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

A generalised Modified Cam clay model for clay and sand incorporating kinematic hardening and bounding surface plasticity G. R. McDowell, K. W. Hau

Abstract This paper proposes a simple non-associated Modified Cam clay model suitable for clay and sand. The yield surface is taken to be that of Modified Cam clay, which is a simple ellipse. The modified model reduces the amount of shear strain predicted, and for clay requires no new parameters because the flow rule uses a well established empirical result. For sand, the critical state frictional dissipation constant is required in addition to the stress ratio at the peak of the yield surface. This permits realistic modelling of the undrained behaviour of sand in states looser and denser than critical. The model resembles more sophisticated models with yield surfaces of more complex shapes, but is much simpler. More realistic behaviour could be obtained by assuming a yield surface with the same form as the potential if required. The model is suitable for incorporating kinematic hardening for the modelling of cyclic loading of clay. In addition, bounding surface plasticity can be included to distinguish between compacted and overconsolidated sand. The contribution in this paper is therefore to provide a generalised simple model based on Modified Cam clay. Keywords Bounding surface plasticity, Clay, Kinematic hardening, Non-associated flow, Plasticity, Sand, Yield surface

1 Introduction McDowell [1] derived a family of yield loci in triaxial stress space, based on the idea that the relative amounts of plastic work dissipated in friction and fracture should be a simple function of stress ratio. He used the normality criterion, together with a simple stress-dilatancy rule, to generate the following family of yield loci for triaxial compression: Received: 28 July 2003

1

η = M[(a + 1) ln (p
o /p
)] a+1

(1)

where η is the stress ratio q/p
, q is deviatoric stress (=σ1
− σ3
where σ1
and σ3
are major and minor principal effective stresses 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. Thus the equation of the yield surface is obtained by selecting an appropriate value for the parameter a in addition to the value of M. McDowell [1] assumed that normality applied, so that the stress-dilatancy rule is given by the equation: dεpv Ma+1 − η a+1 = dεpq ηa

(2)

where dεpv and dεpq are the plastic volumetric and triaxial shear strain increments respectively, plotted along the same axes as the associated stresses p
and q respectively at the current state in stress space, to give the plastic strain increment vector. McDowell [2] compared this model, which requires one new parameter a, to that of Lagioia et al. [3] which requires two parameters to define the shape of the yield surface. McDowell [2] then generalised the model to allow non-associated flow, so that the critical 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 critical state point does not occur at the top of the yield locus [4, 5]. The models proposed by Chandler [4, 5], which make use of the mathematical theory of envelopes and micro structural considerations, are suitable for clays and sands, but require the measurement of microscopic parameters, and have not been adopted widely by geotechnical engineers due to their complexity. McDowell [2] also noted that the model proposed by Yu [6] which gives a yield surface of the same form as (1) uses Rowe’s stress-dilatancy relationship [7], which gives non-associated flow under isotropic conditions: behaviour which is not observed in the literature. The resulting equations for the yield surface and plastic potential, for the model proposed by McDowell [2] for sand are respectively:

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

η = N[(a + 1) ln (p
o /p
)] a+1
  1 η = M (b + 1) ln p
p /p
b+1

K. W. Hau Research Student, University of Nottingham, UK

The parameter N is the stress ratio at the peak of the yield surface, and a controls the shape of the yield surface, whilst b controls the flow rule and p
p is the hardening parameter for the potential. The model can correctly

The authors are grateful to Mr C.D. Khong for discussions on the bounding surface formulation of the CASM model.

1

(3) (4)

12

predict the coefficient of earth pressure at rest Ko,nc (defined as the lateral effective stress divided by axial effective stress) for one-dimensional normal (i.e. plastic) compression, for which dεv dε1 = 1.5 = dεq 2dε1 /3

(5)

and the value of Ko,nc is found empirically [8] to satisfy the equation Ko,nc = 1 − sin φ


(6)

where φ
is the angle of shearing resistance. It is found for clays that the value of φ
in (6) is the critical state angle of frictionφ
crit . Since M=

6 sin φ
crit 3 − sin φ
crit

(7)

(6) and (7) imply that the stress ratio ηo,nc during onedimensional normal compression is given by ηo,nc ≈ 0.6 M

(8)

