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The effect of asperity order on the roughness of rock joints
International Journal of Rock Mechanics & Mining Sciences 38 (2001) 745–752
Technical Note
The effect of asperity order on the roughness of rock joints Z.Y. Yang*, C.C. Di, K.C. Yen Department of Civil Engineering, Tamkang University, Tamsui, Taipei 25137, Taiwan Accepted 21 May 2001
1. Introduction Joint roughness has an essential influence on the shear behavior of rock joints. Patton [1] recognized that the asperity of a rough joint occurs on many scales. He first categorized asperity into first-order (waviness) and second-order (unevenness) categories. The behavior of rock joints is controlled primarily by the second-order asperity during small displacements and the first-order asperity governs the shearing behavior for large displacements. Barton [2] and Hoek and Bray [3] also stated that at low normal stress levels the second-order asperity (with higher-angle and narrow base length) controls the shearing process. As the normal stress increases, the second-order asperity is sheared off and the first-order asperity (with longer base length and lower-angle) takes over as the controlling factor. In engineering practice, the shear strength criterion proposed by Barton [2] for rock joints is widely adopted. In which, the JRC (joint roughness coefficient) value for a given joint profile can be estimated visibly by comparing it with the ten JRC profiles whose JRC ranges are from 0 to 20. This set of profiles has subsequently been adopted as a standard by the ISRM. Accordingly a large amount of test joint results, several empirical formula connected to the mechanical parameters such as the shear stiffness and joint aperture were related to the JRC. However, in practice it may be difficult to determine the proper JRC number and is highly subjective. To minimize subjectivity, alternate methods have been proposed for JRC estimation. Many researchers have thus attempted to calculate the JRC value from the profile geometry. At present, one commonly adopts Tse and Cruden’s [4] empirical statistical relationship between the JRC and Z2 to *Corresponding author. Tel.: +886-2623-4215; fax: +886-26209747. E-mail address: [email protected] (Z.Y. Yang).
calculate typical JRC values. The parameter Z2 is the root mean square (r.m.s) of the tangents of the slope angles along the profile. Following the introduction of fractal geometry by Mandelbrot [5], numerous researchers have applied the concept of fractal dimensions to rock joints. They tried to interpret the JRC using fractal dimensions (D). They found that a good correlation between the JRC and fractal dimension, and thus the fractal dimension can represent the JRC profile. Generally, large D represents a rough joint profile and the JRC is larger. The value of D for one-dimensional joint profile ranges between 1 and 2. The divider and box-counting methods were often used to obtain the fractal dimension of the JRC profiles. However, the obtained values are insensitive to the ten JRC profiles as listed in Table 1 [6–8]. It is difficult to distinguish the degree of roughness between different JRC profiles. Usually, none of the actual rock joint profiles such as the JRC profiles were found to have the property of strict self-similar, but were instead self-affine [9–12]. Odling [11] explains the result in that the divider method (or box-counting method) is only suitable for a self-similar fractal curve. Thus, it is not appropriate for joint profiles that are self-affine fractals. Such an application has commonly given the fractal dimension estimation as very close to unity. Furthermore, in Table 1, Lee et al. [7] have shown a positive correlation between the JRC and D (see also in [8]). This means that the rougher profile with the higher JRC displays a larger D value. The respective D values seemed to conform to the profile order in the roughness scale. Turk et al. [6] also produced a similar trend, except that the profiles nearby JRC=12–14 were not in a proper order (see Table 1). However, there have been many controversial findings for the JRC and D relationship reported in the fractal characterization literatures of the JRC profiles [11,13]. Olding [11] employed the structure function to obtain the fractal dimensions of ten JRC profiles (see also in Table 1). The structure
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Table 1 The fractal dimension and Hurst index of ten JRC profiles JRC profile
0–2 2–4 4–6 6–8 8–10 10–12 12–14 14–16 16–18 18–20
D value
H value
Lee
Seidel
Turk
Sakellariou
Odling
Odling
This paper
1.000446 1.001687 1.002805 1.003974 1.004413 1.005641 1.007109 1.008055 1.009584 1.013435
1.00009 1.00054 1.00072 1.0014 1.0018 1.004 1.0053 1.0081 1.0096 1.012
1 1.0019 1.0027 1.0049 1.0054 1.0045 1.0077 1.007 1.0104 1.017
1.493 1.519 1.360 1.522 1.347 1.166 1.208 1.118 1.218 1.188
1.50 1.54 F F 1.30 1.17 1.20 1.15 1.25 F
0.5 0.46 F F 0.7 0.83 0.80 0.85 0.75 F
0.365 0.575 0.765 0.66 0.755 0.83 0.89 0.90 0.87 0.885
function is simply the mean square height difference of two points on the profiles at a specified separation. This fractal dimension derived from the structural function can relate to a roughness index (H), the Hurst index [14–15], by D ¼ 2 H: It was found that the fractal dimensions between joints obtained by this structure function are more distinguishable than the divider method. It was surprising to find that the fractal dimension is 1.5 (H ¼ 0:5) for the JRC=0–2 profile and 1.15 (H ¼ 0:85) for the JRC=16–18 profile. The JRC=0–2 profile being almost a straight line has a D value much larger (smaller H) than the other which seems to be rougher. The relationship between the JRC and D shows a negative correlation. The fractal dimensions are not in proper order corresponding to their JRC scale (see also). This strange order also took place at the JRC=12–14 profile (see Table 1). A similar conclusion by Sakellariou et al. [10] using spectral analysis is shown in Table 1. In another expression, Turk et al. [6] used the fractal dimension to estimate the asperity angle of a rough joint profile. They found that although the JRC increases from 0 to 20 with increasing roughness, the estimated roughness angles did not show a similar increase corresponding to the JRC values. We also notice that the order of the fractal dimensions in their paper is strange in the profiles for JRC=10–12 and 14–16 (see also Table 1). Also, Huang et al. [16] suggested that the r.m.s. of a rough joint profile might perhaps be a better indication of joint roughness than the fractal dimension. They conjectured that the fractal dimension describes the amount of interlocking between joint asperities, whereas the r.m.s. variance in height is more closely related to the shear strength of the rock joints. Thus, using the fractal dimension as an invariant scaling parameter to describe the joint profile roughness does not seem warranted, but its use as one of the number of parameters can have positive benefits. In view of the abnormal order in the fractal dimension, we presume that something is implied from
the fractal characteristic of the ten JRC profiles. This could imply that the roughness property of the joints that determined by the fractal dimension should take into account not only the primary asperity, but also the secondary asperity. Thus, the JRC or fractal dimension seems not enough to completely describe the roughness properties of a joint profile.
