P ¼ I þ φ I 2 S
ð
ÞΓ A þ S
ð
Þ
À1
ð6:137Þ
Q ¼ φ I 2 S
ð
ÞΓ A þ S
ð
Þ
2 1
ð6:138Þ
where the components of P and Q are given
P ijkl ¼ P 1 δ ij δ kl þ P 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:139Þ
Q ijkl ¼ Q 1 δ ij δ kl þ Q 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:140Þ
with
3P 1 þ 2P 2 ¼ aφ þ 1
ð6:141Þ
P 2 ¼
1
2
bφ þ 1
ð
Þ
ð6:142Þ
3Q 1 þ 2Q 2 ¼ aφ
ð6:143Þ
Q 2 ¼
1
2
bφ
ð6:144Þ
a ¼ 20 1 À 2v 0
ð
Þ
3γ 1 þ 2γ 2
3α þ 2β
ð6:145Þ
b ¼ 7 À 5v 0
ð
Þ
2γ 2
β
ð6:146Þ
Using Eq. (6.136), we arrive at the alternative expression for the ensembleaveraged current stress norm in a matrix pointx:
H
h i m x
ð Þ ¼ σ : T : σ þ 2σ : T
à : σ
T
þ σ
T
: T
ÃÃ : σ
T
ð6:147Þ
where the positive definite fourth-rank tensor T, T
à and T
ÃÃ are defined as
T = P
2 1
À
Á T Á T Á P
2 1
ð6:148Þ
T
à = P
2 1
À
Á T Á T Á P
2 1
Á Q 2 T
Ã
Á P
2 1
ð6:149Þ
T
ÃÃ = P
2 1
Á Q
À
Á T Á T Á P
2 1
Á Q
À
Á þ T
Ã
2 2T
Ã
Á P
2 1
Á Q
À
Á
ð6:150Þ
where the components of T, T
à , and T
ÃÃ are
T ijkl ¼ T 1 δ ij δ kl þ T 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:151Þ
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
ð6:152Þ
6.4 Pairwise Interacting Solution for Two-Phase Composites
297