Thermal conduction in bi-layer materials with an interfacial inclusion

Abstract

We establish an analytical model for a thermal inclusion in bi-layer materials. The inclusion is considered as either an elliptic filler of finite thermal conductivity or a crack of ideal thermal insulation. Unlike most existing studies, this article focuses on bonded media of finite size in both their height and the length directions. The temperature field is formulated by a singular integral equation method. The effects of inclusion size and interfacial cracking on the effective thermal conductivity of the medium are studied. In addition, the crack tip thermal flux intensity factors are also given as they are not available in the open literature.Multi-layered materials are widely used in engineering applications such as thermal barrier coatings, oxidation/corrosion resistance coatings, magnetic storage devices, electronic packaging and solid-oxide fuel cells. The temperature gradient becomes high at the interfaces because of the difference in the thermal and mechanical properties between bonded layers. The high-temperature gradient results in high thermal stresses, which may cause de-bonding of the interface, thus reducing the function and reliability of the systems. A better understanding of temperature and thermal flux distributions in the bonded media is critical in many applications.In the literature, numerous studies have been reported on the investigation of thermal conduction in particle composites and multi-layered media [1][2][3][4][5][6][7][8][9][10]. To the authors' best knowledge, the thermal behavior of bi-layer materials with an interface inclusion has not been investigated for finite media. Motivated by this consideration, this article focuses on a bonded medium of finite size in its length and height directions. Figure 1 shows the material system occupying Àh x h, Àc5y5c, À15z51. The bonded two layers are of the same thickness c and same length h. There is an interface inclusion at the geometric center of the medium. It is assumed that the maximum height of the inclusion 0 is much smaller than its length 2a. The upper surface of the medium is subjected to a temperature T þ and the lower surface of the medium is subjected to a temperature T À . There is no internal heat source inside the medium. In isotropic materials, the thermal fluxes in the x and y directions