Computers and Geotechnics 38 (2011) 648–658

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Computers and Geotechnics

journal homepage: www.elsevier .com/ locate/compgeo

A critical state model for overconsolidated structured clays

Jirayut Suebsuk a, Suksun Horpibulsuk b,⇑, Martin D. Liu c

a Department of Civil Engineering, Rajamangala University of Technology Isan, Nakhon Ratchasima, Thailandb School of Civil Engineering, Suranaree University of Technology, Nakhon Ratchasima, Thailandc Faculty of Engineering, University of Wollongong, Australia

a r t i c l e i n f o

Article history:Received 16 August 2010Received in revised form 30 March 2011Accepted 30 March 2011Available online 6 May 2011

Keywords:Constitutive modelPlasticityBounding surface theoryStructured claysDestructuring

0266-352X/$ - see front matter � 2011 Elsevier Ltd.doi:10.1016/j.compgeo.2011.03.010

⇑ Corresponding author. Tel.: +66 44 22 4322; fax:E-mail addresses: suksun@g.sut.ac.th, suksun@yah

a b s t r a c t

This paper presents a generalised critical state model with the bounding surface theory for simulating thestress–strain behaviour of overconsolidated structured clays. The model is formulated based on theframework of the Structured Cam Clay (SCC) model and is designated as the Modified Structured CamClay with Bounding Surface Theory (MSCC-B) model. The hardening and destructuring processes forstructured clays in the overconsolidated state can be described by the proposed model. The image stresspoint defined by the radial mapping rule is used to determine the plastic hardening modulus, which var-ies along loading paths. A new proposed parameter h, which depends on the material characteristics, isintroduced into the plastic hardening modulus equation to take the soil behaviour into account in theoverconsolidated state. The MSCC-B model is finally evaluated in light of the model performance by com-parisons with the measured data of both naturally and artificially structured clays under compressionand shearing tests. From the comparisons, it is found that the MSCC-B model gives reasonable good sim-ulations of mechanical response of structured clays in both drained and undrained conditions. With itssimplicity and performance, the MSCC-B model is regarded as a practical geotechnical model for imple-mentation in numerical analysis.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Reconstituted clay behaviour has been studied for the last fourdecades. The critical state theory is one of the major developmentsof the constitutive model for clays. The models of the Cam Clayfamily such as the Cam Clay (CC) model [40] and the ModifiedCam Clay (MCC) model [39] are used widely to describe claybehaviour in the reconstituted state. The models give good agree-ment between experimental and model simulation results. How-ever, recent studies [26,4,27,7,8,43,19] have shown that naturalclay is structured and its behaviour is different from its behaviourin the reconstituted state. Cement stabilised clay is also identifiedas artificially structured clay [33,16,18]. The critical state theory,which is widely accepted for simulating clay behaviour [42,35],has been extended to simulate structured clay behaviour by con-sidering the effect of soil structure [14,21,22,41,30,3,25]. Taiebatet al. [46] have proposed a model taking into account the effectof destructuring and anisotropy on the stress–strain response.

The Structured Cam Clay (SCC) model [30] is a simple and pre-dictive critical state model for structured clays. It was formulatedsimply by introducing the influences of soil structure on the volu-metric deformation and the plastic deviatoric strain into the MCC

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+66 44 22 4607.oo.com (S. Horpibulsuk).

model. A key assumption of the SCC model is that both the harden-ing rule and destructuring of structured clay are dependent onplastic volumetric deformation. A simple elliptical yield surfacewith a non-associated flow rule is used for predicting the behav-iour of clays in naturally structured states. The SCC model success-fully captures many important features (particularly volumetrichardening/destructuring) of the naturally structured clay behav-iour with insignificant cohesion.

However, the cohesion and strain softening, which are generallyvery significant for naturally structured stiff clays [4,27,8] andartificially structured clays such as cement-stabilised clays [48,5,21,18], cannot be taken into account by the SCC model. For bettersimulation of cemented clays, the modified effective concept wasintroduced into the SCC model [17].

