Biomedical Engineering Reference
In-Depth Information
The resultant laminate forces and moments are:
In the equations above
h
is the total thickness of the
laminate.
3
x
3
y
g
xy
N
x
N
y
N
xy
k
x
k
y
k
xy
¼½A
ij
:
þ½B
ij
:
(3.2.12.31)
Short-fiber composites
A distinguishing feature of the unidirectional laminated
composites discussed above is that they have higher
strength and modulus in the fiber direction, and thus
their properties are amenable to alteration to produce
specialized laminates. However, in some applications,
unidirectional multiple-ply laminates may not be re-
quired. It may be advantageous to have isotropic laminae.
An effective way of producing an isotropic lamina is to
use randomly oriented short fibers as the reinforcement.
Of course, molding compounds consisting of short fibers
that can be easily molded by injection or compression
molding may be used to produce generally isotropic
composites. The theory of stress transfer between fibers
and matrix in short-fiber composites goes beyond this
text; it is covered in detail by
Agarwal and Broutman
(1980)
. However, the longitudinal and transverse moduli
(E
L
and
E
T
,
respectively) for an aligned short-fiber lamina
can be derived from the generalized Halpin-Tsai equa-
tions (
Halpin and Kardos, 1976
), as:
3
x
3
y
g
xy
M
x
M
y
M
xy
k
x
k
y
k
xy
¼½B
ij
:
þ½D
ij
:
(3.2.12.32)
The
k
vector represents the respective curvatures of
the various planes. The resultant forces and moments of
a loaded composite can be analyzed given the
ABD
ma-
trices. If the laminate is assumed symmetric, the force
equation reduces to
N
x
N
y
N
xy
3
x
3
y
g
xy
¼½A
ij
:
(3.2.12.33)
Once the laminate strains are determined, the stresses
in the
xy
direction for each lamina can be calculated. The
most useful information gained from the
ABD
matrices
involves the determination of generalized in-plane and
bending properties of the laminate.
In a generic laminate, normal stresses
N
x
and/or
N
y
(or thermal stresses or liquid sorption) will cause de-
formations in the directions
x
and/or
y,
but also shear
strains, unless
A
16
and
A
26
of the extensional stiffness
matrix are equal to 0. These coefficient become 0 if the
laminate is balanced, i.e., has the same number of lami-
nae oriented at
F
and
F
.
Moreover, in a generic laminate, normal or shear
stresses will produce bending, and bending or twisting
will cause mid-plane strains. The coupling between bend-
ing and extension can be eliminated if the coefficients of
the
B
ij
matrix are equal to zero, that is, if the laminate is
fabricated symmetric with respect to its midplane.
The equivalent elastic constants (
E
x
,E
y
,G
xy
,
n
xy
)of
a symmetric and balanced laminate can be easily evalu-
ated from the
A
ij
coefficients (
Barbero, 1998
):
¼
1
þðð
2
l=dÞ
h
L
V
f
Þ
1
h
L
V
f
E
L
E
m
(3.2.12.38)
¼
1
þ
2h
T
V
f
1
h
T
V
f
E
T
E
m
(3.2.12.39)
E
f
=E
m
1
E
f
=E
m
þ
2
ðl=dÞ
h
L
¼
(3.2.12.40)
h
T
¼
E
f
=E
m
1
E
f
=E
m
þ
2
(3.2.12.41)
In the previous equations
E
m
is the elastic modulus of
the matrix,
l
and
d
are the fiber length and diameter
respectively, and
V
f
is the fiber volume fraction.
For a ratio of fiber to matrix modulus of 20, the var-
iation of longitudinal modulus of an
aligned
short-fiber
lamina as a function of fiber aspect ratio,
l/d,
for different
fiber volume fractions is shown in
Fig. 3.2.12-4
. It can be
seen that approximately 85% of the modulus obtainable
from a continuous fiber lamina is attainable with an
aspect ratio of 20.
The problem of predicting properties of
randomly
oriented
short-fiber composites is more complex. The
following empirical equation can be used to predict the
modulus of composites containing fibers that are ran-
domly oriented in a plane:
A
11
A
22
A
12
A
22
E
x
¼
1
h
(3.2.12.34)
A
11
A
22
A
12
A
11
E
y
¼
1
h
(3.2.12.35)
v
xy
¼
A
12
A
22
(3.2.12.36)
G
xy
¼
1
h
A
66
(3.2.12.37)
