Biomedical Engineering Reference
In-Depth Information
Fig. 6.3
From left to right: the Maxwell and the Kelvin-Voigt element, followed by a combination
of the two
start to take over the stress from the spring and relax it. Hence, the Maxwell element
can characterize stress relaxation.
For a constant stress variation (i.e. step inputs) we have
σ
1
t
η
ε(t)
=
E
+
(6.7)
which shows a spontaneous elastic strain with the stress. When the stress variation
stops, the spring returns to its initial position, while the damper remains in an irre-
versible state.
A second possible combination is the parallel spring-dashpot, referred to as the
Kelvin-Voigt
element (see Fig.
6.3
). In this model, we cannot account for a constant
strain, given the force on the damper must be infinitely big; hence, this model does
not show stress-relaxation properties.
Assuming a constant stress input, we have
E
1
τ
)
σ
e
(
−
ε(t)
=
−
(6.8)
η
E
, the relaxation time. At the time instant
t
with
τ
0, the damper begins to
change slowly, while the spring reached asymptotically its taut value. Hence, the
Kelvin-Voigt element describes the creep phenomena in viscoelastic materials well.
Finally, both the Maxwell element and the Kelvin-Voigt element do not fully
characterize the true viscoelastic behavior. Hence, combining both elements seems
to be a good solution to overcome their individual limitation:
N
parallel Maxwell-
elements, all in parallel with an extra spring, as shown in Fig.
6.3
.
=
=
6.2 Mechanical Analogue and Ladder Network Models
In this chapter we treat the
symmetric
structure of the respiratory tree [
97
,
135
,
163
,
164
], with morphological values given as in Table
2.1
. For the purpose of this
