2.2.1.1 Paul's equations and the rule of mixtures
Paul's equations (equations 2.1 and 2.2) calculate the effective elastic moduli of
two-phase, irregular geometry composite materials.
1 K* < Kc, +K2c2 (2.1)
C1 C2
K1 K2
1 < G* < G c,1 + G2C2 (2.2)
C1 C2
G, G2
where
K*, G* = Effective bulk and shear moduli of the composite
K1, K2 = Bulk moduli of phase 1 and 2
G1, G2 = Shear moduli of phase 1 and 2
ci, c2 = Volume fractions of phase 1 and 2
The shear and bulk moduli can be related to Young's modulus (E), and Poisson's
ratio by the following equations:
E = (2.3)
1 1
-+
3G 9K
G = (2.4)
3(1 + v)
The right-hand side of the equations 2.1 and 2.2 are referred to as the "Law of
Mixtures."
2.2.1.2 Hashin and Shtrikman's arbitrary phase geometry model
Hashin and Shtrikman (5) derived the equations for an n-phase composite of
arbitrary phase geometry. The following equations are based on a two-phase composite.