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Journal of Elasticity

, Volume 134, Issue 1, pp 1–38 | Cite as

Representation of Hashin–Shtrikman Bounds in Terms of Texture Coefficients for Arbitrarily Anisotropic Polycrystalline Materials

  • Mauricio Lobos FernándezEmail author
  • Thomas BöhlkeEmail author
Article
  • 109 Downloads

Abstract

The present work generalizes the results of Böhlke and Lobos (Acta Mater. 67:324–334, 2014) by giving an explicit representation of the Hashin–Shtrikman (HS) bounds of linear elastic properties in terms of tensorial Fourier texture coefficients not only for cubic materials but for arbitrarily anisotropic linear elastic polycrystalline materials. Based on the HS bounds as given by Walpole (J. Mech. Phys. Solids 14(3):151–162, 1966) and tensor functions for the representation of the crystallite orientation distribution function, it is shown that the HS bounds are represented in terms of the exact same second- and fourth-order texture coefficients which appear in the consideration of the Voigt and Reuss bounds. The derived representations in terms of tensorial texture coefficients are valid for all symmetry classes in elasticity and present expressions highly attractive for the description of physical quantities in terms of tensorial variables. In order to make these results also accessible for the community of quantitative texture analysis, transformation relations between experimentally obtained Bunge’s or Roe’s coefficients and the tensorial texture coefficients are given. The representation of the present work offers a finite and low dimensional parametrization of the fully anisotropic Hashin–Shtrikman bounds, which can be used in inverse materials design problems in order to explore the set of possible materials properties or for the determination of optimal microstructural influence with respect to prescribed material properties. Examples for orthotropic polycrystals of cubic materials and transversely isotropic polycrystals of hexagonal materials (showing the connection and applicability of the results also to fiber orientation distributions) are discussed. Finally, an implementation in Mathematica® 11 of the HS bounds for arbitrarily anisotropic materials is offered, such that readers can reproduce all the results of this work and use them for their own purposes.

Keywords

Orientation average Polycrystals Texture Texture coefficients Effective anisotropic properties Materials design 

Mathematics Subject Classification (2010)

74Q15 74Q20 74P05 15A72 15A21 42C10 

Notes

Acknowledgements

M. Lobos Fernández thanks Prof. John R. Willis for his valuable comments and fruitful discussions during a research stay in Cambridge supported by the Karlsruhe House of Young Scientist (KHYS) of the Karlsruhe Institute of Technology (KIT). T. Böhlke acknowledges the partial support by the German Research Foundation (DFG) within the International Research Training Group “Integrated engineering of continuous-discontinuous long fiber reinforced polymer structures” (GRK 2078). Furthermore, the authors deeply thank the reviewers for their valuable suggestions, which, without a doubt, greatly improved the quality of the present manuscript.

