Ching S Chang

-Pierre-Yves Hicher
email: pierre-yves.hicher@ec-nantes.fr

Ali Daouadji

Investigating instability in granular materials by means of a microstructural model

Keywords: granular material, instability, micromechanics, stress-strain relationship, sand. MOTS-CLÉS : matériau granulaire, instabilité, micromécanique, relation contraintedéformation, sable

Under certain loading conditions, instability in loose sand can develop at a shear stress level much lower than the critical state failure line. To analyse these types of problems, we adopt the micromechanical approach developed earlier for the modelling of granular material behaviour. The stress-strain relationship for a granular assembly is determined by integrating the behaviour of the inter-particle contacts in all orientations. The constitutive model is applied to simulate undrained triaxial and constant-q tests on loose sand. Numerical simulations are compared to experimental results in order to evaluate the model's ability to predict the different modes of instability of the granular assembly. With the model we have also analysed instability at inter-particle level to observe how it is connected to instability at the assembly level.

RÉSUMÉ. Dans certaines conditions de chargement, des ruptures par instabilité peuvent prendre naissance dans les sables lâches pour des niveaux de contraintes très nettement inférieurs à ceux correspondant à la plasticité parfaite. Pour analyser ce type de problèmes, nous adoptons une approche d'homogénéisation développée précédemment pour la modélisation du comportement des matériaux granulaires. La relation contrainte-déformation macroscopique est obtenue en intégrant le comportement défini au niveau du contact intergranulaire. Le modèle est utilisé pour simuler des essais non drainés et des essais à déviateur

constant sur un sable lâche. Les résultats de la simulation à l'échelle macroscopique sont comparés à des résultats expérimentaux de façon à apprécier la capacité du modèle à prédire la naissance de l'instabilité sur ces chemins particuliers. Une analyse à l'échelle des différents plans de contact inter-granulaire a également été entreprise afin de relier les conditions d'instabilité aux différentes échelles.

Introduction

Instability of granular material is an important topic in geotechnical engineering because it may lead to catastrophic events such as the collapse of earth structures. This phenomenon can be categorised into two major aspects: material instability (also known as intrinsic/constitutive instability) and geometrical instability (see, for example, [START_REF] Goddard | Material instability in complex fluids[END_REF]). The development of localisation in a specimen of finite size can be studied as a boundary value problem for which initial inhomogeneities, as well as boundary conditions, play an important role. However, it has been shown by [START_REF] Rudnicki | Conditions for the localization of deformation in pressure sensitive dilatant materials[END_REF] that the occurrence of localisation can be predicted at a constitutive level. More recently, several authors have demonstrated that other modes of instability can occur within granular materials. Darve and co-authors (2004Darve and co-authors ( , 1998Darve and co-authors ( , 2002) ) have studied the mode of instability which they called diffuse failure, using a macroscopic phenomenological approach based on an incrementally non-linear model. [START_REF] Nova | Controllability of the incremental response of soil specimens subjected to arbitrary loading programs[END_REF] has developed similar concepts and demonstrated that any non-associated elastoplastic model could exhibit domains of instability inside the plastic limit. Experimental evidences support these theoretical approaches. For example, in loose sand under undrained conditions, an unstable condition can be obtained at a low shear stress level and, subsequently, the strength is reduced to almost zero, corresponding to a material state known as static liquefaction.

In this paper, we present a study of material instability, using the micromechanical approach developed by [START_REF] Chang | An elastic-plastic model for granular materials with microstructural consideration[END_REF]. Under this approach, the stress-strain relationship for a granular assembly can be determined by integrating the behaviour of the inter-particle contacts in all orientations, based on a static hypothesis which relates the average stress of the granular assembly to the mean field of particle contact forces. Model simulations are compared to experimental results obtained on loose Hostun sand along two different loading paths: undrained triaxial tests and constant -q tests.

Experimental evidences of instability in sand

Undrained triaxial tests

Undrained triaxial tests on Hostun sand with various initial relative densities, D r , are presented in Figure 1 [START_REF] Hicher | Experimental behavior of granular materials[END_REF]. These results show two distinctive trends corresponding to a contractive or a dilative behaviour. For loose sand (small values of D r ), a maximum strength is reached in the q-1 plane, and the peak in the stressstrain curve is followed by a rapid decrease of the deviatoric stress down to a minimum strength. This peak corresponds to the development of material instability as will be analysed later on. The minimum strength can be almost zero for a relative density close to zero. This represents the phenomenon called static liquefaction. For medium dense sand (medium values of D r ), the tendency of softening still occurs at the beginning of the loading, but it is followed by an increase of the deviatoric stress up to the ultimate strength, corresponding to the critical state at large deformations. For dense sand (high values of D r ), the material is strongly dilatant and no strain softening can be observed. Instead, the deviatoric stress continuously increases up to the ultimate strength achieved at large deformations. These different evolutions can be related to the stress paths followed in a p'-q plane; a continuous decrease of the mean effective stress is observed for strongly contractant materials; a decrease followed by an increase of the mean effective stress up to the critical state is observed for dilatant materials. [START_REF] Hicher | Experimental behavior of granular materials[END_REF] If we examine the position of the peak in the stress plane p'-q, we can see that instability occurs at a stress state below the critical state failure line. Depending on the density of the sand, the position of the peak stress can be more or less distant from the failure line.

Constant-q tests

This type of test consists in shearing the specimen to a prescribe stress ratio along a drained compression triaxial path, and then in decreasing the mean effective stress while keeping constant the deviatoric stress. This stress path can simulate the loading condition of a soil element within a slope when a progressive increase in pore pressure occurs.

Several investigations have demonstrated that instability can occur in loose sand during a constant-q stress path [START_REF] Sasitharan | Collapse behavior of sand[END_REF][START_REF] Imposimato | Non-uniqueness of the incremental response of soil specimens under true-triaxial stress paths[END_REF][START_REF] Gajo | Analysis of certain factors affecting the unstable behaviour of saturated loose sand[END_REF][START_REF] Lade | Instability, shear banding, and failure in granular materials[END_REF][START_REF] Chu | Recent progress in experimental studies on instability of granular soil[END_REF]. [START_REF] Darve | Bifurcations in granular media: macro-and micro-mechanics approaches[END_REF] have presented similar test results on Hostun sand and observed a sudden collapse of the sand specimen for stress states located well below the critical state failure line. Typical results of constant-q tests on loose Hostun sand are presented in Figure 2.

After an isotropic consolidation stage to a desired initial effective mean pressure p'o, a drained triaxial compression test was applied to the sample up to a prescribed value of the deviatoric stress q. Then, while keeping q constant, a decrease of the mean effective stress p' was applied by increasing the pore water pressure and maintaining constant the total stresses. At a given point of the test, the axial strain rate started to increase very rapidly and the deviatoric stress could no longer be kept constant. The test is no longer controllable (as defined by [START_REF] Nova | Controllability of the incremental response of soil specimens subjected to arbitrary loading programs[END_REF]) in the sense that the imposed loading program cannot be maintained. As will be shown later, this point corresponds to a loss of stability, since any small change of one control variable, as defined below, will lead to a catastrophic failure.