2. Hurst index of JRC profiles As mentioned, the ten JRC profiles have the selfaffinity property. The profile appearance under a specified elongation in the lengths will look about the same and, by touch, will be perceived to have the same roughness. That is, a self-affine profile must be scaled differently in two perpendicular directions to maintain statistical similarity. To maintain the same roughness after enlarging the JRC profile, the self-affinity transformation must be obeyed [5,11,17–18]. For example, the joint profile can be stretched in the x-coordinate by a factor of r and in the amplitude by rH : There will be no striking differences between the original and enlarged profiles [19]. Here, the Hurst exponent H is an index to describe the degree of joint roughness [15]. This intrinsic value H which is scale invariant during the self-affinity enlargement, can be obtained by trying H from 0 to 1 to satisfy the self-affinity properties [15,18–19] such as: yðxi þ DxÞ yðxi Þ D
yðxi þ rDxÞ yðxi Þ ; rH
ð1Þ
where, the scaling ratio r is a real number and r > 0: This means that the joint profile enlarged by the r ratio in the x-direction and the heights of yðxÞ by the rH ratio still satisfies the self-affine properties. The H value ranges from 0 to 1, and a rougher profile possesses a smaller H:Ho0:5; the increments of asperity heights have a negative correlation and the profile appears more rugged. This trial-and-error procedure is improved more objectively to find the most possible H by trying H from
Z.Y. Yang et al. / International Journal of Rock Mechanics & Mining Sciences 38 (2001) 745–752
4 to 5 [20]. The fractal dimension (D) of a joint profile is obtained by 2 H: The Fourier transform method has been used successfully in the characterization of joint surfaces [21–22]. The joint profile regarded as a periodic function in the whole length can be divided into several sine or cosine waves, each having a definite wavelength, amplitude and phase. In this study, the JRC profile in 10 cm length was first digitized into about 500 data points by hand. In each profile, the 500 selected points were then used to back-calculate the coefficients of the Fourier series. It demonstrated that the first twenty harmonics summed up to simulate the JRC profile are very similar to the original profile [23]. In this paper, the profile geometry represented by the function of the forty Fourier series is divided into many discrete data points at Dx ¼ 0:5 mm intervals to find the Hurst index. The Hurst indexes of ten JRC profiles obtained in this study are shown in Table 1 and Fig. 1. It was found that the trend of H values does not agree with the common sense in classifying the JRC profiles according to their roughness. The increase in the calculated H values, decrease in the D values, for these ten JRC profiles is abnormal to the original JRC increase. The smaller JRC profile, being a smoother profile, possesses a smaller H; whose small H describing a more rugged fractal curve, is controversial. This result is similar to the observations of Odling [11] and Sakellariou et al. [10] as shown in Table 1 or Fig. 1. A positive correlation between JRC and H occurs before the JRC=14–16 profile (see Fig. 1). However, a negative correlation appears after the profiles with the JRC value greater than 14–16. This indicates the fractal parameter, such as the roughness index H; did answer the roughness property for the
Fig. 1. Correlation between the Hurst index and JRC of 10 standard JRC profiles.
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rougher profiles (with larger JRC), but it might mislead the roughness degree for application to smoother JRC profiles. This could imply that the fractal parameter is better for reflecting the roughness property for rougher profiles than smoother profiles.
3. Interpretation of JRC and H characteristics from the experimental result In fact, the appearance of a natural joint look like the resulted profile from numerous secondary asperities (i.e., small-scale unevenness) superimposed onto some primary asperities (i.e., large-scale undulation). In this paragraph, we attempt to investigate the role of the primary and secondary asperities on the shear behaviors. It is believed that secondary asperity is an important factor of the shear strength at lower stress levels, and at higher levels it is depressed. The JRC=14– 16 profile (Fig. 2(a)) was used as a standard. It was first divided into several sine waves, each having a definite wavelength, amplitude and phase. Two synthetic profiles resulted from superimposing the first five and forty harmonics, respectively. We can see that the second profile in Fig. 2(b) which consists of the first five harmonics (5-order profile) well approximates the apparent surface structure or the primary bumps (i.e., the primary asperities) of the JRC profile (see Fig. 2(a)). The JRC value calculated is 7.1, using the sample interval Dx ¼ 0:2 cm in Tse-Cruden formula [4]. In this profile, the primary asperity (the first-order roughness) mainly controls the entire roughness. In the other hand, the third profile in Fig. 2(c), consisting of the first forty harmonics (40-order profile), shows that several secondary asperities appear on those primary asperity surfaces. That is, there are asperities on an asperity. The entire roughness of this profile is contributed by both the primary and secondary asperities. This profile is visually rougher than the previous profile shown in Fig. 2(b), due to the appearance of the secondary asperity. This is also
Fig. 2. (a–c) The original JRC profile and the synthetic profiles consist of the first five and forty harmonics.
