Granular materials contain particles that vary in size, shape, and crystalline integrity. Varied qualities in particles lead to varied structures within which particles lay next to one another. The main typologies of structures that can appear in granular materials were established. It then shows ways to relax this randomness by sorting structural typologies into certain groups or ‘universal fractal’. Through experimental evidence, it then shows how each material — with structures in a certain universal fractal — responds to stress in unique ways and exhibits stress-strain behaviours of nondifferentiable, self-similar trajectories. An awareness of ‘universal fractals’ allow designers to imitate these in engineered systems and decrease the risks associated with behavioural variability and uncertainty.

This is a novel approach in design that involves in understanding how certain sized particulate matters self-form certain fractions of particle assemblages that cause certain repeatable hydromechanical behaviours. The significance of ‘fractals’ was portrayed in ^[1], where the concept was unconsciously used in a geological context. Fagg and Smalley ^[1] correlated the random shape of loessic closed depression landscapes with the random shape of soil particles constructing the landscape. They deployed the Zingg particle-shape system to determine a mean random shape for soil particles and fed that into Monte Carlo analysis to simulate the formation and collapse of metastable loess random structures. The collapse was argued to form land depressions at larger scale. The concept of ‘fractals’ have application beyond geological materials. Some of the problems recently looked at from ‘fractals’ perspective are flow of grains at silos’ outlets ^[2], mass transport in polymer electrolyte membrane fuel cells ^[3], characteristics of fat in food ^[4], breakage of dairy powders and implications on product flowability and solubility ^[5], and variable dissolution rate of pharmaceutical ingredients ^[6].

Ten quartz sands, collected from a range of locations on earth, were progressively crushed in finer grades using a range of mills. Mode and particle size distribution of finer grades were then analysed, alongside with particle shape data, to determine particle size ranges that almost always survive the erosive environment, almost always break in the erosive environment, or continuously break and re-aggregate to same size in the erosive environment. These are the ‘universal fractals’. Six select material grades (from pure sands to silty sands) were then subjected to three programmes of laboratory testing: undrained triaxial static compression, undrained triaxial cyclic compression, and undrained triaxial static anisotropic compression, torsion, and combinations of the two. Testing was done on cylindrical 36 mm in diameter 80 mm in height saturated specimens at a void ratio ranging from 0.77 to 0.80. Figure 1 illustrates the pictures of the three experimental programmes.

Figure 1. Experimental programme conducted on six select material grades: (a) cylindrical test specimens remoulded to the high 0.77 to 0.80 void ratio; (b) undrained triaxial cyclic compression; (c) undrained triaxial static anisotropic compression, torsion, and combinations; (d) undrained triaxial static compression.

Granular materials, in natural and engineered forms, constitute particles. By appropriating the fractality concept, each particle size can be associated with a ‘universal fractal’. Granular materials with a pronounced particle size in any of the ‘universal fractals’ can be expected to exhibit repeatable, reproducible and predictable stress-strain behaviour, flow and steady-states.

Three universal fractals are proposed: (i) inherently breakable — granular matters with a pronounced particle size of and around 30-34 μm, and 104-193 μm; (ii) inherently unbreakable — granular matters with a pronounced particle size of and around 10 to 17 μm, 45 to 47 μm and 56 to 58 μm; (iii) self-healing — where particles break and reaggregate, that is, granular matters with a pronounced particle size of and around 5, 7, 12, 20, 68, 75, 316, 440, 610 and 720 μm.

5. Fractals for Structures

Structure or form refers to mutual arrangement of particles and the surrounding pore space in a granular material. In a three-dimensional Euclidean space, every eight particles build a hexahedron, six-faced cubic cell. Eleven possible cubic cells can form for monodisperse spherical particles. The loosest is a 600-cell providing a high 47.5% porosity — this is a cube packing of eight particles with 6 A-faces, 0 B-faces, and 0 C-faces, where A-faces are square, C-faces are rhombohedral, and B-faces are between a square and a rhombohedron. The densest is a 204-cell providing a low 26% porosity. Structures gradually change from the loosest 600 to 420, 240, 060, and 402 (porosity ranging from the high 47.5% to 37.5%), then 042 and 222 (a porosity between 37.5 to 33.3%), and further to 024, 204 and 204 (porosity between 33.3% and 26%). The three sequence here follow three fractal structures, from loosest to densest, Random Loose Packing RLP, Random Close Packing RCP and Close Packing CP.

Particles that belong to one of the three universal fractals form packings that belong to one of the three main universal fractal structures. For example, granular materials with particles in the ‘self-healing’ fractal size range are likely to self-form structures in the Random Loose Packing RLP range. On the other hand, particles in the ‘inherently breakable’ range appear to tend to self-form structures in universal random close packing (RCP). Experiments show that stress paths for granular materials in each fractal structure converge, lending further evidence to significance of structure in design of granular materials.