Modeling of the Mechanical Behavior
of Untreated Granular Materials
Bachir Melbouci
Laboratory of geomaterials, Environment and Installation (L.G.E.A.)
University Mouloud Mammeri Tizi ouzou Algeria.
bmelbouci@yahoo.fr
El-Mahdi Meghlat
Laboratory of geomaterials, Environment and Installation (L.G.E.A.)
University Mouloud Mammeri Tizi ouzou Algeria.
mahdi_meg@yahoo.fr
Malek Cherchar
Laboratory of geomaterials, Environment and Installation (L.G.E.A.)
University Mouloud Mammeri Tizi ouzou Algeria.
malek_cherchar@yahoo.fr
ABSTRACT
The present article is a synthesis of the studies of modelling carried out on untreated granular materials.
The granular material constitutes one of the essential elements of our environment either in its elaborate
form or in its natural form. The extent of their application fields is very wide; we can mention their use in
the road infrastructure.
The evaluating of the principal parameters affecting the performances of granular materials using behaviour
laws requires an identification of these materials borne out by tests carried out in a laboratory (triaxial
compression test). These tests aim at best simulating their behaviours as well as possible and thus
optimizing and specify the dimensions of geotechnical structure.
The nonlinear model for the elastic and elastoplastic behaviour -thus developed in this research- will be
validated by perfect simulation and the description of the authentic performances of the granular materials
subjected to this study. To this end; a parametric study is carried out on the Pegmatite material in these
various states and various size ranges by varying adequately the geotechnical characteristics and the
confining stresses. We retained three behaviour models for this study:
a) Nonlinear elastic behaviour.
b) Elastic-perfectly plastic behaviour.
c) Elastoplastic behaviour with hardening parameters.
KEYWORDS: mechanical behaviour, untreated granular material, triaxial test, Boyce model, Softening
/Hardening model.
(Editor’s note: A comma is used in this paper for decimal point; e.g. 0,5 means 1/2)
INTRODUCTION
The construction of civil engineering infrastructures such as roads or dams, exploit a great
quantities of granular materials such as gravels and sands. Their use requires a good knowledge
of the mechanical behaviour in the short and the long times for a better control of the
characteristics of the civil engineering structures and in order to optimize their dimensions. To
satisfy certain exploitation and economic requirements, we showed more interest in the use of
local granular materials considering their presence in sufficient quantity, particularly, in great
Kabylia, Algeria.
The need for economy in the projects on the one hand, and to stop the abusive extraction of
alluvial materials on the other, makes it necessary to find alternative solution. The magmatic
rocks like Pegmatite, slightly employed in the traditional uses, could certainly provide the
necessary quantities if we had a better knowledge of there behaviour.
The study of the granular Pegmatite material behaviour developed in this research requires the
implementation of specific resources: experimental, theoretical and numerical for the
characterization of the granular mediums in general and especially local materials of Kabylia.
Three models of the behaviour law are retained:
1) The Boyce model which is a nonlinear elastic model adapted to untreated materials.
2) The Mohr-Coulomb model which is elastic perfectly plastic model.
3) The Strain Softening/Hardening model which is an elastoplastic model.
It should be noted that the two last behaviour laws were used by undertaking a work of simulation
of the triaxial compression test by a software program FLAC 2D established on the finite
difference method and specialized in the resolution of the geotechnics problems.
The study of the pegmatite material according to the triaxial stress path made it possible to
highlight the attrition of the grains which provoke significant deformations in the granular
material by activate the grains rearrangement mechanism. This phenomenon of attrition of the
grains is also at the origin of the progressive decrease of the resistance of materials with the
increase in the confining pressure (Ramamurthy 2001). The choice of a failure criterion of the
Mohr-Coulomb model is thus unsuited for this kind of material.
The exploitation of the two behaviour models, Boyce and Strain Softening/Hardening, we
reproduced the principal results of the triaxial compression tests carried out on the Pegmatite
aggregates; the results are closer for the second model. That is due to the elastoplastic nature of
the Strain Softening/Hardening model, which takes into account the phenomenon of hardening
parameters. However, it is necessary to pay caution concerning the phenomena of crushing of the
large sized grains.
