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doi:10.1016/j.ijr

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International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186

www.elsevier.com/locate/ijrmms

Technical Note

A quantitative comparison of six rock failure criteria

Thomas Benza,�, Radu Schwabb

aInstitute of Geotechnical Engineering, Universitat Stuttgart, Pfaffenwaldring 35, 70569 Stuttgart, GermanybFederal Waterways Engineering and Research Institute, Kussmaulstr. 17, 76187 Karlsruhe, Germany

Received 8 May 2007; received in revised form 8 January 2008; accepted 17 January 2008

Available online 1 April 2008

1. Introduction

This paper presents a quantitative comparison of someof the most common criteria for rock failure: TheMohr–Coulomb (MC) criterion, the Lade–Duncan (Lade)criterion, an approximation to the Wiebols and Cook(WC) criterion, the Hoek–Brown (HB) criterion, and theMogi criterion. Furthermore, the new Hoek–Brown–Matsuoka–Nakai (HBMN) failure criterion proposed byBenz et al. [1] is considered in the comparison, which isconducted using true triaxial test data for eight differentrocks. Of the several possible versions of the Mogicriterion, only its linear form, where its parameters areclosely related to the MC shear strength parameters [2], isconsidered here. The HB criterion is employed in itsgeneralized 2002 version [3]. The HB and the HBMNcriteria use the same set of four material parameters. Twoof them define intact rock behavior and two define rockmass behavior. Therefore, all criteria considered in thisstudy require two material parameters to describe triaxialtest data of intact rock.

In stress space, the six failure criteria differ in two mainaspects. In p2q space, the MC, ML, WC, and the Mogicriteria are linear, whereas the HB and the HBMN are non-linear. In contrast to the ML, WC, Mogi, and the HBMNcriteria, the MC and the HB criteria are not influenced bythe intermediate principal stress. Definitions of the stressinvariants p and q are given in the next section.

The quantitative comparison of the six failure criteriafollows the methodology introduced in Colmenares andZoback [4], which is outlined in more detail in Section 3: Agrid search was performed to determine the materialparameters which result in the smallest misfit for each

e front matter r 2008 Elsevier Ltd. All rights reserved.

mms.2008.01.007

ing author. Tel.: +49711 685 69020;

5 62439.

ess: Thomas.Benz@igs.uni-stuttgart.de (T. Benz).

rock type. Conclusions are then drawn from the magnitudeof misfit as well as from the material parameters thatoptimize each criterion’s data fitting behavior.

2. Six rock failure criteria

2.1. Definition of stress measures

Within this paper, compressive stress is consideredpositive. Tensile stress is negative. Stresses are alwaysconsidered to be effective values without any specialindication by a prime. Principal stresses are denoted byonly one subscript, e.g. si with i ¼ 1; 2; 3. The Roscoe stressinvariants p (mean stress) and q (deviatoric stress), aredefined as:

p ¼s1 þ s2 þ s3

3and

q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1 � s2Þ

2þ ðs2 � s3Þ

2þ ðs3 � s1Þ

2

2

s. (1)

The Roscoe stress invariants relate to the octahedralnormal stress, soct and to the octahedral shear stress,toct as:

soct ¼ p and toct ¼

ffiffiffi2p

3q, (2)

respectively.

2.2. The Mohr–Coulomb criterion

The Mohr–Coulomb (MC) failure criterion [5] is one ofthe earliest and most trusted failure criteria for soils androcks. Failure is assumed when in any (failure) plane theshear stress t reaches the failure shear stress tmax which isgiven by a functional relation of the form:

tmax ¼ cþ sn tanj, (3)

ARTICLE IN PRESST. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186 1177

where c is the cohesion of the material, j is the frictionangle of the material, and sn is the normal stress acting onthe respective failure plane. Alternatively, the MC criterioncan be expressed in principal stresses as follows:

s1 � s3 ¼ 2c cosjþ sinjðs1 þ s3Þ. (4)