Figure 2. Experimental results of constant-q tests on loose Hostun sand

Analysis of undrained compression and constant-q tests in terms of instability condition

The importance of the stress path can be linked to Hill's sufficient condition of stability [START_REF] Hill | A general theory of uniqueness and stability in elasto-plastic solids[END_REF], which states that a material, progressing from one stress state to another, is stable if the second-order work is strictly positive, i.e.:

2 0 ij ij d W d d [1]
Thus, according to Hill's condition, whether a material is stable or not depends not only on the current stress state but also on the direction of the stress increment.

Because q -1 and v -' 3 are conjugate variables with respect to energy, Equation [1] can be re-arranged to give (Darve et al., 2003):

d 2 W = dq d 1 + d ' 3 d v [2]
Thus, as the control variables are dq and d v , the constitutive relation linking the stress increments to the strain increments can be rearrange to give a generalized mixed relation between generalised incremental stresses and generalised incremental strains can be given as: where E i are the pseudo Young moduli and i are the pseudo Poisson's ratios. As no volumetric variation is allowed during undrained tests, Equation [3] is modified as follows:

3 1 3 31 13 33 31 13 3 1 1 2 1 2 2 1 1 2 0 d d E E E E dq [4]
Equation [4] indicates that instability will occur only if dq = 0 (peak shear stress); the previous equation is therefore: Thus, instability can take place only if the determinant of the matrix P of the constitutive relation given by Equation [5] becomes equal to zero.

As no variation of the shear stress is imposed during constant-q tests (dq = 0), Equation [3] can be rewritten as:

3 1 3 31 13 33 31 13 3 1 1 2 1 2 2 1 1 2 0 d d E E E E d v [6]
Taking into account Equation [3], instability will occur in this case only if d v = 0, because d ' 3 >0 is imposed by the loading program. The condition is the same as the one obtained for undrained tests (Equation [4]) and both correspond to:

3 1 13 33 31 1 2 2 1 det E E P [7]
Equation [7] is verified for non trivial solutions if det(P) = 0. As described by [START_REF] Darve | Failure in geomaterials: continuous and discrete analyses[END_REF], the condition of instability for constant-q tests coincides with the condition for undrained tests, so the stress states in the p'-q plane corresponding to the vanishing of the volumetric strain increment during constant-q tests are the same as the stress states corresponding to the peaks of the stress-strain curves for the undrained tests. Figure 3 presents different results obtained on very loose Hostun sand (Dr = 0). One can see that the condition of instability in the p'-q plane is found to be the same for undrained and constant-q triaxial tests, in agreement with the theoretical developments presented above. This condition defines an instability line for a mobilised friction angle equal to 16°, much lower than the friction angle at critical state equal to 30°.

Stress-strain model based on micromechanical approach

In this Section, the microstructural model developed by [START_REF] Chang | An elastic-plastic model for granular materials with microstructural consideration[END_REF] is briefly described. In this model, a granular material is viewed as a collection of particles. The deformation of a representative volume of the material is generated by the mobilisation of particle contacts in all orientations. On each contact plane, an auxiliary local coordinate can be established by means of three orthogonal unit vectors n, s, t. The vector n is outward normal to the contact plane. Vectors s and t are on the contact plane.

Inter-particle behaviour

Elastic stiffness: the contact stiffness of a contact plane n k includes normal stiffness, n k , and shear stiffness, k t . The elastic stiffness tensor is defined by: e e i i j j f k [8] which can be related to the contact normal and shear stiffness:

t t s s k n n k k j i j i t j i n e ij [9]
The value of the stiffness for two elastic spheres can be estimated from Hertz-Mindlin's formulation (1953). For sand grains, a revised form was adopted [START_REF] Chang | Initial moduli of particulate mass with frictional contacts[END_REF], given by: 0 0 2 2

;

n n n n n n t t g g f f k k k k G l G l [10]
where g G is the elastic modulus for the grains, n f is the contact force in normal direction, l is the branch length between two particles, k k n t , 0 0 and n are material constants.

Plastic yield function: the yield function is assumed to be of the Mohr-Coulomb type, defined in a contact-force space (e.g. , ,

n s t f f f ): ( , ) 0 p i n F f T f [11]
where ( ) P is a hardening/softening parameter. The shear force T and the rate of plastic sliding p are defined as:

2 2 s t T f f and 2 2 p p p s t [12]
The hardening function is defined by a hyperbolic curve in p plane, which involves two material constants: p and 0

p k . 0 0 tan tan p p p p n p p k f k [13]
Plastic flow rule: the plastic sliding often occurs along the tangential direction of the contact plane with an upward or downward movement, thus, shear-induced dilation/contraction takes place. The dilatancy effect can be described by:

0 tan p n p n d T d f [14]
where the material constant 0 can be considered in most cases equal to the interparticle friction angle . On the yield surface, under a loading condition, the shear plastic flow is determined by a normality rule applied to the yield function. However, the plastic flow in the direction normal to the contact plane is governed by the stress-dilatancy equation in Equation [14]. Thus the flow rule is non-associated.

Elasto-plastic relationship: with the elements discussed above, the incremental force-displacement relationship of the inter-particle contact can be obtained. Including both elastic and plastic behaviour, this relationship is given by:

p i i j j f k [15]
Detailed expression of the elasto-plastic stiffness tensor can be derived in a straightforward manner from yield function and flow rule, and, therefore, is not given here.

Influence of void ratio on p

The resistance against sliding in a contact plane is dependent on the degree of interlocking by neighbouring particles. The resistance can be related to the packing void ratio e by: tan tan m c p e e [16] where m is a material constant [START_REF] Biarez | Elementary Mechanics of Soil Behaviour[END_REF]) and e c corresponds to the critical void ratio for a given state of stress. For dense packing, e c /e is greater than 1 and therefore the apparent inter-particle friction angle p is greater than the internal friction angle

. When the packing structure dilates, the degree of interlocking and the apparent friction angle is reduced, which results in a strain-softening phenomenon. For loose packing, the apparent friction angle p is smaller than the internal friction angle and increases during the material contraction.

The critical void ratio e c is a function of the mean stress applied to the overall assembly and can be written as follows [START_REF] Schofield | Critical State Soil Mechanics[END_REF]

Micro-macro relationship

The stress-strain relationship for an assembly can be determined by integrating the behaviour of inter-particle contacts in all orientations. In the integration process, a micro-macro relationship is required. Following the Love-Weber approach, the stress increment can be obtained by the contact forces and branch vectors for all contacts [START_REF] Christofferson | A micromechanical description on granular material behavior[END_REF]Rothenburg and Salvadurai, 1981), as follows:

1 1 N ij j i f l V [18]
The mean force on the contact plane of each orientation is:

1 j i j i k k f A l V [19]
where the branch vector k l is defined as the vector joining the centres of two particles, and the fabric tensor is defined as

1 N ik i k A l l .
Using the principle of energy balance, which states that the work done in a representative volume element is equal to the work done on all inter-cluster planes within the element,

j N j i j ij f V u 1 , 1 [20]
From Equations [19] and [20], we obtain the relation between the global strain and the inter-particle displacement:

1 , 1 N j i ik j k u A l [21]

Stress-strain relationship

From Equations [15], [18], and [21], the following relationship between stress increment and strain increment can be obtained: , i j ijmp mp u C

; where

1 1 1 1 N ep ijmp ik mn jp k n C A A V k l l [22]
When the contact number N is sufficiently large in an isotropic packing, the summation of the compliance tensor in Equation [22] and the summation of the fabric tensor in Equation [19] can be written in integral form, given by:

/2 2 1 1 1 0 0 , , , sin 2 ep ijmp ik mn jp k n NV C A A k l l d d [23]
and:

/ 2 2 0 0 , , sin 2

ik i k N A l l d d [24]
The integration of Equations [23] and [24] in a spherical coordinate can be carried out numerically by using Gauss integration points over the surface of the sphere.

