Journal of Elasticity

, Volume 105, Issue 1–2, pp 29–48 | Cite as

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

  • Chi-Sing Man
  • Mojia Huang
Original Paper


The Voigt-Reuss-Hill (VRH) average provides a simple way to estimate the elastic constants of a textured polycrystal in terms of its crystallographic texture and the elastic constants of the constituting crystallites. Empirically the VRH estimates were found in most cases to have an accuracy comparable to those obtained by more sophisticated techniques such as self-consistent schemes. In this paper we determine, in the space of fourth-order tensors with major and minor symmetries, a special set of irreducible basis tensors, with which we obtain a simple explicit formula for the VRH average for elastic polycrystals with arbitrary crystal and texture symmetries. Our formula is correct to first order in the texture coefficients.


Polycrystals Texture Anisotropy Voigt-Reuss-Hill average Irreducible tensor basis Elasticity 

Mathematics Subject Classification (2000)

74A20 74A60 74E10 74E15 74E25 74B05 20C35 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KentuckyLexingtonUSA
  2. 2.Institute for Advanced Study, and Institute of Engineering MechanicsNanchang UniversityNanchangChina

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