In this paper, a multi-scale analysis method for heat transfer in heterogeneous solids is presented. The principles of the method rely on a two-scale computational homogenization approach which is applied successfully for the stress analysis of multi-phase solids under purely mechanical loading. The present paper extends this methodology to heat conduction problems. The flexibility of the method permits one to take into account local microstructural heterogeneities and thermal anisotropy, including non-linearities which might arise at some stage of the thermal loading history. The resulting complex microstructural response is transferred back to the macro level in a consistent manner. A proper macro to micro transition is established in terms of the applied boundary conditions and likewise a micro to macro transition is formulated in the form of consistent averaging relations. Imposition of boundary conditions and extraction of macroscopic quantities are elaborated in detail. A nested finite element solution procedure is outlined, and the effectiveness of the approach is demonstrated by some illustrative example problems.
|International Journal for Numerical Methods in Engineering
|Published - 2008