T ijkl ¼ T 1 δ ij δ kl þ T 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:57Þ
T
Ã
ijkl ¼ T
Ã
1 δ ij δ kl þ T
Ã
2 δ ik δ jl þ δ il δ jk
À
Á
ð6:58Þ
with
3T 1 þ 2T 2 ¼ 200 1 À 2v 0
ð
Þ
2
φ
3α þ 2β
ð
Þ
2
ð6:59Þ
T 2 ¼
1
2
þ 23 À 50v 0 þ 35v
2
0
À
Á φ
β
2
ð6:60Þ
3T
Ã
1 þ 2T
Ã
2 ¼ 200 1 À 2v 0
ð
Þ
2
φ
3α þ 2β
ð
Þ
2
ð6:61Þ
T
Ã
2 ¼ 23 À 50v 0 þ 35v
2
0
À
Á φ
β
2
ð6:62Þ
α and β are given by Eqs. (6.5) and (6.6) φ is the particle volume fraction.
The ensemble-averaged current stress norm at a matrix point must be established
in terms of the macroscopic stress σ in order to express the effective loading function
in terms of the macroscopic stress. In the special case of uniform dispersions of
identical elastic spheres in a homogeneous matrix, the macroscopic stress and the
far-field stress take the form
σ ¼ C 0 : ε À φ ε
Ã0
þ ε
T
À
Á
Â
Ã
ð6:63Þ
σ
0
¼ C 0 : ε À φS : ε
T
þ ε
Ã0
À
Á
Â
Ã
ð6:64Þ
Using Eqs. (6.63), (6.64), and (6.42), the relation between the far-field stress
σ
0 and the macroscopic stress σ takes the form
σ ¼ P : σ
0
À Q : σ
T
ð6:65Þ
where
P ¼ I þ φ I 2 S
ð
Þ A þ S
ð
Þ
2 1
ð6:66Þ
Q ¼ φ I 2 S
ð
Þ A þ S
ð
Þ
2 1
ð6:67Þ
with the components of P and Q are given by
P ijkl ¼ P 1 δ ij δ kl þ P 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:68Þ
Q ijkl ¼ Q 1 δ ij δ kl þ Q 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:69Þ
where
288
6 Unified Micromechanics of Particulate Composites