Biomedical Engineering Reference
In-Depth Information
where
H
0
is called the generalized structure tensor,
1
2
H
0
=
Ω
ρ
(
Θ
,
Φ
)
M
0
⊗
M
0
d
Ω
(6.34)
π
and can be seen to be an average of the structure tensor
M
0
⊗
M
0
, weighted by the
distribution of fibre angles. As noted in [22], the average stretch can be calculated
using the structure tensor. From (6.32) and (6.34),
1
λ
2
t
λ
=
C
0
:
H
0
.
(6.35)
2
a
An important advantage of the Generalized Structure Tensor approach is that
H
0
can
be calculated a priori since it depends on the material structure in
κ
0
. Further, the
structure tensor simplifies greatly for the distributions in (6.23) and (6.25). However,
as discussed below, there are clear limitations to this approach.
Generalized Structure Tensor for materials with conical splay
For the special case where the fibres have rotational symmetry about a mean refer-
ential direction aligned with
a
0
(e.g. direction
e
3
in Fig. 6.3), we can use (6.21) and
(6.23) with (6.34) to show,
π
0
ρ
(
Θ
)
1
4
sin
3
Conical Splay
H
0
=
κ
I
+(
1
−
3
κ
)
a
0
⊗
a
0
,
κ
=
Θ
Θ
.
(6.36)
d
When there is no preferred orientation,
3, reducing
H
0
to one third of the identity tensor. It follows from (6.35) and (6.36) that the average
stretch can be written with respect to the invariants of
C
0
and
a
0
⊗
ρ
is equal to one and
κ
=
1
/
a
0
κ
IV
0
,
1
λ
1
λ
2
t
λ
=
C
0
:
H
0
=
I
0
+(
1
−
3
κ
)
where
2
a
2
a
I
0
=
tr
C
0
,
IV
0
=
C
0
:
(
a
0
⊗
a
0
)
.
(6.37)
A commonly used strain energy function with a conical splay generalized structure
tensor is [35, 54]
e
γ
(
λ
1
2
)
η
γ
2
t
−
1
)
W
aniso
= H(
λ
t
−
1
−
,
(6.38)
where
1.
For the case where the fibre distribution has rotational symmetry, the structure
tensor can be written with respect to a single
dispersion parameter
which represents
the fibre distribution in an integral sense. This parameter can either be thought of as
a material parameter, determined directly from the experimental data, or calculated,
from experimental knowledge of
λ
a
=
ρ
(
Θ
)
. If the first approach is taken, it is desirable to
know what range of
κ
is allowable. Li and Robertson [71] proved that
κ
∈
(
0
,
1
/
2
]
.
