Comparison of two true-triaxial strength criteria

International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

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Technical Note

Comparison of two true-triaxial strength criteria Mingqing You n School of Energy Sciences and Engineering, Henan Polytechnic University, Jiaozuo 454010, China

a r t i c l e i n f o Article history: Received 11 September 2011 Received in revised form 3 June 2012 Accepted 7 June 2012 Available online 30 June 2012

1. Introduction Many true-triaxial criteria or polyaxial criteria have been proposed and studied to describe the effect of the intermediate principal stress on strength of rock [1–9]. Criteria with the form l (s1, s2, s3) ¼F [Z(s1, s2, s3)] that may be determined by fitting pseudo-triaxial experimental data, however, are usually not expected to express the strength under various stress state. Those seemingly good correlations mainly result from the dominant influence of the major principal stress in the metrics of l and Z, for which some physical concepts or names are given, such as maximum shear stress, octahedral shear stress, normal stress or effective mean normal stress, and stress tensor invariants [6]. The true-triaxial strength criterion proposed by You [6] is an explicit form

s1 ¼ sS ðs3 Þ þHðs2 , s3 Þ

ð1Þ

where sS is the conventional triaxial strength at confining pressure (CP) of s3; H is a function to describe the effect of the intermediate principal stress s2. The explicit equation of the principal stresses is simpler and clearer than those criteria with stress tensor invariants [1,4,5,7,8]. The latest one proposed by Rafiai [9] is also with the structure as in Eq. (1). You [10] evaluated the accuracy of 16 conventional triaxial strength criteria totally, of which three criteria are with one parameter, six criteria with two parameters and seven criteria with three parameters. Generally speaking, the more parameters the criterion has, the lower misfit for test data will be obtained. Therefore, the Rafiai criterion, with three parameters to describe the conventional triaxial strength, needs to compare criteria with three parameters, such as the exponential criterion [6], the generalized Hoek–Brown criterion [11], and the Sheorey criterion [12],

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not the Coulomb criterion and the Hoek–Brown criterion with two parameters. The Rafiai criterion with five parameters [9] for true-triaxial strength was thought as the best one, after compared the modified Wiebols–Cook criterion [13] and the exponential criterion [6] using test data of six rocks based on the root mean square error (RMSE). However, the minimum RMSE does not confirm completely the validity of a criterion as illustrated in [10]. The Rafiai criterion needs to be analyzed mathematically and numerically as well.

2. Conventional triaxial strength criteria The exponential criterion describes the relation between conventional triaxial compression strength and confining pressure with low mean misfit for brittle rocks, and ductile rocks as well [6,10]:   ðK 1Þs3 sS s3 ¼ Q 1 ðQ 1 Q 0 Þ exp  0 ð2Þ Q 1 Q 0 where Q0 is the uniaxial compression strength (UCS), QN is the limitation of the differential stress when confining pressure increases up to infinite, and K0 is the increase rate of strength to confining pressure at confining pressure ¼0. The three parameters are independent [14]. The exponential criterion is approach to the linear relation sS ¼ Q0 þK0 s3 within the range s3)(QN–Q0)/(K0  1). The conventional triaxial strength in the Rafiai criterion [9] is expressed as

sS Q0

¼

s3 Q0

þ

1þ As3 =Q 0 r 1 þBs3 =Q 0

ð3Þ

where Q0, A and B are material-dependent parameters; r is a strength reduction factor, and is 0 for intact rock and 1 for heavily joint rock mass. The criterion reduces to the Coulomb criterion when B ¼0. Apart from this case, the differential stress sS  s3 in

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

Eq. (3) approaches to a constant with CP increasing, just like that in the exponential criterion. Therefore, parameters A and B in Eq. (3) may be replaced with the parameters used in the exponential criterion: AB ¼ K 0 1

ð4Þ

A Q ¼ 1 B Q0

ð5Þ

115

The mean misfit, and RMSE as well, is the same in a given numerical accuracy for the parameters in a certain region. Therefore, we may optimize one parameter in the region, such as the parameter Q0 for Yamaguchi marble is set as the real magnitude of UCS, as presented in Table 1.

2.2. Abnormal test data

Hence, Eq. (3) may be transformed as   ðK 1Þs3 1 sS s3 ¼ Q 1 ðQ 1 Q 0 Þ 1 þ 0 Q 1 Q 0

ð6Þ

2.1. Fitting methods and solutions Parameters in the two criteria are determined by fitting Eqs. (2) and (6) to test data. Various solutions will be obtained using different methods. Two methods are usually used: the least square method is to search the parameters for the minimum ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X RMSE ¼ ðsS sTS Þ2 =N ð7Þ and the least absolute deviation method is to search parameters for the minimum mean misfit X ð8Þ mf ¼ absðsS sTS Þ=N where sTS is the test strength, and N is the number of test data. The two methods have been discussed in [10,15]. Test data of eight rocks, cited from [16–21], are used to evaluate the accuracy of the exponential criterion and the Rafiai criterion. The number of test data, maximum confining pressure, UCS and fitting solutions with the least mean misfits is presented in Table 1. Fitting solutions with the least square method for Jinping sandstone [16], Mizuho trachyte and Dunham dolomite [17] are presented in Table 2.

