Biomedical Engineering Reference
In-Depth Information
As noted by Holzapfel and Ogden [54], the limiting case of
2 corresponds
to an isotropic distribution of fibres in a plane perpendicular to
a
0
. They found un-
physical results using a von Mises fibre distribution for pressure inflation at some
values of
κ
=
1
/
approaches 1/2 and fibre orientation
tends towards a planar distribution, there will be no fibres in the
a
0
direction to resist
loading. In this case, it would be more appropriate to consider an alternate
a
0
or use
the planar splay discussed below.
The generalized structural tensor approach has received a great deal of atten-
tion for modelling biological tissues such as pulmonary aveoli [115], abdominal
aneurysms [95] and cerebral arteries [71]. Pandolfi and Holzapfel modelled the hu-
man cornea using a slight generalization of the structure tensor model given in (6.38)
to include two average fibre directions rather than one [51].
κ
>
1
/
3. It should be noted that as
κ
Generalized Structure Tensor for materials with planar splay
The use of generalized structure tensors for a planar distribution of fibres has also
been considered previously [22, 55] If we assume the planar distribution of fibre
angles displays a symmetry to reflections about a line tangent to direction
a
0
in this
plane (e.g. such that
ρ
(
Φ
)=
ρ
(
−
Φ
)
in Fig. 6.3, then it follows from (6.21), (6.25)
and (6.34) that,
Planar Splay with Symmetry
H
0
=
κ
2
D
I
2
D
+(
1
−
2
κ
2
D
)
a
0
⊗
a
0
π
/
2
(6.39)
1
π
sin
2
κ
2
D
=
2
ρ
2
D
(
Φ
)
Φ
d
Φ
.
−
π
/
Namely, as for the case of conical splay, the structure tensor depends only on one
material parameter. When the fibres are isotropically distributed in the plane,
ρ
2
D
is
equal to one and
2, reducing
H
0
to one half of the planar identity tensor.
Following similar approaches to that used in [71], it is straightforward to show
that
κ
2
D
=
1
/
κ
2
D
∈
[
0
,
1
]
.
Comparison of Generalized Structure Tensor (GST) model with integral fibre
distribution model
The generalized structure tensor strain energy function given in (6.31) will be recov-
ered from the integral strain energy function in (6.29) if the angle dependent fibre
strain energy in (6.29) is approximated by the same function, evaluated at the repre-
sentative average stretch. Therefore, in some sense, (6.31) can be considered as an
approximation of the integral fibre distribution model (6.29). However, there is an
additional difference, in that the condition for fibre loading
λ
t
≥
λ
a
is only met in
an averaged sense for the generalized structure tensor model. Efforts have recently
been made to try to quantify the magnitude of the differences in predictions of these
two models [22, 27] though controversy remains [55]. Cortes et al. have compared
the Piolo-Kirchhoff stress from the GST model with the results of the integral model
using material constants chosen for the supraspinatus tendon [22]. Results were com-
pared for uniaxial tension, biaxial loading and simple shear. The formulations were
