Cam Clay for Sand and Clay

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Granular Matter 6, 11–16 c Springer-Verlag 2004DOI 10.1007/s10035-003-0152-8

A generalised Modified Cam clay model for clay and sandincorporating kinematic hardening and bounding surfaceplasticity

G. R. McDowell, K. W. Hau

Abstract This paper proposes a simple non-associatedModified Cam clay model suitable for clay and sand. Theyield surface is taken to be that of Modified Cam clay,which is a simple ellipse. The modified model reduces theamount of shear strain predicted, and for clay requires nonew parameters because the flow rule uses a well estab-

lished empirical result. For sand, the critical state fric-tional dissipation constant is required in addition to thestress ratio at the peak of the yield surface. This per-mits realistic modelling of the undrained behaviour of sand in states looser and denser than critical. The modelresembles more sophisticated models with yield surfaces of more complex shapes, but is much simpler. More realisticbehaviour could be obtained by assuming a yield surfacewith the same form as the potential if required. The modelis suitable for incorporating kinematic hardening for themodelling of cyclic loading of clay. In addition, boundingsurface plasticity can be included to distinguish betweencompacted and overconsolidated sand. The contribution

in this paper is therefore to provide a generalised simplemodel based on Modified Cam clay.

Keywords Bounding surface plasticity, Clay, Kinematichardening, Non-associated flow, Plasticity, Sand, Yieldsurface

1Introduction

McDowell [1] derived a family of yield loci in triaxial stressspace, based on the idea that the relative amounts of

plastic work dissipated in friction and fracture should bea simple function of stress ratio. He used the normalitycriterion, together with a simple stress-dilatancy rule, togenerate the following family of yield loci for triaxial com-pression:

Received: 28 July 2003

G. R. McDowell (&)Senior Lecturer, University of Nottingham, UKe-mail: [email protected]

K. W. Hau

Research Student, University of Nottingham, UKThe authors are grateful to Mr C.D. Khong for discussions onthe bounding surface formulation of the CASM model.

η = M[(a + 1) ln( po/p)]

1a+1 (1)

where η is the stress ratio q/p, q is deviatoric stress (=σ1−σ3 where σ

1 and σ

3 are major and minor principal effectivestresses respectively), p is mean effective stress (= (σ1 +2σ3)/3), p

o is the isotropic preconsolidation pressure and

M is the critical state frictional dissipation constant. Thusthe equation of the yield surface is obtained by selectingan appropriate value for the parameter a in addition to thevalue of M. McDowell [1] assumed that normality applied,so that the stress-dilatancy rule is given by the equation:

dεpvdεpq

= Ma+1 − ηa+1

ηa (2)

where dεpv and dεpq are the plastic volumetric and tri-

axial shear strain increments respectively, plotted alongthe same axes as the associated stresses p and q respec-tively at the current state in stress space, to give the plas-

tic strain increment vector. McDowell [2] compared thismodel, which requires one new parameter a , to that of Lagioia et al. [3] which requires two parameters to definethe shape of the yield surface. McDowell [2] then gener-alised the model to allow non-associated flow, so that thecritical state was permitted to lie to the left of the peak of the yield surface in deviatoric:mean effective stress space;it is well known that for granular materials the criticalstate point does not occur at the top of the yield locus[4,5]. The models proposed by Chandler [4, 5], which makeuse of the mathematical theory of envelopes and microstructural considerations, are suitable for clays and sands,but require the measurement of microscopic parameters,

and have not been adopted widely by geotechnical engi-neers due to their complexity. McDowell [2] also noted thatthe model proposed by Yu [6] which gives a yield surface of the same form as (1) uses Rowe’s stress-dilatancy relation-ship [7], which gives non-associated flow under isotropicconditions: behaviour which is not observed in the liter-ature. The resulting equations for the yield surface andplastic potential, for the model proposed by McDowell [2]for sand are respectively:

η = N[(a + 1) ln( po/p)]

1a+1 (3)

η = M

(b + 1)ln pp/p

1

b+1 (4)

The parameter N is the stress ratio at the peak of theyield surface, and a controls the shape of the yield sur-face, whilst b controls the flow rule and pp is the harden-ing parameter for the potential. The model can correctly