Strength envelopes for Jinping sandstone are different from the exponential criterion and the Rafiai criterion determined with the least absolute deviation method, but nearly the same with the least square method, as shown in Fig. 1. The UCS is 83.4 MPa predicted by the exponential criterion with the least mean misfit, 21.8 MPa higher than the real magnitude of 61.6 MPa indicated with X in Fig. 1, and the huge contribution of 66% to the total misfit of 32.8 MPa. The UCSs from the other fitting solutions are 61.7–69.9 MPa, as presented in Tables 1 and 2. Three fracture strengths from confining depressure test, as plotted with triangles in Fig. 2, close to the fitting solution using the exponential criterion with the least mean misfit. The Rafiai criterion with the least mean misfit predicts UCS perfectly, but shows errors for strengths at confining pressure of 5 MPa and 10 MPa, and large errors for the three strengths from confining depressure test, as shown in Fig. 2. Fitting solutions of two criteria with the minimum RMSE presents uniformly deviation for all test data as shown in Figs. 1b and 2. Fitting solutions for Mizuho trachyte with the least absolute deviation method in Fig. 3a pass through five test data, and misfits are localized at one test datum indicated with letter Y. Fitting solutions with the least square method in Fig. 3b do not pass any test data, but have low deviation for all test data. The magnitude of strength predicted by the exponential criterion with the least absolute deviation method is 405.7 MPa at a confining pressure of 100 MPa, which contributes 31.3 MPa, or 83% for the total misfit of 38.1 MPa. If the test under a

Table 1 Parameters of eight rocks and fitting solutions using two criteria with the least absolute deviation method. Rock

JS JSn MT MTn DD BS YM SL IL VS a b

Tests

9 8 7 6 8 12 11 9 11 7

Max. CP (MPa)

70 Delete datum X 100 Delete datum Y 125 100 200 304 69 60

UCS (MPa)

61.6 100.0 262.0 60.0 81.0 293.0 45.0 33.5b

Exponential criterion

Rafiai criterion

Q0 (MPa)

QN (MPa)

K0

mf (MPa)

Q0 (MPa)

QN (MPa)

K0

mf (MPa)

83.4 83.4 100.0 100.0 262.0 60.0 84.7 293.0 45.0 30.1

270.3 270.3 316.2 314.5 701.7 298.0 260.3 403.2 90.2 115.3

5.48 5.48 7.54 7.58 6.15 5.43 3.92 4.99 3.84 6.91

3.6 1.4 5.4 1.0 2.5 3.9 2.9 3.7a 0.7 0.9

61.7 81.4 100.0 100.0 262.0 60.1 81.0 293.0 45.0 29.0

323.4 356.8 407.6 394.6 934.3 403.0 306.0 411.2 100.0 135.6

8.38 6.04 8.33 8.47 6.44 5.90 4.75 8.10 4.89 9.32

3.1 1.5 3.9 0.9 2.4 3.6 2.3 3.8 1.1 1.4

The least mean misfit is 3.66 MPa at Q0 ¼ 299.8 MPa; and the mean misfit is 3.72 for optimizing at Q0 ¼293 MPa. The strength at confining pressure of 0.5 MPa, not the UCS.

Table 2 Fitting solutions using two criteria with the least square method. Rock

JS JSn MT MTn DD

Tests

9 8 7 6 8

Max. CP (MPa)

70 Delete datum X 100 Delete datum Y 125

UCS (MPa)

61.6 100.0 262.0

Exponential criterion

Rafiai criterion

Q0 (MPa)

QN (MPa)

K0

RMSE (MPa)

Q0 (MPa)

QN (MPa)

K0

RMSE (MPa)

69.9 86.0 105.0 100.7 262.4

251.0 272.0 356.0 314.0 703.6

6.95 5.37 6.33 7.52 6.14

5.1 1.9 7.6 1.6 3.4

66.5 82.9 102.3 100.0 261.4

314.4 356.0 452.0 396.5 935.0

8.37 6.04 7.31 8.52 6.65

4.0 1.9 6.0 1.2 3.5

116

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

250

160 ∆ confining depressure

S – 3(MPa)

S – 3(MPa)

150

100

Rafiai mf = 3.1 MPa

0

120 100 Rafiai RMSE Rafiai mf Exp. mf

80

Exp. mf = 3.6 MPa

X

50

X

60 20

40 3 (MPa)

60

0

80

250

5

10 3 (MPa)

15

20

Fig. 2. Localized amplification of fitting solutions for Jinping sandstone. Three strengths from confining depressure test are presented in triangles. Fitting solution with the minimum RMSE in dash line presents homogeneous deviation for all test data.

200

350 Y

150 300 100

S – 3 (MPa)

S – 3(MPa)

axial compression

140

200

Rafiai RMSE = 4.0 MPa Exp. RMSE = 5.1 MPa

X 50 0

20

40 3 (MPa)

60

250 200

80 Rafiai mf = 3.9 MPa Exp. mf = 5.4 MPa

150

Fig. 1. Fitting solutions using the Rafiai criterion and the exponential criterion for Jinping sandstone with (a) the least absolute deviation method and (b) the least square method.

