Elasto-plastic Modelling of Unsaturated Soils : an Overview

Transcription

1 he 12 th International Conference of International Aociation for Comuter Method and Advance in Geomechanic (IACMAG) 1-6 Octoer, 28 Goa, India Elato-latic Modelling of Unaturated Soil : an Overview Daichao Sheng Centre for Geotechnical and Material Modelling, he Univerity of Newcatle, Autralia. Delwyn G Fredlund Golder Aociate Ltd., Sakatoon, SK, Canada Keyword: unaturated oil, elato-latic model, finite element ABSRAC: hi aer reent an overview of elato-latic modelling of unaturated oil. Alternative tre and train variale that have een ued in contitutive model for unaturated oil are dicued. he ue of an effective tre equation may imlify contitutive relation, ut often reult in a tre ace that deend on the material tate. In thi cae, the contitutive ehaviour i emodied oth in the tre definition and in the contitutive relation. Alternative volumetric model in the literature are then dicued. he focu i laced on the comarion of a new model with other exiting model. hi new model i formulated in term of indeendent tre variale and feature a mooth tranition etween aturated and unaturated tate. he imlication of the volumetric model for yield tre and hear trength function are dicued. he aer alo how how hyterei in oil-water characteritic curve can e incororated into the elato-latic framework, leading to a couled hydro-mechanical model. Finally, the aer demontrate the derivation of the incremental tre-train relation for unaturated oil and dicue riefly the imlementation of thee relation into the finite element method. 1 Introduction Unaturated oil mechanic did not emerge in arallel with aturated oil mechanic. here wa a delay of aout three decade efore the aic rincile of unaturated oil egan to e undertood. Initial tudie of unaturated oil focued on the earch for an effective tre equation for unaturated oil. hen reearcher focued on tudie related to the volume change ehaviour of exanive oil through a erie of international conference from 1965 to here wa alo limited conideration of collaile oil ehaviour during thi eriod. It ecame oviou at each uequent international conference that there wa a deire to exand the coe of interet to include hear trength, contaminant tranort, heat flow and other unaturated oil henomena. he uject of hear trength ehaviour ha roven to e eaier to undertand than volume change ehaviour. he theorie that emerged for volume change and hear trength took the form of extenion of claic form that had een develoed for aturated oil. With time it wa oviou that the claic oil mechanic framework needed to e exanded to more adequately emrace elato-latic formulation for unaturated oil. In 1995 the Firt International Conference on Unaturated Soil (Pari, France) wa held. hi conference witneed an enlargement of the coe of unaturated oil rolem of interet to geotechnical engineer. Suequent international conference on unaturated oil have een held in Beijing, China, Recife, Brazil and Carefree, Arizona. In addition, there have een a erie of regional conference on unaturated oil held in Brazil, Southeat Aia, Euroe and United State. here ha een continuouly increaing interet in undertanding the fundamental ehaviour of unaturated oil, articularly in the hear trength, volume change and aturation change ehaviour aociated with uction change. Significant rogre ha een made on undertanding hear trength, volumetric and oil-water characteritic ehaviour through 197 to 199. On the other hand, contitutive model that are aed on elato-laticity theory and accommodate a more comlete tre-train relationhi tarted to emerge ince early 199, and y now numerou uch model exit in the literature. Unfortunately, the two aroache (i.e. the fundamental trength-volume-aturation-uction modelling and the elato-latic tre-train modelling) have een advancing in an indeendent manner rather than emracing each other. In thi aer, an different aroach i attemted: an overview of elato-latic modelling of unaturated oil i emodied in the dicuion of the key fundamental iue: (1) the volumetric ehaviour related to uction or aturation change, (2) the trength ehaviour related to uction or aturation change, (3) the oil-water characteritic curve, and (4) the imlementation of unaturated oil model. In o doing, the advantage and diadvantage of alternative model are alo dicued. hi review i therefore omewhat 284

