Biomedical Engineering Reference
In-Depth Information
In order to express more specific function forms of (
11.64
), recall that each
eigenvalue
2
2
2
ðl
I
; l
II
; l
III
Þ
of a matrix like
C
satisfies the same characteristic equation
(A54), thus
6
4
2
l
I
C
l
þ
II
C
l
III
C
¼
0
;
(11.65)
where
1
2
½ð
2
trC
2
I
C
¼
trC
;
II
C
¼
trC
Þ
;
III
C
¼
DetC
:
(11.66)
2
I
2
II
2
III
2
I
,
If the expression for
T
2
PK
11
¼
t
11
ðl
; l
; l
Þ
were expanded in a power series in
l
2
4
(
11.65
) above could be used to eliminate any term not proportional to 1,
l
I
or
l
I
. With
T
2
PK
11
T
2
PK
22
T
2
PK
33
of
T
2PK
are expressed as functions
this motivation the eigenvalues
ð
;
;
Þ
2
2
2
of the eigenvalues
ðl
I
; l
II
; l
III
Þ
of
C
as follows:
T
2
PK
11
2
I
4
I
T
2
PK
22
2
II
4
III
¼
a
o
þ
a
1
l
þ
a
2
l
;
¼
a
o
þ
a
1
l
þ
a
2
l
;
(11.67)
T
2
PK
33
2
4
¼
a
o
þ
a
1
l
III
þ
a
2
l
III
:
This system of equations has a unique solution for the three unknown functions
a
o
,
a
1
, and
a
2
. These functions are elementary symmetric functions of the three
eigenvalues
2
I
2
II
2
III
ðl
; l
; l
Þ
or the three (isotropic) invariants of
C
, thus
2
I
2
II
2
III
2
I
2
II
2
III
a
o
¼
a
o
ðl
; l
; l
Þ¼
a
o
ð
I
C
;
II
C
;
III
C
Þ;
a
1
¼
a
1
ðl
; l
; l
Þ
2
I
; l
2
II
; l
2
III
Þ¼
¼
a
1
ð
I
C
;
II
C
;
III
C
Þ;
a
2
¼
a
2
ðl
a
2
ð
I
C
;
II
C
;
III
C
Þ:
(11.68)
In the principal coordinate system it then follows that
T
2
PK
a
2
C
2
¼
a
o
1
þ
a
1
C
þ
;
(11.69)
an expression that is equivalent to (
11.67
) in the principal coordinate system of
C
(or
t
(
C
)), but that also holds in any arbitrary coordinate system. A necessary and
sufficient condition that
the constitutive relation (
11.56
),
T
2PK
¼
t
(
C
) satisfy
T
2PK
Q
T
Q
T
), is that
T
2PK
the material isotropy requirement (
11.57
),
Q
¼
t
(
Q
C
¼
t
(
C
) have a representation of the form (
11.69
) with
a
o
,
a
1
, and
a
2
given by
(
11.68
).
The representation (
11.69
)of
T
2PK
as isotropic function of
C
may also be
expressed as an equivalent isotropic relationship between the Cauchy stress
T
and
the left Cauchy-Green tensor
B
. The algebraic manipulations that achieve this
equivalence begin with recalling from (
11.27
) that
C
F
T
¼
F
, thus (
11.69
) may
be written in the form
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