Third-order and fourth-order bounds on the effective transverse thermal conductivity ke 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 ke 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 ke for an insulating fiber phase such that 0.01 ≤ α ≤ 1, where α = k2 k1. The results obtained in this study for ke translate immediately into equivalent results for the axial shear modulus of fiber-reinforced materials.
All Science Journal Classification (ASJC) codes
- Materials Science(all)
- Mechanics of Materials
- Mechanical Engineering