Heat-conduction theory for composite materials

Abstract

The problem of the macroscopic conduction of heat in a composite medium is considered. The medium consists of inclusions of arbitrary geometry embedded in a matrix. Starting with an analysis of the micro-conduction in a suitably defined representative volume, a fourth order equation governing the macro-conduction is deduced together with appropriate boundary conditions. The theory closely resembles the mechanical theory of non-simple materials of grade 2 developed by Toupin [2]. In particular, the well known characteristic material length encountered in stress-gradient theories is evaluated in terms of constituent material properties. It is further shown, as a basis of the theory, that anatural ratio exists between the dimensions of a local representative volume and those of a unit-cell given by the ratio of the classical Voigt and Reuss bounds on the effective diffusivities. In consequence, application of the theory hinges on the evaluation of certain material constants from given information on the inclusion arrangement in a representative volume of specific dimensions. Differences in the behaviour of media with different inclusion distributions depend entirely on the magnitude of such constants.