The unconfined compressive strength sci postulated by theMC criterion is then obtained by setting s3 ¼ 0:

sci ¼ 2c tanp4þ

j2

� �. (5)

With Eq. (5), the MC criterion (Eq. (4)) can be alsorewritten as:

s1 ¼ sci þ tan2p4þ

j2

� �s3. (6)

2.3. The modified Lade–Duncan criterion

Lade and Duncan [6] proposed the following failurecriterion:

I31I3¼ Kj, (7)

where I1 ¼ ðs1 þ s2 þ s3Þ=3 ¼ p and I3 ¼ s1s2s3 are thefirst and third stress invariants, respectively, and Kj is amaterial parameter. The original Lade–Duncan criteriondoes not take cohesion into account. As cohesion is anessential concept for rock failure criteria that are linear inp2q space, a shift of the Lade–Duncan yield surface alongthe hydrostatic axis p ¼ const: was later introduced by Ewy[7]. The modified Lade–Duncan (Lade) criterion accordingto Ewy can be written as:

I31I3¼ 33ð33 þ ZÞ, (8)

where the stress in calculating the first and third stressinvariants is shifted by a ¼ c= tanj:

I1 ¼ ððs1 þ aÞ þ ðs2 þ aÞ þ ðs3 þ aÞÞ (9)

and

I3 ¼ ðs1 þ aÞðs2 þ aÞðs3 þ aÞ, (10)

where c is the cohesion of the material and j is the frictionangle of the material. By means of Eq. (5), these modifiedstress invariants could likewise be expressed in terms of theunconfined compressive strength sci. In triaxial compres-sion, consistent results from the Lade criterion and the MCcriterion are obtained by defining Z as follows:

Z ¼4 tan2jð9� 7 sinjÞ

1� sinj. (11)

2.4. The modified Wiebols and Cook criterion

Wiebols and Cook [8] derived a failure criterion bycalculating the shear strain energy associated with micro-cracks in rock material. This model requires the knowledgeof the coefficient of sliding friction between crack surfaces,

the number of mean cracks per unit solid angle in a unitvolume, and the critical value of shear strain energy. As thecriterion cannot be evaluated in closed form in the case ofpolyaxial stresses, Zhou [9] proposed a similar but simplercriterion. The latter has been named the modified Wiebolsand Cook (WC) criterion [4] and is explained in more detailhere. The modified Wiebols and Cook criterion postulatesfailure whenever:

q ¼ sci þBffiffiffi3p ð3p� sciÞ þ

C

3ffiffiffi3p ð9p2 � s2ciÞ, (12)

where p and q are the Roscoe invariants as defined inSection 2.1. The adaption of this extended Drucker–Pragercriterion to the Wiebols and Cook criterion is accom-plished by setting:

B ¼

ffiffiffi3pðn� 1Þ

nþ 2�

C

3ð2sci þ ðnþ 2Þs3Þ (13)

and

C ¼

ffiffiffiffiffi27p

2mþ ðn� 1Þs3 � sci

�mþ ðn� 1Þs3 � sci

2mþ ð2nþ 1Þs3 � sci

�n� 1

nþ 2

� �, (14)

where n ¼ tan2ðp=4þ j=2Þ and m ¼ ð1þ 3=5 tanjÞsci.

