Biomedical Engineering Reference
In-Depth Information
Maxwell Model of Viscoelasticity
−
series spring and dashpot
Stress Relaxation
x
−
x1
K(x
−
x1) = F
1
x
Sum
K
F
x1
F/B = dx1/
dt
F
1
1
1/B
s
Integrator
7.5
5
2.5
0
2.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time t [s]
FIGURE 4.23
Stress relaxation of the Maxwell viscoelastic model. This model solves
F
¼
K
½
x
R
F
=
B
, again
with
1 and arbitrary units. The ideal spring instantly responds followed by stress relaxation via the dashpot
to the steady-state force level.
K
¼
B
¼
William Thompson (Lord Kelvin, 1824-1907) used the three-element viscoelastic model
(Figure 4.19c) to describe the mechanical properties of different solids in the form of a tor-
sional pendulum. Figure 4.26 shows the three-element Kelvin model's creep response. This
model has an initial rapid jump in position with subsequent slow creep. Figure 4.27 shows
the Kelvin model stress relaxation test. Initially, the material is very stiff, with subsequent
stress decay to a nonzero steady-state level that is due to the extension of the dashpot.
The three-element Kelvin model is the simplest lumped viscoelastic model that is bounded
both in extension and force.
The three-element viscoelastic model describes the basic features of stress relaxation and
creep. Biological materials often exhibit more complex viscoelastic properties. For example,
plotting hysteresis as a function of frequency of applied strain gives discrete curves for the
lumped viscoelastic models. Biological tissues demonstrate broad, distributed hysteresis prop-
erties. One solution is to describe biomaterials with a distributed network of three-element
models. A second method is to use the generalized viscoelastic model of Westerhof and Noor-
dergraaf (1970) to describe the viscoelastic wall properties of blood vessels. Making the elastic
modulus mathematically complex yields a model that includes the frequency dependent elas-
tic modulus, stress relaxation, creep, and hysteresis exhibited by arteries. Further, the Voight
and Maxwell models emerge as special (limited) cases of this general approach.
