Biomedical Engineering Reference
In-Depth Information
argument
C
is transformed, the value of the function
T
2PK
is transformed in the
same manner. The development objective of this section is to use the isotropy
requirement (
11.57
) to restrict the functional form of the relationship (
11.56
). The
first step in this development is to show that the principal axes of
T
2PK
must
coincide with the principal axes of the right Cauchy-Green tensor
C
if the isotropy
requirement (
11.57
) is satisfied.
To show that this is the case let
c
denotes an eigenvector of
C
corresponding to
the eigenvalue, say
2
I
c
. This eigenvector
c
is used to construct a
reflective symmetry transformation
R
ðcÞ
with the properties specified by (5.1) for the
vector
a
. Replacing
a
in (4.2) by
c
, it follows that
2
I
C
, thus
C
l
c
¼ l
R
ðaÞ
ij
R
ðcÞ
¼
1
2
a
a
; eð
¼ d
ij
2
c
i
c
j
Þ;
(11.58)
R
ðcÞ
in (
11.57
), it follows that
then, setting
Q
¼
T
T
R
ðcÞ
T
2
PK
R
ðcÞ
R
ðcÞ
R
ðcÞ
¼
t
ð
C
Þ:
(11.59)
For a symmetric second order tensor
A
is easy to show, using (
11.58
), that
R
ðcÞ
R
ðcÞ
¼
A
A
2
c
A
c
2
A
c
c
þ
4
ð
c
A
c
Þ
c
c
(11.60)
and, in particular, when
A
¼
C
, it follows from the fact that
c
is an eigenvector of
C
that
T
R
ðcÞ
R
ðcÞ
C
¼
C
:
(11.61)
The substitution of (
11.61
) into (
11.59
) reduces (
11.59
)to
T
R
ðcÞ
T
2
PK
R
ðcÞ
¼
t
ð
C
Þ:
(11.62)
t
(
C
) in the identity (
11.60
), then it follows with a little algebra, and
recalling that
R
ðcÞ
Next let
A
¼
c ¼c
, that
t
ð
C
Þ
c
¼ð
c
t
ð
C
Þ
c
Þ
c
:
(11.63)
c
is a scalar, this result shows that
c
is an eigenvector of
t
(
C
)as
well as
C
. It then follows that any eigenvector of
C
is also an eigenvector of
T
2PK
Since
c
t
ð
C
Þ
t
(
C
). Since
C
and
T
2PK
¼
¼
t
(
C
) have the same set of principal axes, then
of
T
2PK
are functions of the eigenvalues
2
I
2
II
T
2
PK
11
T
2
PK
22
T
2
PK
33
the eigenvalues
ð
;
;
Þ
ðl
; l
;
2
III
l
Þ
of
C
given by (
11.33
), thus
T
2
PK
11
2
2
2
T
2
PK
22
2
2
2
T
2
PK
33
2
2
2
¼
t
11
ðl
I
; l
II
; l
III
Þ;
¼
t
22
ðl
I
; l
II
; l
III
Þ;
¼
t
33
ðl
I
; l
II
; l
III
Þ:
(11.64)
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