Analytical expressions for effective thermal conductivity of composite materials with inclusions of square cross-section

Thermal conductivity of composite materials with periodically distributed cylindrical inclusions of square cross-section is investigated. Analytical interpolation formula for the effective thermal conductivity in transversal direction valid in the entire range of volume fraction of inclusions and value of their thermal conductivity is derived. The methods of asymptotic homogenization, boundary shape perturbation and asymptotically equivalent functions are used. Asymptotic expansions of the effective thermal conductivity in limiting cases of size of inclusions and their thermal conductivity are obtained. The analytical expressions are evaluated numerically and compared to other models. It is shown that the discrepancy between these results is very low in all cases.