
model the behaviour of grains, this method is not classically used to take into account the cohesive
material behaviour. Nevertheless, for clay soils, grains do not correspond to clay particles, but are simply
a means to discretize space, and provide appropriate macroscopic mechanical behaviour through a
convenient fitting with experimental results. It is worth to note that the granular distribution, the initial
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porosity, the shape and the methodology of setting up the particles have also a great influence on the
The main difficulty of DEM is that the parameters needed for the numerical approach are micro-
parameters and not macro-parameters as it can be obtained from laboratory tests. Therefore, first steps of
computation are needed to fit the micro-parameters with respect to results of basic laboratory tests.
The numerical software used (PFC2D, 1997) is a two dimensional software which is an implementation of
the model of Cundall and Strack (1979) based on the principle of molecular dynamics. The algorithm of
computation consists in alternating the application of the Newton's second law of motion for particles
and force-displacement law for grain contacts. The equations of motion are integrated using an explicit
The basic discrete elements used are discs (2D) of various sizes that interact with each other. It is worth
to note that, in order to obtain a more realistic behaviour of soil (particularly to reach high values of
internal friction angle), several adjacent particles are jointed together to make clusters.
A cluster is composed of two discs of diameters ratio D/d = 0.6. Clusters were generated with different
In this study, the contact laws are described by five rheological parameters: normal and tangential
stiffness ( K n and K s ), maximum normal and shear strengths (C n and C s ) and friction coefficient ( ).
It is assumed the parameters of two particles in contact act in series (Fig. 3).
The stiffness model, in correlation with the elastic behaviour material, is a function of the normal and
tangential stiffness. The normal and tangential components of the contact forces ( F n and F s ) are
proportional, respectively, to the overlap between two discs in contact and to the tangential displacement
at contact.
The corresponding basic equations used to define the contact laws between two particles are:
Fin (t ) = K nU i n (t ) (1)
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Fis = K s U is (2)
Where Fin and Fis are respectively the normal and tangential forces at contact (i ) between two particles.
Also ni ,U i n and U is are respectively the normal and relative tangential displacements at contact (i ).
( t ) is a time step during the particle movement. The stiffness contact parameters are computed in
PFC2D as:
(Kni Knj )
Kn = (4)
K ni + K nj
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(Ksi Ksj )
Ks = (5)
K si + K sj
Where Kni , Knj , Ksi and K sj are the stiffness moduli of the particles i and j .The failure behaviour is
defined by a Coulomblike slip model. The adhesion at contact of two particles is defined by normal and
tangential local adhesions C f n andC f s . The adhesions serve to limit the total normal and shear forces that
the contact can carry by enforcing adhesion-strength limits. Hence, the tensile and shear strengths,
C n andC s , of a contact between two particles of diameters di and d j are computed in PFC2D as:
C n = C f n min(di , d j ) (6)
C s = C f s min(di , d j ) (7)
The tangential component of the contact force is limited in magnitude with a Coulomb-like slip model,
with friction angle . At each step of computation the reliable contacts are re-actualized according to
conditions:
F s Fmax
s
(8)
Fn Cn (9)
8
s
Fmax = max( F n ,C s ) (10)
The distinct element scheme is based on Newtons second law of motion. Force displacement law at
contact for the translational motion and Eulers law for the rotational motion.
Fi = mi (xi gi ) (11)
Where mi is the particles mass, xi is its acceleration, gi is the gravity acceleration of particle (i ), M i (3)
, I i(3) and i (3) are respectively the resultant moment, the principal moment of inertia of the particle and
the angular acceleration about the principal axis which is normal to the plane of the disc.
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If the material behaviour is elastic, the energy supplied to the assembly of particles is conserved.
