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65th anniversary of the Chair of pyrometallurgical processes of the South Ural State University (SUSU)
ArticleName Probabilistic-statistical testing method for the techniques of metallographic determination of the amount of non-metallic inclusions in metal
ArticleAuthor A. D. Drozin, N. M. Yaparova

South Ural State University (Chelyabinsk, Russia):

A. D. Drozin, Dr. Eng., Prof., Head of Honour Education Center, e-mail:
N. M. Yaparova, Cand. Phys.-Math., Associate Prof., Head of the Chair of Computational Mathematics and High-Performance Computing, e-mail:


Non-metallic inclusions are harmful impurities in steel. To determine their numbers and sizes, researcher usually takes a sample of metal and makes a polished plane. The plane cuts the inclusions, and the researcher sees with a microscope the sections of these inclusions. On the basis of these data about nonmetallic inclusions, metal is rejected or approved. This approach gives a distorted picture. Inclusions of different sizes can give sections of the same size, and the researcher will consider them to be the same. Conversely, inclusions of the same size can give completely different cuts. Many methods for determining the true number of inclusions in the investigated metal volume based on their cuts by the section plane have been developed. These methods are designed to recreate the three-dimensional picture of distribution of inclusions on the basis of a two-dimensional picture of distribution of their sections. Direct verification of the adequacy of these techniques is hardly possible. In particular, the chemical dissolution of the metal is not suitable for each type of inclusions, and the result gives extremely low numbers of inclusions. Counting millions of inclusions and determining the size of each of them is hardly possible also. We have developed a probabilistic-statistical method of verification, which consists of the following. We set some initial volumetric distribution of inclusions, select some volume of metal and set the table: the size of the inclusion - the number of such inclusions in the sample. Next, using the random number generator, the location of each inclusion is specified. Also, the coordinate of the section plane is randomly assigned. It is calculated which inclusions will be cut by the section plane and calculated its section size. We apply the verifiable method to the obtained section distribution, obtain estimations of inclusion distribution in the sample and compare them with the given (in this case, the true distribution). We perform a large number of such comparisons and after statistical processing we can have concluded whether the test method is suitable.

This research was supported by Ministry of Education and Science of the Russian Federation of applied scientific research within the framework of the basic part of the State task «Development, research and implementation of data processing algorithms for dynamic measurements of spatially distributed objects», Terms of Reference 8.9692.2017 / 8.9 from 17.02.2017.

keywords Metallography, crystallography, non-metallic inclusions, size distribution, stereology, math modeling, testing, algorithm

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