Asymptotic Analysis of Heat Transfer in Composite Materials with Nonlinear Thermal Properties Igor V. Andrianov a , Heiko Topol b , Vladyslav V. Danishevskyy c a Institute of General Mechanics, RWTH Aachen University, Templergraben 64, 52062 Aachen, Germany b Center for Advanced Materials, Qatar University, P.O. Box 2713, Doha, Qatar c School of Computing and Mathematics, Keele University, Staffordshire, ST5 5BG, UK Abstract We study heat transfer through a composite with periodic microstructure. The thermal conduc- tivity of the constituents is assumed to be temperature-dependent, and it is modeled as a poly- nomial in terms of the temperature. The thermal resistance between the constituents is taken to be nonlinear. In order to determine the effective thermal properties of the material, we apply the asymptotic homogenization method. We discuss different approaches to determine these effec- tive properties for the different volume fractions of the inclusions. For high volume fractions of the inclusion, we apply the lubrication theory. In the case of low volume fractions of the inclu- sions, we apply the three-phase model. Comparing some special cases of our results to existing ones in the literature shows a good accuracy. Keywords: heat transfer, composites, nonlinearity, asymptotic homogenizaton method, three-phase model, lubrication theory 1. Introduction 1 Modeling of the thermal properties of composites might be challenging, especially when the 2 size of the heterogeneities is significantly smaller than the macroscopic size of the considered 3 structure. In order to simplify the treatment of heat diffusion problems, different approaches have 4 been developed, in which the original heterogeneous material is replaced by a homogenized or 5 effective material with the same macroscopic properties as the original heterogeneous material. 6 Early works on this topic are, for example, the works of Hershey [1], Hill [2], Kerner [3], Kröner 7 [4], Keller [5], and van der Poel [6]. Examples for works on computational homogenization are 8 article of Özdemir et al. [7], and the work of Geers et al. [8] discusses some trends and challenges 9 in this field. 10 A powerful and wide-spread technique denoted as the asymptotic homogenization method 11 (AHM) has been developed in order to obtain the effective properties of different asymptotic or- 12 ders of heterogeneous materials with periodic microstructures. The theory behind this technique 13 is described, for example, in the books of Bensoussan et al. [9] and Panasenko [10]. The AHM 14 allows to investigate a macroscopic boundary value problem within a single repeated unit cell of 15 the microstructure. In this approach, a small parameter is introduced, which relates the size of the 16 heterogeneities to the size of the macroscopic problem. The original coordinate variables are then 17 replaced by so-called fast coordinate variables, which consider the problem on the micro-scale, 18 and by slow coordinates, which consider the problem on the macro-scale. The AHM has been 19 Preprint submitted to International Journal of Heat and Mass Transfer February 16, 2017