Model performances

The model needs a set of input parameters, such as mean particle size, particle stiffness, inter-particle friction, initial porosity, and parameters defining the critical state of the sand. The mean size of the particle for fine Hostun sand is d = 0.4 mm. The inter-particle elastic constant 0 n k is assumed to be equal to 61000 N/mm. The total number of contacts per unit volume changes during the deformation. Using the experimental data by [START_REF] Oda | Co-ordination number and its relation to shear strength of granular material[END_REF] for three mixtures of spheres, the total number of contact per unit volume can be approximately related to the void ratio by the following expression:

0 0 0 1 1 e e N N V V e e [25]
where e 0 is the initial void ratio of the granular assembly. This equation is used to account for the evolution of the contact number per unit volume. The initial contact number per unit volume can be obtained by matching the predicted and experimentally measured elastic modulus for specimens with different void ratios [START_REF] Hicher | An anisotropic non-linear elastic model for particulate materials[END_REF].

The value of k t0 / k n0 is commonly about 0.4, corresponding to a Poisson's ratio for Hostun Sand = 0.2 and the exponent n = 0.5. From drained triaxial test results, we were able to derive the values of the two parameters corresponding to the position of the critical state in the e-p' plane: = 0.2 and p ref = 0.01 MPa for e ref = e max = 1. In Equation [16], the value of m = 1 was determined from the test results. The values of k p0 are assumed to be same as the elastic stiffness k p0 = k n . The set of parameters for fine Hostun sand is presented in Table 1. The model performances will be demonstrated in the following sections by comparing the predicted and measured stress-strain behaviour.

Table 1. Model parameters for fine Hostun sand

Figure 4. Model predictions for undrained triaxial tests on Hostun sand with various relative densities

Figure 4 presents numerical results for undrained triaxial tests on Hostun sand with various initial void ratios, corresponding to relative densities between 0.05 and 1. Both predicted stress-strain curves and stress paths are in agreement with the experimental curves in Figure 1. Results indicate that the model is capable of capturing the general trend observed for contractive and dilative sands. In order to examine the inception of instability, the predicted shear stress and second-order work are plotted against the shear strain in Figure 5 for a test on loose Hostun sand with an initial confining stress 3c = 300 kPa. For undrained conditions ( 0 v d ), the second-order work is reduced to 2 d W dq d where is the deviatoric strain. As discussed in Section 2.3, since the deviatoric strain increases continuously, d is always positive and the second-order work can become nonpositive if and only if 0 dq (i.e., decrease in q). Figure 5 shows that instability begins at the shear stress peak. The second-order work is positive before the peak stress. After the peak, the second-order work remains negative and approaches zero at critical state. The numerical simulations can be carried out after the peak because the control variable is the vertical strain 1 and not the deviatoric stress q.

It should be noted that the undrained condition imposed in the tests is taken into account in the modelling by the condition of no volume change, i.e., isochoric condition. The instability is therefore not triggered by the pore pressure built up, but rather tests on a dry loose sand specimen have also led to the same instability mode [START_REF] Lanier | Essais à volume constant realisés sur presse tridimensionnelle[END_REF].

Figure 5. Predicted second-order work and stress-strain curves for undrained triaxial test on loose Hostun sand

Constant-q tests

Similarly, we also used the parameters in Table 1 to predict the results of constant-q tests on loose Hostun sand. The predicted and measured results for the confining stress 3c = 300 kPa are presented in Figure 6. The initial part of the v p curve shows that, as the mean stress p decreases, the volume increases. This trend continues until a certain point where the volume starts to decrease. For constant-q tests ( 0 dq

Data p-q num

undrained triaxial tests exp undrained triaxial tests num constant-q tests exp constant-q tests num instability line critical state line q (kPa) p' (kPa)

Figure 7. Comparison of predicted and measured instability condition for loose Hostun sand determined from undrained triaxial and constant-q tests

Instability conditions obtained by numerical simulations of undrained compression and constant-q tests are plotted in the p'-q plane together with the experimental results presented in Section 2.3 (Figure 7). One can see that the model is capable of predicting very accurately the condition of instability associated with these two types of tests. The position of the instability line determined by the model simulations is in very good agreement with the position obtained experimentally.

Local plane behaviour

In order to gain insight of the material behaviour at the inter-particle level, we selected several contact planes oriented in six different directions indicated by the branch vectors, as shown in Figure 8. The six directions are defined by the angle between the vector n perpendicular to the contact plane and the vertical axis. A larger value of indicates a more inclined contact plane.

Figure 8. Orientations of branch vectors for the inter-particle contact planes selected in the analysis

In a loaded assembly, the forces are different on each contact plane. In order to obtain a more direct comparison between the local and the overall stress-strain behaviour, we retained the following variables, local strain and local stress, instead of inter-particle force and inter-particle displacement. For this purpose, we define a local normal stress / 3 n f Nl V and a local shear stress / 3

r f Nl V ,
where l is the branch length and N/V is the total number of contacts per unit volume.

The corresponding local normal strain is defined as / n l and the local shear strain is defined as / r l .

The local stress paths in the six contact planes are plotted in Figure 9. Since the local stresses are related to the stresses applied to the overall assembly by the static hypothesis, the local stresses on each contact plane can be determined from the applied stresses, even though they are different on each plane. At any given moment, the maximum local shear stress is located in the 45° inclined plane. The less-inclined planes (oriented more horizontally) carry normal forces larger than the moreinclined planes (oriented more vertically).

Figure 9. Comparison of local stress paths for various inter-particle contact planes

In Figure 9, the circular points on each curve correspond to the moment when the peak stress point occurs in Figure 5. It indicates that all contact planes experience a reduction of the contact shear force after the occurrence of the peak stress. Since the local stress paths for all contact planes display a softening behaviour, it is of interest to know whether all contact planes experience instability at once.

Figure 10. Comparison of second-order work in various inter-particle contact planes

In order to answer this question, we computed the second-order work for each individual plane. Note that in an undrained test, the overall volume change is zero, but the local normal strain does not need to remain constant ( 0 second-order work for the th contact plane, expressed in terms of local stress and local strain, is given by:

2 d W d d d d [26]
The second-order work for the six contact planes is plotted in Figure 10. For the two less-inclined contact planes at 17.9° and 28.3°, the second-order work is always positive, therefore, these two planes remain stable during the entire undrained triaxial test. On the contrary, for the four more-inclined contact planes oriented between 45° and 72°, the second-order work displays negative values but does not become negative simultaneously in each plane. Instability occurs first in the two contact planes with inclination angles 45° and 54.7°. After a few additional load steps, instability occurs subsequently in the contact planes with inclination angles 70.4° and 72°. Thus, even though the local stress paths in all the planes show a softening behaviour, only a part of the contact planes experience instability, but at different times. The assembly instability occurs only when few contact planes (inclined at 45°-54.7°) start to become unstable. Even though the majority of the inter-particle planes are still stable at this time, the unstable condition of the few planes is severe enough for the overall assembly to experience instability. This can be seen from the negative sum of the second-order work for all the planes at this point.

