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An index for describing the anisotropy of joint surfaces
hr.
J. Rock Mech.
PII: SO148-9062(97)0001&l
Sci. Vol. 34, No. 6, pp. 1031-1044, 1997 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0148-9062/97 $17.00 + 0.00
Min.
0
Pergamon
Technical Note An Index for Describing the Anisotropy
of Joint Surfaces
Z. Y. YANGt s. c. Lot
INTRODUCTION
FRACTAL BROWNIAN MOTION
Joint roughness is of paramount importance for the shear behavior of rock joints. Patton [l] concluded that the strength of joints is controlled by the three factors of friction property, dilation effect, and shear-off effect for a rock material. In particular, the dilation effect and thus the shear strength is dominated by the roughness of joints at a low normal stress level. The formula for shear strength of Barton [2] also demonstrates the importance of the joint roughness coefficient (JRC). Barton introduces 10 standard JRC profiles defined by a JRC number. The JRC is an average index to describe the roughness degree of joint surfaces. Recently, fractal theory [3] has been applied to describe the roughness of joint surfaces by several authors [4-71. They found good correlation between the JRC and fractal dimension, D, and concluded that the fractal dimension can represent the JRC profile. However, both the JRC and D describe the average roughness of a given profile. They are independent of the direction of profiles and only indicate the scalar characteristics. However, it has been observed that the strength of a joint surface depends on the shear direction from several experimental tests [8-lo]. The single value of JRC or D is thus insufficient for describing the directional properties. Aydan et al. [ 1l] propose a parameter related to the measure direction and demonstrates the directional dependence of joint surfaces. Nevertheless, the anisotropic mechanical behavior is not included in his discussion. This paper uses a new index, the Hurst exponent based on the theory of fractional Brownian motion (fBm) [12], to represent the anisotropic characteristic of the joint profile. The Hurst exponent which is dependent on the sequence of sampling, can indicate the past height history of asperities. Thus, it can represent the anisotropic roughness characteristic of a profile according to different sampling orders and shearing directions. The fractal dimension and JRC can be calculated from the Hurst exponent for a profile in different directions. The anisotropic shear strength of joints, based on the Barton empirical formula, is also investigated. tDepartment of Civil Engineering, Taipei 25137, Taiwan.
Tamkang University, Tamsui, 1031
Since Mandelbrot [3] presented the fractal concept, the correlation formula between D and JRC has been applied to the shear strength of rock joints. It has been shown that the fractal dimension is capable of describing a random distribution of joint asperities. The fractal dimension of a joint profile, however, is an average index to represent the surface. It is a non-directional parameter and independent of the sequence of sampling data. Therefore, it does not reflect the anisotropic characteristic of the different shearing directions. An index that is direction-dependent is now needed for joint surfaces in rock mechanics. Surface roughness of joints
There have been numerous experimental and numerical studies concerned with characterizing the surface morphology of joint surfaces. The research that focused on the correlation of D and JRC has not addressed the anisotropic character of joint profiles. Three approaches are usually adopted to calculate the fractal dimension D, i.e. the divider, box and spectrum methods. Early work in fractal measurement with the divider method is presented by Rengers [ 131 in rock mechanics. Rengers distinguishes the effect of positive and negative angle for a joint profile in his study. He shows that the angle distribution depends on the measure direction and the measure intervals. Ueng and Chang [14], and Huang and co-workers [9, lo] also attempt to consider the anisotropic strength by distinguishing the upward and downward angles of asperity regarding the shear direction. Chen [15] and Huang et al. [lo] estimate the part JRC value by ignoring the downward asperity angle to take into account shearing directions. The joint surface is produced by a succession of mechanical events which cumulatively can be modeled as a random process during tectonic circumstances. The fracture surface that is continuous can be regarded as a single-valued function; such surfaces are self-similar or self-affine. The self-similar surface has the same roughness regardless of the sample size. The self-affine surface appears progressively smoother for larger and larger samples. The joint surface, a self-affine curve [ 161, is such that its “shape” appears the same at any magnification, but its value is not exactly the same for
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TECHNICAL
NOTE
any sample size. In addition, Swan and Zongqi [ 171 joint surface can be represented by two different values. Considering the different shearing direction, the Hurst assumed that the peaks of asperities are distributed exponent H and then the fractal dimension D have normally. different values. The Hurst exponent H can represent an Theory of fractal Brownian motion index of the anisotropic characteristics of joint roughness. Through the empirical relation of JRC and In one dimension, a random process y(x) of a variable D, the anisotropic shear strength of joints can be x is called fractal Brownian motion (pm) [ 181, if the determined. increment of Ay = y(x2) - y(x,) has a Gaussian distribution with mean zero p = 0 and the variance Var[y(x2)
- y(xdl
= 0~1x2 - x112H
(1)
where H is referred to as the “Hurst exponent” having the values of 0 < H < 1 and 0 is the standard deviation. The increments of y(x) are statically self-affine with H. That is, Y&O + Ax) - I
SHEAR
STRENGTH
AND
ROUGHNESS
The most widely used formula for the shear strengths of joints is Barton’s empirical equation [2] given as z = rr,,tan[JRC loglo(JJ)
+ 9.1
(4)
and LY(x~+ rAx) - y(xo)l/r”
are statistically indistinguishable for any x0 and r > 0 scaling differently in x- and y-coordinates. The scaling property of fBm represented above is different from the statistical self-similarity that repeats its shape with the same magnifications in both directions. For the exponent H = 0, the variance is independent of scaling. The curve y(x) still looks the same for all r > 0. This means y(x) can be expended or contracted in the x-direction by any factor, thus implying that a sample of y(x) with H = 0 must densely fill up the plane on which y(x) is shown. If H = 1, the opposite case prevails. For other values of the Hurst exponent H, y(x) represents different properties: (a) H = l/2; the y(x) is the classical Brownian motion type that has independent increments. (b) H > l/2; the increments of y(x) have a positive correlation called persistence, e.g. if y(x) increases for x0, it tends to continue to increase for x0 + Ax. (c) H c l/2; the opposite of the above holds, called anti-persistence, and y(x) appears more rugged. Therefore the Hurst exponent H plays a role that can be interpreted as the “roughness” of a joint profile. It also relates fractal dimension D by D=n+l-H
where, n denotes dimensional sional curve, D = 2 - H. The y(x) is fractal when its fractal 1 < D < 2. Therefore, a joint Brownian motion property relation
(2)
number. In a one-dimenone-dimensional fBm of dimension is in the range profile with the fractal satisfies the following
y(x + Ax) - y(x) r y(x + ‘$)
- y(x)
(3)
where r is a real number and r > 0. It means that the joint profile enlarged by the r ratio in the x-direction and the heights of y(x) by the rH ratio still satisfies self-tine properties. The successive height of each asperity is related to the past heights and thus it can “remember” the previous shearing history. Thus, the two opposite directions of a
where, & is the basic friction angle, JCS is the joint wall strength, and JRC is the joint roughness coefficient. The JRC of a given surface can be estimated visibly by comparison with the 10 typical profiles. However, in practice it may be difficult to determine the proper JRC number. Researchers have attempted to calculate the value of the JRC from the geometry of a joint profile. Among these, Tse and Cruden [19] developed a relation between the JRC and the root mean square of the first derivative of the profile Z2 as JRC = 32.2 + 32.471og,,Z2
(5)
where, the root mean square is
In the equation, Ax is the sampling interval, yi+ , and yi are the asperity heights of two adjacent sampling points, and M is the sample number. In this paper, the JRC number calculated by this approach is called the original JRC. Relationship of fractal
dimension D and JRC
Recently, a number of researchers [4-71 have applied the concept of fractal dimension to the rock joints. The empirical relation between the JRC and fractal dimension D determined by Venkatachalam [4] is now adopted in this paper as JRC = 85.2671(0 - 1)“.5679.