For sand, the value of φ
in (6) is less certain. According to Muir Wood [9], for sand the value of Ko,nc will depend on the initial structure of the sand, and is therefore likely to depend on the maximum angle of shearing resistance. However, for a sand which has yielded and is deforming plastically under one-dimensional normal compression (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
crit as for clay. The model permits the separation of the critical state line and isotropic normal compression line in voids ratio:mean effective stress space to be correctly reproduced. However, a simpler approach would to be to allow the yield surface to be of the Modified Cam clay [10] type (i.e. an ellipse). The following section examines how Modified Cam clay can be modified further in a simple way in order to model better the behaviour of clay, and to model the behaviour of sand. 2 A generalised soil model We now generalise the Modified Cam clay model so as to be suitable for clay and sand. The equation of the Modified Cam clay [10] yield surface is: 2  2 q2 p
o p
o
+ p − = M2 2 4

[11] showed that the model could be adjusted to reproduce less shear strain, by writing the flow rule as: M2 − η 2 dεpv p = dεq kη

(11)

This flow rule was also proposed by Ohmaki [12] to correctly predict Ko,nc , and used by Alonso et al. [13] to model the behaviour of partially saturated clays. The plastic potential has the equation: M2 q =− 1−k 2



p
p
p

 k2

p
2 p +

M2 p
2 1−k

except for k =1, when    q = Mp
2 ln p
p /p


(12)

(13)

In (12), (13), p
p is the hardening parameter for the potential. The potentials are shown in Figure 1. McDowell and Hau [11] showed that for clays obeying (6) and (8), combining (5), (8) and (11) and neglecting elastic strains predicts that k ≈ 0.7 M

(14)

Consequently, for soils obeying Jˆ aky’s relationship [8] in (6), a non-associated Modified Cam clay model suitable for clay has a flow rule dεpv M2 − η 2 p = dεq 0.7Mη

(15)

and a plastic potential given by: q2 = −

2 
 0.7M p M2 M2 p
2 p
2 p +
1 − 0.7M pp 1 − 0.7M

(16)

This model requires no new parameters and produces the correct amount of shear strain under one-dimensional conditions. If the shape of the state boundary surface differs significantly from Modified Cam clay, then the potentials in Fig. 1 could be used as yield surfaces with associated flow, with the critical state at the apex of the yield surface, as is observed experimentally. We now generalise the soil model, so as to be able to model the behaviour of sand.

(9)

where p
o is the isotropic preconsolidation pressure, with flow rule given by dεpv M2 − η 2 p = dεq 2η

(10)

However, this model reproduces too much shear strain and therefore overpredicts Ko,nc [11]. McDowell and Hau

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

13

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

For sand, the yield surface now has an equation: 2  2 q2 p
o p
o
+ p − = N2 2 4

(17)

where N is the stress ratio at the peak of the yield surface, and the flow rule and plastic potential are given by (11), (12) respectively. i.e. for sand, the stress ratio at the apex of the yield surface N is required in addition to critical state stress ratio M. The use of the non-associated flow rule with N < M means that the behaviour of sand in undrained tests can be modelled, in the same way as described by McDowell [2]. Figure 2 shows the yield surface and flow rule, and for an undrained test on an isotropically normally consolidated sand, the stress path will follow the yield surface to a critical state (if elastic strains are assumed to be very small). If the shape of the yield surface differs significantly from Modified Cam clay, then the following equations can be used for the yield surface and potential respectively: N2 q =− 1 − ky 2

M2 q =− 1 − kp 2





p
p
o p
p
p

 k2

y

 k2

p

p
o +

N2 p
2 1 − ky

(18)

p
2 p +

M2 p
2 1 − kp

(19)

