Biomedical Engineering Reference
In-Depth Information
Fig. 13.3
Orthotropic architecture of the myocardium. The orthogonal unit vectors
f
0
and
s
0
designate the preferred fiber and sheet directions in the undeformed configuration, respectively.
The third direction
n
0
is orthogonal to the latter by its definition
n
0
:=
(
f
0
×
s
0
)/
|
f
0
×
s
0
|
and Ogden (
2009
). The specific form of this model can be expressed through the
following free-energy function
Ψ
p
(
g
+
Ψ(I
1
,I
4m
,I
4n
,I
4k
),
;
F
,
M
,
N
,
K
)
=
U(J)
(13.13)
where
U(J)
is the purely volumetric part and the orthotropic part is denoted by
Ψ(I
1
,I
4m
,I
4n
,I
4k
)
. The latter is defined as
2
b
exp
b(I
1
−
3
)
+
2
b
i
exp
b
i
(I
4
i
−
1
)
2
−
1
a
a
i
Ψ
=
i
=
m
,
n
2
b
k
exp
b
k
I
4k
−
1
,
a
k
+
(13.14)
in terms of the material parameters
a,b,a
m
,b
m
,a
n
,b
n
,a
k
,b
k
and the invariants
I
1
,I
4m
,I
4n
, and
I
4k
, with
I
1
:=
g
:
b
,
I
4m
:=
g
:
m
,
I
4n
:=
g
:
n
,
I
4k
:=
g
:
k
.
(13.15)
The Eulerian structural tensors
m
,
n
, and
k
are defined as the push-forward of the
Lagrangian structural tensors
FMF
T
,
FNF
T
,
FKF
T
,
m
:=
n
:=
k
:=
(13.16)
and the Lagrangian structural tensors
M
:=
f
0
⊗
f
0
,
N
:=
s
0
⊗
s
0
,
K
:=
sym
(
f
0
⊗
s
0
)
(13.17)
reflect the underlying orthotropic micro-structure of the myocardium through the
vectors
f
0
and
s
0
that denote the preferred fiber and sheet directions of the material
micro-structure in the undeformed configuration as depicted in Fig.
13.3
.Forthe
explicit form of the passive Kirchhoff stress tensor
τ
p
and the corresponding tangent
ˆ