(7)
Generally, large D represents a rough surface and the JRC is larger for the joint profile. The value of D for one-dimensional joints ranges between 1 and 2. Lee et al. [6] investigated the fractal dimension of Barton’s 10 standard profiles by the divider method and concluded that D varies between 1.000446 (for the JRC = l-2 profile) and 1.013435 (for the JRC = 18-20 profile). The difference of D in various profiles was not evident and D was a single statistical value. Odling [20] explains the result in that the divider method is only suited for a
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self-similar fractal curve, and thus is not suited for joint surfaces that are self-affine fractals. Such an application has commonly given the fractal dimension estimation for a joint surface as very close to 1. Kulatilake states that the step of the divider method can be greater than the cross length [21]. EVALUATION.
BY LABORATORY
TEST
RESULTS
Calculating the anisotropic JRC values for a profile
A natural joint surface is usually continuous and undifferentiable, and the height distribution of the asperities is Gaussian, the bell-shaped distribution. Thus, the joint surface may be modeled using the fractional Brownian motion model with a parameter H, 0 < H < 1. For a fractal surface to satisfy the Gaussian process and self-affine properties, the H value can be determined by a trial-and-error approach. In this paper, we enlarge the x-length of a joint surface by the r ratio (r = 0.5, 50 and 500) and the y-height by the r* ratio. If the statistical properties still hold as self-affine, such that the mean value and variance are invariant, H representing the roughness is then obtained. Actually, there are different values, the HR and HL, in the two sampling directions. Then, the fractal dimension D is calculated as 2 - H and the JRC by equation (7). The shear strengths of opposite directions for a joint surface can also be obtained through Barton’s formula. The shear strengths are always different due to the different JRC values and reveal a directional characteristic.
NOTE
Anisotropic joints
1033
roughness and shear strengths of artificial
Test program A: (square sample). An artificial extension joint produced by a double wedge-type guillotine was used to evaluate the application of the model [22]. The rock material was a mixture of plaster, sand and water with a JCS of 7.63 MPa and basic friction angle of 3 1”) and its properties are reproducible. A series of direct shear tests using square samples (10 x 10 cm) was performed and the peak shear strength was obtained. Two typical profiles duplicated as shown in Fig. 1 are measured by a digital profile-meter and their original JRC values are also estimated (sampling interval Ax = 0.23 mm, sample point M E 410). The original JRC is 15.13 for profile A and 14.19 for profile B, respectively. The height distributions of asperities of the profiles are shown as Fig. 2 and are like a bell-shaped or Gaussian distribution. It thus satisfies the assumption of fractal Brownian motion, i.e. the normal distribution and a non-differentiable, and continuous function. Two Hurst exponents H of HR (shearing to the right) = 0.98 and HL (shearing to the left) = 0.89 for profile A are obtained by trying three different scaling values (r = 0.5, 50 and 500). The H value in the opposite direction is different and thus there are two different “roughnesses” for a single profile. The corresponding JRC value calculated by equations (2) and (7) are 9.24 (JR&) and 24.34 (JRC), and both are different from the original
5,
4
- ProfileA:JRC=15.127
-1I 0
10
20
30
40
50 x (mm)
->
Hrt=0.920
<--
HL =0.8@6 JRC2=24.76
60
70
JRCl-20.5
80
90
100
(a) profile A 5
-1 -2 0
I
I
I
10
20
30
I 40
I 50
<--
I 60
H~=0.913
I
I
70
80
JRC2=21.29
I
90
100
x (mm)
(b) profile B Fig. 1. Joint profile plotted by digital data and the Hurst index in two opposite directions. RMhlS34/6K
1034
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NOTE
1r 0.8
k
0.6
P R
!I?I0.4 0.2
A
0 4
-3
-2
-1 Y/U
(a) profile A
1
0.8
0.2
0 4
-3
-1
-2
0
1
Y/G
(b) profile B Fig. 2. Shape of asperity height distribution,
JRC (15.12). The value of roughness starting from the left side is rougher than that from the right side and thus the shear strength is directional. Profile B also
(a) profile A, (b) profile B.
demonstrates the same anisotropic characteristics. Both profiles have H > l/2 and imply the persistence correlation between their asperity heights. It means that
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NOTE
1035
501
PROFILE A Test Data
n 40
-
. -
_ -
JRC=15.127 fBm:-->JRCR=20.50 fBm:G-JRCL=24.76
w-w
10
JCS=76.3(kg/cm2)
+b=3 lo 00 0
10
20
30
40
50
60
70
Normal Stress (kgkmz)
(a) highnormalstress
-
. -
JRC=15.127 fBm:--~JRCR=20.50 fBm:G-m&=24.76
0
20 Normal Sgs
(kghmz)
(b) low normalstress Fig. 3. Difference of shear strength in reversed direction of profile A, (a) high normal stress, (b) low normal stress.
