Review of two hypoplastic equations for clay considering axisymmetric element deformations

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Computers and Geotechnics 35 (2008) 760–774

Review of two hypoplastic equations for clay consideringaxisymmetric element deformations

T. Weifner *, D. Kolymbas

Department of Geotechnical and Tunnel Engineering, Technikerstr. 13, A-6020 Innsbruck, Austria

Received 14 December 2006; received in revised form 29 November 2007; accepted 9 December 2007Available online 11 February 2008

Abstract

In this paper are discussed two recently developed rate-independent hypoplastic material laws for cohesive soils. The performance ofthese versions is shown with numerical simulations carried out by the authors.

For each version a brief introduction is given in which the theoretical background and the main features of the two versions aredescribed.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Clay; Constitutive model; Hypoplasticity; Plasticity; Computer simulation

1. Introduction

Hypoplastic material laws can be successfully employedto model the behaviour of non-cohesive soils. Despite thefact that sand and clay have many common properties,some difficulties to model the behaviour of cohesive soilsarised, e.g. due to their lower friction angle (cf. [9]) andtheir behaviour depending on the consolidation history.Thus, in the last years it was tried to find new hypoplasticversions which are able to model also the behaviour ofcohesive soils. This was tried mainly with two types ofapproaches: at one side with the modification of the exist-ing rate-independent hypoplastic material laws and on theother side with rate-dependent hypoplastic material laws,the so-called visco-hypoplasticity. The first hypoplasticmodel was introduced in 1977 by Kolymbas [11]. This ver-sion was rate-dependent (visco-hypoplastic) and allowedtherefore also to model time-dependent effects. However,his improved version, proposed in 1985, did not considerviscous effects and also the following versions of Wu andKolymbas (1990) [22,23] and von Wolffersdorff (1996)[3,8,20] were rate-independent.

0266-352X/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.12.001

* Corresponding author. Tel.: +43 512 5076671; fax: +43 512 5072996.E-mail address: [email protected] (T. Weifner).

The improved version of Kolymbas introduced in 1985[12] was also used to model cohesive soils, whereas the fol-lowing versions were developed mainly for sand. In 1996Gudehus (1996) and Bauer (1996) proposed a model, inwhich was incorporated later on a predefined Matsuoka–Nakai limit condition, which led to the so-called versionof von Wolffersdorff. This version was used as the basisfor further modifications. The rate-dependent versions ofNiemunis [16] and Gudehus [4] were developed to modelviscous effects, while Herle [9] and Masın [14] developedrate-independent modifications of the version of von Wolf-fersdorff. The performance of these last two versions will bediscussed here, while the two rate-dependent versions ofNiemunis [16] and Gudehus [4] will be discussed in a forth-coming paper [21]. The hypoplastic version of Masın is anattempt to combine the Modified Cam–clay model andhypoplasticity. A brief description of this hypoplastic ver-sion and numerical simulations are shown in Section 2.

Herle carried out a modification of the version of vonWolffersdorff to allow its use for soils with low frictionangles. This version was also calibrated for Weald clay toallow for a direct comparison of the two versions and ispresented in Section 3.

Huang et al. [10] proposed a formulation applicable fornormally consolidated clay, which allows to model an

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Nomenclature

T stress tensortr T trace of stress tensorT* deviatoric part of stress tensorD stretching tensorbT normalised stress tensorbT� normalised deviatoric stress tensorfs factor for barotropy — version of Masınfd factor for pycnotropy — version of Masına barotropy exponent — version of MasınL hypoelastic tensorN non-linear tensork slope of the isotropic compression loading curve

in the lnð1þ eÞ � ln p or e� ln p-diagramj slope of the isotropic compression unloading

and reloading curve (URL) in thelnð1þ eÞ � ln p or e� ln p-diagram

a parameter of limit conditionF stress function according to Matsuoka/Nakaip mean stresspr reference mean stress (1 kPa)m tensor expressing the hypoplastic flow rule —

version of MasınY degree of non-linearity (limit stress condition)

— version of MasınI1; I2; I3 invariants of the stress tensorOCR overconsolidation ratioN parameter determining the position of the iso-

tropic consolidation line (ICL)c1; c2 factors applied to the linear part of stress tensor

to change the shape of response envelopes —versions of Herle and Masın

r ratio of bulk and shear modulus — version ofMasın

e void ratioe100 void ratio corresponding to a mean stress of

100 kPa (referred to the ICL)pe100

mean stress corresponding to e100 (referred tothe ICL)

r ratio of triaxial and isotropic stiffness — versionof Herle

a; b; n exponents — versions of von Wolffersdorff andHerle

fs factor for barotropy and pycnotropy — versionof von Wolffersdorff

�f s modified factor for barotropy and pycnotropy— version of Herle

fd factor for pycnotropy — versions of von Wolf-fersdorff and Herle

hs granulate hardness — versions of von Wolf-fersdorff and Herle

ec0 critical void ratio with ps ¼ 0ed0 minimum void ratio with ps ¼ 0ei0 maximum void ratio for isotropic compression

with ps ¼ 0ec critical void ratioed minimum void ratioei maximum void ratio for isotropic compressionuc critical friction angleL1;L2 tensors used to calculate the hypoelastic tensor

