The Heat Conductivity Problem in a Thin Plate with Contrasting Fiber Inclusions

Based on the asymptotic analysis of an elliptic boundary value problem in a thin domain, a homogenized model of the heat distribution in a composite plate of small relative thickness h is an element of (0,1] is constructed under the assumption that thermal conductivity of the fiber and that of the filler contrast very much. Namely, the plate is assumed to contain several periodic families of fibers, the diameters of the fibers and the distances between the fibers being of the same order h. Fibers in each family have the same thermal conductivity; the values of thermal conductivity of fibers in different families may vary, but should be of the same order in h. Thermal conductivity of the filler is one order smaller in h. The asymptotics is constructed by means of matching the classical asymptotic ansatz for thin plates and fibers. The periodic structure of the composite is crucially used to construct the asymptotic expansion which consists of terms of the following two types: a periodic solution of the three-dimensional problems in the periodicity cell and a solution to a two-dimensional homogenized problem in the longitudinal cross-section of the plate. The asymptotic procedure provides a simple algorithm to compute coefficients in the homogenized second-order differential operator. The asymptotics obtained is justified using the weighted Friedrichs inequality and the error estimates are asymptotically sharp.