T
ÃÃ
ijkl ¼ T
ÃÃ
1 δ ij δ kl þ T
ÃÃ
2 δ ik δ jl þ δ il δ jk
À
Á
ð6:153Þ
with
3T 1 þ 2T 2 ¼
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:154Þ
T 2 ¼
T 2
bφ þ 1
ð
Þ
2
ð6:155Þ
3T
Ã
1 þ 2T
Ã
2 ¼ À
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:156Þ
T
Ã
2 ¼ À
T 2
bφ þ 1
ð
Þ
2
þ
1
2 bφ þ 1
ð
Þ
ð6:157Þ
3T
ÃÃ
1 þ 2T
ÃÃ
2 ¼
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:158Þ
T
ÃÃ
2 ¼
T 2
bφ þ 1
ð
Þ
2
þ
bφ À 1
2 bφ þ 1
ð
Þ
ð6:159Þ
Because σ
T
¼ AC 1 : ε
T is spherical stress, caused by CTE mismatch, Eq. (6.147)
of the ensemble-averaged loading function can be simplified as
H
h i m x
ð Þ ¼ σ À σ
T
À
Á : T : σ À σ
T
À
Á
ð6:160Þ
6.4.3 Ensemble-Average Stress in the Filler Particles
The ensemble-volume averaged strain for two-phase composites can be given by
ε ¼ ε
0
þ φS : ε
T
þ ε
Ã
À
Á
ð6:161Þ
Substituting Eq. (6.121)
ε ¼ ε
0
þ φSΓ : ε
T
þ ε
Ã0
À
Á
ð6:162Þ
With the noninteracting solution ε
Ã0 of the eigenstrain given by Eq. (6.42), we
arrive at
298
6 Unified Micromechanics of Particulate Composites
ÃÃ
ijkl ¼ T
ÃÃ
1 δ ij δ kl þ T
ÃÃ
2 δ ik δ jl þ δ il δ jk
À
Á
ð6:153Þ
with
3T 1 þ 2T 2 ¼
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:154Þ
T 2 ¼
T 2
bφ þ 1
ð
Þ
2
ð6:155Þ
3T
Ã
1 þ 2T
Ã
2 ¼ À
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:156Þ
T
Ã
2 ¼ À
T 2
bφ þ 1
ð
Þ
2
þ
1
2 bφ þ 1
ð
Þ
ð6:157Þ
3T
ÃÃ
1 þ 2T
ÃÃ
2 ¼
3T 1 þ 2T 2
aφ þ 1
ð
Þ
2
ð6:158Þ
T
ÃÃ
2 ¼
T 2
bφ þ 1
ð
Þ
2
þ
bφ À 1
2 bφ þ 1
ð
Þ
ð6:159Þ
Because σ
T
¼ AC 1 : ε
T is spherical stress, caused by CTE mismatch, Eq. (6.147)
of the ensemble-averaged loading function can be simplified as
H
h i m x
ð Þ ¼ σ À σ
T
À
Á : T : σ À σ
T
À
Á
ð6:160Þ
6.4.3 Ensemble-Average Stress in the Filler Particles
The ensemble-volume averaged strain for two-phase composites can be given by
ε ¼ ε
0
þ φS : ε
T
þ ε
Ã
À
Á
ð6:161Þ
Substituting Eq. (6.121)
ε ¼ ε
0
þ φSΓ : ε
T
þ ε
Ã0
À
Á
ð6:162Þ
With the noninteracting solution ε
Ã0 of the eigenstrain given by Eq. (6.42), we
arrive at
298
6 Unified Micromechanics of Particulate Composites