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NOTE
fBm:<--JRCL=21.29 /,
w-w
’ -’
JCS=76.3(kg/cd)
6&=31°
00 0
10
20
50 NolmLtress
60
70
(ggcmq
(a) highnormalstress 25 PROFILE B Test Data
n 20 --a--
s
. -
JRc=14.19 fBm:-*JRcR=30.39
mm:<--JRQ
=21.
R
JCS=76.3(kg/cd
0
0
20
(b) low nod
stress
Fig. 4. Difference of shear strength in reversed direction of profile B, (a) high normal stress, (b) low normal stress.
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TECHNICAL
smoother than the irregularity concerned in the field of fractal dimension, e.g. a sonic wave. Figures 3 and 4 show the directional characteristics of the shear strengths compared with the experimental results. It shows that the predicted strength is different in reverse directions for a given profile. It also demonstrates that the two predicted strengths are different in the middle normal stress level and close at high stress. This means that the roughness is of paramount importance in the range of lower normal stress. All of the experimental data locate below the two predicted envelopes and close to the lower one. The over-estimated shear strength and anisotropic JRC larger than 20 are introduced by transforming from the empirical equation (7). In addition, the same strength in two reverse directions under very low stress is due to the upper limit of 70” in Barton’s formula. Test program B: (circular surface). Huang and Doong [9] performed a series of anisotropic direct shear tests using circular joint surfaces (Fig. 5). The property of the model material is the same as program A, except JCS = 4 MPa and the basic friction angle &, = 24”. In their research, roughness profiles were obtained in six directions (at every 30”) on the model joint surface. In each direction, roughness profiles were measured along nine parallel lines that were spaced at equal distances [21]. Then, 12 directions of shearing tests on each circular profile were performed. It was concluded that the shear strengths of test results for 12 directions and six normal stresses depend not on the normal stress, but
NOTE
1037
also on the shearing direction [9, lo]. It was also shown that the shear strength is quite different in the two opposite directions for a joint profile. The roughness profiles and, thus, directional strengths are considered to support the concept in this paper. Figures 6(a)-1 l(a) show the nine profiles with different lengths for one direction and the corresponding Hurst indices in the two opposite directions. They reveal that the roughness indices are different for every given profile. The corresponding JRC value in every direction can then be calculated by equations (2) and (7). In this paper, the JRC that represented the profile in one direction is averaged over the whole joint surface as
JRC = i (JRCi X Li)/ i Li i=l i=l
(8)
in which, n (n = 9) is the total number of profiles, JRC, is the roughness coefficient of the ith profile calculated from equation (4) and Li is the length of the ith profile. Then, the directional shear strength can be calculated from the average JRC value. In addition, a procedure for evaluating the shape of the asperity distribution is implemented and the results show as Figs 6(b)-11(b). Each of the distributions represents an approximation of a bell-shaped or Gaussian distribution, satisfying the assumption of fBm theory. The differences in directional strengths for the 12 shearing directions are also shown as Figs 6(ckl l(c).
I
120
270;
/ -1’
90 300
Fig.
5. Three-dimensional
plot
of the joint
surface
using
nine
profiles’
digital
data.
1038
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and
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TECHNICAL
NOTE
0.72
0.817
I
0.871
0.845
0.918 0.852
I
0.943
0.872
0. 0.846 L---
I
12
cm
0” e+ 180” (a) 9 profles andthe Hurst indexin 0 ’ -180’ direction
0
1.0
16-
LJ -
cc?0.8
O---B180 O<---180
. -
Original
-
JRC=27.159
--> JRCu=29.769
s
<-- JRQ=27.644
gO6 Lrc . .,I 0.4 $02 z . 0.0
JCS40
-8 -6 -4 -2 0
2
4
6
Y
0
(b) asperityheightdistribution Fig.
6. Height distribution, index in the O”-180”
Hurst index and shear direction. (b) Asperity
kg/cm*
5
No21
(l&n2)
20
25
Stress
(c) comparisonof anisotropicshearstrengths strength height
of profiles distribution.
in the O”-180” direction. (a) Nine profiles and the Hurst (c) Comparison of anisotropic shear strengths.
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TECHNICAL
NOTE
1039
HL. 0.899 I
0.728
0.684
0.812
I
- - 0.921 -
I
0.847 0.883 0.908
I
0.736
I
0.785
ol--1cm
210” f) 30” (a) 9 profilesandHurst indexin 30’ -210’ direction
s5 0.8 s g 0.6 % .g 0.4 0Ii 0.2 z 0.0
n
210~x30
l
210<--30
JCS=40 k&m2
-7 -6 -4 -3 -1
0
1
3
Y (b) asperityheightdistribution
(c) comparisonof anisotropicshearstrengths
Fig. 7. Height distribution, Hurst index and shear strength of profiles in the 30”-210” direction. (a) Nine profiles and Hurst index in the 30”-210” direction. (b) Asperity height distribution. (c) Comparison of anisotropic shear strengths.
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The predicted value by the original JRC is only an average prediction result. However, they all show that the strengths predicted by the Hurst index are quite
NOTE
different for a profile in two opposite directions. They show that the tendency of a directional characteristic in the experimental data is consistent with the predicted HL
HR
I
0.724 0.618 -
0.608
--
0.623
I
0.865
I
0.704 0.78
I
0.938
0.887
I/
w
I 0
I
240” e
1
2cm
60”
(a) 9 profilesandHurst indexin 60 ’ -240 ’ direction
1.0 s g 0.8 3 g 0.6 -8 .y 0.4 1 2 02’ 0.0
5
(b) asperityheightdistribution Fig. 8. Height distribution, Hurst in a 60”-240” direction.
10 NcmnaI Stress ($om2)
20
25
(c) comparisonof anisotropicshearstrengths
index and shear strength of profiles in a 60-240” (b) Asperity height distribution. (c) Comparison
direction. (a) Nine profiles and Hurst of anisotropic shear strengths.
index
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results. The cases of more non-directional characteristics than observed in the experimental results, e.g. Fig. 6(c), also show slight directionality in prediction. The
TECHNICAL
NOTE
1041
anisotropic strength for all 12 directions at six normal stresses is also duplicated as a pole diagram (Fig. 12). This also shows that the shear strength is
HR
0.787
0.802 I
0.685 0.782
.
