Computers and Geotechnics 41 (2012) 90–94

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Computers and Geotechnics

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

Technical Communication

A three-dimensional criterion for asymptotic states

Ting Luo a, Yang-Ping Yao a,⇑, An-Nan Zhou b, Xing-Guo Tian a

a Department of Civil Engineering, Beihang University, Beijing 100191, Chinab School of Civil, Environmental and Chemical Engineering, Royal Melbourne Institute of Technology (RMIT), Melbourne, VIC 3001, Australia

a r t i c l e i n f o

Article history:Received 30 September 2011Received in revised form 5 December 2011Accepted 5 December 2011Available online 22 December 2011

Keywords:Asymptotic stateThree-dimensional criterionDeformation constraints

0266-352X/$ - see front matter � 2011 Elsevier Ltd.doi:10.1016/j.compgeo.2011.12.002

⇑ Corresponding author.E-mail addresses: tluo@buaa.edu.cn (T. Luo), ypy

annan.zhou@rmit.edu.au (A.-N. Zhou), tianxingguo@c

a b s t r a c t

The asymptotic state of a soil is affected by stress states and deformation constraints. Experimentalresults indicate that the stress ratio approaches a constant when soil is loaded along with a strain path.Such a stable state is referred to as the asymptotic state. Indeed, failure can be viewed as a special con-dition of asymptotic states. Therefore, the shear strength of a soil at its failure state can also be inter-preted by the criterion for asymptotic states. A three-dimensional criterion for asymptotic states thatcan take into account the influence of deformation constraints is proposed in this technical note. Byinvolving both triaxial compression and extension tests to calibrate a new fitting parameter, a, the pro-posed criterion can provide an accurate simulation of the influence of Lode’s angle. The performance ofthe proposed three-dimensional criterion for asymptotic states is investigated via numerical examplesand then validated against experimental results.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The shear strength of a soil, in general, is referred to as the stressratio (g, the proportion between deviator stress and mean stress,g = q/p) at failure. In the case of some simple stress states, such astriaxial compression and extension, which can conventionally berealised in the laboratory, the shear strength can be determined di-rectly by laboratory testing. However, in real engineering practice,stress states are complex and difficult to duplicate using laboratoryequipment. In other words, we cannot determine the shear strengthof a soil under random stress states purely by laboratory testing. Amore effective and efficient way is to propose a general shearstrength criterion that can be used to predict the shear strength un-der unconventional stress states. Shear strength under conven-tional stress states that can be measured via laboratory testingcan be employed to calibrate the general shear strength criterion.

Over the past five decades, many general shear strength criteriahave been built to interpret the failure of soils. These general shearstrength criteria are also defined as three-dimensional criteria toemphasise that all three principal stress (r1, r2 and r3) are takeninto account. One of the most widely adopted general shearstrength criteria is proposed by Lade and Duncan [4], which was la-ter modified by Lade [3] for other geomaterials, including rock andconcrete. Based on a completely different theoretical background,Matsuoka and Nakai [5] proposed a general shear strength

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ao@buaa.edu.an (Y.-P. Yao),e.buaa.cn (X.-G. Tian).

criterion (SMP) by developing the concept of a ‘spatial mobilityplane’ that is expressed similarly in Lade and Duncan’s criterion.The two general criteria mentioned above can be calibrated by atriaxial compression test and used to predict the failure of soil un-der general stress states. The calibration of both Lade–Duncan andMatsuoka–Nakai criteria is based only on the triaxial compressiontest, which is convenient when the two criteria mentioned aboveare used in numerical modelling because only triaxial compressiondata would be needed for calibration. However, this conveniencealso sacrifices the accuracy of simulation to some extent.