Suebsuk et al. [45] developed a general constitutive modelbased on the critical state framework for destructured, naturallystructured and artificially structured clays. The proposed modelis the Modified Structured Cam Clay (MSCC) model. It was formu-lated based on the SCC model for cemented clay [17]. In the MSCCmodel, the influence of soil structure and destructuring were intro-duced into the yield function, plastic potential, and hardening ruleto describe the mechanical behaviour of structured clays. The soilstructure increases the mean effective yield stress under isotropiccompression (p0y) and structure strength (p0b). The destructuringmechanism in the MSCC model is the process of reducing the

Nomenclature

a image stress ratiob destructuring index due to the volumetric deformationCSL critical state lineDe additional voids ratio sustained by soil structureDei additional voids ratio sustained by soil structure at the

initial virgin yieldingev volumetric strainep

v plastic volumetric strainep

d plastic deviatoric strained deviatoric straine voids ratioe�IC voids ratio at reference stress (1 kPa) on ICLG0 shear modulus in terms of effective stressh non-dimensional parameter, describing the effect of

material characteristics on the hardening modulusICL intrinsic compression line (of destructured soil)K0 bulk modulus in terms of effective stressj gradient of unloading or swelling line of structured

claysk� gradient of normal compression line of destructured

clays

l0 Poisson ratio in terms of effective stressM gradient of critical state line on q–p0 plane�g modified stress ratio, q=ðp0 þ p0bÞw parameter describing the shape of the plastic potentialp0 mean effective stressp0b mean effective stress increasing due to structure or

structure strengthp0b0 initial of structure strength in the q–p0 planep0c reference size of the loading surfacep00 reference size of yield surface or stress historyp00j reference size of the structural bounding surfacep0j mean effective stress at image stress pointp0y;i initial yield stress of isotropic compression line of struc-

tured soilq deviatoric stressqj deviatoric stress at image stress pointSCL structured compression linen destructuring index due to the shear deformationYSRiso isotropic yield stress ratio, p0y=p0 [6]

J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658 649

structure strength, p0b due to the degradation and the crushing ofstructure. The destructuring behaviour of the MSCC model is di-vided into two parts for volumetric deformation and shearing.The destructuring due to shearing causes a reduction in structurestrength that is directly related to the plastic deviatoric strain, ep

d.The strain softening can be described by the MSCC model whenthe stress states are on the state boundary surface.

In the MSCC model, a yield surface separates elastic behaviour(i.e., stress states inside the yield surface) from elastoplastic behav-iour (i.e., stress states on the yield surface). However, in reality,even in the overconsolidated state, naturally structured clays, arti-ficially structured clays and many geomaterials often exhibit anon-recoverable behaviour upon unloading and repeated loading[4], which results in overestimation of the soil strength and thelack of a smooth transition between elastic and elastoplasticbehaviour. To obtain accurate predictions of the deformational re-sponse, it is necessary to improve the MSCC model to better simu-late the soil behaviour in the overconsolidated state. It is notedherein that the traditional term ‘‘overconsolidated state’’ is alsoused for structured clay to indicate the relative position of the cur-rent stress in relation to its limit state surface or bounding surfacethat the yield stress (stress history) is caused by both mechanicaland chemical effects.

There are two basic theories that have been widely used inmodelling the plastic deformation of soils for loading inside thelimit surface. One is the multi-surface plasticity (MS) theory, whichwas originally applied to metal with the kinemetic hardening (KH)rule [20,34] and then the Mroz’s multi-surface plasticity conceptwas applied to soil stress–strain modelling by Prevost (1977,1978). The other is the bounding surface plasticity (BS) theory,which was formulated with the kinemetic hardening rule[13,9,24]. Those concepts have been extended to many soil modelsfor geomaterials (e.g.,[1,44,22,41,38,32,3]. However, the major set-back of MS theory and KH rule is that the numerical computation iscomplex and requires considerable memory for the configurationof the sub-yield and stress reversal surface. The bounding surfacemodel with the radial mapping rule proposed by Dafalias and Herr-mann [11] and Dafalias [10] removed the above deficiencies. Manyadvanced soil models have been developed based on the radialmapping rule [2,47,52,50,51]. This version of bounding surface

plasticity is introduced in the current modelling of plastic deforma-tion of structured soil inside the bounding surface.

To keep a model simple for practical use, it is impossible to in-clude all features of the soil behaviour in a model. A complex mod-el with many features should be avoided because it may havehidden inaccuracies, numerical instabilities, lack of a convergedsolution and other errors [49]. In this paper, the bounding surfacetheory with radial mapping rule was employed with the MSCCmodel to simulate the stress–strain behaviour of overconsolidatedstructured clays because this theory is simple and predictive. Theproposed model is called the ‘‘Modified Structured Cam Clay withBounding Surface Theory’’ (MSCC-B) model. The MSCC-B model ispresented in a four-dimensional space consisting of the currentstress state, the current voids ratio, the stress history and the cur-rent soil structure. The MSCC-B model requires six parameters inaddition to the standard parameters from the MCC model. The fiveparameters (b, Dei, p0b0, n and w) are the same as those of the MSCCmodel and have clear physical meaning. A new constant materialparameter, h, is proposed in the study to take into account the ef-fect of the material characteristics on the plastic hardening modu-lus in the overconsolidation state.

