scholarly journals Application of Normal Distributions on SO(3) and Sn for Orientation Distribution Function Approximation

1993 ◽  
Vol 21 (2-3) ◽  
pp. 161-176 ◽  
Author(s):  
T. I. Bucharova ◽  
T. I. Savyolova
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Function Approximation ◽  
Rolling Texture ◽  
Polycrystalline Sample ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Normal Distributions ◽  
Pole Figures ◽  
Ill Posed

The orientation distribution function (ODF) in a polycrystalline sample is of special interest in texture analysis. Its determination from pole figures leads to an ill-posed problem, the solution of which is non-unique.In the present paper the properties of normal distributions on the rotation group SO(3) proposed by Parthasarathy (1964), Savyolova (1984) are discussed. A method for ODF determination based on the superposition of the normal distributions is proposed. The parameters of normal distributions are determined from the experimental pole figures. The application of this method is demonstrated for a rolling texture of beryllium.

scholarly journals Approximation of the Pole Figures and the Orientation of Distribution of Grains in Polycrystalline Samples by Means of Canonical Normal Distributions

1993 ◽  
Vol 22 (1) ◽  
pp. 17-27 ◽  
Author(s):  
T. I. Savyolova
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Hexagonal Lattice ◽  
Rolling Texture ◽  
Polycrystalline Sample ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Normal Distributions ◽  
Lattice Symmetry ◽  
Pole Figures

The orientation distribution function (ODF) as determined from experimental pole figures (PF) in a polycrystalline sample by classical spherical harmonics analysis can have ghost effects and regions of negative values. The regions of negative values and the ghosts are a consequence of the loss of information on the “odd” part of ODF.In the present paper the canonical normal distributions (CND) on the rotation group SO(3) and on the sphere S2 in R3 used in texture analysis are discussed.The examples of CND on SO(3), S2 and their PF calculated for hexagonal lattice symmetry and for a rolling texture of beryllium are demonstrated.

scholarly journals Optimal Calculation of ODF With the Canonical Normal Distribution on the Rotation Group

1999 ◽  
Vol 33 (1-4) ◽  
pp. 337-341
Author(s):  
T. I. Savyolova ◽  
E. A. Davidzhan ◽  
T. M. Ivanova
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Physical Properties ◽  
Normal Distribution ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Rotation Group ◽  
Polycrystalline Materials ◽  
Pole Figures ◽  
Optimal Calculation

Macroscopic physical properties of most polycrystalline materials are controlled by orientation distribution of their grains. The orientation distribution function (ODF) of a polycrystal is seldom if ever determined directly from an experiment. Usually experimental data are represented by a set of pole figures (PFs), these latter are some integral projections of the ODF. The main problem of quantitative texture analysis is to recover ODF from its corresponding PFs. With any set of PFs the solution of this problem is non-unique. That is why some assumptions about ODF structure are necessary. We consider ODF as superposition of the canonical normal distribution (CND) on the rotation group SO(3).

scholarly journals The Effect of Smoothing on ODF Reproduction

1996 ◽  
Vol 25 (2-4) ◽  
pp. 149-157 ◽  
Author(s):  
D. I. Nikolayev ◽  
K. Ullemeyer
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Primary Data ◽  
Pole Density ◽  
Data Set ◽  
Normal Distributions ◽  
Ideal Orientation ◽  
Pole Figures ◽  
Characteristic Component

Although the smoothing of experimental pole-density data modifies the primary data set, the effect of such a procedure on the Orientation Distribution Function and on the interpretation of the texture was not considered until recently. The influence of smoothing on texture reproduction will be derived for the case, that the Orientation Distribution Function is approximated by a linear combination of normal distributions (ideal orientation components) and smoothing is carried out with normal distributions, too. The characteristic component parameters are well-suited to indicate changes of the Orientation Distribution Function. The observed texture variations as a consequence of smoothing lead to the conclusion, that moderate smoothing does not falsifythe texture. Several possibilities to control the effect of smoothing will be discussed. Based on the visual comparison of smoothed and unsmoothed pole-figures it is argued, that even extreme smoothing may be useful for some purposes.

scholarly journals Texture Analysis of a Zinc Layer on a Steel Substrate Using Neutron Diffraction

