Biomedical Engineering Reference
In-Depth Information
are given by
z
λ
,
(
r
,
θ
,
z
)
→
(
ρ
,
ϑ
,
ζ
)
:
ρ
=
ρ
(
r
)
,
ϑ
=
θ
,
ζ
=
(7.7)
(
ρ
)
,
Θ
=
π
−
Φ
=
ζ
0
(
ρ
,
ϑ
,
ζ
)
→
(
R
,
Θ
,
Z
)
:
R
=
R
,
Z
Λ
,
(7.8)
π
whereby the deformation gradients for these motions are
diag
∂ρ
∂
diag
∂
r
,
r
,
1
λ
∂ρ
,
(
π
−
Φ
0
)
R
R
1
Λ
F
1
=
,
F
2
=
,
,
(7.9)
πρ
with
Φ
0
and
Λ
the residual stress related opening angle and axial stretch, respec-
the additional axial stretch related primarily to the
in vivo
“prestretch”
1
.
The total deformation gradient is thus computed via
F
tively, and
λ
F
2
F
1
and incompressibility
is assumed to hold during transient motions, but not overall growth and remodelling,
hence det
F
=
=
1 here. Assuming quasi-static motions [22], the Cauchy stress field
σ
associated with either the first
(
F
=
F
1
with
F
2
=
I
)
or the total
(
F
=
F
2
F
1
)
motion
can be computed via
σ
=
div
0
,
(7.10)
+
∂
W
∂
F
F
T
act
σ
=
−
+
σ
,
p
I
(7.11)
where
p
is a Lagrange multiplier that enforces incompressibility,
W
is the net strain
energy function for the passive behaviour of the wall, and
act
accounts for smooth
σ
muscle activity.
Recall the key fact that elastin is produced during the perinatal period and is
normally stable thereafter [10, 26], thus it undergoes large multiaxial stretches as
the artery grows to the adult configuration
β
0
≡
β
s
. On the other hand, collagen
and smooth muscle turn over continuously throughout life [26] and we assume that
they are deposited at a preferred stretch during maturity. These assumptions result in
higher “prestretches” in elastin than in collagen fibres and smooth muscle in matu-
rity. That is, the different timing of deposition and values of deposition stretches for
each constituent play particularly important roles because the associated prestretches
can thereby vary transmurally, which together with the changing mass fractions of
the constituents can strongly influence the unloaded length (i.e., net axial prestress)
and the opening angle (i.e., net residual stress) of the vessel. To study this mathe-
matically, consider the following.
Equilibrium of the
in vivo
configuration
β
0
, both local and global (integral) forms,
requires
r
r
−
r
+
∂
W
∂λ
r
λ
d
r
r
,
(
)=
(
σ
θθ
−
σ
)
p
r
P
(7.12)
rr
r
i
1
One can think of these deformation gradients as inverses of the usual tensors referred to the stress-
free configuration [20].
