The representation of orientation distributions
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I.
INTRODUCTION
IT is widely acknowledged that texture is the prime cause of anisotropy in polycrystalline metals: the nonrandom distribution of the crystallographic orientations of the grains ('preferred orientation', 'texture', or 'fabric') transfers some of the anisotropic properties of single crystals to the aggregate. Whereas some properties show little or no anisotropy even in single crystals, many properties are strongly anisotropic even in materials with a cubic lattice. A nonrandomness of the orientation distribution is virtually everpresent in metals, because all processes involved in producing such materials (casting, deformation, recrystallization) are locally orientation dependent. Texture studies are, in fact, frequently used by metallurgists to help identify the crystallographic details of such processes. Despite this general recognition of the importance of texture for a description of macroscopic properties, it seems that quantitative evaluations of texture are rarely used in engineering practice or even in academic physical metallurgy, outside a small community of texture experts. In general applications, one or two pole figures are given at best, or some idealized orientations ('texture components'), with a qualitative interpretation of the expected effects. This is so even though sophisticated quantitative descriptions of three-dimensional orientation distributions have been available for twenty years. 1.2.3 Why has there been such inertia in using quantitative texture descriptions as a general tool of deformation studies? We submit that much of this is due to choices that were made by the pioneers of this development, with respect to the graphic representation of orientation distributions. Of course, any representation may appear easy once it has become familiar; but some provide a significant activation barrier upon first contact. H. R. WENK is Professor in the Department of Geology and Geophysics at the University of California in Berkeley. U. E KOCKS is with Los Alamos National Laboratory, in the Center for Materials Science, Los Alamos, NM 87545. Manuscript submitted February 10, 1986.
METALLURGICAL TRANSACTIONS A
A number of alternative representations have been suggested in the literature but have, for some reason, not become common. We find a particular combination of these especially easy to visualize for the uninitiated, easy to assess qualitatively, and easy to evaluate quantitatively. These plots use polar rather than Cartesian coordinates (as proposed by Williams 3 and by Pospiech and Lficke,4'5 and equal-area projections (as they are commonly used in geology. 6 We will describe these representations in the present paper, and also review in tutorial form some of the basic concepts of orientation distributions in uniform terminology. In addition to graphic representations of orientation distributions, algebraic ones were introduced right at the beginning of this development; particularly, the definition of a continuous orientation distribution function (ODF), and its expansion in
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