Therefore, it is necessary to introduce a damping term in the equations of motion to obtain a static or a
steady-state solution. The local non-viscous damping proposed in PFC2D is taken into account
if Vi 0
Where .sign (Vi ) = and Vi is the particle velocity in the ei direction.
if Vi < 0
The calculation of stress for a discrete system is well established (Bardet, 1998). The average stress
tensor can be obtained by (14), (Love, 1927), (Weber, 1966), (Christoffersen et al. 1981), (Cambou et
al., 1995)
1
=
V
F
c
(c )
l (c ) (14)
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(c )
Where l is the vector relating the centres of the two particles which are in contact at contact ( c ),
(c )
F the vector contact forces in ( c ) and where the sum covers all contacts in the volume (V ).
This relationship was introduced first by Love then followed used by many other authors. It can be
proved by many ways. Among them, the formulation used in PFC 2D (1997) and proposed by (Thornton
The average stress tensor for the volume V ( p ) occupied by a single particle may be written as
1
ijp =
V (p) V (p)
ij( p )dV p (15)
1
ij( p ) =
V (p ) S(p)
x iti( p )dS ( p ) (16)
Since each particle is loaded by point forces acting at discrete contact locations
1
ij(p ) = (p )
V
xNc
(c ) (c )
i Fj (17)
The minus sign is introduced here to ensure that compressive/tensile forces produce negative/positive
average stresses.
The average stress tensor in a volume V containing a large assembly of particles is defined by
1
ij =
V V
ijdV (18)
Since ij = 0 in the voids, the integral can be replaced by a sum over the N p particles contained within
V as
1
ij =
V
Np
(p )
ij V (p ) (19)
Since measurements are done into circles, and only the particles with centroids that are contained within
the measurement circle are considered in the computation of the average stress tensor, a correction factor
is introduced in order to take into account the additional area that is being neglected. Finally for an
1 n
ij = x c x (p) n (c,p)F c
( ) ( )
(21)
V N p Nc
i i i j
( p )
N
p
Where the summations are taken over the N p discs with centroids contained within the measurement
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- V (p) is the volume of particle ( p ), taken to be equal to the area of particle ( p ) times a unit-thickness;
- xi(p) and x i c are the locations of a particle centroid and its point of contact, respectively;
( )
- ni (c,p) is the unit normal vector directed from a particle centroid to its contact location;
The stress tensor defined by (21) is a useful approximation of the stress tensor used in continuum
mechanics for granular assemblies comprising a large but finite number of particles in a circular area
Although the calculation of stress for a discrete system being well defined, different approaches are
considered to calculate strain (Catherine et al., 2003). Various methods have been developed to link the
DEM and continuum medium approaches to define the strain tensor. As an overview, these methods can
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be classified into two groups. The first group, called kinematic homogenization methods, which are
based on the spatial discretization, uses interpolation functions of nodal displacement field. Such the
case of a graph or a nodal network technique which is constructed by considering the particle centroidal
coordinates. Hence, the incremental displacement gradient is calculated by considering the relative
incremental displacement along each edge of the graph, and assuming a linear variation of displacement
Thomas and Bray (1999) and Dedecker et al. (2000) used approaches based on triangulation of the
granular medium and displacement of the particle centroids. With the linear variation in displacement
assumption, the displacement gradient can be easily calculated. Nevertheless, it appears that the spatial
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discretization approaches are not suited for problems involving strain localizations because the particle
rotations cannot be accounted using these interpolation techniques except some methods proposed by
The second group of method includes an energy based method which equates the strain energy in the
equivalent continuum to the energy stored in the contacts of discrete elements (Calvetti et al., 1997),
(Cambou et al., 2000). It appears these approaches yield to inaccurate estimates of strain.