100 0

20

40

60 80 3 (MPa)

350

(1) Mean misfit, and RMSE as well, using two criteria drops dramatically after one datum is deleted; and the fitting solutions are nearly the same in the experimental range. Therefore, X, i.e. the UCS of Jinping sandstone, may be an abnormal datum from a flaw specimen; and Y, i.e. the strength of Mizuho trachyte at CP of 100 MPa is not in harmony with other data, and is probably enhanced by the friction effect at the specimen ends as discussed in [10]. (2) The fitting solutions with RMSE as shown in Figs. 1b and 3b cannot reveal oddity data, and are heavily influenced by one abnormal datum for both the exponential criterion and the Rafiai criterion as presented in Table 2. Strength envelopes using the Rafiai criterion for two rocks are shown in Fig. 4. Therefore, the RMSE is not a good index to evaluate strength criteria.

120

Y

300 S – 3 (MPa)

confining pressure of 100 MPa were not carried, then the mean misfit should be 1.1 MPa for the other six test data. In fact, test result is an integration of CP, specimen and test machine, etc. Strengths at different confining pressures come from different specimens. Sometimes, there are a few abnormal test data, of which the errors are amplified by the square calculation and influence the fitting solution when using the least square method [10]. We delete the datum X in Jinping sandstone and Y in Mizuho trachyte, and name the left data as JS* and MT*, respectively. The fitting solutions using two criteria with two methods are presented in Table 1 and 2. We may conclude as follows:

100

250 200 Rafiai RMSE = 6.0 MPa Exp. RMSE = 7.6 MPa

150 100 0

20

40

60 80 3 (MPa)

100

120

Fig. 3. Fitting solutions for Mizuho trachyte with (a) the least absolute deviation method and (b) the least square method. Misfits of the former are localized at datum Y; and those of the later are distributed to all the test data.

(3) The fitting solutions for both Jinping sandstone and Mizuho trachyte using the exponential criterion with the least mean misfit are not influenced by the abnormal datum, as presented in Table 1. Furthermore, the criterion exposes the abnormal data with huge misfits, as shown in Figs. 1a and 3a. (4) UCS from the exponential criterion with the least mean misfit is the same as the real magnitude, unless it is an abnormal one like that of Jinping sandstone, as presented in Table 1.

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

117

650

250

Dunham dolomite 550 S – 3 (MPa)

S – 3 (MPa)

200

150

100

350

JS RMSE = 4.0 MPa JS* RMSE = 1.9 MPa

X

Rafiai mf = 2.4 MPa Exp. mf = 2.5 MPa

250

50 0

20

40 3 (MPa)

60

0

80

25

50

75 100 3 (MPa)

125

150

300

350

Y

Bunt sandstone 250 S – 3 (MPa)

300 S – 3 (MPa)

450

250 200

150 Rafiai mf = 3.6 MPa

100

MT RMSE = 6.0 MPa MT* RMSE = 1.2 MPa

150

200

Exp. mf = 3.9 MPa 50

100 0

20

40

60 80 3 (MPa)

100

0

120

20

40

60 80 3 (MPa)

100

120

275 Fig. 4. Fitting solutions using the Rafiai criterion for (a) Jinping sandstone and (b) Mizuho trachyte with the least square method. They are heavily influenced by one abnormal datum.

Yamaguchi marble

The exponential criterion is the ‘‘worse’’ one for Jinping sandstone and Mizuho trachyte on both the mean misfit and RMSE, as shown in Figs. 1 and 3. However, it has the same fitting property with the Rafiai criterion after eliminating one datum that is most probable an abnormal one, as mentioned above. The fitting solutions using the exponential criterion and the Rafiai criterion for Dunham dolomite and Bunt sandstone [18] are almost the same in the test range of confining pressure, as shown in Fig. 5. The slight difference of mean misfits does not verify the better or worse for the two criteria. As presented in Table 2, RSME from the exponential criterion is 0.1 MPa less than that from the Rafiai criterion for Dunham dolomite. The Rafiai criterion is slight better than the exponential criterion for Yamaguchi marble [17] from the view of mean misfit (Fig. 5). Datum Z is on the lower side of both criteria, and does not harmonize with data in its neighborhood. If the datum is not considered, the mean misfit using the exponential criterion for the other 10 test data will decrease from 2.9 MPa to 2.4 MPa, and is same as that from the Rafiai criterion. The strength of Yamaguchi marble at CP of 150 MPa, and 200 MPa as well, is the average value from two tests with difference of 1 MPa [17]. Their reliability is high. The exponential criterion passes through the two data, but the Rafiai criterion fails for one datum (Fig. 5). Envelopes of the exponential criterion are perfect for strengths of Solnhofen limestone [17], Indiana limestone [19] and Vosges sandstone [20], as shown in Fig. 6; and the mean misfits are also lower than that using the Rafiai criterion.

S – 3 (MPa)

225 2.3. Fitting accuracy of the two criteria

Z 175

125

Rafiai mf = 2.3 MPa Exp. mf = 2.9 MPa

75 0

50

100 150 3 (MPa)

200

250

Fig. 5. Fitting solutions using the Rafiai criterion and the exponential criterion. Two criteria have low misfits and their difference is beyond the accuracy of the test data.

The test data of Vosges sandstone in [21] were provided by Be´suelle, who carried the test [20]. The specimens were lubricated using a mixture of vaseline and stearic acid to reduce the friction on the interfaces between the specimen and loading heads. There is no clamping effect at specimen ends; therefore, the differential stress does not increase at high CP. Certainly, the Rafiai criterion has low misfit to test data, but shows little advantage over the exponential criterion.

