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Antonio Gens

UPC Barcelona Technical University of Catalonia, Department of Geotechnical Engineering and Geosciences Barcelona, Spain

Antonio Gens obtained his Civil Engineering degree in 1972 at the Technical University of Madrid and a M.Sc. degree in Soil Mechanics at Imperial College, London, in 1973. After some years working in industry, he returned to Imperial College where he was awarded a Ph.D. in 1982. His supervisors were Professors A.W. Bishop and P.R. Vaughan. Since 1983 he has been a member of the Department of Geotechnical Engineering and Geosciences of the Technical University of Catalonia (UPC) in Barcelona (Spain) where he was appointed full Professor in 1987. He has been Head of the Department of Geotechnical Engineering and Geosciences, Vice-president of the Council of Departments of the Universityand member of the Governing Council of the University.

Antonio Gens has been involved in geotechnical research, consulting and teaching for over 35 years. His research interests have notably included unsaturated soils and the development of coupled multi-physics analysis. He has also made significant contributions in the areas of laboratory testing, field measurements and backanalysis, constitutive and numerical modelling,in situ characterization of soils, soil freezing and ground improvement, slope stability, tailing dams, tunnelling and nuclear waste disposal. He has authored or co-authored more than 300 papers published in refereed Journals and International Conferences, 9 book Chapters and has co-edited 9 books and Special Journal issues.He has delivered over 50 Keynote Lectures, General Reports and Specially Invited Lectures. He has been academic visitor at MIT (USA), University of Newcastle (Australia) and Imperial College (UK). He has also lectured widely to Academic and Professional Organizations in all five

Antonio Gens je diplomiral na Gradbeni fakulteti Tehniške univerze v Madridu leta 1972. Naziv magistra znanosti iz mehanike tal je pridobil na kolidţu Imperial v Londonu leta 1973. Po nekaj letih dela v inţenirski praksi se je vrnil na kolidţ Imperial, kje je 1982 končal doktorski študij. Njegova mentorja sta bila profesorja A. W. Bishop in P. R. Vaughan. Od leta 1983 je član Oddelka za geotehnično inţenirstvo in geoznanosti na Tehniški univerzi Katalonije (UPC) v Barceloni v Španiji, kjer je bil 1987 izvoljen za rednega profesorja. Bil je vodja oddelka za geotehnično inţenirstvo in geoznanosti, podpredsednik skupine oddelkov in član nadzornega sveta Univerze.

Antonio Gens je ţe 35 let aktiven v geotehničnem raziskovanju, svetovanju in poučevanju. Njegov raziskovalni interes je bil usmerjen predvsem v nezasičena tla in razvoj povezanih analiz večih fizičnih pojavov. Pripravil je pomembne prispevke o laboratorijskem testiranju, terenskih meritvah in povratnih analizah ter o konstitutivnem in numeričnem modeliranju, in-situ karakterizaciji tal, zmrzovanju tal, izboljšanju tal, stabilnosti breţin, zemeljskih pregradah, gradnji predorov in shranjevanju jedrskih odpadkov. Kot avtor ali soavtor je nastopil v več kot 300 člankih v geotehničnih časopisih in prispevkih na mednarodnih konferencah ter v devetih knjigah. Je sourednik devetih knjig in posebnih izdaj časopisov. Pripravil je več kot 50 uvodnih predavanj, generalnih poročil in posebnih vabljenih predavanj. Na specializaciji je bil na MIT v ZDA, na Univerzi v Newcastlu v Avstraliji in na kolidţu Imperial v Veliki Britaniji. Predaval je v številnih akademskih in strokovnih organizacijah na vseh celinah.

Kot konzultant je doma in v tujini sodeloval pri

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continents.

He has consulted widely at home and abroad in a wide variety of geotechnical projects involving tunnelling, deep excavations, harbour quays, breakwaters, embankment and concrete dams, tailing dams, power stations, airports, radioactive waste repositories, large-scale scientific facilities, deep and shallow foundations (onshore and offshore), ground improvement, slope stability, underpinning of structures and site investigations.He was a member of the Experts Committee for the investigation of the failure of Nicoll Highway (Singapore), he served as a court-appointed expert for the failure of the Aznalcóllar tailings dam and he has been a member of the UNESCO-sponsored Experts Committee to monitor the construction of a tunnel in the vicinity of the SagradaFamilia church and other UNESCO-listed buildings in Barcelona

Professor Gens’ contributions in the field of soil mechanics and geotechnical engineering have been recognized in a variety of ways. The International Association for Computer Methods and Advances in Geomechanics (IACMAG) awarded him the Chandra Desai Medal in 2001 and the Outstanding Contributions Awardin 2011. He has also been awarded the Case History Award by the American Rock Mechanics Association in 2006 and the R.M. Quigley Award (first prize) by the Canadian Geotechnical Society in 2009. In addition, he has been presented, by the UK’s Institution of Civil Engineers, the Telford Medal twice, in 1994 and in 2007, and the George Stephenson Medal also twice in 2008 and 2012. In 2005 he was made a member of the Real Academia de Doctores (Royal Academy of Doctors in Spain) and in 2007 he was presented the Ildefonso Cerdà Medal, the highest distinction of the Colegio de Ingenieros de Caminos (Institution of Civil Engineers) in Catalonia. In 1998 he delivered the British Geotechnical Society Touring Lecture, in 2000 the 8

th Prague

Geotechnical Lecture, in 2008 the Golders Associates Distinguished Lecture in Canada and in 2013 the Coulomb lecture in Paris. In 2007 he presented the 47th Rankine Lecture in London with the title: “Soil-environment interactions in Geotechnical Engineering”.In 2011, he was elected a Fellow of the UK’s Royal Academy of Engineering.

His involvement with the geotechnical

mnoţici različnih geotehničnih projektov, vključno z gradnjo predorov, globokimi izkopi, pristanišči, vodnimi zajetji, nasutimi in betonskimi pregradami, pogonskimi centralami, letališči, jalovišči in deponijami radioaktivnih odpadkov, znanstveno infrastrukturo večjega obsega, globokim in plitvim temeljenjem (na obali, morju ali oceanu), izboljšanjem zemljine, stabilnostjo breţin, sanacijo temeljenja in geološko-geomehanskimi raziskavami. Bil je član strokovne skupine za raziskovanje rušenja avtoceste Nicoll v Singapurju, delal je tudi kot ekspertni izvedenec za rušitev jeza jalovišča Aznalcóllar. Kot član ekspertne skupine, ki jo je sponzoriral UNESCO, je spremljal gradnjo tunela v bliţini cerkve Sagrada Familia in drugih UNESCOVIH zaščitenih stavb in objektov v Barceloni.

Prispevki profesorja Gensa na področju razvoja mehanike tal in geotehničnega inţenirstva so dobili več priznanj. Mednarodna zveza za računalniške metode in napredke v geomehaniki mu je podelila medaljo Chandra Desai leta 2001 in nagrado za izjemne zasluge leta 2011. Ameriška zveza za mehaniko kamnin (American Rock Mechanics Association) ga je nagradila za primer dobre geotehnične prakse leta 2006, Kanadsko geotehniško društvo (Canadian Geotechnical Society) pa mu je podelilo prvo nagrado R. M. Quigley leta 2009. V letih 1994 in 2007 je dobil Telfordovi medalji od Zbornice gradbenih inţenirjev (Institution of Civil Engineers) iz Velike Britanije, v letih 2008 in 2012 pa medalji Georga Stephensona. Leta 2005 je postal član Kraljevske doktorske akademije (Real Academia de Doctores) v Španiji, kjer je leta 2007 dobil še medaljo Ildefonso Cerdà, ki je najvišja nagrada, ki je dodeli Inţenirska Zbornica Katalonije (Colegio de Ingenieros de Caminos). Pripravil je naslednja vabljena predavanja: leta 1998 "British Geotechnical Society Touring Lecture", leta 2000 "8th Prague Geotechnical Lecture", leta 2008 "The Golders Associates Distinguished Lecture" v Kanadi in leta 2013 "Coulomb lecture" v Parizu. Leta 2007 je predstavil 47. Rankinovo predavanje v Londonu z naslovom: “Interakcija tal in okolja v geotehničnem inţenirstvu”. Leta 2011 je bil izbran za člana Kraljevske inţenirske akademije v Veliki Britaniji (UK’s Royal Academy of Engineering).

