Orientation Analysis

Simple figures illustrate the basic concepts: orientation, Euler angles, Euler space, orientation density function, pole density function. The iteration that decisively influenced the development of orientation analysis follows directly from the relationship between the two density functions. The minimum principle defines the initial function and the structure of the iteration. Using model orientation density function, we prove that this kind of orientation analysis is extremely effective.

Crystallographic structural changes of Al1050 under different types of sheet metal rolling

AA1050 aluminum alloy was studied with symmetric and asymmetric rolling in various technological modes of rolling equipment. The rotational speeds of the rolls and the number of passes varied according to the initially foreseen scheme. Rolling of aluminum sheet alloy was carried out on an experimental - laboratory installation. The crystallographic texture before and after rolling was investigated using X-ray diffraction. X-ray analysis data was processed using MTEX MATLAB toolbox 75. The pole figures and the texture of the samples of aluminum alloy on the Euler space are given. It was shown that pole figures obtained based on X-ray analysis of samples, rolled by the traditional way, had orthorhombic symmetry, and the sample texture had typical components of traditional rolling, like Cu (copper), Su (with lower intensity) B (Brass).

A coordinate-system-independent method for comparing joint rotational mobilities

ABSTRACTThree-dimensional studies of range of motion currently plot joint poses in a ‘Euler space’ whose axes are angles measured in the joint's three rotational degrees of freedom. Researchers then compute the volume of a pose cloud to measure rotational mobility. However, pairs of poses that are equally different from one another in orientation are not always plotted equally far apart in Euler space. This distortion causes a single joint's mobility to change when measured based on different joint coordinate systems and precludes fair comparison among joints. Here, we present two alternative spaces inspired by a 16th century map projection – cosine-corrected and sine-corrected Euler spaces – that allow coordinate-system-independent comparison of joint rotational mobility. When tested with data from a bird hip joint, cosine-corrected Euler space demonstrated a 10-fold reduction in variation among mobilities measured from three joint coordinate systems. This new quantitative framework enables previously intractable, comparative studies of articular function.

Visualization of texture components using MTEX

Knowledge of the appearance of texture components and fibres in pole figures, in inverse pole figures and in Euler space is fundamental for texture analysis. For cubic crystal systems, such as steels, an extensive literature exists and, for example, the book by Matthies, Vinel & Helming [Standard Distributions in Texture Analysis: Maps for the Case of Cubic Orthorhomic Symmetry, (1987), Akademie-Verlag Berlin] provides an atlas to identify texture components. For lower crystal symmetries, however, equivalent comprehensive overviews that can serve as guidance for the interpretation of experimental textures do not exist. This paper closes this gap by providing a set of scripts for the MTEX package [Bachmann, Hielscher & Schaeben (2010). Solid State Phenom. 160, 63–68] that allow the texture practitioner to compile such an atlas for a given material system, thus aiding orientation distribution function analysis also for non-cubic systems.

MISORIENTATION DISTRIBUTION FUNCTION FOR CUBIC CRYSTALS

A calculation method for obtaining the misorientation distribution function (MDF) for cubic crystals which can be used to estimate the presence or absence of special boundaries in the materials is presented. The calculation was carried out for two samples of Al-Mg-Si alloy subjected to various mechanical and thermal treatments: the first sample is subjected to rolling; the second sample is subjected to recrystallization annealing. MDF is calculated for each sample; the results are presented in the Euler space and in the angle-axis space. The novelty of the method consists in the possibility of gaining data on the grain boundaries from X-ray texture analysis without using electron microscopy. A calculation involving only mathematical operations on matrices was performed on the basis of the orientation distribution function restored from incomplete pole figures. It is shown that no special boundaries are observed in the deformed sample, whereas in the recrystallized alloy, special boundaries are detected at Ʃ = 23, 13, and 17. The shortcoming of the proposed method can be attributed to the lack of accurate data on grain boundaries, since all possible orientation in the polycrystal should be taken into account in MDF calculation.

Quantification of the effect of texture on the magnetization behavior

For electrical steels there is a need to describe the intensity of the present texture components in the finally processed material as well as after the various processing steps. Preferable texture components like the cube fibre texture will improve the magnetization behaviour. Furthermore, there is interplay between the various processing steps at fabrication on the resulting final texture in the fully processed material. A texture parameter A can be derived from the orientation distribution function (ODF) for arbitrary texture, which describes the texture for each texture components. Taking into account this fact, we used a so-called A-map. The A-map gives the value of A for each point in the Euler space for two fixed values (constants). This A-map may be used to estimate immediately the effect of a different resulting image of texture due to variation of the technology of fabrication of the material. Any increase of the intensity of texture within the area of the ODF, where the A-values are smaller than a certain value, results in improved magnetization behaviour. Within the paper some examples are given.

A method to obtain orientation curves in Euler space for a seccond diffraction process in polycrystals

A method is presented to obtain the orientation curve in the Eulerian space, of crystallites which diffract in one point of a Debye-Scherer ring in a second diffraction process. The incident beam is therefore the reflected beam of a previous diffraction process, and the sample has a general orientation for a pole figure measurement, given as usual by two angles, χ around the sample Y axis, and φ around the sample normal. Two solutions are found for all secondary reflections. The method proposed here was outlined somewhere else for the measurement of pole figures by neutron diffraction [1], and here important improvements are made, especially regarding the mathematical methods.

Three-dimensional texture visualization approaches: applications to nickel and titanium alloys

This paper applies the three-dimensional visualization techniques explored theoretically by Callahan, Echlin, Pollock, Singh & De Graef [J. Appl. Cryst.(2017),50, 430–440] to a series of experimentally acquired texture data sets, namely a sharp cube texture in a single-crystal Ni-based superalloy, a sharp Goss texture in single-crystal Nb, a random texture in a powder metallurgy polycrystalline René 88-DT alloy and a rolled plate texture in Ti-6Al-4V. Three-dimensional visualizations are shown (and made available as movies as supplementary material) using the Rodrigues, Euler and three-dimensional stereographic projection representations. In addition, it is shown that the true symmetry of Euler space, as derived from a mapping onto quaternion space, is described by the monoclinic color space groupPccin the Opechowski and Guccione nomenclature.

Three-dimensional texture visualization approaches: theoretical analysis and examples

Crystallographic textures are commonly represented in terms of Euler angle triplets and contour plots of planar sections through Euler space. In this paper, the basic theory is provided for the creation of alternative orientation representations using three-dimensional visualizations. The use of homochoric, cubochoric, Rodrigues and stereographic orientation representations is discussed, and illustrations are provided of fundamental zones for all rotational point-group symmetries. A connection is made to the more traditional Euler space representations. An extensive set of three-dimensional visualizations in both standard and anaglyph movies is available.