Biomedical Engineering Reference
In-Depth Information
of the time domain the problem is encountered that it is not possible to carry out
measurements for an unlimited (infinite) period of time.
Both functions
J
(
t
) and
G
(
t
) are continuous with respect to time. Often these
functions are approximated by discrete spectra. Examples of such spectra are
N
e
−
t
/τ
k
]
J
(
t
)
=
J
0
+
f
k
[1
−
+
t
/η
,
(5.36)
k
=
1
with
J
0
,
f
k
,
τ
k
(
k
=
1, ...,
N
) and
η
material parameters (constants) or
M
g
j
e
−
t
/τ
j
,
G
(
t
)
=
G
∞
+
(5.37)
j
=
1
with
G
∞
,
g
j
,
τ
j
(
j
=
1, ...,
M
) material constants.
These discrete descriptions can be derived from spring-dashpot models, which
will be the subject of the next section.
5.3.2
Visco-elastic models based on springs and dashpots: Maxwell model
An alternative way of describing linear visco-elastic materials is by assembling
a model using the elastic and viscous components as discussed before. Two
examples are given, while only small stretches are considered. In that case the
constitutive models for the elastic spring and viscous dashpot are given by
F
=
c
ε
,
F
=
c
η
ε
.
(5.38)
In the Maxwell model according to the set-up of Fig.
5.10
(a) the strain
ε
is addi-
tionally composed of the strain in the spring (
ε
s
) and the strain in the damper (
ε
d
):
ε
=
ε
s
+
ε
d
,
(5.39)
implying that
ε
=
ε
s
+
ε
d
.
(5.40)
F
F
F
F
(a) Spring-dashpot in series
(b) Spring-dashpot in parallel
Figure 5.10
A Maxwell (a) and Kelvin-Voigt (b) arrangement of the spring and dashpot.