2.5. The Mogi criterion

Mogi proposed two functional relationships for rockfailure, of which only the latter (Mogi 1971 criterion [21]) isconsidered here. In this, Mogi relates the octahedral shearstress at failure to the sum of the minimum and maximumprincipal stresses:

toct ¼ fs1 þ s3

2

� �, (15)

where f is a monotonically increasing function. Plotting toctagainst sm;2 ¼ ðs1 þ s3Þ=2 for different experimental datareveals that a linear function f readily gives satisfactoryresults, e.g. [2]. The linear Mogi criterion can be written as:

toct ¼ aþ bs1 þ s3

2

� �. (16)

Considering that in triaxial conditions q ¼ s1 � s3 and thattoct ¼

ffiffiffi2p

=3q (see Eq. (2)), the linear Mogi parameters a

and b relate to the Coulomb shear strength parameters c

and j in triaxial compression and extension as follows:

a ¼2ffiffiffi2p

3c cosj and b ¼

2ffiffiffi2p

3sinj. (17)

This can be proven by substituting Eq. (16) into Eq. (4). Asa result, the linear Mogi criterion is also sometimes referredto as the Mogi–Coulomb criterion [2].

2.6. The Hoek–Brown criterion

At failure, the generalized HB criterion [3] relates themaximum effective stress, s1 to the minimum effective

ARTICLE IN PRESST. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–11861178

stress s3 through the functional relation:

s1 ¼ s3 þ sci mb

s3sci

þ s

� �a

, (18)

where mb extrapolates the intact rock constant mi to therock mass:

mb ¼ mi expGSI� 100

28� 14D

� �, (19)

sci is the uniaxial compressive strength of the intact rockand s and a are constants which depend upon the rockmass’s characteristics:

s ¼ expGSI� 100

9� 3D

� �, (20)

a ¼1

2þ

1

6exp

�GSI

15

� �� exp

�20

3

� �� �. (21)

The geological strength index (GSI), introduced by Hoek[10], provides a system for estimating the reduction in rock

Fig. 1. Failure criterion in principal stress space and in the deviatoric plane. (

(d) Modified Wiebols–Cook; (e) Hook-Brown; (f) Hook-Brown (MN).

mass strength under different geological conditions. GSItakes into account the geometrical shape of intact rockfragments as well as the condition of joint faces. Finally, D

is a factor which quantifies the disturbance of rock masses.It varies from 0 (undisturbed) to 1 (disturbed) dependingon the amount of stress relief, weathering, and blastdamage as a result of nearby excavations.Henceforth the generalized HB criterion is written as:

f HB ¼ s1 � s3 � ~f ðs3Þ with ~f ¼ sci mb

s3sci

þ s

� �a

. (22)

2.7. The HBMN criterion

The HBMN criterion extends the generalized HBcriterion as described above, with the spatial mobilizedplane concept of Matsuoka and Nakai (MN) [11,12]. Thedeviatoric shape of the MN criterion is assigned to the HBcriterion by setting:

f HBMN ¼ q� LMc;HBðp� ptÞ, (23)

a) Mohr–Coulomb; (b) Mogi (linear Mogi–Coulomb); (c) Modified Lade;

ARTICLE IN PRESS

Table 1

Rock types and data source

Rock# Rock type Test reference Data source

1 Dunham dolomite [15] [4]

2 Solenhofen limestone [15] [4]

3 Shirahama sandstone [16] [4]

4 Yuubari shale [16] [4]

5 KTB amphibolite [17] [4]

6 Mizuho trachyte [15] [2]

7 Dense marble [18,19] [2]

8 Westerly granite [20] [2]

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186 1179

where

Mc;HB ¼3 ~f 0

~f 0 þ 3and Me;HB ¼

3 ~f 0

2 ~f 0 þ 3, (24)

and

pt ¼ �~f

Mc;HBþ~f

3þ s3, (25)

and L varies between 1 and d ¼Me;MC=Mc;MC for triaxialcompression and extension, respectively [13]:

LðyÞ ¼

ffiffiffi3p

d

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 � dþ 1

p 1

cos W

with

W ¼1

6arccos �1þ

27d2ð1� dÞ2

2ðd2 � dþ 1Þsin2ð3yÞ

� �

for yp0;

W ¼p3�

1

6arccos �1þ

27d2ð1� dÞ2

2ðd2 � dþ 1Þsin2ð3yÞ

� �

for y40;

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

(26)

where the Lode angle y is defined as:

y ¼1

3arcsin

�3ffiffiffi3p

J3

2J3=22

!(27)

and J2 and J3 are the second and third invariants of thedeviatoric stress tensor, respectively.