2

where ky controls the shape of the yield surface and kp the flow rule, in the same way that two parameters were used to do this in the alternative model described by McDowell [2]. An example of a plot of the yield surface and flow rule is 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 determined from standard triaxial tests. If the sand behaves isotropically elastically along a linear unload-reload line in v−ln p
space (where v is specific volume), then for sand which is yielding an elastic line can be plotted through the current state in v − ln p
space to obtain the preconsolidation pressure p
o . 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 p
o and plotted in q/p
o − p
/p
o space. This can be repeated for yielding at different stress ratios to plot out the state boundary surface. This was done by McDowell et al. [14] for high pressure triaxial tests on silica sand, as described by McDowell [2]. The stress ratio N corresponding to the peak value of q/p
o can then be deduced, and a suitable value of ky can be determined. A series of conventional drained tests to ultimate critical states can be used to deduce the value of the stress ratio M at 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 could be determined as a function of stress ratio, by deducting the elastic strains calculated from unload-reload data from measured total strains at a range of stress ratios, as described by Coop [15]. So far, the yield surface and plastic potential have been given for the specific case of triaxial compression. The model can easily be applied in general stress space, so that the equation of the Modified Cam clay yield surface becomes: 2  2 3 p
o p
o
(20) s s + p − = ij ij 2N2 2 4 and the plastic potential 3 M2 sij sij = − 2 1−k



p
p
p

 k2

p
2 p +

M2 p
2 1−k

(21)

where sij is the deviatoric stress tensor. If the yield surface is assumed to be of the same form as the potential, then its equation can be generalised in the same way. In equations (20) and (21) it has been assumed that the parameters M and N do not vary with Lode angle in principal stress space; the parameters M, N and k would, in general, be determined from triaxial compression tests. However, it is well known that the Mohr Coulomb criterion is more appropriate to failure conditions in soils [16], such that the value of M is greater in triaxial compression than in

14

triaxial extension. The values of M and N could easily be made to be a function of Lode angle θ, where   
1 σ2 − σ3
−1 √ 2
θ = tan −1 (22) 3 σ1 − σ3
For example, the following equation proposed by Sheng et al. [17] for the shape of the failure surface, is useful: 1/4  2α4 M(θ) = Mmax (23) 1 + α4 + (1 − α4 ) sin 3θ where α=

Mmin 3 − sin φ
= Mmax 3 + sin φ


(24)

and Mmax is the value of M in triaxial compression with θ = −30◦ , and Mmin is the value in triaxial extension with θ = +30◦ . This gives a failure surface in the π-plane which has a shape similar to that proposed by Matsuoka and Nakai [18], and is sketched in Figure 4. The effect of dM/dθ will be important for the potential under plane strain conditions, under which it is well known that the shape of the potential is crucial [16]. For axisymmetric problems, the effect of the rate of change of M with Lode angle θ can be neglected for simplicity: this is equivalent to assuming a circular surface in the π-plane with the value of M corresponding to the Lode angle at the current point in stress space [11]. Potts and Zdravkovic [16] have also noted that the shape of the yield surface in the π-plane has a much smaller effect on drained behaviour under plane strain conditions, provided the correct angle of shearing resistance is obtained at failure. For the above models, a suitable hardening rule for the model would be the volumetric hardening rule used in conventional critical state models [10] such that plastic volumetric strain is related to changes in the preconsolidation pressure p
o according to the equation: δεpv =

(λ − κ) δp
o v p
o

(25)

where λ is the slope of the normal compression line in v- ln p
space, and κ is the slope of an unload-reload line in v- ln p
space. It should be noted that it has been assumed that the behaviour inside the state boundary surface is elastic and isotropic, so that by normalising q and p
by p
o in

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

equations (9), (17) and (18), a normalised elastic section through the state boundary surface is obtained in each case. 3 Kinematic hardening and bounding surface plasticity An advantage of using Modified Cam clay as the state boundary surface is that kinematic hardening can be easily incorporated. The three-surface kinematic hardening (3-SKH) model [19] is shown in Figure 5, but full details can be found in Stallebrass and Taylor [20]. The notation in this paper is the same as that used in Stallebrass and Taylor [20], and detailed definitions of parameters can be found in that paper if required. The equations of the yield surface, history surface and bounding surface are given respectively below for triaxial stress space (see Figure 5 for definitions of stress parameters): 2

2

(q − qb ) T 2 S 2 p
o 2 + (p
− p
b ) = 2 M 4 2

(26)

2

T 2 p
o (q − qa )

2 + (p − p ) = a M2 4   2 2 q2 p
o p
o
+ p − = M2 2 4

(27) (9)

and the ratios of the sizes of the surfaces always remain constant. The elastic strains are given by:  e  ∗

