Estimations and bounds of the effective conductivity of composites with anisotropic inclusions and general imperfect interfaces

• The inhomogeneities of composites can be anisotropic and exhibit a spherical or circular form.
• The imperfect interface can be modeled by a general thermal imperfect interface model.
• A unified and exact solution for the thermal fields of the inhomogeneity problem is obtained.
• The differential scheme is extended to predicting the effective thermal conductivity of composites.
• The minimum potential and complementary energy principles and the morphologically representative pattern approach are generalized to derive the first- and second-order bounds.

The present work aims to determine the effective thermal conductivity of composites made of an isotropic matrix phase in which circular or spherical inhomogeneities are embedded. The inhomogeneity phases can be anisotropic and the interface between the inhomogeneity and matrix phase can be modeled by a general thermal imperfect interface model across which both the temperature and normal heat flux across can suffer a discontinuity. To achieve this objective, we derive first a unified and exact solution for the thermal fields of the inhomogeneity problem consisting of a spherical or circular anisotropic inhomogeneity inserted via a general thermal imperfect interface into an infinity isotropic matrix medium subjected to a remote uniform loading at its external surface. Unlike the relevant results in elasticity, the intensity and heat flux fields inside circular and spherical inhomogeneities are shown to remain uniform even in the presence of the general thermal imperfect interface and anisotropy of inhomogeneity. Next, with the help of the foregoing solution results for the heterogeneity problem, the differential scheme is extended to predicting the effective thermal conductivity of composites with taking into account the imperfect interfaces between the constituent phases. Finally, the minimum potential and complementary energy principles and the morphologically representative pattern approach based on the Hashin–Shtrikman variational principles and the variational polarization principles are applied to such inhomogeneous materials and to bracketing their effective thermal properties. By constructing trial appropriate intensity and heat flux fields, the first- and second-order upper and lower bounds are obtained for the effective thermal conductivity of multiphase materials consisting of spherical or circular inhomogeneities embedded in a matrix. The estimations obtained by the differential scheme for the effective conductivity are shown to comply with the first- and second-order upper and lower bounds. Numerical results are provided to illustrate the dependence of the effective conductivity on the sizes of inhomogeneities and to compare the estimations with the relevant upper and lower bounds.