In addition to the stress states, the ultimate state (ultimatestress ratio) of a soil is also relevant to the deformation con-straints but is usually ignored in the current shear strength cri-teria. A series of strain path controlled triaxial tests ([1], Fig. 1)show that the stress ratio at failure depends on the ratio be-tween volumetric strain increment and axial strain increment(dev/de1). Indeed, such a strain ratio can be viewed as an indica-tor of the deformation constraints. For example, zero generalstrain increment ratio (dev/de1 = 0) corresponds to an undrainedcondition. To describe such a relationship between shearstrength and deformation constraints, Gudehus et al. [2] pro-posed the concept of ‘asymptotic states’. When a soil sample isloaded along with a strain path, the stress ratio will finally ap-proach a constant, no matter what the initial stress state is. Sucha final stress state is named an ‘asymptotic state’. Asymptoticstates contain failure states and non-failure states. For example,in the case of undrained triaxial compression tests, an asymp-totic state is equal to a failure state (or critical state). Whilefor isotropic or K0 compression tests, an asymptotic state is a

0

0.5

1

1.5

2

0 1 2 3 4

dεv/dε1 = – 0.67

Axial strain, ε1 : %

dεv/dε1 = – 0.43

dεv/dε1 = – 0.11

Stre

ss r

atio

, η :

-

Fig. 1. Stress ratios at asymptotic states under different deformation constraints,data following Chu and Lo [1].

0

0.5

1

1.5

2

-3 -2 -1 0 1

η

–n

A: Isotropic comp.

B: K0 comp.

C: Undrained triaxial comp.

D: Drained triaxial comp.

A B C D

Eq. (2)

Fig. 2. The stress ratio at the asymptotic state and empirical simulation, datafollowing Chu and Lo [1].

q

n−

p

A

B C

D

A: Isotropic comp.

B: K0 comp.

C: Undrained triaxial comp.

D: Drained triaxial comp.

Fig. 3. Asymptotic surfaces simulated by Eq. (2) in the space of p, q and �n.

T. Luo et al. / Computers and Geotechnics 41 (2012) 90–94 91

non-failure state. But the stress ratio is still a constant. There-fore, failure is a special case of asymptotic states.

In this technical note, a new three-dimensional criterion forasymptotic states is proposed. The key features of the proposed cri-terion include the following: (1) take into account the influence ofdeformation constraints; (2) give an accurate simulation on the ef-fect of Lode’s angle by involving both triaxial compression andextension tests into calibration. This technical note is organised asfollows: (1) in the plane of mean stress (p) and deviator stress (q),a criterion to describe asymptotic states is introduced first; (2) then,this two-dimensional criterion is extended to three-dimensionalstress state; (3) finally, the performance of the proposed three-dimensional criterion for asymptotic states is investigated vianumerical examples and then validated against the experimentalresults.

2. Asymptotic state and two-dimensional criterion in p–q plane

In general, deformation constraints are quite common in bothreal engineering practices and laboratory investigations. For exam-ple, the deformation constraint of the soil beneath the strip foun-dation is that horizontal deformation is not allowed along thelongitudinal direction. For an undrained triaxial test, the volumet-ric strain increment is zero. The deformation constraints men-tioned above can be expressed via a strain increment ratio (n):

n ¼ dev

de1ð1Þ

where n is the strain increment ratio defined by the ratio betweenvolumetric strain increment and axial strain increment (the firstprincipal strain increment), and dev is the volumetric strain incre-ment and is equal to de1 + de2 + de3; de1, de2 and de3 are the first,second and the third principal strain increment, respectively. Differ-ent values of the strain increment ratio indicate different conditionsof deformation constraints and represent different drainage condi-tions. For example, the values n = 0, n > 0 and n < 0 correspond tothe completely undrained condition, the partially positive drainedcondition (water flows out of the soil), and the partially negativedrained condition (water flows into the soil), respectively.

Chu and Lo [1] presented a series of triaxial tests on Sydneysand with different strain ratios (n = �0.11, �0.43, and �0.67).The test data are plotted in the plane of the stress ratio (q/p) andthe axial strain, as shown in Fig. 1. The experimental results indi-cate that the stress ratio at failure state or the stress ratio at anasymptotic state is dependent on the values of the strain incrementratio. In other words, the shear strength of a soil sample is relevantto both the stress path and the strain path. For example, for testswith the same stress path (triaxial compression, r1 > r2 = r3), dif-ferent strain paths can generate different shear strengths (e.g.,the drained strength is higher than the undrained strength).

Chu and Lo [1] employed the following empirical equation (seeFig. 2) to describe the relationship between the stress ratio at theasymptotic state (ga) and the strain increment ratio (n):

ga ¼ M0 1� n3

� �¼ 6 sin /c0

3� sin /c0

� �1� n

3

� �ð2Þ

where M0 is the stress ratio at the asymptotic state for undrainedcompression tests (n = 0). /c0 is the friction angle obtained from un-drained compression tests. Eq. (2) can be visualised in the space ofp, q and n, as shown in Fig. 3. In Figs. 2 and 3, some typical points areindicated, such as isotropic compression (point A), K0 compression(point B), undrained triaxial compression (point C), and drained tri-axial compression (point D).

