A constitutive thermomechanical model for saturated clays

8NG~NSHRING

G~©L©GY ELSEVIER

Engineering Geology 41 (1996) 145-169

A constitutive thermomechanical model for saturated clays J-C. Robinet a, A. Rahbaoui a ,, F. Plas b, p. Lebon b a C R M D - C N R S UMR 131 lb, rue de la Fdrollerie 45071 Orleans, France b Agenee Nationale Pour La Gestion Des DOehets Radioactifs (ANDRA) Route du Panorama Robert Schuman, BP 38, 92266 Fontenay-aux-Roses, France

Received 28 February 1994; accepted 23 June 1995

Abstract

The structural deformation in clays results from microscopic phenomena involving the mechanical contact-stress change, the physico-chemical variation of repulsive forces in expansive clays, and thermal dilatancy of macropores. These textural strains are associated to three plastic mechanisms represented by respectively the yield surfaces frm, fR-a and ft. Under a thermal cycle, the sizes of interlamellar spaces between clay platelets are not modified, hence the temperature cycle is expected to have no effect on repulsive forces and thus the second mechanism is not affected by temperature changes. This paper suggests a formulation of a model of therrno-elasto-plastic behaviour of non-expansive saturated clays characterised by two plastic mechanisms. The mechanical yield surface fTm of the contact-stress mechanism is based on a modified cam-clay model; the thermal softening yield surface f r is a plane separating two thermal domains. In normally consolidated conditions, the resulting response to an increase of temperature is compressive. However, in highly overconsolidated conditions, a small irreversible dilative volumetric strain is observed when the temperature is above a threshold value. In intermediate conditions, the material starts with an expansion and tends to a compression. The constitutive model combines thermo-mechanical hardening, predominant in normally consolidated states (NCS) and absent in overconsolidated states (OCS) where the thermal softening occurs. The characterisation of the model requires information about rheological parameters obtained from oedometric and triaxial paths. Lastly, some numerical simulations of thermo-mechanical tests on remoulded Boom, 'Bassin Parisien' and Pontida clays are presented, which show satisfactory agreement between experiments and model predictions.

1. Introduction

Three classes of pore space are determined by the textural organisation of clay: (1) an interlamellar space limited by fibre-shaped clay flakes (Touret et al., 1990), with average size about 15 to 25 i ; (2) an interparticle porosity a m o n g connected clay flakes, about 200 to 1500 A in high * Corresponding author, Eurogeomat Consulting, 51, Route d'Olivet, 45100 Orlrans, France. 0013-7952/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0000-0000(95)00049-6

porosity clays (Touret et al., 1990), and from 10 to 25 A in low porosity clays (Pusch and GOven, 1988); (3) an inter-aggregate space between aggregates of particles defining a pore size of 1.5 to 16 ~t. This classification induces a complex distribution of water molecules in clay pores. Two main types of water in saturated clays are retained (Pons et al., 1982): free water, mainly in inter-aggregate space, and adsorbed water located in the interparticle and interlamellar spaces. The amounts of free and adsorbed water may vary considerably

146

J-C. Robinet et al./'l~)lghwering Geol~g~v 41 (1996) 145 169

and depend upon the distribution and size of the pores. For highly compacted and particularly for expansive clays, the pore distribution is monomodal (interlamellar) and the water is essentially adsorbed. In contrast, for highly porous clays, the pore distribution is bimodal and free water is predominant (Atabeck et al., 1991). Israelachvili and Adams (1978) performed shear tests on complex clays-water systems and showed that the response is independent of the loading kinetics, and that the maximum shear stress between two clay platelets increases when the number of water molecule layers decreases. The adsorbed water can thus be considered as similar to a plastic granular medium. Hence, the saturated clay may be regarded as a biphasic medium in isothermal conditions (a skeleton surrounded with adsorbed layers as solid phase and free water as liquid phase). Heating increases thermal agitation of adsorbed molecules which can consequently move as free water, with a simultaneous expansion of the medium. This process occurs with a decrease of the adsorbed layers strength (Paaswell, 1967). Exploiting the nuclear magnetic resonance (NMR) in which the dielectric relaxation time T2 of the water protons is proportional to the free water content, Carlsson (1985) has found that heating causes an increase of the free water content with a corresponding decrease of adsorbed water (Fig. 1). We can thus idealise saturated clay sub150

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2. Microscopic mechanisms responsible for thermal volume change

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jected to thermo-mechanical loading as a threephase medium consisting of respectively clay flakes, adsorbed water and free water. In expansive clays, constituted by iono-covalent stacks of three layers with exposed oxygen atoms and hydroxyl groups, the flakes acquire a negative charge caused by isomorphic substitutions. Hence, clays can be classified according to their electric charge deficiency (e.g. 0.4 to 1.2 eV by mesh for smectite). The electric field is intense between clay flakes in which the charge is high and located near the clay surface (Van Damme et al., 1987), so that exchangeable cations and water molecules are fixed close to the particle surface giving rise to two internal forces: attractive van der Waals forces and electrostatic repulsive forces which depend on the distance between flakes (Israelachvili, 1991). The effect of temperature on repulsive forces is not well understood. In highly porous clays, heat first induces a dilation of double layers, then an increase of repulsive forces (Kenny, 1966; Plum and Esrig, 1969). The double-layer repulsion of adjacent stacks is expected to increase in analogy with the heat-induced increase in osmotic pressure in molecular system (Pusch et al., 1991). This was established by Yong et al. (1969) who showed the increase of the swelling pressure with temperature. However, at low porosity, no increase in swelling pressure is detected during temperature change and no effect on interlamellar space (Pusch et al., 1991 ), and thus repulsive forces were not affected. It is suggested that in these conditions, the theory of continuous double layers is no longer applicable and water in the interlamellar space must be considered as a discrete medium (Israelachvili and Pashley, 1983; Fig. 2).

