3T
Ã
1 þ 2T
Ã
2 ¼ 200 1 À 2v 0
ð
Þ
2
φ 1
3α 1 þ 2β 1
ð
Þ
2
ð6:195Þ
T
Ã
2 ¼ 23 À 50v 0 þ 35v
2
0
À
Á φ
β
2
1
ð6:196Þ
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 yield 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
X 2
q¼1
φ q ε
Ã0
q þ ε
T
q
"
#
ð6:197Þ
σ
0
¼ C 0 : ε À
X 2
q¼1
φ q S ε
Ã0
q þ ε
T
q
"
#
ð6:198Þ
Using Eqs. (6.197), (6.198), and (6.185), the relation between the far-field stress
σ
0 and the macroscopic stress σ is given by
σ ¼ P : σ
0
À Q : σ
T
ð6:199Þ
where
P = I þ
X 2
q¼1
φ q I 2 S
ð
Þ A q þ S
À
Á 2 1
ð6:200Þ
Q = φ 1 I 2 S
ð
Þ A 1 þ S
ð
Þ
2 1
ð6:201Þ
with the components of P and Q are
P ijkl ¼ P 1 δ ij δ kl þ P 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:202Þ
Q ijkl ¼ Q 1 δ ij δ kl þ Q 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:203Þ
where
3P 1 þ 2P 2 ¼
X 2
q¼1
a q φ q þ 1
ð6:204Þ
304
6 Unified Micromechanics of Particulate Composites