0.59
0.683 - 0.891
0.881
I
0.847
I
0.813
0.822 0.72 ./--
I ,
0.876 r
I
0
90" e
1
2cm
270"
(a) 9 profilesandHurst indexin 90 ’ -270’ direction
20,
90-->270
0
1.0
Ml-n
9 ii 0.8 Es g 0.6 -8 .y 0.4
I-
- -
%X--270 Origid JRe24.145 original
1 02 L2 * 0.0 -5
-4
-3
-1 Y
(b) asperityheightdistribution
0
2
3
JCS=40 k&n? $h=2-f
(c) comparisonof anisotropicshearstrengths
Fig. 9. Height distribution, Hurst index and shear strength of profiles in the 90”-270” direction. (a) Nine profiles and Hurst index in 90”-270” direction. (b) Asperity height distribution. (c) Comparison of anisotropic shear strengths.
1042
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and
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TECHNICAL
NOTE
HL
HR
0.652
0.633
I
0.639
0.889 I
0.898
0.867 0.892
I
I I
0
1
2cm
300” f) 120” (a) 9 profilesandHurst indexin 120’ -300’ direction
20
1.0
r
1
n 16
t
-8
300<--120
0 -
-ii
300->120
. -
Ortinal
JRC=28.525
-->JRC~=40.336
12
#
.g 0.4
E
1 02 z . nn
1 rA
0
4
5
JCS=4O 0
(b) asperityheightdistribution Fig.
10. Height distribution, index in 120”-300”
5
Nor2 stress(l&Ld)
20
25
(c) comparisonof anisotropicshearstrengths
Hurst index and shear strength of profiles direction. (b) Asperity height distribution.
directional and reveals an anisotropic characteristic in the whole surface. The effect of anisotropy decreases with increasing applied normal stress in both the test and predicted results. The more isotropic characteristic is
0
k&m2
in the 120”-300” (c) Comparison
direction. (a) Nine profiles and Hurst of anisotropic shear strengths.
observed at high normal stress implying that the roughness is not all pervasive. From Fig. 12, it is seen that the predicted results are generally consistent with the experimental data. However, a special phenomenon
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TECHNICAL
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1043
HL
HR
0.716
0.578 0
0.659
I
0.640
0.784
0.893 0.69 0.908
I
0
1
2cm
330” ++ 150” (a) profilesandHurst indexin 150’ -330’ direction
1.0 s ii 0.8 is E 0.6 B .y 0.4
16-
n
330-s150
l
330<--150
___
. -
original
JRC=29.414
->JR’&=38.285
2 E 02 . 0.0 -7 -5 -3 -1
1
3
5
6
(b) asperityheightdistribution Fig.
Il. Height distribution, in the 150”-330”
Hurst direction.
JCS=40
(c) comparisonof anisotropicshearstrengths
index and shear strength of profiles in the 150”-330” (b) Asperity height distribution. (c) Comparison
of isotropy observed in very low normal stress is introduced by the upper limit of a 70” total friction angle in equation (4) of Barton’s formula PI.
k&n*
direction. (a) Profiles and Hurst of anisotropic shear strengths.
index
CONCLUSIONS
The Hurst exponent H is capable of describing the anisotropic characteristic of joint roughness. The
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1044
0 0 A
a,=o.s .J,=l.O
A
a,=s.o
TECHNICAL
NOTE
distribution of asperity heights of a profile is a Gaussian process, and enough sample data or joint lengths are included, is important and needs greater study.
a,=3.2
a,=10.0 Ll
and LO:
120
Acknowledgements-The authors would like to thank T. H. Huang, Professor of National Taiwan University, for providing the experimental data in test program B.
\
arF20.0
T
30 Accepted
for
publication
300 I 270
(a) anisompic chanctcriatic of e.x+mad
0 0 A
o,=l.O
n
O.=S.O
n q
a.=lO.O
data
O&S
q=3.2 120
0"=20.0 150 \
/ 210
240
I 270
300
@)animtmpiccbamctaiaicofprsdining&a
1. Patton, F. D., Multiple modes of shear failure in rock. The 1st Int. Congr. Rock Mech., Lisbon, 1966, pp. 1411-1420. 2. Barton, N., Review of a new shear strength criterion for rock joints. Engng Geol., 1973, 7, 255-279. 3. Mandelbrot, B. B., The Fractal Geometry of Nature. Freeman, San Francisco, 1983, p. 468. 4. Venkatachalam, G., Modeling of rock joints. Proc. of ht. Symp. on Fundamentals of Rock Joints, Bjorkhden, 1985, pp. 453-459. 5. Turk, N., Greig, M. J., Dearman, W. R. and Amin, F. F., Characterization of rock ioint surfaces by fractal dimension. Proc. of’ the 28th Symp. -on Rock Mech, Tucson, 1987, pp. 1223-1236. 6. Lee, Y. H., Har, J. R., Bars, D. J. and Hass, C. J., The fractal dimension as a measure of the roughness of rock discontinuity profiles. Int. J. Rock Mech. & Min. Sci. Abstr., 1990,27,45~64. 7. Maerz, N. H. and Franklin, J. A., Roughness effect and fractal dimension. Proc. of 1st Int. Workshop on Scale Eflect in Rock Masses, Loen, 1990, pp. 121-126. 8. Brown, E. T., Richart, L. R. and Barr, M. V., Shear strength characteristics of Delabole slates. Proc. of Rock Engng, Newcastle, 1977, pp. 33-51. 9. Huang, T. H. and Doong, Y. S., Anisotropic shear strength of rock joints. Proc. of the Int. Symp. on Rock Joints, Loen, 1990, pp. 211-218. 10. Huang, T. H., Doong, Y. S. and Sheng, J., Measurement of rock joint roughness and its directional shear strength. Proc. of the Znt. Conf. on Mech. of Jointed and Faulted Rock, Vienna, 1990, pp. 337-343. 11. Aydan, O., Shim& Y. and Kawamoto, T., The anisotropic of surface morphology characteristics of rock discontinuities. Rock Mech. Rock Engng, 1996, 29(l), 47-59. 12. Hurst, H. E., Black, R. P. and Simaika, Y. M., Long-Term Storage: An Experimental Study. Constable, London, 1965. 13. Rengers, N., Influence of surface roughness on the friction properties of rock planes. Proc. of 2nd Cong. ISRM, Belgrade, 1970, Vol. 1, pp. 229-234. 14. Ueng, T. S. and Chang, W. C., Shear strength of joint surfaces. Proc. of 31st U.S. Symp. on Rock Mech. Balkema, 1990, pp. 245251. 15. Chen, K. L., The anisotropic shear strength and deformability of joints. MS thesis of National Taiwan University, Taipei, 1990. 16. Outer, A. D., Kaashoek, J. F. and Hack, H. R. G. K., Difficulties with using continuous fractal theory for discontinuity surfaces. Int. J. Rock
Fig. 12. Pole diagram comparison of anisotropic shearing strengths. (a) Anisotropic characteristic of experimental data. (b) Anisotropic characteristic of predicting data.
roughness index H can capture the past history of the asperities with different values in the two opposite directions. The value of H shown in this paper is greater than l/2 so the roughness reveals a persistence property. The anisotropic shear strength can be estimated by Barton’s formula with the aid of different H values according to the shearing directions of a given profile. The above conclusions are based on the assumption that the height distribution of asperities is Gaussian and that the joint surface is a fractal curve. Whether the
1996.