60

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"l ~/s Fig. 1. Relaxation time of proton T2 in bentonite clay versus temperature and water content (Carlsson, 1985).

The thermo-mechanical behaviour of expansive clays is described using an extension of the effective stress concept of Terzaghi, as proposed by Graham et al. (1992) on dense clays, and which showed that, at equilibrium of volume, temperature and chemical composition, the behaviour of clay is controlled by the effective contact-stress ~c', pore-

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pressure u and internal stresses #R-A resulting from the repulsive and attractive forces between the flakes, through the following relation: 5#ext = 6~¢' + 6#R-A + 5UI

(1)

in which each term of the right hand side, implicitly depends on temperature. Under isothermal conditions, mechanical consolidation of clay mud at different stresses has no effect on micropore size. The irreversible volumetric strains result essentially from the compression of macropores (Robinet et al., 1993; Fig. 3). It follows that for saturated expansive clays, the repulsive forces, acting in the interlamellar and interparticle spaces remain constant; macroscopic

irreversible strains thus build up in accordance with the conventional theory of consolidation. Texture deformation proceeds essentially from contact-stress change in the normally consolidated state and from repulsive stress change in the overconsolidated state (Fig. 4). Recent work by Pons et al. (1994) and Thomas et al. (1994) show by X-ray diffraction of suspensions of Na-montmorillonite in a closed reactor, that clay gel subjected to a thermal cycle displays a disorganisation and decompression of its structure: heating irreversibly breaks up the initial particles into smaller particles by a decrease of the stacking number (from 10 to 80% according to the chemical nature of solution, Table 1) along the

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J-C. Robinet et al./Engineering Geology 41 (1996) 145 169

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remoulded Boom clay (Belanteur, 1993).

existing lenticular pores (mesopores) with sizes of 200 to 500 A, leaving large pore spaces between them (larger than 1 pan) without a significant change in interlamellar space sizes. An increase of material isotropy was also noted. All these phenomena consequently result in the expansion of the macropores with an accompanying increase in homogeneity of the structure. Fig. 5 shows the variation of the fractal dimension of the structure for different clays, at room temperature and after a temperature cycle. The observed growth in fractal dimension implies a significant swelling for all clays after a temperature cycle. Table 1 gives the structure parameters obtained from small angle X-ray scattering tests. After a thermal cycle, there

Table 1 Structural parameters obtained from small angle X-ray scattering (Pons et al., 1994) Specimen

Average number of flakes by particle Room T/200°C

CaClz 0.1 mol KCI 0.1 mol KC1 0.5 mol KC1 1 mol

45/25 19.6/19.4 No modulation for the parallel flakes 80/30 16/15.8 75/65 15.9/15.5

Average distance between Distance (A_) with maximal flakes (A) probability Room T/200 C Room T/200°C

Parameter of disorder Room T/200°C

21.6/21.6

0,045/0.039

15.6/15.6

0.047/0.079 0.O55/0.076

15.6/15.6

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200 Temperagare (NT)

Fig. 5. Fractal dimension of structure versus temperature and ionic concentration (Pons et al., 1994).

are no changes in the average distance between flakes and, presumably, no variation of repulsive forces. However, a small reversible increase of these forces may have occurred during heating, which would account for the difference in behaviour of the expansive and non-expansive clays (Fig. 6). In dense material, heating under closed conditions will raise the pore-pressure to very high values, which will drive interstitial water molecules into the interlamellar (or rather lenticular) space and this may create unstable stress conditions resulting in disintegration of stacks and an improved degree of homogeneity (Pusch et al., 1991). It has been show also by Pusch (1987) that normally consolidated clay at a density of 2 g/cm 3 undergoes irreversible meso-strnctural changes (closure of lenticular pores) during a temperature cycle. These comprise a denser grouping of the montmorillonite flakes, leaving larger voids

149

between aggregates than in the reference nonheated material. Summarising this discussion, we can suppose that the physico-chemical phenomena which can occur between clay particles are qualitatively independent of the mechanical state of the material even if the global response of each remains different. In other words, both in compacted clays and in suspensions or gels of clay, the physicochemical interactions continue to be similar because they depend essentially on the clay lattice constitution, the chemical nature of the interstitial fluid, and interlayer distance. The expansion of the macropores shown by Pons et al. (1994) and Pusch et al. (1991 ) with a breaking up of particles concerns an overconsolidated material or suspensions that the material can easily expand. However, for highly compacted, and particularly, for normally consolidated material, heating increased sliding between particles, induced a denser grouping of flakes and closed the lenticular mesopores. In any case, although authors differ in their interpretation of the micro-mechanisms responsible for the volume change in clays during temperature increase, almost all agree that the global response is: (1) swelling when the material is in suspension or highly overconsolidated. Its magnitude, which is comparatively small, depends on the overconsolidation ratio. The material swells more if the (OCR) is high until a value where the compaction starts. (2) compaction if the material undergoes a maximum of applied stress or near the (NCS). expansive clay non expansive clay

High OCR

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NCS

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Fig. 6. Schematisation of thermo-mechanical macroscopic strains during thermal cycle.

