Biomedical Engineering Reference
In-Depth Information
According to (
3.92
), each stress vector t
k
in (
3.87
) is yielded by left scalar
multiplication of the C
AUCHY
stress tensor S with the corresponding indexed normal
vector n
k
(this can be shown instantaneously by scalar multiplication of S with
e
k
¼
n
k
according to (
3.89
)
1
. The stress state at a (material) point X is thus defined
by either the three stress vectors t
k
or the six stress coordinates r
ij
of S. This can be
generalized for any arbitrary cutting plane dA with normal vector n which may also
be situated on the surface area (boundary) of the body (cf. Fig.
3.18
). This leads to
C
AUCHY
's lemma
t
n
ð
X
;
t; n
Þ¼
n
S
ð
X
;
t
Þ
ð
3
:
93
Þ
whereby the stress vector t
n
of area element dA is given by scalar multiplication of the
normal vector n with the stress tensor S (S maps n onto t
n
). According to (
3.93
)the
stress vector t
n
is a linear function of the normal vector n,whereasS is independent of n.
Note: Some prefer right scalar multiplication of n in (
3.92
) and (
3.93
), which
however, entails that in (
3.87
) et seqq. the first index of the stress coordinates r
ij
denotes the direction of the stress vector and the second index denotes the
direction of the normal vector of the area element!
First and Second P
IOLA
-K
IRCHHOFF
Stress Tensor.IfP
I
is referred to as the
first P
IOLA
-K
IRCHHOFF
stress tensor and t
0n
is the stress vector corresponding to area
element dA
0
in the ICFG, analogue to (
3.93
) the following C
AUCHY
lemma applies
t
0n
ð
X
;
t; n
0
Þ¼
n
0
P
I
ð
X
;
t
Þ
ð
3
:
94
Þ
A correlation between P
I
and S can thus be established as follows: substitution
of (
3.93
) and (
3.94
)in(
3.85
) using (
3.54
) and a
T
¼
T
T
a (for arbitrary vectors
a and tensors T) and the symmetry of S regarding (
3.128
) leads to
Z
t
0n
dA
0
¼
Z
A
0
n
0
P
I
dA
0
Z
A
0
Z
P
I
T
dA
0
¼
!
Z
A
P
I
T
ð
dA
0
n
0
Þ
|{z}
dA
0
t
n
dA
A
0
A
0
Z
A
n
SdA
Z
A
Z
A
S
dA
Z
A
0
S
T
ð
dAn
Þ
|{z}
dA
JS
F
T
dA
0
ð
3
:
95
Þ
Comparison of the underlined integrands in (
3.95
) for arbitrarily directed area
elements dA
0
leads, after transposition of both sides to a correlation of the first
P
IOLA
-K
IRCHHOFF
stress tensor and the C
AUCHY
stress tensor as follows
P
I
¼
JF
1
S
ð
CofF
Þ
T
S
S
¼
J
1
F
P
I
;
and
respectively
: ð
3
:
96
Þ
Note: Some specify the first P
IOLA
-K
IRCHHOFF
stress tensor by the transposition
of (
3.96
). This is a result of the differently defined lemma of C
AUCHY
(cf. the note
after (
3.93
)).