Sodelovanje prof. Gensa v geotehničnih profesionalnih zdruţenjih je posebnega

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community has been particularly close. He is or has been a member of 7 ISSMGE’s Technical Committees and he is currently a core member of Technical Committees TC-106 (Unsaturated soils) and TC-215 (Environmental Geotechnics). He has recently been elected Vice-President for Europe of the ISSMGE for the period 2013-2017.

pomena. Bil je član sedmih različnih tehničnih komitejev Mednarodnega društva za mehaniko tal in geotehnično inţenirstvo (ISSMGE) in je trenutno oţji član Tehničnega komiteja TC-106 (Nezasičena tla) in TC-215 (Okoljevarstvena geotehnika). Pred kratkim je bil izbran za podpredsednika Mednarodnega društva za mehaniko tal in geotehnično inţenirstvo za Evropo (ISSMGE) za obdobje 2013-2017.

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Antonio Gens The development of unsaturated soil mechanics

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Antonio GENS

Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia, Barcelona, Spain

e-naslov: [email protected]

THE DEVELOPMENT OF UNSATURATED SOIL MECHANICS

ABSTRACT: The paper presents a summary overview of some significant milestones in the development of Unsaturated Soil Mechanics. The concepts of potential and suction are first reviewed with a special focus on matrix suction, the most important component concerning mechanical behaviour of soils. The alternatives for selecting the relevant constitutive variables are then discussed in the context of the work rate equation. The formulation of elastoplastic models for an unsaturated soil is then addressed by describing three different models corresponding to different periods of development and to increasing levels of complexity. Finally, the relevance of those developments is illustrated by the reference to the two examples of the application involving the foundation collapse due to the wetting and the non-isothermal behaviour of a compacted clay barrier for nuclear waste isolation.

POVZETEK: V članku je predstavljen kratek pregled nekaterih pomembnih mejnikov v razvoju mehanike nezasičenih tal. Najprej je predstavljen pregled konceptov potenciala in sukcije, s posebnim poudarkom na matrični sukciji, ki je najpomembnejša komponenta mehanskega obnašanja tal. Sledi predstavitev moţnosti izbire relevantnih konstitutivnih spremenljivk v kontekstu enačbe prirasta dela. Nato so prikazane formulacije elastoplastičnih modelov za nezasičene zemljine. Opisani so trije različni modeli glede na različna obdobja razvoja in naraščanje kompleksnosti. Na koncu je ponazorjena pomembnost razvoja predstavljenih modelov z njihovo uporabo na dveh praktičnih primerih. Prvi zadeva rušenje temelja zaradi vlaţenja temeljnih tal in drugi neizotermalno obnašanje kompaktirane glinene bariere za izolacijo radioaktivnih odpadkov.


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INTRODUCTION

Unsaturated Soil Mechanics has now a rather long but uneven development history. An important step was taken when the key role of suction on the mechanical behaviour of unsaturated soils was explicitly recognized (e.g. Croney, 1952). Subsequently, there was widespread experimental work (e.g. Bishop et al., 1960; Bishop and Blight, 1963). Laboratory results were generally interpreted in terms of a proposed effective stresses for unsaturated soils (Bishop, 1959). However, it was soon noted that such an effective stress was unable to characterize the full range of unsaturated soil behaviour (Jennings and Burland, 1962). Perhaps as a consequence of this there was a relative pause in the basic study of the mechanical behaviour of this type of soils. During that period, unsaturated soils often were lumped together with other materials in soil categories such as “difficult soils”, “regional soils” or “special soils” or other similar groupings.

To change this situation, it was necessary to notice that any soil could be unsaturated and, therefore, there could be no reason why a fundamental approach, already successful in the case of saturated soils, could not be applied also to this type of materials. As a matter of fact, there is nothing special in an unsaturated soil apart from the fact that some part of the pore space is occupied by air (or other non-wetting fluids). Instead of considering unsaturated soils as a separate class of materials, there should be an unbroken continuity with the well-established understanding of saturated soil behaviour. Important steps in this direction were given by the separate consideration of two stress variables in the definition of state surfaces (Matyas and Radhakrisna, 1968), an idea already anticipated in Coleman (1962) and Bishop and Blight (1963). The use of state surfaces and the theoretical and experimental justification of using two independent stress variables were further reinforced by the work of Fredlund and Morgenstern (1977) and Fredlund & Rahardjo (1993). Further evidence on relevant stress variables has been provided by Tarantino et al. (2000).

Research on unsaturated soils has intensified in the last decades resulting in a large amount of theoretical developments, laboratory investigations, methods of suction control and measurement, and, to a lesser extent, field applications. A fundamental tool to organize the observations obtained from this systematic research is the development of constitutive laws capable of reproducing, in a consistent manner, the most important features of the mechanical behaviour of unsaturated soils. Overcoming the limitations of the state surface approach, elastoplasticity has proved a very successful framework for developing constitutive laws appropriate for unsaturated soils although other approaches such as hypoplasticity have also been successfully attempted (Masin & Khalili, 2008). Those laws can be considered a key component of general theoretical coupled formulations that include mechanical deformation, gas flow, liquid flow and, often, thermal aspects (e.g. Olivella et al., 1994, Gawin et al., 1995, Thomas and He, 1995, Khalili and Loret, 2001, Gens 2010a).

In this paper the concept and role of suction, a key variable in Unsaturated Soil Mechanics is discussed followed by a summary review of the, on occasion, controversial issue of what are the relevant variables for the constitutive description of the behaviour of unsaturated soils. To provide an overview of the pattern of development of elastoplastic constitutive formulations, three different models are briefly presented spanning from the early BBM model to some recent developments that include microstructural variables and hydromechanical coupling. Finally two examples of application to engineering problems are introduced, again covering a wide range of complexity of phenomena.

SUCTION

Water potential

The concept of suction is absolutely the key for the adequate understanding of the generalized behaviour of soils. A good understanding of unsaturated soil behaviour requires therefore a good physical grounding for the new variable. It is probably significant that the way in which suction should be viewed and the manner in which it is introduced in the stress variables used in constitutive models has received increasing attention in recent times (Li 2003, Lu 2008, Baker and Frydman 2009, Scholtès et al. 2009, Gens 2010a, 2010b).


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Suction is closely related to water potential that can be defined as the amount of work that must be done per unit mass of pure water in order to transport reversibly and isothermally an infinitesimal quantity of water from a reservoir of pure water at a specified elevation and gas pressure to the soil point under consideration (units: L

2T

-2). Although only total water potential matters concerning water flow, it has

proved useful, especially when considering mechanical behaviour, to distinguish between different components of potential (e.g. Aitchison et al., 1965):

ompg (1)

where Ψg is the gravitational potential, Ψp is the gas pressure potential, Ψm is the matric potential and Ψo is the osmotic potential. Only the internal components of the potential, matric and osmotic, have noticeable effects on unsaturated soil mechanical behaviour. It should be noted that the existence of osmotic potential requires the presence of a semi-permeable membrane, a string limitation in field situations. It should be noted that matric potential is equivalent to Gibb’s (free) energy (under constant pressure and temperature) and, using an adequate definition, it is also equivalent to the chemical potential of water.

Expressing potential per unit volume (and changing the sign), matric, s, and osmotic, so, suctions are obtained, they have units of pressure (ML

-1T

-2). Total suction is the sum of the two components: st =s +

so; total suction is related to relative humidity through Kelvin’s law:

00

/ / expw w t wv v g g

w

s MRH p p

RT

(2)

where pv is the partial vapour pressure, (pv)0 is the equilibrium vapour pressure at the same temperature

and zero suction, Θgw is the vapour concentration, (Θg

w)0 is the reference vapour concentration at the

same temperature, Mw is the molecular mass of water (18.016 kg/kmol), R is the universal gas constant (8.314 J/mol K), and T the absolute temperature in degrees Kelvin. It should be noted that relative humidity is in fact equivalent to the definition of water activity. The rest of this section is focused on matric suction, the most important suction component concerning the mechanical behaviours of soil. Osmotic suction is only relevant in the case of some very active clays, in which a degree of semi-permeable membrane is naturally established.

