d
Ã
¼ À
N
V
K
À1
Á
Z 1
2a
ρ
3
Z
A
H
1 dAdr þ 2
Z 1
2a
ρ
5
Z
A
H
2 dAdrH
2
2
6
4
3
7
5
8
> <
> :
9
> =
> ;
: ε
T
þ ε
Ã0
À
Á
À
N
V
Z 1
2a
ρ
6
Z
A
L Á H
1
À
Á
dAdr
2
6
4
3
7
5 : ε
T
þ ε
Ã0
À
Á þ 0 ρ
8
À Á
ð6:120Þ
where A denotes the spherical surface with radiusr. After lengthy but straightforward
mathematical manipulation, the final expression takes the following form:
ε
Ã
h i ¼ Γ : ε
Ã0
þ Γ
Ã
: ε
T
ð6:121Þ
The components of the positive definite fourth-rank tensor Γ and Γ
à read
Γ ijkl ¼ γ 1 δ ij δ kl þ γ 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:122Þ
Γ ijkl
Ã
¼ γ 1
Ã
δ ij δ kl þ γ 2
Ã
δ ik δ jl þ δ il δ jk
À
Á
ð6:123Þ
where
γ 1
Ã
¼
5φ
8β
2
13 À 14v 0
ð
Þ v 0 À
8α
3α þ 2β
1 À 2v 0
ð
Þ1 þ v 0
ð
Þ
&
'
ð6:124Þ
γ 2
Ã
¼
5φ
16β
2
25 À 34v 0 þ 22v
2
0
À
Á À
6α
3α þ 2β
1 À 2v 0
ð
Þ1 þ v 0
ð
Þ
&
'
ð6:125Þ
With γ 1 and γ 2 are given in Eqs. (4.7) and (4.8).
6.4.2 Ensemble-Average Stress Norm in the Matrix
The total stress at any point x in the matrix is given by the superposition of the farfield stress σ 0 and the perturbed stress σ
0 induced by the particles. Assuming the
elastic eigenstrain in the particle ε
à is uniform, the perturbed stress for any matrix
point x takes the form (Ju and Chen 1994a)
σ
0 xjx 1
ð
Þ ¼ C 0 Á G x 2 x 1
ð
Þ
Â
à : ε
T
þ ε
Ã
À
Á
ð6:126Þ
where ε
T is the eigenstrain caused by the CTE mismatch between the matrix and the
particle. Using the ensemble-volume averaged eigenstrain given in Eq. (6.121), the
stress perturbation can be given by the following relation:
6.4 Pairwise Interacting Solution for Two-Phase Composites
295
Ã
¼ À
N
V
K
À1
Á
Z 1
2a
ρ
3
Z
A
H
1 dAdr þ 2
Z 1
2a
ρ
5
Z
A
H
2 dAdrH
2
2
6
4
3
7
5
8
> <
> :
9
> =
> ;
: ε
T
þ ε
Ã0
À
Á
À
N
V
Z 1
2a
ρ
6
Z
A
L Á H
1
À
Á
dAdr
2
6
4
3
7
5 : ε
T
þ ε
Ã0
À
Á þ 0 ρ
8
À Á
ð6:120Þ
where A denotes the spherical surface with radiusr. After lengthy but straightforward
mathematical manipulation, the final expression takes the following form:
ε
Ã
h i ¼ Γ : ε
Ã0
þ Γ
Ã
: ε
T
ð6:121Þ
The components of the positive definite fourth-rank tensor Γ and Γ
à read
Γ ijkl ¼ γ 1 δ ij δ kl þ γ 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:122Þ
Γ ijkl
Ã
¼ γ 1
Ã
δ ij δ kl þ γ 2
Ã
δ ik δ jl þ δ il δ jk
À
Á
ð6:123Þ
where
γ 1
Ã
¼
5φ
8β
2
13 À 14v 0
ð
Þ v 0 À
8α
3α þ 2β
1 À 2v 0
ð
Þ1 þ v 0
ð
Þ
&
'
ð6:124Þ
γ 2
Ã
¼
5φ
16β
2
25 À 34v 0 þ 22v
2
0
À
Á À
6α
3α þ 2β
1 À 2v 0
ð
Þ1 þ v 0
ð
Þ
&
'
ð6:125Þ
With γ 1 and γ 2 are given in Eqs. (4.7) and (4.8).
6.4.2 Ensemble-Average Stress Norm in the Matrix
The total stress at any point x in the matrix is given by the superposition of the farfield stress σ 0 and the perturbed stress σ
0 induced by the particles. Assuming the
elastic eigenstrain in the particle ε
à is uniform, the perturbed stress for any matrix
point x takes the form (Ju and Chen 1994a)
σ
0 xjx 1
ð
Þ ¼ C 0 Á G x 2 x 1
ð
Þ
Â
à : ε
T
þ ε
Ã
À
Á
ð6:126Þ
where ε
T is the eigenstrain caused by the CTE mismatch between the matrix and the
particle. Using the ensemble-volume averaged eigenstrain given in Eq. (6.121), the
stress perturbation can be given by the following relation:
6.4 Pairwise Interacting Solution for Two-Phase Composites
295