Effective properties of fiber-reinforced materials: I-Bounds on the effective thermal conductivity of dispersions of fully penetrable cylinders

Third-order and fourth-order bounds on the effective transverse thermal conductivity k_e of fiber-reinforced materials (statistically isotropic in the transverse plane) due to Silnutzer and to Milton, respectively, are evaluated for a distribution of fully penetrable cylinders in a matrix. Both of these bounds involve a multidimensional integral which depends upon the three-point matrix probability function of the model. For nontrivial values of the fiber volume fraction φ₂ (i.e. 0 < φ₂ < 1) in the fully penetrable-cylinder model, the third-order bounds always improve upon the second-order bounds due to Hashin and the fourth-order bounds are always more restrictive than third-order bounds. The fourth-order bounds are sharp enough for 0 ≤ φ₂ ≤ 0.3 to give quantitatively useful estimates of k_e even when the conductivity of the fiber phase k₂ is as much as two orders of magnitude greater than the matrix conductivity k₁. For 0.8 ≤ φ₂ ≤ 1, the Milton bounds are sufficiently restrictive to provide reasonable approximations of k_e for an insulating fiber phase such that 0.01 ≤ α ≤ 1, where α = k₂ k₁. The results obtained in this study for k_e translate immediately into equivalent results for the axial shear modulus of fiber-reinforced materials.