Statistical considerations on uniform grain size
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I.
INTRODUCTION
THE microstructure of a metal is characterized not only by 9 [ll the shape but also by the size of its grams. For certain purposes it may be sufficient to describe the grain size as coarse, medium, or fine, although these terms are relative and have no definite meaning. Some indication of their size may be obtained by counting grains, taking into account the magnification at which the sample is observed and assigning numerical values to the different grain sizes. Nevertheless, grain size is one of the most important microstructural features of metals and alloys because of its influence mainly on mechanical properties. Thus the literature t2~contains a large number of expressions for estimating grain size on the basis of examination of metallographic sections of these materials. Theoretically, the grain size can be expressed in terms of line, plane, or space parameters9 In metallography, grain size is usually expressed by twoparameters, which are the average diameter of the grain (D) or its volume (V), although in the latter case it is more common to use the inverse value of the grain volume or the number of grains per unit volume (Nv). The average diameter of the grain is defined as the average orthogonal distance between the infinite pairs of parallel planes tangent to the surface of the grain and is related to the grain volume by the average grain area (A) on a random two-dimensional section: [31
The first works, about related research, led to the idea that the grain size distribution, in recrystallized metals, could be adjusted by means of log-normal distribution. 161 Research on grain size distribution in polycrystalline materials has been done by a great number of authors, as much in real specimens as in computer simulation of grain growth9 The results seem to indicate that these distributions can be analytically adjusted by a modified shape of the Rayleigh distribution9 Nevertheless, all of these results have been evaluated by Pande, ITj coming to the conclusion that this modified Rayleigh distribution is very similar to log-normal distribution, which is the one that represents better the experimental data which appear in the pertinent literature9 The probability function of a log-normal distribution is given by the expression: Y= ~
1 Exp [ - + I n V - l n V t In o'~ In o'g
[1]
Although its use has been generalized as a parameter to describe grain size, the only parameter that can represent grain size, independently of shape, is grain volume. I41 Therefore, from now on whenever we refer to grain size we will be talking about its volume. It is known that the size of the grains making up a material presents a certain distribution and it is common in metallography to use terms like "uniform", "nonuniform", "biased", "duplex", etc. to characterize these distributions, although these qualifiers are generally assigned to the grain size on the basis of a subjective appreciation of microstructures, 151 since there are no accurate definitions for these terms.
C. NUNEZ, Professor and Chairman, an
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