Calculation of Effective Characteristics of a 2D Composite with Rhombic Voids Using an Inhomogeneous Cell Model

One of the most common approximations in the theory of composite materials is the homogenization theory. The main difficulty in its application lies in solving the cell problem, i.e., the boundary value problem on a periodically repeating element of composite material. For the case of inclusions of large size and with characteristics far superior to those of the matrix, the lubrication approach gives good results. However, the use of such an asymptotic approach is unnatural for the case when the characteristics of the matrix exceed the characteristics of the inclusion. In the presented work, the problem of conductivity for a 2D composite with rhombic voids is considered. The solution of the cell problem is carried out by two methods. First, the lubrication approach is used for this purpose. In addition, a modification of the Schwarz alternating method is proposed. This new approach has been called an "inhomogeneous cell model". Both methods made it possible to obtain analytical expressions for the effective conductivity. A comparison of the indicated approximate models is carried out, and it is shown that the obtained solutions exactly satisfy Keller's theorem. The physical foundations of the proposed inhomogeneous cell model are discussed. Its advantage in solving the considered problem is shown.