A significant'size effect' is observed in tensile strength of solid particles, such as ice, rock, ceramics and concrete: the tensile strength is not independent of the fragment size, but decreases with increasing size. The Weibull statistical theory was universally used to calculate the size effect observed in solid particles. Recent developments in fractal theory suggest that fractals may provide a more realistic representation of solid particles. In this paper, the scaling phenomenon of ice mechanics is explained using the fractal model for ice particle fragmentation. The Weibull statistics is modified using the fractal crushing of ice, and is compared with the conventional one. Goodness-of-fit statistics show that the modified Weibull statistics fits the experimental data of ice much better than the conventional one. The modified Weibull statistics has only one parameter, the fractal dimension of the fragment size distribution, which has a general value of 2.50 for the ice fragmentations.

The hydraulic conductivity of unsaturated soils with constant volume is a function of the saturation degree or matric suction, and can be obtained based on currently available procedures. However, each procedure has its limitations and consequently cares should be taken in the selection of a proper procedure. Fractal approach seems to be a potentially useful tool to describe hierarchical systems and is suitable to model the structure and hydraulic properties of porous media. In this paper, the soil-water characteristic curve and hydraulic conductivity function of unsaturated soils were derived from the fractal model for the pore-size distribution, and were expressed by only two independently physical parameters, the fractal dimension and the air-entry value. The predictions of the proposed soil-water characteristic curve and unsaturated hydraulic conductivity were in good accord with the published experimental data. Comparisons between the experimental results and the predictions of both the fractal model and the van Genuchten- Mualem model were performed, and it was found that the predictions of the fractal model were better than that of the van G-M model.

Soils are generally penetrated by a pore system. Most transport processes usually occur in pore space of soils. The description of pore space shape is necessary to quantify how the geometry in-uences the percolation of substances such as water and nutrients. In this paper, the pores in soils are described by fractal geometry, and the fractal dimension is obtained from porosimetric measurements. The water permeability function is studied to correlate with the pore-size distribution of unsaturated soils, and is derived from the fractal model of pore-size distribution. The comparisons between the collected data of permeability and the proposed permeability function show that the calculations of the proposed permeability function are in satisfactory agreements with the measurements.