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A note on the Lade-Duncan failure criterion

Online Publication Date: 01 January 2006To cite this Article: Yang, X. Q., Fung, W. H., Au, S. K. and Cheng, Y. M. (2006) 'Anote on the Lade-Duncan failure criterion', Geomechanics and Geoengineering, 1:4,299 - 304To link to this article: DOI: 10.1080/17486020600970797URL: http://dx.doi.org/10.1080/17486020600970797

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A note on the Lade--Duncan failure criterion

X. Q. YANG*†, W. H. FUNGz, S. K. AU§, and Y. M. CHENG††

†School of Civil Engineering and Architecture, Hubei University of Technology, Wuhan 430068, P.R. ChinazDepartment of Building and Construction, City University of Hong Kong, Hong Kong

§Department of Civil Engineering, Hong Kong University, Hong Kong††Department of Civil and Structural Engineering, Hong Kong Polytechnic University, Hong Kong

(Received 1 December 2005; in final form 16 August 2006)

The non-linear relationship between the internal frictional angle � and the first stress tensor invariant I1, together with the cohesion c of geomaterial, isused to show that the Lade--Duncan failure criterion is a more general failure criterion with a deep physical meaning. It can be used to describe thestrength failure characteristics of sand and normally consolidated soil mass as well as those of cemented geomaterials. As demonstrated for �e� �c onp planes, the Lade--Duncan failure criterion reveals the basic mechanical deformation characteristics for geomaterials. The test results for redsandstone indicate that the differences between the internal frictional angles in the triaxial extension and compression states on p planes will bereduced as the average principal stress increases.

Keywords: Lade--Duncan failure criterion; Ridge function; Friction angle; Stress tensor invariant; Deviatoric stress tensor invariant

1. Introduction

It is well known that the Lade--Duncan failure criterion (Lade

and Duncan 1975, 1978) meets the convexity requirement for a

wide range of friction angles in three-dimensional stress space

and is widely used in geotechnical engineering calculations. In

order to include cohesion, friction, and curved meridian in a

single model, Lade (1982, 1984) proposed a three-parameter

failure criterion to capture the strength characteristics of geo-

material. Lade’s investigations showed that the improved three-

parameter criterion gave a better prediction of the failure

strength of geomaterials.

The Lade--Duncan failure criterion (Lade and Duncan 1975,

1978) does not coincide with the Mohr--Coulomb failure criter-

ion at the triaxial extension and compression states at the same

time. This means that, on a p plane with average principal stress

constant, the internal frictional angle � of the geomaterial

gradually increases from the minimum value in the triaxial

compression state to almost the maximum value in the triaxial

extension state as the corresponding Lode angle �� is gradually

increased from -308 to 308. The failure criterion proposed by

Desai (1980) revealed similar behaviour for geomaterials.

These predictions are supported by the experimental data,

which shows that, in the plane strain state, the values of the

internal friction angle are about 10% larger than the values in

the triaxial compression state. van Eekelen (1980) and

Hashiguchi (1978, 2002) both proposed a more complex

mechanical explanation for the mechanism of the Lade--

Duncan failure criterion (Lade and Duncan 1975, 1978).

A further explanation of the Lade--Duncan failure criterion in

terms of micro- and macromechanical deformation mechanisms,

based on the above work and using actual strength test data for

the failure of red sandstone (Li 1990), is presented in this note.

