Three-phase model for a composite material with cylindrical circular inclusions. Part I: Application of the boundary shape perturbation method

The problem of determining the effective thermal conductivity of a composite material with periodic cylindrical inclusions of a circular cross-section arranged in a square grid is analyzed. Defining mathematical relationships are derived on the basis of a three-phase composite model, asymptotic homogenization technique and application of the boundary shape perturbation method to solve the derived unit cell problems. The analytical expression for the effective coefficient of thermal conductivity is obtained in the zero-order approximation and the correction to this expression is derived in the first-order approximation. This correction allows taking into account the geometry of inclusion, not just its volume fraction. It is shown that a small ε1-order perturbation of the unit cell contour yields the ε12-order contribution to the homogenized relations. The obtained solution is analyzed and compared with known results in some particular cases, and the limits of its applicability are evaluated.