To examine more deeply the behaviour of each individual contact plane, we plotted the local normal stress-strain curves and local shear stress-strain curves for four contact planes. The less-inclined planes (17.9°, 28.3°) are plotted in Figure 11 and the more-inclined planes (54.7°, 70.4°) are plotted in Figure 12. Initially, the specimen is isotropically consolidated to 0.3 MPa and then sheared in undrained condition. In the local normal stress-strain curves in Figures 11 and12, the part corresponding to the isotropic consolidation is plotted as a dotted line up to 0.3 MPa. From that point on, the curve represents the local response during undrained shear loading. The isotropic consolidation is not displayed in the local shear stress-strain curves. The entire curve represents the local response during the undrained shear loading stage. The four circular points marked on each curve correspond to the four loading stages in Figure 5. It can be observed, in the less-inclined planes, that the local normal strain expands as the normal stress is released. Thus, the second-order work due to the normal stress is positive ( 0 d d

). It can also be observed that the local shear stress increases monotonically with the local shear strain until the peak is reached. After the peak, the local shear stress decreases with the local shear strain, producing a positive second-order work ( 0 d d

). Thus, the overall secondorder work is always positive.

On the plane with an inclination angle of 70.4° (Figure 12), the normal behaviour is similar to the one corresponding to the less-inclined planes, i.e., the local normal strain expands as the normal stress is released ( 0 d d

). However, the behaviour is reversed on the plane with an inclination angle of 54.7°, since, when the local normal stress is released, the local normal strain contracts rather than expands, producing large negative second-order work 0 d d .

The local shear behaviour for the more-inclined planes is also reversed. After the peak stress is reached, the shear stress starts to decrease as the shear strain continues to increase, showing a softening pattern and producing a large negative second-order work 0 d d . The overall second-order works for the more-inclined planes are negative. The behaviour is dominated by an instable condition.

Summary and Conclusion

According to Hill's condition, whether a material is stable or not depends not only on the current stress state but also on the direction of the stress increment. Thus, under specific loading conditions, loose sand can remain stable until the critical state failure line is reached. However, under other loading conditions, loose sand can succumb to instability at a shear stress level much lower than the critical state failure line. Therefore, the usual design concept in geotechnical engineering is likely to overestimate the soil strength, which can cause catastrophic collapse of earth slopes.

A micromechanical approach has been adopted for the analysis of this type of instability problem. The model considers the material as an assembly of particles. The stress-strain relationship for the assembly is determined by integrating the behaviour of the inter-particle contacts in all orientations. The inter-particle contact is assumed to have an elasto-plastic behaviour.

The constitutive model has been used to simulate undrained triaxial tests and constant-q tests. The comparison between experimental and predicted results has shown the capability of the model for obtaining the modes of instability at the assembly level. In particular, the position of the instability line, the same for these two types of tests in the p'-q plane, has been predicted with very good accuracy by the model simulations.

Figure 1 .

 1 Figure 1. Experimental results for undrained triaxial tests on Hostun sand with various densities: a) stress-strain curves, and b) stress paths[START_REF] Hicher | Experimental behavior of granular materials[END_REF]

Figure 3 .

 3 Figure 3. Instability line for loose Hostun sand determined from undrained triaxial and constant-q tests

Figure 6 .

 6 Figure 6. Comparison of predicted and measured results for constant-q test on Hostun sand

Figure 11 .

 11 Figure11. Local stress-strain curves for inter-particle contact planes with inclined angles 17.9° and 28.3°

Figure 12 .

 12 Figure 12. Local stress-strain curves for inter-particle contact planes with inclined angles 54.7° and 70.4°

  where and are two material constants, p' is the mean stress of the packing, and( ref e , ref p) is a reference point on the critical state line.

	c e	log	p	or	c e e ref	log	ref p p	[17]
							):


Pierre-Yves Hicherii
email: hattab@lpmm.univ-metz.fr

DILATING BEHAVIOUR OF OVER CONSOLIDATED CLAY MAHDIA HATIABi) and

Keywords: dilatancy, overconsolidation ratio, plastic strain mechanisms, triaxial compression tests

The object of this study was to understand and characterize dilatancy in clay in relation to the overcoQsolidation ratio. This phenomenon was investigated along a large range of loading paths, with strong emphasis placed on constant mean stress paths given that it was possible to measure the volumetric strain created by the sole deviatoric stress. Triaxial tests performed on remolded clay (Kaolinite P300) allowed us to determine three different types of behaviour for specimen submitted to a deviatoric stress: no volume change, dilatancy and contraction. In the (p', q) plane these domains are limited by the maximum strength envelope(, which coincides with the perfect plasticity line M for OCRs < 2. The experimental results were then compared to plastic flow theories used in the Cam Clay model and in Rowe's dilatancy theory. Finally, by measuring the plastic strain increment vectors along different stress paths in the (p', q) plane, we demonstrated that the uniqueness hypothesis of the plastic potential is not valid . The results could be explained by using two plastic strain mechanisms: a deviatoric one and an isotropic one.

INTRODUCTION

Dilatancy was first put forward by [START_REF] Reynolds | On the clilatancy of media composed of rigid particles in comact, with experimental illustration[END_REF]. He demonstrated that the volume change of a regular packing of rigid spheres under shear depended on the initial arrangement of the spheres and that contraction would occur in a loose packing whereas dilatancy would occur in a dense packing. Consequently the shear strength of the sphere assembly depended not only on the friction between the spheres, as one used to assume, but also on the volume change. Therefore, the Coulomb-Rankine relation had to be extended in order to take into account the dilatancy effects.

In the steps of this pioneer work, many other experimental studies on remolded soils, particularly sands, have improved our understanding of this phenomenon. Drained triaxial test results on sands showed clearly that the initial density is an important parameter, which has a great influence on the dilatancy rate. Dilatancy is more pronounced when the sand is initially in a dense state [START_REF] Taylor | Fundamemals of Soil Mechanics[END_REF]. In undrained triaxial tests, the dilatant behaviour of dense sand leads to a pore pressure decrease, while a pore pressure increase is obtained in loose samples [START_REF] Bishop | The Measurement of Soil Proprieties in the Triaxial Test[END_REF]). In the case of clay materials, the results of drained (and undrained) tests showed that the volume change (or pore pressure evolution) during shearing depends mainly on the clay loading history expressed by the overconsolidation ratio. These experimental studies showed that clay behaviour is qualitatively close to sand behaviour. Therefore, the results obtained in sandy materials can highlight different phenomena which can also be found in clayed materials [START_REF] Biarez | ElementaryMechanicsofSoils Behaviour, Sawrated Remoulded Soils[END_REF].