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predict the coefficient of earth pressure at rest K o,nc (de-fined as the lateral effective stress divided by axial effectivestress) for one-dimensional normal (i.e. plastic) compres-sion, for which

dεvdεq =

dε12dε1/3 = 1.5 (5)

and the value of K o,nc is found empirically [8] to satisfythe equation

K o,nc = 1 − sin φ (6)

where φ is the angle of shearing resistance. It is found forclays that the value of φ in (6) is the critical state angleof frictionφcrit. Since

M = 6sinφcrit3 − sinφcrit

(7)

(6) and (7) imply that the stress ratio ηo,nc during one-dimensional normal compression is given by

ηo,nc ≈ 0.6 M (8)

For sand, the value of φ in (6) is less certain. Accord-ing to Muir Wood [9], for sand the value of K o,nc willdepend on the initial structure of the sand, and is there-fore likely to depend on the maximum angle of shearingresistance. However, for a sand which has yielded and isdeforming plastically under one-dimensional normal com-pression (i.e. the state lies on the state boundary surface),it would be expected that the initial structure will have

been eliminated, so that the value of φ

in (6) will be φ

critas for clay.

The model permits the separation of the criticalstate line and isotropic normal compression line in voidsratio:mean effective stress space to be correctly repro-duced. However, a simpler approach would to be to allowthe yield surface to be of the Modified Cam clay [10] type(i.e. an ellipse). The following section examines how Mod-ified Cam clay can be modified further in a simple way inorder to model better the behaviour of clay, and to modelthe behaviour of sand.

2A generalised soil model

We now generalise the Modified Cam clay model so as tobe suitable for clay and sand. The equation of the Modi-fied Cam clay [10] yield surface is:

q 2

M2 +

p − p

o

2

2= p2o

4 (9)

where po is the isotropic preconsolidation pressure, withflow rule given by

dεpv

dε

p

q

= M2 − η2

2η

(10)

However, this model reproduces too much shear strainand therefore overpredicts K o,nc [11]. McDowell and Hau

[11] showed that the model could be adjusted to reproduceless shear strain, by writing the flow rule as:

dεpvdεpq

= M2 − η2

kη (11)

This flow rule was also proposed by Ohmaki [12] tocorrectly predict K o,nc, and used by Alonso et al. [13]to model the behaviour of partially saturated clays. Theplastic potential has the equation:

q 2 = − M2

1 − k

p

pp

2k

p2p + M2 p2

1 − k (12)

except for k =1, when

q = M p

2 ln pp/p

(13)

In (12), (13), pp

is the hardening parameter for the po-tential. The potentials are shown in Figure 1. McDowelland Hau [11] showed that for clays obeying (6) and (8),combining (5), (8) and (11) and neglecting elastic strainspredicts that

k ≈ 0.7 M (14)

Consequently, for soils obeying Jâky’s relationship [8]in (6), a non-associated Modified Cam clay model suitablefor clay has a flow rule

dεpvdεpq

= M2 − η2

0.7Mη (15)

and a plastic potential given by:

q 2 = − M2

1 − 0.7M

p

pp

20.7M

p2p + M2 p2

1 − 0.7M (16)

This model requires no new parameters and producesthe correct amount of shear strain under one-dimensionalconditions. If the shape of the state boundary surfacediffers significantly from Modified Cam clay, then thepotentials in Fig. 1 could be used as yield surfaces withassociated flow, with the critical state at the apex of theyield surface, as is observed experimentally. We now gen-eralise the soil model, so as to be able to model the behav-iour of sand.

Fig. 1. Potentials for non-associated Modified Cam clay model


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Fig. 2. Non-associated model with M=1.2, N=0.7, k =0.8

For sand, the yield surface now has an equation:

q 2

N2 +

p − p

o

2

2= p2o

4 (17)

where N is the stress ratio at the peak of the yield sur-face, and the flow rule and plastic potential are given by(11), (12) respectively. i.e. for sand, the stress ratio atthe apex of the yield surface N is required in addition tocritical state stress ratio M. The use of the non-associ-ated flow rule with N < M means that the behaviour of sand in undrained tests can be modelled, in the same wayas described by McDowell [2]. Figure 2 shows the yieldsurface and flow rule, and for an undrained test on an iso-tropically normally consolidated sand, the stress path willfollow the yield surface to a critical state (if elastic strainsare assumed to be very small). If the shape of the yieldsurface differs significantly from Modified Cam clay, thenthe following equations can be used for the yield surfaceand potential respectively:

