Biomedical Engineering Reference
In-Depth Information
Fig. 2.4 Projection of vector
n in the coordinate axis and in
the oxy plane
r
zx
¼ r
zy
¼ r
zz
¼ 0, the material compliance matrix s is obtained directly from
the three-dimensional compliance matrix s,
2
3
t
yx
E
yy
1
E
xx
0
4
5
t
xy
E
xx
1
E
yy
0
s ¼
ð
2
:
39
Þ
1
G
xy
0
0
For the plane strain deformation theory it is considered e
zx
¼ e
zy
¼ e
zz
¼ 0 and
the material compliance matrix s is defined as,
2
4
3
5
t
yx
E
yy
t
zx
t
yz
E
xx
t
zx
t
xz
1
0
E
xx
E
yy
t
xy
E
xx
t
zy
t
xz
E
yy
t
zy
t
yz
1
0
s ¼
ð
2
:
40
Þ
E
xx
E
yy
1
G
xy
0
0
In the case of an anisotropic material, it is possible to rotate the material
constitutive matrix c and orientate the material directions with a vector. Consider a
known vector n in the Euclidean space
3
, Fig.
2.4
, and the respective projections
on the coordinate axis and in the oxy plane. As it is known,
R
!
!
n
oxy
n
ox
n
oxy
n
oxy
n
n
oxy
h ¼ cos
1
n
o
kk
x ¼ cos
1
kk
and
ð
2
:
41
Þ
With the obtained angle information it is now possible to rotate the material
matrix using the rotational transformation matrix and therefore align the material
ox axis with the known vector n. The rotational transformation matrix that permits
an anticlockwise rotation along the ox axis of a known angle b can be defined as,
