scholarly journals The Use of a Quadratic Form for the Determination of Non-negative Texture Functions

1983 ◽  
Vol 6 (1) ◽  
pp. 1-19 ◽  
Author(s):  
P. Van Houtte
Keyword(s):  
Series Expansion ◽  
Quadratic Forms ◽  
Expansion Method ◽  
Experimental Methods ◽  
Classical Analysis ◽  
Pole Figures ◽  
Zero Range ◽  
Euler Space ◽  
Series Expansion Method

The classical analysis of measured pole figures of textured polycrystals by the series expansion method does not necessarily produce a non-negative texture function. The main reason for this is, that the method is unable to find the terms of odd rank l of the series expansion.A new method is proposed, which introduces the non-negativity condition into the series expansion method by the use of quadratic forms. The method is found to be successful when treating sharp textures, which have a considerable zero range in Euler space. The preliminary determination of this zero range by experimental methods is however not necessary.

scholarly journals A Positivity Method for the Determination of Complete Orientation Distribution Functions

1988 ◽  
Vol 10 (1) ◽  
pp. 21-35 ◽  
Author(s):  
M. Dahms ◽  
H. J. Bunge
Keyword(s):  
Expansion Method ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Positivity Condition ◽  
Odd Order ◽  
Pole Figures ◽  
Zero Range ◽  
Even Order ◽  
Series Expansion Method

A refinement of the zero-range method, a procedure to calculate the odd order coefficients in the series expansion method of texture analysis, is presented. The only assumption in this procedure is the positivity condition. In this respect, it is comparable to the quadratic method. Contrary to this method, however, the even order coefficients are not changed. No zero range in the pole figures and no shape of the existing texture is to be assumed.

The iterative series-expansion method for quantitative texture analysis. II. Applications

1992 ◽  
Vol 25 (2) ◽  
pp. 259-267 ◽  
Author(s):  
M. Dahms
Keyword(s):  
Distribution Function ◽  
Series Expansion ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Pole Density ◽  
General Outline ◽  
Positivity Condition ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) of the crystallites of polycrystalline materials can be calculated from experimentally measured pole density functions (pole figures). This procedure, called pole-figure inversion, can be achieved by the series-expansion method (harmonic method). As a consequence of the (hkl)-({\bar h}{\bar k}{\bar l}) superposition, the solution is mathematically not unique. There is a range of possible solutions (the kernel) that is only limited by the positivity condition of the distribution function. The complete distribution function f(g) can be split into two parts, \tilde {f}(g) and \tildes {f}(q), expressed by even- and odd-order terms of the series expansions. For the calculation of the even part \tilde {f}(g), the positivity condition for all pole figures contributes essentially to an `economic' calculation of this part, whereas, for the odd part, the positivity condition of the ODF is the essential basis. Both of these positivity conditions can be easily incorporated in the series-expansion method by using several iterative cycles. This method proves to be particularly versatile since it makes use of the orthogonality and positivity at the same time. In the previous paper in this series [Dahms & Bunge (1989) J. Appl. Cryst. 22, 439–447] a general outline of the method was given. This, the second part, gives details of the system of programs used as well as typical examples showing the versatility of the method.

scholarly journals Development of the Rolling Texture in Titanium

1988 ◽  
Vol 7 (4) ◽  
pp. 317-337 ◽  
Author(s):  
H. P. Lee ◽  
C. Esling ◽  
H. J. Bunge
Keyword(s):  
Series Expansion ◽  
Characteristic Zero ◽  
Minor Component ◽  
Rolling Texture ◽  
Expansion Method ◽  
Zero Range ◽  
Different Types ◽  
Cold Rolled ◽  
A Minor ◽  
Series Expansion Method

The complete ODF of titanium, cold rolled up to 80% deformation, was calculated using the series expansion method, including the zero range method. The rolling texture obtained after 80% deformation is mainly characterized by the well-known orientation {0001}〈101¯0〉 ± 40°TD but with distinct spread ranges about it. At about 40% deformation several other texture components are found of which the component {0001}〈112¯0〉 must be mentioned. Further features of the obtained textures are a minor component as well as characteristic zero ranges. Texture development as function of the rolling degree can be divided into three ranges judged by increase or decrease of various texture components. In the early stages twinning in two different types of twinning systems is assumed whereas at higher deformation degrees the formation of the rolling texture is ascribed to glide deformation only.

scholarly journals Examination of the Iterative Series-Expansion Method for Quantitative Texture Analysis

1995 ◽  
Vol 23 (2) ◽  
pp. 115-129 ◽  
Author(s):  
D. Raabe
Keyword(s):  
Series Expansion ◽  
Linear Equations ◽  
Three Dimensional ◽  
Expansion Method ◽  
Orientation Space ◽  
First Case ◽  
Pole Figures ◽  
Polycrystalline Aggregates ◽  
Analytical Tools ◽  
Series Expansion Method

Three-dimensional orientation distributions of grains in polycrystalline aggregates are referred to as crystallographic textures. Commonly, they are computed from two-dimensional centro-symmetric pole figures by employment of series expansion techniques or so called direct inversion methods. Both approaches lead to inaccuracies which are due to the absence of the odd coefficients and by truncation errors in the first case and to the under-determination of the set of linear equations combining cells in the pole figures and in the three-dimensional orientation space in the second case. For both types of calculation methods various correction procedures were suggested. In case of the series expansion methods the introduction of the non-negativity condition was reported to considerably improve the obtained solution. However, before large series of experimental data can be processed by such a method, its reliability has to be checked by use of analytical tools. Hence, in the present study a recently introduced iterative series-expansion method which accounts for the non-negativity condition is examined by use of standard functions.

scholarly journals Approximation of the Orientation Distribution of Grains in Polycrystalline Samples by Means of Gaussians

1992 ◽  
Vol 19 (1-2) ◽  
pp. 9-27 ◽  
Author(s):  
D. I. Nikolayev ◽  
T. I. Savyolova ◽  
K. Feldmann
Keyword(s):  
Rolling Texture ◽  
Expansion Method ◽  
Operating Mode ◽  
Orientation Distribution ◽  
New Method ◽  
Gaussian Distributions ◽  
Positivity Condition ◽  
The Central Limit Theorem ◽  
Pole Figures ◽  
Series Expansion Method

The orientation distribution function (ODF) obtained by classical spherical harmonics analysis may be falsified by ghost influences as well as series truncation effects. The ghosts are a consequence of the inversion symmetry of experimental pole figures which leads to the loss of information on the “odd” part of ODF.In the present paper a new method for ODF reproduction is proposed. It is based on the superposition of Gaussian distributions satisfying the central limit theorem in the SO(3)-space as well as the ODF positivity condition. The kind of ODF determination offered here is restricted to the fit of Gaussian parameters and weights with respect to the experimental pole figures. The operating mode of the new method is demonstrated for a rolling texture of copper. The results are compared with the corresponding ones obtained by the series expansion method.

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