In this paper, we use the best-fit approach to define local strain (Liao et al. 1997). This approach is based
on the translations of individual particles, ui( p ) ( ui( p ) means the difference between the translation of
particle ' P ' , and the average translation of all particles contained into the measurement circle). The
position of particle ' P ' is denoted x i( p ) , ( x i( p ) denotes again the difference of the particle position from
the average position of all particles contained into the measurement circle). Assuming that every particle
would translate according to a deformation gradient tensor ij , the predicted translations of particles are
given by
ui( p ) = ij x (j p ) (22)
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A measure of the error in these predicted values is given by
Hence, we are looking for that ij which makes the square-sum of these errors the smallest:
2
F (ij ) = ui( p ) ij x (j p ) (24)
Np
F
=0 (25)
ij
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Substituting (24) into (25) and differentiating, from the following equations the optimum ij values are
obtained.
x 1( p )x 1( p ) x (p ) (p )
x ui( p )x 1( p )
N N p
2 1
p Np
i1
=
( p ) ( p ) i 2
, i = 1, 2 (26)
x 1( p )x 2( p ) x 2 x 2 i 2
u (p ) (p )
x
Np N p
Np
Finally the average strain which corresponds to the symmetric part of ij is deduced.
As mentioned before, the major difficulty in using the DEM for real application is to assign the good
micro-properties which lead to global realistic macroscopic behaviour of the studied material.
In order to predict the results of bending tests, the fitting between the micro and macro parameters is
done by the comparison between the numerical and the experimental results of two basic laboratory tests:
unconfined compression test to fit the stiffness and the compressive failure parameters (Khanal et al.,
2005) and indirect tensile test (Brazilian test) to fit the micro mechanical tensile parameters.
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To insure the reproducibility of the numerical results, we paid a particular attention to how the particles
are set to an initial position in terms of porosity, shapes of cluster and size distributions. For this study,
the particles are set to a fixed porosity by a progressive increase in particles radii with a decrease in the
internal friction. The homogeneity of the numerical sample was checked by measuring the porosity in
Several numerical simulations were carried out systematically for each of the curve presented. The
number of particles used is 18 particles per cm (5000 to 8000 particles) which is enough to obtain
Numerical unconfined compression tests were carried out on a rectangular sample of twinge two, and the
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Brazilian test on a circular sample. In both cases, the loads were applied to the numerical samples by
vertically moving two horizontal rigid walls. The vertical velocity is taken small enough to ensure that no
Quantitative results obtained with an initial set of parameters show that the failure mechanism of the clay
material is well reproduced by the numerical model (behaviour law, cracking zone and failure mode).
Figure 5 shows that the failure mechanisms in experimental and numerical results of unconfined
compression tests are identical. The fitting of the micro parameters was done in order to reproduce the
whole experimental results. Once the unconfined compression and the Brazilian tests are well
reproduced, the micro-parameters retained to fit the experimental results are those given in table 2.
Figures 6 and 7 show the results of the numerical prediction of the curves of the experimental tests.
Figures 5 and 8 show the numerical and experimental failure mode of the unconfined compression and
the Brazilian tests. The experimental curve corresponds to the average tests data.
numerical sample is made of 7146 particles positioned according to the numerical procedure described
previously. The loading was applied to the beam by vertically moving the upper rods.
The experimental results of the beam tests are presented and compared with the numerical results (Fig.
9). Figure 10 shows similar failure modes. The difference between the x-coordinate failure sections
between experimental and numerical tests is not significant while bending moment is constant between
the two rods for transferring loads. As shown in Figure 9, the reproducibility of experimental tests is
conditioned by the possibility of keeping initial water content and dry density constant. These conditions
remain difficult and usually lead to repeat the test several times and to indicate the variability of the
results (see above). Nevertheless, the experimental and numerical load-displacement curves are very
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close.
In order to ensure the convenient failure stress computation, different sections are considered in the area
located between the two points for transferring load. The stresses are calculated according formula (21)
in measurement circles (Fig. 11). The number of particles included in each circle of measurement is
about 50 which makes it possible to obtain a representative value of the average stresses. Figure 12
shows the stress field homogeneity and the independency of the choice of x-coordinate of section in this
area.