References

  1. 1.
    Adams, B.L., Boehler, J., Guidi, M., Onat, E.: Group theory and representation of microstructure and mechanical behavior of polycrystals. J. Mech. Phys. Solids 40(4), 723–737 (1992) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, B.L., Kalidindi, S.R., Fullwood, D.T.: Microstructure Sensitive Design for Performance Optimization. Butterworth-Heinemann, Waltham (2013).  https://doi.org/10.1016/B978-0-12-396989-7.00001-0 Google Scholar
  3. 3.
    Böhlke, T.: Application of the maximum entropy method in texture analysis. Comput. Mater. Sci. 32(3–4), 276–283 (2005).  https://doi.org/10.1016/j.commatsci.2004.09.041 CrossRefGoogle Scholar
  4. 4.
    Böhlke, T.: Texture simulation based on tensorial Fourier coefficients. Comput. Struct. 84(17–18), 1086–1094 (2006).  https://doi.org/10.1016/j.compstruc.2006.01.006 CrossRefGoogle Scholar
  5. 5.
    Böhlke, T., Haus, U.U., Schulze, V.: Crystallographic texture approximation by quadratic programming. Acta Mater. 54(5), 1359–1368 (2006).  https://doi.org/10.1016/j.actamat.2005.11.009 CrossRefGoogle Scholar
  6. 6.
    Böhlke, T., Lobos, M.: Representation of Hashin–Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design. Acta Mater. 67, 324–334 (2014).  https://doi.org/10.1016/j.actamat.2013.11.003 CrossRefGoogle Scholar
  7. 7.
    Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Springer, Berlin, Heidelberg (1985) CrossRefzbMATHGoogle Scholar
  8. 8.
    Bunge, H.J.: Texture Analysis in Materials Science: Mathematical Methods. Butterworth, London (1982) Google Scholar
  9. 9.
    Cheng, L., Assary, R.S., Qu, X., Jain, A., Ong, S.P., Rajput, N.N., Persson, K., Curtiss, L.A.: Accelerating electrolyte discovery for energy storage with high-throughput screening. J. Phys. Chem. Lett. 6(2), 283–291 (2015).  https://doi.org/10.1021/jz502319n CrossRefGoogle Scholar
  10. 10.
    Eschner, T., Fundenberger, J.J.: Application of anisotropic texture components. Textures Microstruct. 28(C), 181–195 (1997) CrossRefGoogle Scholar
  11. 11.
    Forte, S., Vianello, M.: Symmetry classes for elasticity tensors. J. Elast. 43(2), 81–108 (1996).  https://doi.org/10.1007/BF00042505 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Forte, S., Vianello, M.: Symmetry classes and harmonic decomposition for photoelasticity tensors. Int. J. Eng. Sci. 35(14), 1317–1326 (1997).  https://doi.org/10.1016/S0020-7225(97)00036-0 MathSciNetCrossRefzbMATHGoogle Scholar
  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).  https://doi.org/10.1016/j.pmatsci.2009.08.002 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.
    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).  https://doi.org/10.1155/TSM.19.147 CrossRefGoogle Scholar
  16. 16.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10(4), 343–352 (1962).  https://doi.org/10.1016/0022-5096(62)90005-4 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10(4), 335–342 (1962) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127–140 (1963) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Helming, K.: Some applications of the texture component model. Mater. Sci. Forum 157(162), 363–368 (1994).  https://doi.org/10.4028/www.scientific.net/MSF.157-162.363 CrossRefGoogle Scholar
  20. 20.
    Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11(5), 357–372 (1963) ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1990) zbMATHGoogle Scholar
  22. 22.
    Huang, M., Man, C.S.: Explicit bounds of effective stiffness tensors for textured aggregates of cubic crystallites. Math. Mech. Solids 13(5), 408–430 (2007).  https://doi.org/10.1177/1081286507078299 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jain, A., Ong, S.P., Hautier, G., Chen, W., Richards, W.D., Dacek, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., Persson, K.A.: Commentary: the materials project: a materials genome approach to accelerating materials innovation. APL Mater. 1(1), 011002 (2013).  https://doi.org/10.1063/1.4812323 ADSCrossRefGoogle Scholar
  24. 24.
    de Jong, M., Chen, W., Angsten, T., Jain, A., Notestine, R., Gamst, A., Sluiter, M., Krishna Ande, C., van der Zwaag, S., Plata, J.J., Toher, C., Curtarolo, S., Ceder, G., Persson, K.A., Asta, M.: Charting the complete elastic properties of inorganic crystalline compounds. Sci. Data 2, 1–13 (2015).  https://doi.org/10.1038/sdata.2015.9 CrossRefGoogle Scholar
  25. 25.
    Kalidindi, S.R., Knezevic, M., Niezgoda, S., Shaffer, J.: Representation of the orientation distribution function and computation of first-order elastic properties closures using discrete Fourier transforms. Acta Mater. 57(13), 3916–3923 (2009).  https://doi.org/10.1016/j.actamat.2009.04.055 CrossRefGoogle Scholar
  26. 26.
    Kröner, E.: Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls. Z. Phys. 151(4), 504–518 (1958) ADSCrossRefGoogle Scholar
  27. 27.
    Kröner, E.: Bounds for effective elastic moduli of disordered materials. J. Mech. Phys. Solids 25(2), 137–155 (1977).  https://doi.org/10.1016/0022-5096(77)90009-6 ADSCrossRefzbMATHGoogle Scholar
  28. 28.
    Kröner, E.: Self-consistent scheme and graded disorder in polycrystal elasticity. J. Phys. F, Met. Phys. 8, 2261–2267 (1978) ADSCrossRefGoogle Scholar
  29. 29.
    Lobos, M., Böhlke, T.: Materials design for the anisotropic linear elastic properties of textured cubic crystal aggregates using zeroth-, first- and second-order bounds. Int. J. Mech. Mater. Des. 11(1), 59–78 (2015).  https://doi.org/10.1007/s10999-014-9272-z CrossRefGoogle Scholar
  30. 30.
    Lobos, M., Böhlke, T.: On optimal zeroth-order bounds of linear elastic properties of multiphase materials and application in materials design. Int. J. Solids Struct. 84, 40–48 (2016).  https://doi.org/10.1016/j.ijsolstr.2015.12.015 CrossRefGoogle Scholar
  31. 31.
