Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface

The particulate composite under consideration consists of arbitrarily shaped micro-inclusions embedded in a matrix. The interface between the matrix and each inclusion is imperfect and highly conducting, and corresponds to the case of inclusions coated with a highly conducting material. More precisely, the temperature across the interface is continuous while the normal heat flux across the interface suffers a jump related to the surface flux. The purpose of the present work is to elaborate an efficient numerical tool for computing the effective conductivities of the composite by combining the level-set method and the extended finite element method (XFEM). The elaborated numerical procedure has the advantage of avoiding curvilinear coordinates and surface elements in treating imperfect interfaces. It is first validated with the help of some analytical exact and approximate results as benchmarks. It is then applied to determine the inclusion size and shape effects on the effective thermal conductivities of composites with highly conducting coated inclusions and arbitrary geometries. The elaborated numerical procedure is directly applicable to other physically analogous phenomena, such as electrical conduction, dielectrics and diffusion, and to the mathematically identical phenomenon of anti-plane elasticity.