This paper has shown that the technique of acquiring 3-D particle images via x-ray tomography, extracting the individual particles computationally from the image, and analyzing the particle in terms of a spherical harmonic expansion, can completely mathematically characterize the shape of random particles. Using the expansion, one can systematically build aggregate databases, similarly to how cement image databases are being built [43] inside the Virtual Cement and Concrete Testing Laboratory [44]. These databases can be used to analyze the aggregate particles from a certain quarry or made by a certain process, making use of the quantitative shape information that comes from the spherical harmonic expansion.
One must remember the limitations of the technique, balancing the digital resolution of the original tomographic image against the number of Gaussian quadrature points used to determine the surface and spherical harmonic coefficients and to calculate the various particle properties. Careful analysis of these effects leads to estimates of the uncertainties involved in the process.
One can also estimate what is the smallest aggregate particle that can be handled with the x-ray tomographic resolution available. Suppose that the resolution of the x-ray tomographic unit is xr micrometers per voxel edge length. From the ellipsoid work carried out above, the smallest particle, in terms of voxels, that can reasonable approximate the shape of a real particle, is probably about 5-10 voxels on a side. Taking 5as the lower limit, the size of the particle is at most 5 xr micrometers in extent. If xr = 50 µm, then the smallest particle that can be analyzed is 250 µm in size, corresponding to a standard mesh size between No. 100 and No. 50. This is a small sand particle, in terms of usual aggregate classifications, but not the smallest routinely encountered.
In an x-ray tomographic apparatus, the resolution xr can be reduced, but at the expense of using a smaller sample. Roughly speaking, whatever the physical size of sample is, the 3-D tomographic image will be N 3 in size, where N is typically 512 or 1024. So the value of xr is just the sample size divided by N. Decreasing the sample size will decrease the image resolution, but at the expense of having to use smaller aggregates. To acquire images of the entire aggregate range used in concrete, coarse to fine, would probably require at least two samples, one large and one small.
The first and second proposed use of the spherical harmonic technique, as discussed in the introduction, was to analyze and classify the shapes of individual particles and compare their shape quantitatively to performance properties. This can now be done. A range of aggregate images can be captured via x-ray tomography, and the particles acquired and stored in a database automatically using the modified burning algorithm. A spherical harmonic expansion can be generated for each one, and the expansion coefficients analyzed and correlated versus performance properties, since they are an exact measure of shape. Much more research needs to be done in this regard, since this paper only makes the technique available - applications need to be carried out.
The third proposal was to use the analyzed shapes to build models using these particles in a similar way to how spheres and ellipsoids could be handled before. This task is not yet complete. What mainly needs to be done is to be able to take two particles, place them at arbitrary locations and at arbitrary rotations, and then decide whether or not these two particles overlap. The mathematics of placing and rotating particles is contained in Ref. [22], and so just needs to be implemented for this program. Deciding whether two particles overlap, which is necessary for building up real concrete models, is somewhat harder and remains to be worked out. Some insight will come from analyses of ellipsoidal contact functions [17,18,19], which are mathematical functions that give an unambiguous answer, > 1 or < 1, whether two ellipsoids overlap or not. This task is achievable, as in principle all the information about the surface is analytically contained in the spherical harmonic expansion. Progress has been made in accomplishing this task, and will be reported in a forthcoming publication [47].
As has been seen from the literature, the spherical harmonic expansion technique for reconstructing particle shape has been used for satellite orbits and the shape of the earth (~ 107 m in size), for the shape of asteroids (~ 104 m in size), and for the shape of molecular orbitals (~ 10 -10 m in size). The size scale of the application discussed here, aggregate shapes, lies in between these extreme length scales. But in all these applications, only the size scale and physical problem differs - the mathematical techniques are the same. So it is rather satisfying to use these techniques at the size scale of aggregates employed in concrete. The combination of x-ray tomography and spherical harmonic analysis allows the routine 3-D analysis of aggregate shapes, completing the multi-scale picture of "particle" shape for a very wide range of length scales.