IDENTIFICATION OF STUDIED MATERIAL
The granular material retained for this study is the Pegmatite of Kabylia extracted in the form of
blocks then crushed in three size ranges 0/6, 6/10 and 10/14 and taken under three states of
compactness (dense, loose and wet). The values of the geotechnical characteristics selected for
numerical modelling, are the results of the work of Mr. Melbouci carried out on these local
materials of Algeria using the triaxial compression tests under a monotonous loading, (Melbouci
and Roth, 2001; Melbouci, 2002a, 2002b; Melbouci, 2003).
The pegmatite is a magmatic rock having a very coarse grained structure chemically similar to the
granite on which it is closely dependent. Pegmatite’s minerals are very characteristic of the acid
intrusive rocks: the feldspar orthoclase, quartz and mica. Pegmatite is very widespread in the
earth's crust, but it is mainly found in the oldest mountain ranges. Pegmatite presents the same
mineralogical composition as the granite. The principal minerals are:
- The feldspar (Kaolin).
- Black mica (in small quantity).
- The quartz of milky white colour.
- White mica or muscovite (shining).
These minerals can vary in the following proportions:
a) Feldspar (15 to 35%). Their abundance (with quartz) gives the rock a leucocratic aspect. The
potassic feldspars usually appear in big orthoclase crystals and sometimes in perthitic or
microline crystals.
b) Quartz (55 to 75 % out of xenomorphic crystals), has a structure in mosaic.
c) Muscovite or biotitic (5 to 20%). Muscovite is automorphe and is found in larger quantities
than the biotitic spangles. Biotitic is scare and its pleochroïsm ranges from brown dark to brown
green. It can also appear in spangles epigenesist and in oxides which is slightly oriented, by
undergoing a weak deformation.
The various physical and mechanical characteristics of the Pegmatite material are mentioned
respectively in table 1 and 2 according to:
Table I: Physical characteristics of the pegmatite material.
Caract.
material
 s (kN/m3)
n (%)
 d max (kN/m3) emin
emax
Pegmatite
26,7
12,28
17,85
1,17
0,41
Table II : Mechanical characteristics of the pegmatite material.
Caract.
material
MDE (%)
LA (%)
 dopt (kN/m3) wopt (%)
ICBR
Pegmatite
51,9
28
17,8
38,78
9,22
APPLICATION OF NONLINEAR ELASTIC BEHAVIOUR
MODEL
Boyce model
The reversible behaviour of the untreated gravels is rather well described by a nonlinear elastic
model suggested by Boyce (1976, 1980). This model has the advantage of being relatively
simple (three parameters) and derives from a potential. The model of Boyce is expressed in terms
of compressibility modulus K and shearing modulus G.
q
pn 
V  1    
K
 p

q 
pn  q 
 
3G  p 
2



(1)
(2)
with
  1  n 
K
6G
(3)
Where: p and q indicate respectively normal stress and the deviator of stress.
Β: parameter characterizing the dilation of material.
The Boyce model, such as described by the preceding equations is defined only in one field of
stresses such as:
p must be strictly positive;
2
q / p lower than a value such as
q
1      0
 p
Starting a mathematical law known as behaviour law connecting physical parameters together, we
chose a method of programming which will enable us to determine the parameters of the Boyce
law.
For that, we consider the following assumptions:
f ( p,q)
. p 
(4)
q
1  ( ) 2
p
With (  ,  ,  ) positive real parameters.
To find the value of X = (  ,  ,  ) , it is enough to minimize the functional calculus: J (  ,  , )
J (  ,  , )  i  Vi  f ( pi , q i )
2
(5)
And that by twin two numerical methods: Newton and Gauss-Seidel.
Identification of the functions used for the determination of the Boyce law parameters
The relations of elastic behaviour can be found in the following general form:
 v  f  p, q 
and
 q  g p , q
(6)
(7)
Where f and g are two mathematical functions to be determined.