— versions of von Wolffersdorff and Herlen exponent of c1 — version of HerleT max=T min maximum/minimum main stress

T. Weifner, D. Kolymbas / Computers and Geotechnics 35 (2008) 760–774 761

elliptical shape of the stress path under undrained condi-tion similar to the modified Cam–clay model. However,this model does not allow to model stress paths withincreasing q close to the critical state line. An advantageof this model is surely the fact that the volumetric deforma-tion in shear tests can be scaled, which is not possible withall other hypoplastic models known to the authors.

2. The hypoplastic version of Masın

The hypoplastic version of Masın [14] is a combinationof generalised hypoplasticity principles with traditionalCritical State Soil Mechanics [19]. The formulation of theisotropic compression and critical state lines correspondsto the Modified Cam–clay model [18], the limit surface ofMatsuoka and Nakai [13] is used and the non-linear behav-iour is given by the generalised hypoplastic formulation.

The main equation has the following form:

T�¼ fsL : Dþ fsfdNkDk ð1Þ

fs and fd are scalar functions and are called barotropy andpyknotropy factors, respectively:

fs ¼ �tr T

k3þ a2 � 2aa

ffiffiffi3p� ��1

ð2Þ

fd ¼ � 2trT

3pr

explnð1þ eÞ � N

k

� �� ð3Þ

with a:

a ¼ 1

ln 2ln

k� jkþ j

3þ a2

affiffiffi3p

� �� ð4Þ

The factor a is defined like in the equation of von Wolf-fersdorff [20] and depends on the critical friction angle. k isthe slope of the isotropic consolidation line (ICL) in thelnð1þ eÞ � ln p diagram. N is the value of lnð1þ eÞ for amean stress pr of 1 kPa.

The fourth order tensor L is defined as follows:

L ¼ 3 c1Iþ c2a2T� T� �

ð5Þ

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 30 40 60 100 200 400 800

e

trσ/3

Isotropic consolidation

Fig. 1. Simulation (thick lines) and experimental results with Weald clayby Henkel [5] (thin lines) of isotropic compression, loading and unloading,k ¼ 0:06, j ¼ 0:031, N ¼ 0:806.

762 T. Weifner, D. Kolymbas / Computers and Geotechnics 35 (2008) 760–774

The factors c1 and c2 are calculated with r being the ratioof bulk and shear modulus:

c1 ¼2 3þ a2 � 2aa

ffiffiffi3p

9rð6Þ

c2 ¼ 1þ ð1� c1Þ3

a2ð7Þ

The hypoplastic flow rule is incorporated in the secondorder tensor N via the tensor m:

N ¼L : �Ym

kmk

� �ð8Þ

m ¼ � aF

Tþ T� � T

3

6T : T� 1

ðF =a2Þ þ T : T

!" #ð9Þ

Y ¼ffiffiffi3p

a3þ a2

� 1

!ðI1I2 þ 9I3Þð1� sin 2ucÞ

8I3 sin 2uc

þffiffiffi3p

a3þ a2

ð10Þ

Y is the degree of non-linearity and I1, I2 and I3 are thestress invariants:

I1 ¼ tr T; I2 ¼1

2T : T� ðI1Þ2h i

; I3 ¼ det T ð11Þ

The factor F is the same as in the equation of von Wolf-fersdorff [20].

2.1. Calibration of the equation for Weald clay

As it is derived from the Modified Cam–clay model, thisversion requires also five material parameters: uc; k; j;N ; r.

Experimental results of isotropic, drained andundrained triaxial tests carried out with Weald clay byHenkel et al. [1,5,17] were considered.

uc can be determined to be 24� from a drained triaxialtest:

k and j can be obtained from the log–log equation ofButterfield [2] as required by the version.

ln1þ e100

1þ e

� �¼ k ln

pe

pe100

!ð12Þ

With e100 ¼ 0:70, e ¼ 0:50, pe100¼ 100 kPa and

pe ¼ 800 kPa, k ¼ 0:06 can be obtained from the isotropicconsolidation line (ICL).

N can be defined as the value of lnð1þ eÞ for p ¼ 1 kPaand can be therefore determined to be 0.806 with the fol-lowing equation using pe ¼ 800 kPa and e ¼ 0:50:

N ¼ lnð1þ eÞ þ k lnðpÞ ð13Þj ¼ 0:031 can be obtained with e ¼ 0:6 and pe ¼ 100 kPausing the unloading/reloading curve (URL).

