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Material Behavior: Texture and Anisotropy

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Handbook of Geomathematics

Abstract

This contribution is an attempt to present a self-contained and comprehensive survey of the mathematics and physics of material behavior of rocks in terms of texture and anisotropy. Being generally multi-phase and poly-crystalline, where each single crystallite is anisotropic with respect to its physical properties, texture, i.e., the statistical and spatial distribution of crystallographic orientations becomes a constitutive characteristic and determines the material behavior except for grain boundary effects, i.e., in first order approximation. This chapter is in particular an account of modern mathematical texture analysis, explicitly clarifying terms, providing definitions and justifying their application, and emphasizing its major insights. Thus, mathematical texture analysis is brought back to the realm of spherical Radon and Fourier transforms, spherical approximation, spherical probability, i.e., to the mathematics of spherical tomography.

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References

  • Abramson EH, Brown JM, Slutsky LJ, Zaug J (1997) The elastic constants of San Carlos olivine to 17 GPa. J Geophys Res 102:12253–12263

    Article  Google Scholar 

  • Altmann SL (1986) Rotations, quaternions and double groups. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Barruol G, Kern H (1996) P and S waves velocities and shear wave splitting in the lower crustal/ upper mantle transition (Ivrea Zone). Experimental and calculated data. Phys Earth Planet Int 95:175–194

    Article  Google Scholar 

  • Ben Ismaïl W, Mainprice D (1998) An olivine fabric database: an overview of upper mantle fabrics and seismic anisotropy. Tectonophysics 296:145–157

    Article  Google Scholar 

  • Bernier JV, Miller MP, Boyce DE (2006) A novel optimization-based pole-figure inversion method: comparison with WIMV and maximum entropy methods. J Appl Cryst 39:697-713

    Article  Google Scholar 

  • Bernstein S, Schaeben H (2005) A one-dimensional radon transform on SO(3) and its application to texture goniometry. Math Methods Appl Sci 28:1269–1289

    Article  MATH  MathSciNet  Google Scholar 

  • Bernstein S, Hielscher R, Schaeben H (2009) The generalized totally geodesic Radon transform and its application in texture analysis. Math Methods Appl Sci 32:379–394

    Article  MATH  MathSciNet  Google Scholar 

  • Boudier F, Baronnet A, Mainprice D (2009) Serpentine mineral replacements of natural olivine and their seismic implications: oceanic lizardite versus subduction-related antigorite. J Petrol. doi:10.1093/petrology/egp049

    Google Scholar 

  • Bunge HJ (1965) Zur Darstellung allgemeiner Texturen. Z Metallk 56:872–874

    Google Scholar 

  • Bunge HJ (1969) Mathematische Methoden der Texturanalyse. Akademie–Verlag, New York

    Google Scholar 

  • Bunge HJ (1982) Texture analysis in materials science. Butterworths, Boston, MA

    Google Scholar 

  • Bunge HJ, Weiland H (1988) Orientation correlation in grain and phase boundaries. Textures Microstruct 7:231–263

    Article  Google Scholar 

  • Chisholm M (2000) The sphere in three dimensions and higher: generalizations and special cases. Personal Communication

    Google Scholar 

  • Cowley JM (1995) Diffraction physics, 3rd edn. North-Holland Personal Library, Oxford.

    Google Scholar 

  • Crosson RS, Lin JW (1971) Voigt and Reuss prediction of anisotropic elasticity of dunite. J Geophys Res 76:570–578

    Article  Google Scholar 

  • Epanechnikov VA (1969) Nonparametric estimates of a multivariate probability density. Theor Prob Appl 14:153–158

    Article  Google Scholar 

  • Eshelby JD (1957) The determination of the elastic field of a ellipsoidal inclusion, and related problems. Proc R Soc Lond A241:376–396

    MathSciNet  Google Scholar 

  • Faccenda M, Burlini L, Gerya T, Mainprice D (2008) Fault-induced seismic anisotropy by hydration in subducting oceanic plates. Nature 455:1097–1101

    Article  Google Scholar 

  • Fengler MJ, Freeden W, Gutting M (2005) The Spherical Bernstein Wavelet. Schriften zur Funktionalanalysis und Geomathematik 20, TUKaiserslautern

    Google Scholar 

  • Forsyth JB (1988) Single crystal diffractometry. In: Newport RJ, Rainford BD, Cywinski R (eds) Neutron scattering at a pulsed source, Adam Hilger, Bristol, England pp 177–188,