Matric suction

Matric suction is often linked to capillary phenomena. It is well known that, in some saturation ranges, water forms menisci with curved surfaces in the contact zone between particles, akin to those that appear during water ascension inside capillary tubes. Surface tension ensures that the liquid pressure inside the meniscus is lower than the gas pressure outside. Matric suction, s, is then defined as:

a ws p p (3)

where pa is the air pressure and pw is the water pressure. Strictly speaking, matric suction should be defined in terms of gas and liquid pressure but, in this paper, gas and liquid are equated to air and water, respectively.

The meniscus suction gives rise to the interparticle forces that have a stabilising effect on soil structure. The force generated by a liquid meniscus between two spheres of equal radii in contact was calculated by Fisher (1926) and it is plotted in Figure 1 as a function of meniscus angle. The calculation considers the effect of the negative pressure over the area of the meniscus and the effect of the surface tension in the gas/liquid interface. The computed force depends on the value of the surface tension and on the diameter of the spheres.


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Figure 1. Variation of interparticle force with meniscus wetting angle Fisher (1926). s is the surface tension and R the radii of the spheres.

The actual fabric of a soil, however, is very different from a bundle of capillary tubes, especially if fine-grained soils are considered. The capillary model should thus be used prudently and not beyond the range of plausibility. Accordingly, it is necessary to extend the concept of matric suction to incorporate a series of additional phenomena other than capillary forces (Nitao and Bear, 1996). In this context, Philip (1977) developed a unitary approach that included, in addition to capillary condensation, a term representing adsorption phenomena in porous media. The adsorptive component is a product of a number of both long term and short term forces (e.g. electrostatic Coulomb forces, van der Waals forces, hydration forces, hydrogen bonding, charged surface-dipole attraction) that control the molecular interactions of water near a solid surface (Derjaguin et al., 1987). The augmented potential now reads:

( ) ( )m cC A h

(4)

where C is the capillary component of the matric potential and A the adsorptive one; κc is the mean curvature of the interface and h is the thickness of the film of adsorbed water. The capillary component can be derived from Laplace’s equation and, for the adsorptive component Phillip (1977) proposed an empirical expression, assuming that it depended on the film thickness only,

2( ) ; ( )s c

w

RTC A h

h

(5)

in which s is the surface tension, w is the density of the water, ’ is a positive constant, R is the gas constant and T is the absolute temperature in K. This work has been updated and extended to a more general pore model (Figure 2) in Tuller et al. (1999).


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Figure 2. A sketch illustrating liquid-gas interfacial configurations during transition from adsorption (a,b) to

capillary dominated imbibition (c,d) in an idealized pore geometry (modified from Tuller et al., 1999).

It is likely that the capillary component and adsorptive component may have different effects on mechanical behaviour. It is difficult, however, to separate them in practice and the tendency is to lump the two effects together into a single measure of matric suction that is generally expressed as [3]. Therefore, negative pore pressures that sometimes may reach a very high magnitude should not be viewed as a conventional pressure of bulk water but as the expression of the potential of the water attached to the solid phase. At low degrees of saturation, this degree of attachment (potential) can be sometimes very large and, consequently, equivalent pressure values of hundreds of MPa are sometimes quoted. This fact should not cause undue difficulties if viewed in this context.

Indeed, the separation between adsorptive and capillary potential, though useful for conceptual understanding, should not be unduly overstressed. It is likely that the transition between the two regimes is gradual. After all, capillary phenomena arise from interaction between liquid, solid surface and interface that are only noticeable in the proximity of solid surfaces. There is surface tension but not significant capillary effects when water is inside a large vessel. So the development of increasing negative potentials should be viewed as a continuous process largely without jumps with the possible exception of the ordered layers of water that occur in the immediate vicinity of a clay particle surface.

STRESS VARIABLES

The stress variables required to describe the full mechanical behaviour of unsaturated soils and to formulate the corresponding constitutive laws have been a matter of some controversy in the development of Unsaturated Soil Mechanics. In principle, any relevant variable can be selected as constitutive variable but, for engineering purposes, constitutive laws are usually (but not always) defined in stress space, i.e. using scalar, vector or tensor variables that have units of stresses. However, as it will be indicated later, it would possible to adopt other types of variables (e.g. degree of saturation, water content) if they had a greater descriptive or predictive power.

In any case, there is general agreement that at least two constitutive variables are generally required to represent adequately the full range of unsaturated soil behaviour, i.e. including strength and deformation. As stated by Jommi (2000): in fact, no single stress variable has ever been found which, substituted for effective stress, allows a description of all the aspects of the mechanical behaviour of a given soil in the unsaturated range. It is convenient to refer to them as the first and second constitutive variables (FCV and SCV). Intentionally, they are not called constitutive stress variables to remain open to the possibility that variable types other than stresses are adopted. Usually, the FCV tries to account for the overall stress state of the soil whereas the SCV tends to address mainly the effects of suction changes. Thus, if the constitutive variables chosen were net stresses and suction (as in the Barcelona Basic Model, BBM)), the FCV (net stress) responds to loading changes whereas the SCV (suction) deals with environmental actions related to suction variations. The same comment can be made if, as in the model of Gallipoli et al. (2003) described below, average skeleton stress and a bonding parameter are used. In any case, in the


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majority of constitutive models developed, those two constitutive variables normally adopt the following general form:

FCV: 1 ,ij a ij r ijp s S ; SCV: 2 , rs S (6)

where is total stress, Sr is the degree of saturation and ij is Kronecker’s delta.

The FCV is expressed as a modified stress tensor; in contrast, for the SCV, aiming to represent the additional effects of suction, a scalar is usually adopted. However, one of the main effects of suction is the generation of stabilising interparticle forces. It would be reasonable to expect, as pointed out by Li (2003) and Scholtès et al. (2009), that the SCV should incorporate some kind of fabric measure of the soil; the variable would then no longer be a scalar, more likely a tensor. Indeed, if suction effects have a tensorial form, it is questionable that an averaged pressure in the fluid phase is an adequate representation of the role of suction even when defining the FCV (Scholtès et al., 2009). In any case, the use of a tensorial form for the effects of suction is difficult to accomplish due to the complexity of determining the soil fabric and its variation as soil deforms. Accordingly, scalar definitions of the SCV have been dominant hitherto. It can be argued that suction has a different physical nature than stress variables (Lu, 2008) and some objections have been put forward concerning the validity of combining stresses and suction into a single expression (Baker and Frydman, 2009). However, there does not seem to be any a priori reason against a constitutive variable combining different types of variables if an adequate description of soil behaviour is thereby achieved.

A constitutive variable that has been frequently used and for which, sometimes, a special status is claimed is written as:

( )ij ij a ij r a w ijp S p p (7)

As stated before, this variable has been called Bishop’s stress (Bolzon et al., 1996, Tamagnini, 2004) and, also, average skeleton stress (Jommi, 2000, Gallipoli et al., 2003, Wheeler et al., 2003). Other terms that have also been used are Bishop’s effective stress, average stress (Sheng et al., 2004) and generalized effective stress (Laloui and Nuth, 2009). It should be noted, however, that Bishop (1959) used a scaling factor χ(Sl) and not Sl in his expression for effective stress. In the context of unsaturated

soils, equation (7) was probably first used by Schrefler (1984).

An important contribution to this issue was made by Houlsby (1997) who derived, under reasonably general conditions, the rate of work input per unit volume of unsaturated soil (W):

(1 ) / ( ) (1 )a r a a a w r ij r w w a ij ijW p n S p p nS S p S p (8)

where pa is the air density, n is porosity and ij are the strains. In this expression, the work performed by the flow of fluids is not included and it is also assumed that, on average, the contractile skin move with the soil skeleton, perhaps the most questionable assumption. If the air compressibility term is neglected, the work rate input is:

( ) (1 )a w w ij r w r a ij ijW p p nS S p S p (9)

It is now possible to define sets of stress and strain variables that are conjugate to each other. An immediate choice is:

(1 )

( )

ijij r w r a ij

ra w

S p S p

Sn p p

(10)


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where Bishop’s stress is conjugate to strains and a modified suction is conjugate to degree of saturation. However, expression (10) can also be rearranged to give the following conjugates stresses and strains:

( ) ( )

ij a ij ij

a w w r r v

p

p p nS S

(11)

Now, net stress is conjugate to strains and suction is conjugate to the volumetric water content, w = nSr. From this perspective, therefore, the work input expression does not provide a criterion to select a set of constitutive variables in preference to the other. Indeed, it may be stated that, at the present state of development, selection of constitutive variables should be a matter of convenience, depending on the intended application.