Finally, the MSCC-B model is evaluated in the light of the modelperformance. The MSCC-B model was implemented in a single ele-ment elastoplastic calculation for simulating the behaviour ofstructured clays in both naturally and artificially structured states.Test data over a wide range of the mean effective stress and degreeof cementation under both drained and undrained shearing wereused in the verification.

2. MSCC: a generalised critical state model for structured clays

The MSCC model was developed based on the following keyassumptions: (1) the reconstituted or destructured soil behaviourcan be described adequately by the MCC model with the intrinsicproperties obtained from a reconstituted sample, (2) the structuredsoil behaviour is divided into the elastic and virgin yielding regionsby the current yield surface and (3) the destructuring is only re-lated to the plastic strain (insignificant for stress state inside theyield surface).

650 J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658

The material idealisation of structured clay for the MSCC modelis shown in Fig. 1. The current yield surface (the structural yieldsurface) is defined by its current stress state (q, p0), stress history(p00) and structure strength (p0b). The structural yield surface inthe q–p0 plane is assumed to be elliptical in shape and passesthrough the origin of the stress coordinates. The p0b affects the ori-gin of structural yield surface that moves to the left side of themean effective stress axis. The structural yield surface is thus givenin terms of the conventional triaxial parameters as follows:

f ¼ q2 �M2ðp0 þ p0bÞðp00 � p0Þ ¼ 0; ð1Þ

where p00 is the reference size of the yield surface and M is the gra-dient of failure envelope in the q–p0 plane.

The elastic behaviour inside the yield surface is assumed to bethe same as in traditional Cam Clay family. It is defined by the bulkmodulus (K0) and shear modulus (G0). These parameters are deter-mined by the following basic equations:

K 0 ¼ ð1þ eÞj

p0; ð2Þ

Fig. 2. Influence of the w on the shape of plastic potential.

G0 ¼ 3ð1� 2l0Þ2ð1þ l0Þ K 0; ð3Þ

where j is a gradient of unloading or swelling line of structuredclay, l0 is Poisson’s ratio and e is the voids ratio.

The plastic potential with the influence of soil structure takesthe form

g ¼ q2 þ M2

1� wp0 þ p0bp0p þ p0b

!2w

p0p þ p0b� �2

� ðp0 þ p0bÞ2

24

35 ¼ 0; ð4Þ

where p0p is a reference size parameter and w is a parameter describ-ing the shape of the plastic potential. The p0p can be determined forany stress state (q, p0) by solving the above equation. Fig. 2 shows

Fig. 1. The material idealisation of structured clays for the MSCC model.

the shape of the plastic potentials, which pass through a stress state(q1;p

01).

Fig. 3. Idealisation of bounding surface plasticity model for isotropic compressionof structured clays.

Fig. 4. Mapping rule for the MSCC-B model.

Fig. 5. Influence of h on the soil behaviour under the undrained shearing test: (a)the stress path, (b) the stress–strain behaviour and (c) the development of excesspore pressure.

Fig. 6. Influence of h on the a—epv curve under the drained shearing test.

J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658 651

The voids ratio, which is a state parameter for the derivation ofthe hardening rule during virgin compression along a generalstress path, is proposed as follows:

e ¼ e�IC � j ln p0 þ Dei

p0y;ip00

� �b

� ðk� � jÞ ln p00; ð5Þ

where b is a destructuring index due to the volumetric deformation,p0y;i is the initial isotropic yield stress and Dei is an additional voidsratio between the intrinsic compression line (ICL) and the struc-tured compression line (SCL) at the p0y;i (Fig. 1a). Eq. (5) shows thatas the stress increases, De approaches zero. The rate of the removalof soil structure, i.e., represented by De, is dependent on destructur-ing index b (Liu and Carter [28,29]). For the deformation range ofexperimental data and practical applications concerned, Eq. (5) isvalid and effective for determination of volumetric deformation.

