H
h i m x
ð Þ ¼ H
0
þ
Z
r 1 >a
H xjx 1
ð
ÞÀH
0
È
É
P x 1
ð Þdx 1
þ
Z
r 2 >a
H xjx 2
ð
ÞÀH
0
È
É
P x 2
ð Þdx 2
ð6:187Þ
where r 1 ¼ j x 2 x 1 j, r 2 ¼ j x 2 x 2 j P(x 1 ) and P(x 2 ) denote the probability density
functions for finding a particle centered at x 1 and a void centered at x 2 , respectively.
For simplicity, P(x 1 ) and P(x 2 ) are assumed to be statistically homogeneous, isotropic, and uniform. Using the properties of the fourth-order tensor G x 2 x q
À
Á
, we
obtain the ensemble-averaged current stress norm at any matrix point x as
H
h i m x
ð Þ ¼ H
0
þ
N 1
V
Z
r 1 >a
dr 1
Z
A r 1
ð Þ
σ
0 xjx 1
ð
Þ : I d : σ
0 xjx 1
ð
Þ
ð
Þ dA
þ
N 2
V
Z
r 2 >a
dr 1
Z
A r 2
ð Þ
σ
0 xjx 2
ð
Þ : I d : σ
0 xjx 2
ð
Þ
ð
Þ dA
ð6:188Þ
By carrying out the lengthy but straightforward algebra, we have
H
h i m x
ð Þ ¼ σ
0
: T : σ
0
þ σ
T
: T
Ã
: σ
T
À 2σ
0
: T
Ã
: σ
T
ð6:189Þ
where
σ
T
¼ A 1 C 1 : ε
T
ð6:190Þ
The components of the positive definite fourth-rank tensor T and T
à read
T ijkl ¼ T 1 δ ij δ kl þ T 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:191Þ
T
Ã
ijkl ¼ T
Ã
1 δ ij δ kl þ T
Ã
2 δ ik δ jl þ δ il δ jk
À
Á
ð6:192Þ
with
3T 1 þ 2T 2 ¼ 200 1 À 2v 0
ð
Þ
2
X 2
q¼1
φ q
3α q þ 2β q
À
Á 2
ð6:193Þ
T 2 ¼
1
2
þ 23 À 50v 0 þ 35v
2
0
À
Á X 2
q¼1
φ q
β
2
q
ð6:194Þ
6.5 Noninteracting Solution for Three-Phase Composites
303