Biomedical Engineering Reference
In-Depth Information
Fig. 2.3
Left
: Cylindrical shell in reference configuration with middle surface radius R and shell
thickness h.
Right
: Deformed shell
mathematically justified using asymptotic methods to be consistent with three-
dimensional elasticity [
37
]. Ciarlet and Lods showed in [
37
] that the Koiter shell
model has the same asymptotic behavior as the three-dimensional membrane model,
the bending model and the generalized membrane model in the respective regimes
in which each of them holds. Motivated by these remarkable properties of the
Koiter shell model, in [
27
,
28
] Canic et al. derived the Koiter shell equations for
the cylindrical geometry with the purpose of using the equations as a model to
study the mechanical behavior of arterial walls. The models in [
27
,
28
], and a
portion of the text presented in this section, were based on the derivations of the
cylindrical Koiter shell equations, obtained by Tambaca in [
139
]. The cylindrical
Koiter shell equations are a generalization of several classes of models that have
been used in modeling of arterial walls. They include the linear string model
proposed by Quarteroni et al. in [
26
,
133
] as a benchmark problem for testing
numerical schemes for FSI in blood flow, the independent ring model [
133
], and
the cylindrical membrane model.
In [
27
,
28
] Canic et al. have extended the linearly elastic cylindrical Koiter model
to include the viscous effects of Kelvin-Voigt type, observed in the measurements of
the mechanical properties of vessel walls [
3
,
4
,
14
]. It was shown in [
3
,
4
,
14
] that the
Kelvin-Voigt model approximates well the experimentally measured viscoelastic
properties of the canine aorta and of the human femoral and carotid arteries. In [
27
,
28
] it was shown that a reduced FSI model between the linearly elastic cylindrical
Koiter shell and the flow of an incompressible, viscous fluid, approximates well the
experimentally measured data presented in [
3
,
4
,
14
]. The Kelvin-Voight model was
also used in [
130
] to model the arterial walls as a linearly viscoelastic membrane.
We summarize the derivation of the
Koiter shell model
next.
The Cylindrical Koiter Shell Equations: General Framework
Consider a clamped cylindrical shell of thickness h, length L, and reference radius
of the middle surface equal to R.SeeFig.
2.3
. This reference configuration, which
we denote by , can be defined via the parameterization
3
; '
z
;/D .R cos ;Rsin ;
z
/
t
;
' W ! !
R
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