For the ratio r of the bulk to shear modulus, Masın rec-ommends a parameter study to obtain a suitable stiffnessbehaviour for isotropic compression as well as for thedrained and undrained triaxial tests.

2.2. Simulation results

Fig. 1 shows that the simulation of isotropic compres-sion fits the experimental results quite well. However, the

simulation of drained triaxial tests with a normally con-solidated ðr2 ¼ 204 kPaÞ and a heavily overconsolidatedsample (OCR ¼ 24 and r2 ¼ 34:48 kPa) is unsatisfactoryusing r ¼ 0:6 (Fig. 2a). Although the maximum frictionangle is used to calibrate the equation, the critical stateis not captured well; for the NC sample the maximumdeviatoric stress is too high and for the OC sample toolow. Only the critical state in triaxial extension for thenormally consolidated clay is achieved realistically. Thesoftening of the overconsolidated sample is too exagger-ated, the deviatoric stress becomes very small as visible,e.g. in Fig. 2a. The volumetric strain vs. axial strain curvesshow very high contractancy; for the NC sample the vol-umetric strains are overestimated, for the OC sample — incontradiction to the experiments — no expansion of thesample can be observed and the volumetric strains arehigher than for the normally consolidated sample. Inaddition, the volumetric strain vs. axial strain curves showa highly dilatant behaviour, which is unrealistic. The vol-umetric curves for the triaxial extension test are also unre-alistic, because one of the NC samples overestimates thevolumetric strain and shows initial dilatant behaviourwhile there should be only contractancy and the OC sam-ple is dilatant only at the very beginning and becomescontractant thereafter, while the experiment shows onlydilatancy (Fig. 2c).

To overcome these problems the author tried also withr ¼ 0:4 and r ¼ 0:2 to reduce the stiffness in triaxial testswith respect to the isotropic test. Unfortunately, this doesnot change the curves much (Figs. 2b and d, 3a and c).Only r ¼ 0:075 leads to a realistic critical state for the nor-mally consolidated sample (Fig. 3b). But with this calibra-tion the volumetric behaviour of the NC and the OCsample becomes qualitatively similar, i.e. the simulationof the triaxial test with the OC sample shows only contrac-tancy (Fig. 3d). In the simulation of unloading and exten-sion, numerical problems occurred with the determinationof the lateral strain (Figs. 3b and d).

-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ]

Drained triaxial test

204.75

34.48

-200

-100

0

100

200

300

400

0 5 10 15 20 25 30ε1 [ % ]

ε1 [ % ] ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30


204.75

34.48-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε v [

% ]


204.75

34.48

Fig. 2. Experimental results with clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) of drained triaxialcompression and extension tests with varying parameter r. Note on (c) and (d) the unsatisfactory volumetric performance for the OC sample.

1 D. Masın, private communication.


Simulations of undrained triaxial tests with the samenormally consolidated and heavily overconsolidated sam-ples as used for the drained tests were also carried out.The results with the same parameters uc; k; j;N as for thedrained tests and r ¼ 0:6 are shown in Figs. 4a and c and5a.

Considering a normally consolidated sample, the stresspath (Fig. 4c) is quite satisfactory at the beginning, butthe deviatoric stress at the critical state is much too highand a monotonous increase of the deviatoric stress can beobserved (Fig. 4a). The pore pressure is underestimatedat the critical state (Fig. 5a).

For the overconsolidated sample the stress path (Fig. 4c)is unrealistic, the deviatoric stress vanishes for e1 ¼ 20%(Fig. 4a) and in contradiction to the experiment no nega-tive pore pressure is reached (Fig. 5a).

Similar as with drained tests, using r ¼ 0:2 does notaffect the results much as visible in Figs. 4b and d and 5b.

Only reducing r to 0.075 leads to a better performance ofthe material law. Now the normally consolidated sampleshows no monotonous increase of the deviatoric stress, inaccordance to the experimental results, but the maximumdeviatoric stress is still too high (Fig. 5c). As shown in

Fig. 5d the pore pressure is overestimated at the beginningof the test and underestimated at the end.

Considering an overconsolidated sample, no negativepore pressure is achieved (Fig. 5d) and the deviatoric stressvs. axial strain curve shown in Fig. 5c shows softening andthe deviatoric stress vanishes for an axial strain of 30%.

In Fig. 5e the stress paths for both samples are shown.The curves end at the same critical state line as the exper-iments, but at different stress states. The stress path forthe OC sample is unrealistic and the one for the NC sampleconsiderably deviates from the experimental curve.

