Biomedical Engineering Reference
In-Depth Information
The associated, normalized, mutually perpendicular eigenvectors are the prin-
cipal stress directions and specified by
n
1
,
n
2
and
n
3
. It will be clear that
σ
·
n
i
=
σ
i
n
i
for
i
=
1, 2, 3
(8.65)
and also
n
i
·
n
j
=
1for:
i
=
j
=
0for:
i
=
j
.
(8.66)
Based on the above it is obvious, that to every arbitrary stress cube, as depicted
in Fig.
8.8
, another cube can be attributed, which is differently oriented in space,
upon which only normal stresses (the principal stresses) and no shear stresses
are acting. Such a cube is called a principal stress cube, see Fig.
8.14
. Positive
principal stresses indicate extension, negative stresses indicate compression. The
principal stress cube makes it easier to interpret a stress state and to identify the
way a material is loaded. In the following section this will be discussed in more
detail. As observed earlier, the stress state in a certain point is determined com-
pletely by the stress tensor
; in other words, by all stress components that act
upon a cube, of which the orientation coincides with the
xyz
-coordinate system.
Because, actually, the choice of the coordinate system is arbitrary, it can also be
stated that the stress state is completely determined when the principal stresses
and principal stress directions are known. How the principal stresses
σ
σ
i
(with
i
=
1, 2, 3) and the principal stress directions
n
i
can be determined when the stress
tensor
σ
is known is discussed above. The inverse procedure to reconstruct the
z
σ
3
n
3
σ
1
σ
2
n
2
n
1
y
σ
2
σ
1
σ
3
x
Figure 8.14
The principal stress cube.