Based on the voids ratio in Eq. (5), the plastic volumetric strainincrement can be formulated as follows:

depv ¼ ðk� � jÞ þ M

M � �g

� �bDe

� �dp00

ð1þ eÞp00: ð6Þ

The derivation of this equation can be found in the papers by Liuand Carter [30,31]. In the MSCC model, the destructuring law isproposed to describe the reduction in the modified effective stress[45]. Destructuring during shearing consists of two processes: deg-radation of structure and crushing of soil–cementation structure.The degradation of structure occurs when the stress state is onthe yield surface whereas the crushing of the soil–cementationstructure happens at post-failure during strain softening. The p0bis assumed to be constant up to the virgin yielding. During virginyielding (during which plastic deviatoric strain occurs), the p0bgradually decreases due to degradation of the structure until thefailure state. Beyond this state, a sudden decrease in the p0b occursdue to the crushing of the soil–cementation structure and dimin-ishes at the critical state. The reduction in p0b due to the degrada-tion of structure (pre-failure) and the crushing of the soil–cementation structure (post-failure) are proposed in terms of theplastic deviatoric strain as follows:

p0b ¼ p0b0 expð�epdÞ for pre-failure ðhardening and destructuringÞ;

ð7Þ

p0b ¼ p0b;f exp �nðepd � ep

d;f Þh i

for post-failure ðcrushingÞ; ð8Þ

where p0b0 is the initial structure strength, p0b;f is the structurestrength at failure (peak strength), ep

d;f is the plastic deviatoric strainat failure and n is the destructuring index due to shearing in

Table 1Parameters for a parametric study on the effect of h.

Parameter/symbol Physical meaning Value Unit

k⁄ Gradient of ICL in e�ln p0 plane 0.147 –e�IC Voids ratio at reference stress (p0 = 1 kPa) on ICL 1.92 –j Gradient of unloading–reloading line of SCL in e�ln p’ 0.027 –b Non-dimension parameter describing the destructuring by volumetric plastic strain 0.6 –Dei Additional of voids ratio between ICL and SCL at yield stress 1.92 –p0y Mean effective yield stress on SCL 100 kPaM Gradient of critical state line in q–p0 plane 1.2 –G0 Shear modulus 3000 kPaw Non-dimension constant parameter define the shape of plastic potential 2.0 –p0b0 Initial structure strength on in q–p0 plane 20 kPan Non-dimension constant parameter describing the destructuring by shearing 10 –

Table 2Physical properties of the base clays.

Properties Pappadai clay Ariake clay

Reference Cotecchia [6] and Cotecchia andChandler [8]

Horpibulsuk[15,18]

Specific density 2.73 2.70Natural water

content (%)29.8–32.6 135–150

Liquid limit (%) 65 120Plasticity index (%) 35 63Activity 0.42–0.72 1.24–1.47Clay fraction (%) 50 55Silt fraction (%) – 44Sand fraction (%) – 1In-situ voids ratio 0.68–0.90 3.65–4.05Sample type Intact Cemented

652 J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658

post-failure. The formulation and parametric study of the materialparameters of MSCC have been presented clearly in the paper bySuebsuk et al. [45].

3. MSCC-B: a critical state model for overconsolidatedstructured clays

In 1975, Dafalias and Popov firstly proposed the concept ofbounding surface plasticity. Then a bounding surface theory withradial mapping rule was proposed by Dafalias and Herrmann[11] and Dafalias [10]. This version of bounding surface plasticity,

Table 3Testing programme for triaxial tests on the base clays.

Base clay Sample No. YSRiso p0 before shearing(kPa)

Intact Pappadai clay TN14 4.52 500TN15 2.825 800TN18 1.50 1500TN17 1.00 2500TN5 3.22 700TN3 2.167 1042TN11 1.40 1600

Cemented Ariakeclay

AW6-50 1.40 50

AW18-400 5.00 400AW18-500 4.00 500AW18-1000

2.00 1000

AW9-100 3.70 100AW9-200 1.85 200AW18-200 10.00 200AW18-400 5.00 400

Remarks: U = undrained test and D = drained test.* dea/dt.

** mm/min.

i.e., the bounding surface model with radial mapping rule, is intro-duced in the modelling of plastic deformation of structured soil in-side the bounding surface. For structured clay, the boundingsurface is referred to as the structural bounding surface, which isaffected by the soil structure. The variation of the structuralbounding surface is dependent on destructuring as well as harden-ing, both of which are assumed to be determined due to the changein plastic volumetric deformation and plastic deviatoric deforma-tion following that of classic plasticity [42,31].

Based on the bounding surface theory with the radial mappingrule and the framework of the MSCC model, Fig. 3 shows the sche-matic diagram for illustrating the change of stress state (e�ln p0

plane and q–p0 plane) under isotropic compression loading for path1–6. Point 1 is an initial in situ state for a clay whose structuralbounding surface is defined by isotropic yield stress on SCL, p0y;iand structure strength, p0b. As loading continues along stress path1–2, subyielding occurs, and the loading surface expands (insidethe structural bounding surface). Supposing that the loading sur-face and the structural bounding surface coincide at point 2, thatis, p0c;2 ¼ p00j;2 (where p0c;2 and p00j;2 are the reference size of the load-ing and structural bounding surfaces at point 2, respectively), vir-gin yielding commences at this point and continues along thepath 2–3.