The shortcomings of the model are connected to the lowratio k=j ¼ 2.1 Therefore, j was changed to 0.01 (k=j ¼ 6)to improve the behaviour of the model. Now isotropicunloading shows a very stiff behaviour (Fig. 6e). The volu-metric performance is too dilatant for the OC sample,which shows also exaggerated softening (Fig. 6c) with anunrealistic high peak stress (Fig. 6a). In the undrained tri-axial test results (Fig. 6b, d and f), a considerable deviationfor the overconsolidated sample is also visible.

-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ]


204.75

34.48

-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


204.75

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


204.7534.48

Fig. 3. Experimental results with Weald clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) of drainedtriaxial compression and extension tests with the variation of parameter r. Note on (c) and (d) the bad volumetric performance for the OC sample. Forr ¼ 0:075 no limit state can be reached in extension for both samples (b).


2.3. Conclusions

This constitutive version is able to describe the behav-iour in isotropic compression quite well. The behaviour indrained triaxial tests, however, is not satisfactory (Figs.2a to 3c). The parameter r must be chosen by trial anderror, but even using many different values of r, the per-formance of the model in drained triaxial compressioncannot be improved significantly. Only the maximumpeak stress of the experimental curve can be fitted byvarying r, as shown in Fig. 3a and b. In the simulationof the overconsolidated sample the deviatoric stress van-ishes, which causes numerical problems. The volumetricbehaviour in the simulation is rather unrealistic, e.g. thevolumetric strains of the OC sample are higher than forthe NC sample and the behaviour at unloading is strictlydilatant.

Considering the simulation of undrained triaxialtests, the achieved results are even less satisfactory: nei-ther the deviatoric stress curves (Figs. 4a and b and5c) nor the pore pressure curves (Fig. 5a, b and d)fit well, the stress path for the overconsolidated sampleis unrealistic and for the normally consolidated sample

only a qualitative agreement can be achieved by vary-ing r (Fig. 5e).

The model has the advantage that the parametersuc; k; j;N can be easily determined. Only the factor r

should be determined by a parameter study. In the simula-tions, a wide range of the values for r ranging from 0.05 to0.8 were used.

Changing j to 0.01 to improve the behaviour of themodel leads to a very stiff isotropic unloading (Fig. 6e),and the exaggerated softening with an unrealistic high peakstress (Fig. 6a) could not be reduced. Therefore, it has to beconcluded that this material model is unable to model theexperiments of Weald clay with the suggested calibrationmethod in which only the parameter r has to be varied,while the other parameters can be determined without trialand error (cf. [15]). The following shortcomings can besummarized:

� Unsatisfactory stress-dilatancy behaviour for OCsamples.� Problems related to small ratios k=j.� Unrealistic prediction of the effects of

overconsolidation.

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ] ε1 [ % ]

Undrained triaxial test

34.48

204.75

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30


34.475

204.75

0

50

100

150

200

0 50 100 150 200 250 300

q

p’


0

50

100

150

200

0 50 100 150 200 250 300

q

p’


Fig. 4. Experimental results with clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) of undrainedtriaxial compression tests, variation of parameter r. For the NC sample the deviatoric stress is overestimated and the OC sample shows exaggeratedsoftening. Note that the stress path follows the opposite direction, it goes to the right for NC clay and to the left for the OC one.


3. The hypoplastic version of Herle

The hypoplastic version of von Wolffersdorff wasapplied to model the behaviour of sand and it was usedalso to model element tests with fine grained granulateswith high friction angles (cf. [7]). However, for low valuesof uc the simulated material behaviour becomes unaccept-able, because the incremental stiffness differs significantlyfor isotropic and triaxial compression at a particular iso-tropic stress state (cf. [9]). The reason of this deficiency liesin the shape of the response envelopes: for low critical fric-tion angles their shape becomes narrower, but the stiffnessfor isotropic loading remains practically the same. Theshape of the response envelopes is governed exclusivelyby the parameter a, which depends on uc. Thus, a cannotbe modified unless the critical states change, therefore, Her-le chose another solution. He introduced two scalar factorsc1 and c2, which are used as multipliers of the two linearparts L1 and L2 of the basic equation. Eq. (14) shows thebasic equation. It should be remarked that the version ofHerle uses the same expression for N as the version ofvon Wolffersdorff

T�¼ c1fsL1Dþ c2fsL2DþNkDk ð14Þ

Factor c1 scales the size of the response envelopes, e.g.for c1 ¼ 2 they expand, and c2 changes the ratio of thesemi-axes of the response envelope, e.g. c2 ¼ 2 increasesthe size of the longer axis, but does not change the shorteraxis. Applying both factors, namely c2 < 1 and c1 > 1 leadsto the desired effect: the response envelope becomes widerdue to c2 and the size of the major principal axisapproaches its original size by c1.