    Google Scholar 

  • Friedel G (1913) Sur les symetries cristallines que peut reveler la diffraction des rayons Röntgen. CR Acad Sci 157:1533–1536

    Google Scholar 

  • Ganster J, Geiss D (1985) Polycrystalline simple average of mechanical properties in the general (triclinic) case. Phys Stat Sol (b) 132:395–407

    Article  Google Scholar 

  • Gel’fand IM, Minlos RA, Shapiro ZYa (1963) Representations of the rotation and Lorentz groups and their application. Pergamon Press, Oxford

    Google Scholar 

  • Gürlebeck K, Sprössig W (1997) Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley, New York

    MATH  Google Scholar 

  • Hall P, Watson GS, Cabrera J (1987) Kernel density estimation with spherical data. Biometrika 74:751–762

    Article  MATH  MathSciNet  Google Scholar 

  • Hammond C (1997) The basics of crystallography and diffraction. Oxford University Press, Oxford

    Google Scholar 

  • Hanson AJ (2006) Visualizing quaternions. Morgan Kaufmann, San Francisco, CA

    Google Scholar 

  • Helgason S (1984) Groups and geometric analysis. Academic Press, New York and Orlando

    MATH  Google Scholar 

  • Helgason S (1994) New York and Orlando Geometric analysis on symmetric spaces. Mathematical Surveys and Monographs 39, Am Math Soc

    Google Scholar 

  • Helgason S (1999) The Radon transform, 2nd ed. Birkhäuser Boston, Boston

    MATH  Google Scholar 

  • Hielscher R (2007) The Radon transform on the rotation group—inversion and application to texture analysis. PhD Thesis, TU Bergakademie Freiberg

    Google Scholar 

  • Hielscher R, Schaeben H (2008a) A novel pole figure inversion method: Specification of the MTEX algorithm. J Appl Cryst 41:1024–1037

    Article  Google Scholar 

  • Hielscher R, Schaeben H (2008b) MultiScale texture modeling. Math Geosci 40:63–82

    Article  MATH  MathSciNet  Google Scholar 

  • Hielscher R, Prestin J, Vollrath A (2010) Fast summation of functions on SO(3). Mathematical Geosci, to appear

    Google Scholar 

  • Hielscher R, Potts D, Prestin J, Schaeben H, Schmalz M (2008) The Radon transform on SO(3): a Fourier slice theorem and numerical inversion. Inverse Prob 24:025011 (21p)

    Google Scholar 

  • Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Lond Ser A65: 349–354

    Article  Google Scholar 

  • Hill R (1965) A self consistent mechanics of composite materials. J Mech Phys Solids 13:213–222

    Article  Google Scholar 

  • Johnson GC, Wenk HR (1986) Elastic properties of polycrystals with trigonal crystal and orthorhombic specimen symmetry. J Appl Phys 60:3868–3875

    Article  Google Scholar 

  • Katayama I, Hirauchi KI, Michibayashi K, Ando JI (2009) Trench-parallel anisotropy produced by serpentine deformation in the hydrated mantle wedge. Nature 461:1114–1118. doi:10.1038/nature08513

    Article  Google Scholar 

  • Kneller EA, Long MD, van Keken PE (2008) Olivine fabric transitions and shear wave anisotropy in the Ryukyu subduction system. Earth Planet Sci Lett 268:268–282

    Article  Google Scholar 

  • Kostelec PJ, Rockmore DN (2003) FFTs on the rotation group. Santa Fe Institute Working Papers Series Paper, 03-11-060

    Google Scholar 

  • Kreminski R (1997) Visualizing the Hopf fibration. Math Educ Res 6:9–14

    Google Scholar 

  • Kuipers JB (1999) Quaternions and rotation sequences—a primer with applications to orbits, aerospace, and virtual reality. Princeton Univer- sity Press, Princeton, NJ

    MATH  Google Scholar 

  • Kunze K (1991) Zur quantitativen Texturanalyse von Gesteinen: Bestimmung, Interpretation und Simulation von Quarzteilgefügen. PhD Thesis, RWTH Aachen, Germany

    Google Scholar 

  • Kunze K, Schaeben H (2004) The Bingham distribution of rotations and its spherical Radon transform in texture analysis. Math Geol 36:917–943

    Article  MathSciNet  Google Scholar 

  • Mainprice D (1990) A FORTRAN program to calculate seismic anisotropy from the lattice preferred orientation of minerals. Comp Geosci 16:385–393

    Article  Google Scholar 

  • Mainprice D, Humbert M (1994) Methods of calculating petrophysical properties from lattice preferred orientation data. Surv Geophys 15:575–592 (Special Issue Seismic properties of crustal and mantle rocks: laboratory measurements and theoretical calculations)

