TPT November 2010

A rticle

Application of the stereology reconstruction methods in assessment of the spatial grain structure of metals and alloys V V Perchanik, Ye Ya Lezinskaya, D Yu Klyuev (National Metallurgical Academy of Ukraine) N A Koryaka (ITA Representative in CIS, Ukraine)

Distribution of grain sizes in the volume of a metal product is an important characteristic of dispersivity and homogeneity, and hence of properties and endurance of the product and the entire structure during its operation. Depending on the application of the product and the conditions of its using, the requirement to a grain size and variation in grain size is a criterion of stability and reliability of this product. For example, requirements to nuclear power plant (NPP) fuel element cladding tubes, tubes for bellows, capillary tubes used as heaters in incandescent lamps, etc. Due to opacity of metals, a specially treated flat cut (a polished section) offers an initial information about the structure, which allows determination of the averaged grain size in the examined plane using reliable existing standard methods (GОST 5639, ASTM E112, etc). A large number of methods of structure reconstruction by its flat image have been developed since the 1930s, because the flat cut is just an indirect reflection of the spatial metal structure which is responsible for all physical and mechanical properties of metal products. Spherical shape of structural components was the basis of all developed methods of reconstruction. Description of the known methods of reconstruction of the stereological objects by their mapping on a plane has been thoroughly made in a paper [1] which shows that all methods of the structure stereology reconstruction by its mapping can be reduced to solution of an integral equation which characterises the probabilistic relation between mapping parameters and the actual size of circular or spherical elements of the statistical population. Our secondary analysis of a number of methods based on distribution of the chord lengths in a random section has shown that they were erroneous because of an incorrectly determined measure of the geometric probability elements. The formulas of Spector, Bocstiegel, Lord and Willis correspond exactly to each other after a number of their transformations, and the calculations made in accordance with these formulas confirm experimentally the authors’ mistake because reconstruction

results in the structure refining which is physically inexplicable. Nevertheless, the formula of Bocstiegel was used widely in the programs of quantitative evaluation of the metal structure with the use of Epiquant and Quantimet microscopes. More correct methods are those based on the distribution of random sections of diameters of spherical objects (eg Scheil’s method developed in 1931 and improved later by Schwartz and Saltykov [2] ). The experimental statistics of distribution of ‘diameters’ of the flat cut circles (maximum sizes of each grain) is the base of this method. This method is quite correct from the point of view of choice of the geometric probability measure but it does not provide a strict mathematical procedure of reconstruction and has a number of restrictions like the following: • predetermination of the discrete testing intervals in the general volume of the statistical population; • method for derivation of source data and a compulsory account for all elements of distribution of objects in the flat cut; • poorly representing statistics and an intricate calculation method with successive substitutions and accumulation of errors need a radical improvement which does not allow this method to be widely used in industry. Attempts made by Schwartz and Saltykov to ‘improve’ the Scheil’s method have not practically changed this method essence and were unsuccessful. The use of Saltykov’s function of inverse diameters introduces a significant error to the theory of reconstruction as it changes the geometric measure of probability. Numerous serious errors have been made in a group of the reconstruction methods based on the change of the areas of random sections (methods by Johnson and Saltykov). It is of very high interest to get the source information on the structure as the area distribution of the statistical objects because it does not require assumption concerning the shape of the flat cut grains. However, it violates the choice of the geometric measure of probability. Johnson’s method has been introduced into ASTM E112 just for assessment of parameters of the structure in the flat section, without a volumetric reconstruction.

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N ovember 2010

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