To allow that the axis lengths of the response envelopesfor isotropic compression remain unchanged, c2 must berelated to c1 in the following way:

c2 ¼ 1þ ð1� cn1Þ

3

a2ð15Þ

The rate Eq. (16) for _p which appears in the function forthe factor fs must be consistent with Eq. (22) for isotropiccompression, therefore the Eq. (17) for fs used in the ver-sion of von Wolffersdorff must be modified to Eq. (18)for �f s.

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]


204.75

34.48

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]

ε1 [ % ] ε1 [ % ]


204.75

34.48

0

50

100

150

200

0 5 10 15 20 25 30

σ 1-σ

3


204.75

34.48-100

-50

0

50

100

150

0 5 10 15 20 25 30

u


204.75

34.48

0

50

100

150

200

0 50 100 150 200 250 300

q

p’


Fig. 5. Experimental results with clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) of undrained triaxialcompression tests, variation of parameter r. The stress path follows still the opposite direction for the OC sample, for the NC clay it is qualitatively correct.


_p ¼ � 1

3

_ei

ei

hs

n3phs

� �1�n

ð16Þ

fs ¼hs

n1þ ei

ei

ei

e

� �b 3phs

� �1�n

3þ a2 � affiffiffi3p ei0 � ed0

ec0 � ed0

� �a� ��1

ð17Þ

�f s ¼hs

n1þ ei

ei

ei

e

� �b 3phs

� �1�n

� 3c1 þ a2c2 � affiffiffi3p ei0 � ed0

ec0 � ed0

� �a� ��1

ð18Þ

Factor c1 can be determined from the ratio of the stiff-nesses in isotropic and undrained triaxial compressions,r ¼ E�iso=E�cv, as follows:

c1 ¼1þ 1

3a2 � 1ffiffi

3p a

1:5rð19Þ

-150

-100

-50

0

50

100

150

200

250

300

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ] ε1 [ % ]


34.48

204.48

(a) =0.6, =0.01

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30


Drained triaxial test Undrained triaxial test

204.75

34.48

(b) =0.6, =0.01

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [ %

]

ε1 [ % ] ε1 [ % ]

34.48

204.48

(c) =0.6, =0.01

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

204.75

34.48

(d) =0.6, =0.01

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 30 40 60 100 200 400 800

e

trσ/3


(e) =0.6, =0.01

0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

q

p’


(f) =0.6, =0.01

Fig. 6. Experimental results with Weald clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) of drainedtriaxial compression and extension tests after the variation of j. Note on (c) the volumetric performance which is too dilatant for the OC sample, whichshows also exaggerated softening with an unrealistic high peak stress (a). Now isotropic unloading shows a very stiff behaviour (e).


To enable that the critical stress state (T�¼ 0 and fd ¼ 1)

is not influenced by the factors c1 and c2, their influencemust vanish if the critical states are approached. This canbe achieved by applying the exponent n to c1. Eq. (20) pro-vides that n vanishes if the mobilised friction angle sin um

reaches uc.

n ¼ sin uc � sin um

sin uc

� �ð20Þ

with

sin um ¼T max � T min

T max þ T min

: ð21Þ


The modified main equation reads as

T�¼ �f s

1

trT2cn

1F 2Dþ c2a2TtrðTDÞ þ fdaF ðTþ T�ÞkDkh i

ð22Þ

3.1. Calibration for Weald clay

The model needs nine parameters, eight of them are thesame as for the version of von Wolffersdorff, and parameterr is required additionally. The calibration procedure rec-ommended by Herle [6,9] is used in the following.

The critical friction angle uc can be determined from ashear test (e.g. drained triaxial compression) and assumes24� for Weald clay.

For the determination of parameters n and hs it is neces-sary to know k, using the e� ln p plot. Considering the testresults from Henkel et al. [1,5,17], the virgin consolidationline is quite straight and thus, a unique k ¼ 0:0938 can beobtained (e1 ¼ 0:76, e2 ¼ 0:50, p1 ¼ 50 kPa, p2 ¼ 800kPa). From Eq. (23) follows n ¼ 0:15, and Eq. (24) leadsto hs ¼ 472 kPa using p ¼ 50 kPa and e ¼ 0:76

n ¼ln e1

e2

� �ln p2

p1

� � ð23Þ

hs ¼ 3pnek

� �1=nð24Þ

Using p ¼ 300 kPa and e ¼ 0:59 obtained from the exper-imental isotropic consolidation line leads to ei0 ¼ 1:78 withEq. (25). A point of the critical state line can be obtainedfrom the result of a drained triaxial test: Withp ¼ ð3 � 204:75þ 270Þ=3 ¼ 295 kPa and e ¼ 0:59�0:045 � 1:59 ¼ 0:52 follows ec0 ¼ 1:56 using Eq. (25). Fromthe void ratio at plasticity limit, ep ¼ 0:18 � 2:74 ¼ 0:49 andassuming pp ¼ 5 kPa (cf. [3]) follows ed0 ¼ 1:00.

