V ¼ C 1 fφSΓ I À SΓ A þ S
ð
Þ
À1
h
i
I À φSΓ A þ S
ð
Þ
À1
h
i À1
A þ S
ð
Þ
À1 A þ I
ð
Þ
À SΓ A þ S
ð
Þ
À1 A þ I
ð
ÞÀI
h
i
g
ð6:170Þ
By carrying out the lengthy algebra, the components of the positive definite
fourth-rank tensor U and V are explicitly given by
U ijkl ¼ U 1 δ ij δ kl þ U 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:171Þ
V ijkl ¼ V 1 δ ij δ kl þ V 2 δ ik δ jl þ δ il δ jk
À
Á
ð6:172Þ
where
3U 1 þ 2U 2 ¼
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þ 3γ 1 þ 2γ 2
ð
Þ
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þφ 3γ 1 þ 2γ 2
ð
Þ
Á 3k 1
ð6:173Þ
U 2 ¼
β À 4 4 À 5v 0
ð
Þ γ 2
β À 4 4 À 5v 0
ð
Þ φγ 2
Á μ 1
ð6:174Þ
3V 1 þ 2V 2 ¼
30 1 þ v 0
ð
Þ 3γ 1 þ 2γ 2
ð
Þ k
2
1
k 1 À k 0
ð
Þ3α þ 2β
ð
Þ
Â
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þ 3γ 1 þ 2γ 2
ð
Þ
3α þ 2β
ð
ÞÀ10 1 þ v 0
ð
Þφ 3γ 1 þ 2γ 2
ð
Þ
Á φ À 1
!
þ 3k 1 ð6:175Þ
V 2 ¼
4 4 À 5v 0
ð
Þ γ 2 μ
2
1
μ 1 À μ 0
ð
Þ β
Á
β À 4 4 À 5v 0
ð
Þ γ 2
β À 4 4 À 5v 0
ð
Þ γ 2 φ
Á φ À 1
!
þ μ 1
ð6:176Þ
6.5 Noninteracting Solution for Three-Phase Composites
Let us consider an initially perfectly bonded two-phase composite consisting of an
elastic matrix (phase 0) with bulk modulus k 0 and shear modulus μ 0 and randomly
dispersed elastic spherical particles (phase 1) with bulk modulus k 1 and shear
modulus μ 1 . Subsequently, as loadings or deformations are applied, some particles
could experience complete interfacial debonding. These completely debonded particles can be regarded as spherical voids (phase 2).
300
6 Unified Micromechanics of Particulate Composites