scholarly journals Program for Calculation of Augmented Jacobi Polynomials

Texture ◽  
1975 ◽  
Vol 2 (1) ◽  
pp. 57-66 ◽  
Author(s):  
Peter R. Morris
Keyword(s):  
Computer Program ◽  
Jacobi Polynomials ◽  
Orientation Distribution ◽  
Series Expansions ◽  
Legendre Functions ◽  
Crystallite Orientation ◽  
Fourier Sine

The augmented Jacobi polynomials, designated Zℓmn by Roe, or the closely related generalized Legendre functions, Pℓmn(cos⁡ϕ) of Bunge, are required for series expansions of the crystallite orientation distribution. The problem of publishing extensive tabular information has limited compilation of these functions to certain crystal and physical symmetries. This paper gives a brief description and listing of a computer program which permits calculation of Fourier sine (m–n odd) or cosine (m–n even) series expansions of Zℓmn(ξ) for ℓ=0,1,…,32; m=0,1,…,ℓ; n=0,1,…,ℓ.

Crystallite Orientation Analysis for Rolled Cubic Materials

1967 ◽  
Vol 11 ◽  
pp. 454-472 ◽  
Author(s):  
Peter R. Morris ◽  
Alan J. Heckler
Keyword(s):  
Spherical Harmonics ◽  
Jacobi Polynomials ◽  
Numerical Technique ◽  
Orientation Distribution ◽  
Angular Width ◽  
Crystallite Orientation ◽  
Cubic Materials ◽  
Generalized Spherical Harmonics ◽  
Data Points ◽  
Physical Symmetry

AbstractRoe's method for deriving the crystallite orientation distribution in a series of generalized spherical harmonics is applied to the analysis of texture in rolled cubic materials. The augmented Jacobi polynomials, which are the basis of the generalized spherical harmonics, have been derived for cubic crystallographic symmetry and orthotopic physical symmetry through the sixteenth order. Truncation of the series expansions at the sixteenth order should permit treatment of textures having a maximum of 17 times random and a minimum angular width at half maximum of 34°. A numerical technique has been developed which permits approximate evaluation of the integral equations from a finite array of data points. The method is illustrated for commercial steels and is used to elucidate the primary recrystalization texture of a decarburized Fe-3%Si alloy.

Crystallite Orientation Analysis from Incomplete Pole Figures

1974 ◽  
Vol 18 ◽  
pp. 514-534 ◽  
Author(s):  
Peter R. Morris
Keyword(s):  
Orientation Distribution ◽  
Manganese Steel ◽  
Legendre Functions ◽  
Crystallite Orientation ◽  
Associated Legendre Functions ◽  
Sheet Surface ◽  
Pole Figures ◽  
Orthogonality Relations ◽  
Crystallographic Symmetry ◽  
Physical Symmetry

AbstractThe use of incomplete pole figures results in the loss of orthogonality relations among the associated Legendre functions, and necessitates explicitly evaluating integrals of products of these functions. The required indefinite integrals of associated Legendre functions and their products have been evaluated for cubic crystallographic symmetry and orthotropic physical symmetry through sixteenth order.The solution has been particularized for {200}, {222}, and {110} back-reflection pole figures, where data are confined to the region not exceeding 60 degrees from the sheet normal direction.Data obtained from a sample of low—manganese steel sheet are used to illustrate the method, and results are compared to those obtained using complete pole figures obtained with a composite sample of the same material.The method described makes it possible to study crystallite orientation distribution as a function of distance from the sheet surface, by a series of pole figure measurements on the surface after successive material removals by polishing and etching.

A software for diffraction stress factor calculations for textured materials

2012 ◽  
Vol 27 (2) ◽  
pp. 114-116 ◽  
Author(s):  
Thomas Gnäupel-Herold
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Elastic Constants ◽  
Orientation Distribution Function ◽  
Lattice Strain ◽  
Orientation Distribution ◽  
Crystallite Orientation ◽  
Interaction Models ◽  
Grain Interaction ◽  
Textured Materials

A software for the calculation of diffraction elastic constants (DEC) for materials both with and without preferred orientation was developed. All grain-interaction models that can use the crystallite orientation distribution function (ODF) are incorporated, including Kröner, Hill, inverse Kröner, and Reuss. The functions of the software include: reading the ODF in common textual formats, pole figure calculation, calculation of DEC for different (hkl,φ,ψ), calculation of anisotropic bulk constants from the ODF, calculation of macro-stress from lattice strain and vice versa, as well as mixture ratios of (hkl) of overlapped reflections in textured materials.

Crystallite Orientation Distribution in Biaxially Oriented Polyethylene

1968 ◽  
Vol 49 (4) ◽  
pp. 1532-1542 ◽  
Author(s):  
W. R. Krigbaum ◽  
T. Adachi ◽  
J. V. Dawkins
Keyword(s):  
Orientation Distribution ◽  
Crystallite Orientation ◽  
Oriented Polyethylene

The Generalisation and Refinement of the Vector Method for the Texture Analysis of Polycrystalline Materials

1979 ◽  
Vol 23 ◽  
pp. 349-360 ◽  
Author(s):  
Daniel Ruer ◽  
Albert Vadon ◽  
Raymond Baro
Keyword(s):  
Experimental Data ◽  
Texture Analysis ◽  
Orientation Distribution ◽  
Polycrystalline Materials ◽  
Crystallite Orientation ◽  
Vector Method ◽  
Cubic Materials ◽  
First Time

AbstractA so-called “Vector Method” for the texture analysis of cubic materials was presented for the first time at this conference in 1976. Since then this method has been refined and applied successfully to non cubic-materials. It is shown in this paper that the Vector Method provides several advantages over series methods of texture analysis, the most important of which being the relatively small amount of experimental data which are needed for the determination of the entire crystallite orientation distribution.

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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