Biomedical Engineering Reference
In-Depth Information
Fig. 5.17
Plane stresses
acting on an element in the
arterial wall, where.
σ
is the
normal stress, and
τ
is the
shear stress
1
(
ε
σε σε σε τγ
∫
(5.30)
u
=
d
=
+
+
)
e
x
x
y
y
xy
xy
2
0
which in matrix notation is
1
1
T
T
{ } { }
{ }
{ }
(5.31)
u
=
εσ ε ε
=
[]
D
2
2
The total strain energy of a body subjected to plane stress is then
1
T
{ }
{ }
(5.32)
∫∫∫
U
=
ε
[
D
]
ε
dV
e
2
where V is total volume of the body.
5.3.3
Structural Dynamics Equations
Structural analysis mainly concerns the behaviour of a physical structure when sub-
jected to a force. We present an introduction to structural dynamics by exploring
a vessel wall analogous to a simple single degree of freedom (i.e. one displace-
ment direction) of a spring-mass-damper system. Vessel walls, which are elastic in
behaviour, are subjected to hydrodynamic forces from the blood that it transports
causing deformation in the wall structure. If the applied force deforms the vessel
wall sufficiently to overcome the structure's natural frequency, an oscillatory mo-
tion will occur. In response to an applied force, the vessel walls produce three op-
posing forces:
..
d
• an inertia force, which is proportional to acceleration,
i
Fm
=
• a damping force, which is proportional to velocity,
c
F
=
d
• a stiffness force, which is proportional to displacement,
Fk
k
=
d
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