P ¼ 7 þ 5v f
ð
ÞM þ 4 7 À 10v f
ð
Þ
ð 6:253Þ
Q ¼ 8 À 10v i
ð
Þ M þ 7 À 5v i
ð
Þ
ð6:254Þ
S ¼ 35 7 þ 5v f
ð
Þ M 1 À v i
ð
ÞÀP 7 þ 5v i
ð
Þ
ð6:255Þ
φ is given in Eq. (6.239).
Equations (6.241) and (6.248) are verified as equivalent in the determination of
the effective shear modulus of the CSA. Solving the above equations, one can
determine the exact solution for the effective shear modulus of the CSA model.
One of the roots is negative and is extraneous. The positive root provides the value of
the effective shear modulus.
6.6.4 Effective Young’s Modulus and Effective Poisson’s
Ratio
Effective Young’s modulus and Poisson’s ratio for the CSA can be calculated from
the well-known expressions
E
Ã
¼
9k
Ã
μ
Ã
3k
Ã
þ μ Ã
ð6:256Þ
v
Ã
¼
3k
Ã
À 2μ
Ã
6k
Ã
þ 2μ Ã
ð6:257Þ
6.6.5 Numerical Examples
In order to illustrate the effects of imperfect interface and interphase thickness on the
overall effective mechanical and thermal properties of CSA, we consider a special
case of CSA that consists of spherical alumina trihydrate (ATH) particle and
interphase with the following properties:
For particles
E f ¼ 70 GPa
v f ¼ 0:24
α f ¼ 13 Â 10
À6
=
C
For interphase material
312
6 Unified Micromechanics of Particulate Composites