Biomedical Engineering Reference
In-Depth Information
3.3.4
Elastic Modulus of Composite Materials
Properties of a composite material are weighted combinations of the component
properties [68]. The characteristic properties for quantifying each phase (
i
)are
the volume (
v
i
) and weight (
w
i
). The combination of these properties, the density
(
v
i
), is also frequently used to characterize the phases. From these funda-
mental properties one can calculate the weight fraction (
W
F
)orvolumefraction
(
V
F
) for the particle or filler phase (F):
ρ
=
w
i
/
i
w
F
W
F
=
(3.1)
w
M
+
w
F
v
F
V
F
=
(3.2)
v
M
+
v
F
where the subscript M is used to indicate the matrix or reinforced phase.
A rule-of-mixtures approach is a first approximation to describe the composite
in terms of the component phases. For example, the density of the composite (
ρ
c
)
is related to the density and volume fraction of the component phases by
ρ
M
V
M
(3.3)
There is a simple relationship between weight fraction and volume fraction for
two-phase composite (assuming no porosity):
=
ρ
F
V
F
+
ρ
c
w
F
w
M
ρ
F
1
=
ρ
M
w
F
=
1
ρ
M
W
F
1
V
F
(3.4)
ρ
F
+
+
−
For a two-phase material
with
porosity, there is no change in the weight fraction
expression but there is a volume contribution from the weightless pores (where
v
P
is the pore volume):
v
F
V
F
=
(3.5)
v
M
+
v
F
+
v
P
Manymaterial properties of a compositematerial, including the elasticmodulus, are
typically expressed as the volume-fraction-weighted combination of the component
phase properties. In the simplest approximation for elastic modulus, the two
components are volume-fraction-weighted series or parallel springs. The two
combinations (series and parallel) form extreme bounds on the actual composite's
behavior, and these simple bounds for two-phase composites are frequently called
Voigt-Reuss (V-R) bounds. The upper (iso-strain, parallel springs, Voigt) and
lower (iso-stress, series springs, Reuss) bounds are [68]
E
U
=
V
2
E
2
+
(
1
−
V
2
)
E
1
(3.6)
V
2
−
1
E
2
+
(
1
−
V
2
)
E
L
=
(3.7)
E
1
where by convention
E
2
>
E
1
and
V
1
+
V
2
=
1. These bounds are frequently used
to describe the longitudinal (upper bound) and transverse (lower bound) moduli of
oriented fiber-reinforced composites with fibers aligned in one primary direction.
