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.

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 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.

scholarly journals Calculation of Yield Surfaces and Determination of Forming Limit Diagrams of Aluminium Alloys

1993 ◽  
Vol 21 (2-3) ◽  
pp. 93-108 ◽  
Author(s):  
J. J. Fundenberger ◽  
M. J. Philippe ◽  
C. Esling ◽  
P. Lequeu ◽  
B. Chenal
Keyword(s):  
Aluminium Alloys ◽  
Expansion Method ◽  
Strain Ratio ◽  
Orientation Distribution ◽  
Forming Limit ◽  
Deformation Model ◽  
Plastic Strain Ratio ◽  
Yield Loci ◽  
Series Expansion Method

In order to point out the influence of the crystallographic texture on the formability of 2 aluminium alloys, the orientation distribution function (ODF) will be carried out using the series expansion method. Combining the ODF with a Taylor plastic deformation model we are able to calculate the yield loci and to predict the plastic strain ratio which is of high interest in the formability.

scholarly journals ODF Calculation by Series Expansion from Incompletely Measured Pole Figures Using the Positivity Condition Part I—Cubic Crystal Symmetry

1987 ◽  
Vol 7 (3) ◽  
pp. 171-185 ◽  
Author(s):  
M. Dahms ◽  
H. J. Bunge
Keyword(s):  
Series Expansion ◽  
Iterative Procedure ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Crystal Symmetry ◽  
Cubic Crystal ◽  
Positivity Condition ◽  
Pole Figures ◽  
Orientation Distribution Functions

The calculation of orientation distribution functions (ODF) from incomplete pole figures can be carried out by an iterative procedure taking into account the positivity condition for all pole figures. This method strongly reduces instabilities which may occasionally occur in other methods.

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 The Development of Rolling Texture in α-Brass Determined by Neutron Diffraction

Texture ◽  
1974 ◽  
Vol 1 (4) ◽  
pp. 211-231 ◽  
Author(s):  
J. Tobisch ◽  
A. Mücklichf
Keyword(s):  
Neutron Diffraction ◽  
Three Dimensional ◽  
Rolling Texture ◽  
Orientation Distribution ◽  
Zinc Alloy ◽  
Tube Axis ◽  
Pole Figures ◽  
Cold Rolled

The three-dimensional orientation distribution was calculated from neutron diffraction pole figures for a copper 27.2% zinc alloy cold rolled to different degrees of deformation. The results agree qualitatively with those of other authors. There are however differences in the quantitative respect which influence the conclusions to be drawn. For rolling degrees lower than about 70% the texture exhibits an orientation tube similar to that of the copper type, but with a significantly different distribution along the tube axis. For rolling degrees larger than 70% the texture can be described by the orientation {110}〈112〉. The deformation is assumed to occur according to the Wassermann model and the Hu model respectively in these two ranges.

The development of the rolling texture in copper measured by neutron diffraction

1971 ◽  
Vol 4 (4) ◽  
pp. 303-310 ◽  
Author(s):  
H. J. Bunge ◽  
J. Tobisch ◽  
W. Sonntag
Keyword(s):  
Neutron Diffraction ◽  
Texture Component ◽  
Deformation Mechanism ◽  
Three Dimensional ◽  
Rolling Texture ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Anisotropic Hardening ◽  
Pole Figures ◽  
Cold Rolled

Three-dimensional orientation distribution functions of the crystallites in copper sheets, cold rolled to different degrees of reduction, have been determined using neutron diffraction pole figures. The main features of the textures may be represented by the orientation `tube' already described in prior publications. Two ranges of rolling reduction can be distinguished, a lower one (30 to 50%) and a higher one (70 to 95%) the texture changes of which correspond to those calculated after the Taylor theory. In an intermediate range (50 to 70%) a different deformation mechanism occurs which leads to an intermediate (001) [110] texture component. It is supposed that anisotropic hardening may have occurred in this range.

scholarly journals A New Approach to Describing Three-Dimensional Orientation Distribution Functions in Textured Materials—Part II: Model ODF for Rolling Textures

1994 ◽  
Vol 22 (3) ◽  
pp. 169-175 ◽  
Author(s):  
V. N. Dnieprenko ◽  
S. V. Divinskii
Keyword(s):  
Three Dimensional ◽  
Rolling Texture ◽  
Orientation Distribution ◽  
Distribution Functions ◽  
Crystallite Orientation ◽  
New Approach ◽  
Texture Axis ◽  
Pole Figures ◽  
Spread Parameters ◽  
Textured Materials

Sections of a three-dimensional Orientation Distribution Function (ODF) for the α-Fe rolling texture typical for most b.c.c. metals have been constructed on the basis of the proposed new method for ODF simulation through the representation of a crystallite orientation by nine rotations, only three of which are varied for a given component. The description of texture by superposition of partial fibre components in used. A comparison of such a model ODF with an ODF reconstructed from experimental pole figures by series expansion is presented. As a result all really encountered textures can be simulated by variation of the crystallite spread parameters, texture axis positions, and predominant preferred orientations in terms of a common approach.

Acoustoelastic Determination of the Fourth Order ODF Coefficients and Application to R-Value Prediction

1988 ◽  
Vol 142 ◽  
Author(s):  
D. Daniel ◽  
K. Sakata ◽  
J. J. Jonas ◽  
I. Makarow ◽  
J. F. Bussiere
Keyword(s):  
Fourth Order ◽  
Low Carbon Steel ◽  
Expansion Method ◽  
Orientation Distribution ◽  
Rolling Plane ◽  
Low Carbon ◽  
Acoustic Transducers ◽  
Pole Figures ◽  
The Hill ◽  
Good Agreement

AbstractThe fourth order orientation distribution function (ODF) coefficients of textured low carbon steel sheets were determined nondestructively from the anisotropy of the velocity of Lamb (So) and SHO plate waves measured using electromagnetic acoustic transducers (EMATs). The three coefficients (C411, C412, C413) are calculated from five velocity measurements made in three directions in the rolling plane of the sheet using the Hill approximation by an iterative numerical method. The coefficients were also determined from Young's modulus measurements based on a resonance technique and are compared to those obtained ultrasonically. The comparison with coefficients determined from X-ray diffraction pole figures permits adjustment of the C411 coefficient and then very good agreement is obtained. The plastic strain ratios (R-values) of the steel samples are predicted from the adjusted coefficients using a series expansion method based on the Taylor theory of crystal plasticity. These are compared with experimental measurements and again good agreement is displayed.

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