Multifractal Model for Soil Aggregate Fragmentation

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.