Three-phase model for a fiber-reinforced composite material

The fiber-reinforced composite materials with cylindrical inclusions of a square cross-section are analyzed using three-phase composite model combined with the asymptotic homogenization. The problem of thermal conductivity of such composite structure is solved in the zero- and first-order approximations using the boundary shape perturbation method. And the explicit analytical expression for the effective conductivity is derived. Analysis of the effective thermal conductivity is carried out and the formulae for the effective conductivity are obtained over the entire range of geometric and physical properties of the composite material. The obtained asymptotic solution is compared with the known numerical and analytical results. It is shown that three-phase model for a composite structure with cylindrical inclusions of a square cross-section provides accurate solution for the effective thermal conductivity in the entire range of sizes of the inclusions and their thermal conductivities, including the limiting cases.