Civil Engineering Reference
In-Depth Information
Assuming that the fibers are uniform in thickness, the cumulative length
of each component in the direction of the stress is proportional to its volume
fraction. Thus,
L
m
= n
m
L
c
(11.12)
and
L
f
= n
f
L
c
(11.13)
Substituting the values of
L
c
from Equations 11.12 and 11.13 in Equa-
tion 11.11 yields
n
f
¢L
f
L
f
¢L
c
L
c
n
m
¢
L
m
L
m
=
+
(11.14)
¢
L
L
Since
e =
,
Equation 11.14 can be rewritten as
e
c
= n
m
e
m
+ n
f
e
f
(11.15)
s
E
Replacing with gives
e
s
E
c
= n
m
s
E
m
+ n
f
s
E
f
(11.16)
or
n
f
E
f
n
m
E
m
1
E
c
=
+
(11.17)
Equation 11.17 can be rewritten as
E
m
E
f
n
m
E
f
E
c
=
(11.18)
+ n
f
E
m
As with Equation 11.8, Equation 11.18 can be generalized as
X
m
X
f
n
m
X
f
X
c
=
(11.19)
+ n
f
X
m
where
X
is a property such as thermal conductivity, electrical conductivity,
or diffusivity.
The moduli in Equations 11.6 and 11.18 can be plotted as functions of
the volume fraction of the fiber, as shown in Figure 11.14. Clearly, the fibers
are more effective in raising the modulus of the composite when loading
parallel to fibers than when loading perpendicular to fibers.
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