Journal of Mining Science, VoL 32, No. 2, 1996
D I S I N T E G R A T I O N OF R O C K
FRACTAL MODELS OF THE DEVELOPMENT OF CLEAVAGE FRACTURES IN ROCK
D. V. Alekseev, P. V. Egorov, and A. G. Pimonov
UDC 622.235(088.8):519.21
It has been recently established that fractures which form in the course of the disintegration of different types of materials possess fractal dimensions [1-3]. Moreover, the fractal dimension of the process of disintegration manifests itself also in the time-dependent relations observed in the kinetics of the accumulation of microscopic fractures [4, 5]. Cleavage fractures develop and grow because of processes involved in the formation and diffusion of elementary structural defects (point defects, pores, etc.) in the anisotropic field of mechanical stresses [6, 7]. For this reason, it is of paramount importance to learn how the characteristics of fractal fractures depend on the anisotropy of the processes involved in the generation and diffusion of elementary defects. In the present article an attempt is undertaken to study the influence of the anisotropy of the formation and diffusion of elementary structural defects on the fractal dimension of cleavage fractures. The study is based on anisotropic random-walk models. The growth of cleavage fractures is modeled in a two-dimensional rectangular lattice (raster) having a dimension of a × b (7 < a < 65, 7 < b < 41) nodes. An inoculation defect measuring 2 × 2 is placed in the center of the raster. The process in which a fracture grows is the result of the attachment of elementary defects, each occupying a node of the raster adjacent to any occupied node, to the inoculating defect. The emergence of a microscopic defect adjacent to a growing fracture occurs by means of a random-walk process experienced by the elementary defect which, in fact, may travel in any one of four directions towards one of the four adjacent nodes; each of the nodes may be selected with equal probability. In the present study, four different random-walk processes are constructed, and four different characteristic forms of cleavage fractures accordingly created. 1. A wandering elementary defect is placed at one of two diagonally opposite nodes of the raster (upper left and lower right) both of which, moreover, have an equal probability of being selected; this node becomes the initial point of the motion. The defect is permitted to move with a step which is three times as great toward the enveloping growing rectangular fracture (the sides of the rectangle are parallel to the sides of the raster) if the microscopic defect is at least three nodes away from the rectangle. The resultant cleavage fracture which is obtained for this type of random-walk process (it possesses a typical prolate form) is presented in Fig. la. 2. The defect moves with a step three iJmes as great not only toward the enveloping rectangle, but also along any one of its sides if this will not cause it to go outside the rectangle. A typical resultant cleavage fracture for this model is presented in Fig. lb. 3. At the start of motion, an elementary defect is situated at one of four randomly selected corner nodes of the raster, each having the same probability. A resultant cleavage fracture for this model is presented in Fig. lc. 4. The fourth method differs from the preceding three methods by the fact that the microscopic defect, when at least three nodes away from an enveloping growing fracture, is permitted to move with a step three times as great both toward the rectangle as well as along any one of its sides. An example of a model cleavage fracture obtained in this case is presented in
Fig. ld. State Engineering Institute, Kemerovo. Translated from Fiziko-tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 2, pp. 48-53, March-April, 1996. Original article submitted June 20, 1995.
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Fig. 1. Examples of cleavage fractures obtained by means of different growth models [a) first model; b) second model; c) third model; and d) fourth model]. Dissection of the raster along one direction represents one condition under which a cleavage fracture will cease to grow in all four cases. In Fig. 1 (b and d) the lattice is dissected along the vertical and along the horizontal, respectively. Another condition occurs when the growing fracture occupies one of the possible spots from which the elementary defect starts to travel. In Fig. la, it is the upper left corner of the raster which is occupied. It is clear from the figures that all the cleavage fractures which are obtained possess a structure that is typical of fractals. The growth models of cleavage fractures which we have been describing have been realized in the form of a set of subroutines written in the MSX standard Basic computer language using elements of computer graphics. Sequences of pseudorandom numbers that are needed for the statistical modeling process are specified by means of the built-in RND sensor. The initial database for the RND sensor is associated with the start of the modeling period, as determined on the basis of the TIME function. To obtain the statistical data we conducted 10 model experiments on each of the four cleavage fracture growth models. The modeling process was realized on a square raster measuring 30 × 30. The results of the modeling process (number of microscopic defects which have merged together into the mainline cleavage fracture) are presented in Table 1. The fractal dimension of the resulting model cleavage fractures was determined by means of a formula for the "mass of a fractal cluster" [8]:
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(1)
where b is the characteristic dimension of the object and M(b) is its mass (in our case, the number of microscopic fractures in the mainliue cleavage fracutre). The parameter k occurring in (I) and the fractal dimension D were determined by means of the method of least squares using the following linear dependence: In IM(b)] -. In(k) + Din(b).