Unloading starts at point 3 and continues along the stress path3–4, i.e., dp0c < 0 and p0c – p00j. Subsequently, the current loadingsurface enters inside the structural bounding surface. When thestress path changes direction again at point 4 with dp0c > 0;

e beforeshearing

Shearingcondition

Strain or displacementrate

0.875 D 0.2–0.6%/day*

0.8725 D 0.189%/day*

0.839 D 0.4%/day*

0.766 D 0.189%/day*

0.895 U 4%/day*

0.862 U 4.3%/day*

0.856 U 5%/day*

4.098 D 0.0025**

3.673 D 0.0025**

3.668 D 0.0025**

3.657 D 0.0025**

3.650 U 0.0075**

3.589 U 0.0075**

3.707 U 0.0075**

3.673 U 0.0075**

J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658 653

reloading then commences and continues along stress path 4–5.The perfectly hysteresis loop is assumed for the MSCC-B model,and hence p0c;3 ¼ p0c;5. At point 5, the loading surface coincides withthe structural bounding surface, i.e., p0c;5 ¼ p0oj;5. In this case, virginyielding recommences at point 5 and continues for loading alongstress path 5–6. For isotropic condition, the expansion and contrac-tion of the loading surface is only dependent on the plastic defor-mation. The change of stress state under shearing can be explainedin the same way, but p0b0 decreases due to the destructuring law(Eqs. (7) and (8)). The plastic deviatoric strain influences the stresspath direction when g > 0.

Table 4Parameters for MSCC-B model for intact Pappadai clay.

Test type Parameter/symbol Unit Value

Isotropic compression test k* – 0.206e�IC – 3.17j – 0.009b – 0.1Dei – 0.32p0y kPa 2300

Drained or undrained shearing test h – 1000M – 0.83G0 kPa 40,000p0b0 kPa 480n – 10w – 2.0

* Tested result on reconstituted sample.

Table 5Parameters for MSCC-B model for cemented Ariake clay.

Test type Parameter/symbol

Unit Aw = 6% Aw = 9% Aw = 18%

Isotropic compressiontest

k* – 0.44 0.44 0.44

e�IC – 4.37 4.37 4.37j – 0.06 0.03 0.01b – 0.05 0.01 0.001Dei – 2.0 2.5 2.72p00 kPa 70 370 2000

Drained or undrainedshearing test

h – 100 800 1000

M – 1.44 1.73 1.32G0 kPa 4000 8000 40,000p0b0 kPa 50 75 730n – 10 10 10w – 0.9 0.5 0.1

* Tested result on reconstituted sample.

Fig. 7. Comparison of measured and simulated isotropic compression tests of intactPappadai clay.

3.1. Bounding surface theory with radial mapping rule

The bounding surface theory for modelling the elastoplasticdeformation of material with the radial mapping rule [47,52] wasextended for this study. The structural bounding surface is definedby the stress history, p00j, and structure strength, p0b. The clay behav-iour is divided into the virgin yielding (p0c ¼ p00j) and subyielding(p0c < p00j), where p0c is a reference size of loading surface. Similarto the MSCC model, the equation for the structural bounding sur-face is assumed to be elliptical and can be expressed as follows:

f ¼ q2 �M2ðp0 þ p0bÞðp00j � p0Þ ¼ 0: ð9Þ

The loading surface is defined as the surface on which the cur-rent stress state remains. For simplicity, the loading surface is as-sumed to be the same shape as the structural bounding surfaceand with an aspect ratio similarly equal to M.

3.1.1. Plastic potentialThe plastic potential adopted in the MSCC-B model is the same

as that proposed in the MSCC model as shown in Eq. (4).

3.1.2. Destructuring lawThe volumetric hardening and destructuring law presented in

the previous section (Eq. (6)) were adopted in the formulation ofthe hardening modulus for the MSCC-B model with the assumptionthat p00j ¼ p00 and dp00j ¼ dp00. The destructuring law due to shearingpresented in a previous section (Eqs. (7) and (8)) is used in theMSCC-B model.

3.1.3. Mapping ruleThe mapping rule is an essential part of a bounding surface the-

ory based on which the stiffness of soil deformation during sub-yielding state is defined. A detailed discussion on the significanceof mapping rule can be found in a paper by Yang et al. [51]. The ori-ginal radial mapping rule was modified for structured clay in thisstudy since this method is both simple and effective for many con-ventional stress paths. The image stress point on the structuralbounding surface is used for determining the plastic volumetricstrain. The schematic diagram for the mapping rule is shown inFig. 4. An association between any stress point and the image stresspoint is described by the intersection of the structural boundingsurface with the straight line passing through the origin and thecurrent stress state, based on the assumption that the hardeningmodulus at the current stress point (H) is related to the hardeningmodulus at its corresponding image stress point (Hj) as well as tothe ratio of the image stress ratio (a). The variation of a is assumedto be a function of both the current stress state (q, p0) and theimage stress state (qj; p0j) as follows:

Fig. 8. Comparison of measured and simulated isotropic compression tests ofcemented Ariake clay.

