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Material is also included on how to mathematically analyze the shape of aggregate particles, as taken from an x-ray tomograph, using spherical harmonic techniques, which also arise in atomic quantum mechanics.
This section presents a study of the polarizability (intrinsic conductivity) and the intrinsic viscosity for a very wide range of shapes. It is found that for
a very wide range of shapes, the intrinsic conductivity, in the conducting limit, is proportional to the intrinsic viscosity in the vanishing shear limit.
Intrinsic viscosity and polarizability of particles having a wide range of
(1) Intrinsic viscosity and polarizability of particles having a wide range of shapes
This section extends the work of the previous section to the polarizability of objects with a general conductivity compared to the matrix. It is shown that a simple Pade approximant, incorporating knowledge of the intrinsic conductivity in various limits, describes well the intrinsic conductivity of the object for any value of its conductivity. The approach is also shown to work for intrinsic viscosity and intrinsic elastic moduli.
(2) Intrinsic conductivity of objects having arbitrary shape and conductivity
This section is an introduction, without mathematical details, of how to acquire aggregate shapes with x-ray tomography, analyze their shapes with spherical harmonic function techniques, and use these and similar shapes in the simulation of suspension rheology using dissipative particle dynamics techniques.
(3) Acquiring, Analyzing, and Using Complete Three-Dimensional Aggregate Shape Information
This section contains material on how to mathematically analyze the shape of aggregate particles, as taken from an x-ray tomograph, using spherical harmonic techniques. Many mathematical details are included in appendices, including a list of the associated Legendre functions up to n = 8, which we have not found elsewhere. Hopefully these details will allow others to readily use these techniques. These same techniques also arise in atomic quantum mechanics, geodesy, the analysis of the shape of molecular orbitals, and in the reconstruction of the 3-D shape of asteroids.
(4) Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: Application to aggregates used in concrete
The shape of aggregates used in concrete is an important parameter that helps determine many concrete properties. This section discusses the sample preparation and image analysis techniques necessary for obtaining an aggregate particle image in 3-D, using x-ray computed tomography, which is suitable for spherical harmonic analysis. The shape of three reference rocks are analyzed for uncertainty determination via direct comparison to the geometry of their reconstructed images. Shape data on several different kinds of coarse aggregates are compared and used to illustrate potential mathematical shape analyses made possible by the spherical harmonic information.(5) Three-dimensional shape analysis of coarse aggregates: Methodology and preliminary results on several different coarse aggregates and reference rocks
This section is a short magazine article that appeared in Stone, Sand, and Gravel Review. It describes, in a semi-technical fashion, how the Virtual Cement and Concrete Testing Laboratory works, and how it could be used in industry.(6) Tying Together Theory and Tests via Virtual Testing
This paper discusses some of the properties of irregular particles that are of interest to engineers, including volume, density, and surface area. While the motivation, examples and applications are from the construction materials industry, the results should be of interest to others. Measurement techniques used included x-ray computed tomography (CT) and multiple projected images, augmented by traditional laboratory techniques. To compare the results of these techniques a set of 12 rocks were studied with equivalent spherical diameters between 19 mm and 6.3 mm. Microfine versions of these rocks (< 80 µm equivalent spherical diameter) were also studied and compared. The shapes of the rocks were studied by relating three dimensions to their volume and surface area. These three physical dimensions were defined by direct measurement of three unique orthogonal dimensions on the rock surface, and by the use of absolute first moments of volume and principle second moments of volume. These measurementsand calculated moments allowed the development of three-parameter equivalent shape models based on rectangular parallelepipeds and tri-axial ellipsoids.(7) Some Properties of irregular 3-D particles
(1) J.F. Douglas and E.J. Garboczi, Advances in Chemical Physics 91
(2) E.J. Garboczi and J.F. Douglas, Physical Review E 53, 6169-6180 (1996).
(3) E.J. Garboczi, N.S. Martys, H.H. Saleh, R.A. Livingston, Proceedings of the Ninth Annual Symposium for the International Center for Aggregate Research, April 22-25, 2001.
(4) E.J. Garboczi, Cement and Concrete Research 32 (10), 1621-1638 (2002).
(5) S.T. Erdogan, P.N. Quiroga, D.W. Fowler, H.A. Saleh, R.A. Livingston, E.J. Garboczi, P.M. Ketcham, J.G. Hagedorn, and S.G. Satterfield, submitted to Cement and Concrete Research (2005).
(6) E.J. Garboczi, reprinted from Stone, Sand, & Gravel Review January/February, 10-11 (2006).