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118
N
ovember
2010
www.read-tpt.com›
A
rticle
R&D Department
Power Electronics Division
SAET Group
Via Torino 213 – Leinì (TO), Italy
Department of Electrical Engineering,
University of Padova
Via Gradenigo, 6/a, 35131 – Padova, Italy
Use the same principle of superposition and obtain a complete
function of distribution of independent equigranular systems with
their corresponding relative frequencies in the specified intervals:
(17)
where:
f
i
(
l
)
is the function of distribution of chords in a plane within the
i
-th
interval;
f
i
(
δ
) is the function of the distribution of the diameters in a plane
within the
i
-th interval;
l
i
=
δ
i
is the upper boundary of the
i
-th interval;
l
n
=
δ
n
is the upper limit of the statistical population of objects.
By its structure, equation (17) corresponds exactly to equation (14)
and when values
f
i
(
l
) are known, it makes it possible to construct a
system of
n
linear equations which when solved, give reconstruction
of the function of distribution of circle diameters in the plane.
Thus, when the data of distribution of chord sizes in the section
plane (microsection) are available, the spatial structure can be
reconstructed.
Reconstruction of volumetric metal structures is made in two steps:
The first step proceeds from the distribution of chords to the
distribution of circle diameters in a flat cut:
f
i
(
l
)
f
i
(
δ
).
The second step proceeds from the distribution of circle diameters
in a flat cut to the distribution of sphere diameters in volume:
f
i
(
δ
)
f
i
(
D
).
The requirement of the new method of reconstruction is equality of
boundaries of intervals of the size groups, ie
l
i
=
δ
i
=
D
i
.
The intervals of the size groups are chosen arbitrarily, in any quantity
(the number of linear equations depends on it) and quality (their
values can vary according to any law from zero to a maximum value
in the dimensional set of the statistical population).
The first step of reconstruction, when a confidential statistical
population of the chords is used as input data, allows an advance
assessment of asymmetry and the form factor of the flat section
elements.
Besides, the first step of the reconstruction can be easily checked
experimentally and it will allow assessment of error of the spherical
approximation by using model structures in the form of regular
polygons.
The advantages of the new method of reconstruction are as
follows:
• an unlimited number of the size groups;
• a complete independence in the selection of the number and
nature of variation of the size intervals;
• linear intersectionwith the boundaries of grains in a flat section (the
method of chords) is used as the source data of reconstruction;
• optional account of all elements in the examined area provided the
representative statistics by the method of chords is obtainable.
The main advantages of the new method are as follows:
• the proof of a complete identity of relations of the distribution
functions in systems ‘line – plane’ and ‘plane – volume’ by
choosing the geometric probability of the cutting element position
relative to the centre of the object as a unified measure;
• application of the principle of superposition for equigranular
components of distribution in each interval with their corresponding
relative frequencies in the general statistical population.
An important advantage of the new method is the possibility of a
complete automation of the process of obtaining all parameters of
the structure directly from the section using the specially developed
program STRUCTURE 2001
[10]
.
References
[1]
КS Chernyavsky. Stereology in Physical Metallurgy – Мoscow: Меtallurgia
Publishers, 1977 – p45-54.
[2]
SА Saltykov. Stereological Physical Metallurgy – Мoscow: Меtallurgia Publishers,
1970 – p375.
[3]
B Honigman. Growth and Shape of Crystals – Inostrannaya Literatura Publishers,
Мoscow, 1961 – p209.
[4]
FC Hull and WJ Houk – Journal of Metals, April 1953, p565
[5]
D McLin. Grain Borders in Metals – GNTIL On Ferrous and Nonferrous Metallurgy,
Мoscow. 1960 – p322.
[6]
Ye М Grinberg, СI Arkhangelsky, ОV Khromov. Моdelling Grain Structure of
Single-phase Systems and Check Model Conformity. In: Proceedings of the 4
th
International Conference of Young Scientists “Меtallurgy of the 20
th
Century”.
[7]
Patent of Ukraine No. 77135. The Method of Determination of Main Parameters
of the Metal Structure. Bulletin “Promyshlennaya Sobstvennost”. – No. 10.
16.10.2006.
[8]
Patent of Russian Federation No. 2317539 The Method of Determination of Main
Parameters of the Metal Structure. Bulletin of Inventions No. 5, 20. 02. 2008.
[9]
Ye S Ventsel. Probability Theory – Мoscow. Nauka Publishers, 1964 – p265.
[10]
Computer program STRUCTURE 2001; Bulletin of State Department of
Intellectual Property “The Certificate of Author’s Right No. 1577, Ukraine. МОN”
– No. 9. – Kyiv, 2006.