The geometry of irregular particles is of interest to researchers in many fields. A non-exhaustive list includes mining, materials engineering, geology, soil science, food science, astronomy, catalysis, physics, and pharmacology. It is not hard to define various dimensions for irregular objects. But an open question for random objects is how to quantitatively relate these physical dimensions to properties like volume and surface area. For regular objects, this is a question that has been solved for millennia. But for random 3-D objects, many measurements have been either impractical or impossible until recently, so that data have not even been available for testing any proposed theory. Devices have now been developed that can measure almost all the properties that are of interest (Lin and miller, 2005). Hence, it was felt to be timely and useful to collect data on a select set of random-shaped rocks that could be used to test some preliminary theories relating physical dimensions and volume and surface area. The background, and hence the motivation and interests, of the authors arise from the construction industry. The material types studied, therefore, are gravels and (micro) sands. However, as stated above, the results should be of wider interest since the techniques reported may be used on almost any object within a similar size range. The properties of individual particles studied in this paper include volume, density, and surface area. Shape is also studied via quantitatively defining physical dimensions and relating them to volume and surface area using novel three-parameter equivalent shape models.
In general, irregular bodies may be divided into two classes depending upon whether or not they are convex. The definition of a convex body is that a straight line joining (any) two points in the interior or on the boundary of the body always remains within the body. Most construction aggregate particles have some concavities on their surface – especially if they have been crushed – and hence are not truly convex. Visual examination of all the rocks used in this study revealed concavities. Thus, none of these rocks were truly convex. However, the concavities were always small and thus the rocks considered were “almost” convex. In this sense, a standard golf ball is “almost” convex, at least at length scales above the scale of the surface concavities.
A second useful classification uses the concept of “star shaped”. A star-shaped object requires that all lines that connect the center of volume to any point inside the body (or on its surface) lie entirely within the body. This definition is similar to that of convexity, except that one of each pair of points chosen must lie at the center of volume. Almost all rock particles used in construction are star-shaped (Garboczi 2002) and all the rocks studied in this paper were found to be star-shaped. The properties of convexity and being star-shaped are important because they appear as assumptions in various relevant mathematical theorems (Cauchy, 1850).
A set of 12 rocks was chosen for intensive measurement of properties. The 12 test rocks consisted of a sample from a fully crushed granite material taken from a commercial quarry that was prepared in sizes ranging from 37 mm down to below 75 mm. The granite came from a relatively homogenous source, so the 12 rocks were expected to be more uniform than most commercial products. Six rocks were selected at random from the 12.7 mm to 19 mm size range (0.5 in to 0.75 in), labeled 1 ” 6, and six were chosen from the 6 mm to 12.7 mm size range (0.25 in to 0.5 in), labeled 1 - 6. These are sizes as determined by standard sieves. These rocks can be made available for other investigators to test. In addition, portions of microfine aggregates of the same material were studied in order to compare to results on these 12 test rocks. The microfine versions had equivalent spherical diameters less than about 80 mm. The equivalent spherical diameter is the diameter of a sphere that has the same volume as the particle in question. Figures 1a and 1b show digital camera images of the 12 test rocks chosen. In the rest of this paper, the smaller rocks will have the label “0.5,” standing for 0.5 in or 12.7 mm, and the larger rocks will have the label “0.75,” standing for 0.75 in or 19 mm. In the tables, for example, the label “0.5-3” means rock number 3 of the 0.5 set.
Figure 1a: The six 0.5 rocks studied from the sieve size range 6.3 mm to 12.7 mm. The smallest division in the scale bars is 1 mm. The lighter material is feldspar; the darker is hornblende.
Figure 1b: The six 0.75 rocks studied from the sieve size range 12.7 mm to 19 mm. The smallest division in the scale bars is 1 mm. The lighter material is feldspar; the darker is hornblende.
Methods and results are presented in the following order. First, a brief
introduction to the x-ray computed tomography (CT) and spherical harmonic method
is given. Then the measurement of volume via CT is presented, and checked by
traditional volume measurement techniques. The density of the rocks is then
discussed, and CT results are presented for density distributions within each
rock. The surface area is presented, computed from CT measurements and validated
by the technique of multiple projected images, which takes advantage of a
theorem by Cauchy. Various types of both real and equivalent physical dimensions
are presented, which both generate and allow the use of three-parameter
equivalent shape models based on rectangular parallelepipeds and tri-axial
ellipsoids. These models are compared as to which is more effective at
predicting volume and surface area. Finally, CT data, which have been well
validated by this point in the paper, is presented for microfine aggregates, on
which direct physical measurements are not available.
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