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
ð6:92Þ
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
ð6:93Þ
Therefore, we have
σ 1 ¼ C 1 A A þ S
ð
Þ
À1 : ε
0
À C 1 A I 2 S
ð
Þ A þ S
ð
Þ
À1 : ε
T
ð6:94Þ
As result of Eq. (6.91), the average internal stresses of particles can be expressed
as:
σ 1 ¼ C 1 A A þ S
ð
Þ
À1 I À φS A þ S
ð
Þ
À1
h
i À1 : ε
À C 1 A A þ S
ð
Þ
À1 φS A þ S
ð
Þ
À1 A þ I
ð
Þ I À φS A þ S
ð
Þ
À1
h
i À1 þ I À S
ð
Þ
&
'
: ε
T
¼ U : ε À V : ε
T
ð6:95Þ
where
U ¼ C 1 A A þ S
ð
Þ
À1 I À φS A þ S
ð
Þ
À1
h
i À1
ð6:96Þ
V ¼ C 1 A A þ S
ð
Þ
À1 φS A þ S
ð
Þ
À1 A þ I
ð
Þ I À φS A þ S
ð
Þ
À1
h
i À1 þ I 2 S
ð
Þ
&
'
ð6:97Þ
By carrying out the lengthy algebra, the components of the positive definite
fourth-rank tensor U and V are explicitly given by
U ijkl ¼ U 1 δ ij δ kl þ U 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:98Þ
V ijkl ¼ V 1 δ ij δ kl þ V 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:99Þ
where
6.3 Noninteracting Solution for Two-Phase Composites
291