Biomedical Engineering Reference
In-Depth Information
be considered here. If at an arbitrary point, a coordinate axis can be chosen such that
ρ
is independent of
Φ
, then the material is transversely isotropic at that point and
(6.22) reduces to,
π
0
ρ
(
Θ
)
1
2
Conical Splay
ρ
=
ρ
(
Θ
)
,
1
=
sin
Θ
d
Θ
.
(6.23)
For example, Gasser and Holzapfel [35] considered a transversely isotropic
-
periodic von Mises orientation density function for collagen fibres, suitably modified
so the normalization condition given in (6.23) was satisfied,
π
4
b
2
[
(
(
Θ
)+
)]
exp
b
cos
2
1
ρ
(
Θ
)=
√
2
b
,
(6.24)
π
erfi
(
)
where erfi
is the imaginary error function. Other distributions such
as Gaussian, Gamma [99] and Bingham [2] have also been considered for arterial
collagen. The latter has been considered in the context of a microsphere-based con-
stitutive model developed using a computational homogenization scheme [1]. For
materials idealized as having a planar splay of distributions, we assume that suitable
coordinates can be chosen such that the tangent to all fibres at an arbitrary mate-
rial point lie in the plane
(
x
)=
−
i
erf
(
x
)
Θ
=
π
/
2. In this case, the two-dimensional orientation
distribution function
ρ
2
D
and normalization condition can be written as,
π
/
2
1
π
Planar Splay
ρ
=
2
δ
[
Θ
−
π
/
2
]
ρ
2
D
(
Φ
)
,
1
=
2
ρ
2
D
(
Φ
)
d
(6.25)
−
π
/
where
δ
[
f
]
is the well known generalized function referred to as the “delta Dirac
function”.
Integral fibre distribution models
Lanir [67] appears to be the first to introduce fibre dispersion directly into a strain
energy function. Following this work, we assume a distribution of fibre angles exists
at each arbitrary point in reference configuration
κ
0
. For simplicity, most models of
the arterial wall, assume either an isotropic, transversely isotropic or planar distri-
bution of fibres. The total strain energy is assumed to arise from the combined effect
of fibres with a distribution of orientations as
W
f
. Further, at each orientation, the
fibres may have a distribution
D
(
λ
a
)
of critical stretches
λ
a
at which load bearing
commences. Namely,
D
is the recruitment distribution function. The minimum
and maximum critical stretches will be denoted by
(
λ
α
)
λ
a
1
and
λ
a
2
, respectively. As in
an earlier work, we consider the possibility that
λ
a
1
may be greater than one, (e.g.
[119]),
1
2
W
f
(
λ
0
)
W
ansio
=
Ω
ρ
(
Θ
,
Φ
)
d
Ω
where
π
λ
0
W
f
= H(
λ
0
−
λ
a
1
)
D
(
λ
a
)
W
f
(
λ
t
)
d
λ
a
(6.26)
λ
a
1