For a more detailed description of the spatial mobilizedplane concept and the HBMN criterion, the reader isreferred to [1].

2.8. A qualitative comparison in stress space

Fig. 1 shows the shape of all six failure criteriaintroduced above in stress space. The MC, Lade, and thelinear Mogi criteria are all linear functions in p2q. TheWC, HB, and the HBMN criteria, on the other hand,represent non-linear functions in p2q. In triaxial compres-sion, however, the WC criterion responds linear. TheMatsuoka–Nakai extension used in the HBMN criteriondoes not alter the shape of the original HB criterion in p2q

space. The difference between the HB and HBMN criterionis only evident in the deviatoric plane. The MC and theHB criterion, on the other hand, are identical in thedeviatoric plane; both failure criteria are independent ofthe intermediate principal stress. In this respect, the Lade,WC, and linear Mogi criterion differ from the MC and HBcriteria, as the former incorporate an influence of theintermediate principal stress.

The Lade and the HBMN criteria have smooth andconvex contours in the deviatoric plane. The WC criterionis non-smooth and moreover not well defined for tensilestresses. The Mogi criterion is neither smooth nor convexand hence violates Drucker’s convexity postulate [14]. In anumerical scheme, the Lade and the HBMN criteria should

therefore be preferred over the other criteria from atheoretical point of view. Whether or not those criteriashould be the preferred choice from a physical point ofview is clarified in the quantitative comparison presentedhere.

3. Quantitative comparison

The following quantitative comparison uses true triaxialtest data from eight different rocks. All the data werecollected from the literature. References to the data origins(experiments) and to the sources where the data sets areprovided in tabular form are both given in Table 1.Following the idea of Colmenares and Zoback [4], the

quantitative comparison is conducted in the s12s2-plane.Figs. 2–5 present data examples in this plane for thefollowing rock types: KTB amphibolite, Mizuho trachyte,Dense marble, and Westerly granite, respectively. Testseries with the same principal stress s3 are marked with thesame symbol. In the examples, all test series are fittedso that the mean standard deviation misfit of the respectivefailure criterion is minimized.

3.1. Least mean standard deviation misfits

The least mean standard deviation misfit within thisstudy is calculated as follows: Let m be the number oftest series (i.e., tests with identical s3), n be the numberof data points per series, and i and j the indices for testseries and data point per series, respectively. Then, thestandard deviation of a failure criterion in test series i iscalculated as:

si ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xj

ðscalc1;j � stest1;j Þ2

s, (28)

where stest1;j is the tested maximum stress at failure for datapoint j and scalc1;j is the calculated one. Finally the meanstandard deviation misfit is calculated as the arithmeticmean of all calculated standard deviations for a specificrock type:

s ¼1

m

Xi

si. (29)

ARTICLE IN PRESS

00

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

KTB amphibolite - MC KTB amphibolite - Mogi

KTB amphibolite - HB KTB amphibolite - HBMN

KTB amphibolite - Mod. Lade-Duncan KTB amphibolite - Mod. Wiebols-Cook

150012501000750500250 00

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]150012501000750500250

00

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]1500125010007505002500

0

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]150012501000750500250

00

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]150012501000750500250 0

0

250

500

750

1000

1250

1500

σ 1 [

MPa

]

σ2 [MPa]150012501000750500250

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

σ3 in [MPa]

σ3 = 0

σ3 = 30

σ3 = 60

σ3 = 100

σ3 = 150

Fig. 2. Best fit solutions for the KTB amphibolite experimental data in the s12s2-plane.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–11861180

In order to find the least mean standard deviation misfit, agrid search over a greater number of material parametercombinations was conducted. As an example, the misfitcontour plot for the HBMN criterion is shown in theappendix.