150

J-C Robinet et al./Engineering Geology 41 (1996) 145 169

In the present paper we shall try to model these two responses with reference to the above conclusion. If the material is highly overconsolidated, it is certain that the expansion is relatively significant, compared to the slightly overconsolidated, because of the greater possibility for macropores to expand; in other words, the greater the (OCR), the greater the expansion. If the material is normally consolidated, simultaneously with the above expansion (which cannot be seen because the external stress is higher), the external stress is so much higher that the macropores (especially the lenticular pores) will compact, causing an increase of the material density. It is also possible, as mentioned by Paaswell (1967), that an interparticle penetration occurs as a fusion of the soil particles.

3. Thermo-mechanieal characteristic of saturated clays

(as at NCS). During cooling, we obtain again a low compression, with curves which are parallel for the three examples (Fig. 6b). Thus, the compression induced by the thermal cycle decreases with the (OCR), in agreement with earlier results (Demars and Charles, 1982; Fig. 7). Campanella and Mitchell (1968) and Baldi et al. (1990) have observed small consolidation of the material during cooling, independently of the mechanical state. We can thus assume that during cooling, all components of the material elastically contract and thus the response is reversible. The general aspect of the thermo-mechanical behaviour of clays is represented by remoulded non-expansive 'Bassin Parisien' clay in oedometric tests (Robinet et al., 1992a, Robinet et al., 1992b) and lightly expansive Boom clay in drained isotropic heating tests (Baldi et al., 1990), characterised in Table 2 and Table 3. 3.1. Oedometric behaviour

Previous studies on kaolinite and smectite (Robinet et al., 1992a,b) have shown that the behaviour of expansive and non-expansive clays may be represented by the sketch in Fig. 6 at a constant effective stress. The swelling during heating in expansive clays is attributed to the addition of the repulsive forces which are expected to increase reversibly with temperature. However, during cooling the repulsive forces decrease to recover the above excess. The expansive clays show more compaction than the non-expansive material. In (OCS), with a high (OCR) (larger than 8), thermal loading causes an expansion of clay structure especially for expansive clay (Fig. 6a). When the material is normally consolidated, heating results in a significant compression which is relatively small in expansive clay. Cooling in normally consolidated state results also in a compression as at the (OCS) (Fig. 6c). For lightly overconsolidated clays, during a first stage of heating, the material volume (as at high OCR) expands at a certain temperature at which the aggregates stop swelling further because of the presence of constant external stress. This temperature T, as will be explained below, corresponds to the temperature where the yield surface meets the stress-point. Beyond it, the second stage starts with compression

The thermo-mechanical oedometric tests were performed on non-expansive 'Bassin Parisien' clay and lightly expansive Boom clay. The clay mud was carefully prepared, placed in oedometric cells, then into a controlled oven initially at room temperature, and consolidated at a total stress of 1.2 MPa with 1 MPa back-pressure. This thermomechanical state is taken as a zero-point for all specimens. We observe that above the effective stress of 1 MPa, the behaviour is identical with the normally consolidated clays. The consolidation O.OIS

0.010

0.005

I 2 J ~ S 6 7 8 0VERCONSOLIOAIION RATIO Fig. 7. Irreversible compressive strain induced by thermal cycle versus (OCR) (Demars and al., 1982).

J-C. Robinet et al./Engmeering Geology 41 (1996) 145 169 Table 2 Physical characteristic of remoulded 'Bassin Parisien' and Boom clays (Euro-Gromat Consulting, 1992) Clays

B. Parisien (%)

Boom (%)

SiO 2 A1203 FezOa CaO MgO Na20 K20 MnO TiO2 P205 Lost by heating

51.60 21.54 6.42 2.035 1.72 0.62 2.25 0.015 1.12 0.44 12.24

57.58 12.93 7.57 2.22 2.40 0.12 1.96 0.01 0.88 0.18 14.15

Clay

Basic cation exchange capacity (meq/100 g)

Total cation exchange capacity (meq/100 g)

BET specific surface (mZ/g)

B. Parisien Boom

46 59

145 213

33.5 42

Clay

B. Parisien

Boom

~'~ (g/cm 3) WL (%) We (%) Activity by oedometric test (Cg/Cc)

2.68 45-60 15 30 0.21

2.67 60 75 25.-35 0.27-0.32

151

constant during the temperature cycle (Figs. 9 and 10). Before dismantling the cells, the specimens were cooled to room temperature and then unloaded mechanically by stages. The dimensions, weights and water contents of specimens were measured and the mercury porosity and BET tests were performed. At a low effective consolidation stress (<0.1 MPa for 'Bassin Parisien' and Boom clays, <0.21 MPa for illitic and bentonite clays; Plum and Esrig, 1969) a minor dependence of consolidation coefficient on temperature is shown (2 = 1.24 at 24°C and 2 = 1.4 at 50°C for illite; 2=0.28 at 24°C and 2=0.238 at 50°C for bentonite). Thus, in our study range, the 2-coefficient may be considered as constant. However, the thermal cycle at constant axial stress in (NCS) induces a material hardening and during a short initial period of mechanical reloading, the curve displays a slope smaller than 2 (Fig. 9 and Fig. 10). In fact during cooling, the preconsolidation stress at room temperature is higher and the stress-point is slightly decreased because of the reversible contractive strains so that, after reloading, a small overconsolidated 'bent' is obtained. 3.2. Triaxial behaviour

slope 2 is independent of applied stress and temperature (Finn, 1951; Campanella and Mitchell, 1968). Three experiments are carried out on 'Bassin Parisien' clay at 20, 40 and 80°C (Fig. 8). In other experiments clay specimens were consolidated at the desired degree of stress, which was maintained