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Abstr.,
1995, 32, 3-9.
17. Swan, G. and Zongqi, S., Determination of stiffness and other joint properties from roughness measurements. Rock Mech. & Rock Engng, 1983, 16, 19-38. 18. Mandelbrot, B. B. and van Ness, J. W., Fricttonal Browman motions, fractional noises and applications. SIAM Rev., 1968, 4, 422437. 19. Tse, R. and Cruden, D. M., Estimating joint roughness coefficients. Int. J. Rock Mech. & Min. Sci., 1978, 16, 303-307. 20. Odling, N. E., Natural fracture profiles, fractal dimension and joint roughness coefficients. Rock Mesh. and Rock Engng, 1994, 27(3), 135-153. 21. Kulatilake, P. H. S., Shou, G., Huang, T. H. and Morgan, P. M., New peak shear strength criteria for anisotropic rock joints. Int.
J. Rock
Mech.
& Min.
Sci.
& Geomech.
Abstr.,
1995, 32,
673-697. 22. Huang, T. H. and Yang, Z. Y., Generation of artificial extension joints and their mechanical properties. Proc. of the 32nd U.S. Svmo.. , , , Norman. 1991. .no.. 1105-I 114.
J. Rock Mech.
PII: SO148-9062(97)0001&l
Sci. Vol. 34, No. 6, pp. 1031-1044, 1997 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0148-9062/97 $17.00 + 0.00
Min.
0
Pergamon
Technical Note An Index for Describing the Anisotropy
of Joint Surfaces
Z. Y. YANGt s. c. Lot
INTRODUCTION
FRACTAL BROWNIAN MOTION
Joint roughness is of paramount importance for the shear behavior of rock joints. Patton [l] concluded that the strength of joints is controlled by the three factors of friction property, dilation effect, and shear-off effect for a rock material. In particular, the dilation effect and thus the shear strength is dominated by the roughness of joints at a low normal stress level. The formula for shear strength of Barton [2] also demonstrates the importance of the joint roughness coefficient (JRC). Barton introduces 10 standard JRC profiles defined by a JRC number. The JRC is an average index to describe the roughness degree of joint surfaces. Recently, fractal theory [3] has been applied to describe the roughness of joint surfaces by several authors [4-71. They found good correlation between the JRC and fractal dimension, D, and concluded that the fractal dimension can represent the JRC profile. However, both the JRC and D describe the average roughness of a given profile. They are independent of the direction of profiles and only indicate the scalar characteristics. However, it has been observed that the strength of a joint surface depends on the shear direction from several experimental tests [8-lo]. The single value of JRC or D is thus insufficient for describing the directional properties. Aydan et al. [ 1l] propose a parameter related to the measure direction and demonstrates the directional dependence of joint surfaces. Nevertheless, the anisotropic mechanical behavior is not included in his discussion. This paper uses a new index, the Hurst exponent based on the theory of fractional Brownian motion (fBm) [12], to represent the anisotropic characteristic of the joint profile. The Hurst exponent which is dependent on the sequence of sampling, can indicate the past height history of asperities. Thus, it can represent the anisotropic roughness characteristic of a profile according to different sampling orders and shearing directions. The fractal dimension and JRC can be calculated from the Hurst exponent for a profile in different directions. The anisotropic shear strength of joints, based on the Barton empirical formula, is also investigated. tDepartment of Civil Engineering, Taipei 25137, Taiwan.
Tamkang University, Tamsui, 1031
Since Mandelbrot [3] presented the fractal concept, the correlation formula between D and JRC has been applied to the shear strength of rock joints. It has been shown that the fractal dimension is capable of describing a random distribution of joint asperities. The fractal dimension of a joint profile, however, is an average index to represent the surface. It is a non-directional parameter and independent of the sequence of sampling data. Therefore, it does not reflect the anisotropic characteristic of the different shearing directions. An index that is direction-dependent is now needed for joint surfaces in rock mechanics. Surface roughness of joints
There have been numerous experimental and numerical studies concerned with characterizing the surface morphology of joint surfaces. The research that focused on the correlation of D and JRC has not addressed the anisotropic character of joint profiles. Three approaches are usually adopted to calculate the fractal dimension D, i.e. the divider, box and spectrum methods. Early work in fractal measurement with the divider method is presented by Rengers [ 131 in rock mechanics. Rengers distinguishes the effect of positive and negative angle for a joint profile in his study. He shows that the angle distribution depends on the measure direction and the measure intervals. Ueng and Chang [14], and Huang and co-workers [9, lo] also attempt to consider the anisotropic strength by distinguishing the upward and downward angles of asperity regarding the shear direction. Chen [15] and Huang et al. [lo] estimate the part JRC value by ignoring the downward asperity angle to take into account shearing directions. The joint surface is produced by a succession of mechanical events which cumulatively can be modeled as a random process during tectonic circumstances. The fracture surface that is continuous can be regarded as a single-valued function; such surfaces are self-similar or self-affine. The self-similar surface has the same roughness regardless of the sample size. The self-affine surface appears progressively smoother for larger and larger samples. The joint surface, a self-affine curve [ 161, is such that its “shape” appears the same at any magnification, but its value is not exactly the same for
YANG
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and
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TECHNICAL
NOTE
any sample size. In addition, Swan and Zongqi [ 171 joint surface can be represented by two different values. Considering the different shearing direction, the Hurst assumed that the peaks of asperities are distributed exponent H and then the fractal dimension D have normally. different values. The Hurst exponent H can represent an Theory of fractal Brownian motion index of the anisotropic characteristics of joint roughness. Through the empirical relation of JRC and In one dimension, a random process y(x) of a variable D, the anisotropic shear strength of joints can be x is called fractal Brownian motion (pm) [ 181, if the determined. increment of Ay = y(x2) - y(x,) has a Gaussian distribution with mean zero p = 0 and the variance Var[y(x2)
- y(xdl
= 0~1x2 - x112H
(1)
where H is referred to as the “Hurst exponent” having the values of 0 < H < 1 and 0 is the standard deviation. The increments of y(x) are statically self-affine with H. That is, Y&O + Ax) - I
SHEAR
STRENGTH
AND
ROUGHNESS
The most widely used formula for the shear strengths of joints is Barton’s empirical equation [2] given as z = rr,,tan[JRC loglo(JJ)
+ 9.1
(4)
and LY(x~+ rAx) - y(xo)l/r”