Least mean standard deviation misfits have beenpreviously introduced in a study by Colmenares andZoback [4]. However, as Colmenares and Zoback do notmathematically define their misfit calculation, their proce-dure can only be assumed identical to the one outlinedabove. For all five rocks investigated by Colmenares and

Zoback, the misfits of the MC criterion and the HBcriterion are almost identical in both studies (except for arather big departure of the MC misfit in the case of theKTB amphibolite, which is due to the limitation of jp45�

in the study). The HBMN criterion was not considered inthe study by Colmenares and Zoback.

3.2. Summary of results

The results of the misfit study are given in Fig. 6, andTables 2 and 3. Fig. 6 illustrates the least mean standard

ARTICLE IN PRESS

0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

Mizuho trachyte - MC Mizuho trachyte - Mogi

Mizuho trachyte - HB Mizuho trachyte - HBMN

Mizuho trachyte - Mod. Lade-Duncan Mizuho trachyte - Mod. Wiebols-Cook

600500400300200100 0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

600500400300200100

0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

600500400300200100 0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

600500400300200100

0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

600500400300200100 0σ2 [MPa]

0

100

200

300

400

500

600

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 15

σ3 = 30

σ3 = 45

σ3 = 60

σ3 = 75

σ3 = 100

600500400300200100

Fig. 3. Best fit solutions for the Mizuho trachyte experimental data in the s12s2-plane.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186 1181

deviation misfits calculated within this study for all eightrocks. The calculated least mean standard deviation misfitsand the respective best fit parameters are summarized inTables 2 and 3, respectively. An example misfit contourplot is shown in the appendix (Fig. 7). Figs. 2–5 show therespective best fitting solutions in the s12s2-plane for fourof the eight rocks.

The mean standard deviation misfit is a scalar indicatorof how precisely a failure criterion can predict rock failure.An ideal criterion in an ideal test environment would yield

no misfit. Generally, the higher the precision of a criterion,the lower the resulting misfit. With the help of thecalculated misfits, the merits of non-linear and intermediateprincipal stress dependent failure criteria are now analyzedusing direct comparisons:

�

Mohr–Coulomb vs. Hoek–BrownThe only difference between the MC and the HB

criterion is their shape in p2q space. The HB criterion iscurved, whereas the MC criterion is linear. Comparing

ARTICLE IN PRESS

Dense marble - HBMNDense marble - HB

Dense marble - Mogi

0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

Dense marble - MC

Dense marble - Mod. Wiebols-CookDense marble - Mod. Lade-Duncan

40035030025020015010050 0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]

40035030025020015010050

0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]

40035030025020015010050 0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]

40035030025020015010050

0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]

40035030025020015010050 0

σ2 [MPa]

0

50

100

150

200

250

300

350

400

σ 1 [

MPa

]

40035030025020015010050

σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

σ3 in [MPa]

σ3 = 0

σ3 = 4

σ3 = 7

σ3 = 14

σ3 = 21

σ3 = 28

σ3 = 55

Fig. 4. Best fit solutions for the dense marble experimental data in the s12s2-plane.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–11861182

the misfits of the MC to the misfits of the HB crite-rion shows that the non-linear criterion alwaysyields equal or less deviation from the test data.The data examples given in Figs. 2–5 illustrate this factwell.

� Hoek–Brown vs. HBMN and Mohr–Coulomb vs. linear

MogiBoth pairs of failure criteria have identical shapes in

the p2q plane and differ only in their deviatoric shape.In six out of the eight rocks tested, a clear reduction ofthe misfit can be seen when the intermediate principal

stress is considered in the failure criterion. Generally,the shape of the HBMN criterion is closer to experi-mentally observed rock failure behavior than the shapeof the Mogi criterion. However, especially in the case ofMizuho trachyte, it appears that neither the Mogicriterion nor the HBMN criterion fits the experimentaldata well (Fig. 3).

