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Representation of Hashin-Shtrikman bounds in terms of texture coefficients for arbitrarily anisotropic polycrystalline materials. (English) Zbl 1411.74048

Summary: The present work generalizes the results of T. Bö hlke and M. Lobos [“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; doi:10.1016/j.actamat.2013.11.003)] 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 L. J. Walpole [J. Mech. Phys. Solids 14, 151–162 (1966; Zbl 0139.18701)] 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.

MSC:

74Q15 Effective constitutive equations in solid mechanics
74Q20 Bounds on effective properties in solid mechanics
74P05 Compliance or weight optimization in solid mechanics
15A72 Vector and tensor algebra, theory of invariants
15A21 Canonical forms, reductions, classification
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Citations:

Zbl 0139.18701

Software:

Mathematica
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Full Text: DOI

References:

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