For triaxial compression, the first principal strain increment(de1) can be expressed via the deviator strain increment (ded)and the volumetric strain increment (dev) as follows.

de1 ¼ ded þ13

dev ð3Þ

Because deviator strain is more frequently used in soil mechan-ics, the strain increment ratio (n) can be replaced by the generalstrain increment ratio (m) between incremental volumetric strainand incremental deviator strain by combining Eqs. (1) and (3):

m ¼ dev

ded¼ 3n

3� nð4Þ

Indeed, there is no substantial difference between the generalstrain increment ratio (m) and the strain increment ratio (n). Butin critical state soil mechanics, the shear-dilation relationship isusually represented by the proportion between increment of volu-metric strain and that of deviator strain. Therefore, the generalstrain increment ratio (m) can perhaps be more conveniently ap-

p

q

Shearing w

ith dila

tion

Shearing with compression

Eq.(5)

a02

3M

η = (K0 compressio

n) a

0Mη <

(m > 0)

a

0Mη

> (m

< 0)

a

0Mη

= (U

ndrained triaxial)

a 0η = (Isotropic compression) aη increases

m increases

Fig. 4. Stress ratios at asymptotic states with different values of m.

1σ

2σ 3σ

0.75=α1=α

0.5=α0.25=α0=α

SMP

Mises

Fig. 5. Various curves of the proposed criterion with different values of a.

92 T. Luo et al. / Computers and Geotechnics 41 (2012) 90–94

plied to more general situations. The relationship between thestress ratio at an asymptotic state (ga) and the general strain incre-ment ratio (m) can be rewritten as:

ga ¼3M0

3þm¼ 3

3þm6 sin /c0

3� sin /c0

� �ð5Þ

Eq. (5) indeed presents a criterion for the asymptotic state in thecondition of triaxial compression (in the plane of p–q). The stress ra-tio at the asymptotic state is affected by the value of the strain ratio(m). For example, when m =1 (i.e., isotropic compression), ga = 0,which means no shear stress is generated during isotropic compres-sion. When m > 0 (i.e., shearing with negative dilation), the stressratio at the asymptotic state is less than that of the undrained state(i.e., ga < M0). When m < 0 (i.e., shearing with positive dilation), thestress ratio at the asymptotic state is larger than that of the un-drained state (i.e., ga > M0). Eq. (5) can be presented in the planeof mean stress and deviator stress (i.e., a two-dimensional stressspace), as shown in Fig. 4.

3. Three-dimensional criterion for asymptotic state

As mentioned above, the asymptotic state can be influenced bythe following two factors: strain conditions and stress conditions.Eq. (5) makes an empirical interpretation of the influence of strainstates on asymptotic states. In this section, the influence of stressstates on asymptotic states will be discussed in detail.

3.1. Generalised nonlinear strength criterion

To reproduce the influence of Lode’s angle on shear strengthwith more accuracy, a generalised nonlinear strength criterion thatis based on the SMP and Extended Mises criteria was proposed byYao et al. [8]. The generalised nonlinear strength criterion can bewritten as:

aqM þ ð1� aÞqS ¼ Mcp ð6Þ

where Mc is the stress ratio at failure under triaxial compressionstate, qM is the deviator stress corresponding to Extended Mises cri-terion, qS is the deviator stress in the SMP criterion. They can be ex-pressed as:

qM ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI21 � 3I2

qð7Þ

and

qS ¼2I1

3ffiffiffiffiffiffiffiffiffiffiffiffiffiI1I2�I3

I1I2�9I3

q� 1

ð8Þ

where I1, I2, and I3 are the first, second and third stress invariants,respectively, which can be decomposed as follows:

I1 ¼ r1 þ r2 þ r3

I2 ¼ r1r2 þ r2r3 þ r3r1

I3 ¼ r1r2r3

9>=>; ð9Þ

The fitting parameter a can be calibrated by triaxial compres-sion and extension tests:

a ¼ 3ð3þ sin /eÞðsin /e � sin /cÞ2 sin2 /eð3� sin /cÞ

ð10Þ

where /c and /e are the friction angles obtained from triaxial com-pression and extension tests, respectively, which can be expressedas:

/c ¼ arcsin 3Mc6þMc

/e ¼ arcsin 3Me6�Me

)ð11Þ

where Me is the stress ratio at failure under triaxial extension state.(1) When a = 1, we have qM = pMc. The failure curve on the p-

plane is a circle; the proposed generalised nonlinearstrength criterion is identical to the Extended Misescriterion.