2. Alternative expression for Lade--Duncan

failure criterion

The sign of a stress component is defined as positive for

compression, and we assume that the stress is the effective

stress. Under general stress states, invariants of the stress tensor

are expressed as follows:

I1 ¼ �ii ð1Þ

I2 ¼ �ij�ij=2 ð2Þ

I3 ¼ �ij�jm�mi=3 ð3Þ

J2 ¼ sijsij=2 ð4Þ

J3 ¼ sijsjmsmi=3 ð5Þ

where I1, I2, and I3 are the first, second, and third invariants of

the stress tensor respectively, J2 and J3 are the second and third

invariants of the deviatoric stress tensor, respectively, and

sij =�ij -- I1dij/3, where dij is the Kronecker delta, is the

Geomechanics and Geoengineering: An International Journal

Vol. 1, No. 4, December 2006, 299--304

*Corresponding author. Email: xueqiangyang@hotmail.com

Geomechanics and Geoengineering: An International JournalISSN 1748-6025 print=ISSN 1748-6033 online � 2006 Taylor & Francis

http:==www.tandf.co.uk=journalsDOI: 10.1080=17486020600970797

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deviatoric stress tensor. The Lode angle is given by

ys ¼1

3sin�1 � 3

ffiffiffi3p

2� J3

J3=22

!; � �

6� ys �

�

6ð6Þ

Relationships between the different stress tensor invariants

are as follows:

I2 ¼ I 21 =3� J2 ð7Þ

I3 ¼ J3 þ I1I2=3� 2I 31 =27 ð8Þ

Lade and Duncan (1975, 1978) proposed the following fail-

ure criterion for geomaterials

I 31 =I3 ¼ k1 ð9Þ

where k1 is a material parameter. Substituting equations

(7) and (8) in equation (9), and using equation (6), we

obtain

I1ffiffiffiffiffiJ2

p� �3

� 9k1

k1 � 27� I1ffiffiffiffiffi

J2

p � 18k1 sin 3��ffiffiffi3pðk1 � 27Þ

¼ 0 ð10Þ

If we substitute I1/p

J2 = r sinb in equation (10), we obtain the

following alternative expression:

sin3 � � 9k1

r2ðk1 � 27Þ sin � �18k1 sin 3��ffiffiffi3p

r3ðk1 � 27Þ¼ 0 ð11Þ

The following identity has a similar format to equation

(11):

sin3 � � 3

4sin� þ 1

4sin 3� ¼ 0 ð12Þ

Then r and sin3b are obtained from the identity as follows:

r ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3k1

k1 � 27

rð13Þ

sin 3� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 � 27

k1

rsin 3�� ð14Þ

Define A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðk1 � 27Þ=k1

p; then the three roots of I1/

pJ2 in

equation (10) are

I1ffiffiffiffiffiJ2

p ¼

rsin bþ2p

3

� �¼2

ffiffiffi3p

Asin

p

3þ1

3sin�1ðAsin3ysÞ

� �

rsinb¼�2ffiffiffi3p

Asin

1

3sin�1ðAsin3ysÞ

� �

rsin b�2p

3

� �¼�2

ffiffiffi3p

A

sinp

3�1

3sin�1ðAsin3ysÞ

� �

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð15Þ

In equation (15), only the first root can satisfy I1/p

J2� 0 in the

range --�/6 � �� � �/6. Therefore the correct solution to

equation (10) is

I1ffiffiffiffiffiJ2

p ¼ 2ffiffiffi3p

Asin

�

3þ 1

3sin�1 ðA sin 3��Þ

� �;

� �6� �� �

�

6

ð16Þ

Rearrangement of equation (16) gives the following alterna-

tive expression for the Lade--Duncan failure criterion:

ffiffiffiffiffiJ2

p¼ AI1

2ffiffiffi3p gð��Þ ð17Þ

where g(��) is a ridge function given by

gð��Þ ¼1

sinf�=3þ ½sin�1 ðA sin 3��Þ�=3g;

� �6� �� �

�

6

ð18Þ

Based on equations (17) and (18), the ratio k ofp

J2c under

triaxial compression (�� = --308) top

J2e under triaxial exten-

sion (�� = --308) on an octahedral p plane with constant I1 is

k ¼ffiffiffiffiffiffiJ2c

J2e

r¼ gð��=6Þ

gð�=6Þ ¼sin½�=3þ ðsin�1 AÞ=3�sin½�=3� ðsin�1 AÞ=3�

ð19Þ

From equation (19), when k1 = 27, A = 0 and k = 1, and

when k1!1, A!1 and k = 2. Therefore for a Lade--Duncan

failure criterion with k1� 27, A should be in the range 0� A� 1,

and the corresponding k. A value should be in the range 1� k�2. Therefore the ridge function given by equation (18) satisfies

the requirement that k = 1.0,2.0, convexity, and smoothness

in the three-dimensional principal stress space.

