k ¼ k m 1 þ
3 1 À v m
ð
Þ k
Ã
À k m
ð
Þ φ f
3 1 À v m
ð
Þk m þ 1 À φ f
ð
Þ 1 þ v m
ð
Þ k
Ã
À k m
ð
Þ
&
'
ð6:258Þ
μ ¼ μ m 1 þ
15 1 À v m
ð
Þ μ
Ã
À μ m
ð
Þ φ f
15 1 À v m
ð
Þμ m þ 1 À φ f
ð
Þ 8 À 10v m
ð
Þμ Ã À μ m
ð
Þ
&
'
ð6:259Þ
Similarly, the pairwise interacting particles’ solution for the effective properties
of the two-phase composite with the imperfect bond can be modified as
k ¼ k m 1 þ
30 1 À v m
ð
Þ 3γ 1 þ 2γ 2
ð
Þ φ f
3β 1 þ 2β 2 À 10 1 þ v m
ð
Þ 3γ 1 þ 2γ 2
ð
Þ φ f
&
'
ð6:260Þ
μ ¼ μ m 1 þ
30 1 À v m
ð
Þγ 2 φ f
β 2 À 4 4 À 5v m
ð
Þ γ 2 φ f
&
'
ð6:261Þ
where
B
C
Fig. 6.10 Modeling procedures—step 2: equivalence (Nie 2005)
Fig. 6.11 Modeling procedures—step 3: thermo-mechanical properties (Nie 2005)
318
6 Unified Micromechanics of Particulate Composites