Biomedical Engineering Reference
In-Depth Information
Fig. 5.18
Schematic of
a
static vessel wall and
b
deformed vessel wall due to a dynamic external
force F(t). The vessel wall structure exhibits material properties: stiffness coefficient, a function of
its Young's modulus; and damping coefficient, a function of friction and viscous effects
The coefficient
m
is the mass,
c
is the damping coefficient, and
k
is the stiffness
coefficient based on the elastic material properties of the vessel walls, and
d
is the
displacement. The elasticity describes its tendency to deform non-permanently. Put-
ting these forces together we get the general equation of motion (Fig.
5.18
).
.. .
d dd
(5.33)
mc
++=
k
F(t)
The
inertia force
is based on d'Almeberts Principle which states that a mass devel-
ops an inertial force,
F
i
proportional to its acceleration in an opposing direction.
Figure
5.19
shows this principle where if we denote
d
as the displacement, then the
velocity is equal to its derivative with respect to time given as
d
.
, and acceleration is
the second derivative with respect to time given as
..
.
The
damping force
,
F
d
is the product of the velocity and damping coefficient,
c
.
This force attenuates motion and structural deformation. In fluid-structure-interac-
tion models an applied momentary force will induce a displacement leading to an
excitation in the structure causing oscillation. The damping force attenuates this
oscillation, reducing the amplitude with time (Fig.
5.19c
). In blood vessels damp-
ing is caused by friction/viscosity, and thermal effects, which dissipates the energy
stored in the oscillation.
The
stiffness force
represents the restoring body force when it has deformed,
allowing the body to restore to its original position. The magnitude of this force is
proportional to the deformation
d,
and the material property
k
, which is the stiffness
coefficient. Various elastic moduli may be used to describe the stiffness, but typi-
cally for an FSI problem, the Young's modulus is sufficient.
In a fluid-structure-interaction problem the structural equations are described in
Lagrangian coordinates, typically applied to a finite element mesh which differs to
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