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Fig. 3. The appearance of joint profiles after shearing. (a) Lower stress level. (b) Higher stress level.
approved by the higher JRC (JRC=10.4) measurement, using the same sampling interval (Dx ¼ 0:2 cm). Two joint profiles mold made of polystyrene were cut using an automatic wire-cutting machine [19] yielding 100 discrete point data for each profile in 10 cm. Two mother profiles were reprinted from the polystyrene molds using silicon rubber. The joint samples were then cast from the mother profiles, using a mixture of gypsum and water at a ratio of 1 : 0.9 by weight. The gypsum model has a basic friction angle of 401 and an uniaxial strength of 9.8 Mpa. A series of direct shear tests on these two types of joint profiles were performed under a lower normal stress level ðsn ¼ 0:098Þ and a higher level (sn ¼ 0:49 Mpa). It was found that, at the lower stress level (sn /JCS=1/ 100), the secondary and primary asperities have almost no damage at the end of shearing (see photos in Fig. 3(a)). The shear stress-displacement curve and dilation curves of these two profiles (see Fig. 4(a)) are obviously different. It was found that the 40-order profile (with a rougher geometrical roughness induced by the superimposing of secondary asperities) demonstrates the higher strength and dilation angle than the 5-order profile. This indicates a higher roughness property. The JRC value of 12 back-calculated from the peak shear strength test data (see Table 2), according
to the Barton’s formula [2], for the 40-order profile is greater than the 5-order profile (JRC=7.8). This agrees with the previous visual observation. In addition, the behavior of the 40-order profile has an obvious peak strength and demonstrates brittleness. The 5-order (smoother) profile, however, shows a ductile type. This implies that the contribution of the secondary asperity to the entire joint roughness (result in the shear strength and dilation angle) cannot be neglected. Under a very low stress circumstance, the small-scaled secondary asperity superimposed onto the primary asperity plays a predominant role in the joint roughness. The roughness index, such as the JRC and fractal parameters, should be capable of capturing this secondary asperity effect. The sampling interval to measure the joint roughness must be smaller than the size of the secondary asperity and can sense this existence. Controversially, at the higher stress level (sn /JCS=1/ 20), the secondary asperities on up-slope face were seriously sheared off at the end of shearing, but the primary asperities still had nearly no damage (see Fig. 3(b)). The shear stress-displacement and dilation curves of these two profiles (Fig. 4(b)) are actually very close. The back-calculated JRC which responds to the entire joint roughness is also very close (JRC=11.2 and JRC=11.4, see Table 2). This implies that the secondary
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Fig. 4. The shear behaviors of two joint profiles consisting of the first five and forty harmonics. (a) At lower stress level (sn =0.098 MPa). (b) At higher stress level (sn =0.49 Mpa).
Table 2 Comparison of the peak strengths and dilation angles for joint profiles consisting of the first five and forty harmonics Normal stress (MPa)
0.098 0.49
Shear strength (MPa)
Dilation angle (1)
5-order
40-order
5-order
40-order
5-order
40-order
0.143 0.688
0.208 0.696
10.3 8.0
14.9 7.6
7.8 11.2
12 11.4
asperity superimposed onto the primary asperity is negligible. That is, the shear behaviors under the higher stress level are mainly controlled by the large-scale primary asperity. Thus, under higher stress levels, the measure of roughness can only catch the roughness characteristic of these primary asperities. The sampling interval used to measure the joint roughness can be longer. Moreover, we used various sampling intervals to inspect the averaged up-slope angle of the asperities along the shearing direction for these two profiles. This shows that the asperity angle (i) decreases as the sampling interval increases (see Fig. 5), due to the loss
JRC back-calculated
of measurement on the secondary asperity. The asperity angles of both profiles tend to an identical value (i ¼ 13:51) as the sampling interval reaches 0.8 cm, then shifts between 9.41 and 12.81 due to sampling bias, but are very close to one another. This indicates that the secondary asperity is lost using larger sampling intervals. At very small sampling intervals, it is also found that the averaging asperity angle (i ¼ 27:41) of the 40order profile is much greater than the 5-order profile (i ¼ 14:71), due to the existence of the secondary asperities. This effect of the secondary asperity on the entire geometrical roughness of the 40-order profile displays the higher dilation angle and peak strength at
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very low stress level (Fig. 4(a)). It is interesting to find that, at the low normal stress level (sn /JCS=1/100), the dilation angle (14.91) of shear tests for the 40-order profile is rather close to the averaging angle of primary asperity (i ¼ 14:71) measured in the 5-order profile. This dilation angle is, however, a great departure from the average angle (i ¼ 27:41) measured in the 40-order profile. This implies that the primary asperity (i.e., the macroscopic surface structure), but not the secondary asperity in smaller sizes, controls the overall dilation behavior.
Fig. 5. The decrease in asperity angles of the first five and forty harmonic profiles measured in various ruler scales.