Based on equation (17), the value of A for a normally con-

solidated soil mass in which the consolidation pressure varies

only slightly can be assumed to be constant; thereforep

J2 is a

linear function of I1. Equations (9) and (17) are plotted in the

300 X. Q. Yang et al.

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meridian plane in figure 1(a) and in the octahedral p plane in

figure 1(b). Similar results for the Lade--Duncan failure criter-

ion had been reported by van Eekelen (1980) and Jiang and

Pietruszczak (1987).

3. Further development of the Lade--Duncan

failure criterion

An alternative form of the Lade--Duncan failure criterion is

(Hashiguchi 1978, 2002)

I 31

I3

¼ k1

¼ 12þ 81� sin�c1þ sin�c

þ 61þ sin�c1� sin�c

þ 1þ sin�c1� sin�c

� �2

ð20Þ

where �c is the frictional angle in the triaxial compression state.

The frictional angle �e in the triaxial extension state will be

discussed later.

According to the Mohr--Coulomb failure criterion, the ratio k

on an octahedral plane can be written

k ¼ 3þ sin�c3� sin�c

ð21Þ

which can be rearranged to give

sin�c ¼ 3k � 1

k þ 1ð22Þ

Now k should be a function of I1 which satisfies k = 1,2.

The simplest form of this function is

k ¼ 1þ expð��I1=fcÞ ð23Þ

where � is a material parameter and fc is the uniaxial compres-

sion strength of the geomaterial.

Substituting equation (23) into equation (22) gives

sin�c ¼3 expð��I1=f

cÞ

2þ expð��I1=fcÞ ð24Þ

Substituting equation (24) into equation (20) gives

I 31

I3

¼ k1¼12þ81�3expð��I1=f

cÞ=½2þ expð��I1=f

c�

1þ3expð��I1=fcÞ=½2þ expð��I1=f

c�

þ61þ3expð��I1=f

cÞ=½2þ expð��I1=f

c�

1�3expð��I1=fcÞ=½2þ expð��I1=f

c�

þ 1þ3expð��I1=fcÞ=½2þ expð��I1=f

c�

1�3expð��I1=fcÞ=½2þ expð��I1=f

c�

� �2

:

ð25Þ

In the case of a general geomaterial, the principal stress space

should be translated along the hydrostatic axis in order to

include the contribution of the cohesion c and tensile strength

�t in the failure criterion (Lade, 1982; Houlsby, 1986). Thus a

constant stress afc is added to the normal stress before substitu-

tion in equation (25):

Figure 1. Lade--Duncan failure criterion in three-dimensional principal stressspace.

Lade--Duncan failure criterion 301

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�11 ¼ �11 þ �fc

�22 ¼ �22 þ �fc

�33 ¼ �33 þ �fc

ð26Þ

where a is a material parameter. The value of afc reflects the

effect of the tensile strength �t of the geomaterial.

Therefore a more general Lade--Duncan failure criterion for

a geomaterial with cohesion cand tensile strength �t is given by

I3

1

I3

¼ k1 ¼ 12 þ 81� 3 expð��I1=f

cÞ=½2þ expð��I1=f

c�

1þ 3 expð��I1=fcÞ=½2þ expð��I1=f

c�

þ 61þ 3 expð��I1=f

cÞ=½2þ expð��I1=f

c�

1� 3 expð��I1=fcÞ=½2þ expð��I1=f

c�

þ 1þ 3 expð��I1=fcÞ=½2þ expð��I1=f

c�

1� 3 expð��I1=fcÞ=½2þ expð��I1=f

c�

� �2

:

ð27Þ

where the first and the third stress tensor invariants I1 and I3 can be

calculated from the new stress tensors �11, �22, and �33.

