Biomedical Engineering Reference
In-Depth Information
11.11 Transversely Isotropic Hyperelasticity
The specialized constitutive equations transversely isotropic hyperelastic materials
are developed in this section. Recall from Chapter
4
that transversely isotropic
material symmetry is characterized by a unique direction that serves as an axis of
rotational symmetry for the material structure. The plane perpendicular or trans-
verse to the unique direction is a plane of isotropy, hence the descriptive term
transverse isotropy
. The particular material symmetry of an object is only constant
through infinitesimal deformations; larger deformations will change the type of
material symmetry. Thus when the material symmetry of a finitely deformed elastic
object is noted, it is the material symmetry of the reference or undeformed configu-
ration, not the material symmetry of the deformed configuration.
The hyperelastic constitutive equation, the first of (
11.77
), is the starting point of
this development,
T
@
W
2
J
F
F
T
T ¼
;
(11.97)
@
C
where
¼ r
o
. The selected direction is taken as the
e
3
axis and all orthogonal
rotations about that axis by an angle
r
J
f
leave the value of
W
(
) unchanged. Let
R
(
f
)
C
represent an orthogonal transformation about
e
3
by and the angle
f
,
2
4
3
5
;
cos
f
sin
f
0
R
T
R
¼
1
;
R
¼
sin
f
cos
f
0
(11.98)
0
0
1
it follows from
T
dR
d
dR
d
R
T
f
þ
R
¼
0
;
(11.99)
f
that
dR
d
T
R
T
¼
O
;
where
O
f
;
(11.100)
O
hence the representation of the components of
as an axial vector
, where
O
O
e
ijk
O
k
:
O
ij
¼
(11.101)
The requirement of the invariance of
W
ð
C
Þ
under the rotation
R
may be written as
C
0
Þ;
where C
0
¼
R
T
W
ð
C
Þ¼
W
ð
R
C
;
(11.102)
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