Abstract
Based on the perturbation solution to the effective thermal conductivity problem for an
N
-component material, we derive new third- and fourth-order bounds for the effective thermal conductivity of the composite. The new third-order bounds are accurate to third order in |
ϵ
a
—
ϵ
b
|, where
ϵ
a
is the thermal conductivity of phase
a
and require a total of ½
N
(
N
— 1)
2
third-order correlation parameters in their evaluation. For two-phase composites, these bounds require only one geometrical parameter and are identical to Beran’s bounds. For
N
≽ 3, the third-order bounds use the same information as Beran’s bounds, but always more restrictive than Beran’s bounds. The new fourth-order bounds are accurate to fourth-order in |
ϵ
a
-
ϵ
b
| and require an additional ¼
N
2
(N — 1)
2
fourth-order correlation parameters. For
N
= 2, only two parameters are needed, and the fourth-order bounds are shown to be more restrictive than the third-order Beran’s bounds and the second-order Hashin-Shtrikman’s bounds. The application of the third-order bounds to a spherical cell material of Miller is illustrated.
Cited by
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