Analyzing drained triaxial test results at constant lateral stress on sands, [START_REF] Kirkpatrick | Discussion on soil properties and their measurement[END_REF] demonstrated that the change in behaviour from contractant at low stress ratios to dilatant at high stress ratios was obtained for a given value of (a{/ af)c, independent of the initial consolidation pressure and of the initial density. [START_REF] Rowe | The stress dilatancy relation for static equilibrium of an assembly of particles in contact[END_REF] confirmed this result and introduced a relationship between the stress ratio and rthe dilatancy rate. This particular stress ratio was subsequently generalized to introduce the notions of phase transformation [START_REF] Ishihara | Effects of stress history on cyclic behavior of sands[END_REF] and characteristic state [START_REF] Luong | Etat caracteristique du sol[END_REF], which defined a particular friction angle called (/Jc, intrinsic to the material. (/Jc or the corresponding slope 1'/c in the (p', q) plane separates the stress space in two regions: a region of contractant behaviour for stress ratios lower than (/Jc and a region of dilatant behaviour for stress ratio higher than (/Jc. Stress-Strain behaviour, on the other hand, has been also investigated on different strain paths [START_REF] Asaka | A simple stress-strain relation based on stress-path behavior in strainpath controlled triaxialtests[END_REF].

Experimental results on clays, on the other hand, are less numerous. Drained and undrained triaxial tests published by [START_REF] Henkel | The effect of overconsolidation on the behaviour of clays during shear[END_REF] showed that the volume change for clayed materials depended mainly on the overconsolidation ratio (OCR). Normally consolidated or slightly overconsolidated samples showed a con-tractant behaviour, while highly overconsolidated samples showed a dilatant behaviour. Similar results were obtained by [START_REF] Zervoyannis | Etude synthetique des proprietes mecaniques des argiles saturees et des sables sur chemin cedometrique et triaxial de revolution[END_REF] who showed that dilatancy phenomena appeared generally for 2<0CR<2.5 depending on the clay mineralogy. Shimizu (1982) performed triaxial tests at constant mean effective stress on Fujinanori clay with different overconsolidation ratios. He showed that the (p', q) plane could be divided into several regions of contractant and dilatant behaviours according to the stress level q I p' and the OCR. These first experimental clay studies showed that clay behaviour is similar to sand behaviour. Contractancy or dilatancy in sand are directly connected to the relative density, while in clays these volumetric changes can be more precisely related to the overconsolidation ratio .

TESTING PROCEDURE

In soils, as in other porous media, the application of a stress tensor a' generates volume changes. q_' can be divided into two parts: the deviatoric part q • !2. and the

isotropic part p' • L• such as: - (1)
In the special case of an axisymmetric compression test, the isotropic stress is defined by p' = a(+ 2a{/3; and the deviatoric part of the stress tensor by q = a(-a{. a( is the major principal effective stress and a{ is the minor principal effective stress. The relation (1) shows that the stress tensor is a combination of the deviatoric tensor and the isotropic tensor. An increase of the isotropic stress always generates a contractive volumetric strain de~>> 0; while an increase of the deviatoric part of the stress tensor produces a volumetric strain which can be positive or negative, i.e. de~> > 0 for contractant material or detd> < 0 for dilatant material. As it will be demonstrated later in the case of clay, the sign of de~dl depends highly on the overconsolidation ratio (OCR).

Dilatancy will be defined here as the volume increase characterized by dddl < 0, and caused by the increase of the deviatoric part of the stress tensor q. In a classical triaxial test (a{= constant) there is an increase of the mean pressure p' at the same time as of the deviatoric stress q. Thus we obtain a volumetric total strain increment ddT> which is the result of the combination between the two effects of the stress tensor, i.e. de~> and de~d> : [START_REF] Balasubramaniam | Formulation of stress-strain behaviour inside the state boundary surface[END_REF] In these conditions, characterizing dilatancy can be difficult, because we measure the total volumetric strain de~T> experimentally, but we have no access to each part of it.

The approach proposed here is based on the analysis of pure deviatoric stress paths in the triaxial cell. It consists of studying clay samples with several OCR, as shown in Fig. I. The clay samples are first isotropically consolidated up to a given isotropic stress and then isotropically unloaded in order to obtain a given value of the overcon- (3)

The contribution of the isotropic part of the stress tensor on the volumetric strain mechanism was also studied along different stress paths for which the mean stress was not kept constant, as shown in Fig. 2.

TESTING PROGRAMM

Focusing on the stress-strain relationship and volumetric change measurements, the study required a reliable experimental system (especially for the volume change measurement), with an automatic recording system and test control.

As shown in Fig. 3, the system was made of various apparati connected by a closed loop. The test controls were operated by a PC via a driving program. Two acquisition (AC/DC) cards and! an IEEE interface were integrated into the PC. Figure 3 also shows that the tests were conducted by using the hydraulic triaxial cell (Fig. 4). The radial stress noted a 3 was applied by controlling the water pressure inside the cell, and the axial stress a, was applied by controlling the pressure pr in the lower pressure chamber (Bishop and W esley, 197 5). The value of a, is dependent on both a 3 and pr following the (GDS l and GDS2) via an IEEE interface. This operation is represented in Fig. 3 by a continuous line. The imposed stress values can be acquired at each moment for data processing. At the same time, we can calculate the deformations induced by the stress change, as represented in Fig. 3 by a broken line. As can be seen, the volume change was acquired by a third controller (GDS3) linked to the sample by a capillary tube. The axial displacement was measured by a LVDT transducer, the data being sent to the PC via the AC/DC interface. This experimental environment was conducted by a software program called Piltriax (Hattab and H icher, 1995). As an example, the classical path with a 3 constant required only the control of pr. a 3 was maintained constant and a 1 was increased following the relation (5):

a da~= -dpr

At

(5)

The "Piltriax" program insured the triaxial test control. The structure of its organization chart is represented in Fig. 5. The software was constructed so that it could be easy to add any other procedure to the five already implemented.

Another procedure example is "p' =constant" for the stress. The working principle consists of two steps, as explained in Fig. 6(a). The first step was to apply a pressure increment called pr. This step increased pr from pr{to prl . The second step was to bringp{back to p 0 . The pressure could be controlled with an accuracy of 1 kP a. This step required for relation [START_REF] Frost | Effect of end platens on microst ructure evolution in dilatant specimens[END_REF] to be verified all along loading:

a{+ 2a)= 3p 0 (6)
Carrying out the "p' =constant" procedure allowed in obtaining a very good control o f the stress path as shown by the example in Fig. 6(b) . It corresponds to a constant mean stress path with p' = p 0 = 1000 kP a. We can note that, using this program, an accuracy of dp' = ± 1 kPa was reached, which corresponds to the best use of the pressure controller GDS possibilities. This experimental technique as a whole, considering the high degree of automation and of precision, allowed to investigate the mechanical behaviour of clay, following different stress controlled paths, and to obtain very reliable experimental results.

SP ECIMEN PREPARATION

All the tests were performed on a remolded clay called "Kaolinite P 300" . Atterberg's limits were L L = 40% and P L= 20% .

The specimen were prepared from a clay slurry mixed with a water content of twice the liquid limit and consolidated in a double drainage consolidometer at a vertical pressure of 120 kPa. After removing the sample, the specimens were trimmed and installed in the triaxial cell. The slenderness ratio (length to diameter) was chosen equal to 1 (H=D= 35 mm). An anti-frictional system was placed in the two extremities of the sample. T he anti-frictional device was made of two circula r smooth plates lubricated by means of a grease and latex layers «sandwich)).

