DIMENSIONS and EQUIVALENT SHAPE MODELS

 The very definition of a regular body means that it can be specified by a few geometrical parameters related to size; e.g., the diameter of a sphere or the side length of a cube (one dimension), the aspect ratio and the length of one axis of a prolate or oblate spheroid (two dimensions), or the semi-axis lengths of a tri-axial ellipsoid (three dimensions). Other, more complicated shapes could require more dimensions. One way that the notion of their regular shape is quantified is from the analytically simple relations between their dimensions and their geometrical properties like volume and surface area. For irregular bodies, a very great number of parameters have been proposed that purport to relate dimensions and geometry. This attempt is to have a convenient single number with which to refer to an irregular shape. Russ (1999) lists 10 parameters and does not claim the list to be exhaustive. Mather (1966) lists even more. Space does not permit a full description of these efforts. Excellent bibliographic lists may be found in Mather (1966), Davies (1973), Hawkins (1993) and Russ (1999). All these could be described as “equivalent shape” methods. 

The dimensions of irregularly shaped particles are usually defined via equivalent shape methods. Building equivalent shape models follows these steps: (1) one or more geometrical properties are selected, (2) the properties are measured by some means for the irregular particle, and (3) a regular shape is selected. One should note that the geometrical properties chosen must be analytically known for the regular shape. The last step is determining the dimensions of the regular shape by equating the geometrical properties of the irregular shape to the analytically known geometrical properties of the regular shape, and solving for the dimensions of the regular shape. This regular shape then becomes known as the “equivalent regular shape” particle that has the same selected geometrical properties as does the irregular shape.

 One-parameter equivalent shapes are defined by only one geometrical property, two-parameter models by two geometrical properties, and so forth. An example of a one-parameter equivalent shape model is an equivalent volume sphere, whose diameter is determined by forcing the sphere to have the same volume as the irregular particle. This is probably the most highly used one-parameter model, at least in construction materials. Another one-parameter model, an equivalent surface area cube, would be defined by using a cube and forcing its surface area to be equal to that of the irregular particle. If measurements of both the surface area and the volume were available, then the specific surface area, the ratio of surface area to volume, could be used to define an effective sphere or cube. One-parameter models are of limited usefulness in describing the shape of irregular particles, for the following reason. If one constructs an equivalent volume sphere, then an immediate question becomes: is any other geometrical property of the equivalent shape besides the volume at all close to that of the original irregular particle? For example, is the surface area of the equivalent volume sphere equal or close to that of the irregular particle?  Usually the answer to this question is negative. One can use two or more parameter models. In this paper, three-parameter models were investigated. These will be shown to be adequate for many engineering purposes, in the sense that they give good predictions of other geometrical properties besides those that define their three parameters.

 Two three-parameter models are defined by choosing two regular objects with three dimensions:  rectangular parallelepipeds (boxes), and tri-axial ellipsoids. All geometrical properties of both objects are totally defined by three lengths:  a, b, and c, where the dimensions of the box are 2a, 2b, and 2c, and the semi-axes of the tri-axial ellipsoid are a, b, and c.  One should note that these objects include the cube and sphere as degenerate cases. Their volume and surface area are as follows. The ellipsoid and box volumes are

                  (6)

Note that both volumes are a constant times the triple product of a, b, and c. The surface areas of these objects, however, are more complex (Lawden, 1989; Maas, 1994; Legendre, 1825):

                 (7)

where F is an elliptic integral of the first kind and E is an elliptic integral of the second kind (Arfken, 1970),

               (8)

and it is assumed that a > b > c. An approximate formula that eliminates the use of elliptic integrals and that is accurate ( ≈0.1 % error relative to the exact results) over a very wide range of ellipsoids is

             (9)

where p = ln(2)/ln(p/2) ≈1.5349 and k = 0.0942 (Thomsen, 2004). We use this approximate formula for the rest of this paper. For a box, the surface area is given by

                    (10)

Using the two geometrical properties of volume and surface area is not enough to determine the three dimensions of an equivalent box or ellipsoid. In fact, we want to determine these three dimensions another way, and then predict the volume and surface area of the irregular object by computing the volume and surface area of the equivalent object. We next consider three different ways to determine the three dimensions of the equivalent box or ellipsoid.

 One possible approach is to directly measure three dimensions, called length, L, width, W, and thickness, T, in some reasonable fashion. A common procedure is to measure the longest line within the body and to call this the “length” of the body L; then to find the longest such line that is orthogonal to L and to term it W. A similar procedure yields T, which has to be orthogonal to both L and T (ASTM D4791). This procedure can be done directly using digital calipers, for example. The spherical harmonic-based mathematical approximation of the particle can also be used in a simple algorithm to find approximations for L, W, and T simply by searching for pairs of surface points that satisfy the length and direction criteria. This is a well-defined and unique way to obtain three orthogonal dimensions from an irregular body. Note that this procedure does not depend on the assumption of an equivalent regular shape, so these measured dimensions could be used to generate box or ellipsoid three-parameter equivalent shape models. 

