On the relation between the number-weighted and volume-weighted grain volume distribution parameters
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INTRODUCTION
POLYCRYSTALLINE material microstructures always contain grains with a range of sizes. Since the grain volume and its distribution significantly affect the mechanical and physical properties of the materials, estimation of the grain volume distribution parameters is important.[1] Generally, grain volume distribution is represented by the number-weighted distribution of the grain volume (fN(v)). Distributing the number of grains according to volume involves collecting information about the fraction of the total number of grains which have a certain volume or which have sizes within a certain narrow range and, in doing so, giving each grain the same weight. By definition, fN(v) can be characterized by the number-weighted mean grain volumeVN and the coefficient of variation CVN(v). From a practical point of view, the volume-weighted distribution of grain volume (fv(v)), which reports the fraction size of the total volume of all grains that belong to a certain narrow grain size range, has been of interest recently.[2–6] Accordingly, fv(v) may be characterized by the volume-weighted mean grain volume VV and the coefficient of variation CVV(v). Grain volume distribution can be measured directly, using serial section analysis[1] or by grain separating and weighing,[7] but with great difficulty. Although stereological methods with grain shape assumptions[8] can be used to calculate the three-dimensional grain size distribution from two-dimensional measurements on plane sections through the material, the shape assumptions often lead to unrealistic results. Recently, a point-sampled intercept method requiring only a single section has been used successfully to deHAIBO YU, formerly Doctoral Student, Department of Materials Science and Engineering, Beijing University of Science and Technology, is with the Department of Applied Physics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China. GUOQUAN LIU, Professor, is with the Graduate School, Department of Materials Science and Engineering, Beijing University of Science and Technology, Beijing, 100083, People’s Republic of China. XIAOYAN SONG, formerly Doctoral Student, Department of Materials Science and Engineering, Beijing University of Science and Technology, is with the Department of Materials Science and Engineering, Hebei University of Technology, Tianjin, 300130, People’s Republic of China. Manuscript submitted November 26, 1996. METALLURGICAL AND MATERIALS TRANSACTIONS A
termine the unbiased volume-weighted mean grain volume and the coefficient of variation,[4,6] while an unbiased, socalled ‘‘disector’’ method requiring a pair of serial sections has been proposed as an efficient three-dimensional probe to measure the number-weighted mean grain volume.[4] With the aid of the point-sampled intercept method, the disector method can further determine the coefficient of variation for the number-weighted grain volume distribution.[4,5] On the other hand, if the relationships between the volume-weighted and the number-weighted grain vo
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