Walpole bounds on the effective elastic moduli of isotropic multicomponent composites

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Abstract
Explicit results are presented for Walpole bounds on the effective elastic properties of overall isotropic multiphase composites of randomly oriented crystals of materials with either cubic or hexagonal symmetry. The Walpole bounds are considerably narrower than the widely used Voigt and Reuss bounds, but slightly further apart than the Hashin–Shtrikman bounds. The difference between the Walpole and Hashin–Shtrikman bounds arises because of the way the comparison material that optimizes the bounds is chosen. The usual Hashin–Shtrikman (and Voigt-Reuss) approach to estimating the overall elastic moduli of multiphase composites from the single-crystal elastic stiffnesses of the component phases calculates polycrystalline averages for each phase separately, then uses the arithmetic means of the polycrystalline bounds as ‘‘the’’ elastic properties of the (isotropic) components of the multiphase aggregate. This technique is not appropriate for all isotropic multiphase composites, depending on how the composite is formed. The Walpole averaging technique performs the averaging for all the components simultaneously. We demonstrate that the widely used Voigt–Reuss–Hill average can differ from the arithmetic mean of the Walpole or Hashin–Shtrikman bounds up to 10 or 20% in composites with large differences in the elastic properties of the components.

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