Biomedical Engineering Reference
In-Depth Information
a second order tensor in a space of three dimensions
T
is a special case of (A.80)
written in the form
T
¼
T
ij
e
i
e
j
¼
T
ab
e
a
e
b
(A.147)
or, executing the summation in the Latin system,
T
¼
T
11
e
1
e
1
þ
T
22
e
2
e
2
þ
T
33
e
3
e
3
þ
T
23
ð
e
2
e
3
þ
e
3
e
2
Þ
þ
T
13
ð
e
1
e
3
þ
e
3
e
1
Þþ
T
12
ð
e
1
e
2
þ
e
2
e
1
Þ:
(A.148)
If a new set of base vectors defined by
1
p
ð
e
1
¼
e
1
; e
2
¼
e
2
; e
3
¼
e
3
; e
4
¼
e
1
e
2
e
3
e
2
e
3
þ
e
3
e
2
Þ
1
p ð
1
p ð
^
e
5
¼
e
1
e
3
þ
e
3
e
1
Þ; ^
e
6
¼
e
1
e
2
þ
e
2
e
1
Þ
(A.149)
is introduced as well as a new set of tensor components defined by
p
T
23
; T
5
¼
p
T
13
; T
6
¼
p
T
12
;
T
1
¼
T
11
; T
2
¼
T
22
; T
3
¼
T
33
; T
4
¼
(A.150)
then (A.148) may be rewritten as
T
¼ T
1
^
e
1
þ T
2
^
e
2
þ T
3
^
e
3
þ T
4
^
e
4
þ T
5
^
e
5
þ T
6
^
e
6
;
(A.151)
or
T
¼ T
i
e
i
¼ T
a
e
a
;
(A.152)
which is the definition of a vector in six dimensions. This establishes the one-to-one
connection between the components of the symmetric second order tensors
T
and
the six-dimensional vector
T
.
The second point to be made is that fourth order tensors in three dimensions, with
certain symmetries, may also be considered as second order tensors in a six-dimensional
space. The one-to-one connectionbetween the components of the fourth order tensors in
three dimensions
C
and the second order tensors in six dimensions vector
C
is described
as follows. Consider next a fourth order tensor
c
in three dimensions defined by
C
¼
C
ijkm
e
i
e
j
e
k
e
m
¼
C
abgd
e
a
e
b
e
g
e
d
;
(A.153)
and having symmetry in its first and second pair of indices,
C
ijkm
¼
C
jikm
and
C
ijkm
¼
C
ijmk
, but not another symmetry in its indices; in particular
C
ijkm
is not
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