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ovember
2010
117
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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)
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
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:
(15)
and the discrete function of distribution:
(16)
for the equigranular system of circles.
Figure 3
:
Distribution of equigranular spherical systems in an unequigranular
statistical population
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