Biomedical Engineering Reference
In-Depth Information
denote partitions of the consistent tangent-moduli as specified in Waffenschmidt
et al. [
16
]. This results in the following symmetric linearised system of equations
K
˕˕
r
e
r
e
ʔ
˕
e
ʔ
ˆ
e
0
0
K
˕ˆ
e
e
+
·
=
(45)
K
ˆ˕
K
ˆˆ
e
e
for the determination of the unknown increments of the element degrees of freedom
ʔ
˕
e
and
ʔ
ˆ
e
. The assembly of all elements results in the global linearised system
of equations in the
k
th iteration step
r
+
K
·
ʔ
d
=
0
with
ʔ
d
=
d
k
+
1
−
d
k
,
(46)
with
K
being the global tangent stiffness matrix,
d
the global incremental vector of
degrees of freedom,
r
the global residual vector including the internal and external
system loads.
ʔ
5 Residual Stresses
The incorporation of residual stresses is of key importance within the modelling and
simulation of soft biological tissues. Different concepts have been discussed in the
literature to account for these equilibrated stress contributions present in the absence
of external loading. The procedure employed here to incorporate such effects is
based on a multiplicative composition of the total deformation gradient as discussed
by Johnson and Hoger [
7
] and Holzapfel et al. [
6
]. To be specific, we consider a
stress-free reference configuration
B
0
, a load-free residually stressed configuration
B
res
and a current configuration
B
t
, cf. Fig.
1
. In this regard, we—on the one hand—
introduce a deformation gradient-type tensor
F
res
which transforms line elements
from the stress-free reference configuration
B
0
to the residually stressed configuration
B
res
and—on the other hand—another deformation gradient-type tensor
F
p
which
transforms line elements from the residually stressed configuration
B
res
to the current
configuration
B
t
. The resulting deformation gradient tensor can then be written in
terms of cylindrical base vectors as
F
=
F
p
·
F
res
=
ʻ
i
e
i
⊗
E
i
|
i
∈{
r
,ʸ,
z
}
.
(47)
For a perfect cylindrical geometry, the radial and the circumferential principal
stretches can be expressed in terms of spatial cylindrical coordinates
{
r
,ʸ,
z
}
as
k
R
i
. In these rela-
tions, we prescribe the inner radius of the closed configuration
r
i
, the inner radius of
the opened configuration
R
i
, the opening angle
r
i
]+
ʻ
r
=
R
/
[
rk
ʻ
z
]
and
ʻ
ʸ
=
rk
/
R
, where
R
=
ʻ
z
[
r
2
−
ʱ
, and the axial residual stretch
ʻ
z
, and
=
ˀ/
[
ˀ
−
ʱ
]
introduce the opening angle parameter as
k
2
2
. Composition (
47
) can