Th is kind of procedure was selected after considering the existing studies about the influence of the slenderness ratio and anti-frictional system o n the strain homogeneity throughout the specimen [START_REF] Hicher | Comportement mecaniquedesargilessawrees sur divers chemins de sollicitations monotones et cycliques application a une modelisation elastoplastique et viscoplastique[END_REF][START_REF] Lade | Effects of localization in triaxial tests on clays[END_REF], and recently J .D. purpose of the anti-friction system was to maintain the homogeneity of the sample up to large strains, by significantly reducing the friction at the sample ends. This allowed to reduce the slenderness ratio down to 1 and to prevent the rapid development of shear bands which are usually present in large strains, especially in overconsolidated clay.

BEHAVIOUR ALONG ISOTROPIC STRESS PATHS

In order to create initially a normally consolidated or overconsolidated state, the clay specimen were subjected to isotropic loading and unloading as represented in Fig. I.

The specimen were isotropically loaded up to the maximum consolidation pressure p 0 = 1000 kPa, and then isotropically unloaded to the pressure p{= p 0 /0CR, by degrees of 100 kPa/h, with volume changes recorded every 5 minutes, a similar operation mode for isotropic paths was previously performed by Shimizu (1982). Figure 7 shows an example of the isotropic consolidation of the clay with a final OCR of 8. This kind of consolidation process appears particulary interesting, because it ensures the detection of accidental leaks, particularly obvious in unloading, linked to an accidental porosity of the membrane. Since the volume changes were obtained by measuring the amount of pore water which circulated from and to the specimen, detecting accidental leakage through the membrane is highly significant. This approach imposed a very hig!h level of precision in volume change measurements (Hattab and Richer, 1995). A series of 16 tests were performed on normally and overconsolidated clay samples with an OCR between 1 and 50 (Hattab, 1995). From the isotropic consolidation stages, values of the compressio•n index Cc= 0.1 9 and of the swelling index Cs = 0.066 were measured. They both are usual values for a clay with low plasticity index (Fig. 8).

BEHAVIOUR ALONG PURELY DEVIATORIC STRESS PATHS

Figure 9 presents triaxial test results at constant mean stress in the planes (e~, q) and (e,, ev). For all tests, the maximum consolidation pressure was p 0 = 1000 kPa. The results show that the overconsolidation ratio influences not only the volumetric strain evolution with a dilatant behaviour for high OCR and a contractant one for low OCR, but also the strength evolution.

Evolution of the Maximum Strength Qmax

In the (q, p') plane, normally consolidated clay samples are characterized by a maximum strength envelope corresponding to a straight line of slope M= 1.05. This line corresponds to a perfect plastic state also called critical state as defined in the Cam Clay model [START_REF] Roscoe | On the yielding of soils[END_REF][START_REF] Schofield | Critical State Soil Mechanics[END_REF].

Overconsolidated clay samples with OCR> 2 are characterized by a maximum strength envelope corresponding to a curve called ( which is situated above M (Fig. 10). The respective position of M and ( indicates that the friction angle at peak is an increasing function of OCR (Fig. 10). This maximum strength envelop (can be consider as a precise representation of Hvorslev surface in the (p' -q) plane. It is not a straight line as proposed by [START_REF] Parry | Triaxial compression and extension tests on remoulded saturated clay[END_REF]. The ( and M curves remain valid for other axisymmetric stress paths, as the conventional triaxial tests (cr3 = constant). In Fig. 11 drained test results by Wahyudi (1991) on normally consolidated samples giving the same value of M are presented. In the same figure, drained and undrained triaxial tests with OCR = 5 and 8 end up on the (envelope.

Figure 12 presents the results of more complex paths constituted by two parts: the first part corresponds to a constant mean stress loading and the second part to a 11 = q / p' =constant stress path. One can see that, for OCR = 8 and 11 = 1.26, a maximum strength Qmax located on the (curve is obtained. For 11 = 0.84 lower than M, loading can be continued without any occurrence of the sample failure.

Most of the tests were performed at the same maximum isotropic consolidation stress= 1000 kPa. Few others Po= 1500 kPa Po= 10 MPa In order to investigate the influence of Po on the results, all the tests were plotted in an adapted plane (qmaxl Po, p' I p 0 ), (Fig. 13). One can see that for OCR= 50 and 30, the stress paths end on the same ( line. For OCR = 1.5, the maximum strength is located on the same straight line M. For higher consolidation pressures, the line M is probably no longer straight, there is a decrease in the slope as shown by [START_REF] Biarez | ElementaryMechanicsofSoils Behaviour, Sawrated Remoulded Soils[END_REF]. Thus, the two curves (and M are intrinsic characteristics of the clay. It represents, in the plane (p' , q), the boundaries which delimit the size of the domain accessible by different loading paths. Clay behaviour inside this domain will be analyzed below.

Volumetric Strain Evolution

The volumetric strain evolution was investigated mainly in the (e, p') plane. The evolution of the void ratio e, for normally consolidated clay samples during constant p' and constant a { loading is presented in Fig. 14(b). Isotropic loading (Cc) and unloading (Cs) are also plotted on this plane. We can note here that the points representing the end of loading are located on a straight line, parallel to the isotropic consolidation line (Cc) . Since the volumetric strain reached a constant value for large deformations, this line corresponds to a perfect plastic state (or critical state). This remains valid for all OCRs apart from two cases: OCR = 50 and 20 as it appears in Fig. 14(a). In these two last cases, we can observe in Fig. 9 that the volumetric strain at the end of the test was not stabilized. We can therefore assume that the perfect plastic state was not reached.

The perfect plastic line cuts the isotropic unloading line for OCR=2.5. In these conditions for OCRs<2.5, the clay sample exhibits a decrease of the void ratio during a constant p' loading, corresponding to a contractant behaviour, while for OCRs > 2.5 the samples present an increase of the void ratio, corresponding to a dilatant behaviour.

In Fig. 14(c), the results obtained for two OCRs are presented: 8 and 5, for p' = constant and a{= constant loading paths as well as for undrained tests. We can see that the same perfect plastic line is obtained for all the tests . The p ' constant tests and undrained tests are represented, respectively, by vertical and horizontal straight lines, whereas a {= constant tests follow, at first, the reloading line (C, ) and, after a certain point, leave this line to exhibit a dilatant behaviour up to the perfect plastic line. These results will be analyzed later in terms of plastic mechanisms.

Several more complex loading paths were applied on samples overconsolidated at OCR equal to 8 (Fig. 12). The behaviour can be observed in Fig. 15, where, (lp), (1), ( 2), ( 3) and (4) correspond to the following stress paths: (lp) is a constant mean stress path as already discussed before.

(1) is a stress path combining different loading stages: at first a constant mean stress path beyond the perfect plastic line M, and then a constant 17 = q I p' = 1.26 path up to failure. In Fig. 12 one can see that this path has reached the (curve. In the (p' , e) plane (Fig. 15), the path followed, during the first part of the test, a vertical line towards the increasing values of e and then, during the second part of the loading, reached the perfect plastic line with no volume change (e=constant). ( 2) is the same path as in the case (1), i.e. the stress path combines a first part at constant mean stress and a second part at constant 17 = 0.84 in which the maximum strength envelop (or M is never reached. In the (p', e) plane, the corresponding path reached a line parallel to the isotropic consolidation line and the perfect plastic state line, (3) is a constant mean stress up to 17 = 0.84 followed by an unloading test with l]=constant=0.84. In the (e, p') plane, the path obtained during the unloading part reached the Cs line and then followed it until the end of the test .