A similar process to that given in equations (9)--(18) can be

carried out using the following alternative form of equation (27):

ffiffiffiffiffiJ2

p¼ A I1

2ffiffiffi3p gð��Þ ð28Þ

where

A ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik1 � 27

k1

sð29Þ

and

gð��Þ ¼1

sinf�=3þ ½sin�1 ðA sin 3��Þ�=3g;

� �=6 � �� � �=6:

ð30Þ

Therefore the more general Lade--Duncan failure criterion

contains only two unknown material parameters, � and �,

which can describe both the failure strength characteristics of

the geomaterial, such as friction, tensile strength, or cohesion,

and the curved meridian in the three-dimensional principal

stress space. Equations (27) and (28) are shown in the meridian

plane and in the octahedral plane in figure 2 and figure 1(b),

respectively.

Lade (1982, 1984) modified the Lade--Duncan failure criter-

ion as follows:

I3

1

I3

� 27

!I1

pa

� �m

��1 ¼ 0 ð31Þ

where pa is the atmospheric pressure (expressed in the same

units as the stress) and�, m, and �1 are material parameters. The

corresponding equations are obtained as follows:

I3

1

I3

¼ k¢1 ¼ 27þ �1

I1

pa

� ��m

ð32Þ

A¢ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik¢

1 � 27

k¢1

sð33Þ

g¢ð��Þ ¼1

sinf�=3þ ½sin�1 ðA¢ sin 3��Þ�=3Þ;

� �=6 � �� � �=6

ð34Þ

ffiffiffiffiffiJ2

p¼ A¢I1

2ffiffiffi3p g¢ð��Þ ð35Þ

Comparison of equations (32) and (27) shows that both k¢1 and

k1 are non-linear functions of I1. Their common characteristics is

k¢1 values decrease as I1 is gradually increased. Although equa-

tion (32) is simpler than equation (27), three parameters have to

be identified. Relatively speaking, equation (27), established

using equations (22) and (23), appears to have a more direct

physical meaning for geomaterials than equation (31).

4. Optimization fitting technique

Equation (24) can be rearranged to give

exp � �I1

fc

� �¼ 2 sin�c

3� sin�cð36Þ

The material parameter � is then expressed as

Figure 2. Lade--Duncan failure criterion in normal stress--shear stress space.

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� ¼ lnð3� sin�cÞ � lnð2 sin�cÞI1=fc

ð37Þ

Then the average value � for the n-set true triaxial

failure test data (�11/fc, �22/fc, �33/fc) of a geomaterial is

given by

� ¼Xn

i¼1

lnð3� sin�ciÞ � lnð2 sin�ciÞI1i=fc

" #�n ð38Þ

The corresponding values ofp

J2m are calculated from the

test data as follows:

ffiffiffiffiffiffiffiJ2m

p¼ 1ffiffiffi

6p ½ð�11 � �22Þ2 þ ð�22 � �33Þ2

þ ð�11 � �33Þ2�1=2

ð39Þ

The error between the experimental value ofp

J2m calculated

using equation (39) and the theoreticalp

J2 value calculated

using equation (28) for the n-set test result is established as a

target function:

err ¼Xn

i¼1

ffiffiffiffiffiffiffiJ2mp�

i�

ffiffiffiffiffiJ2

p� i

�2 ð40Þ

For the tensile strength afc, it is well known that a = 0 for

sand and that � = 1.0 for metal. Therefore a should hence be

in the range 0 � � � 1.0 for general geomaterials. Therefore

based on the n-set of true triaxial failure test data (�11/fc, �22/fc,

�33/fc) of a geomaterial, the correct calculation procedures are

follows.