q 2 = − N2

1 − ky

p

po

2ky

p2o +

N2 p2

1 − ky(18)

q 2 = − M2

1 − kp

p

pp

2kp

p2p + M2 p2

1 − kp(19)

where ky controls the shape of the yield surface and kp theflow rule, in the same way that two parameters were usedto do this in the alternative model described by McDowell[2]. An example of a plot of the yield surface and flow ruleis drawn in Figure 3 for the case with ky = 0.7, kp = 0.8,M = 1.2, N = 0.8.

The parameters ky , kp, M and N can be determinedfrom standard triaxial tests. If the sand behaves isotropi-cally elastically along a linear unload-reload line in v−ln p

space (where v is specific volume), then for sand which isyielding an elastic line can be plotted through the cur-rent state in v − ln p space to obtain the preconsolidationpressure po. The current values of q and p

can then be

Fig. 3. Yield surface and flow rule for sand with M=1.2,

N=0.8, ky =0.7, kp =0.8

normalised by po and plotted in q/p

o − p/po space. Thiscan be repeated for yielding at different stress ratios toplot out the state boundary surface. This was done byMcDowell et al. [14] for high pressure triaxial tests on sil-ica sand, as described by McDowell [2]. The stress ratioN corresponding to the peak value of q/po can then bededuced, and a suitable value of ky can be determined.A series of conventional drained tests to ultimate criticalstates can be used to deduce the value of the stress ratio Mat a critical state. For convenience, the value of ky could

be taken to be equal to the value of kp. Alternatively,the direction of the plastic strain increment vector couldbe determined as a function of stress ratio, by deduct-ing the elastic strains calculated from unload-reload datafrom measured total strains at a range of stress ratios, asdescribed by Coop [15].

So far, the yield surface and plastic potential have beengiven for the specific case of triaxial compression. Themodel can easily be applied in general stress space, sothat the equation of the Modified Cam clay yield surfacebecomes:

3

2N2sijsij +

p − p

o

2

2= p2o

4 (20)

and the plastic potential

3

2sijsij = −

M2

1 − k

p

pp

2k

p2p + M2 p2

1 − k (21)

where sij is the deviatoric stress tensor. If the yield surfaceis assumed to be of the same form as the potential, then itsequation can be generalised in the same way. In equations(20) and (21) it has been assumed that the parametersM and N do not vary with Lode angle in principal stressspace; the parameters M, N and k would, in general, be

determined from triaxial compression tests. However, itis well known that the Mohr Coulomb criterion is moreappropriate to failure conditions in soils [16], such thatthe value of M is greater in triaxial compression than in


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triaxial extension. The values of M and N could easily bemade to be a function of Lode angle θ , where

θ = tan−1

1√ 3

2σ2 − σ3σ1 − σ3

− 1

(22)

For example, the following equation proposed by Shenget al. [17] for the shape of the failure surface, is useful:

M(θ) = Mmax

2α4

1 + α4 + (1 − α4)sin3θ

1/4(23)

where

α = MminMmax

= 3 − sinφ3 + sin φ

(24)

and M max is the value of M in triaxial compression withθ = −30◦, and M min is the value in triaxial extensionwith θ = +30◦. This gives a failure surface in the π -plane

which has a shape similar to that proposed by Matsuokaand Nakai [18], and is sketched in Figure 4. The effectof dM/dθ will be important for the potential under planestrain conditions, under which it is well known that theshape of the potential is crucial [16]. For axisymmetricproblems, the effect of the rate of change of M with Lodeangle θ can be neglected for simplicity: this is equivalent toassuming a circular surface in the π-plane with the valueof M corresponding to the Lode angle at the current pointin stress space [11]. Potts and Zdravkovic [16] have alsonoted that the shape of the yield surface in the π-plane hasa much smaller effect on drained behaviour under planestrain conditions, provided the correct angle of shearing

resistance is obtained at failure.For the above models, a suitable hardening rule forthe model would be the volumetric hardening rule usedin conventional critical state models [10] such that plasticvolumetric strain is related to changes in the preconsoli-dation pressure po according to the equation:

δεpv = (λ− κ)

v

δpo po

(25)

where λ is the slope of the normal compression line inv- ln p space, and κ is the slope of an unload-reload linein v- ln p space.