(4) is an isotropic loading path. In the (e, p') plane it corresponds to a classical consolidation line following at first a line of slope Cs and then a line of slope Cc when a normally consolidated state was obtained. All these results allow to characterize the perfect plastic state, defined by the perfect plastic line in the (e, p') plane. This straight line is parallel to the isotropic line Cc. Besides, one can define a domain of dilatancy for high OCRs and a contraction domain for low OCRs. OCR= 2.5 (where no volume change was observed during a p' constant path) is, for the studied clay, the point where a change takes place between the two behaviour areas . In the dilatant domain, the relation e-p' is governed, at small stress ratios, by the line Cs which is followed along stress paths corresponding to either an increase or a decrease of the mean stress p'. For higher stress ratios, the dilatant behaviour will superimpose its influence on this general trend.

CHARACTERIZING DILATANCY ALONG CONSTANT MEAN STRESS LOADING

In the (q, ev) plane, we can study the volumetric strain evolution as a function of the deviatoric stress for different ,OCRs (Fig. 16). For OCR values between 1 and 2.5 we observe a contractant behaviour, while a dilatant behaviour characterizes all the specimen with OCR> 2.5. The maximum slope defined by (ev/q)rnax is an increasing function of OCR. Dilatant behaviour for OCR greater than 2.5 or contractant behaviour for OCR smaller than 2.5 starts after an initial no-volume phase, except for q (kPa) 1200~----------------------------. We called qc the deviator which started to induce a volumetric strain, positive or negative, after the novolume phase. If we report the corresponding points in the plane (p', q), we obtain several domains (Fig. M* is a straight line for OCR::::: 3. Its slope is equal to 0.45 and it is located underneath the critical state line M (equal to 1.05) (Fig. 17).

For OCR between 2.5 and 3, there is a transition behaviour and the locus of the stress states characterizing the beginning of dilatancy is a curve called m* 2 which connects the M* line and the ( curve.

For OCR lower than 2.5, we obtain a curve m* 1 characterizing the beginning of contraction. The shape of this curve is similar to the yield surface in Cam Clay Model.

We can see in fig. 18 that we obtain for high OCRs a scattering in the values of qc/P' around the M* line.

These results are due to the small initial stress values for the highest OCRs. For this purpose, we performed some other tests with a higher value of the maximum consoli- dation stress p 0 . Therefore, the mean stress p( for a given OCR is also increased (for example p(= 200 kPa, OCR = 50). In these conditions we obtained an experimental point qc/ p' closer to M* .

PLASTIC FLOW ANALYSIS

Having defined the conditions by which the maximum strength develops and how the dilatancy evolves according to the overconsolidation ratio, we will now study the conditions of development of plastic strains, and mainly the evolution of the ratio between the plastic volumetric strain increment dee and the plastic deviatoric strain increment deP which gives the condition of plastic flow along triaxial stress paths.

Di/atancy in Cam Clay Model

T he Cam Clay Model was developed in the 1960s by Roscoe et al. [START_REF] Roscoe | On the yielding of soils[END_REF][START_REF] Schofield | Critical State Soil Mechanics[END_REF]). The model is based on elastoplasticity concepts and on the clay behaviour in triaxial compression tests. Equation [START_REF] Hanab | Etude expcrimentale du comportement dilatant des argiles surconsolidees[END_REF] gives the flow rule:

M-.!L = dee p' deP (7)
The perfect plastic state, called critical state was introduced:

qc dee=O and M-I = 0 Pc The beginning of loading is elastic (dee=deP= O). Then, a plastic contraction is induced following a straight line with slope= 1, before reaching the perfect plastic state for M-q/p' =0.

2. The intermediate state, obtained for OCR= 2. 7. The perfect plastic state follows the elastic behaviour immediately: we obtain an elastic, perfectly plastic behaviour.

3. For higher OCRs the behaviour is at first elastic until the maximum strength is reached, corresponding to the actual position of the yield surface. Later dilatancy appears following a straight li ne with slope= I, its development evolves with a decrease of the deviatoric stress until the plastic perfect state is reached.

Plastic Flow during Constant Mean Stress Pachs for Different OCRs

We found that the p lastic flow occurred differently according to the overconsolidation ratio. The analysis was conducted in the (dee/deP, M-qlp') plane (Fig. 20).

For normally consolidated samples, the plastic flow follows a straight Une in this p lane. It can be defined by:

q dee M -- = a• -- p'
deP [START_REF] Henkel | The effect of overconsolidation on the behaviour of clays during shear[END_REF] q (kPa) 1200 ~-------------------- We obtain therefore a plastic flow ru le slightly different from the one assumed in the Cam Clay model where a is equal to 1.

As one can see in Figs. 20 and 21 for overconsolidated samples, plastic flow is dependant entirely on the OCR values. For all OCRs, a plastic volumetric flow equal to 0, was observed in the beginning of each loading. The limit of this first phase corresponds, in (p', q) plane to the straight line M* for OCR~ 3, to the curve m* 2 for OCR = 2.7 and 2.5, and to the curve m* , for OCR = 2 and OCR= 2.24 (Fig. 17). We obtained a similar shape of the elastic domain as the one assumed by the Cam Clay model in the case of smal l OCRs (OCR= 2, and OCR= 2.24) as we can see in case 1, Fig. 19, and also for OCR = 2.5 for which we noted that dee was equal to 0 all along the loading; this corresponds for the Cam Clay model to case 2 (Fig. 19).

For OCR~ 3, Cam Clay Model assumes that the material behaves elastically (dee = 0) until the yield surface is reached beyond M (Fig. 19, case 3). In our experiments, the plastic flow was obtained earlier, for M-q/p' = 0.6 as we can see in Figs. [START_REF] Rowe | The stress dilatancy relation for static equilibrium of an assembly of particles in contact[END_REF] and 21.

1f we analyze now how the plastic flow develops when dee /de~' ~O, we can see in Figs. 20 and21 that: 1-For OCRs ~ 2.5, the dilatancy rate, defined by the value of dee/deP first increased up to a certain stress ratio and then decreased to 0. This stress ratio is an increasing function of the OCR and is equal to 0 for OCR = 2.5, in agreement, for this last point, with the Cam Clay Model as we discussed before (Fig. 19, case 2). 2-For OCRs < 2.5 we can observe that the volumetric plastic flow occurred following two stages of contractancy: the first one corresponds to an increase of dee/ deP and the second one to a decrease of dee/deP. The Cam Clay model seems agree in this case. These results confirm the fact that the Cam Clay model can simulate correctly the behaviour of slightly overconsolidated clays (OCR< 2 to 3) but is far from reality for higher OCRs.

Study of Stress Path Influence on Plastic Flow Phenomenon

Given one stress state located within either the dilatant or the contractant domain, the application of a stress increment da', defined by dq and dp' induces a plastic strain increment dc;P defined by dee and deP. We will examine here the influence of the initial stress state and of the stress increment on the orientation of the plastic increment given by the ratio dee/deP. The first point concerns the evolution of the plastic strain increment direction as a function of the stress increment direction. Thus we will investigate it in seven concourse points following different stress paths. As we can see in Fig. 22, P5/8, P4/5, and P3/5 are concourse points in (p', q) plane between p ' constant and a { constant stress paths. P5, P3, P2, and Pl are concourse points between a path M-q/p' --_(~)_ ----------I at '7 = q I p' constant and different p' constant paths (Fig. 22).

