Biomedical Engineering Reference
In-Depth Information
The most prominent assumption made in the mechan-
ics-of-materials approach is that strains in the fiber di-
rection of a unidirectional fibrous composite are the
same in the fibers and the matrix. This assumption
allows the planes to remain parallel to the fiber di-
rection. It also allows the longitudinal normal strain to
vary linearly throughout the member with the distance
from the neutral axis. Accordingly, the stress will also
have a linear distribution.
Some other important assumptions are as follows:
1.
The lamina is macroscopically homogeneous, linearly
elastic, orthotropic, and initially stress-free.
2.
The fibers are homogeneous, linearly elastic, isotro-
pic, regularly spaced, and perfectly aligned.
3.
The matrix is homogeneous, linearly elastic, and
isotropic.
In addition, no voids are modeled in the fibers, the matrix
or between them.
The mechanical properties of a lamina are determined
by fiber orientation. The most often used laminate co-
ordinate system has the length of the laminate in the
x
direction and the width in the
y
direction. The principal
fiber direction is the 1 direction, and the 2 direction is
normal to that. The angle between the
x
and 1 directions
is
f
. A counterclockwise rotation of the 1-2 system
yields a positive
f
.
The mechanical properties of the lamina are de-
pendent on the material properties and the volume
content of the constituent materials. The equations for
the mechanical properties of a lamina in the 1-2 di-
rections are:
Macromechanics of a lamina
The generalized Hooke's law relating stresses to strains is
s
i
¼ C
ij
3
j
ij ¼
1
;
2
;
.
;
6
(3.2.12.7)
where s
i
¼
stress components,
C
ij
¼
stiffness matrix, and
e
j
¼
strain components. An alternative form of the stress-
strain relationship is
3
ij
¼ S
ij
s
i
ij ¼
1
;
2
;
.
;
6
(3.2.12.8)
where
S
ij
¼
compliance matrix.
Given that
C
ij
¼ C
ji
,
the stiffness matrix is symmetric,
thus reducing its population of 36 elements to 21 in-
dependent constants. We can further reduce the matrix
size by assuming the laminae are orthotropic. There are
nine independent constants for orthotropic laminae. In
order to reduce this three-dimensional situation to a two-
dimensional situation for plane stress, we have
s
3
¼
0
¼
s
23
¼
s
13
(3.2.12.9)
thus reducing the stress-strain relationship to
3
1
3
2
g
12
S
11
S
12
0
S
21
S
22
0
00
S
66
s
1
s
2
s
12
¼
(3.2.12.10)
The stress-strain relation can be inverted to obtain
s
1
s
2
s
12
Q
11
Q
12
0
3
1
3
2
g
12
¼
Q
21
Q
22
0
(3.2.12.11)
0
0
Q
66
E
1
¼ E
f
V
f
þ E
m
V
m
(3.2.12.1)
where
Q
ij
are the reduced stiffnesses. The equations for
these stiffnesses are
E
2
¼
E
f
E
m
þ V
f
E
m
(3.2.12.2)
V
m
E
f
E
1
1
v
21
v
12
Q
11
¼
(3.2.12.12)
v
12
¼ V
m
v
m
þ V
f
v
f
(3.2.12.3)
v
12
E
2
1
v
12
v
21
v
21
E
l
1
v
12
v
21
Q
12
¼
¼
¼ Q
21
v
21
E
1
¼ v
12
E
2
(3.2.12.4)
(3.2.12.13)
G
12
¼
G
f
G
m
þ V
f
G
m
(3.2.12.5)
E
2
1
v
21
v
21
V
m
G
f
Q
22
¼
(3.2.12.14)
V
m
¼
1
V
f
(3.2.12.6)
Q
66
¼ G
12
(3.2.12.15)
where
E
is Young's modulus,
G
is the shear modulus,
V
is
the volume fraction, n is Poisson's ratio, and subscripts
f
and
m
represent fiber and matrix properties, respectively.
These equations are based on the law of mixtures for
composite materials.
The material directions of the lamina may not coincide
with the body coordinates. The equations for the trans-
formation of stresses in the 1-2 direction to the
x-y
direction are