It should be immediately added however, that the choice of constitutive variables has an influence on the nature of the constitutive law developed. To examine this issue, it was proved to be convenient to classify the constitutive laws according to the choice of the FCV (Gens, 1996, 2010a); a similar classification has been explored in Nuth and Laloui (2008).

Class I models adopt a FCV with 1=0, i.e. net stresses are used. In general, for those models:

- the stress paths of conventional tests and typical loading situations are easily represented, this type of models are well suited to illustrate conceptual constitutive developments

- the constitutive variable is independent of the material state - if suction is the SCV, the two constitutive variables are independent (except for the gas

pressure). However, the conjugate strain variables are not independent (equation (11)). - continuity of constitutive variables and behaviour is not ensured at the saturated/unsaturated

transition. Some procedures have been proposed to achieve continuity in numerical analysis (Vaunat et al., 1997).

- the hydraulic component of the model is independent of the mechanical one, specific mechanical effects of degree of saturation are not included

- an independent function is required to model the increase of shear strength with suction.

It should be pointed out that a recent constitutive model of this Class, the SFG model (Sheng et al., 2008), is able to overcome several of the shortcomings listed above.

In class II models, 1(s) is a function of suction but does not include degree of saturation. In that case,

- representation of stress paths of conventional tests and typical loading situations is not straightforward

- the constitutive variable is still independent of the material state if no additional parameters are included

- the two constitutive variables are linked - continuity of variables and behaviour is not automatically assured at the saturated/unsaturated

transition - the hydraulic component of the model is independent of the mechanical one, the specific

mechanical effects of degree of saturation are not included - the increase of strength with suction is a result of the constitutive variable definition - strength and elastic behaviour can be unified with an adequate selection of the FCV (Khalili et

al., 2004).

Finally, in class III models, 1(s, Sr) is a function of both suction and degree of saturation. Constitutive models using Bishop’s stress as the FCV belong to this class of models.

- representation of stress paths of conventional tests and typical loading situations is complex and sometimes impossible if degree of saturation data is not available

- the constitutive variable depends on degree of saturation and, therefore, on the material state


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- the two constitutive variables are linked - continuity is automatically ensured at the saturated/unsaturated transition - the hydraulic component of the model is closely coupled to the mechanical component and the

specific mechanical effects of degree of saturation can be included. Hydraulic and hysteresis effects can be incorporated in a natural way but the success of the mechanical model is dependent on having an adequate hydraulic relationship. In particular, care should be taken that the product Srs that enters the definition of Bishop’s stress should not take unrealistic large values at low degrees of saturation (Loret and Khalili, 2002). In that range, suctions can have extremely large magnitudes and the product sSl may reach absurdly high values that disrupt any model prediction.

- the increase of strength with suction is a result of the constitutive variable definition - strength and elastic behaviour can be unified with an adequate selection of the FCV.

A similar classification could be performed concerning the SCV but, in this case, the variation is significantly less as most constitutive models use suction. Wheeler et al. (2003) adopted modified suction defined as ns where n is porosity and s suction, consistent with equation (10). Some recent models have directly selected degree of saturation, Sr, or effective degree of saturation, Se (Zhou et al., 2012a, 2012b, Alonso et al. 2013). Effective degree of saturation is defined as (Romero & Vaunat 2000, Tarantino & Tombolato 2005):

0

res

r re res

r r

S SS

S S

(12)

where Sr0 is the degree of saturation at zero suction (generally taken as 1) and Sr

res is the residual degree

of saturation.

ELASTOPLASTIC CONSTITUTIVE MODELS

Starting in the sixties, behaviour of unsaturated soils was frequently described by state or constitutive surfaces that established relationships between two stress variables (generally net stress and suction) and volume change. Sometimes, state surfaces for other variables such as degree of saturation were also derived (Matyas and Radhakrishna, 1968, Aitchison and Martin, 1973, Fredlund and Morgenstern, 1976, Lloret and Alonso, 1985). State surfaces, however, were limited when trying to model the general behaviour of unsaturated soils. Issues of irreversibility, yield and stress path dependency are not easily accommodated in this type of framework. Also, state surfaces usually addressed partial aspects of behaviour without establishing consistent links between different features of behaviour.

However, there were clear advantages in trying to develop constitutive models within the general framework of elastoplasticity. One of the objectives of this development was to try to place Unsaturated Soil Mechanics closer to the mainstream of Saturated Soil Mechanics that, for constitutive descriptions of soil behaviour, was largely dominated by the concepts underlying Critical State Soil Mechanics and elastoplasticity. The aim was to construct a complete and consistent framework capable of providing predictions under all circumstances. The development of such a framework should offer some significant benefits:

- to provide a consistent framework for an integrated understanding of the various features of unsaturated soil behaviour

- to assist in the identification of basic parameters and reference states governing soil behaviour - to furnish bases for further development of more complex constitutive laws and to enable the

performance of numerical analysis for application to engineering problems

Thus, the formulation of consistent and complete constitutive models within the framework of elastoplasticity constitutes one of the most significant developments in Unsaturated Soil Mechanics. Those developments are so numerous (Gens, 2010a) that it is futile to attempt a useful review in a limited space. Instead, in this section, three examples of constitutive laws are summarily described. They have


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been chosen to illustrate the development of increasingly complex models accounting for additional variables and phenomena. They display, in a necessarily partial way, the range of options, variables and generality available for modelling unsaturated soils.

The models presented are:

i) The Barcelona Basic Model (BBM). This is probably the earliest elastoplastic model for unsaturated soils using net stresses and suction as constitutive variables. It is arguably the most widely used in analysis. It represents a very basic model at the early stage of development of the subject but many of the concepts underlying the model have been adopted by subsequent developments.

ii) The model of Gallipoli et al. (2003) that illustrates the increasing tendency to incorporate microstructural considerations into the model. It uses Bishop’s stress and a microstructurally-informed parameter as constitutive variables.

iii) The model of Zhou et al. (2012a, 2012b) that incorporates a key feature of unsaturated soil behaviour: the coupling between mechanical and hydraulic behaviour. In addition to adopt a modified Bishop’s stress as the FCV, it uses the effective degree of saturation (defined above) as SCV.

The Barcelona Basic Model (BBM)

The BBM was fully described in Alonso et al. (1990), and, in a more summary form, in Gens et al. (1989). Indeed, the use of the concepts of the model to represent qualitatively unsaturated soil behaviour was first introduced in Alonso et al. (1987) before the mathematical formulation was formally presented.

A starting point for the development of the model is depicted schematically in Figure 3a where the idealized results of consolidation tests under constant suction are shown. It can be observed that the yield point is located beyond the saturated consolidation line (corresponding to the line s=0) and reaches further to the right as suction increases. The stabilising forces due to suction allow the material to sustain a higher applied stress at a given void ratio. The higher the suction, the higher is the stress that can be sustained before yield. The various yield points are plotted in isotropic p-s (mean net stress-suction) space (Figure 3b). Joining them together, a yield curve, denoted as LC (Loading-Collapse), results. The name refers to the fact that this yield curve controls irreversible behaviour due to either loading or to collapse (actually compression) due to increasing soil saturation (Figure 4).

S1

S = 0 Yield

S = 0

S2

S3

MEAN NET STRESS, p

VO

ID R

AT

IO, e

S1

S = 0 Yield

S = 0

S2

S3

S1

S = 0 Yield

S = 0

S2

S3

MEAN NET STRESS, p

VO

ID R

AT

IO, e

MEAN NET STRESS, p

VO

ID R

AT

IO, e

MEAN NET STRESS, p

VO

ID R

AT

IO, e

MEAN NET STRESS, p

SU

CT

ION

, s

S1

S3

S2

Yield curve

LCElastic

domain

MEAN NET STRESS, p

SU

CT

ION

, s

MEAN NET STRESS, p

SU

CT

ION

, s

S1

S3

S2

Yield curve

LCElastic

domain

S1

S3

S2

S1

S3

S2

Yield curve

LCElastic

domain

Figure 3. a) Idealised scheme of consolidation lines at different suction values. b) Definition of the LC (Loading-Collapse) yield curve in the isotropic plane).


