Thermal conduction of a circular inclusion with variable interface parameter

An imperfect bonding problem associated with a solitary circular inclusion embedded in an infinite matrix under a remotely applied uniform intensity is considered. Specifically, we study the effect of imperfect interfaces which are either of weakly or of highly conducting type and that the interface parameter could vary arbitrarily along the interface. By using the orthogonality properties of the trigonometric series, we show that the solution field is governed by a linear set of algebraic equations with an infinite number of unknowns. The governing matrix for the unknowns is primarily composed of elements which are simple combinations of the Fourier coefficients of the interface parameter. Solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of circular inclusions with equal size. The effective properties solely depend on two particular constants among an infinite number of unknowns. It is demonstrated that, even for a composite with isotropic dispersions of inclusions, the composite may become effectively anisotropic due to the presence of a variable interface parameter. Further, we present two microstructure independent properties regarding the effective conductivity of the considered system. We first show that the effective conductivity tensor for a composite with variably imperfect interfaces is always diagonally symmetric. This is accomplished by means of a reciprocal relation that is established in such systems. Next, we present dual relations for the effective conductivities of two-dimensional composites with variably imperfect interfaces. The latter result is a direct consequence of the existence of a dual relation for the local fields in such composites, as pointed out by Benveniste and Miloh (Benveniste, Y., Miloh, T., 1999. J. Mech. Phys. Solids 47, 1873-1892).