Fig. 9. Comparison of measured and simulated CID test of intact Pappadai clay: (a)stress ratio–deviatoric strain curve and (b) volumetric strain–deviatoric straincurve.

Fig. 10. Comparison of measured and simulated CID test of cemented Ariake claywith 6% cement content: (a) stress ratio–deviatoric strain curve and (b) volumetricstrain–deviatoric strain curve.

654 J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp0=p0jÞ þ ðq=qjÞ

2

s; ð10Þ

with 0 6 a 6 1. a for any p0 and q can be computed from Eq. (10)when p0j and qj are known. The p0j and qj values for any p00j are deter-mined from the structural bounding surface (Eq. (9)). The parameter

a becomes the unique value (a ¼ p0

p0j¼ q

qj¼ p00

p00j

) at the critical state

when p0b is zero due to the complete removal of cementation struc-ture (destructuring).

3.2. Hardening modulus

3.2.1. Hardening modulus at an image stress point (Hj)Based on the state boundary surface theory, the variation of the

structural bounding surface (dp00j) is related to the plastic volumet-ric strain increment (dep

v ). It is developed based on Eq. (6) asfollows:

dp00j ¼ð1þ eÞp00j

ðk� � jÞ þ MM � �g

� �bDe

depv : ð11Þ

The differential form of the structural bounding surface is

@f@p0

dp0j þ@f@q

dqj þ@f@p00j

dp00j ¼ 0: ð12Þ

Differentiating Eq. (9) with respect to p00j;

@f@p00j

¼ �M2ðp0 þ p0bÞ: ð13Þ

Substituting Eqs. (11) and (13) into Eq. (12),

@f@p0

dp0j þ@f@q

dqj �M2ðp0 þ p0bÞð1þ eÞp00j

ðk� � jÞ þ MM � �g

� �bDe

depv ¼ 0:

ð14Þ

The plastic strain increment in the subyielding state is definedin terms of the hardening modulus, yield function and plastic po-tential in the same way as that in the virgin yielding state as pre-viously successfully done by Dafalias and Herrmann [11], Bardet[2] and Yu et al. [52]. Thus,

depv ¼

1Hj

@F@p0

dp0j þ@F@q

dqj

� �@g@p0

j

!: ð15Þ

Substituting Eq. (15) into Eq. (14),

Hj ¼ M2ðp0 þ p0bÞð1þ eÞp00j

ðk� � jÞ þ MM � �g

� �bDe

@g@p0

j

!; ð16Þ

where@g@p0

j

are given as

@g@p0

j

¼ M2

1� w

2p0j þ p0bp0p þ p0b

!p0p þ p0b� �2

wðp0j þ p0bÞ� 2 p0j þ p0b

� �0BBBB@

1CCCCA: ð17Þ

3.2.2. Hardening modulus at the stress point (H)A specific feature of the bounding surface theory is that the

hardening modulus (H) is not only dependent on the location of

J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658 655

the image point but also on a function of the distance from thestress point to the structural bounding surface with the followingrequirements [12,23,52]:

H ¼ þ1 if a ¼ 0; ð18aÞH ¼ Hj if a ¼ 1: ð18bÞ

The restriction imposed by Eqs. (18a) and (18b) ensures that theclay behaviour is almost purely elastic when the stress state is faraway from the structural bounding surface and that the stress pointand the structural bounding surface move together when the cur-rent stress state lies on the structural bounding surface. The harden-ing modulus for the MSCC-B model under monotonic loadingcondition is proposed to satisfy the restriction Eqs. (18a) and(18b) as follows:

H ¼ Hj þ ðh � Hj;iÞ �ð1� aÞ2

a; ð19Þ

where h is the non-dimensional parameter describing the effect ofmaterial characteristics on H and Hj,i is the initial hardening modu-lus at the image stress point, which is calculated by Eq. (16).