On stress paths where a change in the mean effective stress p' occurred, plastic volumetric strain increment were measured by using Eq. ( 2). deSd> was assumed to be purely plastic, while de~> was assumed to be purely elastic for overconsolidated specimens. de<il could be calculated from the relation between e (void ratio) and p'(mean effective stress), obtained from isotropic unloading (Fig. 8).

From where:

or:

de(i>= Cs dp ' 2.3 p' c dp' de~>= s 2.3 (1 + eo) p' ( 11) (12)
The concourse points P 1, P2, P3 and P5 show clearly the dependency of dee/deP with the stress increment directions (Figs. 22 and25). In points P5 and P3 (Fig. 22) the two vectors deP are located on both sides of the vertical direction, which means plastic dilatancy for p' = const path and plastic contraction for '7 = const path. In points P 1 and P2, the directions of the plastic strain increments are located on the same side of the vertical direction (Fig. 22). However, the directions are clearly different.

From these results, we can conclude that, when the directions of da' at a given stress point are significantly different (PI, P2, P3 , P4), we clearly obtain different orientations of plastic strain increments. Therefore, the uniqueness of the plastic potential cannot be verified. In P4/5 case we can note a small difference, the stress path direction seems to influence here also the value of dee/deP (Figs. 22 and 23). In this case the directions of da' are closer, which lead to also closer directions for the strains increments compared to the results presented M -q/p' ( M )

1 . 00 --------------- For points P5/8 and P3/5, dee/deP is practically identical for the two loading paths (Fig. 22). When the stress points approach the straight line M, the plastic volumetric strain increment is close to zero for any stress path, which indicates that the plastic potential can be considered as unique (Figs. 23 and 24). This shows that the hypothesis which is often assumed (e.g. [START_REF] Pender | A model for the behaviour of overconsolidated soil[END_REF] that the direction of the stress path has a slight effect on the direction of the plastic strain increment vector, is not always verified .

The results presented above along constant mean stress paths allow to define a deviatoric plastic mechanism which is induced by a purely deviatoric stress increment. The introduction of an isotropic stress component, using different loading paths, shows that the unicity of the plastic potential is not verified. We observe a combined effect of two mechanisms: a deviatoric type mechanism due to the deviatoric part of the stress tensor, and an isotropic type mechanism due to the isotropic part of the stress tensor. By combining these two effects, we obtain, at a given stress point, an influence of the stress increment vector direction on the plastic flow as indicated in Figs. 23 and 24. In particular, the dilatancy rate due to the deviatoric loading can be modified by the amplitude of the isotropic stress increment, and therefore, the dilatant behaviour of a clay specimen depends strongly on both the overconsolidation ratio and the loading path.

CONCLUSION

The initial state of a clay sample can be characterized by its overconsolidation ratio . It is a very important parameter which controls the amount of volumetric strain during loading. When subjected to a pure deviatoric loading, overconsolidated samples dilate after having reached a given value of the deviatoric stress, whereas a normally consolidated or a slightly overconsolidated sample is always contractant.

The analysis of the clay behaviour undertaken in this study, highlighted some mechanical characteristics suitable for all the stress paths investigated above. These parameters control the relationship between void ratio and mean effective stress and correspond to the slopes of the one-dimensional consolidation and swelling lines (Cc and Cs), as well as the positions of the one-dimensional compression line and the perfect plastic line (PP) in the (e-log p') plane. A second set of parameters control the boundaries of the domain in the stress plane (p' -q). M corresponds to the slope of the perfect plastic line in this plane and ( represents the envelop of peak strength above the perfect plastic line for overconsolidated samples. Inside these upper boundaries, the stress plane can be divided into different regions correspondjng to a contractant, dilatant or no-volume change behaviour along pure deviatoric stress paths.

The st udy of the plastic flow inside these different regions leads to the following conclusions. In the contractant domain, the plastic flow represented by the ratio dee/deP follows a relationsh ip close to the one proposed by the Cam Clay model: M-q =a • dee with a= 1.5 p' deP For highly overconsolidated samples, the dilatancy appears for a value of the stress rat ion q I p' smaller than M and the plastic strain ratio dee/dl:P increases first up to a maximum value and then decreases to 0 when the perfect plastic state is reached.

The constant mean stress path generates a plastic strain mechanism due to a pure deviatoric stress incremenl. When an isotropic stress part is introduced using different loading paths, the unicity of the potential cannot be verified. The results show a combined mechanism effect: a deviatoric mechanism and an isotropic mechanism, these two kjnds of mechanism are strongly dependent and lead to a dependency of the plastic strain increment vector with the direction of the stress increment. When the stress point comes close to the perfect plastic state, the plastic potential can be considered as unique.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Constant mean stress paths on overconsolidated clay samples

Fig. 4 .Fig. 6 (Fig. 6 (

 466 Fig. 4. Bishop's and Wesley's hydraulic triaxial apparatus

Fig. 7 .

 7 Fig. 7. Isotropic loading and unloading OCR = 8 and Pt)= 1000 kPa

Fig. 8 .

 8 Fig. 8. Results of isotropic consolidation in e-p' plane

Fig. 9 .Fig. 10 .Fig. 11 .Fig. 12 .

 9101112 Fig. 9. Constant mean stress tests on clay samples with different OCRs

Fig. 13 .

 13 Fig. 13. Strength envelop normalized b y the preconsolidation pressure

Fig

  Fig. 14. e-p' evolution for different stress paths

Fig. 16 .

 16 Fig. 16. Volume changes as function of deviatoric stress in constant mean stress tests

Fig. 17 .

 17 Fig.17. Volume cha nge condition in the (p ' , q) pla ne

  ). The dilatancy domain is obtained for high OCRs (OCR::::: 2.5) and is limited by the maximum strength envelope; the contraction domain for low OCRs (OCR s 2.5) is limited by the critical state line M . T hese two domains are limited by a third domain of no volume change. The boundaries of this third domain correspond to the beginning of dilatancy or contraction and can be defined by M*, m* 2 and m* I • This result is in agreement with a similar zone, defined like a nucleus within which only elastic volumetric strains take place, proposed by Balasu bramaniam et al.( 1993). However, this zone seems to be more reduced than the no volume change domain obtained in Fig.17. Tests were carried out on overconsolidated soft Bangkok clay which more submitted to a variety of stress paths.

Fig. 18 .

 18 Fig. 18. Stress conditions corresponding to the beginning of volume cha nge for different OCRs

( 8 )Fig. 19 .Fig. 20 .

 81920 Fig. 19. Plastic How in Cam Clay model

24 OCRFig. 2 1

 242 Fig. 2 1. Evolutio n of plastic strain incr ement di rec tions during constant mean stress tests

Fig. 22 .

 22 Fig. 22. Infl uence of stress increme n t d irectio n on plastic st rain increment di rection a t concourse points

Fig. 23 .

 23 Fig. 23. Plastic flow evolution along different stress paths for OCR = 5 clay sample

Fig. 24 .Fig. 25 .

 2425 Fig. 24. Plastic flow evolution along d iffcrent stress paths for OCR = 8 clay sample