Biomedical Engineering Reference
In-Depth Information
10.4.2 Implementation
The constitutive model is implemented in the Abaqus user subroutine
UANISO-
HYPER_INV
and
VUANISOHYPER_INV
, a family of subroutines designed for
anisotropic, hyperelastic material models, in which the strain-energy function
Ψ
is formulated as a function of the strain invariants, written for Abaqus/Standard
and Abaqus/Explicit, respectively. This subroutine can handle and update solution-
dependent internal variables and requires that the derivatives of the strain-energy
function are defined with respect to the scalar invariants
I
1
,
I
2
,
I
3
,
I
fi
4
,
I
fi
6
,
I
sm
4
,
which are provided as input. It is called at each integration point during each load in-
crement to calculate the total strain energy
Ψ
and its first and second derivatives with
respect to the invariants
∂Ψ/∂ I
i
and
∂
2
Ψ/∂ I
i
∂ I
j
for
i,j
1
,
2
,
3
,
4
fib
,
6
fib
,
4
smc
.
Through the input file, a local coordinate system must be set, containing the local
directions
α
fib
for the collagen fibers and
α
smc
for the smooth muscle cells. When
defining the material, memory must be allocated for nine solution-dependent state
variables, namely the damage driving forces
β
mat
,
β
fib
1
,
β
fib
2
, and
β
smc
, and the
damage thresholds
Ψ
ma
0
,
Ψ
fib
0
,
Ψ
fib
0
, and
Ψ
sm
0
. The ninth state dependent variable
is the relative sliding
u
rs
in the actin-myosin complex, which needs to be stored
because of its viscous nature.
The anisotropic, hyperelastic, user-defined material model must be specified with
all the material parameters described above, choosing the options '
formulation
= invariant
', '
local directions = 3
' and '
type = incompress
-
ible
'. A conceptual drawback of the
UANISOHYPER_INV
subroutine is that it
does not provide access to the time step of the solution process, which should
be known for correct programming of the viscous evolution law described in
Eq. (
10.13
). This implies that the exact time step is only known if a fixed time
increment is set, by adding the option '
direct
' to the keyword '
static
'inthe
input file. Otherwise, only the minimum and maximum allowable time step can be
externally prescribed.
=
10.4.3 Parameter Selection
Table
10.1
gives an overview of all parameters of the material model. The first set of
parameters are related to the extracellular matrix with two embedded fiber families.
For the rat abdominal aorta, the main direction of the collagen fibers
α
fib
is set to
5
◦
, i.e., it is almost aligned with the circumferential direction, see O'Connell et al.
(
2008
). The four remaining parameters are set to
κ
±
=
0
.
16
(
−
)
,
k
1
=
32
.
51 kPa,
k
2
=
−
=
3
.
05
(
23
.
63 kPa, by using experimental data from extension-inflation
tests, as described in Famaey et al. (
2012a
). Alternatively, a parameter set from
human arteries can be found in Stålhand (
2009
).
The next set of parameters are the rate constants of the chemical model defining
the fractions
n
III
and
n
IV
in Eq. (
10.12
). They are chosen according to Hai and Mur-
phy (
1988
). These values led to the fractions of
n
III
=
)
and
c
0
.
164 and
n
IV
=
0
.
547, which
