A mixture theory for thermal diffusion in unidirectional composites with cylindrical fibers of arbitrary cross section

Abstract

A binary mixture theory is developed for heat conduction in a unidirectional, fibrous composite containing a two dimensional periodic array of cylindrical fibers of arbitrary cross section. The case considered concerns a class of problems for which conduction occurs primarily in the direction of the fiber axis. Model construction is based upon an asymptotic scheme wherein the ratio of transverse-to-longitudinal diffusion times is assumed to be small: this premise, and the problem class studied, are appropriate for many composites designed primarily for thermal protection. The resulting theory, which retains information on the temperature distribution in the microstructure, contains a mixture interaction coefficient: the latter is determined from the solution of a time-independent boundary value problem in a unit cell. A variational principle-based finite element method is proposed for the solution of this boundary value problem. Consequently, the theory is closed in the sense that no unknown coefficients exist, i.e. the theory is completely determined by the material properties and geometries of the constituents.Numerical analyses are carried out for several microstructural geometries of practical interest. The results indicate that, for achievable volume fractions, the concentric circular cylindrical approximation often used in practice provides an adequate measure of global and averaged local temperature fields for composites containing circular fibers in a hexagonal array, and square fibers arranged in a square array. It is found, however, that such an approximation may not be accurate for composites containing rectangular fibers in a similar unit cell. Here a parametric study reveals that the interaction coefficient is a strong function of the unit-cell aspect ratio.