2.4. A general triaxial strength criterion As illustrated in [15], the differential stress, or the maximum shear stress in rock specimen approach to a constant when

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

S – 3 (MPa)

118

425

f ð1Þ ¼ 0

400

f ð0Þ ¼ 1

0

ð14Þ

Clearly, it is

375

f ðxÞ ¼ expðxÞ ¼

350 Solnhofen limestone

325

f ðxÞ ¼

Rafiai mf = 3.8 MPa

300

Exp. mf = 3.7 MPa 0

50

100

150 200 3 (MPa)

250 300

350

130 110 90 70

Vosges sandstone

f ðxÞ ¼

Rafiai mf = 1.4 MPa

50 30 15

30 45 3 (MPa)

60

ð15Þ

1 1þ x

ð16Þ

1 1þ x þ x2 =4

ð17Þ

of which the strength envelope is plotted in Fig. 8. The UCS from the criterion is also exactly the same with the real magnitude, but the envelope does not pass through the test strength at a confining pressure of 150 MPa, which is high reliability, as mentioned above. Certainly, it is not easy to choose the best

Exp. mf = 0.9 MPa 0

1 1 þ x þ x2 =2þ x3 =3! þ   

for the exponential criterion and the Rafiai criteria, respectively. Several functions satisfy Eqs. (12)–(14) are shown in Fig. 7, and fitting solutions using the general criterion for yield strengths of Yamaguchi marble are presented in Table 3. The maximum magnitude of variable x at s3 ¼200 MPa, i.e. the range of f(x) to describe strengths, varies from 2.41 to 3.41. Clearly, these functions have different envelopes, but they may fit the test data with different parameters. QN in the exponential criterion is the lowest one, as the exponent function approaches zero in the highest rate. The Rafiai criterion is on the contrary. The mean misfits for six functions are from 2.22 MPa to 3.06 MPa, as presented in Table 3. The best one is the function

275

S – 3 (MPa)

ð13Þ

75

100 90

0.8

70 Indiana limestone

60

y = 1/(1+x+x2/4)

0.6

Rafiai mf = 1.1 MPa

50

y = 1/(1+x)

y

S – 3 (MPa)

1 80

Exp. mf = 0.7 MPa

y = 1– atn(πx/2)/(π/2) 0.4

40 0

20

40 3 (MPa)

60

80

0.2

Fig. 6. Fitting solutions using the Rafiai criterion and the exponential criterion. The envelopes of the exponential criterion are pretty to the test data in the range of high CP.

confining pressure increases to infinite. A general equation for conventional triaxial strength is constructed on the assumption

sS s3 ¼ Q 1 ðQ 1 Q 0 Þf ðxÞ

ð10Þ

where x¼

ðK 0 1Þs3 Q 1 Q 0

ð11Þ

and f (x), a monotonically decreasing function, satisfies f ð0Þ ¼ 1

y = exp(–x) 0.5

1.0

1.5

2.0

2.5

3.0

3.5

x Fig. 7. Functions to construct the conventional triaxial strength criterion. They approach to zero with various rates, and describe the same test data with different variable x.

ð9Þ

or the alternative form

sS s3 ¼ Q 0 þ ðQ 1 Q 0 Þ½1f ðxÞ

0 0.0

y = 1/(1+x+x2)

ð12Þ

Table 3 Fitting solutions using the general criterion for yield strengths of Yamaguchi marble. f(x)

Q0 (MPa)

QN (MPa)

K0

Max. x

mf (MPa)

exp(–x) 1/(1 þx) 1/(1 þx þ x2/4) 1/(1 þx þ x2/2) 1/(1 þx þ x2) 1–atn(px/2)/(p/2)

84.7 81.0 81.0 83.7 87.9 88.2

260.3 306.1 281.4 275.3 271.1 286.6

3.92 4.74 4.42 3.99 3.43 3.39

3.33 3.32 3.41 3.12 2.65 2.41

2.85 2.26 2.22 2.50 3.06 2.96

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

275

119

500 Stress at 1=3.5%

175 Gen. HB mf = 3.52 MPa Exp. mf = 2.85 MPa

125

f(x) = 1/ (1 + x/2)2 mf = 2.22 MPa

1 – 3 (MPa)

S – 3 (MPa)

Fracture strength

400

225

300

200

50

100 150 3 (MPa)

200

0

250

Fig. 8. Fitting solutions for yield strengths of Yamaguchi marble using three criteria. Strength criteria with power form, such as the generalized Hoek–Brown criterion, cannot describe yield strengths of rocks in ductile deformation.

