3U 1 þ 2U 2 ¼
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þ
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þφ
Á 3k 1
ð6:100Þ
U 2 ¼
β À 8 À 10v 0
ð
Þ
β À 8 À 10v 0
ð
Þ φ
Á μ 1
ð6:101Þ
3V 1 þ 2V 2 ¼
30 1 þ v 0
ð
Þk
2
1
k 1 À k 0
ð
Þ3α þ 2β
ð
Þ
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þ
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þφ
Á φ À 1
!
þ 3k 1
ð6:102Þ
V 2 ¼
8 À 10v 0
ð
Þ μ
2
1
μ 1 À μ 0
ð
Þ β
Á
β À 8 À 10v 0
ð
Þ
β À 8 À 10v 0
ð
Þ φ
Á φ À 1
!
þ μ 1
ð6:103Þ
6.4 Pairwise Interacting Solution for Two-Phase
Composites
In this section, we extend the noninteracting solution for two-phase composites
developed in Sect. 6.3 to account for the interparticle interactions.
6.4.1 Approximate Solution of Two-Phase Interaction
If we neglect the interparticle interaction effects, then the ensemble-volume averaged
perturbed strain ε
0p
q can be dropped. The resulting noninteracting approximation for
the particles becomes
ÀA : ε
Ã0
¼ ε
0
À ε
T
þ S : ε
T
þ S : ε
Ã0
ð6:104Þ
Within the present two-sphere context, the integral equation governing the
distributed eigenstrain ε
à for a given particle configuration and remote strain field
ε
0 can be written as
ÀA : ε
à i
ð Þ x
ð Þ ¼ ε
0
À ε
T x
ð Þ þ
Z
Ω i
G x 2 x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à i
ð Þ x
0
ð Þ
h
i
dx
0
þ
Z
Ω j
G x 2 x
0
ð
Þ: ε
T x
0
ð Þ þ ε
à j
ð Þ x
0
ð Þ
h
i
dx
0
i 6 ¼ j, i, j ¼ 1, 2
ð
Þ
ð6:105Þ
where assuming ε
T is uniform in the particles.
292
6 Unified Micromechanics of Particulate Composites