Texture Analysis in Materials Science

Texture Analysis in Materials Science

Mathematical Methods

1st Edition - December 15, 1982

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  • Author: H.-J. Bunge
  • eBook ISBN: 9781483278391

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Description

Texture Analysis in Materials Science Mathematical Methods focuses on the methodologies, processes, techniques, and mathematical aids in the orientation distribution of crystallites. The manuscript first offers information on the orientation of individual crystallites and orientation distributions. Topics include properties and representations of rotations, orientation distance, and ambiguity of rotation as a consequence of crystal and specimen symmetry. The book also takes a look at expansion of orientation distribution functions in series of generalized spherical harmonics, fiber textures, and methods not based on the series expansion. The publication reviews special distribution functions, texture transformation, and system of programs for the texture analysis of sheets of cubic materials. The text also ponders on the estimation of errors, texture analysis, and physical properties of polycrystalline materials. Topics include comparison of experimental and recalculated pole figures; indetermination error for incomplete pole figures; and determination of the texture coefficients from anisotropie polycrystal properties. The manuscript is a dependable reference for readers interested in the use of mathematical aids in the orientation distribution of crystallites.

Table of Contents


  • Contents

    List of Symbols Used

    1. Introduction 1

    2. Orientation of Individual Crystallites

    2.1. Various Representations of a Rotation

    2.1.1. Eulebian Angles

    2.1.2. Rotation Axis and Rotation Angle

    2.1.3. Crystal Direction and Angle

    2.1.4. Sample Direction and Angle

    2.1.5. Representation of the Orientation in the Pole Figure

    2.1.6. Representation of the Orientation in the Inverse Pole Figure

    2.1.7. Representation by Miller Indices

    2.1.8. Matrix Representation

    2.1.9. Relations between Different Orientation Parameters

    2.1.10. The Invariant Measure

    2.2. Some Properties of Rotations

    2.3. Ambiguity of Rotation as a Consequence of Crystal and Specimen Symmetry

    2.4. Orientation Distance

    2.5. Orientation for Rotational Symmetry

    3. Orientation Distributions

    4. Expansion of Orientation Distribution Functions in Series of Generalized Spherical Harmonics (Three-dimensional Textures)