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The displacement of the yield curve is associated with plastic volumetric deformations that are equal to the plastic volumetric strains that would be obtained from a consolidation test on a saturated sample (s = 0) going from (po*)1 to (po*)2. Assuming a hardening law of the Cam-clay type, the plastic volumetric

strain, dvp, can be expressed as:

0

0

(0)

1

p

v

dpd

e p

(13)

where is the slope of the saturated virgin consolidation line, is the slope of the unloading/reloading line (assumed independent of suction), e is the void ratio and po* is the hardening parameter, the intersection of the LC curve with the s=0 line in the isotropic plane.

Loading

Collapse

Loading

Collapse

Yield curve

LC1

Elastic

domain

Yield curve

LC2

C

LS1

*

1op

1

op

*

2op MEAN NET STRESS, p

SU

CT

ION

, s

Yield curve

LC1

Elastic

domain

Yield curve

LC2

Yield curve

LC1

Yield curve

LC1

Elastic

domain

Yield curve

LC2

Yield curve

LC2

C

L

C

LS1

*

1op

1

op

*

2op

S1

*

1op

1

op

*

2op MEAN NET STRESS, p

SU

CT

ION

, s

MEAN NET STRESS, p

SU

CT

ION

, s

Figure 4. Displacement of the LC yield surface on loading at constant suction (path L) and wetting at constant applied stress (path C)

The extension of the model to include deviatoric stresses and strains is illustrated in Figure 5a. In the q-p space (q is the triaxial deviatoric stress and p the mean stress), the curve indicated by the label s = 0 represents the yield surface for saturated conditions. To emphasize the compatibility with saturated models, the ellipse of the Modified Cam-clay (MCC) model was chosen but other yield surface shapes could have been equally used. As suction increases, the ellipse size also increases. On the right hand side, po increases in accordance with the shape of the LC yield curve. On the left hand side, ps increases linearly with suction. The implicit assumption here is that the friction angle (or slope of the critical state line) is not affected by suction but the apparent cohesion increases proportionally to suction. The formulation is completed with a flow rule that may be chosen to be associated or non-associated. The model is summarised in Figure 5b that shows a three-dimensional view of the yield surface in p-q-s space. It is a graphic way to illustrate the generalization of the saturated framework via the inclusion of an additional variable, suction. The BBM model defined in this way collapses into the MCC model when suction becomes zero and saturated conditions are reached.

With these simple (even simplistic) assumptions the model can describe a large number of typical features of the mechanical behaviour of unsaturated soils in a natural unforced way (Alonso et al., 1987, 1990). Some examples are: the variation of wetting-induced swelling or collapse strains depending on the magnitude of applied stresses, the reversal of volumetric strains observed sometimes during wetting-induced collapse, the increase of shear strength with suction, the stress path independency associated with wetting paths and the opposite when the stress path involves drying or the apparent increase of preconsolidation stress with suction. Simplicity also implies that only a limited number of additional parameters are required.


15

*

op op

q

CSL (s)

CSL (s=0)

s

s=0

MEAN NET STRESS, p

sp*

op op

q

CSL (s)

CSL (s=0)

s

s=0*

op op

q

CSL (s)

CSL (s=0)

s

s=0

MEAN NET STRESS, p

sp

a) b)

Figure 5. a) Yield surfaces in the q-p plane for saturated (s=0) and unsaturated conditions. b) Three-dimensional view of the yield surface in (p, q, s) stress space.

Other models were quickly developed that, while keeping the same core of basic assumptions, sought to improve some of the shortcomings of the original BBM. Thus Josa et al. (1992) used non linear relationships for the variation of void ratio with lnp so that collapse strains did not increase indefinitely but they went through a maximum before reducing to zero at high stresses. Wheeler and Sivakumar (1995) used model functions more closely based on experimental results and Cui et al. (1995) adopted a saturated yield function typical of anisotropically consolidated soils.

A microstructure-informed constitutive model

This model (Gallipoli et al. 2003) attempts to incorporate in a more direct manner the effect of suction on behaviour using a very idealized picture of the microstructure of a granular soil. There are two ways in which suction affects the behaviour of an unsaturated soil:

- by changing the average skeleton stress through changes in the average fluid pressure acting in the soil pores

- by providing an additional bonding force at particle contacts, often ascribed to capillary phenomena in the water menisci.

For a given suction, two regions can be envisaged in an unsaturated soil: the first one, corresponding to the smaller pores, may be totally saturated, whereas the second one, corresponding to the larger pores, contains gas and liquid in the form of interparticle menisci (Figure 6). The relative proportions of each region vary as suction changes. An important insight (Karube and Kato, 1994, Wheeler and Karube, 1996, Wheeler et al., 2003) is to realize that a variation in suction affects the average skeleton stress due to pore pressure changes in the saturated regions whereas the modifications of the capillary-induced bonding occur in the menisci/gas region. The selection of constitutive variables for the model attempts to introduce explicitly those physical insights in an approximate manner.

A first constitutive variable is Bishop’s stress (Equation (7)) that aims to represent the average skeleton stress. The second constitutive variable must be related to the bonding effect of the capillary forces at interparticle contacts. These interparticle forces provide a stabilising effect with respect to relative movement and slippage between particles. Features of unsaturated soil behaviour must be the result of the bonding and de-bonding phenomena associated with the formation and disappearance of water menisci of interparticle contacts. Those bonding/debonding effects cannot be easily accounted for by using the average skeleton stress alone and a suitable constitutive variable, related to the bonding

phenomena, must be selected for this purpose.


16

The overall magnitude of the bonding force will be the product of i) the number of water menisci per unit volume of the solid fraction and ii) the mean intensity of the interparticle forces created by the menisci. As a first approximation, it is assumed that the number of menisci is proportional to the percentage of volume occupied by the gas/menisci region, roughly 1- Sl. Thus the bonding-related stress variable is:

(1 ) ( )lS f s (14)

where f(s) is the mean interparticle force that is presumed to be dependent on suction. The function f(s) is

evaluated using Fisher’s (1926) solution, plotted in Figure 1.

Meniscus liquid

Gas Bulk liquid

Figure 6. Schematic view of an unsaturated soil showing saturated regions and menisci/gas regions (Wheeler et al., 2003).

Interparticle forces provide the bonding effect that underlies the capacity of unsaturated soils to form looser arrangements that would not be stable if the material were saturated. It is reasonable to expect

that, if the variable is representative of the interparticle forces, it should be directly related to the additional void ratio that an unsaturated soil is capable of sustaining. To test this hypothesis, results from isotropic compression tests performed under several suctions by Sivakumar (1993) and Sharma (1998) are considered. Sivakumar (1993) tested compacted Speswhite kaolin and Sharma (1998) a compacted mixture of bentonite and kaolin.

Figure 7 shows that the experimental results can be unified by considering that the ratio e/es depends on

the bonding variable through:

1 1 exp( )s

ea b

e (15)

where e is the void ratio at a specific applied average skeleton (Bishop’s) stress and suction and es is the void ratio in the saturated virgin consolidation line at the same applied average skeleton stress; a and b

are fitting parameters.


17

The ratio e/es, therefore, represents the excess void ratio sustained by the unsaturated soil and it is a

simple function of the bonding variable, In Gallipoli et al. (2003), it is further shown that the same function of the bonding variable applies also to the critical state of kaolin (Wheeler and Sivakumar 2000) and of other soils such as a lateritic gravel (Toll, 1990). It is perhaps somewhat unexpected that a framework based on a capillary model is able to represent so successfully the behaviour of soils containing a significant proportion of clayey particles. However, the approach appears to have a wide range of applicability; subsequent results presented by Gallipoli et al. (2008) demonstrate that the same approach applies successfully to the consolidation and critical state data of Jossigny silt (Cui, 1993) and of Barcelona clayey silt (Barrera, 2002).

b) d)

c) a)

Figure 7. a) Normal isotropic compression lines at constant suction in terms of average skeleton stress (Sharma, 1998). b) Excess void ratio over the stable saturated void ratio (e/es) as a function of the

bonding variable (data by Sharma, 1998). c) Normal isotropic compression lines at constant suction in terms of average skeleton stress (Sivakumar, 1993). d) Excess void ratio over the stable saturated void

ratio (e/es) as a function of the bonding variable (data by Sivakumar, 1993). (Gallipoli et al. 2003).