In addition, the JRC and Hurst index (H) of the two profiles are also examined using different sampling intervals, as shown in Fig. 6. It was found that the JRC value for both profiles decreases and smoothens as the sampling interval increases (see Fig. 6(a)), similar to the decreasing manner of the asperity angles (i) shown in Fig. 5. Using a smaller sampling interval, the JRC is efficient for distinguishing the roughness degree of these two profiles. However, as JRC is applied to the 5-order profile (a joint profile only consisted of the large-scale asperity), the JRC measure is insensitive to the sampling intervals. This indicates that the JRC does not response mainly the asperity size. Conversely, as the sampling interval increases, the Hurst index (H) somewhat decreases that reflects a progressively irregular trend for both types of joint profiles (see Fig. 6(b)). It is noticed that the increase in the roughness degree interpreted by H with respect to the sampling intervals is opposite to that by the JRC. That is, using larger sampling intervals, H can distinguish between the difference in these two profiles. For the 5-order profile (the joint profile only consisted of the major asperities), the Hurst index (H) is somewhat more efficient than the JRC at the sampling interval greater than a specified value. In fact, this H parameter did reflect mainly the number of asperity appearances (see later). The 40-order profile, having a smaller H (rougher profile), thus indicates that there are many asperity components appearing in the joint length. Therefore, to capture the different-scaled asperity characteristics of a natural joint surface, combining the JRC and H is a potential approach.
Fig. 6. (a,b) Variations of JRC and H at different sampling intervals.
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4. The characteristics of JRC and fractal index (H) Several tooth-shaped joint profiles were examined to investigate the role of JRC and H (or D). As shown in Fig. 7(a), the three profiles were designed at the same amplitude/wavelength ratio of 0.125 in a profile (i.e., asperity angle=7.1251). The profile numbered with the larger asperity base length was assigned to the joint consisting of primary asperities. The profile numbered is for the profile consisting of secondary asperities. Obviously, this appearance of the three profiles is macroscopically different. However, the calculated JRC value is identical (JRC=12.65), obtained by employing Tse-Cruden’s formula [4] at a 0.2 mm sampling interval. The fractal roughness parameter for H indeed shows a very distinct difference between these profiles (H ¼ 0:305; 0.55, and 0.83). This indicates that the JRC value is inefficient for distinguishing these roughness differences, but H is efficient. On the other hand, Fig. 7(b) shows three profiles with the same wavelength (base length of asperity), but the ratio of amplitude and wavelength is different (amplitude/ wavelength=0.0625, 0.125, 0.25). It was found that the calculated JRC value is much different, but the H is consistent. This means that the H value is incapable of characterizing the differences, but JRC is efficient. The fractal parameter H is actually better for representing the appearing asperity frequency, but not the asperity slope angle. This conclusion is analogous to the interlocking numbers of joint asperities conjectured by Huang et al. [16]. In contrast, the JRC gives a good response to the roughness property in the asperity slope angle. In fact, natural joint results from the secondary asperity (unevenness) superimposed onto the primary asperity (waviness). As shown in Fig. 8, we designed the secondary asperities with high frequency (H ¼ 0:305; JRC=12.651) to superimpose onto a tooth-shaped waviness profile consisting of the primary asperities (JRC=32.2, H ¼ 0:83). The new JRC of the superimposed profile (see profile ) has only a small increase from JRC=32.2 up to JRC=33.256. This indicates that the primary waviness in the profile predominates the entire roughness macroscopically. However, the new fractal value (H) has a remarkable change and is sensitive to this superimposed effect by the secondary asperity. Therefore, it is inadequate to describe the roughness characteristic of the primary asperity (or waviness) in the macroscopic scale using a single JRC expression. One single roughness parameter, such as the JRC or fractal dimension is thus not enough to describe the roughness characteristic of a rock joint. In general terms the roughness of the joint profile can be characterized by a large-scale waviness and a small-scale roughness. During shear displacement, the waviness is too large to
Fig. 7. The JRC and Hurst index (H) for profiles with the same JRC or H value. (a) Profiles with the same JRC (asperity angle=7.1251). (b) Profiles with the same H (tooth base length=0.04 cm).
Fig. 8. The change of JRC by superimposing the secondary asperity onto the primary asperity.
be sheared off and cause dilation. However, the small-scale roughness tends to be damaged during shear displacement unless the joint walls have high strength or the stress level is low. At higher stress levels, the overall shear behavior is mainly dominated by the large-scale asperity, but not the small-scale asperity in the profile geometry. However, at very low stress levels, the contribution of the entire roughness for a profile by the small-scale asperity is sequentially noteworthy. Thus, the primary or secondary asperities (mainly corresponding to the JRC or fractal parameters) play a different role in various stress levels. The fractal parameters such as H and D will better reflect the secondary asperity or roughness in the microscopic scale. To respond to the small-scale asperity property, the fractal index D or H is better than the JRC.
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5. Conclusions The Fourier series function is applied to resolve the original JRC profile. Then, two model joints that consist of the first five and forty harmonics are tested to investigate the role of primary and secondary asperity in the shear behavior. From the experimental observation, at very low stress levels the secondary asperity has a remarkable effect on the joint strength, but not on the dilation. The dilation behavior is mainly controlled by the large-scale primary asperity. A single roughness parameter, such as the JRC or D; is not enough to describe the roughness behavior contributed by the numerous scaled asperities. This study found that the fractal parameter, D or H; is better used to reflect the roughness property for the rougher profiles than smoother ones. In fact, the H (or D) is represented for the frequency of asperity appearance than the asperity size. Thus, the fractal parameter (D or H) better reflects the roughness property contributed by the secondary asperity. In contrast, the JRC produces a good response to the roughness property in the asperity size or slope angle. To capture the actual roughness characteristics of a natural joint surface, contributed by both the primary and secondary asperities, combining the JRC and H (or D) has a positive benefit.