2 different from reviou review uch a thoe y Gen (1995), Jommi (2) and Gen et al. (26). 2 Stre and train variale Contitutive relation ued to rereent mechanical ehaviour of material are uually decried in tre ace. he choice of the tre ace i thu a fundamental iue in contitutive modelling. Idealitically the definition of tree hould e indeendent of the ehaviour or the tate of the material, o that the tre ace doe not change with the material tate. here i little argument that total tree hould e ued for decriing the tre ace of ingle hae material uch a metal and dry and. It i alo generally acceted y the oil mechanic community that effective tree (i.e., difference etween the total tree and ore reure) can e ued to decrie the mechanical ehaviour of aturated oil. he definition of the total tree for dry oil and the effective tree for aturated oil are indeendent of the oil ehaviour or oil tate. he tre ace in thee cae are thu earated from the material tate. Alo, the o-called total tre in dry and i actually the difference etween the total tre (i.e., force er unit cro-ection area) and the atmoheric oreair reure. For oil that are unaturated with more than one ore fluid, the choice of the tre ace aear to hitorically have ecome more comlicated in that the tre ace i ued in a manner where it i deendent on the material tate. Biot (1941) wa roaly the firt to ugget the need to ue two indeendent tre tate variale for an unaturated oil. In 195 and 196, however, great effort were made to identify a ingle effective tre that can e ued to decrie the deformation and trength characteritic of unaturated oil, a the one ued for aturated oil (e.g. Aitchon and Donald, 1956; Biho, 1959). he following effective tre equation wa rooed y Biho (1959) ( ) σ = σ u δ + χ u u = σ + χ (1) ij ij a ij a w ij where σ ij i the total tre, σ ij i the Biho effective tre, σ ij i the net tre, ua i the ore air reure, ua i the ore water reure, i the oil uction, χ i a arameter that may deend on the degree of aturation or on the oil uction. he oil uction in thi aer refer to the matric or matrix uction which conit of the caillary and adortive otential. When the water hae in oil ore i continuou, the caillary otential ( ψ c ) i dominant in the matric uction ψ c = ua uw. When the ore water exit a adored water film in the oil, the adortive otential ( ψ a ) i dominant in the matric uction and the true water reure i not well defined. In thi cae an aarent water reure ( u w ) can e ued to quantify the adortive otential ψ a = ua uw, i.e. the aarent water reure rereent the negative adortive otential meaured in exce of air reure. With uch a definition, the matric uction can e ued for a relatively large range of aturation, from fully aturated to very dry tate. Even though the new definition of effective tre in the 196 ha led to ome ucce in decriing the hear trength of unaturated oil, it ha not led to great ucce in modelling the general mechanical ehaviour of unaturated oil, not at leat u to the late 199 and early 2. Some limitation were early reorted in uing a inge Biho effective tre in exlaining volume collae during wetting of unaturated oil (Jenning and Burland, 1962). More imortantly, ecaue the arameter χ uually deend on material roertie and even material tate, the tre ace defined y equation (4) uually deend on the material ehaviour and change with material tate. herefore, the contitutive ehaviour of the material i emodied in oth the contitutive relationhi and the tre ace where the contitutive relation are etalihed. In 196 and 197, it wa realied that it wa oile to ue two indeendent et of tre variale to model unaturated oil ehaviour rather than comining them into one ingle effective tre a in the Biho effective tre. For examle, Coleman (1962) uggeted the ue of the net axial and radial tree and the net ore water reure to rereent triaxial tre tate. Biho and Blight (1963) ued the concet of indeendent tre tate variale when lotting volume change in an unaturated oil. Matya and Radhakrina (1968) ued the indeendent tre variale (called tate arameter ) to decrie the volumetric ehaviour of unaturated oil. Volume change wa reented a 3D urface with reect to the indeendent tree. Numerou other reearcher have uequently reented the volume change ehaviour a urface defined y indeendent tre tate variale (Aitchion and Martin, 1973; Alono and Lloret, 1982). Fredlund and Morgentern (1976) reented volume-ma contitutive relation for a comacted kaolin, unaturated oil. he reult confirmed that two indeendent contitutive urface were required when maing change in volume-ma oil roertie for an unaturated oil. Fredlund and Morgentern (1977) further rovided a theoretical ai and jutification for the ue of two indeendent tre tate variale. he jutification wa aed on the ueroition of coincident equilirium tre field for each of the hae of a multihae 285

3 ytem, within the context of continuum mechanic. hree oile comination of indeendent tre tate variale were hown to e jutifiale from the theoretical continuum mechanic analyi. However, it wa the net tre and the matric uction comination that roved to e the eaiet to aly in engineering ractice: σij uaδij σij = ua u w (2) he net normal tre rimarily emrace the activitie of human which are dominated y alying and removing total tre (i.e., excavation, fill and alied load). he matric uction tre tate variale rimarily emrace the imact of the climatic environment aove the ground urface. Fredlund et al (1978) alo reented a hear trength equation uing the indeendent tre variale. he lieration of the uction from a ingle effective tre ha layed a very oitive role for the knowledge exloion in unaturated oil mechanic in 199 and 2. A numer of event have witneed the knowledge exloion ince 199: the etalihment of the International Conference on Unaturated Soil, the develoment of new contitutive model and teting technique for unaturated oil, and the ulication of text ook uch a Soil Mechanic for Unaturated Soil (Fredlund and Rahardjo, 1992). In the context of contitutive modelling, Alono et al. (199) for the firt time rovided a comlete elatolatic framework for modelling unaturated oil ehaviour. he model of Alono et al. (199) ued the net tre and uction a the tre variale. hi model wa later referred to a the Barcelona Baic Model. A numer of other elatolatic model were develoed in 199 (e.g., Kohgo et al., 1993; Modarei and Aou-Bekr, 1994; Wheeler and Sivakumar, 1995; Cui and Delage, 1996; Bolzon et al., 1996; Loret & Khalili, 22, ang and Graham, 22; Oka et al., 26). All thee model deal with tre-train relation only. More recent model have incororated uction-aturation relationhi with hyterei into tre-train relationhi (Vaunat et al., 2; Wheeler et al., 23; Gallioli et al., 23; Sheng et al., 24; amagnini, 24; Sun et al., 26; Santagiuliana and Schrefler, 26). A common feature of thee model i that the uction i conidered a an additional tre variale, or at leat a an additional hardening arameter (Loret & Khalili, 22). However, there i little conenu on whether an indeendent tre (e.g. net tre or total tre) or an effective tre variale hould e ued. A different argument wa ut forth y Houly (1997) uing the work-conjugate variale in the work inut for a oil element. In term of work inut, one difference in uing a different et of tre variale i the change of their workconjugate train variale. For examle, the work-conjugate train variale to the two et of indeendent tre variale (the net tree and the uction), are the oil keleton train and the volumetric water content: σij uaδij εij ua u w θ (3) where ε ij i the oil keleton train and θ i the volumetric water content. It i noted that the definition of the tre variale in equation (3) are the ame a thoe rooed y Fredlund and Morgentern (1978) and are indeendent of each other and indeendent of material tate. heir work-conjugate train variale are not indeendent (i.e. the oil keleton train and the volumetric water content are deendent uon each other). It i alo noted that the net tre ecome the total tre when the oil i aturated and the ore air i under atmoheric air reure. When the air reure i contantly at atmoheric reure (which i aroximately true for mot initu condition), the matric uction i nothing ele ut negative ore water reure. Early model uing the net tre and uction thu uffer a major drawack, i.e. the continuity at the tranition etween aturated tate and unaturated tate, ecaue the tre variale ued for the aturated oil ehaviour i the erzaghi effective tre. On the other hand, the work-conjugate tre variale related to the two et of indeendent train variale (i.e., the oil keleton train ε ij and the degree of aturation S r ) are the average tree and a (modified) uction (n). hee train variale are conidered to e indeendent ecaue a change in one of them doe not necearily reult in a change in the other. he oroity n lay a role to cale the work inut (due to a change in aturation) er unit void volume to that er unit volume of the oil matrix. herefore, the econd alternative tre variale take the following form (Sheng et al. 24): σij uaδij + Sr( ua uw) εij ( ua uw) S r (4) 286

