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Multifractal Model for Soil Aggregate Fragmentation

Dep. of Land Resource Science, Univ. of Guelph, Guelph, ON, N1G 2W1, Canada

*Corresponding author.

ABSTRACT

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, , the strength of dry aggregates of normalized equivalent cubic length x_* was expressed as a probability of failure, {P(x_*)}. A method was developed for calculating {P(x_*)} from tensile strength data. At intermediate levels of stress (0.3 0.9 MPa), {P(x_*)} 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 . 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_*)} 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 x 10^–2 to 3.46 at x_* = 7.5 x 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.

Received for publication June 5, 1992.

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