[7]
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Technical Note
The effect of asperity order on the roughness of rock joints Z.Y. Yang*, C.C. Di, K.C. Yen Department of Civil Engineering, Tamkang University, Tamsui, Taipei 25137, Taiwan Accepted 21 May 2001
1. Introduction Joint roughness has an essential influence on the shear behavior of rock joints. Patton [1] recognized that the asperity of a rough joint occurs on many scales. He first categorized asperity into first-order (waviness) and second-order (unevenness) categories. The behavior of rock joints is controlled primarily by the second-order asperity during small displacements and the first-order asperity governs the shearing behavior for large displacements. Barton [2] and Hoek and Bray [3] also stated that at low normal stress levels the second-order asperity (with higher-angle and narrow base length) controls the shearing process. As the normal stress increases, the second-order asperity is sheared off and the first-order asperity (with longer base length and lower-angle) takes over as the controlling factor. In engineering practice, the shear strength criterion proposed by Barton [2] for rock joints is widely adopted. In which, the JRC (joint roughness coefficient) value for a given joint profile can be estimated visibly by comparing it with the ten JRC profiles whose JRC ranges are from 0 to 20. This set of profiles has subsequently been adopted as a standard by the ISRM. Accordingly a large amount of test joint results, several empirical formula connected to the mechanical parameters such as the shear stiffness and joint aperture were related to the JRC. However, in practice it may be difficult to determine the proper JRC number and is highly subjective. To minimize subjectivity, alternate methods have been proposed for JRC estimation. Many researchers have thus attempted to calculate the JRC value from the profile geometry. At present, one commonly adopts Tse and Cruden’s [4] empirical statistical relationship between the JRC and Z2 to *Corresponding author. Tel.: +886-2623-4215; fax: +886-26209747. E-mail address: [email protected] (Z.Y. Yang).
calculate typical JRC values. The parameter Z2 is the root mean square (r.m.s) of the tangents of the slope angles along the profile. Following the introduction of fractal geometry by Mandelbrot [5], numerous researchers have applied the concept of fractal dimensions to rock joints. They tried to interpret the JRC using fractal dimensions (D). They found that a good correlation between the JRC and fractal dimension, and thus the fractal dimension can represent the JRC profile. Generally, large D represents a rough joint profile and the JRC is larger. The value of D for one-dimensional joint profile ranges between 1 and 2. The divider and box-counting methods were often used to obtain the fractal dimension of the JRC profiles. However, the obtained values are insensitive to the ten JRC profiles as listed in Table 1 [6–8]. It is difficult to distinguish the degree of roughness between different JRC profiles. Usually, none of the actual rock joint profiles such as the JRC profiles were found to have the property of strict self-similar, but were instead self-affine [9–12]. Odling [11] explains the result in that the divider method (or box-counting method) is only suitable for a self-similar fractal curve. Thus, it is not appropriate for joint profiles that are self-affine fractals. Such an application has commonly given the fractal dimension estimation as very close to unity. Furthermore, in Table 1, Lee et al. [7] have shown a positive correlation between the JRC and D (see also in [8]). This means that the rougher profile with the higher JRC displays a larger D value. The respective D values seemed to conform to the profile order in the roughness scale. Turk et al. [6] also produced a similar trend, except that the profiles nearby JRC=12–14 were not in a proper order (see Table 1). However, there have been many controversial findings for the JRC and D relationship reported in the fractal characterization literatures of the JRC profiles [11,13]. Olding [11] employed the structure function to obtain the fractal dimensions of ten JRC profiles (see also in Table 1). The structure
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Table 1 The fractal dimension and Hurst index of ten JRC profiles JRC profile
0–2 2–4 4–6 6–8 8–10 10–12 12–14 14–16 16–18 18–20
D value
H value
Lee
Seidel
Turk
Sakellariou
Odling
Odling
This paper
1.000446 1.001687 1.002805 1.003974 1.004413 1.005641 1.007109 1.008055 1.009584 1.013435
1.00009 1.00054 1.00072 1.0014 1.0018 1.004 1.0053 1.0081 1.0096 1.012
1 1.0019 1.0027 1.0049 1.0054 1.0045 1.0077 1.007 1.0104 1.017
1.493 1.519 1.360 1.522 1.347 1.166 1.208 1.118 1.218 1.188
1.50 1.54 F F 1.30 1.17 1.20 1.15 1.25 F
0.5 0.46 F F 0.7 0.83 0.80 0.85 0.75 F
0.365 0.575 0.765 0.66 0.755 0.83 0.89 0.90 0.87 0.885
function is simply the mean square height difference of two points on the profiles at a specified separation. This fractal dimension derived from the structural function can relate to a roughness index (H), the Hurst index [14–15], by D ¼ 2 H: It was found that the fractal dimensions between joints obtained by this structure function are more distinguishable than the divider method. It was surprising to find that the fractal dimension is 1.5 (H ¼ 0:5) for the JRC=0–2 profile and 1.15 (H ¼ 0:85) for the JRC=16–18 profile. The JRC=0–2 profile being almost a straight line has a D value much larger (smaller H) than the other which seems to be rougher. The relationship between the JRC and D shows a negative correlation. The fractal dimensions are not in proper order corresponding to their JRC scale (see also). This strange order also took place at the JRC=12–14 profile (see Table 1). A similar conclusion by Sakellariou et al. [10] using spectral analysis is shown in Table 1. In another expression, Turk et al. [6] used the fractal dimension to estimate the asperity angle of a rough joint profile. They found that although the JRC increases from 0 to 20 with increasing roughness, the estimated roughness angles did not show a similar increase corresponding to the JRC values. We also notice that the order of the fractal dimensions in their paper is strange in the profiles for JRC=10–12 and 14–16 (see also Table 1). Also, Huang et al. [16] suggested that the r.m.s. of a rough joint profile might perhaps be a better indication of joint roughness than the fractal dimension. They conjectured that the fractal dimension describes the amount of interlocking between joint asperities, whereas the r.m.s. variance in height is more closely related to the shear strength of the rock joints. Thus, using the fractal dimension as an invariant scaling parameter to describe the joint profile roughness does not seem warranted, but its use as one of the number of parameters can have positive benefits. In view of the abnormal order in the fractal dimension, we presume that something is implied from
the fractal characteristic of the ten JRC profiles. This could imply that the roughness property of the joints that determined by the fractal dimension should take into account not only the primary asperity, but also the secondary asperity. Thus, the JRC or fractal dimension seems not enough to completely describe the roughness properties of a joint profile.