Thus, we set a total characterization of the behaviour of the studied materials starting from
observations made on material behaviour and results obtained by defining the functions f and g.
APPLICATION OF THE ELASTOPLASTIC
BEHAVIORS MODEL
Actually, very few materials have an elastic behaviour. Beyond a certain threshold of stresses, all
or part of the deformation is then irreversible. As a complement to the elastic strain, the
elastoplastic models integrate a degree of permanent plastic deformation characterized by a flow
rule when the failure criterion is reached.
1. Numerical modeling
The numerical modelling of the triaxial compression test was carried out with the computer code
FLAC 2D based on the finite differences method (Itasca, 2005). Calculations were carried out in
two dimensions (in a plane space), where the grid of the sample of the triaxial compression tests
is considered as one mesh. The mesh dimensions are equal to a unit measurement whose
slenderness ratio is equal to two per convenience in such manner to superimpose displacements
and deformations.
The initial and boundary conditions are as follows: on the left vertical boundary of the numerical
model are imposed, conditions of zero displacements according to the horizontal direction (U=0),
while on the lower boundary the displacements are equal to zero on the vertical boundary (V=0).
A uniform and constant pressure is applied to the higher limit and the right limit as well as an
increasing axial stress in a monotonous way until the rupture of the sample (figure 1).
JOB TITLE : Simulation de l'essai triaxial de compression
FLAC (Version 5.00)
1,2
LEGEND
2,2
1.000
X
22-Nov-05 8:20
step
8400
-4.167E-01 <x< 9.167E-01
-1.667E-01 <y< 1.167E+00
0.800
Grid plot
0
0.600
2E -1
Gridpoint Numbers
Grid
point
Numbers
Fixed
Gridpoints
X X-direction
Y Y-direction
B
Both
LGEA
direction
UMMTO
Net Applied
Forces
0.400
0.200
1,1
2,1
B
-0.300
-0.100
0.000
Y
0.100
0.300
0.500
0.700
0.900
triaxial compression test.
Figure 1: Grid considered and boundary conditions of the triaxial compression
test.
2. Elastoplastic behavior
For modelling the granular material behaviour we have selected two criteria of failure provided in
FLAC 2D. The first, the elastic perfectly plastic models which is the Mohr Coulomb model ; the
second is Strain Softening/Hardening elastoplastic model which significantly takes into account
phenomenon such as hardening parameters, the contractance and dilatancy that produces a
granular material during a monotonous loading to the triaxial compression test.
The Mohr-Coulomb model
In the elastoplastic model of Mohr-Coulomb, the strain increments are decomposed as follows:
d i  d ie  d ip
; i = 1,3
(8)
Where the superscripts e and p refer to elastic and plastic parts, respectively, and the plastic
components are nonzero during plastic flow only.
The failure envelope for this model corresponds to a Mohr-Coulomb criterion (shear yield
function) with tension cut-off (tensile yield function). The shear flow rule is non-associated and
the tensile flow rule is associated.
Mohr-Coulomb failure criterion
The failure criterion may be represented in the plane (σ1, σ3) as illustrated in Figure 2. The
failure envelope is defined from point A to point B by the Mohr-Coulomb yield function
f S   1   3 N  2C N  0
(9)
and from B to C by a tension yield function of the form  t
f t  t  3 0
(10)
3
s
f =0
f t= 0
B
C
t
A
2c
N
+
 3 - 1 = 0
c
tan 
1
-
Figure. 2 : Mohr-Coulomb failure criterion in FLAC, according to Itasca.
where φ is the friction angle, c, the cohesion, σt, the tensile strength and
N 
1  sin 
1  sin 
 cm  2 C N 
t 
C
tan 
or
or
sin  
C 
N 1
N 1
 cm
(11)
(12)
2 N
or
C   t tan 
(16)
Mohr-Coulomb flow rule
In the plastic part, a flow law defines the behaviour of material when the deformations are
irreversible. The shear potential function gs corresponds to a non-associated flow rule and has the
form
g s  1   3 N
(13)
and the associated flow rule for tensile failure is derived from the potential function gt , with
g t   3
where  is the dilation angle and
N 
(14)
1  sin 
1  sin 
(15)
The Mohr-Coulomb Parameters
The Mohr-Coulomb model is associated to a linear elastic law, thus the required parameters are
seven (07). The table 3 represents an example of these values for size range 6/10 at the dense
state with confining stress 100 kPa, to the Pegmatite granular material.