One of the interests of the numerical study is the possibility to make a fine analysis of the results and to
obtain the diagram of normal stresses under different loading (Fig. 13). The numerical diagrams show
that there is no equality between stresses in compression and these in tension particularly in extreme
fibres (the average stresses are computed in measurement circles). The difference between them
increases with the loading rate (maximum of c t = 5 kPa ). For each load, a non linearity tensile
stress diagram is observed. Hence, we conclude that at failure, the compressive stress of clay is higher
than the tensile stress. For Troloppe and Chan (1960), non-regular stress-strain behaviour for compacted
of the beams and the bimodular elasticity theory), used as models used for the interpretation of the
It is also shown (Fig. 13) that the diagram is linear only in compression zone and the corresponding
strains remain in elastic domain. However, these diagrams indicate a plastic redistribution of stress in the
tension zone.
In that way, the DEM numerical analysis can be used to predict various bending beam tests. Based on the
numerical tests, this approach could be used to give an accurate analysis of the different tensile tests.
This method can be an interesting tool to help us understanding the mechanical behaviour of clay layers.
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Ajaz and Parry (1975) proposed an alternative suitable analytical model (differential model) to
interpret and analyse the bending tests on clay soils. This model does not assume a preferred stress-strain
In each section of beam, we can express the compression and tensile forces Fc and Ft as functions of
bd
b.c,t dy =
d c
Fc = 0 c
0
f (c,t )d c,t (28)
0 b(h d ) 0
Ft = d h
b c,t dy =
t t f (c,t )d c,t (29)
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Based on the static equilibrium, we can show that the moment M is given by
b h c
M =
(c + t )2 t
f (c,t ) c,t d c,t (30)
Finally, after differentiating the latter equation with respect to c and t , the model leads to the
following non linear system, where the two differential equations are functions of the bending moment
which depends on the compression and tensile strains under the form
1
c = (31)
c + t c
1
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t = (32)
c + t t
Where is given by
2
M (c + t )
= (33)
bh 2
Hence, the compression and tensile stresses are calculated by differentiating with respect to c and t .
It is obvious that the compression and tensile stress values are calculated respectively with the two
equations at the top of the bending sample and at its bottom. Only one differentiation is needed to
calculate the required values of compression and tensile stresses from experimental values of M, c and
t. The authors gave the evolution of the variable term and used a graphical differentiation by drawing
tangents at the desired points. Table 3 summarizes the results of numerical and analytical models. The
numerical results are the normal stresses at the lower and the upper fibres of the beam deduced from the
computed values in the measurement circles (Fig. 11). For the differential method, the process requires
the graphical determination of the derivative of function over compression and tensile strain at failure.
It appears that the numerical and the differential method give the same qualitative results in the tension
zone, but it is not the case for the stress diagram in compression zone (Fig. 13 and Fig. 14). We can also
17
note that the differential method gives a higher stresses values compared with those obtained by the
numerical model. According to several authors (Ajaz and Parry, 1975; Satyanarayana and Satyanarayana,
1972; Ramanathan and Raman, 1975), the ratio between the compression and the tensile stresses varies
between 1 and 3 on experimental beam tests. This ratio increases with the increase of water content and
becomes more and more sensitive. For our numerical simulations, this ratio is equal to 1.26 for the
differential method and equal to 1.11 for the numerical DEM simulation. Due to weak plasticity, the silty
clay of Nabeul can be considered close to a brittle material (Fig. 7 and Fig. 9). In that way, we think that
the mechanical behaviour of the clay is closer to the numerical DEM results. Despite the restrictions that
result from these preliminary results, this shows the importance of using DEM approach for studying the
6 CONCLUSION
The numerical study carried out with the PFC2D code improves the potential of the DEM which can
help in predicting many experimental tests. It is shown that the main condition for an efficient use of the
DEM is to make a good fitting of micro-parameters from basic convenient experimental tests. Its
The analytical differential method is used for comparison because it is not based on a preferred behaviour
law assumption. It seems that it leads to the same numerical stress diagram in tension zone. Nevertheless,
the tensile strength determined by the differential method is also overestimated comparing with the
numerical value (the ratio is 1.2). From the numerical investigation, it is concluded the linear stress-strain
relationship in compression zone during the bending test until the failure. While, non-linear tensile
stresses are highlighted, especially when failure is approached. This result is relevant of the plastic
may conduct to an overestimation of the tensile strength. The bending experimental protocol for the kind
of material is only used for the low beam twinges. Consequently, the elastic beam theory can not be
applied.