2. Hurst index of JRC profiles As mentioned, the ten JRC profiles have the selfaffinity property. The profile appearance under a specified elongation in the lengths will look about the same and, by touch, will be perceived to have the same roughness. That is, a self-affine profile must be scaled differently in two perpendicular directions to maintain statistical similarity. To maintain the same roughness after enlarging the JRC profile, the self-affinity transformation must be obeyed [5,11,17–18]. For example, the joint profile can be stretched in the x-coordinate by a factor of r and in the amplitude by rH : There will be no striking differences between the original and enlarged profiles [19]. Here, the Hurst exponent H is an index to describe the degree of joint roughness [15]. This intrinsic value H which is scale invariant during the self-affinity enlargement, can be obtained by trying H from 0 to 1 to satisfy the self-affinity properties [15,18–19] such as: yðxi þ DxÞ yðxi Þ D
yðxi þ rDxÞ yðxi Þ ; rH
ð1Þ
where, the scaling ratio r is a real number and r > 0: This means that the joint profile enlarged by the r ratio in the x-direction and the heights of yðxÞ by the rH ratio still satisfies the self-affine properties. The H value ranges from 0 to 1, and a rougher profile possesses a smaller H:Ho0:5; the increments of asperity heights have a negative correlation and the profile appears more rugged. This trial-and-error procedure is improved more objectively to find the most possible H by trying H from
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4 to 5 [20]. The fractal dimension (D) of a joint profile is obtained by 2 H: The Fourier transform method has been used successfully in the characterization of joint surfaces [21–22]. The joint profile regarded as a periodic function in the whole length can be divided into several sine or cosine waves, each having a definite wavelength, amplitude and phase. In this study, the JRC profile in 10 cm length was first digitized into about 500 data points by hand. In each profile, the 500 selected points were then used to back-calculate the coefficients of the Fourier series. It demonstrated that the first twenty harmonics summed up to simulate the JRC profile are very similar to the original profile [23]. In this paper, the profile geometry represented by the function of the forty Fourier series is divided into many discrete data points at Dx ¼ 0:5 mm intervals to find the Hurst index. The Hurst indexes of ten JRC profiles obtained in this study are shown in Table 1 and Fig. 1. It was found that the trend of H values does not agree with the common sense in classifying the JRC profiles according to their roughness. The increase in the calculated H values, decrease in the D values, for these ten JRC profiles is abnormal to the original JRC increase. The smaller JRC profile, being a smoother profile, possesses a smaller H; whose small H describing a more rugged fractal curve, is controversial. This result is similar to the observations of Odling [11] and Sakellariou et al. [10] as shown in Table 1 or Fig. 1. A positive correlation between JRC and H occurs before the JRC=14–16 profile (see Fig. 1). However, a negative correlation appears after the profiles with the JRC value greater than 14–16. This indicates the fractal parameter, such as the roughness index H; did answer the roughness property for the
Fig. 1. Correlation between the Hurst index and JRC of 10 standard JRC profiles.
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rougher profiles (with larger JRC), but it might mislead the roughness degree for application to smoother JRC profiles. This could imply that the fractal parameter is better for reflecting the roughness property for rougher profiles than smoother profiles.
3. Interpretation of JRC and H characteristics from the experimental result In fact, the appearance of a natural joint look like the resulted profile from numerous secondary asperities (i.e., small-scale unevenness) superimposed onto some primary asperities (i.e., large-scale undulation). In this paragraph, we attempt to investigate the role of the primary and secondary asperities on the shear behaviors. It is believed that secondary asperity is an important factor of the shear strength at lower stress levels, and at higher levels it is depressed. The JRC=14– 16 profile (Fig. 2(a)) was used as a standard. It was first divided into several sine waves, each having a definite wavelength, amplitude and phase. Two synthetic profiles resulted from superimposing the first five and forty harmonics, respectively. We can see that the second profile in Fig. 2(b) which consists of the first five harmonics (5-order profile) well approximates the apparent surface structure or the primary bumps (i.e., the primary asperities) of the JRC profile (see Fig. 2(a)). The JRC value calculated is 7.1, using the sample interval Dx ¼ 0:2 cm in Tse-Cruden formula [4]. In this profile, the primary asperity (the first-order roughness) mainly controls the entire roughness. In the other hand, the third profile in Fig. 2(c), consisting of the first forty harmonics (40-order profile), shows that several secondary asperities appear on those primary asperity surfaces. That is, there are asperities on an asperity. The entire roughness of this profile is contributed by both the primary and secondary asperities. This profile is visually rougher than the previous profile shown in Fig. 2(b), due to the appearance of the secondary asperity. This is also
Fig. 2. (a–c) The original JRC profile and the synthetic profiles consist of the first five and forty harmonics.
