International Journal of Rock Mechanics and Mining Sciences
Technical NoteThe effect of asperity order on the roughness of rock joints
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 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], [7], [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], [10], [11], [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 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=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=0.5) for the JRC=0–2 profile and 1.15 (H=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.
Section snippets
Hurst index of JRC profiles
As mentioned, the ten JRC profiles have the self-affinity 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
Interpretation of JRC and 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
The characteristics of JRC and fractal index ()
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=7.125°). The profile numbered
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
References (23)
Review of a new shear strength criterion for rock joints
Eng Geol,
(1973)- et al.
Estimating joint roughness coefficients
Int J Rock Mech Min Sci Geomech Abstr
(1979) - et al.
The fractal dimension as a measure of the roughness of rock discontinuity profiles
Int J Rock Mech Min Sci Abstr
(1990) - et al.
On the fractal character of rock surfaces
Int J Rock Mech Min Sci Geomech Abstr
(1991) - et al.
An index for describing the anisotropy of joint surfaces
Int J Rock Mech Min Sci
(1997) - et al.
Applicability of fractal characterization and modelling to rock joint profiles
Int J Rock Mech Min Sci Geomech Abstr
(1992) - et al.
Analysis and synthesis of surface profile using Fourier series
Int J Machine Tool Design Research
(1977) - Patton FD. Multiple model of shear failure in rock. Proceedings of the First Congress of ISRM, Lisbon, 1966. p....
- et al.
Rock slope engineering
(1981) The fractal geometry of nature
(1983)
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