Compact reconstruction of orientation distributions using generalized spherical harmonics to advance large-scale crystal plasticity modeling: Verification using cubic, hexagonal, and orthorhombic polycrystals

2018 ◽  
Vol 155 ◽  
pp. 418-432 ◽  
Author(s):  
Adnan Eghtesad ◽  
Timothy J. Barrett ◽  
Marko Knezevic
Keyword(s):  
Spherical Harmonics ◽  
Crystal Plasticity ◽  
Large Scale ◽  
Generalized Spherical Harmonics ◽  
Orientation Distributions

A generalized spherical harmonics-based procedure for the interpolation of partial datasets of orientation distributions to enable crystal mechanics-based simulations

Materialia ◽  
2019 ◽  
Vol 6 ◽  
pp. 100328 ◽  
Author(s):  
Timothy J. Barrett ◽  
Adnan Eghtesad ◽  
Rodney J. McCabe ◽  
Bjørn Clausen ◽  
Donald W. Brown ◽  
...  
Keyword(s):  
Spherical Harmonics ◽  
Generalized Spherical Harmonics ◽  
Orientation Distributions

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.

scholarly journals Addition Formula and Related Integral Equations for Heine—Stieltjes Polynomials

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1231
Author(s):  
Hans Volkmer
Keyword(s):  
Integral Equations ◽  
Explicit Form ◽  
Spherical Harmonics ◽  
Reproducing Kernel ◽  
Addition Formula ◽  
Symmetric Products ◽  
Special Cases ◽  
Related Integral ◽  
Generalized Spherical Harmonics ◽  
The Relationship

It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found.

Crystal-Plasticity Simulation of the Evolution of the Matt Surface in Pack Rolling of Aluminium Foil

2014 ◽  
Vol 794-796 ◽  
pp. 553-558 ◽  
Author(s):  
Olaf Engler ◽  
Galyna Laptyeva ◽  
Holger Aretz ◽  
Gernot Nitzsche
Keyword(s):  
Crystal Plasticity ◽  
Large Scale ◽  
Rolling Texture ◽  
Spatial Arrangement ◽  
Alloy Sheet ◽  
Aluminium Foil ◽  
Strain Anisotropy ◽  
Recent Project ◽  
Final Rolling ◽  
Different Characteristics

Aluminium foil is rolled double-layered during the final rolling pass. When the sheets are later separated, the inside surface is dull and the outside surface is shiny. The matt inner side is characterized by significant surface corrugations which are believed to be a precursor for the initiation of fracture upon a subsequent forming operation. Therefore, understanding of the development of the matt side of Al foil will help to control and, eventually, improve the properties of Al foil. It was the goal of the present study to correlate the development of the matt side with the spatial arrangement of the crystallographic orientations of the foil rolling texture. This approach builds on a recent project to correlate the phenomenon of roping in AA 6xxx alloy sheet for car body applications to the occurrence of band-like clusters of grains with similar crystallographic orientation. Large-scale orientation maps obtained by electron back-scattered diffraction (EBSD) were input into a visco-plastic self-consistent crystal-plasticity model to analyse the strain anisotropy caused by the spatial distribution of the various rolling texture components. The new model is applied to several Al foils with different characteristics of the matt side.

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.

Modeling Cyclic Deformation of HSLA Steels Using Crystal Plasticity

2004 ◽  
Vol 126 (4) ◽  
pp. 339-352 ◽  
Author(s):  
C. L. Xie ◽  
S. Ghosh ◽  
M. Groeber
Keyword(s):  
Cyclic Loading ◽  
Crystal Plasticity ◽  
Cyclic Deformation ◽  
Deformation Behavior ◽  
Kinematic Hardening ◽  
Orientation Imaging Microscopy ◽  
Plasticity Model ◽  
Hsla Steels ◽  
Crystal Plasticity Model ◽  
Orientation Distributions

High strength low alloy (HSLA) steels, used in a wide variety of applications as structural components are subjected to cyclic loading during their service lives. Understanding the cyclic deformation behavior of HSLA steels is of importance, since it affects the fatigue life of components. This paper combines experiments with finite element based simulations to develop a crystal plasticity model for prediction of the cyclic deformation behavior of HSLA-50 steels. The experiments involve orientation imaging microscopy (OIM) for microstructural characterization and mechanical testing under uniaxial and stress–strain controlled cyclic loading. The computational models incorporate crystallographic orientation distributions from the OIM data. The crystal plasticity model for bcc materials uses a thermally activated energy theory for plastic flow, self and latent hardening, kinematic hardening, as well as yield point phenomena. Material parameters are calibrated from experiments using a genetic algorithm based minimization process. The computational model is validated with experiments on stress and strain controlled cyclic loading. The effect of grain orientation distributions and overall loading conditions on the evolution of microstructural stresses and strains are investigated.

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.

The Large-Scale Magnetic Field and Sunspot Cycles

2000 ◽  
Vol 179 ◽  
pp. 161-162
Author(s):  
V. I. Makarov ◽  
A. G. Tlatov
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Legendre Polynomials ◽  
Large Scale ◽  
Magnetic Moments ◽  
Radial Component ◽  
Wolf Number ◽  
Spherical Functions ◽  
Large Scale Magnetic Field ◽  
The Magnetic Field

Extended abstractWe report on the correlation between the large scale magnetic field and sunspot cycles during the last 80 years that was found by Makarovet al. (1999) and Makarov & Tlatov (2000) in H-αspherical harmonics of the large scale magnetic field for 1915–1999. The sum of intensities of the low modes 1 = 1 and 3, A(t), was used for comparison with the Wolf number, W(t). It was shown that the large scale magnetic field cycles, A(t), precede the sunspot cycles, W(t), by 5.5 years.Let us consider the behaviour in time of the harmonics with low numbers 1 = 1 and 1 = 3. The radial component B(r) of the magnetic field may be expanded in terms of the spherical harmonicswhereθandϕare the latitude and longitude,are Legendre polynomials andandare coefficients of expansion on the spherical functions.The magnetic moments of a dipole (1 = 1) and an octopole (1 = 3) are determined by the following equations:Let us enter the parameter describing their intensity,

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