Modeling Material Microstructure - Continuum Mechanics of Anisotropic Materials - page 182

The ratio of the effective bulk modulus to the matrix bulk modulus, in the
limiting cases when as the ratio of inclusion bulk modulus to the matrix bulk
modulus tends to zero and infinity, are given by

K eff
K m ¼

K eff
K m ¼

3 f s

2

limit
K i =K m ! 0

1 3 f s and limit

K i =K m !1

1 þ ; (7.22)

respectively. The ratio of effective shear modulus to the matrix shear modulus, in
the limiting cases when as the ratio of inclusion shear modulus to the matrix shear
modulus tends to zero and infinity, are given by

G eff
G m ¼

G eff
G m ¼

f s
8

f s

15 15

limit
G i =G m ! 0

1 and limit

G i
G m >1

1 þ

7 ;

(7.23)

respectively. These results illustrate certain intuitive properties of effective moduli.
As the moduli of the inclusion decrease (increase) relative to the moduli of the
matrix material, the effective elastic constants decrease (increase) relative to the
elastic constants of the matrix material. If the inclusions are voids, the formulas
( 7.20 ) simplify to:

K eff
K m ¼

G eff
G m ¼

f s 15 ð 1 n m Þf s

1

G m ;

1 : (7.24)

K m
K m þð 4 = 3 Þ

n m

1 7 5

1/3 these two formulas reduce to the first of ( 7.22 ) and the
first of ( 7.23 ) respectively. The first equation of ( 7.24 ) is given by Nemat-Nasser
and Hori ( 1999 ) as their equation (5.2.6b).
If the material of the inclusion is a fluid, ( 7.20 ) simplifies to the following:

In the case when n m ¼

K f

K m

f s

1

K eff
K m ¼

G eff
G m ¼

15 ð 1 n m Þf s

1 þ

K m þð 4 = 3 ÞG m ;

1 ; (7.25)

K f K m

7 5 n m

1 þ

where K f represents the bulk modulus of the fluid.
Example Exercise 7.4.1
Problem : Calculate the effective bulk modulus K eff , shear modulus G eff , and
Young's modulus E eff , for a composite material consisting of a steel matrix material
and spherical inclusions. The spherical inclusions are made of magnesium, have a
radius r , and are contained within unit cells that are cubes with a dimension of 5 r .
The Young's modulus of steel (magnesium) is 200 GPa (45 GPa) and the shear
modulus of steel (magnesium) is 77 GPa (16 GPa).

Solution : The isotropic bulk modulus K of a material may be determined from
the Young's modulus E and the shear modulus G by use of the formula