The triaxial tests were performed at ISMES on non-expansive remoulded Pontida clay. Baldi et al. (1990) showed that the temperature has no effect on the critical state. At constant total mean stress, heating generates an increase of compressive axial strain which eventually can lead to failure in undrained conditions of the specimen if temperature is significant (Figs. 22 and 24). This failure

Table 3 Composition (wt%) of natural Boom clay (CCE-ARCHIMEDE Project, Progress Report Jan. to June 1993) Mineral

Quartz

Microcline

Plagioclase

Pyrite

Carbonate

Sulphate

Clayey fraction

Content ( Weight %)

20-25

4-5

4 5

4-5

Traces

Traces

65

Clayey fraction

Interstratified disordered illite/smectite

Kaolinite

Chlorite

Content (wt%)

50

20

Traces

Interstratified Illite + mica ordered illite/smectite Traces 25

J-C. Robinet et al./Engineering Geology 41 (1996) 145 169

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4. Constitutive equations for saturated nonexpansive clays The following indices are used in the notation: v for volume, t for total, T for temperature, m for mechanical, p for plastic, e for elastic, w for water, s for skeleton and c for threshold. The elastic (e) and plastic (p) terms will refer to the thermo-

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J-(Z Robinet et aL/Engineering Geology 41 (1996) 145 169

154

mechanical strains, and the reversible (r) and irreversible (ir) terms to the thermal strains. The present investigation is limited to nonexpansive clays which behaviour is only slightly sensitive to the mechanical unloading. This is because the repulsive forces mechanism is almost non-existent owing to their physical stability (e.g. kaolinite). However, the thermal expansion concerns both expansive and non-expansive clays due to a significant presence of mesopores. Thus, we deal with the two principal mechanisms concerning the expansion of the material during heating and mechanical consolidation. It has already been pointed out that the irreversible dilative volumetric strain of clay is the result of the macropore expansion (lenticular porosity) during heating. We consider the total thermal volumetric dilative strain to be defined by: e~rv= -aTbx~l exp(-bTT)J ~irret'ersible part

aT

(2)

reversible part

in which a~ and br are the thermal parameters which depend on the nature of clay. We presume that they are associated to the microscopic variables like interlamellar distance d, quantity of transferred adsorbed water, inter-aggregate pore sizes and others. They also characterise the degree of mesopores and macropores expansion. The above proposed expression is supported by the macroscopic response of specimen volume to heating (Baldi et al., 1988). In fact, the experimental results are fitted well by an exponential expression which gives the values of ar and br (Fig. 11 ). The determination of those parameters is given by applying a heating in isotropic drained test at high overconsolidation ratio in order to obtain only the thermal response. ~/is the overconsolidation ratio and ~ is the global coefficient of cubic thermal expansion of soil. We also consider the existence of a threshold temperature Tc beyond which the irreversible dilative strains can be produced. Indeed, at constant total stress, the first threshold temperature equals initially to room temperature Tc = T0. During heating to a temperature T 1> To, both reversible and irreversible volumetric strains are produced and the threshold temperature becomes Tc = T1. While cooling, only the reversible part of the strain is

100

RemouldedPontidacta Y i

i

i

"2 6o

~, E

40

I .0.25

.0.20

-0.15

Volumetric

l

L -0.10

I

t -0.05

2O 0.00

strain(%)

Fig. 11. Drained heating test at constant effective stress at ( O C R ) = 12--remoulded Pontida clay (Baldi et al., 1988).

recovered and the threshold temperature remains equal to T1. During reheating, the irreversible strain occurs when temperature exceeds the value of T1; in other words, the irreversible part of Eq. 2 is inactive if temperature does not beyond T~.

4.1. Yield surface of thermal softening The irreversible mechanism due to the temperature effect on the material volume change is expansive and causes by the macropores and mesopores expansion. The magnitude of swollen volume depends on the overconsolidation ratio because the macropores expand easily when the material is mechanically less loaded. This dependence was previously noted by Plum and Esrig (1969) and Demars and Charles (1982). The thermal expansion of the microstructure induces an irreversible dilation which is the source of the first irreversible mechanism. This mechanism is associated with a yield surface f r which remains unchanged at a temperature Tc during a thermal unloading (cooling). We consider the yield surface: /T= T--To=0

(3)

to be a plane in the (P', Q, T) diagram dividing two temperature ranges (Fig. 16). When the temperature is less than T,., only the reversible thermal strains are developed. Beyond this temperature,

J-C. Robinet et al./Engineering Geology 41 (1996) 145-169

both reversible and irreversible thermal strains develop. This description remains valid for all mechanical state even in (NCS), where the hardening strains are predominate. The f r yield surface expresses the capability of lenticular pores to expand. The reversible part characterising the thermal expansion of soil is given by: .