4 a < 1 < 2 < 3 a < 1 < 2 < 3 e ln or ln Figure 1. Qualitative rediction of void ratio veru mean tre under contant uction according to Equation (4) or (5). where S r i the degree of aturation and n i the oroity. he tre variale defined in equation (6) are deendent on one another and a well a on the material tate (S r ). However, the work-conjugate train variale are indeendent of one another. he average tre ecome the erzaghi effective tre when the oil ecome aturated. herefore, the aove tre variale can e ued conitently for oth aturated and unaturated tate. Conequently thee tre variale have a ignificant advantage in term of imlementation into finite element code. Such a et of tre variale have alo een ued in more recent model uch a thoe y Sun et al. (26) and Santagiuliana and Schrefler (26). More recently Nuth and Laloui (28) and Laloui and Nuth (28) refer to thi et of tree a the generalied tree for unaturated oil and have rovided further exerimental evidence to endore it ue. he more comlex tre variale defined y equation (4) tend to lead to imler contitutive equation, wherea the imler tre variale defined y equation (3) tend to lead to more comlex contitutive equation. he comlex tre variale in equation (4) deend on material tate and are not controllale in laoratory teting. herefore, trictly eaking, it i not oile to develo a comletely new contitutive relationhi in term of thee variale, unle an exiting framework i ued. However, it i oile to tranform an exiting contitutive relationhi formed in term of the imler tre variale (equation (3)) to the comlex tre ace. Such a tranformation can often overcome the dicontinuity rolem at the tranition etween aturated and unaturated tate, a wa done y Sheng et al. (23a, 23) for the Barcelona Baic Model. he choice of the tre variale alo ha a ignificant influence on the yield and failure urface, which i later dicued in thi aer. 3 Volumetric tre-train model Soil uction affect the volumetric ehaviour, yield tre and hear trength of an unaturated oil. Suction generate interarticle force normal to article contact, while ore reure generate iotroic tree around oil article. A uch, oil uction lay a more comlex role than the reure or mean tre. A ointed out y Li (23), ome meaure of oil faric hould e taken into conideration to accommodate the effect of uction. However, meaure of oil faric are difficult to achieve and have not yet een ued in contitutive modelling. Intead, uction i uually treated a a imilar quantity to the mean tre. Under uch a framework, the only extra contitutive law that i aolutely required to extend a aturated oil model to unaturated oil i the volume tre uction relationhi. he effect of uction on the yield tre and hear trength can e incororated into the model aed on the volume tre uction relationhi. he change in the ecific volume (v) of an unaturated oil in reone to uction () or mean tre () change i tyically modelled in one of the following way: dn d dλ v = (Net tre, Alono et al., 199) (5) d d v d d λv ln d 287

5 .84 Data of Jenning & Burland (1962) e e λ v Prediction uing equation (9) and 1+ a λv = λv 1+ λ =.1, = 3kPa, = v a.8 λ v Shrinkage limit ( re ) a (kpa) water content Figure 2. Schematic view of void ratio veru uction under zero net mean tre ( a : uction correonding to full aturation, re : reidual uction). dn d dλ v d d v d = d λv ln d (Effective tre) (6) d d dv = λv λv (Net tre, Fredlund & Rahardjo, 1993) (7) where N i the ecific volume of the oil when the mean tre i unity, λ v i a material arameter rereenting the tre comreiility under contant uction, i the net mean tre, i the effective mean tre, λ v i a material arameter rereenting the uction comreiility under contant mean tre. he arameter λ v in equation (5) and (6) and the arameter λ v in equation (7) are uually conidered a function of uction. Over a certain tre range, the arameter λ v i uually aroximated y one or two contant, deending on the reconolidation reure. Equation (5) ue the net tre and i ued in e.g. the Barcelona Baic Model of Alono et al (199) and alo in many other model. Equation (6) ue the effective tre and i ued y Kohgo et al. (1994) and Loret and Khalili (22). Equation (7) ue the net tre and earate the comreiility due to a tre change ( λ v ) from that due to a uction change ( λ v ). All thee equation are of coure aed on the Cam Clay elatolaticity for aturated oil: d d d( uw ) dv = λ = λ λ + ( u ) + ( u ) v v v w w (Saturated oil) (8) Equation (5) to (8) are confuingly imilar, ut they ear different imlication. For examle, equation (5) and (7) do not recover equation (8) for aturated tate. A imle verification for thi i to conider the cae where the ore air reure (u a ) remain atmoheric. Under thi condition, equation (5) and (7) are only valid for zero ore water reure when the oil ecome aturated. he atmoheric air reure can e treated a zero ore air reure and uction a a negative ore water reure for all aturated tate. When the ore air reure remain contantly at the atmoheric reure, the net tre ecome the total tre and uction ecome the negative ore water reure. Such a concet will rovide a continuou tranition etween aturated and unaturated tate. Equation (6) fully recover the model for aturated oil, ut contain the effective tre arameter χ. hi arameter often deend on the material a well a the material tate, leading to the quetionale outcome that the tre ace where the material ehaviour i modelled change with the material ehaviour and even the material tate. Equation (7) earate the comreiility due to a tre change ( λ ) from that due to a uction v 288

