Determination of the effective conductive properties of composites with curved oscillating interfaces by a two-scale homogenization procedure

Rough surfaces and interfaces appear in many situations of practical interest in physics and mechanics of solids. When the interfaces between the constituent phases of composites are rough instead of being smooth as assumed usually, all well-known micromechanical schemes resorting to Eshelby's tensor are no longer applicable to computing their effective properties. The present work proposes a two-scale homogenization procedure aiming at determining the effective thermal properties of a two-dimensional composite in which the curved interfaces between the constituent phases oscillate periodically and quickly. An asymptotic analysis method is first used to homogenize a rough interface zone as an equivalent interphase, and the effective thermal conductivity tensor of this interphase at the mesoscopic scale is exactly determined. Then, by applying two micromechanical schemes, closed-form expressions for the effective thermal properties of composites at the macroscopic scale are derived. Finally, the analytical results obtained are compared with the relevant bounds and with the corresponding numerical results provided by the finite element method. The two-scale homogenization procedure turns out to be accurate and efficient.