The criterion with the overall least misfit in the currentbenchmark test is the HBMN criterion, which gave theleast misfit in five out of eight tests (Fig. 6).

ARTICLE IN PRESS

0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

Westerly granite - MC Westerly granite - Mogi

Westerly granite - HB Westerly granite - HBMN

Westerly granite - Mod. Lade-Duncan Westerly granite - Mod. Wiebols-Cook

140012001000800600400200 0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

140012001000800600400200

0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

140012001000800600400200 0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

140012001000800600400200

0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

140012001000800600400200 0

σ2 [MPa]

0

200

400

600

800

1000

1200

1400

σ 1 [

MPa

]

140012001000800600400200

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

σ3 in [MPa]

σ3 = 0

σ3 = 2

σ3 = 20

σ3 = 38

σ3 = 60

σ3 = 77

σ3 = 100

Fig. 5. Best fit solutions for the Westerly granite experimental data in the s12s2-plane.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186 1183

4. Summary and conclusions

The material parameters of six failure criteria wereoptimized for eight rocks so that least mean standarddeviation misfits could be obtained. A comparison of thesemisfits revealed the enhancements that are possible indefining rock failure criteria as non-linear in p2q space anddependent on intermediate principal stress. The newHBMN criterion which incorporates both of these featuresin one failure criterion consequently gave the overall leastmisfit in this study.

Finally, it should be noted that the HBMN criterionuses exactly the same material parameters as the HBcriterion. Nevertheless, the best fit parameters forboth models given in Table 3 are somewhat different:The unconfined compressive strength sci and the intactrock parameter mi are generally smaller within theHBMN criterion. In the case of the unconfined com-pressive strength this is certainly a desirable consequence.The intact rock parameters of both models equallyfit the domains proposed by Hoek et al. [3]. In practicalapplications, the HBMN criterion can be used in

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56

2126 2727

37

21 1923 25 11

97

1213

12

13

9

11

1312

113

89

64

93 9076

26

1821

2522

19

41 40

32

26 25 26

95

52

38

7787

59

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

MC

Mog

i

HB

HB

MN

Lad

eW

C

56 37 13

Fig. 6. Least mean standard deviation misfit (MPa).

Table 2

Least mean standard deviation misfit (MPa)

Rock# 1 2 3 4 5 6 7 8

Mohr–Coulomb 56 37 11 13 113 26 41 95

Mogi (linear) 26 19 12 11 93 25 26 77

Hoek–Brown 56 37 9 13 89 18 40 52

Hoek–Brown (MN) 21 21 7 9 64 21 32 38

Mod. Lade–Duncan 27 23 13 13 90 22 25 87

Mod. Wiebols–Cook 27 25 12 12 76 19 26 59

Rock# refers to Table 1.

Boldface values indicate minimum misfits for the respective rock types.

Table 3

Best fit parameters

Rock# 1 2 3 4 5 6 7 8

Mohr–Coulomb sci (MPa) 440 375 93 114 380 135 100 460

j (1) 33 29 39 29 45 36 40 45

Mogi (linear) sci (MPa) 384 334 73 99 208 150 88 257

j (1) 29 25 33 25 44 29 35 48

Hoek–Brown sci (MPa) 400 368 63 94 250 100 87 290

mi (–) 8 4.8 19.0 7.3 30.5 14.5 16.5 32.0

Hoek–Brown (MN) sci (MPa) 335 320 49 76 230 124 76 205

mi (–) 6.5 4.0 16.0 6.8 21.5 8.8 12.5 32.0

Mod. Lade–Duncan sci (MPa) 374 335 69 99 267 154 85 224

j (1) 27 22 33 24 40 27 34 47

Mod. Wiebols–Cook sci (MPa) 351 326 62 82 251 152 84 209

j (1) 30 24 37 29 44 28 37 52

Experiment sci (MPa) – – – – 176 100 p61 201

Rock# refers to Table 1.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–11861184

the same way as the HB criterion, not only describing thebehavior of intact rock, but also the behavior of rockmasses.