6 e or lne < 1 < 2 < 3 3 = 1 2 ln Figure 3. Qualitative rediction of normal comreion line under contant uction according to Equation (9) or (11). change ( λ ). A uch, the arameter v the other two equation. λ v i not necearily a function of uction, which i an advantage over A chematic view of the rediction according to equation (5) and (6) i hown in Figure 1. hee two model oth have difficultie in exlaining the curvature of the normal comreion line at oitive uction for oil dried from lurry. Let u conider the cae where a lurry oil i firt dried to a ecific uction and then iotroically comreed at that uction. he iotroic comreion line for thi oil i uually curved in the e ln ace, a hown y Jenning and Burland (1962). o aroximate the curve uing equation (5) or (6), we would have to ue the overconolidation concet, o that the curve i aroximated y an initial elatic ortion followed y an elatolatic ortion, a illutrated in Figure 1. However, the oil ha never een overconolidated, ince the roce of drying and loading doe not involve any tre or uction decreae. More recently, Sheng, Fredlund and Gen (26) rooed the following model for the volumetric ehaviour of unaturated oil: d d d v = λv λv ( ) (Net tre, Sheng et al., 26) (9) + + λ v can e indeendent of uction, and the loe λ v varie etween where the loe and zero for uction aove the reidual uction (Figure 2). hi model 1. recover the equation for aturated tate, i.e. equation (8), 2. earate the comreiility due to tre and uction change, and λ v for aturated tate 3. can redict the mooth curvature of the normal comreion line under contant uction for oil dried from lurry, without the ue of the overconolidation concet (ee Figure 3). Equation (9) i very imilar to equation (8), ut with the negative ore water reure relaced y the uction, and the total mean tre relaced y the net mean tre. It i alo reaonale to tate that a change in uction doe not necearily have the ame effect a a change in mean tre once the oil ecome unaturated. Sheng, Fredlund and Gen (26) howed that equation (9) can cature a numer of imortant feature in unaturated oil ehaviour and can well rereent exerimental data. Another iue related to all the aove volumetric model ertain to the uction range and oil tye where the model can e alied. Firt, all thee model aly only to a continuum and ecome invalid once deiccation and cracking occur. Second, for dry granular oil where the water hae ecome dicontinuou, the concet of caillary uction may not fully aly. In thi cae, it i etter to view uction a imly the affinity of the oil for water or the energy required to remove the water from the oil. More tudy i required on the ehaviour of oil with water content le than reidual value. 289

7 A dry and ehave in a imilar manner to a aturated and under fully drained condition. hi henomenon may not e ale to e redicted uing the concet of oil uction. In addition, the ae model for aturated oil, (i.e., equation (8) i known to e more alicale to clayey oil than to granular oil). Neverthele, other contitutive model ued for aturated oil can e generalied to accommodate unaturated oil in a imilar manner. For examle, Sheng et al. (27) howed that the following equation well redict the volume change of aturated or dry and. de d = λv e + re (Saturated and, Sheng et al. 27) (1) where e i the void ratio, and re i a hifting tre deending on the initial void ratio of the oil a well a λ v. he hift tre can alo e interreted a the tre level where ignificant article cruhing occur. Note that the arameter λ v in equation (1) i the loe in the doule logarithmic ln e ln( + re ) ace. If equation (1) i ued for a aturated and, Equation (9) can e modified a follow to uit the unaturated and d d λ λ () de = e d λv > + re + re v v re re re re (11) where re i the reidual uction (ee Figure 2). A threhold uction ( re ) i introduced in the equation (11) and aove thi value uction ha no effect on the volume change. Becaue the reidual uction for and i relatively mall (<1kPa) comared to article cruhing reure of and (1-1MPa), the effect of uction on volume change i relatively limited. Setting re = in Equation (1) recover equation (8). herefore, equation (11) can alo e ued for clay (with re =). 4 Yield urface and hardening law Soil uction i an additional tre variale and therefore it i neceary to determine the variation of the yield tre with uction, or the extenion of the yield urface in the tre uction ace. he yield urface for an iotroic hardening oil uually rereent the contour of the latic volumetric train (i.e. the hardening arameter). A uch, the variation of the yield tre with uction can e derived from the volumetric model. For examle, for the volumetric model defined y equation (5), it i oile to how that the following function rereent the contour of latic volumetric train in the ace (ee Sheng, Fredlund and Gen, 26): c c λv κ c a λv κ r r = > a a (12) where c i the yield tre at uction, c i the yield tre at uction a, r i a reference mean tre, λ v i the loe of the normal comreion line for aturated tate, λ v i the loe of the normal comreion line for unaturated tate (uction ), and κ i the loe of the unloading-reloading line for aturated tate. Equation (12) define the o-called loading-collae (LC) yield urface in the Barcelona Baic Model (BBM) y Alono et al. (199). A chematic view of the loading collae yield urface defined y equation (12) i hown in Figure 4a. A numer of oervation can e made here. Firt, the yield urface i uually hown in the literature for uction aove the aturation uction only. Since the net tre ecome the total tre for aturated tate, the yield urface actually follow the 45 o line for < a. Second, the yield tre c increae with increaing uction only if (1) λv < λv and > r, or (2) λ v > λ v and < r. hee two alternative condition are rerequiite to modelling the wetting-induced collae. hird, an additional yield tre,, rereenting the aarent tenile trength for > mut e defined (ee Figure 4a). he following aarent tenile trength i ued in the BBM: 29

8 c c Elatic zone Elatic zone a c 45 o 45 o a c (a) net mean tre uction ace () effective mean tre - uction Figure 4. Schematic view of loading-collae yield urface in mean tre uction ace. < = α a a (13) On the other hand, if the volumetric model i aed on the effective tre (i.e. equation (6) i ued), the correonding yield tre ecome: c λv κ c λv κ r r c = > a a (14) = (15) In equation (14), r i a reference mean tre and r = 1 if the oil ecific volume at r i given y arameter N(), ee equation (6). he yield urface c i hown chematically in Figure 4. Becaue of the ue of Biho effective tre, it i uually aumed that an aarent coheion i zero. In addition, the loadingcollae yield urface extend to the aturated zone following a vertical line. However, the ame condition ( λv < λv and > r, or λv < λv and < r ) aly to the imulation of wetting-induced collae. he loading-collae yield urface given in Figure 4 can not e ued for a oil dried from lurry. Drying a lurry oil i imilar to comreing the oil. herefore, the tre tate hould alway e on the current yield urface, which i clearly not oile in Figure 4a. Intead, Figure 4a would redict a urely elatic reone for drying a lurry oil under contant tre. In Figure 4, the tre tate could e on the current yield urface if the effective mean tre increae at a fater rate that the yield tre a uction increae, which would then lead to additional condition regarding the definition of the effective tre concet. he model of Sheng, Fredlund and Gen (26) rovide a mooth tranition etween aturated and unaturated tate. he yield tree, and c, can e derived from the volumetric model, i.e. equation (9) or (11), rovided an exlicit function i given to the arameter λ v i given. For examle, the following yield tree can e derived from equation (9) for a lurry oil: 291