3.3. Model parameters

There are 12 parameters for the MSCC-B model. Six parameters(k�, e�IC , j, M, p00, G0 or l0) are the basic parameters adopted from theMCC model. Five parameters (b, Dei, p0b0, n and w) are the structuralparameters describing the hardening and destructuring behaviourfor the original MSCC model. The last parameter is h, which is new-ly presented in this paper to describe the effect of the materialcharacteristics on the hardening modulus in the overconsolidatedstate. The parameters denoted by a superscript ⁄ are the

Fig. 11. Comparison of measured and simulated CID test of cemented Ariake claywith 18% cement content: (a) stress ratio–deviatoric strain curve and (b) volumetricstrain–deviatoric strain curve.

parameters tested with reconstituted samples [4]. The details ofthe parameter determination can be found in the papers by Horpi-bulsuk et al. [17] and Suebsuk et al. [45].

The effect of h on the shear behaviour is illustrated in Figs. 5 and6 using the model parameters listed in Table 1. Fig. 5a shows theeffect of h on the undrained stress path. It is clear that h signifi-cantly affects the hardening and destructuring behaviours forstructured clay in the overconsolidated state. An increase in h in-creases both the stiffness and strength of the clay, as shown inFig. 5a and b. The smoothness of the excess pore pressure–devia-toric strain curve is illustrated in Fig. 5c for various h. At the samedeviatoric stress, a decrease in h leads to a larger deformation pro-duced by the shearing. Fig. 6 presents the influence of h on the rela-tionship between a—ep

v under drained shearing. The parameter hinsignificantly affects the soil stiffness when it is smaller than 1.

The parametric studies (Figs. 5 and 6) show that the destructur-ing depends on both ep

v and ed. Its behaviour can be predicted byEqs. (6)–(8). The plastic volumetric strain occurs at the early stageof loading. If a < 1, the stress state stays inside the structuralbounding surface, and the destructuring is caused by the changein plastic volumetric strain due to dp0c and the degradation of soilstructure due to the deviatoric strain. The destructuring graduallyoccurs as shown by the smooth stress–strain relationship. Whena = 1, the loading surface and structural bounding surface coincide,and the MSCC-B and MSCC models are the same.

4. Performance of the MSCC-B model

In this section, the MSCC-B model is verified by a comparisonbetween measured and simulated data. The tested results of intact

Fig. 12. Comparison of measured and simulated CIU test results of intact Pappadaiclay: (a) deviatoric stress–deviatoric strain curve and (b) excess pore pressure–deviatoric strain curve.

656 J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658

Pappadai clay [6,8] and cemented Ariake clay [15,18] were used forthis verification.

The physical properties of both clays are presented in Table 2.The testing programmes of the Pappadai and cemented Ariakeclays are summarised in Tables 3. The model parameters were ob-tained with the following steps:

(a) The isotropic compression test results of structured samplesand reconstituted (remoulded) samples were taken to deter-mine compression parameters. The k� and e�IC were deter-mined from the intrinsic compression line, ICL [4]. Thethree additional structural parameters (p0y;i, Dei and b) weredetermined from the structured compression line, SCL[30,45]. The j was determined from the unload–reloadingline of the structured sample.

(b) Undrained and drained shearing test results of isotropic con-solidated structured clay were taken to determine theparameters (M, p0b0, G0, w, n and h) at various isotropic yieldstress ratios (YSRiso). The gradient of critical state line Mand initial structure strength p0b0 was obtained from thestress path (vide Fig. 1b). The shear modulus G0 was approx-imated from the q–ed curve. The parameters w and n aredependent on the type of clays and the degree of cementa-tion [45]. The parameter w were estimated from the shapeof the structural bounding surface. For natural clay, the plas-tic potential is approximately elliptical, and w can be takenas 2. For cemented clay, it is usually less than 2 [45]. Theparameter n was estimated from the stress–strain response,and its value is 1 6 n 6 100 [45]. The parameter h wasobtained by plotting the simulated a–ep

v curve comparedwith measured data.

Fig. 13. Comparison of measured and simulated CIU test of cemented Ariake claywith 9% cement content: (a) deviatoric stress–deviatoric strain curve and (b) excesspore pressure–deviatoric strain curve.

Tables 4 and 5 present the input parameters selected for intactPappadia clay and cemented Ariake clay, respectively.

4.1. Isotropic compression

Fig. 7 shows the comparison between the measured and simu-lated data for the isotropic compression of intact Pappadai clay.The tested data can be captured well by the model simulation.The smooth e� ln p0 curve for pre- and post-yield states isseen. The meta-stable voids ratio, which cannot be degraded withthe increase in confining stress, is predicted by the MSCC-B model.The lower the b, the smaller the destructuring. Fig. 8 compares themodel simulation results of the isotropic compression curves withthe measured data for the cemented Ariake clay at different ce-ment contents. The simulations were performed with a single setof intrinsic parameters. The results show that samples with highercement content exhibit smaller destructuring, which is controlledby b. The MSCC-B model simulation broadly matches the experi-mental results better than the original MSCC model, which cannotsimulate the subyielding behaviour. From the comparisons, it isseen that the destructuring by shearing does not affect the isotro-pic compression response, and thus the simulation of isotropiccompression is only controlled by six parameters (k�, e�IC , j, p0y;i, band Dei).

