Optimized energy bounds on mixed-mode elastic constants of unidirectional transversely isotropic multicomponent materials

From the minimum energy variational principles, appropriate constant strain and self-equilibrated piece-wise constant stress trial fields, the free parameter-dependent combination bounds for the mixed longitudinal-transverse stress–strain mode effective moduli of the n-component macroscopically transversely isotropic unidirectional composites have been established. Appropriate optimization procedures have been constructed to derive the new original and iteration separation bounds on all of the moduli of the n-component materials. The new systems of the bounds are much tighter, significantly simpler than our previous ones, and most of them (9 from 12) appear to reduce to the benchmark ones for the specific two-component ones derived indirectly with the help of Hill-type relations, connecting the effective moduli in the case. However, the remaining 3 bounds are still less restrictive than the Hill-type ones for the two-component composites. Some illustrating numerical examples are given.