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This chapter describes research on aggregate shape and properties, and on the shape and properties of general particles. The effect of the shape of inclusions on the properties of the composite material in which they are embedded depends on what property is being considered. For example, if insulating inclusions are put into a conducting matrix, and the particles are fairly prolate, then the overall conductivity is very insensitive to the shape of the particles. On the other hand, if the matrix is a viscous fluid, and the particles are solid (rigid), then the viscosity of the suspension depends very sensitively on the inclusion shape.

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.

(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

(5) Three-dimensional shape analysis of coarse aggregates: Methodology and preliminary results on several different coarse aggregates and reference rocks

(6) Tying Together Theory and Tests via Virtual Testing

(7) Some Properties of irregular 3-D particles

References

(1) J.F. Douglas and E.J. Garboczi, *Advances in Chemical Physics* **91**
83-153 (1995).

(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).