strength criterion under the test data. In other words, there are many strength criteria to describe the test data with low misfits. The fitting solution for the marble using the generalized Hoek– Brown criterion is sS s3  s3 0:296 ¼ 1 þ 23:6 ð18Þ 79:9 79:9 with a mean misfit of 3.52 MPa, which is higher than any forms of the general criterion listed in Table 3. The criterion with power forms cannot describe strength under high confining pressure (Fig. 8). 2.5. Estimation of fracture strength under ductile deformation The test data of Yamaguchi marble mentioned above are yield strengths, i.e. stresses at the permanent strain of 0.2% [17]. Fracture strengths are different from the yield strengths, as shown in Fig. 9. The UCS and the fracture strength at a confining pressure of 55 MPa from three criteria are 82.0 MPa and 285.0 MPa, exactly the same with the real magnitudes, respectively. However, the fitting solutions of three criteria are different at high confining pressure. Limitations of differential stress predicated by the exponential criterion, the Rafiai criterion and the generalized Hoek–Brown criterion are 338.8 MPa, 483.8 MPa and infinite, respectively. There is a lack of fracture strength at confining pressures over 55 MPa due to high ductility of the marble. The axial stress at e1 ¼3.5% is digitized from stress–strain curves in Mogi [17], and plotted with triangle in Fig. 9. The six data close to the fitting curve using the exponential criterion. The specimen does not fracture and may support higher stress than that at e1 ¼3.5%. On the other hand, there is more or less clamping effect at the ends of specimen that has become ductile; hence, the real bearing capacity should be lower than the stress at e1 ¼ 3.5%. In summary, the exponential criterion may demonstrate a reasonable estimation of fracture strength for the marble under ductile deformation. 2.6. Strength criterion in tension–compression region As illustrated in [10], none of the 16 criteria provides reliable tensile strength for all rocks. Perhaps, it is greedy to expect that the tensile strength is predicted by extending a criterion that is optimized from compression test data. The tensile strength is an independent parameter of rock, and must be determined by test.

Gen. HB mf =1.73 MPa Exp. mf =1.63 MPa Rafiai mf =1.84MPa

100

75 0

Yield strength

0

50

100 150 3 (MPa)

200

250

Fig. 9. Fitting solutions for six fracture strengths of Yamaguchi marble. Yield strength, i.e. axial stress at permanent strain of 0.2%, with solid square, and axial stress at e1 ¼ 3.5% with blanket triangle are presented as the reference. Limitations of differential stress predicated by the exponential criterion, the Rafiai criterion and the generalized Hoek–Brown criterion are 338.8 MPa, 483.8 MPa and infinite, respectively.

S

Q0 S = Q0 + K03

Exponential criterion

Tension cut-off Eq. (19) Eq. (20)

–TE

–T

0

3

Fig. 10. Strength criteria in the range of tension–compression. T is the uniaxial tensile strength, and TE is the tensile strength predicted by the exponential criterion. Two polynomials are tangent to s3 ¼  T and sS ¼ Q0 þK0 s3 at the uniaxial tensile strength and UCS, respectively.

Results of both direct tension and Brazilian splitting test have great dispersion; the average magnitude of Brazilian splitting strength is within the strength range limits of the uniaxial tension for a given rock; the average strength of uniaxial tension will be higher than that of Brazilian splitting test at a small ratio of a structural parameter of rock material to the core radius, and vice versa [22–24]. Therefore, we do not make the distinction between two tensile strengths below. One datum of tensile strength, which is much less than compression strength, could not produce large misfit and significantly influence the fitting solution of strength criterion. Of course, we may take the tensile strength as an anchor point for the criterion as Carter [25]; however, this will reduce the accuracy and the application range of the criterion for compression strength. One simple method to describe the strength in the tension– compression region is tension cut-off, i.e. s3 ¼–T, as shown in Fig. 10, where T is the uniaxial tensile strength, which is less than

120

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

TE derived from the exponential criterion. Another method is to construct a function that are tangent to s3 ¼–T and sS ¼Q0 þK0s3 at the uniaxial tensile strength and UCS, respectively. Fig. 10 shows two polynomials with the property mentioned above  2  4 s3 s1 s1 ¼ 1 þ ð2bÞ þ ðb1Þ ð19Þ T Q0 Q0

T

¼ 1 þ ð3bÞ



s1 Q0

4

 6 s1 þ ðb2Þ Q0

0.8 HE (2,3)/QE

s3

1

ð20Þ

where Q0 b¼ 2TK 0

0.6

0.4

0.2 Exponential criterion

ð21Þ

0

and Q0 and K0 are the strength and the influence coefficient of the confining pressure at a confining pressure of 0, respectively. Clearly, more tension–compression tests need to carry for rocks as those in [26–28] before optimizing one feasible criterion.

0

0.4

0.8 1.2 1.6  = (2–3)/(S–3)

2

1

2.7. Discussion

0.8 HR (2,3)/QR

Both the Rafiai criterion and the exponential criterion can describe the relation between strength and confining pressure for intact rocks. The exponential criterion may expose oddity data with huge misfits, such as X in Fig. 1, Y in Fig. 3, and Z in Fig. 5; the envelopes in the range of high confining pressure are also reasonable for Yamaguchi marble in Fig. 5, Solnhofen limestone and Indiana limestone in Fig. 6. The exponential criterion has a clear physical background: rock has friction and cohesion [29], but friction and cohesion do not work for one point at the same time [30,31], as illustrated in [14,15]; therefore, the author recommends the exponential form instead of the others in Table 3 to construct the conventional triaxial strength criterion.

0.6

0.4

0.2 Rafiai criterion 0

0

0.2

0.4 0.6  = (2–3)/S

0.8

1

Fig. 11. Effect of the intermediate principal stress on strength from the exponential criterion and the Rafiai criterion.

