Biomedical Engineering Reference
In-Depth Information
with
a
2 is frequently used for fibres in structurally motivated constitutive models
[52, 71, 110]. In this case, the corresponding Cauchy stress tensor is
=
N
i
=
1
H(
λ
(
i
)
e
γ
(
λ
(
i
)
2
2
−
1
)
)
λ
(
i
)
2
t
(
λ
(
i
)
2
t
a
(
i
)
⊗
a
(
i
)
.
σ
aniso
=
2
η
−
1
−
1
)
(6.20)
t
t
The symbol
in (6.19) and (6.20) is the unit step function. Namely, the anisotropic
contribution from fibre family
i
is zero if
H
(
i
)
≤
λ
1. In [110], (6.19) is used for both
t
(
i
)
α
=
collagen and smooth muscle, through
λ
1 in that work.
6.3.4 Anisotropic mechanism - fibre distribution models
In this section, we consider materials for which the fibres do not have a small number
of distinct orientations in the reference configuration. Rather, they appear to be dis-
persed about distinct directions. With this in mind, we define an
orientation density
function
(ODF) denoted by
which characterizes the three-dimensional dis-
tribution of fibre angles in the reference configuration
ρ
(
M
0
)
κ
0
, Fig. 6.3, (see, e.g. [35]).
In the most general case,
M
0
is an arbitrary unit vector which is characterized by
two Euler angles
and can be written in terms of a
three-dimensional Cartesian coordinate system with basis
Θ
∈
[
0
,
π
]
and
Φ
∈
[
−
π
/
2
,
π
/
2
]
{
e
1
,
e
2
,
e
3
}
,
M
0
(
Θ
,
Φ
)=
sin
Θ
cos
Φ
e
1
+
sin
Θ
sin
Φ
e
2
+
cos
Θ
e
3
,
(6.21)
so that we can also write
ρ
=
ρ
(
Θ
,
Φ
)
. The orientation density function is non-
negative and defined such that
ρ
(
Θ
,
Φ
)
d
Ω
represents the normalized number of
fibres in
κ
0
with orientations in the range
[(
Θ
,
Θ
+
d
Θ
)
,
(
Φ
,
Φ
+
d
Φ
)]
,where
d
Ω
=
sin
Θ
d
Θ
d
Φ
. For definiteness,
ρ
(
Θ
,
Φ
)
will be defined to satisfy the normalization
condition
1
2
1
=
Ω
ρ
(
Θ
,
Φ
)
d
Ω
.
(6.22)
π
The range of
arises from the restriction that the distribution function be insensitive
to fibre reflections (
Φ
ρ
(
M
0
)=
ρ
(
−
M
0
)
). Two limiting cases of fibre distribution will
Fig. 6.3.
Schematic of geometric variables used in dispersion model for collagen fibre distribution