    Google Scholar 

  • Mainprice D, Humbert M, Wagner F (1990) Phase transformations and inherited lattice preferred orientation: implications for seismic properties. Tectonophysics 180:213–228

    Article  Google Scholar 

  • Mainprice D, Tommasi A, Couvy H, Cordier P, Frost DJ (2005) Pressure sensitivity of olivine slip systems: implications for the interpretation of seismic anisotropy of the Earths upper mantle. Nature 433:731–733

    Article  Google Scholar 

  • Mao HK, Shu J, Shen G, Hemley RJ, Li B, Singh, AK (1998) Elasticity and rheology of iron above 220 GPa and the nature of the Earths inner core. Nature 396:741–743

    Article  Google Scholar 

  • Matthies S (1979) On the reproducibility of the orientation distribution function of texture samples from pole figures (ghost phenomena). Phys Stat sol(b) 92:K135–K138

    Google Scholar 

  • Matthies S, Humbert M (1993) The realization of the concept of a geometric mean for calculating physical constants of polycrystalline materials. Phys Stat Sol (b) 177:K47–K50

    Article  Google Scholar 

  • Matthies S, Vinel GW, Helming K (1987) Standard Distributions in Texture Analysis vol I. Akademie Verlag, New York

    Google Scholar 

  • Meister L, Schaeben H (2004) A concise quaternion geometry of rotations. Math Methods Appl Sci 28:101–126

    Article  MathSciNet  Google Scholar 

  • Morawiec A (2004) Orientations and rotations. Springer, Berlin

    MATH  Google Scholar 

  • Morris PR (2006) Polycrystal elastic constants for triclinic crystal and physical symmetry. J Appl Cryst 39:502–508. doi:10.1107/S002188980 6016645

    Article  Google Scholar 

  • Muller J, Esling C, Bunge HJ (1981) An inversion formula expressing the texture function in terms of angular distribution function. J Phys 42: 161–165

    MathSciNet  Google Scholar 

  • Nye JF (1957) Physical properties of crystals — their representation by tensors and matrices. Oxford University Press, Oxford

    MATH  Google Scholar 

  • Nikiforov AF, Uvarov VB (1988) Special functions in mathematical physics. Birkhäuser Boston, Boston

    Google Scholar 

  • Nikolayev DI, Schaeben H (1999) Characteristics of the ultrahyperbolic differential equation governing pole density functions. Inverse Prob 15:1603–1619

    Article  MATH  MathSciNet  Google Scholar 

  • Pellenq RJM, Mainprice D, Ildefonse B, Devouard B, Baronnet A, Grauby O (2009) Atomistic calculations of the elastic properties of antigorite at upper mantle conditions: Application to the seismic properties in subduction zones. EPSL submitted

    Google Scholar 

  • Prior DJ, Mariani E, Wheeler J (2009) EBSD in the Earth Sciences: applications, common practice and challenges. In: Schwartz AJ, Kumar M, Adams BL, Field DP (eds) Electron backscatter diffraction in materials science. Springer, Berlin

    Google Scholar 

  • Randle V, Engler O (2000) Texture analysis: macrotexture, microtexture, and orientation mapping. Gordon and Breach Science, New York, NY

    Google Scholar 

  • Raterron P, Merkel S (2009) In situ rheological measurements at extreme pressure and temperature using synchrotron X-ray diffraction and radiography. J Synchrotron Radiat 16:748–756

    Article  Google Scholar 

  • Reuss A (1929) Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Z Angew Math Mech 9:49–58

    Article  MATH  Google Scholar 

  • Roe RJ (1965) Description of crystallite orientation in polycrystal materials III. General solution to pole figure inversion. J Appl Phys 36: 2024–2031

    Google Scholar 

  • Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27:832–837

    Article  MATH  MathSciNet  Google Scholar 

  • Sander B (1930) Gefügekunde der Gesteine mit besonderer Bercksichtigung der Tektonite. Springer, Berlin, pp 352

    Google Scholar 

  • Savyolova TI (1994) Inverse formulae for orientation distribution function. Bunge HJ (ed.) Proceedings of the Tenth International Conference on Textures of Materials (Materials Science Forum 15762): 419-421

    Google Scholar 

  • Schaeben H (1982) Fabric-diagram contour precision and size of counting element related to sample size by approximation theory methods. Math Geol 14:205–216 [Erratum: Math Geol 15: 579–580]