3. True-triaxial strength criteria 3.1. Characteristic of the exponential criterion Based on the experimental data in [3], the maximum influence of the intermediate principal stress on strength may increase with the minor principal stress for limestone, sandstone, amphibolite and trachyte, but it is not true for other four rocks [6]. The reasonable assumption is that the maximum influence of the intermediate principal stress is a constant. Therefore, the truetriaxial criterion of Eq. (1) is written as [6]

s1 ¼ sS þHE ðs2 , s3 Þ ¼ sS þ Q E go expð1goÞ

ð22Þ

s s o¼ 2 3 sS s3

ð23Þ

where QE is a material-dependent parameter; but g may be a constant about 1.7. Therefore, there are four parameters totally in the exponential criterion for true-triaxial strength. Function HE plotted in Fig. 11 reaches its maximum QE at go ¼1. The inflexion of Eq. (22) occurs at go ¼2, after which HE becomes concave. We believe that the strength decreases with intermediate principal stress when it is high enough to induce the failure in the direction of s2  s3, and the phenomenon is more significant for higher s2. As indicated in [6], the influence of parameter g on the mean misfit is much lower than a few data with huge error. We can set g ¼1.7 to simplify the strength criterion. In such situation, the strength reaches its maximum around s2  s3 ¼0.6(sS  s3), and the intermediate principal stress reaches its maximum s2 ¼ s1 near the inflexion point of the Eq. (22). Therefore, function HE at g ¼ 1.7 is convexity in the range of s2. That is a reasonable outcome.

The criterion with g ¼ 1.7 is feasible for the extension strength of rocks, such as Carrara marble and Dunham dolomite [15,32]. By contrast, the Mogi–Coulomb criterion [3], the HBMN criterion [5], and the 3D Hoek–Brown criterion [7] are against the test results to present the same magnitude for extension strength and compression strength. The increasing rate of conventional triaxial strength with confining pressure decreases from K0 to 1 as confining pressure increases from zero to infinite. As dsS @s1 @s1 ¼ þ ds3 @s2 @s3

ð24Þ

at s2 ¼ s3, and the partial derivative is @s1 geQ E ¼ @s2 sS s3

ð25Þ

at s2 ¼ s3 from Eq. (22), where ‘‘e’’ is the base of the natural logarithm. It decreases from geQE/Q0 to geQE/QN as CP increases from zero to infinite, and is less than the derivative of sS to CP in the experiment range for 10 rocks discussed in [6,32]. That is a reasonable result. The magnitude of geQE/QN is about 1/2 for five rocks when parameter g is set as 1.7 [32]. If the variable

d¼

s2 s3 sS

ð26Þ

rather than the variable o of Eq. (23) were used to express the effect of s2 in Eq. (22), then the partial derivative of s1 to s2 at

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

s2 ¼ s3 would approach to zero as confining pressure and sS increasing. That is not a reasonable result. 3.2. The Rafiai criterion Rafiai [9] proposed another function to describe the effect of the intermediate principal stress in the true-triaxial criterion of Eq. (1): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ðs2 s3 Þ ðs2 Ds3 Þ HR ðs2 , s3 Þ ¼ Q 0 C ð27Þ exp 

sS

sS

where C and D are material-dependent parameters. As presented in [9], the criterion fits test data with lower RMSE than does the exponential criterion. However, RMSE is not the unique standard in evaluating strength criteria for rock. Using the variable d of Eq. (26), Eq. (27) may be transformed as  pffiffiffi pffiffiffi D1 HR ðs2 , s3 Þ ¼ Q 0 C exp s3 d expðdÞ ð28Þ

sS

121

decreases from 125 MPa to 100 MPa at the same s2 of 125 MPa. That is generally not true for rock. The Rafiai criterion therefore does not completely describe the strength property in the experimental range of stress state. The outer envelope of the Rafiai criterion, i.e. the maximum strength for all s3 at a given s2, may be solved mathematically or determined numerically, as shown in Fig. 13 for Orikabe monzonite. Stress state is oddity in the region between the outer envelope and the curve of conventional triaxial strength: at a given s2, strength increases as s3 decreases from the s2. Perhaps, the scale of the figures in [9] is too small to display this nonphysical property. The non-physical outcome in the Rafiai criterion exists for all rocks, and it is not related to the accuracy of data. Whether or not this error is large enough to limit the application of the Rafiai criterion in the estimation of geo-stresses from borehole collapse remains to be seen. 3.3. Fitting solutions

For a given s3, function HR reaches its maximum value of QR at d ¼1/2, where   pffiffiffiffiffiffi D1 s3 ð29Þ Q R ðs3 Þ ¼ 0:4289Q 0 C exp

sS

is the maximum influence of s2. It is almost a linear relation to s3 in the experimental range, and will be discussed in detail in Section 3.4. Function HR(s2, s3) is plotted in Fig. 11 after unified against the magnitude of QR(s3). The magnitude of d is about 1 for the maximum s2 at s2 ¼ s1. Therefore, the envelopes in Fig. 11 are roughly the entire process of the influence of s2 on strength. Clearly, the Rafiai criterion presents a sharp rising and a wide plateau for the true-triaxial strength changing with the intermediate principal stress. The true-triaxial criterion with Eq. (27) leads the partial derivative @s1 ¼1 @s2

RMSE using the exponential criterion for KTB amphibolite at

g ¼1.7 is 0.2 MPa higher than that using the Rafiai criterion; the difference of RMSE is about 2 MPa using two criteria for Mizuho trachyte and Dunham dolomite [9]. The Rafiai criterion is the better one. However, test data of the two rocks used in Rafiai [9] seem to be digitized magnitudes cited from [3], of which some are different from the original results presented in Mogi [17] published in 2007. The errors are in not only the major principal stress, but also the intermediate principal stress. For example, the real strength of Mizuho trachyte is 437 MPa at a confining pressure of 100 MPa, much higher than the digitized one of 419 MPa; and the maximum of s2 is 384 MPa in Mogi [17], but 411.0 MPa in Rafiai [9]. Fitting solutions for the original test data [17] are shown in Figs. 14 and 15. Parameters for the conventional strength criteria have been presented in Tables 1 and 2.