It should be noted that it has been assumed that

the behaviour inside the state boundary surface is elas-tic and isotropic, so that by normalising q and p by po in

Fig. 4. Failure surface given by equation (23)

equations (9), (17) and (18), a normalised elastic sectionthrough the state boundary surface is obtained in eachcase.

3Kinematic hardening and bounding surface plasticity

An advantage of using Modified Cam clay as the stateboundary surface is that kinematic hardening can be eas-ily incorporated. The three-surface kinematic hardening(3-SKH) model [19] is shown in Figure 5, but full detailscan be found in Stallebrass and Taylor [20]. The notationin this paper is the same as that used in Stallebrass andTaylor [20], and detailed definitions of parameters can befound in that paper if required. The equations of the yieldsurface, history surface and bounding surface are givenrespectively below for triaxial stress space (see Figure 5for definitions of stress parameters):

(q − q b)2M2

+ ( p − pb)2

= T 2S 2 p2o

4 (26)

(q − q a)2

M2 + ( p − pa)

2= T 2 p2o

4 (27)

q 2

M2 +

p − p

o

2

2= p2o

4 (9)

and the ratios of the sizes of the surfaces always remainconstant. The elastic strains are given by:

δεevδεeq

=κ

∗

/p

00 1/3Gec

δp

δq

(28)

and the flow rule for plastic strains on the yield surface is:δεpvδεpq

=

λ∗ − κ∗ p( p − pb) +

q(q−qb)M2

( p − pb) + H 1 + H 2

×

( p − pb)

2 ( p − pb)(q−qb)M2

( p − pb)(q−qb)M2

(q−qb)M2

2 δp

δq

(29)

Fig. 5. The 3-SKH model in triaxial stress space


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where the terms H 1 and H 2 are introduced so that themodel does not predict infinite shear strains at a numberof points on the kinematic surfaces [21], and to ensure thatthere is a smooth change in stiffness when the surfaces arein contact. The term H 2 decays to zero as the yield surface

approaches the history surface, and the H 1 term becomeszero when the stress state is on the bounding surface withall three surfaces in contact. The forms of the moduli canbe found in Stallebrass and Taylor [20], and are based onbounding surface plasticity theory [22] such that the mod-ulus deteriorates as the bounding surface is approached.McDowell and Hau [11] have modified the three-surfacekinematic hardening model developed by Stallebrass [19]to include the flow rule in (11) for plastic strains on theyield surface. The flow rule for plastic strains is:

δεpvδεpq

= λ∗ − κ∗

p( p − pb) + q(q−qb)

M2

· 2k ( p − pb) + 2kH 1 + 2kH 2

×

2k ( p − pb)

2 2k ( p

− pb)(q−qb)M2

( p − pb)(q−qb)M2

(q−qb)M2

2 δp

δq

(30)

The moduli H 1 and H 2 in the original model have alsobeen scaled by 2/k so that only the shear strains have beenreduced; Al-Tabbaa [21] found that volumetric strainswere predicted well by the two surface model based onModified Cam clay for kaolin subjected to drained cyclicloading. This simple approach ensures that the singularitypoints on the yield surface are the same as in the originalmodel, and that these singularities are removed by theH 1 and H 2 terms. McDowell and Hau [11] showed thatby allowing the critical state constant M to be a func-tion of Lode angle in stress space according to (23), itwas possible to gain improved predictions for the behav-iour of clay under cyclic loading. Figure 6 shows an exam-ple of a prediction given by their new non-associated flowmodel, compared with that given by the original 3-SKHmodel, for a conventional cyclic triaxial test on kaolin. Thevalue of k was chosen to obtain the correct value of K o,nc.The stress history for the sample is described by Stalle-

brass [19]. It should be noted that the implementation of kinematic hardening with non-elliptical yield surfaces isnumerically cumbersome. However, the use of the Modi-fied Cam clay yield, history and bounding surfaces withnon-associated flow which is symmetrical about the cen-tre line of the yield surface, is very easy to implement andtherefore very attractive. The authors have successfullyused this approach to model the behaviour of pavementsubgrades subjected to repeated wheel loads. This requiresthe extension of the model to general stress space. This isthe subject of a later publication, but the equation of theyield surface is given, for example, below:

f = 3

2

(sij − sijb) : (sij − sijb)M2

+ ( p − pb)2 − T 2S 2 p2o = 0

(31)