δεv κ /p 0 δp (28) = δεeq 0 1/3G
ec δq and the flow rule for plastic strains on the yield surface is:  p λ ∗ − κ∗ δεv  =

δεpq b) p
(p
− p
b ) + q(q−q (p
− p
b ) + H1 + H2 2 M   2 
b) (p
− p
b ) (p
− p
b ) (q−q 2  M2  δp ×
(q−q b ) b) δq (p − p
b ) (q−q M2 M2 (29)

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

15

where the terms H1 and H2 are introduced so that the model does not predict infinite shear strains at a number of points on the kinematic surfaces [21], and to ensure that there is a smooth change in stiffness when the surfaces are in contact. The term H2 decays to zero as the yield surface approaches the history surface, and the H1 term becomes zero when the stress state is on the bounding surface with all three surfaces in contact. The forms of the moduli can be found in Stallebrass and Taylor [20], and are based on bounding surface plasticity theory [22] such that the modulus deteriorates as the bounding surface is approached. McDowell and Hau [11] have modified the three-surface kinematic hardening model developed by Stallebrass [19] to include the flow rule in (11) for plastic strains on the yield surface. The flow rule for plastic strains is: 

δεpv δεpq



where sij is the deviatoric stress tensor, sx = σx
− p
, sy = σy
− p
, sz = σz
− p
, sxy = τxy ,

λ ∗ − κ∗  =

b) p
(p
− p
b ) + q(q−q · k2 (p
− p
b ) + k2 H1 + k2 H2 M2   
2
2

2
(q−q b ) (p − p ) (p − p ) 2 b b k k  M  δp 2 (30) ×
(q−q b ) δq (p − p
) (q−q2b ) 2 b

M

M

The moduli H1 and H2 in the original model have also been scaled by 2/k so that only the shear strains have been reduced; Al-Tabbaa [21] found that volumetric strains were predicted well by the two surface model based on Modified Cam clay for kaolin subjected to drained cyclic loading. This simple approach ensures that the singularity points on the yield surface are the same as in the original model, and that these singularities are removed by the H1 and H2 terms. McDowell and Hau [11] showed that by allowing the critical state constant M to be a function of Lode angle in stress space according to (23), it was possible to gain improved predictions for the behaviour of clay under cyclic loading. Figure 6 shows an example of a prediction given by their new non-associated flow model, compared with that given by the original 3-SKH model, for a conventional cyclic triaxial test on kaolin. The value of k was chosen to obtain the correct value of Ko,nc . The stress history for the sample is described by Stallebrass [19]. It should be noted that the implementation of kinematic hardening with non-elliptical yield surfaces is numerically cumbersome. However, the use of the Modified Cam clay yield, history and bounding surfaces with non-associated flow which is symmetrical about the centre line of the yield surface, is very easy to implement and therefore very attractive. The authors have successfully used this approach to model the behaviour of pavement subgrades subjected to repeated wheel loads. This requires the extension of the model to general stress space. This is the subject of a later publication, but the equation of the yield surface is given, for example, below:

f=

Fig. 6. New model prediction of response to conventional cyclic loading

3 (sij − sijb ) : (sij − sijb ) 2 + (p
− p
b ) − T 2 S 2 p
2 o =0 M2 2 (31)

syz = τyz , sxz = τxz ,

(32)

the subscript b relates to the yield surface, and the relationship between q and sij is:   3 2 sx + s2y + s2z + 2s2xy + 2s2yz + 2s2xz q= (33) 2 The equation of the potential is: g=

3 (sij − sijb ) : (sij − sijb ) 2 M2 
2 2 p − p
b + T Sp
o k  1 2T Sp
p +
1−k 2T Spp 2

(p
− p
b + T Sp
o ) =0 (34) 1−k For granular materials, it is usually found that on unloading from the state boundary surface followed by reloading, the strains are small [14, 23]. Consequently kinematic hardening is unnecessary for overconsolidated sands. However, the behaviour of compacted sands could easily be modelled by permitting an initial loading surface inside the bounding surface, and incorporating a modulus which deteriorates as the loading surface approaches the state boundary surface. There are numerous possibilities for the form of the modulus. One possibility is the formulation proposed by Yu and Khong [24], which is formulated for the CASM [6] Model. It is also readily useable with a Modified Cam clay yield surface, or with a yield surface given by equation (18). Figure 7 shows the bounding surface plasticity model for non-associated Modified Cam clay: the modulus on the loading surface H is given by: −

m

H = Hi +

h (1 − β) p
β

(35)