Each of the 12 test rocks was measured directly using digital calipers, with an estimated uncertainty (dominated by the angular uncertainty in estimating orthogonality between the lines) of about + 1.0 mm. The spherical harmonic-based code was also run to obtain computational estimates of L, W, and T. The results for these two procedures are shown in Table 6, showing in general excellent agreement, within experimental uncertainty, between direct measurement and mathematical computation. Only a few pairs of dimensions are not equal within + 1.0 mm.

 The good agreement between the measured and CT-SH results is yet again evidence that the spherical harmonic mathematical representation gives a faithful representation of the true rock shape. One possible drawback to the use of L, W, and T defined in this fashion is that small protuberances on the rock can artificially extend the values of L, W, and T. These values will still be well defined and unique, but would not necessarily be representative of the rock shape as a whole.

Table 6. Length (L), width (W), and thickness (T) information from two procedures. Note that, by definition, L > W > T. All dimensions are in mm, with an experimental uncertainty of + 1.0 mm.

Other sets of three orthogonal dimensions can be defined via moments. It is known that all bodies possess a center of mass and a center of volume. These will be identical if the body is homogeneous in density. Except in the section on density distributions inside the 12 rocks, we assume that the rocks are uniform in density, so that what follows is on the basis of volume, not mass. The coordinates of the center of volume of a solid body is given by:

                            (11)

where V = the volume of the body, and the integral is taken over the entire volume of the body. Eq. (11) serves also to define the <...>_V notation indicating a volume average. In the spherical harmonic-based mathematical representation of the rock, the origin is taken at the center of volume, so that X₁ = X₂ = X₃ = 0. Measuring the coordinates of a point in the particle from this origin, one can then define the n’th moment of x_i:

                                        (12)

The most well known set of second moments (n = 2) is probably the moment of inertia tensor, defined as a combination of various simpler moments: 

                         (13)

 where r² = (x₁)² + (x₂)² + (x₃)² , d_ij is the Kronecker delta function (1 if i=j, 0 otherwise), and x₁ = x, x₂ = y, and x₃ = z. Since we have eliminated mass from the problem, it is probably more accurate to call this tensor the moment of volume tensor. The moment of volume tensor can be diagonalized, and the diagonal elements are then denoted the principal moments of volume (PMV). The directions associated with these moments are orthogonal. These then are three numbers that totally define the solid body’s reaction to an applied torque, assuming that the body is rigid (Goldstein, 1950).

 The relative values of the three PMV values can give a qualitative indication of shape, if one restricts the class of objects to be star-shaped. If, for example, the three values are almost identical then one knows that the object is highly symmetrical, like a sphere or cube. Without the star-shaped restriction, one could have something like a 3-D addition sign or three mutually intersecting barbells, which are not realistic aggregate shapes. If two of the three PMV values are close to each other and larger than the third, the object is “plate like”. If two values are close to each other but smaller than the third, the object is “needle like”. A graphical representation of the relationships among these numerical constants, and how they may be used to study shape, is given in Taylor (2002). Other uses of various moments have been described in the literature (Mansfield, 2002). 

One can define any other order moment, such as <x₁x₂x₃> or <(x₁)⁴>, but extracting three unique numbers from them is more difficult, so they are not considered further in this paper. Linear moments, like <x₃>, are zero by definition, since they simply give the center of volume coordinates that are taken to be zero. However, one can define the three absolute first moments, by

                                         (14) 

These kinds of moments have been referred to as the mean absolute deviation (Press et al., 1989), where one computes the average deviation of the data from the mean. In this case, the mean is the center of volume, which is taken to be zero. The AFM, as we have defined them, are not zero in general, because the absolute value is taken. The three AFMs will also give three numbers that are somehow characteristic of the body, but their mechanical interpretation is not obvious, like in the case of the PMV.  

The above moments could also be defined by just integrating over the surface of the particle, by defining the average of a function P over the surface to be:

                       (15)

One might think that if we desire length parameters that define the overall shape of the body, then integrating over the volume of the object would not be helpful and it would be more useful to only integrate over the surface. This is conceptually a similar idea in some ways to the use of LADAR (Cheok, 2004), which only defines surface points, to that of x-ray CT, which gives volume information as well. This concept is discussed briefly below.

We now have geometric parameters, the three AFM values and the three PMV values, which can be used to determine the dimensions of a three-parameter equivalent body like the box or the ellipsoid.  One last ingredient has to be supplied, however: the analytical AFM and PMV formulas for these two regular objects. The principal moments of volume for boxes and ellipsoids can be found in any classical mechanics textbook. They are simple formulas: 

                       (16)

where a = 1/5 for the ellipsoid and 1/3 for the box. I_a means the PMV along the axis of the shape with half length = a, etc. To extract values of the length parameters from measured values of the PMV, one simply inverts these expressions to obtain

            (17)

The AFM values are not common parameters, but fortunately the integrals required are simple, giving the three AFM values as:  

                                 (18)

 where b = 3/8 for ellipsoids and ½ for boxes. The length parameters a, b, and c are then given by inverting these simple expressions.