(4)

d~,j - - 3 Tt~ij

where ct is a global coefficient of cubic thermal expansion of soil grouping that of the skeleton, free water and adsorbed water. For a 'freeadsorbed' water system, the coefficient of cubic thermal expansion of mesopores (lenticular pores) and macropores is assumed to be given by the following expression, deduced from the experimental values of Juza (1966) (Fig. 12):

given in Table4, and u is the water-pressure expressed in MPa. Thus, we can formulate the following equation: ct=n~W+(1 - n ) ~ s

+ (dE -- c2T) (ln(u)) 2

(5)

in which ci and vli (i=0, l, 2) are the constants

Ool M P a

7 m 3"

(6)

where ~s is the cubic thermal coefficient of skeleton and n the porosity of material. The irreversible part of thermal volumetric strain is expressed as above by: ~ v = - - a T b T t l e x p ( - - b T T ) T if T= Tc and T > 0

(7) The thermo-mechanical state where T> Tc cannot exist because the material's temperature T beyond Tc moves t h e f t plane parallel to the (P', Q) plane in order to reach a new value of To. The total thermal volumetric strain can then be written as: "t - - "r "ir 6"TV - - ~ ' T V "4- t Z T V

7W(u, T) = (do + coT) + (dl - cl T)ln(u)

155

( 8 )

Some authors consider this expansion to be so small as to be negligible. However, even if the mechanically induced strains are usually much larger than thermal strains, the latter are sometimes of crucial importance. In fact, as will be shown, the yielding and failure at high temperature occur in the undrained state at a lower value of deviator stress than at room temperature. Also, strength appears to be affected by temperature. It decreases with temperature in (OCS), and is constant or decreasing in (NCS). The introduction of the irreversible thermal strain remains the principal difference between the present model and the Hueckel-Borsetto model (1990).

2 0 0 MPa z --

~

I000

4

2

o

2s

4.2. Yield surface o f coupled thermo-mechanical hardening, f T,,

MPa

5'o

¢5

~o

TEHP~RATURE

l~.s

).,IP a

1do

L

('C|

Fig. 12. Variation of water expansion coetficient versus temperature and pressure (Juza et al., 1966).

The textural deformation of non-expansive clays at normally consolidated conditions are the results of the variations of contact-stress induced by the increase of external stresses or/and the thermal dilatancy responsible of the preconsolidation stress decrease. The yield surface of the thermoplastic compression of clay is chosen to have an elliptic shape as in the modified cam-clay model proposed by

J-C Robinet et al./Engineering Geology 41 (1996) 145 169

156

Table 4 Parameters of thermal expansion of clay water Thermal parameters of clay water

co(C -2)

c~(C 2)

c2(C z)

do(C-l)

dl(C-1)

d2(C-1)

5.17x10 6

4.6x10 "

5.01x10 s

1.54x10 a

1.327x10 a

4.668x10 s

Roscoe and Burland (1968), where the thermomechanical preconsolidation stress P'cr depends not only on mechanical volumetric plastic strain but also on thermal volumetric irreversible strain as proposed by the model of Hueckel and Borsetto (1990), in which the yield surface depends explicitly on temperature. In general, the yield surface of thermo-mechanical compression is defined in the (P', Q, e, T) diagram. It can be schematised by two plane projections:

4.2.1. Yield surface projection on the ( T, P') plane In order to obtain the thermo-mechanical dependence of preconsolidation stress on temperature and stress solicitations, the choice of a general path is more complicate because we cannot distinguish between the mechanical and thermal effects. We have thus decoupled those mechanisms by keeping constant the mechanical solicitations during temperature change. In Fig. 13 two thermomechanical paths under drained isotropic conditions are chosen to illustrate the required approach. The first (OAB) concerns specimens maintained at room temperature; loading starts at point O, increases up to point A and then is removed (point B). Thus at room temperature,

the volumetric plastic strain is: ePmv(OAB)= ( f ~ k ) ln(P'c/P'~o) (1 +eo)

where Pco' and Pc' are the initial and actual preconsolidation stresses equal, in the case of cycle (OAB), to: P;=P'coand Pk=P'c

(10)

Thus, the mechanical hardening is described by: ' Pmy) with " Pc' = P°0exp([3me

l +e 0 tim --

2 --

k

(11)

Let us consider a second cycle (OCDEF) which is postulated to have the same final plastic void ratio. A specimen, initially at room temperature (point O), is subjected to a thermo-mechanical compression during heating to T1 at normally consolidated state (point C). It is then loaded to point D, unloaded to point E and cooled to room temperature (point F). We thus obtain the same value for the plastic volumetric strain, and therefore, the points A and D belong to the same thermo-mechanical yield surface. The thermo-mechanical compression between

T

d

C,E

Ln P"

(9)

._

~

O,B,F

Fig. 13. Schematic representation of two isotropic paths.

~

D _,,

LnP"

J-C.

Robinet et al./Engineering Geology41 (1996) 145-169

points O and C is given in general by: £1~mv = f i T )

(12)

where f is a hardening function characterising the evolution of yield surface with temperature. It also quantifies the distance between the consolidation curves at different temperatures. During cooling (as in high overconsolidation), Eq. 12 has no meaning because there are no plastic strains. At temperature T1, as in the above expression, the mechanical plastic volumetric strain between points C, D and E is: 1 v p, , £mv(CDE) = ~mln(D/Pc)

(13)

The introduction of temperature T1, at the condition of the same volumetric strain, lowers the preconsolidation stress at Pb, called a thermomechanical stress and denoted as: P'~=P'cr. Considering also that P~=P;o, the thermomechanical strain of the (OAB) path is the sum of the thermal strain (OC) and the mechanical strain (CDE):

157

The first exponential term is identical to the known mechanical expression; however, the second term symbolises the changed size of the elliptic yield surface with temperature. This rule differs from the thermo-mechanical hardening rule proposed by Hueckel and Borsetto (1990) who consider that the yield surface can change infinitely with temperature even during cooling when no irreversible strain is produced. The present rule also corresponds to the introduction of the irreversible volumetric thermal strain which is not taken account of by Hueckel-Borsetto's model. The simulation of the variation of thermo-mechanical preconsolidation stress with temperature is illustrated in Fig. 14 at different values of initial preconsolidation stress of Pontida clay, when the specimen temperature is beyond the threshold, To.

