Biomedical Engineering Reference
In-Depth Information
with inertial terms or resonance and should not be confounded with other linear ter-
minology. The dynamic behavior is of special interest because viscoelastic materials
are used in situations in which the damping of vibration or the absorption of sound
is necessary. At first sight, this may not be an obvious relation to the lungs, but bear
in mind that lung parenchyma is very similar to polymers, used for such purposes.
The frequency of the sinusoidal load applied to an object or structure may be so
slow that inertial terms do not appear (i.e. the subresonant regime). For respiratory
impedance, these frequencies are below 1 Hz. By contrast, the frequency of the sinu-
soidal load may be high enough such that resonance occurs. For respiratory system,
these resonant frequencies alternate starting from about 10 Hz. A separate study of
these will be done in the next chapter. However, at a sufficiently high frequency,
dynamic behavior is manifested as wave motion. This distinction between ranges of
frequency does not appear in the classical continuum description of a homogeneous
material, because the continuum view deals with differential elements of material.
The lungs are non-homogeneous materials, and the degree of heterogeneity is an
important classification index for correlating the impedance to structural changes in
pathology [
145
].
The stress-strain plot for a linearly viscoelastic material under sinusoidal load
is elliptical, as demonstrated in previous sections of this chapter, and the shape of
the ellipse is independent of stress. By contrast, an elastic-plastic material exhibits
a threshold. Below the threshold yield stress, the material is elastic, and its stress-
strain plot is a straight line. Above the yield stress, irreversible deformation occurs
in the elastic-plastic material.
Let the history of strain to be purely sinusoidal. In complex exponential form,
ε(t)
ε
0
e
iωt
, with
ω
the angular frequency in radians per second. We make use
of the Boltzmann superposition integral, with the lower limit taken as
=
−∞
since in
strict mathematical terms a sinusoid has no starting point:
t
τ)
dε
σ(t)
=
E(t
−
dτ
dτ
(6.43)
−∞
To achieve explicit convergence of the integral, we must decompose the relaxation
function into the sum
E(t)
=
E(t)
)
called the
equilibrium modulus
(in the context of polymers) and substitute the strain history in
(
6.43
). Recalling that
E
e
>
0 for solids and
E
e
=
+
E
e
with
E
e
=
lim
t
−∞
E(t)
=
E(
∞
0 for liquids, it follows that
iωε
0
t
−∞
E
e
ε
0
e
iωt
E(t
τ)e
iτω
dτ
σ(t)
=
+
−
(6.44)
Make the substitution of a new time variable
t
=
t
−
τ
, and obtain
ε
0
e
iωt
E
e
+
E
t
cos
ωt
dt
ω
∞
0
iω
∞
0
E
t
sin
ωt
dt
+
=
σ(t)
(6.45)
If the strain is sinusoidal in time, so is the stress, but they are no longer in phase.
The stress-strain relation becomes
σ(t)
=
E
∗
(ω)ε(t)
=
(E
S
+
jE
D
)ε(t)
(6.46)