are statistically indistinguishable for any x0 and r > 0 scaling differently in x- and y-coordinates. The scaling property of fBm represented above is different from the statistical self-similarity that repeats its shape with the same magnifications in both directions. For the exponent H = 0, the variance is independent of scaling. The curve y(x) still looks the same for all r > 0. This means y(x) can be expended or contracted in the x-direction by any factor, thus implying that a sample of y(x) with H = 0 must densely fill up the plane on which y(x) is shown. If H = 1, the opposite case prevails. For other values of the Hurst exponent H, y(x) represents different properties: (a) H = l/2; the y(x) is the classical Brownian motion type that has independent increments. (b) H > l/2; the increments of y(x) have a positive correlation called persistence, e.g. if y(x) increases for x0, it tends to continue to increase for x0 + Ax. (c) H c l/2; the opposite of the above holds, called anti-persistence, and y(x) appears more rugged. Therefore the Hurst exponent H plays a role that can be interpreted as the “roughness” of a joint profile. It also relates fractal dimension D by D=n+l-H
where, n denotes dimensional sional curve, D = 2 - H. The y(x) is fractal when its fractal 1 < D < 2. Therefore, a joint Brownian motion property relation
(2)
number. In a one-dimenone-dimensional fBm of dimension is in the range profile with the fractal satisfies the following
y(x + Ax) - y(x) r y(x + ‘$)
- y(x)
(3)
where r is a real number and r > 0. It means that the joint profile enlarged by the r ratio in the x-direction and the heights of y(x) by the rH ratio still satisfies self-tine properties. The successive height of each asperity is related to the past heights and thus it can “remember” the previous shearing history. Thus, the two opposite directions of a
where, & is the basic friction angle, JCS is the joint wall strength, and JRC is the joint roughness coefficient. The JRC of a given surface can be estimated visibly by comparison with the 10 typical profiles. However, in practice it may be difficult to determine the proper JRC number. Researchers have attempted to calculate the value of the JRC from the geometry of a joint profile. Among these, Tse and Cruden [19] developed a relation between the JRC and the root mean square of the first derivative of the profile Z2 as JRC = 32.2 + 32.471og,,Z2
(5)
where, the root mean square is
In the equation, Ax is the sampling interval, yi+ , and yi are the asperity heights of two adjacent sampling points, and M is the sample number. In this paper, the JRC number calculated by this approach is called the original JRC. Relationship of fractal
dimension D and JRC
Recently, a number of researchers [4-71 have applied the concept of fractal dimension to the rock joints. The empirical relation between the JRC and fractal dimension D determined by Venkatachalam [4] is now adopted in this paper as JRC = 85.2671(0 - 1)“.5679.
(7)
Generally, large D represents a rough surface and the JRC is larger for the joint profile. The value of D for one-dimensional joints ranges between 1 and 2. Lee et al. [6] investigated the fractal dimension of Barton’s 10 standard profiles by the divider method and concluded that D varies between 1.000446 (for the JRC = l-2 profile) and 1.013435 (for the JRC = 18-20 profile). The difference of D in various profiles was not evident and D was a single statistical value. Odling [20] explains the result in that the divider method is only suited for a
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TECHNICAL
self-similar fractal curve, and thus is not suited for joint surfaces that are self-affine fractals. Such an application has commonly given the fractal dimension estimation for a joint surface as very close to 1. Kulatilake states that the step of the divider method can be greater than the cross length [21]. EVALUATION.
BY LABORATORY
TEST
RESULTS
Calculating the anisotropic JRC values for a profile
A natural joint surface is usually continuous and undifferentiable, and the height distribution of the asperities is Gaussian, the bell-shaped distribution. Thus, the joint surface may be modeled using the fractional Brownian motion model with a parameter H, 0 < H < 1. For a fractal surface to satisfy the Gaussian process and self-affine properties, the H value can be determined by a trial-and-error approach. In this paper, we enlarge the x-length of a joint surface by the r ratio (r = 0.5, 50 and 500) and the y-height by the r* ratio. If the statistical properties still hold as self-affine, such that the mean value and variance are invariant, H representing the roughness is then obtained. Actually, there are different values, the HR and HL, in the two sampling directions. Then, the fractal dimension D is calculated as 2 - H and the JRC by equation (7). The shear strengths of opposite directions for a joint surface can also be obtained through Barton’s formula. The shear strengths are always different due to the different JRC values and reveal a directional characteristic.
NOTE
Anisotropic joints
1033
roughness and shear strengths of artificial
Test program A: (square sample). An artificial extension joint produced by a double wedge-type guillotine was used to evaluate the application of the model [22]. The rock material was a mixture of plaster, sand and water with a JCS of 7.63 MPa and basic friction angle of 3 1”) and its properties are reproducible. A series of direct shear tests using square samples (10 x 10 cm) was performed and the peak shear strength was obtained. Two typical profiles duplicated as shown in Fig. 1 are measured by a digital profile-meter and their original JRC values are also estimated (sampling interval Ax = 0.23 mm, sample point M E 410). The original JRC is 15.13 for profile A and 14.19 for profile B, respectively. The height distributions of asperities of the profiles are shown as Fig. 2 and are like a bell-shaped or Gaussian distribution. It thus satisfies the assumption of fractal Brownian motion, i.e. the normal distribution and a non-differentiable, and continuous function. Two Hurst exponents H of HR (shearing to the right) = 0.98 and HL (shearing to the left) = 0.89 for profile A are obtained by trying three different scaling values (r = 0.5, 50 and 500). The H value in the opposite direction is different and thus there are two different “roughnesses” for a single profile. The corresponding JRC value calculated by equations (2) and (7) are 9.24 (JR&) and 24.34 (JRC), and both are different from the original
5,
4
- ProfileA:JRC=15.127
-1I 0
10
20
30
40
50 x (mm)
->
Hrt=0.920
<--
HL =0.8@6 JRC2=24.76
60
70
JRCl-20.5
80
90
100
(a) profile A 5
-1 -2 0
I
I
I
10
20
30
I 40
I 50
<--
I 60
H~=0.913
I
I
70
80
JRC2=21.29
I
90
100
x (mm)
(b) profile B Fig. 1. Joint profile plotted by digital data and the Hurst index in two opposite directions. RMhlS34/6K
1034
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and LO:
TECHNICAL
NOTE
1r 0.8
k
0.6
P R
!I?I0.4 0.2
A
0 4
-3
-2
-1 Y/U
(a) profile A
1
0.8
0.2
0 4
-3
-1
-2
0
1
Y/G
(b) profile B Fig. 2. Shape of asperity height distribution,
JRC (15.12). The value of roughness starting from the left side is rougher than that from the right side and thus the shear strength is directional. Profile B also
(a) profile A, (b) profile B.