Using those two stress variables, a constitutive model can be developed following the same steps as in the construction of the BBM (Gallipoli et al., 2003). The resulting model has a somewhat higher degree of complexity but it is capable of reproducing rather involved features of behaviour using a single yield surface. An example is provided in Figure 8 where the results of a test performed by Sharma (1998)


18

involving two cycles of wetting and drying are plotted together with the computed results using the model. A good agreement can be noted, in spite of the rather intricate behaviour observed. It is interesting to note that the additional compression that is observed in both drying stages would not be obtained by the BBM model unless an additional yield surface with a complex hardening behaviour was specified. In the new model, the behaviour is readily reproduced employing a single LC yield surface and no additional assumptions.

Figure 8. Experimental results and model predictions for wetting-drying cycles of a specimen of compacted bentonite-kaolin mixture subjected to a constant isotropic net stress of 10 kPa (experimental data from Sharma, 1998).

A coupled Hydro-mechanical constitutive model

A conspicuous omission in the BBM and other early model formulations was the lack of a specific model to describe the variation of water content or degree of saturation due to changes of stresses and/or suction. If the mechanical model is defined in terms of net stresses, the consequences are limited as there is no direct coupling between hydraulic and mechanical model. Thus in the BBM, hydraulic behaviour was simply defined in terms of a state surface. However, when degree of saturation or other hydraulic parameters enter the definition of the constitutive model, the hydraulic-mechanical coupling must be carefully considered. In this context, Houslby’s expressions (10,11) linking mechanical and hydraulic variables, are especially relevant.

The issue of the hydraulic component of the constitutive model was first addressed by Wheeler (1996) and Dangla et al. (1997). A number of models coupling hydraulic and mechanical models have been developed (e.g. Vaunat et al. 2000, Wheeler et al. 2003, Sheng et al. 2004). In this section a model recently proposed by Zhou et al. (2012a, 2012b) is summarily described. Two aspects are explicitly considered: the hysteretic relationship between the effective degree of saturation and suction for an arbitrary drying/wetting path and the change in the degree of saturation due to the stress change under constant suctions. The retention curve of a soil, in terms of effective degree of saturation, Se (equation

(12)), is governed by the suction as well as the volumetric strain due to net stresses, v (Sheng and Zhou, 2011). The change of the degree of saturation can be expressed conceptually as follows:


19

ee e vd d d

SS s D

s

(16)

where D is a general function that defines the effect of volumetric strain on the effective degree of saturation that is caused by a net stress change. The first term on the right-hand side of equation (16) accounts for the change of the effective degree of saturation under a constant net stress. A standard laboratory water retention test can be used to calibrate this term directly. The second term on the right-hand side accounts for the change of the effective degree of saturation due to the net stress under a constant suction. A suction-controlled compression test can be used to determine this term. The popular van Genuchten expression (van Genuchten, 1980) is used to define the main drying curve:

dd

edd

1

nm

sS

a (17)

where da , dm and dn are three fitting parameters. The same expression but with different parameters is

used for the main wetting branch. In addition, a simple boundary scanning rule is adopted here to describe the non-linear scanning (wetting/drying) loops between the main wetting and drying branches. Similar approaches can be found in Li (2005) and Pedroso et al. (2008). Figure 9a shows the hysteresis loops inside the domain between the main drying and wetting curves that are derived from the model. Viaene et al. (1994) presented a wetting-drying test on a sand sample. As shown in Figure 9b the hydraulic model is capable to reproduce satisfactorily the observed behaviour.

a) b)

Figure 9. a) Main drying and wetting retention curves and computed hysteretic loops. b) Hysteresis of a sand sample: observations (Viaene et al. 1994) and computed results.

A second differential feature of this model is the adoption of the effective degree of saturation instead of suction as the SCV. For the FCV, effective degree of saturation is also incorporated in a modified form of Bishop’s stress:

eij ij ijS s (18)

The basic assumption is that In the space of lnv p , the normal compression lines (i.e., volume change

equation) for both saturated and unsaturated soil are assumed to take the following form:

e lnN S p (19)


20

where, p’ is the mean modified Bishop stress and N is the intercept of the normal compression lines with

the v-axis when ln p’=0. A suitable variation of with effective degree of saturation has been postulated (Zhou et al., 2012a). The volume change equation (19) can be rewritten in an incremental form as:

e

e ee

dd ln d

Spv S p S

p S

(20)

Equation (20) indicates that the volume change can consist of two parts: the first part is due to the stress change and the second part is due to the change of the effective degree of saturation. If Se is fixed, (such as for loading under the saturated or the driest condition), the second term on the right hand side of the above equation will degenerate to zero. In this case, equation (20) defines the normal compression lines without the bonding/debonding effect. It also indicates that the bonding effect due to drying can be weakened or eliminated by raising the effective degree of saturation. In other words, an unsaturated soil can be de-structured/debonded by soaking it in water, and a saturated soil can be structured/bonded by desaturation.

Equation (20) can be understood by inspecting Figure 10a. In the loading process under a constant suction, the second term of equation (20), which describes the debonding effect, is always larger than

zero since dSe is larger than or equal to zero. As shown in Figure 10b, although Se) is less than (the slope of the compression line under saturated conditions), the apparent slope of the compression curve for an initially unsaturated soil can still be larger than that for its saturated counterpart because of the debonding effect. As a consequence, the prediction of a typical drying loading process takes the desired form shown in Figure 10b.

a) b)

Figure 10. a) Volume change in a process involving a change of degree of saturation. b) Bonding-debonding effects during a drying-loading path.

The model can be readily extended to deviatoric stresses following the same procedure as in the development of the BBM. Therefore the hardening rule is expressed as:

ep

v lnS

pN

(21)

and the yield surfaces for triaxial stress states are depicted in Figure 11. It can be noted that the additional third axis now corresponds to effective degree of saturation, Se.


21

Figure 11. Yield surfaces under triaxial stress states.

As an example of the performance of the model, the experimental data of Cunningham et al. (2003) are examined. They presented the results of a series of suction-controlled isotropic compression tests under varying suctions on reconstituted silty clay that comprised a mix of 20% pure Speswhite kaolin, 10% London clay and 70% silica silt. The test data as well as the model simulations are shown in Figure 12a. The comparisons indicate that the predicted curves match the experimental data well. It is important to note that there is only one initial state (s = 0) used for all predictions. The initial states for other suction levels (s = 400kPa, 650kPa and 1000kPa, respectively) are calculated through the drying process from the initial state at zero suction. In addition, Cunningham et al. (2003) undertook a series of unconfined shearing tests in which the samples were dried on the bench to a variety of predetermined moisture contents, corresponding to suctions in the range of 270-1220kPa. The samples were then sheared under constant moisture content (undrained) conditions to failure. Throughout shearing, suction probes were used to monitor the suction changes within the specimens. Figure 12b compares the test data with the model simulations, where it can be seen that the proposed model captures the key features of the observations of the unsaturated reconstituted soil behaviour under undrained loading.

a) b)

Figure 12. Comparison of model computations and experimental observations from Cunningham et al. (2003). a) Isotropic compression tests. b) Unconfined undrained compression tests.

EXAMPLES OF APPLICATION

Two quite diverse examples of applications are presented here to demonstrate the performance of the developments reviewed above in engineering problems. The first one involves the collapse of foundations due to saturation and the second one concerns the behaviour of a compacted bentonite barrier under non-isothermal conditions.


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Foundation collapse due to wetting

Pereira Barreto town is located on the right hand bank of the Tietê River, near the confluence with the Paraná River in the North West of the São Paulo State in Brazil. A 50m high dam was constructed just downstream of the town causing a general rise of the groundwater level in the area. Because it was known that some of the buildings were founded in potentially collapsible soil, a comprehensive site investigation was performed including soil borings, in situ tests and laboratory tests. A monitoring system was also installed before the reservoir level rise; it is one of the rare occasions in which the development of collapse settlements has been measured in full.

A typical soil profile is presented in Figure 13; part of the town is founded on a collapsible colluvium layer that overlies a residual soil. The colluvium is a lateritic fine sandy soil without active clay minerals. A thin gravel layer separates colluvium and residual soil. The initial degree of saturation of the materials is quite low, in the region 30%-40%, consistent with the climate conditions of the site.

Figure 13. Soil profile at Pereira Barreto town.


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Figure 14. Building PB-1, Pereira Barreto town: Soil profile and water level elevation, foundation and observed and computed foundation settlements.