Table III: The Mohr-Coulomb Parameters values.
K (kPa)
G (kPa)
C (kPa)
 (KN/m3)
 (°)
 (°)
t
7,86.103
3,63.103
0
1443
29
29
0
K- bulk modulus; G- shear modulus ; C- cohesion ;  -density ;  - friction angle
 - dilation angle; σt – tensile stress.
3. The Strain Softening/Hardening model
This model is based on the FLAC Mohr-Coulomb model with non-associated shear and
associated tension flow rules, as described earlier. The difference, however, lies in the possibility
that the cohesion, friction, dilation and tensile strength may harden or soften after the onset of
plastic yield.
In the Mohr-Coulomb model, those properties are assumed to remain constant. Here, the user can
define the cohesion, friction and dilation as piecewise-linear functions of a hardening parameter
measuring the plastic shear strain. A piecewise-linear softening law for the tensile strength can
also be prescribed in terms of another hardening parameter measuring the plastic tensile strain.
The code measures the total plastic shear and tensile strains by incrementing the hardening
parameters at each time step and causes the model properties to conform to the user-defined
functions.
The yield and potential functions, plastic flow rules and stress corrections are identical to those of
the Mohr-Coulomb model, as discussed in Section 4.
In the softening/hardening model, the user defines the cohesion, friction, dilation and tensile
strength variance as a function of the plastic portion, p, of the total strain. Examples of these
functions are sketched in Figure 3, and may be approximated in FLAC as sets of linear segments.



(a)

(b)
Figure 3: Variation of the friction angle according to the plastic deformation
a) Softening behaviour b) hardening behaviour.
Hardening and softening behaviours for the cohesion, friction and dilation in terms of the shear
parameter are provided by the user in the form of tables. Each table contains pairs of values: one
for the parameter and one for the corresponding property value, (Table 4). It is assumed that the
property varies linearly between two consecutive parameter entries in the table.
Hardening behaviour for the cohesion, friction and dilation can be produced by an increase in
these properties with increasing plastic strain measure.
Table IV: Values of the variation of the friction angle according to the deformation for
Pegmatite of size range 6/10.
Déformations (%)
0
1.5
2.5
6.5
12.5
Friction angle (°)
29
31
33
35
37
STRAIN SOFTENING/HARDENING MODEL
PARAMETERS
The Strain Softening/Hardening model is also associated to a linear elastic law. It is includes
eleven (11) parameters. The particularity of the Strain Softening/Hardening model is the
possibility to control the variation of the friction angle, dilation angle and cohesion according to
the plastic strain. The table 5 represents an example of these values for size range 6/10 at the
dense state with a confining stress 100 kPa, to the Pegmatite granular material.
Table V: Values of the parameters of the Strain Softening/Hardening law

K (KPa)
G (KPa)
(KN/m3)
C (KPa)
 (°)
ftable
 (°)
dtable
t
7,86.103
3,63.103
1443
0
29
Table1
29
Table2
0
K –bulk modulus; G – shear modulus ; C - cohesion;  -density ;  - friction angle
 - dilation angle; σt – tensile stress; ftable et dtable – hardening parameters.
It is necessary to note here that the two remaining parameters of this law ttable and ctable
corresponding to the evolution of cohesion and tensile stress are equal to zero.
COMPARAISON OF THEORETICAL AND
EXPERIMENTAL RESULTS
Throughout the numerical simulation, we followed the evolution of the deviator stress q, the
normal stress p, the axial strain and the volumetric strain. Thus, we can classify the results
obtained according to several parameters which are:
- The law of behaviour used,
- The state of compactness (loose, dense and wet),
- The size range (0/6, 6/10 and 10/14),
- The confining stresses (100, 200, 300 and 400 kPa),
- Axial deformation according to volumetric strain.