ð30Þ

1400

at s2 ¼ s3. Therefore, the initial part of failure envelope for the same s3 is on the upper side of the conventional strength, as shown in Fig. 12. Practically, sS is 449 MPa at a confining pressure of 125 MPa, i.e. s2 ¼125 MPa and s3 ¼ 125 MPa; and the strength is 463 MPa at s2 ¼125 MPa and s3 ¼100 MPa from the Rafiai criterion for Mizuho trachyte. The strength increases up 14 MPa when s3

1 (MPa)

400

3 = 100 MPa 2=3

3 = 75 MPa

350

1300

3 = 200 MPa

C 1200 1 (MPa)

500 450

Outer envelope

A

3 = 140 MPa

1100

B

1000 3 = 140 MPa

2 = 3

3 = 50 MPa

900

3 = 25 MPa

800 100

3 = 200 MPa 2 = 3

300 250 200 25

50

75 2 (MPa)

100

125

Fig. 12. Non-physical property of the Rafiai criterion: strength may increase as s3 decreases at a given s2. The failure envelope is for Mizuho trachyte cited from [9].

200

300 2 (MPa)

400

500

Fig. 13. Strengths of Orikabe monzonite [17] and fitting solutions using the Rafiai criterion [9]. From datum C to the convention triaxial strength, it decreases about 70 MPa, while s3 increases from 200 MPa to 251 MPa and s2 keeps a constant of 251 MPa. This abnormal phenomenon probably results from ends effect in the true-triaxial compression.

122

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

1000

600 Exponential criterion

Exponential criterion

125

2 = 3

900

105 800

100

65

700

75

45

600

400 3 = 25 MPa 2 = 1

500 400

 = 1.7 QE = 175.3 MPa mf = 14.9 MPa

300

60

1 (MPa)

1 (MPa)

2 = 3

500

85

3 = 45 MPa 300 2 = 1  = 1.7

200 0

100

200

300 400 2 (MPa)

500

600

QE= 64.1 MPa

200

RSME = 12.6 MPa

1000 Rafiai criterion 900

2 = 3

100

105

800

1 (MPa)

(mf = 9.9 MPa)

125 0

85

700

65

600

45

100

200 2 (MPa)

300

400

600 Rafiai criterion 100 3 = 25 MPa 2 = 1

500 400

500

C = 1.52 D =1.87 mf = 16.8 MPa

300

75 400

100

200

300 400 2 (MPa)

500

600

Fig. 14. Fitting solutions using the exponential criterion and the Rafiai criterion for Dunham dolomite with the least absolute deviation method. The parameters for the conventional triaxial strength criterion are presented in Table 1.

60

1 (MPa)

200 0

2 = 3

3 = 45 MPa 300 2 = 1 C = 0.184

As mentioned above, different fitting methods result in different solutions. The least mean misfit using the exponential criterion is 14.9 MPa at QE ¼175.3 MPa; and the minimum RMSE is 19.0 MPa at QE ¼172.2 MPa. The parameter g is set as a constant of 1.7. Similarly, the least mean misfit using the Rafiai criterion is 16.8 MPa at C ¼1.52 and D ¼1.87; and the minimum RMSE is 21.6 MPa at C ¼1.47 and D ¼2.18. The exponential criterion with four parameters is better than the Rafiai criterion with five parameters for Dunham dolomite, no matter evaluating with the mean misfit and the RMSE. Envelopes of the solutions with the least mean misfits are plotted in Fig. 14. The exponential criterion is suitable to describe the strengths of Dunham dolomite. On the other hand, the nonphysical outcome of the Rafiai criterion, as illustrated in Section 3.2, is also clearly exhibited. The test data of Dunham dolomite in Fig. 14 do not include the strengths at s3 ¼ 145 MPa for the following considerations: (1) there is no conventional triaxial strength at a confining pressure of 145 MPa; (2) there is a few abnormal data at s3 ¼145 MPa; and (3) exclusion the strengths at s3 ¼145 MPa, there are still 47 data, in which 39 data are true-triaxial strengths at six levels of s3.

D = 7.09

200

RSME = 9.7 MPa (mf = 9.1 MPa) 100 0

100

200 2 (MPa)

300

400

Fig. 15. Fitting solutions using the exponential criterion and the Rafiai criterion for Mizuho trachyte with the least square method. The parameters for the conventional triaxial strength criterion are presented in Table 2.

