Int. J, Rock .%lech. Min. Sci. & Geomech. Abstr. Vol. 27. No. 3. pp. 219-221. 1990 0148-9062 90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright (~ 1990 Pergamon Press pie Technical Note Scale Invariant Behaviour and Fragmented Rock M. M. POULTONt N. MOJTABAI~ I. W. FARMER§ of Massive INTRODUCTION One of the major requirements in rock mechanics is an accurate description of the discontinuity structure of rock masses and of the way in which this affects rock fragmentation processes, particularly during excavation (U.S. National Committee for Rock Mechanics [I]). Although attempts have been made to utilize empirically derived rock classification systems to assess cuttability, rippability and blast design (Sandbak [2]; Smith [3]; Singh et al. [4]) these have not been widely used. More rigorous studies, based on numerical models of flawed rock (for instance Kuszmaul [5]) are likely to be many years in development. One technique which is worth further study is the description of scale invari- ance of both discontinuity structures and fragmented rock. SCALE INVARIANCE Scale invariance requires a power law relation between a number and a size. Such a relation is by definition a fractal. There are numerous examples of scale invariant processes in nature which can be quantified by the fractal concept (Mandelbrot [6]). An example which has direct application to rock mechanics has been illustrated by Turcotte [7]. He showed that fragmented objects, including rocks, coal, soils and meteoric fragments which have been broken in various ways, were all characterized by a size-frequency relation of the form: N(r)~tr - o ( 1 ) where N(r) is the number of fragments with a characteristic linear dimension greater than r and the relation is a fractal defined by the fractal dimension D. Fractal dimensions for fragmented geologic materials, summarized by Turcotte [7] ranged from 1.44 to 3.54. ")'Department of Mining and Geological Engineering. University of Arizona. Tucson. AZ 85721. U.S.A. SNew Mexico Institute of Technology. Socorro. NM 87801, U.S.A. §fan Farmer Associates.Newcastleupon Tyne NE4 7AN, U.K. It has been conventional in the past to describe fragmentation from blast products by a negative expo- nential dependence: N(r) N = exp- (r/ro)", (2) where N is the total number of fragments and r0 is the mean fragment size. This is essentially the same as equation (I). A similar relation may be obtained through the Weibull distribution: N(r) N = l-exp-(r/ro)", (3) which again reduces to a form similar to equation (1). PARTICLE SIZE AND JOINT FREQUENCY DATA Plots from Turcotte [7] and data collected by Mojtabai [8] clearly show the number vs size relation of equation (I). Particle size distribution data from Mojtabai [8] for quartz monzonite broken by flyer plate impact indicate increasing fractal dimensions with increasing energy and pulse duration of impact. Data plotted in Fig. I shows a range from D = 1.8 to 3.03 for in-pressure ranges of 1.6 to 4.7 GPa and pulse durations of I-6/~sec. The advantage of fractal charac- terization of particle size data is that if the fractal dimension is constant for a given lithology in a given area then only one size interval needs to be measured to determine the full size range. The data show quite clearly a correlation between energy input and pressure and pulse duration and fractal dimension. Thus higher fractal dimensions are associated with higher pressure pulses of longer duration. Londe [9], Priest and Hudson [10], Rocha [I I] and Snow [12] amongst others have noted the negative exponential distribution of discontinuity frequencies in various rocks. Fractal dimensions can be defined for fractal spacing and length data in the manner discussed above. Fractal dimensions computed from various sources are listed in Table I. Whilst it is difficult to draw firm conclusions it is possible, for rocks formed under similar tectonic or depositional regimes, to correlate higher fractal dimensions with higher strengths. 219