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


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.


Orientation average Polycrystals Texture Texture coefficients Effective anisotropic properties Materials design 

Mathematics Subject Classification (2010)

74Q15 74Q20 74P05 15A72 15A21 42C10 



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.


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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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