    4.1. Determination of the Coefficients Cµvl

    4.1.1. Individual Orientation Measurements

    4.1.2. Interpolation of the Function f(g)

    4.2. The General Axis Distribution Functions A(h,y)

    4.2.1. Determination of the Coefficients Cµvl by Interpolation of the General Axis Distribution Function

    4.2.2. Pole Figures Ph(y)

    4.2.3. Inverse Pole Figures Ry(h)

    4.2.4. Comparison of the Representations of a Texture by Pole Figures and Inverse Pole Figures

    4.3. The Angular Distribution Function Why(Θ)

    4.3.1. Integral Relation between Pole Figures and Inverse Pole Figures

    4.4. Determination of the Coefficients Cµvl by the Method of Least Squares

    4.5. Measures of Accuracy

    4.5.1. A Special Accuracy Measure for Pole Figures of Materials with Cubic Symmetry

    4.5.2. A Method for the Adaption of Back-reflection and Transmission Range

    4.6. Truncation Error

    4.6.1. Decrease of the Truncation Error by Smearing

    4.7. Determination of the Coefficients Cf from Incompletely Measured Pole Figures

    4.8. Texture Index

    4.9. Ambiguity of the Solution

    4.9.1. Non-random Textures with Random Pole Figures

    4.9.2. The Refinement Procedure of KBIGBAUM

    4.9.3. The Extremum Method of TAVARD

    4.10. Comparison with ROE'S Terminology

    4.11. The Role of the Centre of Inversion

    4.11.1. Right-and Left-handed Crystals

    4.11.2. Centrosymmetric Sample Symmetries

    4.11.3. Centrosymmetric Crystal Symmetries

    4.11.4. Friedel's Law

    4.11.5. Black—White Sample Symmetries

    4.11.6. Determination of the Odd Part of the Texture Function

    5. Fiber Textures

    5.1. Determination of the Coefficients Cµvl

    5.1.1. Individual Orientation Measurements

    5.1.2. Interpolation of the Function R(h)

    5.2. The General Axis Distribution Function A(h Φ)

    5.2.1. Pole Figures Ρh(Φ)

    5.2.2. Inverse Pole Figures RΦ(h)

    5.3. Determination of the Coefficients Cµvl According to the Least Squares Method

    5.4. Measures of Accuracy 130

    5.4.1. A Special Measure of Accuracy for Pole Figures of Materials with Cubic Symmetry

    5.5. Truncation Error

    5.5.1. Decrease of the Truncation Error by Smearing

    5.6. Determination of the Coefficients Cf from Incompletely Measured Pole Figures

    5.7. Texture Index

    5.8. The Approximation Condition for Fibre Textures

    5.9. Calculation of the Function Β(Φ, β) for Various Crystal Symmetries

    5.9.1. Orthorhombic Symmetry

    5.9.2. Cubic Symmetry

    5.10. The Role of the Centre of Inversion

    5.10.1. Right- and Left-handed Crystals

    5.10.2. Centrosymmetric Sample Symmetries

    5.10.3. Centrosymmetric Crystal Symmetries

    5.10.4. Friedel's Law

    5.10.5. Black—White Sample Symmetries

    5.10.6. Determination of the Odd Part of the Texture Function

    6. Methods not Based on the Series Expansion

    6.1. The Method of Perlwitz, LÜCKE and Pitsch

    6.2. The Method of Jetter, Mchargue and Williams

    6.3. The Method of Ruer and Baro

    6.4. The Method of IMHOF

    7. Special Distribution Functions

    7.1. Ideal Orientations

    7.2. Cone and Ring Fibre Textures

    7.3. 'Spherical' Textures

    7.4. Fibre Axes

    7.5. Line and Surface Textures (Dimension of a Texture)

    7.6. Zero Regions

    7.7. Gaussian Distributions

    7.8. Polynomial Approximation (Angular Resolving Power)

    8. Texture Transformation

    9. A System of Programs for the Texture Analysis of Sheets of Cubic Materials

    9.1. The Subroutines

    9.2. The Mainline Programs

    9.3. The Library Program

    9.4. Calculation Times and Storage Requirements

    9.5. Supplementary Programs

    9.6. A Numerical Example

    9.7. Listings of the ODF and Library Programs

    10. Estimation of the Errors

    10.1. A Reliability Criterion for Pole Figures of Materials with Cubic Symmetry

    10.2. The Error Curve ΔFvl

    10.3. The Error Curve ΔCµvl

    10.4. Error Estimation According to the HARRIS Relation

    10.5. Comparison of Experimental and Recalculated Pole Figures

    10.6. Negative Values

    10.7. Estimation of the Truncation Error by Extrapolation

    10.8. The Integration Error

    10.9. The Statistical Error

    10.10. The Indetermination Error for Incomplete Pole Figures

    11. Some Results of Texture Analysis

    11.1. Three-dimensional Orientation Distribution Functions (ODF)

    11.1.1. Determination of the Coefficients Cµvl from Individual Orientation Measurements

    11.1.2. The Rolling Textures of Face-centred Cubic Metals and Alloys

    11.1.3. The Theoretical Rolling Texture for {111} <110> Slip

    11.1.4. The Rolling Textures of Body-centred Cubic Metals

    11.1.5. Textures of Tubes

    11.1.6. Orthorhombic Crystal Symmetry

    11.1.7. Hexagonal Crystal Symmetry

    11.1.8. Trigonal Crystal Symmetry (Separation of Real Coincidences)

    11.1.9. Transformation Textures

    11.1.10. Cubic-triclinic Symmetry

