Biomedical Engineering Reference
In-Depth Information
Fig. 3.15 On the definition
of stress measures: nominal
stress P
I
and true stress r
diagonal e
ij
(i = j) are referred to as shearing strain and represent the deviation
from a right angle of two line elements under a transition from the ICFG to the
CCFG at small displacement derivations.
By comparing (
3.78
) with (
3.70
) and (
3.80
) with (
3.72
) and (
3.73
), it can be
seen that the coordinates of E emerge from the coordinates of G by cancelling the
quadratic and bilinear terms of the displacement gradient (cancelation of the
squared bracket terms in (
3.72
) and the respective parenthesis terms in (
3.73
)).
3.2.4 Stress
3.2.4.1 One-Dimensional Case
Considering a (one-dimensional) bar fixed at the left end and loaded at its right end
with the force K (see Fig.
3.9
), generally two definitions of stress are possible,
assuming a constant stress distribution over the cross section area: (a) referring the
(longitudinal) force K to the cross section area A
0
in the ICFG (Fig.
3.15
), the
nominal or first P
IOLA
-K
IRCHHOFF
-stress definition results
P
I
¼
K
A
0
:
ð
3
:
81
Þ
This stress definition for example, is used in conjunction with laboratory tensile
testing. If however, the (longitudinal) force K is referred to the current cross
sectional area A in the CCFG, the true or C
AUCHY
-stress definition results
r
¼
K
A
:
ð
3
:
82
Þ
In Fig.
3.15
, qualitative stress stress-strain-curves referring to P
I
and r are
depicted where, due to increasing strain, the decreasing cross section area A results
in increasing r compared to P
I
:
Comparison of (
3.81
) and (
3.82
) leads to the relation
