Biomedical Engineering Reference
In-Depth Information
of the elastic fibres separately from that of collagen fibres [118, 119]. The layered
nature of the wall can be handled by idealizing the wall as a membrane in which
the effects of the three layers are collapsed into one surface. At the other extreme,
we can consider the wall as a three layered composite with a separate constitutive
equation of the form (6.3) for each layer. The first approach is taken in Sect. 6.3.5
while the second is used in the angioplasty study in Sect. 6.5.
6.3.1 Isotropic mechanism
In the cerebral artery, the elastin is largely confined to a single elastic lamina which
we model as isotropic. In this section, it is not necessary to exclude the possibility that
other structural components contribute to the isotropic mechanism. In discussions
of damage to individual components (Sect. 6.4) and in the discussion of the multi-
layered wall models introduced in Sect. 6.5, we will be more specific about this
issue. While the IEL is known to be in a wavy or buckled state when the arterial
wall is unloaded, Fig. 6.1, for simplicity, we assume the load necessary to reach this
uncrimped state is negligible compared with the load levels of interest. We therefore
assume the unloaded configuration for the isotropic mechanism corresponds to the
unloaded configuration of the body,
κ
0
, Fig. 6.2. The strain energy of this isotropic
mechanism will then be a function of the deformation gradient relative to
κ
0
denoted
as
F
0
. After imposing invariance requirements, the incompressibility condition and
material isotropy, we can write the strain energy as a function of the first and second
principal invariants of
C
0
=
F
0
F
0
or alternatively of
b
0
=
F
0
F
0
, since they have
the same principal invariants,
2
(
tr
b
0
.
1
2
W
iso
=
W
iso
(
I
0
,
II
0
)
,
where
I
0
=
tr
b
0
,
II
0
=
tr
b
0
)
−
(6.6)
The third principal invariant is a constant (one) since the motion is necessarily iso-
choric for incompressible materials and therefore does not arise in (6.6). It then fol-
lows from (6.5), (6.6) (see, e.g. Spencer [108]) that the Cauchy stress tensor can be
written as a function of
b
0
,
2
∂
W
iso
∂
b
0
−
∂
W
iso
∂
II
0
b
−
1
σ
iso
=
.
(6.7)
0
I
0
F
−
1
0
F
−
T
0
Defining the second Piola-Kirchhoff tensor
S
through
S
=
σ
, it follows from
(6.7) that,
pC
−
1
0
S
=
−
+
S
E
,
S
E
=
S
iso
+
S
aniso
,
2
∂
I
II
0
C
0
W
iso
∂
I
0
∂
W
iso
∂
−
∂
W
iso
∂
S
iso
=
I
0
+
,
(6.8)
II
0
where the Lagrange multipliers
p
and
p
introduced in (6.1) and (6.8) differ by a
factor 2
II
0
∂
W
iso
/
∂
II
0
.
