On the computation of the Hashin–Shtrikman bounds for transversely isotropic two-phase linear elastic fibre-reinforced composites

Abstract

Although various forms for the Hashin–Shtrikman bounds on the effective elastic properties of inhomogeneous materials have been written down over the last few decades, it is often unclear how to construct and compute such bounds when the material is not of simple type (e.g. isotropic spheres inside an isotropic host phase). Here, we show how to construct, in a straightforward manner, the Hashin–Shtrikman bounds for generally transversely isotropic two-phase particulate composites where the inclusion phase is spheroidal, and its distribution is governed by spheroidal statistics. Note that this case covers a multitude of composites used in applications by taking various limits of the spheroid, including both layered media and long unidirectional composites. Of specific interest in this case is the fact that the corresponding Eshelby and Hill tensors can be derived analytically. That the shape of the inclusions and their distribution can be specified independently is of great utility in composite design. We exhibit the implementation of the computations with several examples.