9 Initial elatic zone 6 (kpa) 5 c cn c cn (kpa) Figure 5. Initial yield urface for a oil that wa conolidated to 3kPa at zero uction and it evolution when the oil i then comreed at different uction level ( a =1kPa). c < c = + 1 ( + 1ln ) + c a a a a 1 a (16) hi yield tre decreae with increaing uction. herefore, drying a lurry oil will alway e on the current yield urface. For a lurry oil, i zero. he aarent tenile trength caued y uction i then given a c < = + 1 ( + 1ln ) + a a a a 1 a (17) However, for an unaturated oil that i comreed or comacted at a uction aove the aturation uction, the yield tre in the model y Sheng, Fredlund and Gen (26) change to: c cn < cn = cn ( + 1ln ) c c a a a a + 1 a (18) where c i the initial reconolidation reure at zero uction (Not at a ), cn reure at zero uction (ee Figure 5). i the new reconolidation he yield tree given y equation (16)-(18) are illutrated in Figure 5. he curve rereent the aarent 292

10 tenile trength of the oil caued y uction. he curve c rereent the yield tre if the oil i air-dried. For examle, for a lurry oil that wa firt conolidated to 3 kpa and then air-dried at zero mean tre, the uction that caue latic yielding i 73kPa. If the air-dried oil i comreed under contant uction, the new yield tree are then rereented y the curve cn. herefore, c rereent the yield tre for an air-dried lurry oil and cn rereent the yield tre for a comacted oil. he yield tre increae with increaing uction along the curve cn, not c. herefore, wetting-induced collae i only relevant for comacted oil with thi model. It hould alo e noted that: (1) the tranition etween aturated and unaturated tate i continuou and mooth along all the three yield tree; (2) like the model in Figure 4, the yield urface c and cn are non-convex in the ace; (3) the model i tre ath deendent and different tre ath reult in different yield urface. If the Modified Cam Clay model i ued a the ae model for the aturated oil, the ellitic yield urface can e extended to the uction axi according to equation (16)-(18): 2 2 f q M y = ( )( ) (19) where f i the yield function, q i the deviator tre, M i a hear trength arameter and rereent the loe of the critical tate line in q ace. he yield urface according to equation (19) i hown in Figure 6 for two tye of unaturated oil. 293

11 q kpa kpa kpa (a) Yield urface for air-dried oil from lurry. q kpa kpa (kpa) () Yield urface for comacted oil. Figure 6. Modified Cam Clay yield urface extended to uction axi ( a =1kPa). In the three model mentioned aove, the evolution of the yield urface i governed y the latic volumetric train. he hardening law are written reectively a λ v κ d ln dλv dε v = + d for equation (12) (2) v v d λ v κ d ln dλv dε v = + d v v d λ v κv λv κv dεv = d + d + + for equation (14) (21) for equation (16) and (18) (22) where ε v i the latic volumetric train, κ v i the elatic counterart of λ v for unloading and reloading, and κ v i the elatic counterart of λ v. 294

12 5 Shear trength with uction he hear trength of an unaturated oil i uually a function of uction. Fredlund et al. (1978) rooed the following relationhi which conveniently earate the hear trength due to tre from that due to the matric uction: τ = c + ( σn ua) tanφ + ( ua uw) tanφ = c + ( σn ua) tanφ (23) where τ i the hear trength, c i the effective coheion and i uually zero unle the oil i cemented, the normal tre on the failure lan, φ i the effective friction angle of the oil, uction, and c i the aarent coheion which include the friction due to uction. If (23), the effective tre rincile for aturated oil i recovered. σ n i φ i the frictional angle due to φ i et to φ in equation he aove hear trength equation wa originally ulihed in a linear form ut it wa later realized that exerimental reult howed the latter ortion of the aove equation to e nonlinear. he hear trength due to oil uction commence to deviate from the effective angle of internal friction at aroximately the air entry value of the oil. he oil uction veru hear trength relationhi then aear to have a gradual curvature until reidual uction condition are reached. Once reidual uction condition are reached the hear trength enveloe remain aroximately contant (or level at the ame value) a uction are increaed. However, it i oile for the hear trength to decreae for and oil and increae for clay oil a uction are increaed eyond reidual condition. here are a numer of model availale in the literature for determining the friction angle φ (Vanaalli et al., 1996; Fredlund et al, 1996; Oerg and Salfor, 1997; and Bao et al, 1998). In elato-latic model, the hear trength of an unaturated oil i uually emodied in the aarent tenile trength function. For examle, the aarent coheion in the BBM i given a tan φ < a, c = c = tanφ = αtan φ a, c = (24) he friction angle φ i then given y tanφ < tanφ = α tanφ a a (25) In thi cae, the friction angle φ i indeendent on uction. In the model y Sheng, Fredlund and Gen (26), the aarent coheion due to uction i tan φ < a, c = c = tanφ = + 1 tanφ a + ( a + 1) ln a, c = a + 1 (26) 295