4.2. Drained triaxial shearing

The results of the model simulation of isotropically consoli-dated drained compression (CID) shearing of the intact Pappadaiclay are shown in Fig. 9. For moderate to large deviatoric strain

ig. 14. Comparison of measured and simulated CIU test of cemented Ariake clayith 18% cement content: (a) deviatoric stress–deviatoric strain curve and (b)

xcess pore pressure–deviatoric strain curve.

Fwe

J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658 657

(ed = 0.1–20%), a constant shear modulus is assumed throughoutthis simulation. The MSCC-B model gives more realistic predictionsthan those predicted by the original MSCC model. The smoothstress–strain relationship can be captured by the MSCC-B model.By comparison of the stress ratio–deviatoric strain curve and volu-metric strain–deviatoric strain curve, it is seen that the MSCC-Bmodel provides a good simulation.

Figs. 10 and 11 present the comparison of measured and simu-lated data for the cemented Ariake clay at different cement con-tents. The test data covers the range of cement contents from 6%to 18% with the variation of confining stress from 50 kPa to1000 kPa. The smooth stress–strain response was simulated alongthe loading path. The MSCC-B model gives good results for thestress–strain curve and the volumetric deformation curve.

Fig. 15. Comparisons of experimental and simulated on isotropic compression testresults of intact Pappadai clay by MSCC and MSCC-B models.

Fig. 16. Comparisons of experimental and simulated on CID test results of intactPappadai clay by MSCC and MSCC-B models.

ig. 17. Comparisons of experimental and simulated on CIU test results of intactappadai clay by MSCC and MSCC-B models.

FP

4.3. Undrained triaxial shearing

Fig. 12 compares the measured and simulated results of the in-tact Pappadai clay with various YSRiso from 7.50 to 1.50. Figs. 13and 14 compare the measured and simulated results of the cemen-ted Ariake clay at various YSRiso and cement contents. It is foundthat the deviatoric stress and excess pore pressure versus deviator-ic strain can be predicted well by the MSCC-B model.

From the simulation of compressibility and shearing for natu-rally and artificially structured clays, it is concluded that theMSCC-B model can describe the influence of artificial cementationstructure on soil behaviour.

4.4. Performance of the MSCC-B model compared with the originalMSCC model

To demonstrate the improvement of MSCC-B model, simula-tions of structured soil behaviour made by the MSCC-B modelare compared with those of the original MSCC model as shown inFigs. 15–17 for Pappadai clay. It is seen that there are the followingimprovement in model simulation for the MSCC-B model.

(1) The plastic deformation of soil within the yield surface iscaptured. This is especially important for highly overconsol-idated soils.

(2) The peak strength and the smooth transition from hardeningto softening behaviours, for both drained or undrained situ-ations, have been better represented by the boundary sur-face theory. This improves the overall accuracy of stressand strain relationship.

658 J. Suebsuk et al. / Computers and Geotechnics 38 (2011) 648–658

5. Conclusion

Based on an extensive review of the available experimentaldata, a critical state model with bounding surface theory todescribe the mechanical behaviour of naturally and artificiallystructured clays in the overconsolidated state was proposed. A fun-damental hypothesis of the development is that the hardening anddestructuring of structured clay are dependent on both the plasticvolumetric strain and deviatoric strain. The key feature of theMSCC-B model is that the plastic deformation for stress state insidethe yield surface can be well-captured.

The hardening modulus for structured clay was formulatedbased on the bounding surface theory with the radial mappingrule. It is suitable for describing the compression and shearingbehaviours of natural clays and cement-stabilised clays undermonotonic loading in the overconsolidated state. The MSCC-Bmodel can simulate the stress–strain-strength relationships ofstructured clays well over a wide range of YSRiso. Generally speak-ing, a reasonable good agreement between the model performanceand experimental data is achieved. The model can be used as apowerful tool for simulating structured clay behaviour in the over-consolidated state.

Acknowledgements

The first author thanks the Office of the Higher Education Com-mission, Thailand, for Ph.D. study financial support under the pro-gramme Strategic Scholarships for Frontier Research Network. Thefinancial support and facilities provided by the Suranaree Univer-sity of Technology are appreciated. The authors are grateful tothe reviewers for their constructive and useful comments, whichimprove the quality of this paper.

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