Polycrystal elastic constants for triclinic crystal and physical symmetry

2006 ◽  
Vol 39 (4) ◽  
pp. 502-508 ◽  
Author(s):  
Peter R. Morris
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Weight Functions ◽  
Higher Symmetry ◽  
X Ray ◽  
Pole Figures ◽  
Generalized Spherical Harmonics ◽  
Physical Symmetry ◽  
Symmetry Operations

The problem of obtaining the Voigt average for the elastic stiffnesses with texture-describing weight functions has been solved for triclinic crystal and physical symmetries. The average is obtained by expanding theTijklmnpq, which relate the elastic stiffnesses in the rotated reference frame, c^{\,\prime}_{ijkl}, to those of the principal elastic stiffnesses,cmnpq, in generalized spherical harmonics, multiplying by the orientation distribution function and integrating over all orientations. The condition imposed to assure a unique expansion results in the absence of terms with oddL, so that the results are completely determinable from conventional X-ray pole figures. This is the most general case, from which all higher-symmetry solutions may be obtained by application of symmetry operations. The Reuss average for elastic compliances may be obtained in a similar fashion.

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.

On the Symmetry of Orientation Distribution in Crystal Aggregates

1969 ◽  
Vol 13 ◽  
pp. 435-454
Author(s):  
David W. Baker
Keyword(s):  
Distribution Function ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Spherical Harmonic Analysis ◽  
Euler Angles ◽  
Cartesian Coordinate System ◽  
X Ray ◽  
Pole Figures ◽  
Grain Orientations ◽  
Cartesian Coordinate

In a number of laboratories data from the X-ray pole-figure goniometer are now processed numerically to obtain a suite of complete and normalized pole- figures. From these pole-figures the preferred orientat ion of the grains in a monomineralic aggregate is determined using spherical harmonic analysis and represented as a frequency distribution of the Euler angles Ψ, θ, ø, termed the “orientation distribution function” (ODF),When the frequency density of grain orientations is plotted in a Cartesian coordinate system with Ψ, θ, ø, as axes,it must have a translation periodicity Of 2π or less along each of the axes, because the Euler Angles are angles of rotation.

Texture evolution in metals under mechanical stress: Application of a tensile stage on a laboratory X-ray system

2015 ◽  
Vol 1754 ◽  
pp. 123-128 ◽  
Author(s):  
J. te Nijenhuis ◽  
N. Dadivanyan ◽  
D.J. Götz
Keyword(s):  
Distribution Function ◽  
Mechanical Stress ◽  
Orientation Distribution Function ◽  
Mechanical Load ◽  
Texture Evolution ◽  
Orientation Distribution ◽  
Uniaxial Tensile ◽  
X Ray ◽  
Pole Figures ◽  
Main Component

ABSTRACTThe evolution of texture in copper has been studied in situ as a function of the applied mechanical stress. A uniaxial tensile stage was integrated onto a Eulerian cradle in a laboratory X-ray diffraction system, providing a platform for pole figure measurements on samples under an externally applied mechanical load. Thin strips of rolled copper were investigated at various stages of elongation. The pole figures were of good quality such that the orientation distribution function could be well determined. Changes in the orientation distribution function as a function of strain along the β-fiber could be clearly observed; the initial main component S is replaced by the Copper component at higher stages of elongation.

Estimation of the Minimum Grain Number for the Orientation Distribution Function Calculation from Individual Orientation Measurements on Fe–3%Si and Ti–4Al–6V Alloys

1995 ◽  
Vol 28 (5) ◽  
pp. 582-589 ◽  
Author(s):  
T. Baudin ◽  
J. Jura ◽  
R. Penelle ◽  
J. Pospiech
Keyword(s):  
Experimental Data ◽  
Distribution Function ◽  
Neutron Diffraction ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Grain Number ◽  
X Ray ◽  
Pole Figures ◽  
Function Calculation ◽  
Single Orientation

The calculation of characteristics describing texture as well as relations between orientations and morphological features of microstructure are based on single orientation measurements. For such experimental data, it is essential to estimate the number of necessary measurements of single orientations for a statistically significant representation of the investigated quantity, which, in the present paper, is the orientation distribution function (ODF). In a previous article [Pospiech, Jura & Gottstein (1993) Mater Sci. Forum, 157–162, 407–412], this number has been estimated by a criterion that is used here for a cubic and a hexagonal material. This approach is very useful since it allows one to estimate the minimum orientation number with or without referring to an ODF calculated from pole figures measured by X-ray or neutron diffraction.

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.

scholarly journals Properties of Projection Lines in the Space of the Orientation Distribution Function

1989 ◽  
Vol 10 (3) ◽  
pp. 243-264 ◽  
Author(s):  
A. Morawiec ◽  
J. Pospiech
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Orientation Distribution Function ◽  
Analytical Description ◽  
Orientation Distribution ◽  
Orientation Space ◽  
The Relationship ◽  
Symmetry Operations

The relationship between the orientation distribution function (ODF) and the pole figure is based on the geometry of projection lines in the orientation space.The paper presents an analytical description of the projection lines and their transformations by symmetry operations. Using simple algebraical rules some properties of the projection lines as well as some properties of the associated projection lines (coupled due to the centrosymmetry of the pole figure) have been derived.

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