13 2 Shear trength (kpa) Soil uction (kpa) Figure 7 Shear trength veru oil uction u to 1, kpa for oil with air entry value of 1, 1, and 1 kpa, redicted y equation (27). Shear Strength (kpa) Guadalix Red Silty Clay (σ - u a ) = 12 kpa Exerimental Reult Predicted Shear Strength Suction (kpa) Figure 8. Predicted hear trength value for Guadalix Red ilty Clay according to equation (27) (data taken from Ecario and Jucá, 1989). herefore, the friction angle φ i given y: tanφ < tanφ = a a tanφ + ln a + 1 a a (27) In thi cae, the friction angle φ i a function of uction a well a the aturation uction. he redicted hear trength variation with uction i hown in Figure 7 and comared with exerimental data for Guadalix Red ilty Clay in Figure 8. It i hown that the rediction of equation (27) i very reaonale, at leat qualitatively. On the other hand, if Biho effective tre i ued, the hear trength i uually aumed to e unique in the effective tre ace: ( ) τ = c + σ tanφ = c + σ u tan φ + χ( u u ) tanφ (28) n n a a w 296

14 S r Main drying curve λ w κ w Scanning curve Main wetting curve κ w a ae we re ln Figure 9. Degree of aturation veru uction (dahed line rereent imlification). he aove equation alo imlie that tan = tan. Recently Nuth and Laloui (28) rovided ome φ χ φ exerimental evidence for the uniquene of c and φ in Biho effective tre ace with χ = Sr. 6 Hyterei of oil-water characteritic Extenive reearch ha een done on the oil-water characteritic curve for a oil, firt in the field of oil hyic and later within geotechnical engineering (ee, e.g., Hillel, 1982; Fredlund and Rahardjo, 1993). he oil-water characteritic curve (SWCC) i uually reented in ace a the volumetric water content (θ) veru oil uction or in ace a the degree of aturation (S r ) veru uction. A numer of emirical θ relation exit in the literature, and the one that are commonly ued include that of Gardner (1958), van Guenuchten (198) and Fredlund and Xing (1994). hee relation are uually written a continuou function and do not exlicitly conider the hyteretic ehaviour during a drying-wetting loo. However, in an elato-latic modelling framework which mut redict the reone for all oile wetting and drying ath, an incremental form of etween d and dθ or etween d and ds r i referred. Recently Li (25) reented an incremental oil-water characteritic relationhi etween d and ds r. hi incremental SWCC model include mooth hyteretic reone to aritrary wetting/drying ath, and can e incororated into elato-latic model for unaturated oil. However, the model y Li (25) follow the ounding urface framework which i different from the claic elato-laticity framework dicued in thi aer. More recently, Pedroo and Sheng (28) have develoed an incremental aturation-uction relationhi that incororate the hyteretic ehaviour. heir model i formulated in the ame framework a elato-laticity and can e conveniently incororated into an elato-latic tre train relation. he detail of the model of Pedroo and Sheng (28) can e found in the roceeding of thi conference. In thi aer, a very imle model reented y Sheng, Fredlund and Gen (26) i decried. hi imle model doe not conider the hyteretic ehaviour within the main drying and main wetting curve (ee Figure 9). A a imle aroximation, a iece-wie linear relationhi etween the degree of aturation S r and logarithmic oil uction can e aumed: d dsr = λw (29) where the loe λ w may change with uction. For oil uction elow the aturation uction, the oil i aturated and the degree of aturation remain eentially contant. For oil uction larger than the reidual uction, the water content gradually decreae to zero at a uction of 1 6 kpa (Fredlund and Rahardjo, 1993). he loe λ w i aumed to e contant etween the air entry and the reidual uction for a drying oil (Wheeler et al., 23). herefore, we have for increaing uction, a hown in Figure 9: < a κ w a < ae λw = (3) λw ae < re κ w re where ae i the air entry value, and re i the reidual uction (ee Figure 9). he aove equation i only valid for the main drying curve. For the main wetting curve and the canning curve, the loe mut e adjuted accordingly (ee Figure 9). he oil uction veru water content relationhi i affected y the mean net tre 297

15 SI: Drying y 6 (kpa) 5 ELASIC ZONE SD: Wetting Figure 1. Elatic zone encloed y the yield urface and the drying and wetting urface. rimarily through it influence on the air entry uction and the rate of deaturation (Vanaalli et al., 1999). hi effect i not conidered in thi aer. Hyterei etween the drying and wetting oil-water characteritic curve i uually conidered to e too imortant to ignore. herefore, a wetting curve mut e added and thi curve i controlled y the water entry value we and ha a imilar loe to that otained for drying, λ w (ee Figure 9). A erie of arallel line having a loe κ w, are ued to rereent recoverale change in S r etween the drying (deortion) and the wetting (adortion) curve. hee curve are called "canning curve". For the uroe of thi tudy, the loe of the canning curve i aumed to e identical to the loe of the drying curve for uction elow the air entry value and uction aove the reidual value. he loe of the wetting curve for uction aove the water entry value i alo aumed to e κ w (ee Figure 9). he imlification adoted here are imilar to thoe rooed in the model y Wheeler et al. (23). In the imlified model, the maximum uction that correond to full aturation i the aturation uction ( a ), not the air entry value ( ae ). Hyterei of oil-water characteritic curve can alo e exlained within the ame framework of elato-laticity (Sheng et al., 24). Under uch a framework, an unaturated tate alway lie within the main drying and wetting curve. Drying or wetting from within the hyterei loo will only caue recoverale water content change until the uction reache the main drying or wetting curve. Once oil uction reache the main drying or wetting curve, further drying or wetting will caue irrecoverale water content change. herefore, the drying and wetting curve define the oundarie of recoverale water content change and are imilar to the normal comreion line. he canning curve define the recoverale water content change and are imilar to the unloading-reloading line. On the lane, two additional oundarie can e added, rereenting the main drying and wetting curve, reectively (Figure 1). 7 Incremental tre-train relation he ultimate goal of contitutive modeling i to develo an incremental tre-train relationhi that can e imlemented in a numerical method uch a the finite element method to olve oundary value rolem. For unaturated oil, thee incremental relation can e written in the following form d e e σ D W dε = d G R dθ (31) 298