ec0 could be also calculated using the void ratio at liquidlimit, eL ¼ 1:18 and pL ¼ 5 kPa (cf. [3]). In this caseec0 ¼ 2:14, which is too high, compared with ei0 ¼ 1:78.Herle obtains ec0 ¼ 3:35 in [9] for Rio de Janeiro clay usingthe critical state line; with the void ratio at liquid limitðeL ¼ 1:2 � 2:7 ¼ 3:24Þ resulting in ec0 ¼ 4:87, which is,indeed, lower than ei0 ¼ 5:65 (cf. Herle [9]), but quite differ-ent from ec0 ¼ 3:35 obtained with the CSL, which was usedfor the simulation. It seems to be more appropriate to usethe CSL for the determination of ec0 rather than eL, if dataare available

ei0

ei

¼ ec0

ec

¼ exp3ps

hs

� �n� �ð25Þ

The basic values a ¼ 0:15 and b ¼ 1 are chosen as theydo not affect the behaviour of the normally consolidatedsamples (Herle [9]) and will be varied in the sequel to fitalso the test results of the OC samples. As a first approxi-mation r ¼ 1 holds, assuming the same stiffness in isotropicand undrained triaxial compressions.

3.2. Simulation

The simulation results can be observed in Fig. 7. Thestiffness in the isotropic compression test is slightly toohigh (Fig. 7a). Considering the NC sample and the drainedtriaxial tests, it can be observed that the peak stress andalso the volumetric strains are overestimated (Fig. 7c ande). In the drained and undrained triaxial tests the initialstiffness for the NC sample is correct, therefore, r ¼ 1 issuitable. In the pore pressure diagram and in the p0 � q-dia-gram the differences between simulation and experimentare less pronounced for the NC sample than for the OCone (Fig. 7f and b).

In the following it was tried to vary some parameters toobtain a better agreement between simulation and experi-ment. As a first step, the isotropic stiffness is reduced byassuming a higher value for k, with the consequence thatn increases due to a higher ratio e1=e2. Appropriate valuesare hs ¼ 327 kPa and n ¼ 0:15 using k ¼ 0:1055 (e1 ¼ 0:79,e2 ¼ 0:50). uc is reduced to 22� to achieve the correct peak(i.e. critical) stress for the normally consolidated sample indrained triaxial compression. The results are shown inFig. 8. Now the isotropic compression test (Fig. 8a) showsthe correct stiffness and the stress–strain curve for thedrained triaxial test is satisfactory (Fig. 8c), while the vol-umetric strains are overestimated (Fig. 8e). The initial stiff-ness in undrained compression is correct (Fig. 8d),therefore in the following r ¼ 1 was still used (Table 1).

In the sequel, a and b were varied to improve the perfor-mance of the simulation of the overconsolidated samples.First b was increased, which led to the exaggerated soften-ing of the NC sample in undrained triaxial compressionand to a quite stiff behaviour in isotropic compression,but the behaviour of the OC sample remained more or lessunchanged. Therefore, b ¼ 1 was kept and a ¼ 0:75 waschosen. The results are summarized in Fig. 9. The overcon-solidated sample shows a much too high peak stress in thedrained triaxial test, but the volumetric behaviour showsqualitatively the right tendency (first contractant and thendilatant). In the undrained test the pore pressure curve isqualitatively correct at the beginning positive pore pressureand then negative (Fig. 9f), but the stiffness and the peakstress are too low (Fig. 9d). In the stress paths, the overcon-solidated sample shows a better agreement with the exper-iment than the normally consolidated one (Fig. 9b).Considering the normally consolidated samples, the behav-iour is worse than with the parameters from Table 2(Fig. 8). The undrained test shows softening (Fig. 9d),and the volumetric strains for the drained test are overesti-mated (Fig. 9e). In addition, in the drained triaxial com-pression test for e1 > 20% the stiffness increases Fig. 9c.(Tables 3–5).

