Three-dimensional mathematical analysis of particle shape using X-ray tomography and spherical harmonics: Application to aggregates used in concrete

The properties of composites made by placing inclusions in a matrix are often controlled by the shape and size of the particles used. Mathematically, characterizing the shape of particles in three dimensions is not a particularly easy task, especially when the particle, for whatever reason, cannot be readily visualized. But, even when particles can be visualized, as in the case of aggregates used in concrete, three-dimensional (3-D) randomness of the particles can make mathematical characterization difficult. This paper describes a mathematical procedure using spherical harmonic functions that can completely characterize concrete aggregate particles and other particles of the same nature. The original 3-D particle images are acquired via X-ray tomography. Three main consequences of the availability of this procedure are mathematical classification of the shape of aggregates from different sources, comparison of composite performance properties to precise morphological aspects of particles, and incorporation of random particles into many-particle computational models.