Refinement of the Maxwell Formula for Fiber-Reinforced Composites

The effective properties of the fiber-reinforced composite materials with fibers of square cross-section are investigated. The novel formula for the effective coefficient of thermal conductivity refining the classical Maxwell formula (MF) is derived. The methods of asymptotic homogenization, boundary shape perturbation and Schwarz alternating process are applied. It is shown that the principal term of the asymptotic expansion of the refined formula in powers of small size of inclusions coincides with the classical MF. The corrections to the MF are obtained for different values of geometrical and physical properties of the constituents of the composite material. The analytical and numerical analyses are carried out and illustrated graphically. In particular, the derived refined formula and the MF are compared for the limiting values of the geometric dimensions and physical properties of the composite. It is shown that the refined formula is applicable for the inclusions with any conductivity in the entire range of the geometric sizes of inclusions, including the limiting cases of inclusions with zero thermal conductivity and maximally large inclusions.