demonstrates the same anisotropic characteristics. Both profiles have H > l/2 and imply the persistence correlation between their asperity heights. It means that
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TECHNICAL
NOTE
1035
501
PROFILE A Test Data
n 40
-
. -
_ -
JRC=15.127 fBm:-->JRCR=20.50 fBm:G-JRCL=24.76
w-w
10
JCS=76.3(kg/cm2)
+b=3 lo 00 0
10
20
30
40
50
60
70
Normal Stress (kgkmz)
(a) highnormalstress
-
. -
JRC=15.127 fBm:--~JRCR=20.50 fBm:G-m&=24.76
0
20 Normal Sgs
(kghmz)
(b) low normalstress Fig. 3. Difference of shear strength in reversed direction of profile A, (a) high normal stress, (b) low normal stress.
YANG
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and LO:
TECHNICAL
NOTE
fBm:<--JRCL=21.29 /,
w-w
’ -’
JCS=76.3(kg/cd)
6&=31°
00 0
10
20
50 NolmLtress
60
70
(ggcmq
(a) highnormalstress 25 PROFILE B Test Data
n 20 --a--
s
. -
JRc=14.19 fBm:-*JRcR=30.39
mm:<--JRQ
=21.
R
JCS=76.3(kg/cd
0
0
20
(b) low nod
stress
Fig. 4. Difference of shear strength in reversed direction of profile B, (a) high normal stress, (b) low normal stress.
YANG
and
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TECHNICAL
smoother than the irregularity concerned in the field of fractal dimension, e.g. a sonic wave. Figures 3 and 4 show the directional characteristics of the shear strengths compared with the experimental results. It shows that the predicted strength is different in reverse directions for a given profile. It also demonstrates that the two predicted strengths are different in the middle normal stress level and close at high stress. This means that the roughness is of paramount importance in the range of lower normal stress. All of the experimental data locate below the two predicted envelopes and close to the lower one. The over-estimated shear strength and anisotropic JRC larger than 20 are introduced by transforming from the empirical equation (7). In addition, the same strength in two reverse directions under very low stress is due to the upper limit of 70” in Barton’s formula. Test program B: (circular surface). Huang and Doong [9] performed a series of anisotropic direct shear tests using circular joint surfaces (Fig. 5). The property of the model material is the same as program A, except JCS = 4 MPa and the basic friction angle &, = 24”. In their research, roughness profiles were obtained in six directions (at every 30”) on the model joint surface. In each direction, roughness profiles were measured along nine parallel lines that were spaced at equal distances [21]. Then, 12 directions of shearing tests on each circular profile were performed. It was concluded that the shear strengths of test results for 12 directions and six normal stresses depend not on the normal stress, but
NOTE
1037
also on the shearing direction [9, lo]. It was also shown that the shear strength is quite different in the two opposite directions for a joint profile. The roughness profiles and, thus, directional strengths are considered to support the concept in this paper. Figures 6(a)-1 l(a) show the nine profiles with different lengths for one direction and the corresponding Hurst indices in the two opposite directions. They reveal that the roughness indices are different for every given profile. The corresponding JRC value in every direction can then be calculated by equations (2) and (7). In this paper, the JRC that represented the profile in one direction is averaged over the whole joint surface as
JRC = i (JRCi X Li)/ i Li i=l i=l
(8)
in which, n (n = 9) is the total number of profiles, JRC, is the roughness coefficient of the ith profile calculated from equation (4) and Li is the length of the ith profile. Then, the directional shear strength can be calculated from the average JRC value. In addition, a procedure for evaluating the shape of the asperity distribution is implemented and the results show as Figs 6(b)-11(b). Each of the distributions represents an approximation of a bell-shaped or Gaussian distribution, satisfying the assumption of fBm theory. The differences in directional strengths for the 12 shearing directions are also shown as Figs 6(ckl l(c).
I
120
270;
/ -1’
90 300
Fig.
5. Three-dimensional
plot
of the joint
surface
using
nine
profiles’
digital
data.
1038
YANG
and
LO:
TECHNICAL
NOTE
0.72
0.817
I
0.871
0.845
0.918 0.852
I
0.943
0.872
0. 0.846 L---
I
12
cm
0” e+ 180” (a) 9 profles andthe Hurst indexin 0 ’ -180’ direction
0
1.0
16-
LJ -
cc?0.8
O---B180 O<---180
. -
Original
-
JRC=27.159
--> JRCu=29.769
s
<-- JRQ=27.644
gO6 Lrc . .,I 0.4 $02 z . 0.0
JCS40
-8 -6 -4 -2 0
2
4
6
Y
0
(b) asperityheightdistribution Fig.
6. Height distribution, index in the O”-180”
Hurst index and shear direction. (b) Asperity
kg/cm*
5
No21
(l&n2)
20
25
Stress
(c) comparisonof anisotropicshearstrengths strength height
of profiles distribution.
in the O”-180” direction. (a) Nine profiles and the Hurst (c) Comparison of anisotropic shear strengths.
