Biomedical Engineering Reference
In-Depth Information
2
4
3
5
¼
2
4
3
5
2
4
3
5
2
4
3
5
r
1
00
0 r
2
0
00r
3
n
11
n
12
n
13
r
xx
r
xy
r
xz
n
11
n
21
n
31
n
21
n
22
n
23
r
yx
r
yy
r
yz
n
12
n
22
n
32
n
31
n
32
n
33
r
zx
r
zy
r
zz
n
13
n
23
n
33
ð
2
:
34
Þ
The principal stresses and principal directions characterize the stress in P and
are independent of the orientation of the coordinate system.
2.1.2 Constitutive Equations
The following relation between the stress rate and the strain rate is assumed,
dr ¼ c de
ð
2
:
35
Þ
The material constitutive matrix is defined by c and if material nonlinear
relations exists between r and e, then c ¼ c
ep
. With Eq. (
2.35
) the following
relation can be established,
de ¼ c
1
dr
ð
2
:
36
Þ
being s
¼
c
1
and defined for the three-dimensional case as,
2
4
3
5
t
yx
E
yy
t
zx
E
zz
1
E
xx
000
t
xy
E
xx
t
zy
E
zz
1
E
yy
000
t
xz
E
xx
t
yz
E
yy
1
E
zz
000
s ¼
ð
2
:
37
Þ
1
G
xy
0
0
0
00
0
0
0
0
1
G
yz
0
1
G
zx
0
0
0
0
0
The material constitutive matrix c is obtained by inverting the material com-
pliance matrix s, which is here defined for an three-dimensional anisotropic
material. The elements on matrix s are obtained experimentally. E
ii
is the Young
modulus in direction i, t
ij
is the Poisson ratio which characterizes the deformation
rate in direction j when a force is applied in direction i, G
ij
is the shear modulus
which characterizes the variation angle between directions i and j. Due to sym-
metry the following relation can be established,
E
i
t
ji
¼ E
j
t
ij
ð
2
:
38
Þ
For the two-dimensional case the plane stress and plane strain [
5
] deformation
theory assumptions can be presumed. Considering the plane stress assumptions,