Fig. 6. New model prediction of response to conventionalcyclic loading

where sij is the deviatoric stress tensor,

sx = σ

x

− p, sy = σ

y

− p, sz = σ

z

− p, sxy = τ xy,

(32)syz = τ yz , sxz = τ xz,

the subscript b relates to the yield surface, and the rela-tionship between q and sij is:

q =

3

2

s2x + s

2y + s

2z + 2s

2xy + 2s

2yz + 2s

2xz

(33)

The equation of the potential is:

g = 3

2

(sij − sijb) : (sij − sijb)M2

+ 1

1 − k p − pb + T Spo

2TSp

p

2k

2TSpp2

−( p − pb + TSpo)

2

1 − k = 0 (34)

For granular materials, it is usually found that onunloading from the state boundary surface followed byreloading, the strains are small [14,23]. Consequentlykinematic hardening is unnecessary for overconsolidatedsands. However, the behaviour of compacted sands couldeasily be modelled by permitting an initial loading surfaceinside the bounding surface, and incorporating a moduluswhich deteriorates as the loading surface approaches thestate boundary surface. There are numerous possibilities

for the form of the modulus. One possibility is the formula-tion proposed by Yu and Khong [24], which is formulatedfor the CASM [6] Model. It is also readily useable witha Modified Cam clay yield surface, or with a yield sur-face given by equation (18). Figure 7 shows the boundingsurface plasticity model for non-associated Modified Camclay: the modulus on the loading surface H is given by:

H = H i + h

p(1 − β )m

β (35)

where H i is the modulus at the image point, and h and m are two material parameters (see [24]), the p term givesthe dependence of the modulus on stress level, and

β = p

pi=

q

q i= pol po

(36)


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Fig. 7. Bounding surface plasticity model in triaxial stressspace

so that the modulus decays to the value on the boundingsurface as the bounding surface is approached. For a densesand, the size of the loading surface will be much smallerthan that of the bounding surface, so that β will be verysmall and the response will be stiff. For loose sand at thesame stress level, the bounding surface will be smaller andthe value of β larger so that more plastic strain will occur.

4Conclusions

A generalised Modified Cam clay model has been devel-oped for clay and sand. The new model reduces theamount of shear strain produced by Modified Cam clay,without the need for any new parameters. For sand, thestress ratio at the peak of the yield surface is needed inaddition to the critical state frictional dissipation con-stant. The simple non-associated flow rule makes it pos-sible to model the undrained behaviour of sand. If thestate boundary surface differs significantly from Modi-fied Cam clay, then the yield surface can be assumed tobe of the same form as the potential. Kinematic hard-ening has been incorporated inside the state boundarysurface. For granular materials, kinematic hardening isunnecessary for overconsolidated sands; however, asimple bounding surface plasticity approach can be incor-

porated to distinguish between the behaviour of com-pacted and overconsolidated sand. Thus, this model iscapable of capturing many of the essential features of soilbehaviour.

References

1. G. R. McDowell, Soils and Foundations 40(6) (2000),p. 133

2. G. R. McDowell, Granular Matter 4(2) (2002), p. 653. R. Lagoia, A. M. Puzrin & D. M. Potts, Computers and

Geotechnics 19(3) (1996), p.1714. H. W. Chandler, J. Mech. Phys. Solids 33(3) (1985),

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mechanics 22 (1998), p. 6217. P. W. Rowe, Proc. Roy. Soc. A, 269 (1962), p. 5008. J. Jâky, J. Union of Hungarian Engineers and Architects

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geotechnical engineering: theory. London: Thomas Telford(1999)

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(1974), p. 5919. S. E. Stallebrass, Ph.D Thesis, City University, London

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(1997), p. 23521. A. Al-Tabbaa, Ph.D. Thesis, University of Cambridge

(1987)22. Y. F. Dafalias & L. R. Hermann, Soil Mechanics – Tran-

sient and cyclic loads (eds. G. Pande & O. C. Zienkiewicz),p. 235. John Wiley and Sons, Inc, London (1982)

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