As expected, the ground water table rise brought about collapse settlements that affected some buildings causing various degrees of damage. There were about 20 sites monitored, here only the measurements of two buildings, PB-1 and PB-3, are discussed; they are representative of the range of foundation behaviour encountered. Figure 14 shows the settlements of two points of the foundation of building PB-1; settlements of more than 100 mm were measured. The foundation was just over 1 m deep in the colluvium. In the same Figure, the evolution of the water level as measured in a nearby observation point is also plotted; it can be noted that it reached the lower part of the colluvium layer. The same information on building PB-3 is provided in Figure 15. Foundation is slightly shallower and settlements were significantly smaller, around 20 mm. In this case, the water level reached only the gravel layer.


24

Figure 15. Building PB-1, Pereira Barreto town: Soil profile and water level elevation, foundation and observed and computed foundation settlements.

Block samples of the colluvium were obtained from the site and tested in suction controlled oedometers. Two types of tests were run, loading tests under constant suction (Figure 16a) and wetting tests from two suction values (200 kPa and 60 kPa) performed under several applied stresses (Figure 16b). The results obtained correspond to a typical behaviour of an unsaturated soil. The consolidation lines (Figure 16a) plot further to the right as suction increases, the saturated consolidation line constitutes the lower behaviour bound. In Figure 16b, the results corresponding to the collapse wetting tests performed from a suction of 60 kPa are shown. After collapse, the void ratio – stress state of the soil lies close to the saturated consolidation line. Similar results but higher collapse values were obtained when flooding the samples at a suction of 200 kPa. The test results have been modelled using the BBM. The computed collapse strains corresponding to wetting tests from 60 kPa and 200 kPa suction values are compared with the experimental results in Figure 17. It can be noted that the prediction is good at low stresses but collapse strains are overestimated at higher stress values. This is a consequence of the inability of the standard BBM to reproduce a maximum of collapse. However, since the agreement was adequate for the relevant stress level range, the BBM, calibrated in this manner, was used to perform the calculations. The challenge is to reproduce the range of observed field behaviour using a single set of parameters obtained from the laboratory testing programme.


25

a) b)

Figure 16. a) Stress paths and results of oedometer loading tests under constant suction. b) Stress paths and results of oedometer wetting collapse tests.

a) b)

Figure 17. Computed and observed collapse strains. a) Wetting from 60 kPa suction. b) Wetting from 200 kPa suction.

The settlements computed in this way have been added to Figures 14 and 15. The agreement is quite satisfactory. In the case of building PB-1 (Figure 14), large settlements are predicted, that only start when the water level is already quite high. It is also interesting to note that a rather large reactivation of settlement is computed when a rather modest increase of water level occurs, just as observed. The more muted settlement response of the foundation of building PB-3 (Figure 15) is also successfully reproduced. The different behaviour is due to the different foundation arrangements and, especially, to differences in soil profile. In the latter case, the water level does not reach the collapsible colluvium layer, but the variation of suction due to the presence of a higher water table is sufficient to cause some settlements,


26

although of smaller magnitude. The capability of the unsaturated formulation to provide a satisfactory explanation of the various settlement observations is therefore demonstrated.

Bentonite barrier under non-isothermal conditions

In several countries employing nuclear power, deep geological storage or disposal is an intensively studied option for the long-term confinement of heat-emitting high level nuclear waste (HLW) (Gens, 2003). The construction of deep repositories will involve the excavation of a network of tunnels in a suitable host rock a few hundred meters below surface. The canisters containing nuclear waste will be placed in either horizontal drifts or vertical boreholes. In most designs the canisters are surrounded by an engineered barrier made up of compacted, and therefore unsaturated, expansive clay; normally a bentonite is used.

The major actions that affect the bentonite barrier are heating from the canister and hydration from the surrounding rock. Fig. 18 shows, in a schematic way, some of the thermo-hydraulic processes occurring in the bentonite barrier and immediate adjacent rock. At the inner boundary, the barrier receives a very strong heat flux from the canisters. The dominant heat transfer mechanism is conduction that occurs through the three phases of the material. A temperature gradient will therefore develop in the near field and heat dissipation will be basically controlled by the thermal conductivity of clay barrier and host rock. The heat supplied by the cansiter results in a large temperature increase in the inner zone of the barrier, and strong water evaporation that induces drying of bentonite. Degree of saturation and water pressure will reduce significantly in this region. Vapour arising from the drying of the bentonite will move outwards until finding a cooler region where it will condensate, causing a local increase in water saturation. Vapour diffusion is a significant mechanism of water transfer mechanism and it also contributes, but to a much lesser extent, to heat transport.

Due to the high suction present initially in the unsaturated material that constitutes the backfill, hydration will take place with water moving from the host rock towards the barrier. The distribution of water potential inside the clay barrier is strongly affected by the phenomena of bentonite drying and vapour transport, as described above. Hydration should eventually lead to saturation of the barrier, but full saturation times can be very long due to the low permeability of the bentonite and/or the host rock.

In addition to the thermo-hydraulic behaviour, there are also important mechanical phenomena occurring. Drying of the bentonite will cause shrinking of the material whereas hydration will produce swelling that may be quite strong in highly compacted bentonite barriers. Because the barrier is largely confined between heater and rock, the main result of hydration is the development of swelling pressures in a process quite akin to a swelling pressure test. The magnitude of the stresses developed is critically dependent on the emplacement density of the bentonite and may reach values of several MPa. A crucial feature of the THM behaviour described is that all those phenomena are strongly coupled, interacting with each other in a complex manner; therefore a coupled THM formulation and computer code have to be used. They are described in Olivella et al. (1994, 1996).


27

Figure 18. Thermo-hydraulic processes occurring in the engineered barrier and adjacent rock

This application refers to a large scale heating test (FEBEX test) performed in the Grimsel underground laboratory (Switzerland) to simulate the thermo-hydro-mechanical behaviour of the bentonite barrier under repository conditions. To this end, the FEBEX test simulates, at full scale, the Spanish repository concept for HLW that envisages placing the waste-containing canisters in horizontal drifts surrounded by an engineered barrier made up of compacted bentonite. To install the experiment, a 2.28 m diameter circular tunnel was excavated in the GTS underground laboratory using a TBM machine. The tunnel is 70.4 m long and the final 17.4 m section was selected for the performance of the test. In the test area, two 4,300 W heaters, simulating the thermal effect of the canisters, were placed in the axis of the horizontal drift. The heaters were 4.54 m long and 0.90 m in diameter and were intended to simulate the release of heat by HLW. The space between the rock surface and the heaters was backfilled using blocks of compacted bentonite. Finally, the test area was sealed by a 2.7m long concrete plug. Fig. 19 shows a schematic layout of the test.

The host rock is good quality Central Aare granite crossed, in the test zone, by a lamprophyre dyke. The clay is a bentonite (wL = 98-106, wP = 50-56) obtained from a quarry in the volcanic zone of Serrata de Níjar in Southern Spain. It has high smectite content, in the range of 88%-96%., The bentonite blocks used to construct the barrier were compacted to a dry density of 1.7 g/cm

3 at an average water content of

14.4%. Because there were small gaps between blocks and between the clay barrier and the rock, the overall dry density of the emplaced barrier was 1.6 g/cm

3. The behaviour of the unsaturated compacted

bentonite was modelled by the BBM where the behaviour inside the yield locus has been modified to take into account the large swelling potential of the material.


28

Figure 19. Layout of the FEBEX in situ test

After test installation, four months were allowed to elapse before switching the heaters on. During this time there was some hydration of the barrier from the rock but, obviously, no thermal loading. During the heating stage, the test was temperature-controlled with a prescribed maximum temperature of 100ºC at the contact between liner and bentonite. To achieve this, the sequence of operations was as follows: i) the heaters were switched at a constant power of 1200 W each during 20 days with an aim to identify the thermal response of the system and to adjust the control algorithm; ii) afterwards, the power of each heater was increased to 2000 W and kept constant until a temperature of 100ºC at the contact between liner and bentonite was reached (this happened after 33 days); iii) subsequently, the system operated under temperature control, i.e. power was adjusted to keep a constant 100ºC maximum temperature on the liner/barrier contact.

Model predictions and observations are compared for the first five years of the test in Figures 20 (temperature evolution) and 21 (relative humidity evolution). Relative humidity is plotted because it was the variable directly observed; of course, relative humidity and suction are directly related via Kelvin’s equation (2). It can be seen that a good reproduction of the observations is achieved. Only a small selection of the results can be shown here, more comparisons are presented in Gens et al. (2009). A more demanding test of the modelling process was provided by the observations performed during dismantling.