(i) a is defined in the range 0 � a � 1.0;

(ii) Based on the test data (�11/fc, �22/fc, �33/fc), sin�c can be

obtained using equation (20).

(iii) The values of � and err are then calculated using equa-

tions (38) and (40), respectively.

(iv) Different values of a correspond to different values of �and err, and the minimum value errmin is the final target

value. The values of a and � corresponding to errmin are

called the identified material parameters.

The above calculation procedures can be implemented easily

using a simple computer code. They allow a relatively reason-

able estimate of the tensile strength of a geomaterial, which is

usually difficult to measure, to be obtained.

5. Calibration and verification

The true triaxial failure strength test data for red sandstone (Li

1990) is given in table 1. Using the computer code developed by

the authors, the material parameters � ¼ 0:0982 and a = 0.025

are identified, which give errmin = 0.0152. The maximum

relative error for predicting the failure strength of red sandstone

is �5.56% (table 1). The results of the calculation show that the

predictions obtained using equation (27) are reasonable for

materials such as cemented soils, concrete, mortar, ceramics,

rock, etc.

The values of k and sin�c for red sandstone are

k ¼ 1þ exp½�0:0982ð0:075þ I1=fcÞ� ð41Þ

sin�c ¼ 3k � 1

k þ 1¼ 3 exp½�0:0982ð0:075þ I1=fcÞ�

2þ exp½�0:0982ð0:075þ I1=fcÞ�ð42Þ

According to the Mohr--Coulomb failure criterion (van

Eekelen 1980, Hashiguchi 2002), the following relationship

holds in the triaxial extension state:

Table 1 Comparison of true triaxial failure strength test results for red sandstone with theoretical predictions

�33/fc, �22/fc, �11/fc I1/(3fc) �� (deg)p

J2m/fcp

J2/fc [(p

J2 --p

J2m)/p

J2m] · 100 (%)

0.00, 0.00, 1.00 0.3333 -30.00 0.5774 0.5585 -3.270.00, 0.24, 1.38 0.5400 -20.63 0.7373 0.7298 -1.020.00, 0.44, 1.34 0.5933 -11.21 0.6830 0.6801 -0.430.00, 0.65, 1.28 0.6433 0.52 0.6400 0.6364 -0.510.00, 0.79, 0.92 0.5700 22.50 0.4979 0.5033 1.090.08, 0.08, 1.80 0.6533 -30.00 0.9930 0.9621 -3.120.08, 0.24, 2.20 0.8400 -26.12 1.1805 1.1361 -3.760.08, 0.40, 2.42 0.9667 -22.76 1.2688 1.2108 -4.570.08, 0.54, 2.01 0.8767 -16.81 1.0081 1.0296 2.140.08, 0.84, 1.78 0.9000 -3.50 0.8516 0.8863 4.080.08, 0.98, 1.86 0.9733 0.37 0.8900 0.9110 2.350.16, 0.16, 2.18 0.8333 -30.00 1.1663 1.1545 -1.010.16, 0.39, 2.22 0.9233 -24.15 1.1288 1.1916 5.560.16, 0.59, 2.54 1.0967 -20.24 1.2683 1.2843 1.260.16, 0.79, 2.67 1.2067 -16.04 1.3058 1.3016 -0.320.16, 0.98, 2.71 1.2833 -11.64 1.3018 1.2876 -1.090.16, 1.19, 2.25 1.2000 -0.48 1.0450 1.0867 3.99