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Fig. 3. The appearance of joint profiles after shearing. (a) Lower stress level. (b) Higher stress level.
approved by the higher JRC (JRC=10.4) measurement, using the same sampling interval (Dx ¼ 0:2 cm). Two joint profiles mold made of polystyrene were cut using an automatic wire-cutting machine [19] yielding 100 discrete point data for each profile in 10 cm. Two mother profiles were reprinted from the polystyrene molds using silicon rubber. The joint samples were then cast from the mother profiles, using a mixture of gypsum and water at a ratio of 1 : 0.9 by weight. The gypsum model has a basic friction angle of 401 and an uniaxial strength of 9.8 Mpa. A series of direct shear tests on these two types of joint profiles were performed under a lower normal stress level ðsn ¼ 0:098Þ and a higher level (sn ¼ 0:49 Mpa). It was found that, at the lower stress level (sn /JCS=1/ 100), the secondary and primary asperities have almost no damage at the end of shearing (see photos in Fig. 3(a)). The shear stress-displacement curve and dilation curves of these two profiles (see Fig. 4(a)) are obviously different. It was found that the 40-order profile (with a rougher geometrical roughness induced by the superimposing of secondary asperities) demonstrates the higher strength and dilation angle than the 5-order profile. This indicates a higher roughness property. The JRC value of 12 back-calculated from the peak shear strength test data (see Table 2), according
to the Barton’s formula [2], for the 40-order profile is greater than the 5-order profile (JRC=7.8). This agrees with the previous visual observation. In addition, the behavior of the 40-order profile has an obvious peak strength and demonstrates brittleness. The 5-order (smoother) profile, however, shows a ductile type. This implies that the contribution of the secondary asperity to the entire joint roughness (result in the shear strength and dilation angle) cannot be neglected. Under a very low stress circumstance, the small-scaled secondary asperity superimposed onto the primary asperity plays a predominant role in the joint roughness. The roughness index, such as the JRC and fractal parameters, should be capable of capturing this secondary asperity effect. The sampling interval to measure the joint roughness must be smaller than the size of the secondary asperity and can sense this existence. Controversially, at the higher stress level (sn /JCS=1/ 20), the secondary asperities on up-slope face were seriously sheared off at the end of shearing, but the primary asperities still had nearly no damage (see Fig. 3(b)). The shear stress-displacement and dilation curves of these two profiles (Fig. 4(b)) are actually very close. The back-calculated JRC which responds to the entire joint roughness is also very close (JRC=11.2 and JRC=11.4, see Table 2). This implies that the secondary
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Fig. 4. The shear behaviors of two joint profiles consisting of the first five and forty harmonics. (a) At lower stress level (sn =0.098 MPa). (b) At higher stress level (sn =0.49 Mpa).
Table 2 Comparison of the peak strengths and dilation angles for joint profiles consisting of the first five and forty harmonics Normal stress (MPa)
0.098 0.49
Shear strength (MPa)
Dilation angle (1)
5-order
40-order
5-order
40-order
5-order
40-order
0.143 0.688
0.208 0.696
10.3 8.0
14.9 7.6
7.8 11.2
12 11.4
asperity superimposed onto the primary asperity is negligible. That is, the shear behaviors under the higher stress level are mainly controlled by the large-scale primary asperity. Thus, under higher stress levels, the measure of roughness can only catch the roughness characteristic of these primary asperities. The sampling interval used to measure the joint roughness can be longer. Moreover, we used various sampling intervals to inspect the averaged up-slope angle of the asperities along the shearing direction for these two profiles. This shows that the asperity angle (i) decreases as the sampling interval increases (see Fig. 5), due to the loss
JRC back-calculated
of measurement on the secondary asperity. The asperity angles of both profiles tend to an identical value (i ¼ 13:51) as the sampling interval reaches 0.8 cm, then shifts between 9.41 and 12.81 due to sampling bias, but are very close to one another. This indicates that the secondary asperity is lost using larger sampling intervals. At very small sampling intervals, it is also found that the averaging asperity angle (i ¼ 27:41) of the 40order profile is much greater than the 5-order profile (i ¼ 14:71), due to the existence of the secondary asperities. This effect of the secondary asperity on the entire geometrical roughness of the 40-order profile displays the higher dilation angle and peak strength at
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very low stress level (Fig. 4(a)). It is interesting to find that, at the low normal stress level (sn /JCS=1/100), the dilation angle (14.91) of shear tests for the 40-order profile is rather close to the averaging angle of primary asperity (i ¼ 14:71) measured in the 5-order profile. This dilation angle is, however, a great departure from the average angle (i ¼ 27:41) measured in the 40-order profile. This implies that the primary asperity (i.e., the macroscopic surface structure), but not the secondary asperity in smaller sizes, controls the overall dilation behavior.
Fig. 5. The decrease in asperity angles of the first five and forty harmonic profiles measured in various ruler scales.