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

p
q p
ol = = p
i qi p
o

(36)

16

References

Fig. 7. Bounding surface plasticity model in triaxial stress space

so that the modulus decays to the value on the bounding surface as the bounding surface is approached. For a dense sand, the size of the loading surface will be much smaller than that of the bounding surface, so that β will be very small and the response will be stiff. For loose sand at the same stress level, the bounding surface will be smaller and the value of β larger so that more plastic strain will occur. 4 Conclusions A generalised Modified Cam clay model has been developed for clay and sand. The new model reduces the amount of shear strain produced by Modified Cam clay, without the need for any new parameters. For sand, the stress ratio at the peak of the yield surface is needed in addition to the critical state frictional dissipation constant. The simple non-associated flow rule makes it possible to model the undrained behaviour of sand. If the state boundary surface differs significantly from Modified Cam clay, then the yield surface can be assumed to be of the same form as the potential. Kinematic hardening has been incorporated inside the state boundary surface. For granular materials, kinematic hardening is unnecessary for overconsolidated sands; however, a simple bounding surface plasticity approach can be incorporated to distinguish between the behaviour of compacted and overconsolidated sand. Thus, this model is capable of capturing many of the essential features of soil behaviour.

1. G. R. McDowell, Soils and Foundations 40(6) (2000), p. 133 2. G. R. McDowell, Granular Matter 4(2) (2002), p. 65 3. R. Lagoia, A. M. Puzrin & D. M. Potts, Computers and Geotechnics 19(3) (1996), p.171 4. H. W. Chandler, J. Mech. Phys. Solids 33(3) (1985), p. 215 5. H. W. Chandler, Int. J. Engng. Sci. 28(8) (1990), p. 719 6. H. S. Yu, J. for Numerical and Analytical Methods in Geomechanics 22 (1998), p. 621 7. P. W. Rowe, Proc. Roy. Soc. A, 269 (1962), p. 500 8. J. Jˆ aky, J. Union of Hungarian Engineers and Architects (1944), p. 355 9. D. M. Wood, Soil Behaviour and Critical State Soil Mechanics. Cambridge University Press (1990) 10. K. H. Roscoe & J. B. Burland, Engineering Plasticity (eds. J. Heyman & F.A. Leckie), p. 535. Cambridge University Press (1968) 11. G. R. McDowell and K. W. Hau, G´eotechnique 53(4) (2003), p. 433 12. S. Ohmaki, 1st Int. Symp. Num. Mod. Geomech., Zurich (1982), p. 250 13. E. E. Alonso, A. Gens & A. Josa, G´eotechnique 40(3) (1990), p. 405 14. G. R. McDowell, Y. Nakata & M. Hyodo, G´eotechnique 52(5) (2002), p. 349 15. M. R. Coop., G´eotechnique 40(4) (1990), p. 607 16. D. M. Potts & L. Zdravkovic, Finite element analysis in geotechnical engineering: theory. London: Thomas Telford (1999) 17. D. Sheng, S. W. Sloan & H. S. Yu, Comput. Mech. 26 (2000), p. 185 18. H. Matsuoka & T. Nakai, Proc. Jap. Soc. Civ. Engrs 32 (1974), p. 59 19. S. E. Stallebrass, Ph.D Thesis, City University, London (1990) 20. S. E. Stallebrass & R. N. Taylor, G´eotechnique 47(2) (1997), p. 235 21. A. Al-Tabbaa, Ph.D. Thesis, University of Cambridge (1987) 22. Y. F. Dafalias & L. R. Hermann, Soil Mechanics – Transient and cyclic loads (eds. G. Pande & O. C. Zienkiewicz), p. 235. John Wiley and Sons, Inc, London (1982) 23. C. R. Golightly, Ph.D. dissertation, University of Bradford (1990) 24. H. S. Yu & C. D. Khong, Proceedings of the 3rd International Symposium on Deformation Characteristics of Geomaterials. A. A. Balkema, Rotterdam (2003, in press)