σ
0 xjx 1
ð
Þ ¼ C 0 Á G x 2 x 1
ð
ÞÁΓ
Â
à : ε
T
þ ε
Ã0
À
Á
ð6:127Þ
where ε
Ã0 denotes the solution of the eigenstrain for the single inclusion problem,
which is given by Eq. (6.42). Therefore
σ
0 xjx 1
ð
Þ ¼ À C 0 Á G Á Γ A þ S
ð
Þ
À1
h
i
: ε
0
þ C 0 Á G Á Γ A þ S
ð
Þ
À1 A þ I
ð
Þ
h
i
: ε
T
ð6:128Þ
Following the same procedure used for the noninteracting solution, we obtain the
ensemble-averaged current stress norm at any matrix point x as:
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
ð6:129Þ
where σ
T is given by Eq. (6.56) and the components of the positive definite fourthrank 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
ð6:130Þ
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
ð6:131Þ
with
3T 1 þ 2T 2 ¼ 200 1 À 2v 0
ð
Þ
2 3γ 1 þ 2γ 2
ð
Þ
2
3α þ 2β
ð
Þ
2
φ
ð6:132Þ
T 2 ¼
1
2
þ 23 À 50v 0 þ 35v
2
0
À
Á 4γ
2
2
β
2
φ
ð6:133Þ
3T
Ã
1 þ 2T
Ã
2 ¼ 200 1 À 2v 0
ð
Þ
2 3γ 1 þ 2γ 2
ð
Þ
2
3α þ 2β
ð
Þ
2
φ
ð6:134Þ
T
Ã
2 ¼ 23 À 50v 0 þ 35v
2
0
À
Á 4γ
2
2
β
2
φ
ð6:135Þ
α and β are given in Eqs. (6.5) and (6.6).
The ensemble-averaged current stress norm at a matrix point can also be
expressed in terms of the macroscopic stress σ. Following the same procedure as
in former section, the relation between the far-field stress σ
0 and the macroscopic
stress σ takes the form
σ = P : σ
0
À Q : σ
T
ð6:136Þ
where
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6 Unified Micromechanics of Particulate Composites
0 xjx 1
ð
Þ ¼ C 0 Á G x 2 x 1
ð
ÞÁΓ
Â
à : ε
T
þ ε
Ã0
À
Á
ð6:127Þ
where ε
Ã0 denotes the solution of the eigenstrain for the single inclusion problem,
which is given by Eq. (6.42). Therefore
σ
0 xjx 1
ð
Þ ¼ À C 0 Á G Á Γ A þ S
ð
Þ
À1
h
i
: ε
0
þ C 0 Á G Á Γ A þ S
ð
Þ
À1 A þ I
ð
Þ
h
i
: ε
T
ð6:128Þ
Following the same procedure used for the noninteracting solution, we obtain the
ensemble-averaged current stress norm at any matrix point x as:
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
ð6:129Þ
where σ
T is given by Eq. (6.56) and the components of the positive definite fourthrank 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
ð6:130Þ
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
ð6:131Þ
with
3T 1 þ 2T 2 ¼ 200 1 À 2v 0
ð
Þ
2 3γ 1 þ 2γ 2
ð
Þ
2
3α þ 2β
ð
Þ
2
φ
ð6:132Þ
T 2 ¼
1
2
þ 23 À 50v 0 þ 35v
2
0
À
Á 4γ
2
2
β
2
φ
ð6:133Þ
3T
Ã
1 þ 2T
Ã
2 ¼ 200 1 À 2v 0
ð
Þ
2 3γ 1 þ 2γ 2
ð
Þ
2
3α þ 2β
ð
Þ
2
φ
ð6:134Þ
T
Ã
2 ¼ 23 À 50v 0 þ 35v
2
0
À
Á 4γ
2
2
β
2
φ
ð6:135Þ
α and β are given in Eqs. (6.5) and (6.6).
The ensemble-averaged current stress norm at a matrix point can also be
expressed in terms of the macroscopic stress σ. Following the same procedure as
in former section, the relation between the far-field stress σ
0 and the macroscopic
stress σ takes the form
σ = P : σ
0
À Q : σ
T
ð6:136Þ
where
296
6 Unified Micromechanics of Particulate Composites