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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