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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.
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)
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