On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials

  • Mauricio FernándezEmail author


The present work continues the investigation first started by Lobos et al. (J. Elast. 128(1):17–60, 2017) concerning the orientation average of tensorial quantities connected to single-crystal physical quantities distributed in polycrystals. In Lobos et al. (J. Elast. 128(1):17–60, 2017), central orientation density functions were considered in the orientation average for fourth-order tensors with certain index symmetries belonging to single-crystal quantities. The present work generalizes the results of Lobos et al. (J. Elast. 128(1):17–60, 2017) for the orientation average of tensors of arbitrary order by presenting a clear connection to the Fourier expansion of central orientation density functions and of the general orientation density function in terms of tensorial texture coefficients. The closed form of the orientation average based on a central orientation density function is represented in terms of the Fourier coefficients (referred to as texture eigenvalues) and the central orientation of the central orientation density function. The given representation requires the computation of specific isotropic tensors. A pragmatic algorithm for the automated generation of a basis of isotropic tensors is given. Applications and examples are presented to show that the representation of the orientation average offers a low-dimensional parametrization with major benefits for optimization problems in materials science. A simple implementation in Python 3 for the reproduction of all examples is offered through the GitHub repository


Orientation average Polycrystals Central functions ODF Isotropic tensors 

Mathematics Subject Classification

74A40 74A60 74Q15 42C10 82D25 15A69 



  1. 1.
    Adams, B.L., Lyon, M., Henrie, B.: Microstructures by design: linear problems in elastic–plastic design. Int. J. Plast. 20(8–9), 1577–1602 (2004). CrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, B.L., Kalidindi, S.R., Fullwood, D.T.: Microstructure Sensitive Design for Performance Optimization. Butterworth-Heinemann, Waltham (2013). CrossRefGoogle Scholar
  3. 3.
    Andrews, D.L., Ghoul, W.A.: Eighth rank isotropic tensors and rotational averages. J. Phys. A, Math. Gen. 14(6), 1281–1290 (1999). ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Andrews, D.L., Thirunamachandran, T.: On three-dimensional rotational averages. J. Chem. Phys. 67(11), 5026 (1977). ADSCrossRefGoogle Scholar
  5. 5.
    Auffray, N.: On the algebraic structure of isotropic generalized elasticity theories. Math. Mech. Solids 20(5), 565–581 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Böhlke, T.: Application of the maximum entropy method in texture analysis. Comput. Mater. Sci. 32(3–4), 276–283 (2005). CrossRefGoogle Scholar
  7. 7.
    Böhlke, T.: Texture simulation based on tensorial Fourier coefficients. Comput. Struct. 84(17–18), 1086–1094 (2006). CrossRefGoogle Scholar
  8. 8.
    Böhlke, T., Bertram, A., Krempl, E.: Modeling of deformation induced anisotropy in free-end torsion. Int. J. Plast. 19(11 SPEC.), 1867–1884 (2003). CrossRefzbMATHGoogle Scholar
  9. 9.
    Böhlke, T., Risy, G., Bertram, A.: A texture component model for anisotropic polycrystal plasticity. Comput. Mater. Sci. 32(3–4), 284–293 (2005). CrossRefGoogle Scholar
  10. 10.
    Bunge, H.J.: In: Texture Analysis in Materials Science: Mathematical Methods, Butterworth, London (1982) Google Scholar
  11. 11.
    Cao, T., Cuffari, D., Bongiorno, A.: First-principles calculation of third-order elastic constants via numerical differentiation of the second Piola-Kirchhoff stress tensor. Phys. Rev. Lett. 121(21), 1 (2018). CrossRefGoogle Scholar
  12. 12.
    Fernández, M., GitHub repository (2019).
  13. 13.
    Fullwood, D.T., Niezgoda, S.R., Adams, B.L., Kalidindi, S.R.: Microstructure sensitive design for performance optimization. Prog. Mater. Sci. 55(6), 477–562 (2010). CrossRefGoogle Scholar
  14. 14.
    Gel’fand, I.M., Minlos, R., Shapiro, Z.: Representations of the Rotation and Lorentz Groups and Their Applications. Pergamon Press, Oxford (1963). zbMATHGoogle Scholar
  15. 15.