Appendix A. Misfit contours

See Fig. 7.

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Fig. 7. Hoek–Brown (MN) misfit contours.

T. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–1186 1185

ARTICLE IN PRESST. Benz, R. Schwab / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1176–11861186

References

[1] Benz T, Schwab R, Kauther RA, Vermeer PA. A Hoek–Brown

criterion with intrinsic material strength factorization. Int J Rock

Mech Min Sci 2008;45(2):210–22.

[2] Al-Ajmi AM, Zimmerman RW. Relation between the Mogi and the

Coulomb failure criteria. Int J Rock Mech Min Sci 2005;42:431–9.

[3] Hoek E, Carranza-Torres C, Corkum B. Hoek–Brown failure

criterion—2002 Edition. In: Proceedings of the 5th North American

symposium of NARMS-TAC, Toronto; 2002.

[4] Colmenares LB, Zoback MD. A statistical evaluation of intact rock

failure criteria constrained by polyaxial test data for five different

rocks. Int J Rock Mech Min Sci 2002;39:695–729.

[5] Mohr O. Welche Umstande bedingen die Elastizitatsgrenze und den

Bruch eines Materials? VDI-Zeitschrift 1900;44:1524.

[6] Lade P, Duncan J. Elasto-plastic stress–strain theory for cohesionless

soil. J Geotech Eng Div ASCE 1975;101:1037–53.

[7] Ewy R. Wellbore-stability predictions by use of a modified Lade

criterion. SPE Drill Completion 1999;14(2):85–91.

[8] Wiebols GA, Cook NGW. An energy criterion for the strength of

rocks in polyaxial compression. Int J Rock Mech 1968;5:529–49.

[9] Zhou S. A program to model the initial shape and extend of borehole

breakout. Comput Geosci 1994;20(7–8):1143–60.

[10] Hoek E. Strength of rock and rock masses. ISRMNews J 1994;2(2):4–16.

[11] Matsuoka H. Stress–strain relationships of sands based on the

mobilized plane. Soils Found 1974;14(2):47–61.

[12] Matsuoka H, Nakai T. A new failure criterion for soils in three

dimensional stresses. In: IUTAM conference on deformation and

failure of granular materials, Delft; 1982. p. 253–63.

[13] Bardet JP. Lode dependences for isotropic pressure-sensitive elasto-

plastic materials. Trans ASME 1990;57(9):498–506.

[14] Drucker DC. On uniqueness in the theory of plasticity. Quart Appl

Math 1956;XIV:35–42.

[15] Mogi K. Fracture and flow of rocks under high triaxial compression.

J Geophys Res 1971;76(5):1255–69.

[16] Takahashi M, Koide H. Effect of the intermediate principal stress on

strength and deformation behavior of sedimentary rocks at the depth

shallower than 2000m. In: Maury V, Fourmaintraux D, editors.

Rock at great depth, vol. 1. Rotterdam: Balkema; 1989. p. 19–26.

[17] Chang C, Haimson BC. True triaxial strength and deformability

of the German Continental deep drilling program (KTB) deep hole

amphibolite. J Geophys Res 2000;105:18999–9013.

[18] Michelis P. Polyaxial yielding of granular rock. J Eng Mech 1985;

111(8):1049–66.

[19] Michelis P. True triaxial cyclic behavior of concrete and rock in

compression. Int J Plast 1987;3(2):249–70.

[20] Haimson BC, Chang C. A new true triaxial cell for testing mechanical

properties of rocks, and its use to determine strength and deform-

ability of Westerly granite. Int J Rock Mech Min Sci 2000;37:

285–97.

[21] Mogi K. Fracture and flow of rocks under high triaxial compression.

J Geophys Res 1971;76(5):1255–69.