(14)

4.2.2. Projection of yield surface frr. on the Q-P' plane In order to take account of the thermo-plastic deviatoric strain in yield surface, the modified camclay form associated with the Hujeux parameters (Heujeux, 1985) were used. This will attenuate the over-estimation of the shear strength at higher overconsolidation ratios. The yield surface is expressed as:

and thus the thermo-mechanical hardening rule is obtained as:

fTm(P', Q,P'cT) = Q2 _ )l/12yZp,(bP,cx_ p,)R2(O) = 0 (18)

flmeVrmv(OAD) = flmcPmv(OAD)= tim f ( T )

+ ln(P'~T/P'o)

P'¢T= P'¢oexp(fl~e~m~)exp(--fir. f ( T) )

(15)

Assuming that: i

fiT

fir)

ir

• •

80

where fiT is defined as a coefficient of thermoplastic compressibility. The former expression considers that thermal consolidation in NCS is the exponential function of temperature elevation as it is recognised by several tests on saturated clays. It also gives the expression of preconsolidation stress directly in terms of the irreversible thermal dilative strains. Then, Eq. 15 can be written as:

-

£

P'eT P'¢oexp(flmC:~,,)exp(flTC~v) =

thermal

i f T = T c and 2/'>0

part

(17)

•

".

t ". i •

i

r - r -

;

I

".

h

i'

i ;

i L

; '..

20

i , . . . . .

~

i

i'-

'.

0.0

part

i

(16)

> 0

= - #meOW_

mechanical

i

:

i i -

~

. . . .

i

i

i

i

h

I

~

i "r --"*--

J

:1

2.0

i q - c -

Preconsolidatlon

------

~

i . . . .

I

I

4.0

1.0

stress(MPa)

Fig. 14. Evolution of thermo-mechanical preconsolidation stress with temperatureat differentinitialvalues.

158

J-C. Robinet et al./Engineering Geology 41 (1996) 145 169

P'~r is given

where

by Eq. 17 and:

E'PTrn

(19)

) , - - -

(a -~-~'PTm)

is a hardening parameter reflecting the thermoplastic deviatoric strain contribution (Heujeux, 1985). e~rM is the thermo-mechanical deviatoric strain and a and b are the Heujeux parameters depending on material. The critical state is specified with the slopes M~ in compression and ME in extension: Mc = 6sinqt/(3 - sinqt) ME = -- 6sin¢/(3 + sin¢) (20)

(21

7~

1 + ~)sin(O - g ) + ~x/3 sinO

where - -

(23)

where the elastic strains are expressed as follows:

CTrnij-~-G-~--9KOiJ

(24)

in which K and G are respectively the bulk and shear modulus of the material, and the plastic strains are expressed by: ~Pmij : t~ ~ ' i j

,/3 2

•t - ~"~ ETmij -Tmij Jr E"P Tmij

@

The function

R(O)-

shrinking occurs when the temperature exceeds T,., but no change was noticed during cooling. Total thermo-mechanical strains are given as the sum of elastic and plastic components:

( 25 )

where h is the plastic multiplier and g the potential of plastic flow, taken to be implicitly dependent on temperature change:

g(P',Q,P'oT)=Q2- MzP'(bTP'cT- P')=O ME

(26)

(22)

Mc

5.

is used to introduce the Mohr-Coulomb failure and O is the angular stress invariant. Fig. 15 shows the evolution of thermo-mechanical yield surface frm v e r s u s temperature. The

Experimental

Considering the insufficiency of the existing database, the number of the results to compare with simulations is limited. The major calibration

Q = M P"

! 1

I 2

results and predictions

I I ! 3 4 Iso~ropic sb'ess (MPa)

Critical S ~ te Line

t ! 5

~ 6

Fig. 15. Evolution of cam-clay yield surface size versus temperature.

159

J-C Robinet et al./Engineering Geology 41 (1996) 145-169

of the model has been carried out on remoulded non-expansive Pontida clay with the parameters given in Table 5. In the following simulations, during heating, the yield surface of the initially overconsolidated specimen shrinks. When the state-point and the yield surface meet, the material becomes normally consolidated in both cases whether the yield surface moves (case of undrained or oedometric test) or not (case of drained triaxial or drained isotropic test). Moreover, cooling has no effect on yield surface size because the temperature is less than Tc and no irreversible strains occur. We can summarise this situation with the following differential equation: PeT =

__~PcT d~rnv

-~

~3P;T j r

OO~mv

(27)

OT

Thermo -mechanical variation

Thermalvariation

We will use this expression to explain the evolution of yield surface size. 5.1. Simulation of oedometric paths in thermomechanical loading The temperature increase in two mechanical states has been schematised in Fig. 16 with applied heating in normally consolidated conditions. From the mud state at point A, the material is loaded along the line Q--K'P' up to a point B and then heated from To to T1 between points B and C (on this line, the ratio Q/I~' is equal to - 3 / 2 ) with a significant hardening (P'cTO. During reloading, the material path tends towards the line Q = K'P' and then continues to point D where it shows an elastic unloading to point E. Thermal path in overconsolidated state is illustrated in Fig. 17. After initial loading between points A and B, the material is unloaded to point C, where it is heated from To to T~ along the line (CD), similar to the line (BC) above. During

heating, the yield surface shrinks and the irreversible thermal strains are generated as shown in the (e-log(a1)) curve. At point D, the stress state of the material is always within the yield surface (P'cTO); during reloading, the path becomes parallel to line (BC), until it reaches the yield surface at point E. With reloading maintained, the stresspoint tends to the line Q = K'P' and the hardening material evolves to point F (P'cTO" Fig. 18 presents the oedometric paths simulations and confirms the previous macroscopic description induced by microscopic phenomena. It also shows that heating-cooling cycle 20-95-20°C causes an obvious compression of the material in NCS, and a low expansion in OCS. This expansion always exists, but in NCS it is very small compared to the thermo-mechanical compression.