 We note that we could use the idea of defining moments over just the surface, as in eq. (15). The test case of the box serves to check the usefulness of this idea. One can easily perform the integral in eq. (15) for the AFM of a box defined over its surface. The results are given by,

             (19)

Because of their algebraic form, it is not possible to invert these expressions to get values for a, b, and c in terms of the surface AFM’s. The cases of the ellipsoid for AFM and both the box and ellipsoid for PMV are probably similar; hence we do not consider these surface-defined moments any further.  It is possible that future work may show the utility of these surface-defined moments. 

Using the three choices of dimensions that have been defined, we can now define six three-parameter equivalent shape models for an irregular object: (1) a box with dimensions equal to L, W, and T, (2) an ellipsoid with semi-axes equal to ½L, ½W, and ½T, (3) a box of dimensions defined from the PMV, (4) an ellipsoid of semi-axes defined from the PMV, (5) a box of dimensions defined from the AFM, and (6) an ellipsoid with semi-axes defined from the AFM. An example for clarity: the model in (4) for a given rock means that the model ellipsoid has the same PMV as does the rock. These lead to six different ways of computing volumes and surface areas using eqs. (6), (9), and (10). The usefulness of any of these three parameter equivalent shape models should be judged by whether the volume and surface area of the equivalent shape could be a good predictor of the actual values of these geometric properties of the 12 irregular test particles.

 Figures 6-9 contain 12 curves (= 6 volume and 6 surface area) of approximate surface area and approximate volume vs. the surface area and volume as measured by x-ray CT and spherical harmonic surface reconstruction. Figures 6 and 7 shows these values for the equivalent box models. Figures 8 and 9 show these same values but using the choice of an equivalent ellipsoid model. In Figs. 6-9, the direct measurements of L, W, and T were also used under both the box and ellipsoid assumptions to compute the volume and surface area.

 In Figs. 6-9, for all curves, there was an excellent linear correlation between the various approximations and the x-ray CT data, which is considered to be exact for the purpose of these graphs because it has been shown to agree well with the experimental values. Table 7 shows the parameters of the various lines in Figs. 6-9: the slope, the y-intercept, the y-intercept as a percentage of the maximum abscissa value, and the value of the R² coefficient. If these approximate relations were exact, they would pass through the origin, so the y-intercept expressed as a percentage of the maximum abscissa value is a check on how “realistic” is the linear relation. If the y-intercept is a fairly substantial percentage of the maximum abscissa value, then even if the correlation is linear, it is not physical since the linear relation should go through the origin. All the linear relations shown in the top of Table 7 have small values of this parameter. Another factor is the slope. The closer this slope is to unity, the more useful and physical is the approximate relation relating the effective object, box or ellipsoid, to the real aggregate. In Table 7, the AFM box model prediction for the surface area and the PMV box model prediction for the volume both have slopes quite close to unity, 1.01 and 1.02, respectively (highlighted in gray).

 It is important to note that none of the relations in Table 7 are exact. Therefore, if one simply wants a linear relation that gives good predictability of surface area and volume, the L, W, and T parameters can be used to define either the box or the ellipsoid equivalent shape models. This choice will offer simplification in calculations, because L W and T are relatively easy to obtain. The possibility then exists of routinely predicting surface area and volume from simple length measurements like in ASTM D4791.

Figure 6: Box surface area for the 12 test rocks.          Figure 7: Box volume for the 12 test rocks.

Figure 8:  Ellipsoid SA for the 12 test rocks.         Figure 9: Ellipsoid volume for the 12 test rocks.

Table 7: Linear fit parameters for the 12 test rocks for various choices of estimating volume and surface area from various choices of dimensions.

For these 12 test rocks, it is of interest to more closely compare the various dimensions used in the various models. We have three sets of measurements, direct measurement, AFM, and PMV, each giving a set of lengths, widths, and thicknesses. If we define length > width > thickness, then we can normalize by the thickness and define each dimension set by only two numbers. We lose the absolute volume and surface area information, but can focus more easily on shape. We call this pair of dimensions L and W. Table 8 lists these normalized dimensions and the absolute percentage deviation of the AFM and PMV dimensions from the direct measurements. Only the AFM and PMV labels are used, since whether we derive dimensions from the box or ellipsoid approximations, the normalized values of L and W are the same, since the AFM and PMV values for the tri-axial ellipsoid and box differ only by a multiplicative constant, which disappears when taking ratios. Because the direct measurements were made on the actual geometry of each rock, any large deviation of the moment-derived values from the directly measured values will tend to show that the moment-derived dimensions are not geometrically realistic. Of course, these dimensions could only be exactly equal for regular objects. It is clear from the averages of each D column that the dimensions derived from the PMVs tend to be more geometrically realistic than those coming from the AFM values, since the average D values for the PMV dimension ratios are much less than those for the AFM ratios.

Table 8:  Comparing the normalized L and W parameters for direct measurement, AFM, and PMV. The parameter D is defined as the absolute percentage difference from the direct measurement results.