6.6.1 Effective Bulk Modulus
The effective bulk modulus k
à for the CSA as obtained by Hashin (1962) is given as
k
Ã
¼ k i þ
φ
1
k f Àk i
þ
3 1Àφ
ð
Þ
3k i þ4μ i
ð6:238Þ
where
φ ¼
r
r þ δ
3
ð6:239Þ
r is the radius of the filler particle
δ is the thickness of interphase
Hashin and Shtrikman’s formulae (1963) and Walpole’s formulae (1966) provide
upper and lower bounds for the bulk modulus of the CSA. Equation (6.238) is same
as the highest lower bound value.
6.6.2 Effective Coefficient of Thermal Expansion (ECTE)
The effective coefficient of thermal expansion α
à for the CSA as obtained by Levin
(1967) is given as
α
Ã
¼ α i þ
α f À α i
1
k f
À
1
k i
1
k
à 1
k i
ð6:240Þ
6.6.3 Effective Shear Modulus
Based on the generalized self-consistent scheme (GSCS) model proposed by Hashin
(1962), Christensen and Lo (1979) have given the condition for determining the
effective shear modulus of CSA as follows:
A
μ
Ã
μ i
2
þ B
μ
Ã
μ i
þ D ¼ 0
ð6:241Þ
where
310
6 Unified Micromechanics of Particulate Composites
The effective bulk modulus k
à for the CSA as obtained by Hashin (1962) is given as
k
Ã
¼ k i þ
φ
1
k f Àk i
þ
3 1Àφ
ð
Þ
3k i þ4μ i
ð6:238Þ
where
φ ¼
r
r þ δ
3
ð6:239Þ
r is the radius of the filler particle
δ is the thickness of interphase
Hashin and Shtrikman’s formulae (1963) and Walpole’s formulae (1966) provide
upper and lower bounds for the bulk modulus of the CSA. Equation (6.238) is same
as the highest lower bound value.
6.6.2 Effective Coefficient of Thermal Expansion (ECTE)
The effective coefficient of thermal expansion α
à for the CSA as obtained by Levin
(1967) is given as
α
Ã
¼ α i þ
α f À α i
1
k f
À
1
k i
1
k
à 1
k i
ð6:240Þ
6.6.3 Effective Shear Modulus
Based on the generalized self-consistent scheme (GSCS) model proposed by Hashin
(1962), Christensen and Lo (1979) have given the condition for determining the
effective shear modulus of CSA as follows:
A
μ
Ã
μ i
2
þ B
μ
Ã
μ i
þ D ¼ 0
ð6:241Þ
where
310
6 Unified Micromechanics of Particulate Composites