T 2 ¼
T 2
P 2
q¼1
b q φ q þ 1
! 2
ð6:218Þ
3T
Ã
1 þ 2T
Ã
2 ¼ À
3T 1 þ 2T 2
P 2
q¼1
a q φ q þ 1
! 2
ð6:219Þ
T
Ã
2 ¼ À
T 2
P 2
q¼1
b q φ q þ 1
! 2 þ
1
2
P 2
q¼1
b q φ q þ 1
!
ð6:220Þ
3T
ÃÃ
1 þ 2T
ÃÃ
2 ¼
3T 1 þ 2T 2
P 2
q¼1
a q φ q þ 1
! 2
ð6:221Þ
T
ÃÃ
2 ¼
T 2
P 2
q¼1
b q φ q þ 1
! 2 þ
P 2
q¼1
b q φ q À 1
2
P 2
q¼1
b q φ q þ 1
!
ð6:222Þ
Because σ
T
¼ AC 1 : ε
T caused by the CTE mismatch is spherical stress, so
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
ð6:223Þ
6.5.3 Ensemble-Average Stress in Filler Particles
The ensemble-volume averaged strain for three-phase composites takes the form
ε ¼ ε
0
þ
X 2
q¼1
φ q S ε
Ã0
q þ ε
T
q
ð6:224Þ
With the noninteracting solution ε
Ã0 of the eigenstrain given by Eq. (6.185), we
arrive at
306
6 Unified Micromechanics of Particulate Composites
T 2
P 2
q¼1
b q φ q þ 1
! 2
ð6:218Þ
3T
Ã
1 þ 2T
Ã
2 ¼ À
3T 1 þ 2T 2
P 2
q¼1
a q φ q þ 1
! 2
ð6:219Þ
T
Ã
2 ¼ À
T 2
P 2
q¼1
b q φ q þ 1
! 2 þ
1
2
P 2
q¼1
b q φ q þ 1
!
ð6:220Þ
3T
ÃÃ
1 þ 2T
ÃÃ
2 ¼
3T 1 þ 2T 2
P 2
q¼1
a q φ q þ 1
! 2
ð6:221Þ
T
ÃÃ
2 ¼
T 2
P 2
q¼1
b q φ q þ 1
! 2 þ
P 2
q¼1
b q φ q À 1
2
P 2
q¼1
b q φ q þ 1
!
ð6:222Þ
Because σ
T
¼ AC 1 : ε
T caused by the CTE mismatch is spherical stress, so
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
ð6:223Þ
6.5.3 Ensemble-Average Stress in Filler Particles
The ensemble-volume averaged strain for three-phase composites takes the form
ε ¼ ε
0
þ
X 2
q¼1
φ q S ε
Ã0
q þ ε
T
q
ð6:224Þ
With the noninteracting solution ε
Ã0 of the eigenstrain given by Eq. (6.185), we
arrive at
306
6 Unified Micromechanics of Particulate Composites