Biomedical Engineering Reference
In-Depth Information
Fig. 3.18
On C
AUCHY
's
lemma
2
4
3
5
S
¼
r
ij
e
i
e
j
¼
r
11
e
1
e
1
þ
r
12
e
1
e
2
þ
r
13
e
1
e
3
þ
r
21
e
2
e
1
þ
r
22
e
2
e
2
þ
r
23
e
2
e
3
þ
r
31
e
3
e
1
þ
r
32
e
3
e
2
þ
r
33
e
3
e
3
r
11
r
12
r
13
or
½
S
¼
r
21
r
22
r
23
h
e
i
e
i
i:
r
31
r
32
r
33
ð
3
:
89
Þ
In (
3.89
), e
i
e
j
are the basis dyads and the r
ij
are the nine stress coordinates
defined previously in (
3.87
). According to (
3.89
)
2
, the (three) normal stress
components r
11
, r
22
and r
33
are arranged on the principal diagonal and the (six)
shear stress components r
12
, r
13
-r
32
are arranged on the secondary diagonals.
C
AUCHY
'
S
Lemma. Comparing the right-hand side of (
3.87
) with the expression
in (
3.89
)
1
shows visual agreement of the structural arrangement of the stress
coordinates r
11
, r
12
etc. Furthermore, it can be seen that after identical transfor-
mation of the first term in (
3.89
)
1
and comparison with (
3.88
) the following
relation yields
S
¼
r
ij
e
i
e
j
e
i
ð
r
ij
e
j
Þ
|{z}
t
i
¼
e
i
t
i
ð
3
:
90
Þ
The right-hand side (underlined) of (
3.90
) represents a ''dyadic left multipli-
cation'' of t
i
with e
i
. Left scalar multiplication of both underlined terms in (
3.90
)
with e
k
yields the following k-th stress vector t
k
e
k
S
¼ð
t
i
e
i
Þ
e
k
¼ð
e
k
e
i
Þ
|{z}
d
ki
t
i
¼
d
ki
t
i
¼
t
k
:
ð
3
:
91
Þ
Due to e
k
¼
n
k
(k = 1, 2, 3) (see Fig.
3.18
) and using (
3.86
), the underlined
term in (
3.91
) can be expressed as
t
k
X
;
t; n
k
ð
Þ
n
k
SX
; ðÞ
k
¼
1
;
2
;
3
ð
Þ
ð
3
:
92
Þ