    11.1.11. Representation of the Orientation Distribution Function by Rotation Axis and Rotation Angle

    11.2. Fibre Textures

    11.2.1. The Drawing Texture of Aluminium Wires

    11.2.2. Transformation Texture in Au—Ge TAYLOR Wires

    11.2.3. Hexagonal Crystal Symmetry (Titanium)

    11.2.4. Orthorhombic Crystal Symmetry (Separation of Partial Coincidences)

    11.2.5. Triclinic Crystal Symmetry (Application of the Refinement Procedure)

    11.2.6. Orientation Distribution of the Number of Crystallites and the Mean Grain Size

    11.2.7. Shape of the Spread about Preferred Orientations

    12. Orientation Distribution Functions of Other Structural Elements

    12.1. Orientation Distribution Functions of the Grain Surfaces

    12.1.1. Orientation Distribution of the Crystallographic Planes in the Outer Surface of an Arbitrary Section

    12.2. Orientation Distribution Functions of the Grain Boundaries

    12.2.1. The Distribution Function f(Δg) of the Orientation Differences 282

    12.2.2. The Distribution Function ϕ(y) of the Grain Boundaries in the Sample Fixed Coordinate System

    12.2.3. The Distribution Function ϕ(h) of the Grain Boundaries in the Crystal Fixed Coordinate System

    12.3. Orientation Distribution Functions of the Grain Edges

    12.3.1. The Distribution Function ϕ(y) of the Grain Edges in the Sample Fixed Coordinate System

    12.3.2. The Distribution Function ϕ(h) of the Grain Edges in the Crystal Fixed Coordinate System

    13. Physical Properties of Polycrystalline Materials

    13.1. Physical Properties of Single Crystals

    13.1.1. Representation by Tensors

    13.1.2. Representation by Surfaces

    13.1.3. Representation by Functions of the Orientation g

    13.2. The Problem of Averaging

    13.3. The Calculation of the Simple Mean Valued Ē

    13.3.1. Tensor Representation

    13.3.2. Surface Representation

    13.3.3. Representation by Orientation Functions

    13.4. Average Values of Special Properties

    13.4.1. Magnetization Energy in a Homogeneous Magnetic Field

    13.4.2. The Remanence in Ferromagnetic Materials

    13.4.3. Tensor Properties of Second Order

    13.4.4. Elastic Properties

    13.4.5. Plastic Anisotropy

    13.4.6. The Reflectivity of Crystallites for X-rays

    13.5. Determination of the Texture Coefficients from Anisotropic Polycrystal Properties

    13.6. Determination of Single Crystal Properties from Polycrystal Measurements

    13.7. Textures With Equal Physical Properties

    13.7.1. Fibre Textures of Ferromagnetic Cubic Materials

    13.7.2. Magnetic Anisotropy of an Fe—Si Sheet

    13.7.3. Tensor Properties of Second Rank for Fibre Textures

    13.8. Physical Meaning of the Coefficients Cµvl

    14. Mathematical Aids

    14.1. Generalized Spherical Harmonics

    14.2. Spherical Surface Harmonics

    14.3. FOUBIER Expansion of the Ρmnl(Φ)

    14.4. CLEBSCH—GORDAN Coefficients

    14.5. Symmetric Generalized Spherical Harmonics

    14.5.1. Transformation of the Coefficients Anvl

    14.5.2. The Fundamental Integral

    14.5.3. Convolution Integrals

    14.6. Symmetric Spherical Surface Harmonics

    14.7. The Symmetric Functions of the Various Symmetry Groups

    14.7.1. 'Lower' Symmetry Groups (Non-cubic)

    14.7.2. 'Higher' Symmetry Groups (Cubic)

    14.7.3. Subgroups

    14.7.4. Explicit Representation of Symmetric Generalized Spherical Harmonics

    14.7.5. Representation of the Cubic Spherical Surface Harmonics by Products of Powers of Cubic Polynomials

    14.7.6. Space Groups in the EULER Space

    14.7.7. Cubic Symmetry

    14.8. CLEBSCH—GORDAN Coefficients for Symmetric Functions

    15. Numerical Tables

    References

    Appendix 1 Tables 9.2-9.14

    Appendix 2 Listings of the ODF and Library Programs

    Appendix 3 Tables for Chapter 15

    15.1. Fourier Coefficients

    15.1.1. Qmnl

    15.1.2. a'mnsl

    15.1.3. a'mnsl

    15.2. Symmetry Coefficients Bmμl

    15.2.1. Cubic, Fourfold Axis

    15.2.2. Cubic, Threefold Axis

    15.2.3. Tetragonal, Orthogonal to Cubic

    15.2.4. Cubic, ROE'S Notation

    15.3. Generalized Legendre Functions Pmnl(Φ)

    15.4. Cubic Surface Harmonics kml(Φβ)

    15.4.1. In Steps of Φ and β

    15.4.2. For Low-index Directions

    15.5. Cubic Generalized Spherical Harmonics

    15.5.1. Cubic-orthorhombic Τmnl(φ1Φφ2)

    15.5.2. Cubic-Cubic Tµµ'l(φ1Φφ2)

    Appendix 4 Graphic Representations

    A4.1. The Generalized Legendre Functions Pmnl(Φ)

    A4.2. Cubic Spherical Harmonics kml(Φβ)

    A4.3. Cubic Generalized Spherical Harmonics

    A4.3.1. Cubic-orthorhombic Generalized Spherical Harmonics

    A4.3.2. Cubic-cubic Generalized Spherical Harmonics

    Subject Index

Product details

  • No. of pages: 614
  • Language: English
  • Copyright: © Butterworth-Heinemann 1982
  • Published: December 15, 1982
  • Imprint: Butterworth-Heinemann
  • eBook ISBN: 9781483278391

About the Author

H.-J. Bunge

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