Hyperspace bounds on mixed mode elastic constants of n-component transversely-isotropic unidirectional materials

We consider the general elastic unidirectional composites composed of n transversely-isotropic components (with unidirectional cylindrical boundaries between the phases), which are just macroscopically isotropic in the transverse plane. All 6 mixed-mode elastic constants appear in the complex mixed-mode longitudinal- transverse stress-strain relations (not well-separated into the pure hydrostatic and deviatoric modes as in the usual problems concerning the 3D-or 2D-isotropic bulk and shear moduli). They are subjected to various combination bounds, derived from the minimum energy principles, involving many effective properties simultaneously in the multidimensional-elastic-constant hyperspace. Exploring the possible trial mixed stress- strain modes, we derive from the combination bounds the new sets of bounds, with additional and more refined ones, on all 6 effective mixed-mode longitudinal-transverse constants of the n-component transversely-isotropic unidirectional composites, which improve significantly over the previous results. Refined optimization techniques have been used in conjunction with the iteration procedure. Using Hill relations relating the exact values of some of the effective elastic constants in the two-component case, the bench-mark Paul-Hill-type bounds have also been deduced for all those 6 mixed elastic constants for comparisons with our new bounds (three of them appear to coincide with those of ours, while the others comparatively are near-to-close in numerical examples) in that specific case. Illustrating numerical examples are provided for some two-and multi-component cases. Related to the subject - the matrix approach on bounding elastic anisotropic composites in the literature is critically examined.