ε ¼ I À
X 2
q¼1
φ q S A q þ S
À
Á À1
"
#
: ε
0
þ φ 1 S A 1 þ S
ð
Þ
À1 A 1 þ I
ð
Þ: ε
T
1
ð6:225Þ
The volume-averaged stress tensor for the particles is defined as
σ 1 ¼
1
V 1
Z
V 1
σ 1 x
ð Þdx ¼
1
V 1
Z
V 1
C 1 : ε
0
þ ε
0 x
ð Þ À ε
T x
ð Þ
Â
Ã
dx
¼ C 1 : ε
0
þ ε
0
1 À ε
T
1
Â
Ã
ð6:226Þ
where ε 1
0 can be recast as
ε
0
1 ¼ S :
1
V 1
Z
V 1
ε
T x
0
ð Þ þ ε
Ã0
Â
Ã
dx
0
8
<
:
9
=
;
¼ S : ε
T
1 þ ε
Ã0
1
À
Á
ð6:227Þ
With the help of Eqs. (6.227) and (6.185), Eq. (6.226) can be rephrased as
σ 1 ¼ C 1 A 1 A 1 þ S
ð
Þ
À1 : ε
0
À C 1 A 1 I À S
ð
Þ A 1 þ S
ð
Þ
À1 : ε
T
1
ð6:228Þ
As results of Eq. (6.225), the average internal stress of particles can be expressed
as
σ 1 ¼ C 1 A 1 A 1 þ S
ð
Þ
À1 I À
X 2
q¼1
φ q S A q þ S
À
Á À1
"
# À1
: ε
ÀC 1 A 1 A 1 þ S
ð
Þ
À1 φ 1 S A 1 þ S
ð
Þ
À1 A 1 þ I
ð
Þ I À
X 2
q¼1
φ q S A q þ S
À
Á À1
"
# À1
þ I À S
ð
Þ
8
<
:
9
=
;
: ε
T
1
¼ U : ε À V : ε
T
1
ð6:229Þ
where
U ¼ C 1 A 1 A 1 þ S
ð
Þ
2 1 I À
X 2
q¼1
φ q S A q þ S
À
Á À1
"
# À1
ð6:230Þ
6.5 Noninteracting Solution for Three-Phase Composites
307