Direct fractal measurement of fracture surfaces

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Abstract

To overcome the difficulties in estimation of fractal dimension for fracture surfaces, a new method of fractal measurement—the projective covering method (PCM) is proposed in this paper. Based on the measurements using a laser scanner, the fractal dimension Ds ∈ [2, 3) of fracture surface is directly estimated. The research results agree with the theory of fractal geometry and measurement data.

Introduction

For many years, fracture surfaces have been described by statistical parameters following the metrology used in tribology and contact mechanics (Johnson, 1985) . The parameters can be classified into three categories according to the type of characteristic that they measure:

  • 1.

    Amplitude parameters, such as centerline average value, mean square value, root mean square (RMS) value, mean square of the first derivative, RMS of the first derivative (Z2) , RMS of the second derivative (Z3) , percentage excess of distance (Z4) .

  • 2.

    Spacing parameters, such as autocorrelation function, spectral density function, structure function (SF) , roughness profile index (Rp)

  • 3.

    Hybrid parameters, micro-average i angle etc.

These parameters are not only quite complicated, but also suffer from scale effect, i.e., the estimated values of roughness depend on the length of sample, the digitizing intervals and the resolution of instrument. Based on extensive experiments, Barton and Choubey (1977) proposed a conceptual model to quantify the surface roughness of rock joints. Accordingly, the roughness is classified into ten groups, and the joint roughness coefficients (JRC) ranges from 0–20. This model has been recommended for years by ISRM and adopted for a good period of time in the practice of rock engineering.

Since Mandelbrot (Mandelbrot, 1967, Mandelbrot, 1983) introduced fractal geometry, many investigations have tried to interpret JRC by fractal dimension (Lee et al., 1990; Mearz and Franklin, 1990; Turk et al., 1990; Muralha, 1992; Wakabayashi and Fukushige, 1992; Xie, 1993, Xie, 1996; Xie and Pariseau, 1994) . Fractal theory describes an object with irregular shape, or a physical quantity or natural phenomenon with irregular distribution in a quantitative manner. The property of fractal geometry can be mathematically expressed by the concept of self-similarity and self-affinity, which suggests that when the shape of an object is magnified more and finer structure can be recognized. Fractal dimension is scale-invariant providing geometric structure at all scales.

In fact, natural fracture surfaces rarely show self-similar fractal property (Xie et al., 1996, Xie et al., 1997a) . Different definitions of fractal dimension, fractal measurement techniques and scale parameters may produce different values of fractal dimension even for the same fracture surface. Perhaps, it is the reason that many controversial findings have been reported in recent literatures of fractal characterization of Bartons standard JRC profiles (Miller et al., 1990; Odling, 1994; Outer et al., 1995) . A relation between roughness and the fractal dimensions is not straightforward and cannot be estimated without conditions for sampling parameters, resolution of instrument and measurement methods. Any conclusion on the fact that fractals do or do not exist in rough surfaces should be taken with care and the fractal characterization of fracture surface as fractal regime is at least very doubtful (Outer et al., 1995) .

The most critical problem, however, is that a real fracture usually extends in a spatial plane. In general, it is very difficult to make a direct measurement for a rough surface. Most of the fractal characterization of a rough surface, however, had to employ indirect methods, such as slit island (SI) , spectrum, and variogram to measure a sectional profile. Fractal dimension measured by these methods ranges D ∈ [1, 2) . Mandelbrot, 1983 suggested that fractal dimension of a topographic surface can be obtained by adding 1.0 to the fractal dimension from a single profile of that surface. Investigation (Wang et al., 1996; Xie et al., 1997b) shows, however, that fractal dimensions vary from one sectional profile to another and also they differ in different directions over the fracture surface. Since the anisotropy and heterogeneity of fracture surface structure, fractal measurement based on profiles is questionable as follows: (1) whether a rough surface could be simulated by a profile; (2) which sectional profile and along which direction of the surface could be warrentedly consulted.

To find out a solution, a new fractal measurement method—Projective Covering Method (PCM) — is proposed for direct estimation of real fractal dimension Ds ∈ [2, 3) for a fracture surface. The primary results promote the validity of PCM as applied for description of roughness of rock fracture surfaces.

Section snippets

The projective covering method

As is well known, covering method is one of the most common methods for fractal measurement. It is suitable not only for simple fractals but also for complex fractals (Falconer, 1990: Feder, 1988) . However, it appears impossible using such a method to cover a fractal surface in a direct manner. The real fractal dimension Ds ∈ [2. 3) have been replaced by using approximate fractal dimensions 1 < D < 2 which is obtained from the sectional profile measurement. In the present work, we propose a

Direct measurement of fractal dimensions of fracture surfaces

In order to make a direct measurement of fractal dimensions of fracture surfaces, the measurement technique should be taken into account in the first place. To date, the techniques developed for measuring of rough surface can be classified into mechanical and optical ones.

To avoid damages and errors to the measured surfaces caused by scratches of mechanical probe sliding along the surfaces, the laser scanner (Kwasniewski and Wang, 1993) , a non-contact optical instrument, is employed in the

Comparison with fractal measurement of profiles

To verify the projective covering method, for same fracture surfaces, the fractal dimensions along individual profiles in x- and y-directions, respectively, are measured by the divider method. Fractal dimension D for a profile is calculated following Eq. (4).Fractal dimensions measured in this way has the value D ∈ [1,2) for a single profile. In order to compare fractal dimension of a fracture surface measured by PCM with that measured from profiles, let us elaborate the dimension formulae of

Conclusions

In this paper, a new measurement method—the projective covering method is proposed which makes it possible to cover two-dimensional fractal objects for direct estimation of real fractal dimensions Ds ∈ [2, 3) of fracture surfaces. The results agree well with the theory of fractal geometry.Chhabra and Jensen, 1989

Acknowledgements

The presented work is supported by the National Distinguished Youth Science Foundation of China, the Trans-Century Programme for the Telents by the State Education Commission and China Postdoctoral Research Foundation. Dr M. A. Kwaśniewski has kindly provided help and facilities for the measurement.

References (24)

  • Barton, N., Choubey, V., 1977. The shear strength of rock joints in theory and practice. Rock Mech. 10 (2) ,...
  • Chhabra, A., Jensen, R. V., 1989. Direct dimension of the f (α) singularity spectrum. Phys. Rev. Lett. 62 (12) ,...
  • Falconer, K., 1990. Fractal Geometry—Mathematical Foundations and Applications. John Wiley and Sons Ltd, Chichester,...
  • Feder, J., 1988. Fractals. Plenum Press, New York, pp....
  • Johnson, K.L., 1985. Contact Mechanics. Cambridge University Press, pp....
  • Kwaśniewski, M.A., Wang, J.A., 1993. Application of laser profilometry and fractal analysis to measurement and...
  • Lee, Y.H., Carr, J.R., Barr, D.J., Haas, C.J., 1990. The fractal dimension as a measure of the roughness of rock...
  • Maerz, N.H., Franklin, J.A., 1990. Roughness scale effect and fractal dimensions. In: Pinto da Cunha, A. (Ed.) , Scale...
  • Mandelbrot, B.B., 1967. How long is the coast line of Britain? Statistical self-similarity and fractal dimensions....
  • Mandelbrot, B.B., 1983. The fractal Geometry of Nature. W.H. Freeman, San Francisco, pp....
  • Miller, S.M., McWilliams, P.C., Kerkering J.C., 1990. Ambiguities in estimation of rock fracture surfaces. In:...
  • Muralha, J., 1992. Fractal dimension of joint roughness surface. Paper submitted for the ISRM Symposium. Fractured and...
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