Biomedical Engineering Reference
In-Depth Information
Fig. 6.16
Several example of the response from nonlinear materials under cyclic load. Observe
that the loop is no longer elliptical. Such response is very similar to pressure-volume loops in
diagnosed patients
and
1
4
cos 3
ωt
3
4
cos
ωt
cos
3
ωt
=
+
(6.65)
then we can express the stress as a sum of sinusoids:
N
σ(t)
=
a
n
sin
nω
n
t
+
φ
n
(6.66)
n
=
1
Consequently, the effect of the nonlinearity is to generate higher harmonics (integer
multiples) of the driving frequency. If the driving signal contains several frequen-
cies, the material responds at new frequencies corresponding to sums and differ-
ences of the drive frequencies.
As for a viscoelastic material obeying nonlinear superposition, analysis shows
that higher harmonics are generated as well. In the electrical engineering commu-
nity, this sort of response is called harmonic distortion. Response of a nonlinear
material to cyclic load may also be visualized via a plot of stress versus strain, as
shown schematically in Fig.
6.16
. In a nonlinear material, the plot is no longer ellip-
tical in shape, as it is in a linear material, because harmonic distortion generates a
non-elliptic plot. Many types of curves are observed in healthy and pathologic lung
function pressure-volume loops. The area within the closed curve, as in the case of
linear materials, represents the energy per volume dissipated per cycle, also known
as
work of breathing
.
6.5 Implications in Pathology
Materials can be time dependent in ways other than viscoelastic response. For ex-
ample, a structural member, such as an alveolar wall, can gradually become stiffer
and stronger if additional substance is added in response to heavy loading [
7
,
43
].
By contrast, it can become less dense and weaker if material is removed in response
to minimal loading. Biological materials intrinsically behave in this way, perform-
ing an
adaptation
to maintain functionality of the system. For instance, a change in
stiffness of the lung parenchyma can be assumed to depend on its porosity [
86
]:
σ
ij
=
(ξ
0
+
e)C
ij kl
(e)ε
kl
(6.67)
