TPT November 2010

A rticle

In accordance with this principle, for any fixed i -th interval with limits δ i –1 = D i –1 and D i = δ i , the constant value of the function of distribution f êi ( δ ) is defined by the following correlation:

(13)

It describes the components of a discrete set of distribution of the flat section diameters f ê 3 ( δ ) with a known relative frequency of distribution of spheres f ê ( D ) in a general statistical population. Reconstruction of such unequigranular statistical population of spherical objects is made by solving a system of n linear equations relative to the functions of distribution of spheres in a volume f i ( D ) with known functions of distribution of diameters of circles in a random flat section f i ( δ ). A complete function of distribution in any i -th interval is a sum of constant functions of distribution of independent equigranular systems from i to n taken with their corresponding relative frequencies in these discrete intervals.

(14)

population where the component of the ê -th unequigranular system is highlighted in the i -th interval. The second step of reconstruction solves the problem of determining the statistical relationship between the length l of a random chord and the circle diameter δ k in a plane of section of a spherical model of an equiaxed structure. Figure 4 is a schematic representation of mutual positions of elements in the section resulting in a statistical population of linear elements 0 ≤ l ≤ δ ê when the objects being analysed are randomly arranged. To solve this problem, use logic and sequence of the previous derivation. Replace the spheres visible in the secant plane by nonintersecting circles. Use the condition of equiprobability of position h of the secant line L relative to the centres of circles of equal diameters δ ê . Obtain density of distribution of lengths of random chords: Figure 4 : Schematic representation of section of a circle of diameter δ k by a random line L at a distance h from the circle center 0

where f i ( δ ) is the function of distribution of circle diameters in a flat section within the i -th interval; f i ( D ) is the function of distribution of sphere diameters within the i -th interval; δ i = D i is the top limit within the i -th interval; δ n = D n is the top limit of the statistical population of objects. Figure 3 shows the density of distribution of the flat section diameters for equigranular spherical systems in a unequigranular statistical

Figure 3 : Distribution of equigranular spherical systems in an unequigranular statistical population

(15)

and the discrete function of distribution:

(16)

for the equigranular system of circles.

117

www.read-tpt.com

N ovember 2010

›