Biomedical Engineering Reference
In-Depth Information
ʨ
ani
, representing the individual families of fibres. In the fol-
lowing, we assume the isotropic part to be specified by a compressible neo-Hookean
format
of an anisotropic part
ʨ
iso
=
μ
e
2
)
+
ʻ
e
2
[
I
1
−
3]
−
μ
e
ln
(
J
2
[
ln
(
J
)
]
,
(18)
with
I
1
F
denoting the first invariant. The elastic constants are represented
by the Lamé-parameters
=
F
:
μ
e
and
ʻ
e
=
ʺ
e
−
2
/
3
μ
e
in terms of the shear modulus
μ
e
ʺ
e
. The anisotropic contribution of the local free energy (
1
)
is based on an orthotropic exponential model with two families of fibres including
fibre dispersion according to Gasser et al. [
5
] or Menzel et al. [
12
], i.e.
and the bulk modulus
exp
k
2
2
1
N
k
1
2
k
2
ʨ
ani
=
E
i
−
,
(19)
i
=
1
= κ
+[
−
κ]
−
with the strain-like quantity
E
i
I
1
1
3
I
4
i
1 and the invariant
F
t
I
4
i
=
a
0
i
·
·
F
·
a
0
i
for
N
=
2 fibre families. The term
E
i
, where
• =
2 is the Macaulay bracket, reflects the assumption that fibres
can support tension only. Consequently,
[
|•| +•
]
/
ʨ
ani
>
0 only if the fibre-related strain
is positive, i.e.
E
i
>
0. Fibre dispersion is introduced by means of the parameter
κ ∈[
0
,
1
/
3
]
, where
κ =
0 corresponds to no dispersion, i.e. transverse isotropy, and
where
3 renders an isotropic distribution. Table
1
summarises the structural
and elastic material quantities included in constitutive Eqs. (
18
) and (
19
) together
with their units. It is important to note that the fibre orientations may be defined
arbitrarily, but the present formulation uses only
one
non-local damage variable so
that both fibre families undergo identical degradation. This is physically meaningful
as long as both families of fibers possess one and the same stretch history, otherwise
a second non-local damage variable should be included in the formulation.
κ =
1
/
Ta b l e 1
Constitutive parameters as used in Sects.
3.1
-
3.3
Type
Symbol
Description
Unit
Structural
a
0
i
Fibre orientation vectors
-
κ
Dispersion parameter
-
Elastic
μ
e
Shear modulus
kPa
ʺ
Bulk modulus
kPa
e
k
1
Elastic constant
kPa
k
2
Elastic constant
-
kPa
−
1
mm
2
Regularisation
c
d
Degree of regularisation
kPa
−
1
ʲ
Penalty parameter
d
kPa
−
1
Damage
ʷ
d
Saturation parameter
ʺ
d
Damage threshold
kPa
