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