Influence of the behavior law used
The analysis of the results obtained seem to indicate that the two models: The Boyce model and
the Strain Softening/Hardening model describes well the principal phenomena observed in the
experimental testing and give acceptable results. However, the results obtained for the Strain
Softening/Hardening model are more appreciable than those obtained by the Boyce model. In the
fact the Strain Softening/Hardening, which is an elastoplastic model, takes into account the
hardening parameters, therefore the state of strain increments are irreversible. On the other hand
the choice of a failure criterion of the Mohr-Coulomb is unsuited for this kind of materials
(Figure 4).
Our investigation showed that much iteration are necessary for obtaining the results of the Strain
Softening/Hardening model, while for the Boyce law few iterations are obtained to plot the
curves of pegmatite behaviour.
500
450
400
350
q (Kpa)
300
250
200
150
Expérimentale
Strain softening/Hardening
Loi de Boyce
Mohr-Coulomb
100
50
0
0
2
4
6
8
10
12
14
16
18
20
Figure 4: Comparison of experimental and numerical curves for size range 0/6 at
the dense state with confining stress 100 kPa.
Compactness influence
Theoretical curves, obtained for the two behaviour models and the experimental curves of the
pegmatite material at the different states (dense, loose and wet) are closer ; and this in spite of
not taking into account of some parameters influencing on the compactness of granular material
such as void ratio and water content.
According to the obtained results, the slope representing the elastic field of material varies
according to compactness; this variation is important at dense state and decreases for the wet state
followed by the loose state. However, on the one hand the slope obtained by the Boyce law is
always over-estimated comparing with the other curves and on the other hand this same slope of
the experimental curve always remains lower than that obtained by the Strain
Softening/Hardening model.
The figures 5, 6 and 7 below show that the compressive strength or stress difference at failure is
always underestimated for the results obtained by the Boyce law, whereas the Strain
Softening/Hardening model tends to smooth the experimental curve.
700
600
q (Kpa)
500
400
300
200
Expérimentale
Strain Softening/Hardening
Loi de Boyce
100
0
0
5
10
15
20
25
є1 (%)
Figure 5: Comparison between strain softening/hardening and Boyce models and
measured response in triaxial tests at dense state with confining stress 300 KPa
(size range 6/10).
700
600
q (Kpa)
500
400
300
200
Expérimentale
Strain Softening/Hardening
Loi de Boyce
100
0
0
5
10
15
20
25
30
є1 (%)
Figure 6: Comparison between strain softening/hardening and Boyce models and
measured response in triaxial tests at wet state with confining stress 300 kPa (size
range 6/10).
300
250
q (Kpa)
200
150
100
Expérimentale
Strain Softening/Hardening
Loi de Boyce
50
0
0
2
4
6
8
10
12
14
16
18
є1 (%)
Figure 7: Comparison between strain softening/hardening and Boyce models and
measured response in triaxial tests at loose state with confining stress 300 kPa
(size range 6/10).
Confining stress influence
Figures 8, 9, 10 and 11 illustrate the influence of the confining stresses. A well approximation
between the numerical and experimental curves is obtained for the low confining stresses. As far
as the confining pressure increase, these curves diverge.
According to the results, compressive strength or stress difference at failure is always
underestimated for the results obtained by the Boyce law. This result is significant for small
confining stresses and the theoretical curve approaches the experimental results as the stress of
confining increases. The results of the Strain Softening/Hardening model always tend to smooth
the experimental curve.
The slope obtained by the Boyce law is always over-estimated comparing with the other curves.
On the other hand this slope, in the case of the experimental curve, remains always lower than
that obtained by the Strain Softening/Hardening model.
300
250
q (Kpa)
200
150
100
50
Expérimentale
Strain Softening/Hardening
Loi de Boyce
0
0
5
10
15
20
25
30
є1 (%)
Figure 8: Comparison between the models and measured response in triaxial
tests at wet state with confining stress 100 kPa (size range 6/10).