Especially to eliminate the last-mentioned anomaly, ec0

was varied by trial and ec0 ¼ 1:68 was found to be suitable.A small reduction of the critical friction angle was also nec-essary (uc ¼ 21�). n ¼ 0:15 and hs ¼ 427 kPa were chosen tofit the stiffness in the isotropic compression test. The results

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

20 30 40 60 100 200 400 800

e

trσ/3


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

q

p’


-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

ε1 [ % ]


204.75

34.48

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30

σ 1-σ

3

ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


34.48

204.75

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]


34.48

204.75

a b

c d

e f

Fig. 7. Experimental results with Weald clay (thin lines), NC (r3 ¼ 204:75 kPa), OC (r3 ¼ 34:48 kPa, OCR ¼ 24) and simulation (thick lines) with Eq.(22) and parameters from Table 1.


are shown in Fig. 10. Now the stiffness in isotropic loading iscorrect, while in unloading the behaviour is slightly too soft.The peak stress is still too high for the OC sample, while thestress–strain behaviour of the NC sample is correct(Fig. 10c). The volumetric behaviour of both samples is alsoimproved (Fig. 10e). In the undrained tests the OC sampledoes not approach the correct peak stress and the stiffnessis very low, while the NC sample shows a very good agree-ment with the experimental results (Fig. 10d). The pore pres-sure curves fit the experimental results also quite well and thestress paths are qualitatively correct for the OC sample, while

for the NC sample the shape of the curve is different from theexperimental result, but the peak stress state is quite close tothe experimental one.

It was also tried to increase ed0 instead of a to obtaindilatancy for the OC sample. This leads, however, to theproblem that the minimum void ratio is reached in isotro-pic unloading (for ed0 ¼ 1:40) and also during the drainedtriaxial test (for ed0 ¼ 1:50). Even with ed0 ¼ 1:50 cannotbe achieved a considerable dilatancy in the drained triaxialtest for the OC sample. In Fig. 11 are shown the simulationresults for ed0 ¼ 1:30 and b ¼ 0:40. There are not many dif-

Table 1Material constants, Eq. (22), Weald clay

uc (�) hs (kPa) ed0 ec0 ei0 n a b r

24 472 1 1.56 1.78 0.15 0.15 1 1

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

20 30 40 60 100 200 400 800

e

trσ/3


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

q

p’


-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ] ε1 [ % ]

ε1 [ % ] ε1 [ % ]


204.75

34.48

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]


34.48

204.75

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u


204.75

34.48

a b

c d

e f



ferences with respect to Fig. 10, thus a higher value of ed0

can be used to keep b at a lower value.

3.3. Conclusions

The material law is able to model isotropic compression,drained and undrained triaxial tests with normally consol-

idated clay quite well, and also for highly overconsolidatedclay the results show qualitatively correct tendencies: thevolumetric behaviour in the drained triaxial compressionand extension and the pore pressure curves are qualita-tively correct. The softening in the stress–strain diagramis too pronounced for the OC sample (deviatoric peakstress of 74 kPa instead of 54 kPa) and the axial strainsat the critical state are very high (Fig. 10c). It is alsoremarkable that the achieved uc in the simulations isslightly different from the input parameter uc. The factorsc1 and c2 have no influence on the incremental stress forcritical states as they become c1 ¼ c2 ¼ 0. However, the

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

20 30 40 60 100 200 400 800

e

trσ/3


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

q

p’


-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

ε1 [ % ]


204.75

34.48

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30

σ 1-σ

3

ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


34.48

204.75-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]


34.48

204.75

a b

c d

e f


Table 2Material constants, Eq. (22), Weald clay, after variation of uc; hs; n


22 327 1 1.56 1.78 0.165 0.15 1 1

Table 3Material constants, Eq. (22), Weald clay, after the variation of a


22 327 1 1.56 1.78 0.165 0.75 1 1

Table 4Material constants, Eq. (22), after the variation of ec0;uc; hs; n


21 427 1 1.68 1.78 0.15 0.75 1 1

Table 5Material constants, Eq. (22), after the variation of ed0 and ec0


21 427 1.30 1.68 1.78 0.15 0.40 1 1


0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

20 30 40 60 100 200 400 800

e

trσ/3


0

20

40

60

80

100

120

140

160

0 50 100 150 200 250 300

q

p’


-200

-100

0

100

200

300

400

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ]


204.75

34.48

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


204.75

34.48

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]


204.75

34.48

e f

a b

c d



stiffness in the model of Herle is different to the stiffnessobtained with the version of von Wolffersdorff beforereaching the critical state. Therefore, with the model ofHerle is reached a higher deviatoric stress, as long as thestiffness in triaxial tests (i.e. stress increments) is higher.It must be also pointed out that the parameters must bevaried by trial and error to fit the experimental curves sat-isfactorily. With the proposed determination the results areless favourable, to the author’s opinion a variation of theparameter is appropriate, because it shows the potentials

of the material law and in the future it may be possibleto improve the calibration procedure to obtain the rightparameters a priori.

The possible case that the minimum void ratio ed0 isreached, which leads to a complex number for fd , wasnot encountered during the simulations using ed0 obtainedby the initial calibration. Only by using a very high value ofed0 (e.g. 1.40) this problem can arise. It seems therefore,that this problem can be avoided in many cases with aproper choice of the parameters (especially ed0, ec0 and ei0).

