Thermal Conductivities of Composite Materials

Abstract

A field solution for long wavelengths of the equation of heat conduction is obtained for a composite material with orthorhombic symmetry. This yields exact expressions for the three principal thermal conductivities. They are valid for any number of ortho rhombic constituents and even for continuously varying thermal conductivities within the elementary cell. For a lamellar composite, the average thermal conductivities of the material are expressed directly as integrals over the space-dependent thermal conductivi ties. For a filamentary composite, such explicit expressions are obtained only in the Wigner-Seitz approximation and only for a stepwise variation of the thermal conductivities within the ele mentary cell. In the same manner, the average thermal conduc tivity of a composite with cubic symmetry, e.g. isotropic spherical inclusions in a cubic lattice, is calculated. Analogies with previous results on diffusion coefficients and elastic shear moduli are discussed.

Methods for predicting the thermal conductivity of composite systems: A review

“…Listed alphabetically in the Bibliography are a group k , = kdm kc"-*' Behrens Theoretical Model (6) The theoretical results were based on a field soliltion to the equation ofheat conduction for a composite material with orthorhombic symmetry. Solutions were presented for elliptic filaments in a square lattice: Bruggeman Theoretical Model (7) Using different assumptions than Maxwell for permeability and field \trength, Briiggeman derived the following implicit equation for dilute suspensions of spheres in a homogeneoil5 medium.…”

A variational asymptotic micromechanics model for predicting conductivities of composite materials

“…As shown in Figure 3, VAMUCH results are perfectly located between the variational bounds of [Donea 1972], while the Springer-Tsai model [Springer and Tsai 1967] and the lower bound of [Hashin 1983] underpredict the results. We have also found out that VAMUCH results are the same as those obtained by Behrens [1968], Hatta and Taya [1986], and the upper bound of [Hashin 1983], and these results are not shown in the plot for clarity.…”

“…To validate the present theory, we compare the VAMUCH prediction with other models in the literature [Springer and Tsai 1967;Behrens 1968;Donea 1972;Hashin 1983;Hatta and Taya 1986]. As shown in Figure 3, VAMUCH results are perfectly located between the variational bounds of [Donea 1972], while the Springer-Tsai model [Springer and Tsai 1967] and the lower bound of [Hashin 1983] underpredict the results.…”

“…The effective thermal conductivities computed by different models are plotted in Figure 4. We found out that the results of the Hashin upper bound [Hashin 1983] are the same as those of [Behrens 1968] and [Hatta and Taya 1986]. Hence only the Hashin upper bound is plotted in the figure.…”

“…The very reason that the heterogenous material can be homogenized leads us to believe that w should be asymptotically smaller than ψ, that is, w ∼ η ψ. Substituting Equation (8) into Equation (7) and making use of Equation (5), we can obtain the leading terms of the functional according to the variational asymptotic method [Berdichevsky 1977] as:…”