ε ¼ I À φSΓ A þ S
ð
Þ
À1
h
i
: ε
0
þ φ SΓ A þ S
ð
Þ
À1 A þ I
ð
Þ
h
i
: ε
T
ð6:163Þ
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
À ε
T
Â
Ã
ð6:164Þ
where ε
0 can be recast as
ε
0
¼ S :
1
V 1
Z
V 1
ε
T x
0
ð Þ þ ε
à x
0
ð Þ
Â
Ã
dx
0
8
<
:
9
=
;
¼ SΓ : ε
T
þ ε
Ã0
À
Á
ð6:165Þ
Using Eq. (6.165), we have
σ 1 ¼ C 1 : ε
0
þ C 1 SΓ 2 I
ð
Þ
½
Š: ε
T
þ C 1 SΓ
ð
Þ: ε
Ã0
ð6:166Þ
With the noninteracting solution ε
Ã0 of the eigenstrain, we arrive at
σ 1 ¼ C 1 I À SΓ A þ S
ð
Þ
À1
h
i
: ε
0
þ C 1 SΓ A þ S
ð
Þ
À1 A þ I
ð
ÞÀI
h
i
: ε
T
ð6:167Þ
Combing with Eq. (6.163), the averaged internal stresses of particles can be
expressed as
σ 1 ¼ C 1 I À SΓ A þ S
ð
Þ
À1
h
i
I À φSΓ A þ S
ð
Þ
À1
h
i À1 : ε À C 1 fφSΓ I À SΓ A þ S
ð
Þ
À1
h
i
I À φSΓ A þ S
ð
Þ
À1
h
i À1
A þ S
ð
Þ
À1 A þ I
ð
ÞÀ SΓ A þ S
ð
Þ
À1 A þ I
ð
ÞÀI
h
i
g : ε
T
¼ U : ε À V : ε
T
ð6:168Þ
where
U ¼ C 1 I À SΓ A þ S
ð
Þ
À1
h
i
I À φSΓ A þ S
ð
Þ
À1
h
i À1
ð6:169Þ
6.4 Pairwise Interacting Solution for Two-Phase Composites
299