(2)
The fractal dimension D of the cleavage fractures and the coefficient k were calculated by means of the following formulas: .n ~ ~n(b/)ln[M(b/)] - ~ . in(b/)~ InIM(b/)] D
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i=1
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TABLE 1. Number of Microscopic Defects in Model Cleavage Fractures Experiment 1 2 3 4 5 6 7 8 9 10 Mean value...
qumber of microscopic defects for different cleavage fracture growth models i
2
3
4
84 74 76 73 74 64 89 73 85 75
135 128 141 163 168 172 143 169 131 131
143 130 96 136 115 145 140 138 109"
142 148 110 129 129 132 153 117 140 151
76,70
148,10
129,50
135,10
7,30
17,82
16,94
14,35
14~
Standard deviation...
TABLE 2. Estimated Fractal Dimension D of Model Cleavage Fractures Experiment
6 7 8 9 10
Fractal dimension for different models of cleavage Racmre growth i 2 3 4
1,01961 1,19736
1,30969 1,34595 1,39302 1,47440 1,36804 1,48010 1,30078 1,40610 1,25280 1,36090
1,28338 1,15764 1,17755 1,38058 1,27091 1 ,&S409 1,39288 1,36258 1,45797 1,16592
1,29340 1,30136 1,19365 1z30817 1,28401 1,26574 1,45861 1,26086 1,34031 1,38917
1,06861
1,36918
1,30035
1,30953
0,07154
0,07278
0,10608
0,07319
1,06355 0,93082 1,12434 1,12761 1,07229 1,03961 1,04350 1,06744
Mean value...
Standard deviation...
The process of calculating the estimates (3) and (4) was implemented in the form of separate Basic programs together with cleavage fracture growth models. The results of the calculations are presented in Tables 2 and 3. It is clear from Table 2 that the models of fracture growth which we have been considering may be clearly divided into two types. The fractal dimension of model 1, which is the more anisotropic of the two models, is closest to 1 and differs from the mean fractal dimensions of models 2-4 by a quantity that exceeds the sum of the standard deviations. Consequently, the anisotropy of the processes involved in the formation and diffusion of elementary structural defects influences the fractal dimension of the resulting cleavage fractures at least in the two-dimensional case. Therefore, further studies of the influence of the two-dimensional anisotropy of the development of elementary structural defects would be extremely valuable, for example, using occupation percolation-type models, as well as three-dimensional models in order to draw comparisons with the fra'ctal dimensions of actually observed fractures.
CONCLUSIONS 1. The development of cleavage fractures is numerically modeled on the basis of random-walk models in a rectangular lattice.
121
TABLE 3. Estimated Value of the Coefficient k Experiment I 2 3 4 5 6 7 8 9 10
Mean value...
Coefficient k for different models Of cleavage fracture growth 1
2
3
4
2,42038 3,26266 1,65819 1,70840 2,17791 1,90282 2,50712 2,00917 2,56605 1,42501
1,68360 1,49876 1,38634 1,19406 1,77000 1,24879 1,90284 1,43259 2,01672 1,49000
2,20503 2,95643 1,91776 1,38698 1,68246 1,67674 1,50039 1,75052 1,15174 2,62898
2,03046 1,85115 2,13404 1,68865 1,86306 2,18048 1,27735 1,81365 1,55825 1~43255
2,16377
1,56237
1,88570
1,78298
0,54304
0,27310
0,56185
0,29672
Standard deviation...
2. Estimates of the fractal dimension of the resulting model cleavage fractures are obtained in the case of four models that differ in terms of the degree of anisotropy of the formation and wandering of the elementary defects. 3. tt is established that if the process through which elementary defects are formed is highly anisotropic, t~s will lead to a decrease in the fractal dimension of the resulting two-dimensional fracture.
REFERENCES 1.
2. 3. 4. 5. 6. 7. 8.
122
Fractals in Physics [-Russian translation], Mir, Moscow (1989). F. M. Borodich, "Energy of disintegration of a fractal fracture propagating in concrete or rock," Dokl. Akad. Nauk SSSR, 325, No. 6 (1992). A. S. Balankin and A. L. Bugrimov, "Fractal dimension of fractures formed in brittle failure of model lattices and solids," Pis'ma Zh. Tekh. Fiz., 17, No. 17 (1991). D. V. Alekseev and P. V. Egorov, "The persistence of the accumulation of fractures in loading of rock and the concentration criterion of failure," Dokl. Akad. Nauk SSSR, 333, No. 6 (1993). D. V. Alekseev, P. V. Egorov, and A. G. Pimonov, "On the kinetics of the accumulation of fractures and the concentration criterion of failure," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 1 (1994). P. G. Cheremskii, V. V. Slezkov, and V. I. Betekhtin, Pores in Solid Bodies [in Russian], l~nergoatomizdat, Moscow (1990). Disorder and Fracture: Proc. NATO Adv. Studi Inst. Corgese, May 29-June 9, 1989, New York-London (1990). E. Feder, Fractals ~ussian translation], Mir, Moscow (1991).