16 a in Sheng, Fredlund and Gen (26), or in the following form e e dε dσ D W = d G R dsr (32) a in Sheng et al. (24), deending the tre variale choen. It i noted that in the dilacement finite element method, the ore reure and dilacement are firt olved from equilirium and continuity equation. herefore, the train and uction increment are known, and the tre and water content increment mut e found from the contitutive equation. In uch a context, equation (31) and (32) have to e reformulated o that all known increment are on one ide of the equation. he model y Sheng, Fredlund and Gen (26) can e ued to demontrate the derivation of the incremental tre-train equation. he Modified Cam Clay model i ued a the ae model for aturated oil. he yield function then take the form of 2 2 f = q M ( )( ) (33) y he conitency condition for thi yield function can e written a: d f d f d f d f f = d σ σ y y ε v y y εv (34) he train decomoition and the flow rule can e written a: e e dε = dε + dε = dε + & Λ g (35) σ where g i the latic otential function, and Λ & i the latic multilier to e olved from equation (34). he elatic tre-uction-train relation can e written a: ( ) ( ) ( ) -1-1 e e e dε = D dσ + W d, or e e e e -1 e e e dσ = D dε D W d = D dε W d (36) where D e i the elatic tre-train tiffne matrix, = ( ) 1 W e D e W e. e W i the elatic uction-train vector, and he latic multilier can e found y relacing equation (35) and (36) into equation (34): f e f f y f e D dε + + W σ Λ σ y & = (37) f e g f y g D σ σ y ε v d he tre-train relation i then derived: 299

17 D D dε + D + W σ σ σ e y d d σ σ = D ε f e g f g D σ σ ε e g f e e g f f y f e y y v d (38) he uction-water content relation i given y d d d θ = λ + dε = λ + d wn Sr v wn Sr m ε (39) where m =(1, 1, 1,,, ). he loe λ w hould e relaced y κ w for uction change along canning curve. herefore, uing the following notation D e g f D D σ σ e e e = D, G = w or G = w f e g f y g D σ σ y ε v e g f f y f e D + W σ e y σ W =, R = S r m f e g f g D σ σ ε the final incremental tre-train relationhi can e written: y y v λ n κ n dσ De We dε = dθ R G d (4) where D e i a 6 6 matrix, R i a row vector of 6 element, W e i a column vector of 6 element, and G i a calar. he rate of oil uction i ket on the right-hand ide a the train rate, in conitent with the dilacement finite element method where ore reure and dilacement are firt olved from equilirium and continuity equation. he incremental tre-train relationhi defined y equation (4) can e imlemented into the finite element method to olve oundary value rolem. he imlementation follow Sheng et al. (23a, 23). Due to the non-convexity of the yield urface, ecial technique are required regarding the integration of the rate equation. 21

18 Elatic zone Elatic zone Elatic zone unaturated unaturated trial tre 45 o aturated aturated (a) net tre (Alono et al. 199) () effective tre (Sheng et al. 23) (c) net tre (Sheng, et al. 26) Figure 11. Non-convexity of yield urface for unaturated oil in a uction-tre ace. 8 Finite Element Imlementation he main challenge in imlementing an unaturated oil model into finite element code arie from the nonconvexity of the yield urface around the tranition etween aturated and unaturated tate. he non-convexity exit irreective of the tre variale ued in the model and i demontrated in Figure 11. For given train and uction increment, the current tre tate and internal variale mut e udated in accordance with equation (31) or (32). hi udate i generally carried out uing numerical tre integration cheme. Both imlicit and exlicit cheme have een ued to integrate elatolatic model. Imlicit cheme, where all gradient are etimated at an advanced tre tate, cannot e ued for elatolatic model with nonconvex yield urface, ecaue the extraolated gradient cannot e determined due to the uncertainty of whether an advanced oition i inide or outide the yield urface. On the other hand, exlicit cheme can roceed in an incremental fahion, ut require the interection etween the current yield urface and an elatic trial tre ath to e determined. A key iue in integrating the incremental tre-train relationhi uing an exlicit method i thu to find the interection etween the elatic trial tre and the current yield urface. he mot comlicated ituation occur when the yield urface i croed more than once. However, it i not oile to know a riori how many time the yield urface i croed, ecaue the ize of the yield urface will change after the firt interection due to hardening. herefore, for non-convex yield urface, the key tak i to find the very firt interection for any oile ath. In order to determine whether the yield urface i croed, a ecant trial tre increment can e comuted, aed on an elatic tre-uction-train relationhi. hi elatic trial tre i given a follow: tr e e Δ = Δ + Δ σ D : ε W (41) e where the tre i either the net tre or effective tre (deending on the model), D i the fourth order e elatic tiffne tenor (in tenor notation) and W i a econd order tenor defined according to a ecific law e for unaturated oil. For aturated oil model, the term W Δ deend on the tre variale ued. If the e effective tre i ued, the term W Δ ecome zero and can e diregarded. On the other hand, if the net e tre i ued, the term W Δ ecome m Δuw, where m i the econd order identity tenor and u w the ore water reure. In equation (41), Δε i the train increment rovided from the finite element routine rior to the comutation of the reidual etween internal and external force. For unaturated oil, the increment of uction Δ i alo inut for the tre-udate algorithm. If the elatic modulu i linear, i.e. it i indeendent of the tree, uction 211

