K*L = K1 + 2 (2.5)
1 3c,
K2 -K, 2K + 4G,
K*u = K2 + (2.6)
1 3c2
K, K2 2K2 + 4G2
where
K* < K* < K*u (2.7)
G L = G1 + c2 (2.8)
1 6(K1 + 2G, )c1
+
G2 G, 5G, (3K, + 4G )
Glu = G + c (2.9)
1 6(K2 + 2G2 )c2
G G2 5G2(3K+ 4G2)
G*L < G* < G*u (2.10)
and the following conditions must be met:
K2 > K, and G2 > G, (2.11)
where
K*L = Effective bulk modulus of the composite, lower bound
G*L = Effective shear modulus of the composite, lower bound
K*u = Effective bulk modulus of the composite, upper bound
G*u = Effective bulk modulus of the composite, upper bound
2.2.1.3 Hashin's composite spheres model
The composite spheres model consists of a gradation of infinitely-packed spherical
particles in a continuous matrix phase (3). The model assumes that the ratio of particle
diameter to the diameter of the surrounding concentric matrix (a/b) is constant for all