Multifractal Model for Soil Aggregate Fragmentation
1993
Perfect, E. | Kay, B. D. | Rasiah, V.
Dry aggregate size and strength distributions are important soil structural characteristics. We present a theoretical model based on multifractals for predicting one characteristic from the other. For a specified stress, σ, the strength of dry aggregates of normalized equivalent cubic length x* was expressed as a probability of failure, {P(x*)}σ. A method was developed for calculating {P(x*)}σ from tensile strength data. At intermediate levels of stress (0.3 ≤ σ ≤ 0.9 MPa), {P(x*)}σ decreased with decreasing x*. A Pareto distribution was used to model this scale dependency. The distribution's parameters, q and r, determine the probability of failure of the largest aggregate and the rate of change in scale dependency, respectively. The r increased and the q decreased logarithmically with increasing σ. The fractal dimension, D, was used to characterize the number-size distribution of dry aggregates after fragmentation. For mass-conserving cubic fragmentation, D is related to {P(x*)}σ by the multifractal spectrum, D ≅ log {8(2′ − qx*⁻ʳ)}/log {2}. Previously published dry-sieving data were reanalyzed. The number-size distribution determined by visual counting gave a spectrum of fractal dimensions as predicted by the theory. Values of D ranged from 2.53 at x* = 4.7 × 10⁻² to 3.46 at x* = 7.5 × 10⁻¹. The multifractal spectrum was used to estimate q and r inversely. Further research is required to determine the level of stress associated with these values.
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