TPT November 2010

A rticle

Saltykov’s formula which allows determination of the number of particles per unit of volume in accordance with the area method is based on the assumption of a normal distribution of logarithms of diameters of spherical particles. This formula allows, regardless of the mutual influence of discrete intervals, ‘reconstruction’ of structures in any size group. Such method, as compared with other methods, allows quantitative evaluation of the volumetric structures with a minimum error. Deficiency of the known methods of reconstruction results from the absence of the commonality of the mathematical support of the random processes, limitation of their application field and absence of technical means for processing a great number of the statistical population objects which reduces accuracy of reconstruction. It should be mentioned that the grain shape affects the results of reconstruction of the spatial metal structure. Grains of metals and their alloys have a shape of various types of polyhedrons [3] which hinders reconstruction because the view of flat sections (polygons) does not represent fully the spatial form of the grain. Numerous experiments in modelling of spatial shapes of grains of metallic materials [4-6] and the data of crystallographic analysis show that the shape of grains of single-phase metallic materials is close to the cuboctahedron or truncated dodecahedron. Grains of metals with face-centred lattice have a shape close to octahedron, truncated hexahedron, truncated octahedron or truncated dodecahedron. Authors of the recent paper [6] simulated the single-phase structure by the compression of spherical pallets with various distribution of their diameters and volumes and confirmed the earlier suggestion that “compressed spherical pallets can take form of cuboctahedron with slightly curved faces”. Based on the available data on the shape of the grains of polycrystalline single-phase metals and alloys, a new method of reconstruction has been developed for the quantitative assessment of volumetric structures in accordance with characteristics of the flat section [7, 8] . It is based on the following assumptions: 1. The close-packed structure of the convex equiaxed polyhedrons can be modelled by a system of nonintersecting spheres randomly arranged in space. 2. The position of the secant element relative to the centre of this object is the probability measure which determines in an only way the geometric characteristics of any statistical community object. The functional relations of the spatial structure parameters with the characteristics of its mapping in the plane were obtained based on the selection of a common measure of distribution of geometric probabilities, ie the equiprobability of location of a random section relative to the centre of a sphere or a circle which are the components of the statistical population of the structural objects. The problem is solved in two steps. The first step solves the problem of sectioning of the spherical objects of the equigranular structure which consists of spheres of the same diameter D ê . These spheres are randomly spread in the space, the close packing being impossible and not binding. It is essential that there are no mutually crossing objects of statistical community.

Figure 1 : Schematic section of a sphere having diameter of D ê by a random plane Q at a distance h from the centre of sphere O

Cutting of a equigranular structure by a random plane results in appearance of the circles in this plane with a diameter of 0 ≤ δ ≤ D ê . They are randomly located in the flat section and form a new statistical population of circular objects with diameter of δ (Figure 1). In accordance with the principle of equiprobability of position of the secant plane Q relative to the centre of sphere O , the density of distribution of parameter h from 0 to h max = 0,5 D ê is a constant: (1) A functional relationship is between the sphere diameter D ê , the circle δ and the position of the secant plane:

(2)

which is described by the following correlation in a differential form:

(3)

In the theory of probability [9] , the law of distribution of the monotone continuous function is related with an argument by the following relation:

(4)

The value p ( h ) is determined from the condition of regulation of density of distribution of parameter h :

(5)

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N ovember 2010

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