5.2. Simulation of triaxial thermo-mechanical tests in drained conditions In OCS at constant mean effective stress, when the temperature reaches Tc, the expansive irreversible strains develop and the material swells. Simultaneously, the yield surface size shrinks and follows the equation: P'eT = 0P'eT.~

(28)

0T 1

Since the stress-point is unchanged (P' =const. and then Q = const.) yield surface reaches Tc, and the material behaves as normally consolidated. From this state, thermal softening must be compensated by strain hardening in order to keep the same value of thermo-mechanical preconsolidation stress. Hence, in addition to the irreversible expansive strains, a volumetric compressive thermo-

Table 5 Mosal parameters of Pontida clay Parameters

eo

2

k

E i (MPa)

M

v

ar

br(C-1)

a

fir

Pontida clay

0.82

0.103

0.0164

40.0

1.429

0.30

2.802 × 10 -3

2.5 × 10 -3

9 x 10 -4

2 × 102

b = 0 . 9 7 e x p ( - 0.065~/).

J-C Robinet et al./Engineering Geology 41 (1996) 145 169

160

Q

Q =MP

"

J .~

Q=K'P"

P~' c I

P'cz

7"1

e A

C B

D

log

h

Fig. 16. Schematic path of heating in oedometric test at normally consolidated state.

p,

J-C. Robinet et al./Engineering Geology 41 (1996) 145-169

161

O=MP" Q=K'P"

/ /

C

Ts

~S

~

D P'cTO ]

P'¢TI

e A

D

F

log ( a~) Fig. 17. Schematic path of heating in oedometric test at overconsolidated state.

162

J-C Robinet el aL/'Engineering Geology 41 (1996) 145 169 0.6

2o°9 40oc\

0.5

80

"~

~...,

0.4

0.3 :x

0.2

0.1 .

0.0

.

0.1

.

.

1.0 Vertical

.

.

.

L

.

I

_

=

i

10.0

stress(MPa)

Fig. 18. Oedometricsimulations of isothermal paths. 0.8-

mechanical strain is produced: EPmv~-----~

~T

/

~

T>O

(29)

A'AAA

•

•

0.6

and the total volumetric strain will be compactive. Fig. 19 shows two drained tests performed on Pontida silty clay (Hueckel and Baldi, 1990). Initially, both specimens are at ( O C R ) = 5 under confining stress of P' =0.2 MPa. The first one was accomplished at room temperature and the second at 98°C. The temperature increase appears to have an effect on initial mechanical parameters, especially on the thermo-mechanical preconsolidation effective stress. Fig. 20 illustrates the previous paths in the (P', Q, T) diagram and its projection on the (P', Q) plane. Fig. 21 compares the experimental data (Batdi et al., 1988) and the numerical triaxial simulation. At a normally consolidated state with constant deviatoric stress Q = 1.2 MPa, heating from 22°C to 100°C increases the compactive axial strain but the yield surface is unchanged. When thermal consolidation was completed, mechanical loading

t,

R~oula~ t~ac¢~ a~/

O.4

~

•

Y3~C(~iaat

•

~*c(~aaot i9,)o)

1990)

0.2

0.0 ~0

J

r

,

~

~0

&O

1ZO

1~0

Axial

' 20.0

$train~A)

Fig. 19. Simulationand experimentsof two axial drained paths at overconsolidatedstate.

was applied, always at 100°C, and the specimen was expected to be hardened because it was normally consolidated.

J-C. Robinet et al./Engineering Geology 41 (1996) 145-169

163

Q

O ~P" B

r

#r)

Pro1

J

0 B

C ,C w

A

w

y

Fig. 20. Schematic triaxial drained path of initial overconsolidated material at room and high temperatures.

Pp ,

J-(] Robinet et al./Engineering Geology 41 (1996) 145 169

164

~a

6,0

heated to an adequate temperature before the mechanical loading (path A'B'C'). The water-pressure change is given by the variation of effective mean stress:

40tZ~/0.~

u=-P'

.

:/

0.0

• - -

10.0

Axial

This pressure increase during heating is responsible for failure at lower values of the effective stress because of the macropores dilation. Fig. 24 illustrates two undrained tests (Hueckel and Pelligrini, 1991) with their numerical predictions. One was performed at room temperature, the second was heated by steps to 92°C at constant deviatoric stress Q = 1.2 MPa until failure.

F2qzar,.~ (na~ a,~ m0) simu/~/on

20.0

30.0

strain(%)

6. C o n c l u s i o n

Fig. 21. Simulation and experiment of triaxial drained test at normally consolidated state (Baldi et al, 1990), heating from 22 to 100°Cat Q = 1.2 MPa.