    Google Scholar 

  • Schaeben H (1997) A simple standard orientation density function: The hyperspherical de la Vallée Poussin kernel. Phys Stat Sol(b) 200:367–376

    Google Scholar 

  • Schaeben H (1999) The de la Vallée Poussin standard orientation density function. Textures and Microstruct 33:365–373

    Article  Google Scholar 

  • Schaeben H, Hielscher R, Fundenberger, J-J, Potts D, Prestin J (2007) Orientation density function-controlled pole probability density function measurements: automated adaptive control of texture goniometers. J Appl Cryst 40:570–579

    Article  Google Scholar 

  • Schaeben H, Sprößig W, van den Boogaart KG (2001) The spherical X-ray transform of texture goniometry. In: Brackx F, Chisholm JSR, Soucek V (eds) Clifford analysis and its applications. Proceedings of the NATO Advanced Research Workshop Prague, Oct. 30 Nov. 3, 2000, 283–291

    Google Scholar 

  • Schwartz AJ, Kumar M, Adams BL (2000) Electron back scatter diffraction in materials science. Kluwer Academic, Dordrecht

    Google Scholar 

  • Scott DW (1992) Multivariate density estimation—Theory, practice, and visualization. Wiley, New York

    Book  MATH  Google Scholar 

  • Tommasi A, Mainprice D, Cordier P, Thoraval C, Couvy H (2004) Strain-induced seismic anisotropy of wadsleyite polycrystals: constraints on flow patterns in the mantle transition zone. J Geophys Res 109:B12405 1–10

    Google Scholar 

  • Vajk KM (1995) Spin space and the strange properties of rotations. MSc Thesis, UC Santa Cruz, CA, USA

    Google Scholar 

  • Van den Boogaart KG (2002) Statistics for Individual Crystallographic Orientation Measurements. PhD Thesis, TU Bergakademie Freiberg

    Google Scholar 

  • Van den Boogaart KG, Hielscher R, Prestin J, Schaeben H (2007) Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis. J Comput Appl Math 199:122–140

    Article  MATH  MathSciNet  Google Scholar 

  • Van Houtte P (1980) A method for orientation distribution function analysis from incomplete pole figures normalized by an iterative method. Mater Sci Eng 43:7–11

    Article  Google Scholar 

  • Van Houtte P (1984) A new method for the determination of texture functions from incomplete pole figures - comparison with older methods. Textures Microstruct 6:137–162

    Article  Google Scholar 

  • Varshalovich D, Moskalev A, Khersonski V (1988) Quantum theory of angular momentum. World Scientific, Singapore

    Google Scholar 

  • Vilenkin NJ (1968) Secial functions and the theory of group representations. American Mathematical Society, Providence, RI

    Google Scholar 

  • Vilenkin NJ, Klimyk AU (1991) Representation of Lie groups and special fucntions, vol 1. Kluwer Academic

    Google Scholar 

  • Voigt W (1928) Lehrbuch der Kristallphysik. Teubner-Verlag

    Google Scholar 

  • Vollrath A (2006) Fast Fourier transforms on the rotation group and applications. Diploma thesis, Universität zu Lübeck

    Google Scholar 

  • Watson GS (1969) Density estimation by orthogonal series. Ann Math Stat 40:1496–1498

    Article  MATH  Google Scholar 

  • Watson GS (1983) Statistics on spheres. Wiley, New York

    MATH  Google Scholar 

  • Wenk HR (1985) Preferred orientation in deformed metals and rocks: an introduction to modern texture analysis. Academic Press, New York

    Google Scholar 

  • Zuo L, Xu J, Liang, Z (1989) Average fourth-rank elastic tensors for textured polycrystalline aggregates without symmetry. J Appl Phys 66:2338–2341

    Article  Google Scholar 

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Authors and Affiliations

  1. Applied Functional Analysis, Technical University Chemnitz, Chemnitz, Germany

    Ralf Hielscher, David Mainprice & Helmut Schaeben

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  1. Ralf Hielscher
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  2. David Mainprice
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  3. Helmut Schaeben
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Editors and Affiliations

  1. Geomathematics Group, Technische Universität Kaiserslautern, 3049, 67653, Kaiserslautern, Germany

    Willi Freeden

  2. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    M. Zuhair Nashed

  3. Computational Mathematics, Technische Universität Braunschweig, Pockelsstraße 14, 38106, Braunschweig, Germany

    Thomas Sonar

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Hielscher, R., Mainprice, D., Schaeben, H. (2010). Material Behavior: Texture and Anisotropy. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds) Handbook of Geomathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01546-5_33

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