1993 ◽  
Vol 21 (2-3) ◽  
pp. 71-78
Author(s):  
H.-G. Brokmeier
Keyword(s):  
Distribution Function ◽  
Neutron Diffraction ◽  
Texture Analysis ◽  
Orientation Distribution Function ◽  
Steel Substrate ◽  
Orientation Distribution ◽  
Pole Figures ◽  
Iron And Zinc

This paper describes the application of neutron diffraction to investigate the texture of a zinc layer 8 μm in thickness. In a nondestructive way both the texture of the zinc layer as well as the texture of the steel substrate were studied. Therefore, pole figures of iron ((110), (200) and (211)) and of zinc ((0002), (101¯0), (101¯1); and (101¯3)/(112¯0)) were measured; additionally the orientation distribution function of iron and zinc were calculated.

A New Method for Orientation Distribution Function Analysis

1985 ◽  
Vol 29 ◽  
pp. 443-449
Author(s):  
Munetsugu Matsuo ◽  
Koichi Kawasaki ◽  
Tetsuya Sugai
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Function Analysis ◽  
Orientation Distribution ◽  
Symmetry Operation ◽  
Composite Function ◽  
Odd Order ◽  
Pole Figures ◽  
Generalized Spherical Harmonics

AbstractAs a means for quantitative texture analysis, the crystallite orientation distribution function analysis has an important drawback: to bring ghosts as a consequence of the presence of a non-trivial kernel which consists of the spherical harmonics of odd order terms. In the spherical hamonic analysis, ghosts occur in the particular orientations by symmetry operation from the real orientation in accordance with the symmetry of the harmonics of even orders. For recovery of the odd order harmonics, the 9th-order generalized spherical harmonics are linearly combined and added to the orientation distribution function reconstructed from pole figures to a composite function. The coefficients of the linear combination are optimized to minimize the sum of negative values in the composite function. Reproducibility was simulated by using artificial pole figures of single or multiple component textures. Elimination of the ghosts is accompanied by increase in the height of real peak in the composite function of a single preferred orientation. Relative fractions of both major and minor textural components are reproduced with satisfactory fidelity In the simulation for analysis of multi-component textures.

Quantitative X-Ray Analysis of Deformation Microtexture within Individual Grains

2005 ◽  
Vol 495-497 ◽  
pp. 983-988
Author(s):  
N.Yu. Ermakova ◽  
Nikolay Y. Zolotorevsky ◽  
Yuri Titovets
Keyword(s):  
Distribution Function ◽  
Uniaxial Compression ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
X Ray ◽  
Pole Figures ◽  
Single Grain ◽  
Individual Grains

The method is described which enables to determine the microtexture that is the orientation distribution within individual grains of a polycrystal. The microtexture is evaluated on the base of X-ray pole distributions measured for separate reflections, referred to as microscopic pole figures (MPF). The procedure for treatment of experimental MPF and the following computation of orientation distribution function is described in detail. Precision of the microtexture evaluation and possible ways of its improvement are discussed. As an example of the method application, orientation distribution within a single grain of aluminum polycrystal deformed by uniaxial compression up to 50% has been examined.

scholarly journals A Test of the Refinement Procedure for Determining the Crystallite Orientation Distribution: Polyethylene Terephthalate

Texture ◽  
1972 ◽  
Vol 1 (1) ◽  
pp. 9-16 ◽  
Author(s):  
W. R. Krigbaum ◽  
Anna Marie Harkins Vasek
Keyword(s):  
Distribution Function ◽  
Polyethylene Terephthalate ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Fiber Texture ◽  
Fiber Axis ◽  
Crystallite Orientation ◽  
Plane Normal ◽  
Pole Figures ◽  
Refinement Procedure

A test of the refinement procedure for improving the crystallite orientation distribution function is presented for a fiber texture sample of polyethylene terephthalate. This is a particularly difficult example because the triclinic unit cell offers no simplification due to symmetry, and the pole figures are sharply peaked. The analysis employed 17 observed pole figures and an additional 29 unobserved pole figures reconstructed from the crystallite orientation distribution function. After three cycles of refinement, in which the maximum value of the coefficient was increased from 6 to 16, the standard deviations, σq and σw, of the plane-normal and crystallite orientation distributions were reduced by about a factor of 3. The refined crystallite orientation distribution function indicates that the c-axis tends to align along the fiber axis for this polyethylene terephthalate sample.

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