where N is the total number of particles dispersed in volume V. According to the
assumption of statistical isotropy and uniformity, Eq. (4.45) can be recast into a more
convenient form
H
h i m x
ð Þ ffi H
0
þ
N
V
Z
r>a
dr
Z
A r
ð Þ
H r
ð Þ À H
0
È
É
dA
ð6:48Þ
with the help of Eqs. (6.35), (6.37), (6.44), and (6.46), plus using the following
identities:
Z
A r
ð Þ
n i n j dA ¼
4πr
2
3
δ ij
ð6:49Þ
Z
A r
ð Þ
n i n j n k n l dA ¼
4πr
2
15
δ ij δ kl þ δ ik δ jl þ δ il δ jk
À
Á
ð6:50Þ
Z
r>a
dr
Z
A r
ð Þ
H
1 r
ð ÞdA ¼ 0
ð6:51Þ
Z
r>a
dr
Z
A r
ð Þ
H
2 r
ð ÞdA ¼ 0
ð6:52Þ
Z
r>a
dr
Z
A r
ð Þ
G r
ð ÞdA ¼ 0
ð6:53Þ
Equation (6.48) can be recast as
H
h i m x
ð Þ ¼ H
0
þ
N
V
Z
r>a
dr
Z
A r
ð Þ
σ
0 x
ð Þ : I d : σ
0 x
ð Þ
½
Š dA
ð6:54Þ
After some lengthy yet straightforward derivations, one can obtain the ensembleaveraged current stress norm at any matrix point x as
H
h i m x
ð Þ ¼ σ
0
: T : σ
0
þ σ
T
: T
Ã
: σ
T
À 2σ
0
: T
Ã
: σ
T
ð6:55Þ
where
σ
T
¼ AC 1 : ε
T
ð6:56Þ
The components of the positive definite fourth-rank tensors T and T
à read
6.3 Noninteracting Solution for Two-Phase Composites
287