19 and internal variale, it i trivial to comute the elatic trial increment. Otherwie, for ome non-linear relation, a ecant analytical modulu may e conidered. Finding the interection etween the elatic trial tre increment and the current yield urface can e cat into the rolem of finding multile root of a nonlinear equation. f α ( α ) =. he root (α ) mut e comuted inide the interval [,1]. A thi function involve the evaluation of the yield function along the train and uction ath, it i given a fα( α ) = f( σ α, α, z k ) (42) where f ( σ, z, k ) i the yield function, z k indicate a et of internal variale and the intermediate tre-uction tate σ α and α are calculated according to α tr σα = σcurrent + Δσ and α current = + αδ (43) in which σ current and current are the current tre and uction tate. Note that in equation (42) the internal variale z k are ket contant during the olution for the interection. hee variale only change during hardening/oftening when a ortion of the trial tre-uction ath i located outide the yield urface. he technique rooed herein follow the Kronecker-Picard (KP) formula for the determination of the numer of root of a nonlinear equation (Kavvadia et al., 1999). hi formula, given y ([ fα( a) gα( ) fα( ) gα( a) ]) 2 γ fα( x) hα( x) g( x) 1 γ N = dx arctan π + a fα( x) + γ gα( x) π fα( a) fα( ) + γ gα( a) gα( ) (44) require that f α ( α ) mut e continuouly or iecewie differentiale to the econd order for value of α from a to. In equation (44), g α and h α rereent the firt and econd derivative of the function f α with reect to α, reectively, and γ i a mall oitive contant which doe not affect the reult comuted with the KP formula (Kavvadia et al., 1999). he firt and econd derivative of f α with reect to α can e directly determined a follow: f f dσ f d f f gα ( α) : : α σ dα dα σ α α α α α tr = = + = Δ σ + Δ α α α α (45) f f f f hα ( α) = = Δ : : Δ + 2 Δ : Δ + Δ α tr tr tr 2 σ σ σ 2 2 α σσ σ α α α (46) he numer of root etimated according to equation (44) i ued to divide the interval of α into uinterval until each uinterval contain at mot one root. Firt, N i comuted for the interval [a, ]. If N i larger than one, the a,( a )/2 ( a+ )/2,. he numer of root for interval [a, ] i divided into two equal uinterval, [ + ] and [ ] each uinterval i then comuted and any uinterval that contain more than one root i further divided into two equal u- uinterval. hi roce continue until each uinterval contain at mot one root. A hown y Kavvadia et al. (1999), the uage of equal-ize interval (equiroale art) i not much wore than an algorithm which would conider the tatitical ditriution of the root inide [ a, ], uch a the algorithm reented in Kavvadia et al. (1999). Once the root are racket, the olution of each root can e found y uing exiting numerical method uch a the Newton-Rahon method. It hould e noted that the Newton-Rahon method, although fat, may not converge in ome circumtance ecaue it doe not contrain the olution to lie within ecified ound. herefore, more advanced method can e ued. For examle, the Pegau method ued in Sloan et al. (21) i rout and cometitively fat. he method y Brent (1971) rovide another attractive alternative. he Brent method doe not ue any derivative, doe not require initial guee and guarantee the convergence a long a 212

20 the value of the function are comutale within a given region containing a root. hi characteritic of the Brent method i due to the comination of the iection method, the ecant method and invere quadratic interolation. herefore, it ha the reliaility of the iection method and the efficiency of the le reliale ecant method or invere quadratic interolation. he evaluation of the integral in equation (44) with the KP formula i generally not trivial and o a numerical integration or quadrature technique ha to e ued. For examle, the Gau-Legendre method (Forythe et al., 199) can e ued here. In addition, for highly non-linear yield function, an adative integration cheme may alo have to e ued. Detailed information in thi regard can e found in Pedroo et al (27) and Sheng et al. (27a). 9 Concluion A numer of concluion can e drawn from thi tudy: 1. he ue of an effective tre for unaturated oil can lead to a mooth tranition etween aturated and unaturated tate. However, the key iue i that the effective tre uually ecome a material roerty and even deend on the material tate when the oil ecome unaturated. herefore, a contitutive relation etalihed in uch an effective tre ace i le meaningful, ince the tre ace i contantly changing with the material tate. 2. he ue of indeendent tree for modelling unaturated oil often lead to dicontinuou model at the tranition etween aturated and unaturated tate. However, thi rolem can e avoided if the volumetric tre train model i continuou acro the tranition. 3. Mot exiting model have difficultie in modelling oil dried from lurry. he tre tate during the drying of a lurry oil hould alway e on the current yield urface. he normal comreion line under contant uction for oil dried from lurry are naturally curved and thi curvature i not due to over-conolidation. 4. Mot elatolatic model have emodied hear trength criteria. For examle in Barcelona Baic model, the friction angle φ due to uction i aumed to e contant. In the model y Sheng, Fredlund and Gen (26), the friction angle φ i a function of uction and the air entry value. 5. Hyterei in the oil-water characteritic curve can e formulated in the ame framework of elatolaticity, which lead to a conitent formulation of tre train and uction aturation relation. 6. Unaturated oil model have inevitaly non-convex yield urface at the tranition etween aturated and unaturated tate. hi non-convexity can ignificantly comlicate the imlementation of thee model into finite element code. 7. An exlicit tre integration cheme incororating an efficient root earch algorithm can e ued integrate an unaturated oil model with a non-convex yield urface. 1 Reference Aitchion GD and Donald IB, 1956, Effective tree in unaturated oil, in Proc. 2 nd Autralian New Zealand Conf. Soil Mechanic, Aitchion GD and Martin R, 1973, A memrane oedometer for comlex tre train tudie in exanive clay, in Proc. 3 rd Int. Conf. Exanive Soil (Haifa, Irael), Vol. 2, Alono EE, Gen A and Joa A, 199, A contitutive model for artially aturated oil, Geotechnique, 4(3), Alono EE and Lloret A, 1982, Behaviour of artially aturated oil in undrained loading and te y te emankment contruction, in Proc. IUAM Conf. Deformation and Failure of Grannular Material (Delft, he Netherland), Bao C, Gong B, and Zhan L, 1998, Proertie of unaturated oil and loe taility of exanive oil, Keynote Lecture, Proceeding of the 2nd International Conference on Unaturated Soil (UNSA 98). Beijing, China, Vol.1, Biot, MA, 1941, General theory of three-dimenional conolidation, Journal of Alied Phyic, 12(2), Biho AW, 1959, he rincile of effective tre, eknik Ukelad, 16(39), Biho AW, Blight GE, 1963, Some aect of effective tre in aturated and artly aturated oil, Geotechnique, 13,