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and LO:
TECHNICAL
NOTE
1039
HL. 0.899 I
0.728
0.684
0.812
I
- - 0.921 -
I
0.847 0.883 0.908
I
0.736
I
0.785
ol--1cm
210” f) 30” (a) 9 profilesandHurst indexin 30’ -210’ direction
s5 0.8 s g 0.6 % .g 0.4 0Ii 0.2 z 0.0
n
210~x30
l
210<--30
JCS=40 k&m2
-7 -6 -4 -3 -1
0
1
3
Y (b) asperityheightdistribution
(c) comparisonof anisotropicshearstrengths
Fig. 7. Height distribution, Hurst index and shear strength of profiles in the 30”-210” direction. (a) Nine profiles and Hurst index in the 30”-210” direction. (b) Asperity height distribution. (c) Comparison of anisotropic shear strengths.
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TECHNICAL
The predicted value by the original JRC is only an average prediction result. However, they all show that the strengths predicted by the Hurst index are quite
NOTE
different for a profile in two opposite directions. They show that the tendency of a directional characteristic in the experimental data is consistent with the predicted HL
HR
I
0.724 0.618 -
0.608
--
0.623
I
0.865
I
0.704 0.78
I
0.938
0.887
I/
w
I 0
I
240” e
1
2cm
60”
(a) 9 profilesandHurst indexin 60 ’ -240 ’ direction
1.0 s g 0.8 3 g 0.6 -8 .y 0.4 1 2 02’ 0.0
5
(b) asperityheightdistribution Fig. 8. Height distribution, Hurst in a 60”-240” direction.
10 NcmnaI Stress ($om2)
20
25
(c) comparisonof anisotropicshearstrengths
index and shear strength of profiles in a 60-240” (b) Asperity height distribution. (c) Comparison
direction. (a) Nine profiles and Hurst of anisotropic shear strengths.
index
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results. The cases of more non-directional characteristics than observed in the experimental results, e.g. Fig. 6(c), also show slight directionality in prediction. The
TECHNICAL
NOTE
1041
anisotropic strength for all 12 directions at six normal stresses is also duplicated as a pole diagram (Fig. 12). This also shows that the shear strength is
HR
0.787
0.802 I
0.685 0.782
.
0.59
0.683 - 0.891
0.881
I
0.847
I
0.813
0.822 0.72 ./--
I ,
0.876 r
I
0
90" e
1
2cm
270"
(a) 9 profilesandHurst indexin 90 ’ -270’ direction
20,
90-->270
0
1.0
Ml-n
9 ii 0.8 Es g 0.6 -8 .y 0.4
I-
- -
%X--270 Origid JRe24.145 original
1 02 L2 * 0.0 -5
-4
-3
-1 Y
(b) asperityheightdistribution
0
2
3
JCS=40 k&n? $h=2-f
(c) comparisonof anisotropicshearstrengths
Fig. 9. Height distribution, Hurst index and shear strength of profiles in the 90”-270” direction. (a) Nine profiles and Hurst index in 90”-270” direction. (b) Asperity height distribution. (c) Comparison of anisotropic shear strengths.
1042
YANG
and
LO:
TECHNICAL
NOTE
HL
HR
0.652
0.633
I
0.639
0.889 I
0.898
0.867 0.892
I
I I
0
1
2cm
300” f) 120” (a) 9 profilesandHurst indexin 120’ -300’ direction
20
1.0
r
1
n 16
t
-8
300<--120
0 -
-ii
300->120
. -
Ortinal
JRC=28.525
-->JRC~=40.336
12
#
.g 0.4
E
1 02 z . nn
1 rA
0
4
5
JCS=4O 0
(b) asperityheightdistribution Fig.
10. Height distribution, index in 120”-300”
5
Nor2 stress(l&Ld)
20
25
(c) comparisonof anisotropicshearstrengths
Hurst index and shear strength of profiles direction. (b) Asperity height distribution.
directional and reveals an anisotropic characteristic in the whole surface. The effect of anisotropy decreases with increasing applied normal stress in both the test and predicted results. The more isotropic characteristic is
0
k&m2
in the 120”-300” (c) Comparison
direction. (a) Nine profiles and Hurst of anisotropic shear strengths.
observed at high normal stress implying that the roughness is not all pervasive. From Fig. 12, it is seen that the predicted results are generally consistent with the experimental data. However, a special phenomenon
YANG
and
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TECHNICAL
NOTE
1043
HL
HR
0.716
0.578 0
0.659
I
0.640
0.784
0.893 0.69 0.908
I
0
1
2cm
330” ++ 150” (a) profilesandHurst indexin 150’ -330’ direction
1.0 s ii 0.8 is E 0.6 B .y 0.4
16-
n
330-s150
l
330<--150
___
. -
original
JRC=29.414
->JR’&=38.285
2 E 02 . 0.0 -7 -5 -3 -1
1
3
5
6
(b) asperityheightdistribution Fig.
Il. Height distribution, in the 150”-330”
Hurst direction.
JCS=40
(c) comparisonof anisotropicshearstrengths
index and shear strength of profiles in the 150”-330” (b) Asperity height distribution. (c) Comparison
of isotropy observed in very low normal stress is introduced by the upper limit of a 70” total friction angle in equation (4) of Barton’s formula PI.
k&n*
direction. (a) Profiles and Hurst of anisotropic shear strengths.
index
CONCLUSIONS
The Hurst exponent H is capable of describing the anisotropic characteristic of joint roughness. The
YANG
1044
0 0 A
a,=o.s .J,=l.O
A
a,=s.o
TECHNICAL
NOTE
distribution of asperity heights of a profile is a Gaussian process, and enough sample data or joint lengths are included, is important and needs greater study.
a,=3.2
a,=10.0 Ll
and LO:
120
Acknowledgements-The authors would like to thank T. H. Huang, Professor of National Taiwan University, for providing the experimental data in test program B.
\
arF20.0
T
30 Accepted
for
publication
300 I 270
(a) anisompic chanctcriatic of e.x+mad
0 0 A
o,=l.O
n
O.=S.O
n q
a.=lO.O
data
O&S
q=3.2 120
0"=20.0 150 \
/ 210
240
I 270
300
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Fig. 12. Pole diagram comparison of anisotropic shearing strengths. (a) Anisotropic characteristic of experimental data. (b) Anisotropic characteristic of predicting data.
roughness index H can capture the past history of the asperities with different values in the two opposite directions. The value of H shown in this paper is greater than l/2 so the roughness reveals a persistence property. The anisotropic shear strength can be estimated by Barton’s formula with the aid of different H values according to the shearing directions of a given profile. The above conclusions are based on the assumption that the height distribution of asperities is Gaussian and that the joint surface is a fractal curve. Whether the
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