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116
N
ovember
2010
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A
rticle
The density of distribution of diameters of circles
δ
in the plane of
section of the equigranular system of spheres with diameter of
D
ê
is derived from equations (2-5):
(6)
which is a continuous function of hyperbolic type within
0
≤
δ
≤
D
ê
approaching asymptotically to
δ
max
=
D
ê
(Figure 2)
.
The law of distribution is normalised within 0 to
δ
max
:
(7)
The main numerical characteristics of the statistical population of the
flat section of the equigranular system are as follows:
• average diameter of the objects (mathematical expectation)
(8)
• absolute measure of scatter of the random quantity
δ
relative to
its average value
δ
ñð
(mean-square deviation):
(9)
• relative index irregularity of the plane section objects (variation
coefficient):
(10)
Cutting of an equigranular system of the spheres of diameter
D
ê
by a
random plane results in a statistical population of flat-cut circles with
diameter of
0
≤
δ
≤
D
ê
the average size of which is 22% smaller
than the size of balls in accordance with equation (8).
In stereometric metallography, dimensional groups of various widths
are set for the total volume of the statistical population. They replace
continuous distribution by discrete one, ie by passing from the
density of distribution to the function. The need of such replacement
is caused by the practice of construction of a set of distributions of the
random variable with a well-defined (non-zero) event probability.
A universal characteristic of the random variable
δ
is the so-called
integral function of distribution defined as a discrete function:
(11)
where:
D
ê
is the upper limit of the statistical population set
0
≤
δ
≤
D
ê
;
i
is the number of any
i
-th interval, with a width varying from
δ
i
–1
=
D
i
–1
to
δ
i
=
D
i
.
It should be mentioned that the change of the interval width within
the limits of the total set of statistical population is not limited and can
be selected in accordance with any series: arithmetic one (uniform),
geometric (progressively changing), logarithmic, etc.
As any probability, the distribution function
f
i
(
δ
) is a dimensionless
and constant quantity within the width of one interval. This function
is normalised within the limits of a set of statistical population of the
flat-cut circles:
(12)
The direct problem was solved using equation (11). This problem
determines the probability of appearance of circles of a certain
diameter within any accepted interval in a flat section of the statistical
population of spheres of a same diameter which are randomly
arranged in space.
The inverse problem of determination of sphere diameters within
an equigranular statistical population is solved using equation (11)
with specified values
f
i
(
δ
) and known values of circle diameters in
a flat section at the lower and upper limits of this
i
-th interval. It is
the inverse problem that is of a very high importance for structure
reconstruction by the flat section parameters.
Next, when considering a unequigranular system of spherical objects
representing a statistical set of spheres of various diameters and
defining it as an aggregate of independent equigranular systems,
observance of the principles of superposition for the function of
distribution of any equigranular system in any discrete interval has
been proved.
Figure 2
:
Density of distribution of diameters of the flat sections of the equigranular
system of the spheres