scholarly journals Generalized Spherical Harmonics for Cubic-Triclinic Symmetry

1997 ◽  
Vol 29 (3-4) ◽  
pp. 235-239
Author(s):  
Peter R. Morris
Keyword(s):  
Spherical Harmonics ◽  
Orientation Distribution ◽  
Explicit Representation ◽  
Crystal Symmetry ◽  
Crystallite Orientation ◽  
Cubic Crystal ◽  
Triclinic Symmetry ◽  
Generalized Spherical Harmonics

An explicit representation is suggested for orthogonal generalized spherical harmonics with cubic-crystal and triclinic-sample symmetries. The representation employs sums and differences of orthogonal generalized spherical harmonics with cubic-crystal symmetry, previously described by Bunge for orthorhombic (or higher) sample symmetry, and is illustrated, for T∴iμυ, i = 4, 9, μ = 1, υ = 1 to 5. This representation facilitates crystallite orientation distribution (COD) analysis (aka ODF analysis) for these symmetries, using the Bunge formalism.

scholarly journals Generalized Spherical Harmonics for Cubic-Triclinic Symmetry

1995 ◽  
Vol 24 (4) ◽  
pp. 221-224
Author(s):  
Peter R. Morris
Keyword(s):  
Spherical Harmonics ◽  
Orientation Distribution ◽  
Explicit Representation ◽  
Crystal Symmetry ◽  
Crystallite Orientation ◽  
Cubic Crystal ◽  
Triclinic Symmetry ◽  
Generalized Spherical Harmonics

An explicit representation is suggested for orthogonal generalized spherical harmonics with cubic-crystal and triclinic-sample symmetries. The representation employs sums and differences of orthogonal generalized spherical harmonics with cubic-crystal symmetry, previously described by Bunge for orthorhombic (or higher) sample symmetry, and is illustrated, for T:.ιμν, ι=4, 9, μ=1, ν=1 to 5. This representation facilitates crystallite orientation distribution (COD)analysis (aka ODF analysis) for these symmetries, using the Bunge formalism.

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.

scholarly journals Tensorial Representation of the Orientation Distribution Function in Cubic Polycrystals

1992 ◽  
Vol 19 (3) ◽  
pp. 147-167 ◽  
Author(s):  
Maurizio Guidi ◽  
Brent L. Adams ◽  
E. Turan Onat
Keyword(s):  
Distribution Function ◽  
Spherical Harmonics ◽  
Orientation Distribution Function ◽  
Orientation Distribution ◽  
Basis Functions ◽  
Coordinate Frame ◽  
Crystallite Orientation ◽  
Fourier Representation ◽  
Irreducible Tensors ◽  
Generalized Spherical Harmonics

A precise definition for the crystallite orientation distribution function (codf) of cubic polycrystals is given in terms of the set of distinct orientations of a cube. Elements of the classical Fourier representation of the codf, in terms of (symmetrized) generalized spherical harmonics, are reviewed. An alternative Fourier representation is defined in which the coefficients of the series expansion are irreducible tensors. Since tensors can be defined without the benefit of a coordinate frame, the tensorial representation is coordinate free. A geometrical association between irreducible tensors and a bouquet of lines passing through a common origin is discussed. Algorithms are given for computing the irreducible tensors and basis functions for cubic polycrystals.

scholarly journals Attenuation and Phase Velocity of Elastic Wave in Textured Polycrystals with Ellipsoidal Grains of Arbitrary Crystal Symmetry

Acoustics ◽  
2020 ◽  
Vol 2 (1) ◽  
pp. 51-72
Author(s):  
Gaofeng Sha
Keyword(s):  
Phase Velocity ◽  
Orientation Distribution ◽  
Wave Mode ◽  
Homogenization Method ◽  
Crystal Symmetry ◽  
Polycrystalline Materials ◽  
Phase Velocity Dispersion ◽  
Static Phase ◽  
Generalized Spherical Harmonics ◽  
The Impact

This study extends the second-order attenuation (SOA) model for elastic waves in texture-free inhomogeneous cubic polycrystalline materials with equiaxed grains to textured polycrystals with ellipsoidal grains of arbitrary crystal symmetry. In term of this work, one can predict both the scattering-induced attenuation and phase velocity from Rayleigh region (wavelength >> scatter size) to geometric region (wavelength << scatter size) for an arbitrary incident wave mode (quasi-longitudinal, quasi-transverse fast or quasi-transverse slow mode) in a textured polycrystal and examine the impact of crystallographic texture on attenuation and phase velocity dispersion in the whole frequency range. The predicted attenuation results of this work also agree well with the literature on a textured stainless steel polycrystal. Furthermore, an analytical expression for quasi-static phase velocity at an arbitrary wave propagation direction in a textured polycrystal is derived from the SOA model, which can provide an alternative homogenization method for textured polycrystals based on scattering theory. Computational results using triclinic titanium polycrystals with Gaussian orientation distribution function (ODF) are also presented to demonstrate the texture effect on attenuation and phase velocity behaviors and evaluate the applicability and limitation of an existing analytical model based on the Born approximation for textured polycrystals. Finally, quasi-static phase velocities predicted by this work for a textured polycrystalline copper with generalized spherical harmonics form ODF are compared to available velocity bounds in the literature including Hashin–Shtrikman bounds, and a reasonable agreement is found between this work and the literature.

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.

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.

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.

Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths

1964 ◽  
Vol 54 (2) ◽  
pp. 571-610
Author(s):  
George E. Backus
Keyword(s):  
Spherical Harmonics ◽  
Rayleigh Waves ◽  
Seismic Station ◽  
Great Circle ◽  
Explicit Formulas ◽  
Phase Velocities ◽  
Rapid Accumulation ◽  
Generalized Spherical Harmonics ◽  
Geographical Position ◽  
The Difference

ABSTRACT If the averages of the reciprocal phase velocity c−1 of a given Rayleigh or Love mode over various great circular or great semicircular paths are known, information can be extracted about how c−1 varies with geographical position. Assuming that geometrical optics is applicable, it is shown that if c−1 is isotropic its great circular averages determine only the sum of the values of c−1 at antipodal points and not their difference. The great semicircular averages determine the difference as well. If c−1 is anisotropic through any cause other than the earth's rotation, even great semicircular averages do not determine c−1 completely. Rotation has negligible effect on Love waves, and if it is the only anisotropy present its effect on Rayleigh waves can be measured and removed by comparing the averages of c−1 for the two directions of travel around any great circle not intersecting the poles of rotation. Only great circular and great semicircular paths are considered because every earthquake produces two averages of c−1 over such paths for each seismic station. No other paths permit such rapid accumulation of data when the azimuthal variations of the earthquakes' radiation patterns are unknown. Expansion of the data in generalized spherical harmonics circumvents the fact that the explicit formulas for c−1 in terms of its great circular or great semicircular integrals require differentiation of the data. Formulas are given for calculating the generalized spherical harmonics numerically.

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