The practical calculation also manifests that the exponential criterion is slightly better than the Rafiai criterion in fitting 54 test data totally on the mean misfit and the RMSE as well. The Rafiai criterion fits strengths of Mizuho trachyte with a RSME about 3 MPa less than does the exponential criterion. The RMSE mainly comes from a few data with huge errors. For example, three data at s3 ¼100 MPa provide 48% of square errors for the fitting solution using the exponential criterion (Fig. 15). The mean misfit for 31 test data totally is 9.9 MPa using the

M. You / International Journal of Rock Mechanics & Mining Sciences 54 (2012) 114–124

180 QE 150

Dunham dolomite

QE;QR(MPa)

QR 120 90 60

QE

30

QR

Mizuho trachyte

0 0

25

50 75 3 (MPa)

100

125

Fig. 16. Maximum influence QE and QR of the intermediate principal stress from the exponential criterion and the Rafiai criterion, respectively.

exponential criterion, and 9.1 MPa using the Rafiai criterion. The difference is merely 0.8 MPa. The exponential criterion successes to describe strengths at s3 ¼45 MPa, 60 MPa and 75 MPa, but the envelope for s3 ¼100 MPa is entirely on the low side of the test data (Fig. 15). The Rafiai criterion, however, does not express the initial processes of strength when s2 increases from s3 ¼45 MPa, 60 MPa and 75 MPa, respectively. It fits the strengths at s3 ¼100 MPa with a significant unreasonable envelope. There are only experimental data at s3 ¼25 and 50 MPa for Yuubari shale [3,33], so the average of the real conventional triaxial strengths is used as sS to calculate the true-triaxial strength. The RMSE is 7.60 MPa using the experimental criterion at QE ¼30.3 MPa and a constant g of 1.7 for pffiffiffi26 test data. It is 7.25 MPa using the Rafiai criterion at Q 0 C ¼ 54:0 MPa and D ¼2.20 in Eq. (27). The exponential criterion with four parameters fits test data roughly equivalent to the Rafiai criterion with five parameters. For Orikabe monzonite, RMSE is 28.0 MPa using the Rafiai criterion, and 30.2 MPa using the exponential criterion, as presented in Rafiai [9]. As shown in Fig. 13, the fitting solutions using the Rafiai criterion are very pretty fit to the strengths at s3 ¼140 MPa and 200 MPa. However, the low RMSE does not support the advantage of the Rafiai criterion for the abnormal test data as revealed below. Test data A and B have nearly the same s1 and s2 but a difference 60 MPa of s3. That means the increase of s2 is equality to that of confining pressure, and the minor principal stress does not influence the strength. From datum C to the convention triaxial strength, it decreases about 70 MPa, while s3 increases from 200 MPa to 251 MPa and s2 keeps a constant of 251 MPa. This is totally different from the common knowledge. This abnormal phenomenon probably results from ends effect in the true-triaxial compression.

123

As mentioned above, the Rafiai criterion is not superior to the exponential criterion in fitting test data. The maximum influences of s2 from the Rafiai criterion increases about linearly with s3 for Mizuho trachyte (Fig. 16). In fact, it only has a significant increase at s3 ¼100 MPa for the test data shown in Fig. 15. If the six test data at s3 ¼100 MPa are ignored and the other 25 test data are fitted using the exponential criterion, then the mean misfit drops to 5.4 MPa. The author has the following opinion. In the true-triaxial compression test in Haimson [34], thin copper shims and stearic acid-based lubricant was applied to loading heads in order to minimize friction effect. However, the metal plates that provided the maximum stress acted directly on the specimen ends in Mogi [17], and might result in the clamping effect at the ends of the specimen. The extension in lateral direction of specimen at the failure increases with the axial deformation, i.e. with the minor principal stress. The minor principal stress of 100 MPa is the same with the UCS for Mizuho trachyte; therefore, the friction effect at the specimen ends may influence the strength. On the other hand, s3 of 125 MPa is only half of the UCS for Dunham dolomite, and then the friction effect should not influence the strength significantly. The exponential criterion with four parameters is better than criteria with two or three parameters, such as the Mogi–Coulomb criterion [3], MNHB criterion [5] and 3D HB criterion [7], and has some advantage over the Rafiai criterion with five parameters. Certainly, fitting equations may be constructed mathematically one after one. If the influence of the intermediate principal stress is really enhanced with the minor principal stress, then we may modify the exponential criterion by using a function g(s3) with one parameter to replace the constant QE in Eq. (22).

4. Conclusions Both the exponential criterion and the Rafiai criterion may fit test data of rocks with low RSME and mean misfit; however, the Rafiai criterion has the following unreasonable outcome: the strength may increase while s3 decreases at the same s2. The exponential criterion with four parameters may fit most test data and expose a few abnormal data with huge errors. The two criteria for conventional triaxial strength are the special cases of a general criterion with three parameters proposed in this paper. The dominant characteristic is that the differential stress approaches to a constant as confining pressure increases to infinite. Since the exponential criterion has a clear physical background, and seems to give a reasonable estimation for fracture strength of rock in ductile deformation, it is recommended by the author. Tensile strength is an independent parameter of rock and determined by test, not predicted by extending a compression strength criterion. And then strength criterion with one parameter may be constructed in the tension–compression region, such as a polynomial tangent to s3 ¼  T and sS ¼Q0 þK0 s3 at the uniaxial tensile strength and UCS, respectively.

3.4. Discussion

Acknowledgment

The maximum influences of s2 on strength from the exponential criterion, i.e. QE, are 175.3 MPa and 64.1 MPa for Dunham dolomite and Mizuho trachyte, respectively. However, it varies with s3 from the Rafiai criterion, i.e. QR(s3) of Eq. (29) as shown in Fig. 16. QR increases from 138.6 MPa at s3 ¼0 MPa to 161.0 MPa at s3 ¼ 125 MPa for Dunham dolomite, and is totally lower than the magnitude of 175.3 MPa from the exponential criterion. This results from the sharply rising of Eq. (27) in the Rafiai criterion.

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