Lade--Duncan failure criterion 303

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J2

p

I1

� ��� ¼ 30 ˆ

¼ 2 sin�effiffiffi3pð3þ sin�eÞ

ð43Þ

Substituting equation (28) into equation (43), we obtain

sin�e ¼3A � gð��Þ�¼308

4� A � gð��Þ�¼308

ð44Þ

Equations (41), (42), and (44), and the relationship (�e -- �c) , I1

are further plotted in figures 3, 4, and 5, respectively. Figure 3

shows that the ratio k decreases from 2.0 to 1.0 as the value of

I1/fc gradually increases. Figure 4 shows that the internal

frictional angle �, including �e and �c, decreases from 908 to

08 at the ultimate ideal plastic state as the average principal

stress increases. The value of (�e -- �c), which is initially zero,

increases gradually to its maximum value at I1/fc < 1.0, and

then decreases from the peak value and approaches zero as the

average principal stress is increased further. It is reasonable

that geomaterials exhibit the maximum differences in both

their strength characteristics and their deformation character-

istics at the peak value point. Figures 3, 4, and 5 show that

geomaterials approach a homogeneous state as the average

principal stress is increased.

6. Concluding remarks

As stress approaches the triaxial extension state in geomater-

ials, cracks and fractures develop by a micromechanical defor-

mation mechanism, and induce a higher internal frictional angle

� as a result of macromechanical deformation. As demonstrated

for �c � �e on p planes (figures 4 and 5), the Lade--Duncan

failure criterion can describe the basic physical mechanical

deformation characteristics for geomaterials. This is the reason

why this criterion is widely used in geotechnical engineering.

The original Lade--Duncan failure criterion (equation (9)) is

a more general failure criterion with a deeper physical meaning,

which can describe not only the strength failure characteristics

for sand and normal consolidation soil mass satisfactorily, but

can also capture these characteristics for cemented

geomaterials.

In geomaterials, the difference (�e -- �c) is greater at lower

stress levels and decreases with increasing average principal

stress, in agreement with engineering experience.

References

Desai, C. S., A general basis for yield, failure and potential functions inplasticity. Int. J. Numer. Anal. Methods Geomech., 1980, 4, 361--375.

Hashiguchi, K., Constitutive equation of soils: theories based on elastoplasti-city. Soils Found., 1978, 18(4), 131--142.

Hashiguchi, K., A proposal of the simplest convex-conical surface for soils.Soils Found., 2002, 42(3), 107--113.

Houlsby, G. T., A general failure criterion for frictional and cohesive materials.Soils Found., 1986, 26(2), 97--101.

Jiang J. and Pietruszczak, S., Convexity of yield loci for pressure sensitivematerials. Comput. Geotech., 1987, 5, 51--63.

Lade, P. V., Three-parameter failure criterion for concrete. J. Eng. Mech. Div.ASCE, 1982, 108(5), 850--563.

Lade, P. V., Failure criterion for frictional materials. In Mechanics ofEngineering Materials, edited by C.S. Desai and R.H. Gallagher, pp.385--402, 1984 (John Wiley : New York).

Lade, P. V. and Duncan, J. M., Elastoplastic stress--strain theory for cohesion-less soil. J. Geotech. Eng. ASCE, 1975, 101(10), 1037--1053.

Lade, P. V. and Duncan, J. M., Three-dimensional behavior of remolded clay.J. Geotech. Eng. ASCE, 1978, 104(2), 193--209.

Li, X. C. 1990. Research about effect of the intermediate principal stress on rockstrength and verification about strength theories. Master’s thesis, Rock andSoil Mechanics Institute, Academy of Science of China.

van Eekelen, H. A. M., Isotropic yield surfaces in three dimensions for use insoil mechanics. Int. J. Numer. Anal. Methods Geomech., 1980, 4, 89--101.

0

0.5

1

1.5

2

2.5

–2 0 2 4 6 8cfI1

Figure 3. Relation between k and I1/fc for red sandstone.

0

20

40

60

80

100

–2 0 2 4 6 8

Triaxial compression frictional angleTriaxial extension frictional angle

φ (°)

cf

I1

Figure 4. Relation between � and I1/fc for red sandstone.

0

2

4

6

8

10

12

–2 0 2 4 6 8

(φe − φc)(°)

cf

I1

Figure 5. Relationship curve between �e -- �c and I1/fc for red sandstone.

304 X. Q. Yang et al.