Evaluation of some approximate estimates for the effective tetragonal elastic moduli of two-phase fiber-reinforced composites

This paper examines three sets of approximate formulae for the overall tetragonal effective elastic properties of two-phase fiber-reinforced unidirectional composites with isotropic phases. The fibers are of circular cross-sections and periodically distributed in a matrix in a square pattern. The formulae by Kantor and Bergman, Luciano and Barbero, and estimates based on non-interacting Maxwell's type approximations are rewritten in unified notations. The latter approximations coincide with the most of well-known estimates of the effective medium theories (composite cylinder model, generalized self-consistent model and the Mori-Tanaka method), as well as with one of the Hashin-Shtrikman variational bounds. The approximate estimates are compared with the exact periodic solutions to determine the range of their applicability. The simplest and most accurate formulae are identified and suggested as a set of approximate expressions for accurate estimates of the effective elastic properties of composite materials with a square symmetry.