    Glüge, R., Kalisch, J., Bertram, A.: The eigenmodes in isotropic strain gradient elasticity. In: Generalized Continua as Models for Classical and Advanced Materials, pp. 163–178 (2016) CrossRefGoogle Scholar
  16. 16.
    Guidi, M., Adams, B.L., Onat, E.T.: Tensorial representation of the orientation distribution function in cubic polycrystals. Textures Microstruct. 19(3), 147–167 (1992). CrossRefGoogle Scholar
  17. 17.
    Hearmon, R.F.S.: ‘third-order’ elastic coefficients. Acta Crystallogr. 6(4), 331–340 (1953). MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jakata, K., Every, A.G.: Determination of the dispersive elastic constants of the cubic crystals Ge, Si, GaAs, and InSb. Phys. Rev. B, Condens. Matter Mater. Phys. 77(17), 1 (2008). CrossRefGoogle Scholar
  19. 19.
    Jerphagnon, J., Chemla, D., Bonneville, R.: The description of the physical properties of condensed matter using irreducible tensors. Adv. Phys. 27(4), 609–650 (1978). ADSCrossRefGoogle Scholar
  20. 20.
    Kalidindi, S.R., Houskamp, J.R., Lyons, M., Adams, B.L.: Microstructure sensitive design of an orthotropic plate subjected to tensile load. Int. J. Plast. 20(8–9), 1561–1575 (2004). CrossRefzbMATHGoogle Scholar
  21. 21.
    Kearsley, E.A., Fong, J.T.: Linearly independent sets of isotropic Cartesian tensors of ranks up to eight. J. Res. Natl. Bur. Stand. B, Math. Sci. 79B(1), 49 (1975). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Korobov, A.I., Prokhorov, V.M., Mekhedov, D.M.: Second-order and third-order elastic constants of B95 aluminum alloy and B95/nanodiamond composite. Phys. Solid State 55(1), 8–11 (2013). ADSCrossRefGoogle Scholar
  23. 23.
    Lobos Fernández, M.: Homogenization and materials design of mechanical properties of textured materials based on zeroth-, first- and second-order bounds of linear behavior. Doctoral thesis, Karlsruhe Institute of Technology (2018).
  24. 24.
    Lobos Fernández, M., Böhlke, T.: Representation of Hashin–Shtrikman bounds in terms of texture coefficients for arbitrarily anisotropic polycrystalline materials. J. Elast. 134, 1–38 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lobos, M., Yuzbasioglu, T., Böhlke, T.: Homogenization and materials design of anisotropic multiphase linear elastic materials using central model functions. J. Elast. 128(1), 17–60 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mackey, J.E., Arnold, R.T.: Some combinations of third-order elastic constants for strontium titanate single crystals. J. Appl. Phys. 40(12), 4806–4811 (1969). ADSCrossRefGoogle Scholar
  27. 27.
    Man, C.-S., Huang, M.: A representation theorem for material tensors of weakly-textured polycrystals and its applications in elasticity. J. Elast. 106(1), 1–42 (2012). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Schouten, J.A.: Der Ricci-Kalkül. Springer, Berlin (1924). CrossRefzbMATHGoogle Scholar
  29. 29.
    Takahashi, S., Motegi, R.: Measurement of third-order elastic constants and applications to loaded structural materials. SpringerPlus 4(1), 1–20 (2015). CrossRefGoogle Scholar
  30. 30.
    Vilenkin, N.J.: Special Functions and the Theory of Group Representations, vol. 22. Am. Math. Soc., Providence (1968) CrossRefGoogle Scholar
  31. 31.
    Voigt, W.: Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Teubner, Leipzig (1910) zbMATHGoogle Scholar
  32. 32.
    Zheng, Q.-S., Fu, Y.-B.: Orientation distribution functions for microstructures of heterogeneous materials (II)—crystal distribution functions and irreducible tensors restricted by various material symmetries. Appl. Math. Mech. 22(8), 885–902 (2001) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zheng, Q.-S., Spencer, A.J.M.: On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int. J. Eng. Sci. 31(4), 617–635 (1993). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zou, W.-N., Zheng, Q.-S., Du, D.-X., Rychlewski, J.: Orthogonal irreducible decompositions of tensors of high order. Math. Mech. Solids 6(3), 249–267 (2001). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.EMMA - Efficient Methods for Mechanical Analysis, Institute of Applied Mechanics (CE)University of StuttgartStuttgartGermany

Personalised recommendations