Polarization estimates for longitudinal elastic moduli of unidirectional multi-component materials

We consider the elastic unidirectional composite composed of $n$ transversely-isotropic components, all of which are distributed statistically-isotropic in the transverse plane. Based on the minimum energy or complementary energy principles, the polarization (extended Hashin–Shtrikman-type) strain and stress trial fields are constructed to derive new direct upper or lower polarization estimates for some macroscopic (effective) longitudinal elastic moduli of the $n$-component materials. The more-refined non-trivial polarization trial fields can utilize the statistical isotropy assumption in the transverse plane of the composite to give definitely better estimates from above or below on the effective moduli than those obtained earlier using only the much simpler basic constant strain and piece-wise constant stress trial ones effective for the unidirectional composites which are macroscopically isotropic in the transverse plane (Pham, 2020). The primary upper or lower bounds appear optimal in the two-component case (attained by the coated cylinder assemblage model). The new and other available estimates, and/or exact models are compared in a number of two- and three-component illustrating numerical examples.