    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).  https://doi.org/10.1007/s10659-016-9615-0 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    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, Karlsruhe, Germany (2018, in press) Google Scholar
  33. 33.
    Man, C.S.: On the constitutive equations of some weakly-textured materials. Arch. Ration. Mech. Anal. 143, 77–103 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Man, C.S., Huang, M.: A simple explicit formula for the Voigt–Reuss–Hill average of elastic polycrystals with arbitrary crystal and texture symmetries. J. Elast. 105(1–2), 29–48 (2011).  https://doi.org/10.1007/s10659-011-9312-y MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    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).  https://doi.org/10.1007/s10659-010-9284-3 MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mardia, K.V., Jupp, P.E.: Directional Statistics. Wiley, London (2008) zbMATHGoogle Scholar
  37. 37.
    Matthies, S., Muller, J., Vinel, G.: On the normal distribution in the orientation space. Textures Microstruct. 10(1), 77–96 (1988) CrossRefGoogle Scholar
  38. 38.
    Mehrabadi, M.M., Cowin, S.C.: Eigentensors of linear anisotropic elastic materials. Q. J. Mech. Appl. Math. 43(1), 15–41 (1990).  https://doi.org/10.1093/qjmam/43.1.15 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Milton, G.W.: The Theory of Composites, vol. 6. Cambridge University Press, Cambridge (2002).  https://doi.org/10.1017/CBO9780511613357 CrossRefzbMATHGoogle Scholar
  40. 40.
    Müller, V., Böhlke, T.: Prediction of effective elastic properties of fiber reinforced composites using fiber orientation tensors. Compos. Sci. Technol. 130, 36–45 (2016).  https://doi.org/10.1016/j.compscitech.2016.04.009 CrossRefGoogle Scholar
  41. 41.
    Nadeau, J., Ferrari, M.: On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters. Int. J. Solids Struct. 38(44–45), 7945–7965 (2001).  https://doi.org/10.1016/S0020-7683(00)00393-0 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Niezgoda, S.R., Kanjarla, A.K., Kalidindi, S.R.: Novel microstructure quantification framework for databasing, visualization, and analysis of microstructure data. Integr. Mater. Manuf. Innov. 2(1), 3 (2013).  https://doi.org/10.1186/2193-9772-2-3 CrossRefGoogle Scholar
  43. 43.
    Nomura, S., Kawai, H., Kimura, I., Kagiyama, M.: General description of orientation factors in terms of expansion of orientation distribution function in a series of spherical harmonics. J. Polym. Sci., Part A-2, Polym. Phys. 8(3), 383–400 (1970).  https://doi.org/10.1002/pol.1970.160080305 ADSCrossRefGoogle Scholar
  44. 44.
    Ponte Castañeda, P., Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 171–302 (1997) CrossRefzbMATHGoogle Scholar
  45. 45.
    Reuss, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle Z. Angew. Math. Mech. 9(1), 49–58 (1929) CrossRefzbMATHGoogle Scholar
  46. 46.
    Roe, R.J.: Description of crystallite orientation in polycrystalline materials. III. General solution to pole figure inversion. J. Appl. Phys. 36(6), 2024 (1965).  https://doi.org/10.1063/1.1714396 ADSCrossRefGoogle Scholar
  47. 47.
    Schaeben, H.: Texture approximation or texture modelling with components represented by the von Mises-Fisher matrix distribution on SO(3) and the Bingham distribution on S4+. J. Appl. Crystallogr. 29(5), 516–525 (1996).  https://doi.org/10.1107/S0021889896002804 CrossRefGoogle Scholar
  48. 48.
    Schaeben, H., van den Boogaart, K.G.: Spherical harmonics in texture analysis. Tectonophysics 370(1), 253–268 (2003).  https://doi.org/10.1016/S0040-1951(03)00190-2 ADSCrossRefGoogle Scholar
  49. 49.
    Schouten, J.A.: Der Ricci-Kalkül. Springer, Berlin (1924).  https://doi.org/10.1007/978-3-662-06545-7 CrossRefzbMATHGoogle Scholar
  50. 50.
    Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002) CrossRefzbMATHGoogle Scholar
  51. 51.
    Varshalovich, D., Moskalev, A.N., Khersonskii, V.K.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988) CrossRefGoogle Scholar
  52. 52.
    Voigt, W.: Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Teubner, Leipzig (1910) zbMATHGoogle Scholar
  53. 53.
    Walpole, L.J.: On bounds for the overall elastic moduli of inhomogeneous systems—I. J. Mech. Phys. Solids 14(3), 151–162 (1966) ADSCrossRefzbMATHGoogle Scholar
  54. 54.
    Wassermann, G., Grewen, J.: Texturen metallischer Werkstoffe, 2nd edn. Springer, Berlin, Heidelberg (1962).  https://doi.org/10.1007/978-3-662-13128-2 CrossRefGoogle Scholar
  55. 55.
    Wigner, E.P.: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum. Vieweg+Teuber, Wiesbaden (1931) CrossRefzbMATHGoogle Scholar
  56. 56.
    Willis, J.R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25(3), 185–202 (1977).  https://doi.org/10.1016/0022-5096(77)90022-9 ADSCrossRefzbMATHGoogle Scholar
  57. 57.
    Willis, J.R.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 1–78 (1981) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Yabansu, Y.C., Kalidindi, S.R.: Representation and calibration of elastic localization kernels for a broad class of cubic polycrystals. Acta Mater. 94, 26–35 (2015).  https://doi.org/10.1016/j.actamat.2015.04.049 CrossRefGoogle Scholar
  59. 59.
    Zheng, Q.S., Fu, Y.B.: Orientation distribution functions for microstructures of heterogeneous m materials (II)—crystal distribution functions and irreducible tensors restricted by various material symmetries. Appl. Math. Mech. 22(8), 885–902 (2001) MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.EMMA—Efficient Methods for Mechanical Analysis, Institute of Applied Mechanics (CE)University of StuttgartStuttgartGermany
  2. 2.Chair for Continuum Mechanics, Institute of Engineering MechanicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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