Biomedical Engineering Reference
In-Depth Information
P
I
¼
K
A
0
A
A
K
A
0
A
A
0
K
A
¼
A
A
0
ð
3
:
83
Þ
r
|{z}
r
which however, is no longer valid in the three-dimensional case, cf. (
3.96
). Var-
ious other stress measures exist which only make sense in the three-dimensional
case. In conclusion, in the one-dimensional case where the stress distribution
across the (bar) section is not constant, the (longitudinal) stress P
I
and r and the
(tensile) force K are related by
K
¼
Z
A
0
P
I
dA
0
¼
Z
A
ð
3
:
84
Þ
rdA
where generally, the force K is the integral of the stresses acting at the area
element dA
0
and dA, respectively, and the rule ''stress equals force divided by
area'' no longer holds true. From (
3.84
) the expressions (
3.81
) and (
3.82
) for
constant stress distribution are implied. The previous guidelines can then be
expanded to the three-dimensional case.
3.2.4.2 Stress State and Stress Vectors
Inside a body (mechanical device, human body) subjected to external load (forces,
moments, distributed loads) interparticle contact stress results which can be made
'visible' and mathematically accessible by means of the E
ULER
cut principle.
If, for example, a body is loaded by a gravity force G and a contact force K and
supported by bearings with bearing reaction forces L
1
and L
2
and, the body is cut
into two pieces at any arbitrary point, cf. Fig.
3.16
a, distributed area forces in
terms of stress vectors t must be assigned in the cutting plane A (at both of the
resulting body parts
K
1
and
K
2
--the latter is not depicted in Fig.
3.16
a). Fur-
thermore, by division into two separate bodies, the gravity force G is divided into
G
1
and G
2
. The stress vectors act upon the current area element dA (in the CCFG)
and are referred to as true or C
AUCHY
stress. Under formal aspects and beneficial
for later guidelines, a stress vector t
0
can be defined which relates to an area
element dA
0
in the ICFG and is referred to as nominal or first P
IOLA
-K
IRCHHOFF
stress. Both stress vectors differ in every point of the area in magnitude, direction
and sense and thus create an irregular 'stress topology' over area A and A
0
.
Integration of all stress vectors t
0
and t, acting upon dA
0
and dA, respectively,
yields the resulting section force K
S
, cf. Fig.
3.16
b and c.
K
S
¼
Z
A
0
t
0
dA
0
¼
Z
A
tdA
:
ð
3
:
85
Þ
