Minimum energy bounds on longitudinal elastic constants of transversely isotropic unidirectional composites

We consider then -component transversely isotropic unidirectional elastic composites, the longitudinal axis of which is parallel to those of the transversely isotropic components as well as the generators of the cylindrical phase boundaries between them. From the minimum energy and complementary energy principles, with appropriate constant strain and piece-wise constant stress trial fields, optimization and iteration techniques, a set of bounds for the macroscopic (effective) longitudinal elastic constants of the composites (including the simple lower arithmetic average estimate for longitudinal Young modulusE eff  ≥ E V ) are constructed. Numerical examples are provided to illustrate the obtained results.