700
600
q (Kpa)
500
400
300
200
Expérimentale
Strain Softening/Hardening
Loi de Boyce
100
0
0
5
10
15
20
25
30
є1 (%)
Figure 9: Comparison between the models and measured response in triaxial
tests at wet state with confining stress 200 kPa (size range 6/10).
500
450
400
350
q (Kpa)
300
250
200
150
100
Expérimentale
Strain Softening/Hardening
Loi de Boyce
50
0
0
5
10
15
20
25
30
є1 (%)
Figure 10: Comparison between the models and measured response in triaxial
tests at wet state with confining stress 300 kPa (size range 6/10).
700
600
q (Kpa)
500
400
300
200
Expérimentale
Strain Softening/Hardening
Loi de Boyce
100
0
0
5
10
15
20
25
є1 (%)
Figure 11: Comparison between the models and measured response in triaxial
tests at wet state with confining stress 400 kPa (size range 6/10).
Size range influence
The figures 12, 13 and 14 illustrate the influence of the size range of granular material used in
this study. A well approximation between the numerical and experimental curves is obtained for
small size range of granular material. The difference between these curves increases using larger
size range in which fluctuations appear. This is due to the relative displacements of the grains
rearrangement constituting the granular materials (Shinohara et al., 2000).
This study show that the compressive strength or stress difference at failure is always
underestimated for the results obtained by the Boyce law. The compressive strength value is weak
for the small size ranges and increases when we use increasingly significant size ranges. The
results of the Strain Softening/Hardening model always tend to smooth the experimental curve.
The slope of the elastic field of the Pegmatite material is always over-estimated for the results
obtained by the Boyce law. This slope, in the case of the experimental curve, remains always
lower than that obtained by the Strain Softening/Hardening model.
1000
900
800
700
q (Kpa)
600
500
400
300
200
Expérimentale
Strain Softening/Hardening
Loi de Boyce
100
0
0
2
4
6
8
10
12
14
16
18
20
є1(%)
Figure 12: Comparison between the models and measured response in triaxial
tests at wet state with confining stress 200 kPa (size range 0/6).
400
350
300
q (Kpa)
250
200
150
100
Expérimentale
Strain Softening/Hardening
Loi de Boyce
50
0
0
2
4
6
8
10
є1 (%)
12
14
16
18
20
Figure 13: Comparison between the models and measured response: triaxial tests
on wet state specimen with a confining stress 200 kPa (size range 6/10).
350
300
q (Kpa)
250
200
150
100
Expérimentale
Strain Softening/Hardening
Loi de Boyce
50
0
0
2
4
6
8
10
12
14
16
18
20
є1 (%)
Figure 14: Comparison between the models and measured response: triaxial tests
on wet state specimen with a confining stress 200 kPa (size range 10/14).
Axial strain and volumetric strain
The numerical and experimental results illustrated by the curves of volume change present a
contracting behaviour followed by a phase of dilatancy this is for low confining stresses and the
small size ranges (figure 15). On the other hand, in the presence of great confining stresses and
by using larger size ranges 10/14, we note that dilatancy tends to decrease as the size of the grains
is decrease significantly (figure16).
0
1
2
3
4
5
6
7
8
9
0
-0.5
єv (%)
-1
-1.5
-2
Expérimentale
Strain Softening/Hardening
Loi de Boyce
-2.5
є1 (%)
10
Figure 15: Comparison of volumetric strain data for size range 0/6 at the dense
state with confining stress 100 kPa.
0
2
4
6
8
10
12
14
16
18
20
0
Expérimentale
Strain Softenng/hardening
Loi de Boyce
-1
єv (%)
-2
-3
-4
-5
-6
є1 (%)
Figure 16: Comparison of volumetric strain data for size range 10/14 at the
dense state with confining stress 400 kPa.