5.3. Simulation of undrained triaxial tests The total volumetric strain is maintained constant: "t -e "r "P +egv ~'v = ~Tmv + ~Tv ~- tZTmv

=0

(31)

(30)

At constant total mean stress, water pressure builds up during heating owing essentially to expansion of the liquid phase of the soil. If the material is at (OCS) (Fig. 22), the thermo-elastic volumetric strain ratio ~eT,,~will be negative and then the stress-point decreases towards the lower effective mean stresses (BC). At the same time, the yield surface shrinks due to the irreversible thermal volumetric strain ~r~,. If heating is sufficient, the yield surface reaches stress-point at point C (more probable situation, cf. Hueckel and Pelligrini, 1991). The specimen becomes normally consolidated and the stress-point, always on the yield surface, moves towards lower values of effective mean stress with building-up the thermoplastic strain e~m~,(CD) until failure. Fig. 23 shows the change of clay behaviour with temperature under undrained conditions. The material which is initially overconsolidated (path ABC), becomes normally consolidated if it is

The thermal strain of non-expansive clay results from the expansion of lenticular pores which is accompanied by the irreversible division of clay particles into thinner ones with an undisturbed interlamellar space when the material is in suspension, gel, or highly overconsolidated. This expansion was associated with the irreversible macroscopic dilatancy. The internal organisation of the structure is modified by the variation of the contact-stress resulting from the increase of external mechanical and/or thermal loading. The inventory of microscopic phenomena governing the thermo-mechanical macroscopic behaviour of nonexpansive clays was used to establish a thermoelasto-plastic modelling. The constitutive model is characterised by two plastic mechanisms associated to the yield surfaces fTm and fT- For thermal and mechanical normal states, thermal loading produces the activation of both plastic mechanisms of j)m and fT. This coupling is showing up as a volumetric thermo-mechanical strain of hardening. However, in overconsolidated states, thermal loading is only accompanied by activation of the thermal mechanism fT which generates a volumetric irreversible thermal strain of softening. In both cases, the size of yield surface fT,, depends on temperature increase. The originality of this model, as compared to previous ones, is the introduction of irreversible thermal strains which control thermal softening,

J-C. Robinet et al./Engineering Geology 41 (1996) 145-169

165

Q

Q - M P"

P~cl

TI

Y

Q

Q =MP"

°

.J D

.f*

B

A

y

Fig. 22. Schematic path of heating in triaxial undrained test at initial overconsolidated state.

p,

J-C Robinet et al ,,"Engineering Geology 41 (1996) 145 169

166 Q

Q =MP"

l" m

/

P'~2

P'co

m

P"

TI

g

P'cT2

Q

B

--C

C'

A

A'

y

Fig. 23. Schematic triaxial undrained path of initial overconsolidated material at room and high temperatures.

The latter strain, which is emphasised in drained and overconsolidated conditions, is observed in tests performed by several authors. The model

parameters are quantified from oedometric and triaxial tests. The comparison of the experimental data on remoulded Pontida and 'Bassin Parisien'

J-C Robinet et al./Engineering Geology 41 (1996) 145-169

7. Notation

3.0•

2.2~etaL

1991)

?.0

/

1.0

GO

i

GO

0.5

1.0

1.5

2.0

Mean e f f e c t i v e stress(l~a)

2.5

3.0

1.0

lr2*c I

167

E

O.8

O.6

to92a6" teommeea ~ 22 *C ~ l u e d ~

0.2,0, - -

0.0 0.0

a~¢

el al. 1991)

wilh heating to 92 ° C ( l l u e c k d el: al. 1991) simlllalions

]

,

i

4.0

&O

12.0

Axial

strain(°/d

Fig. 24. Simulation and experiments of two triaxial undrained tests at normally consolidated state.

clays with the model predictions shows that all observed phenomena are correctly described. However, the present model must be validated in several tests performed on different saturated clays. In summary, the objective has been to quantify the heating/cooling effect on the saturated clays at all mechanical states, to give a general tendency of microscopic phenomena and the constitutive equations governing its behaviour.

NCS: Normally consolidated state OCS: Overconsolidated state OCR, ~/: Overconsolidated ratio fTm: Thermo-mechanical yield surface or contact yield surface Thermal yield plan of softening fT: Yield surface of internal stresses fR-A: g: Potential of plastic flow Total stress tensor ~ext: Applied external stresses tensor ~¢': Effective stress or contact-stress tensor U: Water pressure ~R - A: Tensor of internal stresses Tensor of total strains ~: Tensor of total thermal dilative strains Tensor of reversible thermal dilative strains Tensor of irreversible thermal dilative CT • strains ~Tm: Tensor of thermo-mechanical strains ~m: Tensor of thermo-elastic strains Tensor of thermo-plastic strains eo, e: Initial and actual void ratio T: Temperature of material (°C) To: Room temperature (°C) L: Threshold temperature (°C) p': Effective mean stress Q: Deviatoric stress M: Slope of critical state line Angle of shear resistance & Elastic tensor K: Bulk modulus G: Shear modulus 2: Elastoplastic isotropic modulus k: Elastic isotropic modulus /~: Coefficient of plastic compressibility h: Plastic multiplier a, b: Parameters of the Hujeux model aT, br: Thermal parameters of irreversible expansion of clay 7: Hardening parameter Global thermal expansion coefficient of clay as: Thermal expansion coefficient of skeleton Thermal expansion coefficient of clay

168

J-C Robinet et al./Engineering Geology 41 (1996) 145 169

water fiT: Coefficient of irreversible thermal expansion P'0: Initial pre-consolidation stress P;r: Thermo-mechanical pre-consolidation stress n: Porosity ~.': Rate symbol ]: Unit tensor

Acknowledgment The first two authors express their most sincere thanks to the 'Agence Nationale pour la Gestion des D6chets Radioactifs, ANDRA' (France) who supported this work. C E N - S C K (Belgium) is acknowledged for supplying the clays.

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