(2) When a = 0, qS = pMc. The failure curve on the p-plane is acurved triangle, i.e., the SMP strength curve.

(3) When a is between 0 and 1, the shape of the proposed crite-rion is the smooth curve between the Extend Mises criterionand the SMP criterion (see Fig. 5), and the curvature of theproposed criterion is determined by triaxial compressionas well as by triaxial extension test results.

Compared with the former three-dimensional criteria, such asLade–Duncan and Matsuoka–Nakai (SMP) criteria, the generalisednonlinear strength criterion can provide more accurate simulationfor different Lode’s angles. The former three-dimensional criteriaare only calibrated by the experimental data from triaxial com-pression tests. Such criteria indeed provide different extrapolationformula from one known state, i.e., triaxial compression, whereasthe generalised nonlinear strength criterion employs experimentalresults obtained from both triaxial compression and extensionstates to perform the criterion calibration. Eq. (6) can be regardedas an interpolation formulation to interpret the shear strength withdifferent Lode’s angles based on two boundary conditions (i.e., tri-axial compression state and triaxial extension state). In general, theaccuracy of interpolation is always much higher than that ofextrapolation. However, the interpolation method usually requiresmore calibration effort than the extrapolation method.

Fig. 7. The evolution of asymptotic state curves with different values of a0.

T. Luo et al. / Computers and Geotechnics 41 (2012) 90–94 93

3.2. Extension of the criterion for asymptotic states from two-dimensional to three-dimensional

A failure state due to shearing, as mentioned above, can beviewed as a special case of asymptotic states. The interpolationmethod for the criterion of asymptotic states is assumed to beidentical to that for shear strength. In other words, the interpola-tion method presented in Eq. (6) can be applied to Eq. (5) to realisethe extension of the two-dimensional criterion for asymptoticstates to the three-dimensional criterion, i.e.:

3a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI2

1 � 3I2

qI1

þ 2ð1� a0ÞffiffiffiffiffiffiffiffiffiffiffiffiffiI1 I2�I3

I1 I2�9I3

q� 1¼ 3M0

3þmð12Þ

where the fitting parameter a0 can be expressed based on the Eq.(10):

a0 ¼3ð3þ sin /e0Þðsin /e0 � sin /c0Þ

2 sin2 /e0ð3� sin /c0Þð13Þ

where /c0 and /e0 are the friction angles obtained from undrainedtriaxial compression and extension tests, respectively. The addi-tional subscript ‘0’ here means undrained state, i.e., m = 0. Eq. (12)presents a three-dimensional criterion for asymptotic states in theplane of I1, I2 and I3 (or in the plane of r1, r2 and r3). It can alsobe rewritten in the plane of mean stress (p), deviator stress (q),and Lode’s angle (h) as follows:

a0qpþ 6qð1� a0Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi108p3�9q2p�q3 cos 3h

3p3�qp2 cos 3h

q� q¼ 3

3þm6 sin /c0

3� sin /c0

� �ð14Þ

where h is Lode’s angle, which can be written as:

0 6 h ¼ 13

arccos12ð2I3

1 � 9I1I2 þ 27I3ÞðI2

1 � 3I2Þ3=2

!6

p3

ð15Þ

h = 0 and h = p/3 correspond to triaxial compression and extension,respectively. To calibrate Eq. (14), we need to perform triaxial com-pression and extension tests to determine /c0 and /e0. Eq. (14) canbe visualised in the space of r1, r2 and r3 in Fig. 6.

m < 0

m = 0

m > 0

m = ∞

3σ1σ

2σ

Fig. 6. Three-dimensional criterion for asymptotic state surfaces in the space of r1,r2 and r3.

