ABSTRACT
The exact micromechanical models for the effective elastic properties of composites are available only for the infinitely dilute systems, where the volume fraction of the dispersed phase is very small. For infinitely dilute isotropic composite of randomly oriented platelets, the exact expressions for the bulk modulus (K) and shear modulus (G) are given as follows [12,13]:
K
K K K K G
= + − +
+
1 3 4
3 4 φ ( )( )
( ) (11.1)
G
G
G G G G
G G Gm
= + − +
+
1 φ ( )( ) ( )
(11.2)
where
G
G K G K Gg
=
+
+
( ) ( ) 9 8
6 2 (11.3)
As noted in Chapter 7, the isotropic materials are fully characterized by two independent elastic constants [13], such as bulk modulus (K) and shear modulus (G). Once the bulk modulus (K) and shear modulus (G) of a composite are known, the Young’s modulus (E) and Poisson’s ratio (ν) can be determined from the following standard relations for isotropic materials:
E KG
K G =
+
9 3
(11.4)
ν =
−
+
3 2 6 2 K G K G
(11.5)
Equations 11.1 and 11.2 are restricted to infinitely dilute dispersions of thin platelets having random orientation in a continuous matrix phase. They cannot be applied at finite concentrations of dispersed phase (platelets), as the interaction between the particles is ignored in their derivation. Pal [14] has recently developed models for the moduli of concentrated composite solids of thin platelets having random orientation in a continuous matrix phase. The models are derived using a differential scheme along with the exact solution of an infinitely dilute composite. The models are closed-form expressions valid over the full range of the filler concentration.