Exactly after five years of heating, Heater 1 (the one closest to the concrete plug) was switched off. After allowing 33 days for cooling, the bentonite was carefully removed up to the end section of Heater 1, leaving the rest of the barrier in place. Subsequently, a new shotcrete plug was constructed and the test on the remaining part of the experiment continued. A detailed description of the procedures during dismantling is presented in Bárcena et al. (2003).


29

0 300 600 900 1200 1500 1800

Time (days)

10

20

30

40

50

60

70

80

90

100

Tem

pera

ture

(ºC

)

r = 0.48 m

r = 0.82 m

r = 1.10 m

in-situ testSection D1 (x= 4.42 m)

test model

0 300 600 900 1200 1500 1800

Time (days)

10

20

30

40

50

60

70

80

90

100

Tem

pera

ture

(ºC

)

r = 0.48 m

r = 0.82 m

r = 1.10 m

in-situ testSection D2 (x= 14.38 m)

test model

Figure 20. Evolution of temperatures in the bentonite barrier. Sections D1 and D2. Observations and computed results.

0 300 600 900 1200 1500 1800

Time (days)

0

10

20

30

40

50

60

70

80

90

100

Rela

tive h

um

idit

y (

%)

r = 1.10 m

r = 0.60 m

in-situ test

Section C (x= 1.80 m)

test model

0 300 600 900 1200 1500 1800

Time (days)

0

10

20

30

40

50

60

70

80

90

100

Rela

tive h

um

idit

y (

%)

r = 1.07 m

r = 0.82 m

r = 0.52 m

in-situ test

Section H (x= 9.50 m)

test model

Figure 21. Evolution of relative humidity in the bentonite barrier. Sections C (cool) and H (hot). Observations and computed results.


30

This partial test dismantling allowed the direct and detailed observation of the state of the barrier after five years of heating and hydration (Fig. 22a). In addition, large amounts of quantitative data could be obtained; this information is very valuable to get an independent check on the performance of the numerical model as well as on the reliability of predictions. At selected sections, a large number of specimens were cored out from the barrier (Fig. 22b) and dry density and water content were determined immediately in a field laboratory in order to minimize disturbance and humidity loss. This comprehensive set of values of dry density and water content helped to provide a complete picture of the state of the bentonite barrier. Thanks to the high density of determinations, contours of dry density and water content could be plotted in the sampled sections. It was observed that the process of hydration and associated density changes has been basically axisymmetric around the barrier.

a) b)

Figure 22. a) State of the bentonite barrier after five years of heating and hydration. b) Bentonite sampling during dismantling.

Fig. 23 shows the all measured values of water content and dry density as a function of distance to the tunnel axis for a section near the centre of Heater 1; it is therefore representative of a hot region of the test. It can be observed that, as expected, water content increases as the rock is approached, consistent with the process of natural hydration from the rock. It is also interesting to observe that the part of the barrier close to the heater is still below its initial value of water content, a consequence of the strong drying that has occurred in this area. In spite that a certain amount of water has already reached the inner region, it is still insufficient, after five years, to compensate for the initial drying. It is also noticeable that the barrier as a whole is very far from full saturation at the end of the heating stage. The distribution of dry density also shows the expected patterns. Close to the rock the clay has expanded, exhibiting values of dry density well below its initial value. In contrast, in the zone near the heater the dry density has increased. Because of the confined nature of the test, the variation of dry density in the inner part is compensated by the reduction of dry density in the outer part. It should be noted that the change of dry density (i.e. porosity) is the combined effect of expansion due to temperature increase (thermal effect), suction changes (hydraulic effect) and stress increase due to the development of swelling pressure (mechanical effect).

Water content and dry density values are also plotted for a cool Section (Fig. 24) where the temperature increase is very limited. There are some differences in the patterns of observations. Water content again increases as the rock is approached but, now, there is also a net (but small) gain of water content near the centre of the tunnel. There has been no drying in this region but the amount of hydration is very small, even after five years, because of the thickness of the barrier in this section. The dry density also shows a significant reduction close to the rock but a somewhat smaller increase in the inner part where the change appears to be more uniform. Again, the net volume change of the section is quite small.


31

0.4 0.6 0.8 1.0 1.2

Distance to axis (m)

1.45

1.50

1.55

1.60

1.65

1.70

Dry

den

sity

(M

g/

m3)

Post-Morten Analysis Section 27

x = 6.85 (m)

Test

Model

0.4 0.6 0.8 1.0 1.2


0.10

0.15

0.20

0.25

0.30

0.35

Wate

r co

nte

nt


x = 6.85 (m)

Test

Model

1

15 189 22 27 31

8 1310 14

16

Section number

11

15 189 22 27 31

8 1310 14

16

Section number

1

15 189 22 27 31

8 1310 14

16

Section number

11

15 189 22 27 31

8 1310 14

16

Section number

Initial valueInitial value


Figure 23. Values of water content and dry density during dismantling in Section 27 (hot Section). Observations and computed results.

0.0 0.2 0.4 0.6 0.8 1.0 1.2


1.45

1.50

1.55

1.60

1.65

1.70

Dry

den

sity

(M

g/

m3)

Post-Morten Analysis

Section 15

x = 3.27 (m)

Test

Model


0.0 0.2 0.4 0.6 0.8 1.0 1.2


0.10

0.15

0.20

0.25

0.30

0.35

Wate

r co

nte

nt


x = 3.27 (m)

Test

Model

1

15 189 22 27 31

8 1310 14

16

Section number

11

15 189 22 27 31

8 1310 14

16

Section number

1

15 189 22 27 31

8 1310 14

16

Section number

11

15 189 22 27 31

8 1310 14

16

Section number


Figure 24. Values of water content and dry density during dismantling in Section 15 (cool Section). Observations and computed results.


32

It is interesting to observe that the results of the numerical analysis show a very good agreement with the measurements; in this case the computed results are obviously predictions. The patterns of the hot and cool sections are reproduced very well and the quantitative agreement between observations and predictions is quite close. These comparisons, therefore, increase the confidence that formulation, constitutive law and computer code manage to reproduce closely the relevant features of the engineering system and associated coupled phenomena.

CONCLUDING REMARKS

The advances of Unsaturated Soils Mechanics in the last few decades have been abundant and wide-ranging and it is still an area of very active research at present. In this paper some key aspects of such developments have been summarily reviewed and discussed:

- Water potential and suction underlie much of our understanding of the mechanics of unsaturated soils. Matric suction is the most important potential component regarding mechanical behaviour of unsaturated soils. It has been argued that a simple capillary model is insufficient to fully explain matric suction; an adsorptive component must also be incorporated.

- It is unambiguously stated that at least two constitutive variables are required to represent adequately the full range of unsaturated soil behaviour. A number of alternatives are possible with different advantages and disadvantages. Some of the most widely used ones turn to be equivalent from the point of view of the work rate equation. At the end, the choice of constitutive variables is largely a matter of convenience.

- The development of elastoplastic constitutive models able to integrate a wide range of features of unsaturated soils’ behaviour is one of the hallmarks of modern Unsaturated Soil Mechanics. Those advances have been illustrated by reference to three different models of different degree of complexity. Starting from the simple BBM formulated in terms of net stresses, developments lead to the use of Bishop’s stresses, incorporation of microstructural variables and coupling between hydraulic and mechanical behaviour.

- Those developments together with advanced coupled formulations and computer codes (not described in this paper) allow tackling engineering problems involving unsaturated soils in a more rigorous and rational manner. Two examples of different degrees of complexity have been presented to support this point.

It is also noteworthy that the field of Unsaturated Soil Mechanics is proving to have a wider field of applicability than could be conventionally expected. Unsaturated Soil Mechanics concepts and tools have turned out to be very useful in problems such as porous materials partially saturated with oil and water (De Gennaro et al., 2004) or in the study of freezing soil (Nishimura et al. 2009). This cross-fertilization between different areas of research augurs well for the future of the subject. It is likely that the discipline will be able to provide effective answers to increasingly complex civil and environmental engineering problems as well as in other related areas.

ACKNOWLEDGEMENTS

Much of the work reported here has been performed with colleagues at the Technical University of Catalonia (UPC) and elsewhere. To all of them, my grateful acknowledgement is due.

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