by which Hashin-Shtrikman bound on isotropic composite is attained.

by which Hashin-Shtrikman bound on isotropic composite is attained.

Voigt and Reuss composites provide upper and lower bounds respectively on the bulk and shear modulus of isotropic composites. See R. Hill, Elastic properties of reinforced solids: Some theoretical principles, Journal of the Mechanics and Physics of Solids, 11, 357-372 1963.

For Young's modulus, equality with the Voigt formula only occurs when the Poisson ratios are equal. Otherwise the modulus can exceed that in the Voigt formula.

The Voigt formula is often used in an elementary one dimensional view ignoring Poisson's ratio.

Three dimensional analysis reveals effects associated with mismatch in Poisson's ratio and with mutual constraint of laminations.