Biomedical Engineering Reference
In-Depth Information
2.1.1.5 Principal Stress
Another way of describing the Cauchy stress tensor, which completely defines the
stress state in an interest point, is through,
2
3
2
3
t
ð
e
1
Þ
t
ð
e
2
Þ
t
ð
e
3
Þ
r
xx
r
xy
r
xz
4
5
¼
4
5
K ¼
r
yx
r
yy
r
yz
ð
2
:
22
Þ
r
zx
r
zy
r
zz
where e
1
, e
2
and e
3
are the versors of the coordinate system and t
ð
e
i
Þ
is the stress
vector on a plane normal to e
1
passing through the interest point, Fig.
2.2
(a).
Following Cauchy's stress theorem, if the stress vectors of three orthogonal planes,
with a common point, are known, then the stress vector on any other plane passing
through that point can be found through the coordinate transformation equations
[
5
]. Thus, the stress vector t
ð
n
Þ
in a point belonging to an inclined plane,
Fig.
2.2
(b), can be defined by,
2
3
r
xx
r
xy
r
xz
4
5
t
ð
n
Þ
¼ n
r
ij
¼ n
1
½
n
2
n
3
r
yx
r
yy
r
yz
ð
2
:
23
Þ
r
zx
r
zy
r
zz
where n is the inclined plane normal vector. The relation in Eq. (
2.23
) leads to the
transformation rule of the stress tensor. The initial stress tensor r
ij
, defined in the
x
i
coordinate system, can be transformed in a new stress tensor r
ij
, defined in
another x
i
coordinate system by the relation,
K
0
¼ A K A
T
ð
2
:
24
Þ
being A the rotation matrix. Developing Eq. (
2.24
),
2
3
2
3
2
3
2
3
r
0
xx
r
0
xy
r
0
xz
a
11
a
12
a
13
r
xx
r
xy
r
xz
a
11
a
21
a
31
4
5
r
0
yx
r
0
yy
r
0
yz
4
5
4
5
4
5
¼
a
21
a
22
a
23
r
yx
r
yy
r
yz
a
12
a
22
a
32
r
0
zx
r
0
zy
r
0
zz
a
31
a
32
a
33
r
zx
r
zy
r
zz
a
13
a
23
a
33
ð
2
:
25
Þ
The a
ij
coefficients can be understood as the projection of the x
i
coordinate
system versors in the x
i
coordinate system versors. Therefore, the angle between
the versors of each coordinate system can be defined as,
a
ij
ffi
c
ij
¼ cos
1
ð
2
:
26
Þ
Through Eq. (
2.26
) and Fig.
2.3
it is possible to comprehend better the physical
meaning of the a
ij
coefficients and the respective angles.