On the other hand, the DEM gives a power tool to obtain the tensile stress function of the strain, the
Finally, it seems that the contact law presented in this paper is well established for soils with low
plasticity (quasi-brittle material). Nevertheless, for materials with high plasticity, another study should be
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n. 3, pp.586-591.
materials, Cambou B.(ed.), No. 385 in CISM Courses and Lectures. Springer: Wien, New York, pp.
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Bathurst R.J. and Rothenburg L. (1988), Micromechanical aspects of isotropic granular assemblies with
linear contact interaction, Journal of Applied Mechanics, vol. 55, pp. 17-23
Berenbaum R. and Brodie I. (1959) Measurement of the tensile strength of brittle materials, British
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material: relation between structure evolution and loading path, Mechanics of CohesiveFrictional
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Claesson J. and Bohloli B. (2002) Brazilian test: stress field and tensile strength of anisotropic rocks
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Cundall P.A. and Strack O.D.L. (1979) A discrete numerical model for granular assemblies,
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Jack A. Harison, Bobby O. Hardin and Kamyar M.(1994) Fracture toughness of compacted cohesive
soils using ring test, Journal of Geotechnical Engineering, vol.120, n 5, pp. 872-889.
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Kaklis K. N., Agioutantis Z., Sarris E. and Pateli A. (2005) A theoretical and numerical study of discs
with flat edges under diametral compression (Flat Brazilian test), 5th GRACM International Congress
Khanal M., Schubert W. and Tomas J. (2005) DEM simulation of diametrical compression test on
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based on the hypothesis of best fit, International Journal of Solids and Structures, n34, pp. 4087-
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Narain J. and Rawat C. (1970): Tensile strength of compacted soils. J. of the soils Mechanics and
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PFC2D user manual, release 3.00 (1997). Itasca Consulting Group, Inc., Minneapolis.
Prentis H. M. (1951) The distribution of concrete stress in reinforced and prestressed concrete beams
when tested to destruction by a pure bending moment, Mag. Conc. Res., Vol 2, n 5, pp. 73-77.
Ramanathan B. and Raman V. (1974) Split tensile strength of cohesive soils, Geotechnical
Thomas P.A. and Bray J.D. (1999) Capturing the Nonspherical Shape of Granular Media with Disk
Thornton C., Barnes D.J. (1986) Computer simulated deformation of compact granular assemblies,
Troloppe D. L. and Chan C. K. (1960) Soil structure and step-strain phenomenon, J. of the soils
Mechanics and Foundations Division, Am. Soc. Civ. Engrs., 86, n2, pp. 1-39.
Weber J. (1966) Recherches concernant les contraintes intergranulaires dans les milieux pulvrulents.
Figure 2: Cracks on the lower part of the beam clay tested in LIRIGM Laboratory
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23
Normal Stiffness Kn
Tangential Stiffness Ks
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24
(a) (b)
29
Figure 9: Experimental and numerical bending tests simulations
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30
Figure 10: Numerical prediction of failure mode in bending test
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31
between 2 m until 80 m
Plasticity Index PI 13 %
Micro-parameters
kn 14 MN/m
ks 14 MN/m
C nf 150 kN/m
C sf 150 kN/m
0.4
hal-00154652, version 1 - 14 Jun 2007
porosity 0.17