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

20 30 40 60 100 200 400 800

e

trσ/3


0

20

40

60

80

100

120

140

160

180

0 50 100 150 200 250 300

q

p’


-150

-100

-50

0

50

100

150

200

250

300

0 5 10 15 20 25 30

σ 1-σ

3

σ 1-σ

3

ε1 [ % ]


34.48

204.75

0

20

40

60

80

100

120

140

160

180

0 5 10 15 20 25 30ε1 [ % ]


204.75

34.48

-6

-4

-2

0

2

4

6

0 5 10 15 20 25 30

ε v [

% ]

ε1 [ % ]


204.75

34.48

-100

-50

0

50

100

150

0 5 10 15 20 25 30

u

ε1 [ % ]


204.75

34.48

c d

a b

e f



Acknowledgements

This paper is a partial research result of the first authorwhich was financially supported by the Austrian ScienceFund (FWF), Grant No. 14969, and supervised by the sec-ond author.

The financial support by the Austrian Science Fund isthankfully acknowledged.

References

[1] Bishop AW, Henkel DJ. The measurement of soil properties in thetriaxial test. Edward Arnold; 1962.

[2] Butterfield R. A natural compression law for soils. Geotechnique1979;29(4):469–80.

[3] Gudehus G. A comprehensive constitutive equation for granularmaterials. Soil Found 1996;36(1):1–12.

[4] Gudehus G. A visco-hypoplastic constitutive relation for soft soils.Soil Found 2004;44(4):11–25.


[5] Henkel DJ. The effect of overconsolidation on the behaviour of claysduring shear. Geotechnique 1956;6:139–50.

[6] Herle I. Hypoplastizitat und Granulometrie einfacher Korngeruste.Publ. series of: Institut fur Bodenmechanik und Felsmechanik derUniversitat Fridericiana in Karlsruhe, No. 142, 1997.

[7] Herle I. Granulometric limits of hypoplastic models. TASK Quart2000;4(3):389–407.

[8] Herle I, Gudehus G. Determination of parameters of a hypoplasticconstitutive model from properties of grain assemblies. MechCohesive–Frictional Mater 1999;4(5):461–86.

[9] Herle I, Kolymbas D. Hypoplasticity for soils with low frictionangles. Comput Geotech 2004;31(5):365–73.

[10] Huang W, Wu W, Sun D, Sloan S. A simple hypoplastic model fornormally consolidated clay. Acta Geotech 2006;1:15–27.

[11] Kolymbas D. A rate-dependent constitutive equation for soils. MechRes Commun 1977;4:367–72.

[12] Kolymbas D. Eine konstitutive Theorie fur Boden und andere kornigeStoffe. Publ. series of: Institut fur Bodenmechanik und Felsmechanikder Universitat Fridericiana in Karlsruhe, No. 109, 1988.

[13] Matsuoka H, Nakai T. Stress–strain relationship of soil based on theSMP. In: Proceedings of IXth ICSMFE, 1977, p. 153–62.

[14] Masın D. A hypoplastic constitutive model for clay. Int J NumerAnal Meth Geomech 2005;29:311–36.

[15] Masın D. Hypoplastic models for fine-grained soils. PhD thesis,Prague: Charles University; 2006.

[16] Niemunis A. Extended hypoplastic models for soils. Publ. series of:Institut fur Grundbau und Bodenmechanik der Ruhr-UniversatBochum, vol. 34, 2003.

[17] Parry RHG. Triaxial compression and extension tests on remouldedsaturated clay. Geotechnique 1960;10(4):166–80.

[18] Roscoe KH, Burland JB. On the generalised stress–strain behaviourof ‘wet’ clay. In: Heyman J, Leckie F, editors. Engineering plasticity.Cambridge University Press; 1968. p. 535–609.

[19] Schofield AN, Wroth CP. Critical state soil mechanics. McGraw-Hill;1968.

[20] von Wolffersdorff P-A. A hypoplastic relation for granular materialswith a predefined limit state surface. Mech Cohesive–Frictional Mater1996;1:251–71.

[21] Weifner T, Kolymbas D. Review of visco-hypoplastic equations. ActaGeotech 2008 [submitted for publication].

[22] Wu W. Hypoplastizitat als mathematisches Modell zum mechanis-chen Verhalten granularer Stoffe. Publ. series of: Institut furBodenmechanik und Felsmechanik der Universitat Fridericiana inKarlsruhe, No. 129, 1992.

[23] Wu W, Kolymbas D. Numerical testing of the stability criterion forhypoplastic constitutive equations. Mech Mater 1990;9:245–53.

Documents

Review of two hypoplastic equations for clay considering axisymmetric element deformations