3.3. Summary of the proposed criterion

To predict the stress ratio at asymptotic states for a providedLode’s angle (h) and general strain increment ratio (m), we needto measure /c0 and /e0 to determine the fitting parameter (a0)and the value of M0. Indeed, the stress ratio at asymptotic statesfor the proposed criterion is determined by the above three param-eters (i.e., h, M0 and a0) and a general strain increment ratio (m).Parameter M0, which is usually a constant for a soil, is perhapsthe most basic material parameter in soil mechanics. In this sec-tion, the influence of the fitting parameter (a0), the general strainincrement ratio (m) and Lode’s angle (h), on the proposed criterionwill be investigated in detail.

Fig. 7 shows the shape of the asymptotic state curves in the p-plane with different values of parameter a0 with the same value ofm. When a0 = 0, the asymptotic state curve is identical to the curvecalculated using the SMP criterion. The asymptotic state curve is acircle (i.e., Mises criterion) provided that a0 = 1. When a0 has a va-lue between 0 and 1, the asymptotic state curves calculated by Eq.(14) are sandwiched between the SMP criterion and the circle. In-deed, the parameter a0 governs the shape of the asymptotic statecurve.

The effects of the different values of the general strain incre-ment ratio (m) on the asymptotic state curves are plotted in thep-plane, as shown in Fig. 8 (a0 is set to 0, 0.5 and 1). The size ofthe asymptotic state curve is controlled by the values of the gen-eral strain increment ratio (m) with the constant M0.

Fig. 8. The evolution of asymptotic state curves with different values of m.

Fig. 9. The evolution of asymptotic state curves with different values of Lode’sangle.

Fig. 10. Calibration and validation of proposed criterion in general stress states(data following Nakai and Matsuoka [6,7]).

94 T. Luo et al. / Computers and Geotechnics 41 (2012) 90–94

Fig. 9 shows the asymptotic state curves in the plane of stressratio and Lode’s angle. M0 is set to 1.3. The value of m varies from�0.6 to 1.2, while the value of a0 from 0 to 1. From Fig. 9, we canfind that the influence of the values of a0 becomes more and morespecific due to the increase of Lode’s angle as well as the decreaseof the value of m.

4. Experimental validations

Nakai and Matsuoka [6,7] presented a series of true triaxial testsand undrained triaxial tests on Toyoura sand, as shown in Fig. 10,where the symbol d stands for undrained triaxial test results, andthe symbol j stands for drained true triaxial test results. The aver-age particle size of Toyoura sand is 0.2 mm, and the specific gravityis 2.65. The maximum and minimum void ratios are 0.95 and 0.58,respectively. In the undrained triaxial compression tests, the stress

ratio (r1/r3) at critical state is equal to 3.82. Based on this value,the parameter M0 can be determined as 1.45 for Toyoura sand,and the corresponding friction angle /c0 is 35.80�. In the undrainedtriaxial extension test, the stress ratio (r1/r3) at the critical state is4.4, and the corresponding friction angle /e0 is 39.05�. Parametera0 (=0.26) can be determined using Eq. (12), /c0 and /e0. In thedrained triaxial compression test, the ratio between incrementalvolumetric strain and incremental deviator strain is equal to�0.41, which can be used to calibrate the general strain incrementratio m (=�0.41).

As shown in Fig. 10, the predicted shear strength curve (solidcurve) by Eq. (14) and that by SMP criterion (dash curve) are com-pared with the test data of Toyoura sand. The comparison indicatesthat Eq. (14) gives a good prediction on the drained test (i.e.,m = �0.41) results under the general stress state. It is necessaryto state that, in the prediction, all the parameters are calibratedby undrained compression/extension tests. Indeed, we use theparameters obtained from the undrained state to successfully pre-dict the shear strength (or asymptotic state) in the drained state.

5. Conclusions

A three-dimensional criterion for asymptotic states that can beused to predict the asymptotic state in general stress states for agiven deformation constraint (m) is proposed in this paper. Twomaterial parameters are introduced in the proposed criterion:stress ratio at critical state under undrained triaxial compressioncondition and a fitting parameter. Both of these parameters canbe conventionally calibrated by undrained triaxial compressionand extension tests. The comparison between the predictions andlaboratory measures in general stress states confirms the validityof the proposed criterion.

Acknowledgement

The authors wish to acknowledge the support provided by theNational Natural Science Foundation of China (Grant Nos.10872016, 11072016 and 90815024).

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