In addition, the JRC and Hurst index (H) of the two profiles are also examined using different sampling intervals, as shown in Fig. 6. It was found that the JRC value for both profiles decreases and smoothens as the sampling interval increases (see Fig. 6(a)), similar to the decreasing manner of the asperity angles (i) shown in Fig. 5. Using a smaller sampling interval, the JRC is efficient for distinguishing the roughness degree of these two profiles. However, as JRC is applied to the 5-order profile (a joint profile only consisted of the large-scale asperity), the JRC measure is insensitive to the sampling intervals. This indicates that the JRC does not response mainly the asperity size. Conversely, as the sampling interval increases, the Hurst index (H) somewhat decreases that reflects a progressively irregular trend for both types of joint profiles (see Fig. 6(b)). It is noticed that the increase in the roughness degree interpreted by H with respect to the sampling intervals is opposite to that by the JRC. That is, using larger sampling intervals, H can distinguish between the difference in these two profiles. For the 5-order profile (the joint profile only consisted of the major asperities), the Hurst index (H) is somewhat more efficient than the JRC at the sampling interval greater than a specified value. In fact, this H parameter did reflect mainly the number of asperity appearances (see later). The 40-order profile, having a smaller H (rougher profile), thus indicates that there are many asperity components appearing in the joint length. Therefore, to capture the different-scaled asperity characteristics of a natural joint surface, combining the JRC and H is a potential approach.
Fig. 6. (a,b) Variations of JRC and H at different sampling intervals.
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4. The characteristics of JRC and fractal index (H) Several tooth-shaped joint profiles were examined to investigate the role of JRC and H (or D). As shown in Fig. 7(a), the three profiles were designed at the same amplitude/wavelength ratio of 0.125 in a profile (i.e., asperity angle=7.1251). The profile numbered with the larger asperity base length was assigned to the joint consisting of primary asperities. The profile numbered is for the profile consisting of secondary asperities. Obviously, this appearance of the three profiles is macroscopically different. However, the calculated JRC value is identical (JRC=12.65), obtained by employing Tse-Cruden’s formula [4] at a 0.2 mm sampling interval. The fractal roughness parameter for H indeed shows a very distinct difference between these profiles (H ¼ 0:305; 0.55, and 0.83). This indicates that the JRC value is inefficient for distinguishing these roughness differences, but H is efficient. On the other hand, Fig. 7(b) shows three profiles with the same wavelength (base length of asperity), but the ratio of amplitude and wavelength is different (amplitude/ wavelength=0.0625, 0.125, 0.25). It was found that the calculated JRC value is much different, but the H is consistent. This means that the H value is incapable of characterizing the differences, but JRC is efficient. The fractal parameter H is actually better for representing the appearing asperity frequency, but not the asperity slope angle. This conclusion is analogous to the interlocking numbers of joint asperities conjectured by Huang et al. [16]. In contrast, the JRC gives a good response to the roughness property in the asperity slope angle. In fact, natural joint results from the secondary asperity (unevenness) superimposed onto the primary asperity (waviness). As shown in Fig. 8, we designed the secondary asperities with high frequency (H ¼ 0:305; JRC=12.651) to superimpose onto a tooth-shaped waviness profile consisting of the primary asperities (JRC=32.2, H ¼ 0:83). The new JRC of the superimposed profile (see profile ) has only a small increase from JRC=32.2 up to JRC=33.256. This indicates that the primary waviness in the profile predominates the entire roughness macroscopically. However, the new fractal value (H) has a remarkable change and is sensitive to this superimposed effect by the secondary asperity. Therefore, it is inadequate to describe the roughness characteristic of the primary asperity (or waviness) in the macroscopic scale using a single JRC expression. One single roughness parameter, such as the JRC or fractal dimension is thus not enough to describe the roughness characteristic of a rock joint. In general terms the roughness of the joint profile can be characterized by a large-scale waviness and a small-scale roughness. During shear displacement, the waviness is too large to
Fig. 7. The JRC and Hurst index (H) for profiles with the same JRC or H value. (a) Profiles with the same JRC (asperity angle=7.1251). (b) Profiles with the same H (tooth base length=0.04 cm).
Fig. 8. The change of JRC by superimposing the secondary asperity onto the primary asperity.
be sheared off and cause dilation. However, the small-scale roughness tends to be damaged during shear displacement unless the joint walls have high strength or the stress level is low. At higher stress levels, the overall shear behavior is mainly dominated by the large-scale asperity, but not the small-scale asperity in the profile geometry. However, at very low stress levels, the contribution of the entire roughness for a profile by the small-scale asperity is sequentially noteworthy. Thus, the primary or secondary asperities (mainly corresponding to the JRC or fractal parameters) play a different role in various stress levels. The fractal parameters such as H and D will better reflect the secondary asperity or roughness in the microscopic scale. To respond to the small-scale asperity property, the fractal index D or H is better than the JRC.
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5. Conclusions The Fourier series function is applied to resolve the original JRC profile. Then, two model joints that consist of the first five and forty harmonics are tested to investigate the role of primary and secondary asperity in the shear behavior. From the experimental observation, at very low stress levels the secondary asperity has a remarkable effect on the joint strength, but not on the dilation. The dilation behavior is mainly controlled by the large-scale primary asperity. A single roughness parameter, such as the JRC or D; is not enough to describe the roughness behavior contributed by the numerous scaled asperities. This study found that the fractal parameter, D or H; is better used to reflect the roughness property for the rougher profiles than smoother ones. In fact, the H (or D) is represented for the frequency of asperity appearance than the asperity size. Thus, the fractal parameter (D or H) better reflects the roughness property contributed by the secondary asperity. In contrast, the JRC produces a good response to the roughness property in the asperity size or slope angle. To capture the actual roughness characteristics of a natural joint surface, contributed by both the primary and secondary asperities, combining the JRC and H (or D) has a positive benefit.
[7]
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[11]
[12]
[13]
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[17]
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