Biomedical Engineering Reference
In-Depth Information
Fig. 6.10
Four typical
relaxation functions
G
(
t
). The
vertical scale is the relaxation
function
G
(
t
) that has the
dimensions of stress. The
horizontal scale is time
It follows then that the relaxation function is just the value of the resulting stress
divided by
E
o
,
G
dev
ð
t
Þ¼ð
1
=
E
o
Þ
T
12
:
This result can then be used as the basis of an experiment to determine the
relaxation function
G
dev
(
t
). Typical relaxation functions are of the form of decaying
functions of time like that shown in Fig.
6.10
. The decaying with time is consistent
with the hypothesis of fading memory.
Example 6.5.2
The Maxwell model is a lumped viscoelastic model (see section 1.8) that is a
combination of a spring and a dashpot in series (Fig. 1.9a). When a force applied to
a Maxwell model is changed from zero to a finite value at an instant of time and held
constant thereafter, there is an instantaneous initial elastic extension and then there is
a continued deformation forever as the damper in the dashpot is drawn through the
dashpot cylinder. Thus a Maxwell model exhibits the characteristics of a fluid with an
initial elastic response. In this model, in general, the deflection represents the strain
and the force represents the stress. In this particular illustration the deflection
represents the shearing strain and the force represents the shearing stress. The
differential equation for a Maxwell model is formed by adding together the rate of
strain of the spring, obtained by differentiating Hooke's law in isotropic shear,
E
12
¼
(1/
G
o
)
T
12
, with respect to time, (
d
/d
t
)
E
12
¼
(1/
G
o
)(
d
/d
t
)
T
12
, and the dashpot
(
d
/d
t
)
E
12
¼
(1/
t
r
G
o
)
T
12
where
t
r
G
o
represents the viscosity of the dashpot, thus
d
E
12
d
t
¼
1
t
r
G
o
T
12
þ
1
G
o
d
T
12
d
t
:
The problem is to determine the response of the Maxwell model to a step loading
in shearing strain
E
12
. The magnitude of the step loading is
E
o
. This example is
formally similar to the previous example, the only change being in the model. The
solution of the problem requires the solution of a first-order ordinary differential
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