6.5.1 Effective Elastic Modulus of Multiphase Composites
Effective elastic moduli of multiphase composites containing randomly located,
uni-directionally aligned elastic ellipsoids were explicitly derived by Ju and Chen
(1994a). For a multiphase composite, the effective (noninteracting) elasticity tensor
C Ã reads
C Ã ¼ C 0 Á I þ B Á I Á B
ð
Þ
2 1
h
i
ð6:177Þ
where B takes the form
B ¼
X n
q¼1
φ q S þ A q
À
Á À1
ð6:178Þ
Since phase 2 contains spherical voids, we have A 2 = 2 I. The effective bulk
modulus k à and shear modulus μ à for the three-phase composite (n ¼ 2) can be
explicitly expressed as
k à ¼ k 0 1 þ
30 1 À v 0
ð
Þ
P 2
q¼1
φ q
3α q þ2β q
1 À 10 1 þ v 0
ð
Þ
P 2
q¼1
φ q
3α q þ2β q
0
B
B
B
@
1
C
C
C
A
ð6:179Þ
μ
Ã
¼ μ 0 1 þ
15 1 À v 0
ð
Þ
P 2
q¼1
φ q
β q
1 À 2 4 À 5v 0
ð
Þ
P 2
q¼1
φ q
β q
0
B
B
B
@
1
C
C
C
A
ð6:180Þ
with
α q ¼ 2 5v 0 À 1
ð
Þþ10 1 À v 0
ð
Þ
k 0
k q À k 0
À
μ 0
μ q À μ 0
ð6:181Þ
β q ¼ 2 4 À 5v 0
ð
Þþ15 1 À v 0
ð
Þ
μ 0
μ q À μ 0
ð6:182Þ
6.5 Noninteracting Solution for Three-Phase Composites
301