Modeling of a fracturing model in Excel

This is taken from Turcotte's book on fractals in the geoscience. The simple rule is that you start with a cube and break it into 8 blocks. Of those 8, 2 diagonal blocks go on to survive and stay intact, while the remaining 6 fracture again to each create 8 more blocks. This process is repeated again and again. The fracture geometry is fractal. A plot of particle size versus the number of particles with that size shows a nice log-log linear relationship. A similar relationship is found in naturally fragmented material (see Turcotte.)

N = number of reiterations of cube breakage block side length # of blocks that go on to fracture again # of blocks that survive with block side length log of block size log of # of surviving blocks
1 0.5 6 2 -0.30103 0.30103
2 0.25 36 12 -0.60206 1.07918125
3 0.125 216 72 -0.90309 1.8573325
4 0.0625 1296 432 -1.20412 2.63548375
5 0.03125 7776 2592 -1.50515 3.413635
6 0.015625 46656 15552 -1.80618 4.19178625
7 0.0078125 279936 93312 -2.10721 4.9699375
8 0.00390625 1679616 559872 -2.40824 5.74808875
9 0.001953125 10077696 3359232 -2.70927 6.52624
10 0.000976563 60466176 20155392 -3.0103 7.30439125
11 0.000488281 362797056 120932352 -3.31133 8.0825425
12 0.000244141 2176782336 725594112 -3.6123599 8.86069375
13 0.00012207 13060694016 4353564672 -3.9133899 9.638845
14 6.10352E-05 78364164096 2.6121E+10 -4.2144199 10.4169963
15 3.05176E-05 4.70185E+11 1.5673E+11 -4.5154499 11.1951475
16 1.52588E-05 2.82111E+12 9.4037E+11 -4.8164799 11.9732988
17 7.62939E-06 1.69267E+13 5.6422E+12 -5.1175099 12.75145
18 3.8147E-06 1.0156E+14 3.3853E+13 -5.4185399 13.5296013

ChartObject Log-log plot of block size vs. # of clasts with that size

Is this how materials actually fracture?


Copyright by Harmon D. Maher Jr.. This material may be used for non-profit educational purposes if proper attribution is given. Otherwise please contact Harmon D. Maher Jr.


Last Updated on 2/5/02