CONCLUSION
The knowledge of the behaviour of the geotechnical materials constitutes an essential aspect to
specify the dimensions of the geotechnical structures. Numerical modelling as well as the
numerical methods gives us this possibility to approach and to know the behaviour of these
materials by comparing the numerical results with the experimental ones. In order to test the
validity of the models, the triaxial compression tests under a monotonous loading were carried
out on pegmatite granular material. The results of the numerical simulation carried out by the
exploitation of two behaviour models, the Strain Softening/Hardening and Boyce models; enable
us to highlight the following conclusions
From the obtained results, it's seem to indicate that the two models: The Boyce model and the
Strain Softening/Hardening model describes well the principal phenomena observed in the
experimental testing for granular material and give acceptable results. However, the results
obtained for the Strain Softening/Hardening model are more appreciable than those obtained by
the Boyce model. In the fact the Strain Softening/Hardening, which is an elastoplastic model,
takes into account the hardening parameters, therefore the state of strain increments are
irreversible. But the Boyce law is an elastic non-linear model which the strain increments are
reversible. In the end, our investigation showed that the choice of a failure criterion of the MohrCoulomb is unsuited for the granular material
We can note that much iteration are necessary for obtaining the results of the Strain
Softening/Hardening model, while for the Boyce law few iterations are obtained to plot the
curves of pegmatite behaviour. Therefore a quick convergence is obtained for the last model
whereas more accentuated precision is obtained for the Strain Softening/Hardening model.
The compressive strength or deviator stress at failure is always underestimated for the results
obtained by the Boyce model in the different state of compactness, in the different size range and
in the different confining stress of the pegmatite granular material. However, for the Strain
Softening/Hardening model the results tend to smooth the experimental curve.
Furthermore, the slope of the elastic field of the Pegmatite material is always over-estimated for
the results obtained by the Boyce law. This slope, in the case of the experimental curve, remains
always lower than that obtained by the Strain Softening/Hardening model; always in the different
state of compactness, in the different size range and in the different confining stress of the
pegmatite granular material.
The numerical and experimental results defined by the curves of volume change present a
contracting behaviour followed by a phase of dilatancy this is for low confining stresses and the
small size ranges. On the other hand, in the presence of great confining stresses and by using
larger size ranges, we note that dilatancy tends to decrease as the size of the grains is decrease
significantly.
REFERENCES
1. Boyce, H. (1980) “A not linear model for the elastic behaviour of granular materials
under repeated loading”. International symposium on soils under cyclic and transient
loading, Swansea, 1980, pp. 285 - 291.
2. Itasca Consulting Group, Inc (2005) Flac 5.0 Manual. Minneapolis.
3. Melbouci, B et J. C. Roth (2001) « Etude du comportement de la pegmatite de Kabylie au
triaxial ». Revue Française de Génie civil, Vol. 5, N°4/2001, pp. 495-504.
4. Melbouci, B. (2002a) « Pegmatite de la grande Kabylie : caractéristiques et
comportement en géotechnique routière ». Thèse de doctorat d’état 2001, université
Mouloud Mammeri Tizi-Ouzou.
5. Melbouci, B. (2002b) « Identification du matériau pegmatite de la région de Kabylie ».
Revue Algérie Equipement n° 36- 2002, pp. 8-11.
6. Melbouci, B. (2003) « Etude du comportement mécanique du matériau local pegmatite ».
13 ème congrès régional d’Afrique. Marrakech. 8 – 11 décembre 2003, pp. 663-672.
7. Ramamurthy, T. (2001) “Shear strength response of some geological materials in triaxial
compression”. International Newspaper of Rock Mechanics and Mining Sciences, pp.
683 - 697.
8. Shinohara, K., Mr. Oida & B. Golman (2000) “Effect of particles shape one triaxial
angle of internal friction by compression test”. Powder Technology, Elsevier, pp. 131 136.
NOTATION
E : Young modulus.
K : volumetric modulus.
G : Shear modulus
 Vi : Volumetric deformation
 : Dilatancy parameter
 : Precision parameter
 : Friction angle
 : Poisson’s Coefficient.
p : normal stress.
q : Deviator stress
 q : Deviator deformation
 et  : Adjustment parameters
(  ,  ,  ) : parameters of Boyce law
 : dilation angle
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