Biomedical Engineering Reference
In-Depth Information
c
2
F
1
F
c
η
c
1
F
2
Figure 5.14
The 3-parameter standard linear visco-elastic model.
Similar to the procedure as used in Section
5.3.2
the total strain of the Maxwell
element is considered as an addition of the strain in the dashpot (
ε
d
) and the strain
in the spring (
ε
s
=
ε
−
ε
d
). The following relations can be proposed for variables
that determine the standard linear model:
F
=
F
1
+
F
2
F
1
=
c
2
ε
F
2
=
c
η
ε
d
F
2
=
c
1
(
ε
−
ε
d
) .
(5.94)
Elimination of
ε
d
,
F
1
and
F
2
from this set of equations leads to
+
τ
R
F
F
=
c
2
ε
+
(
c
1
+
c
2
)
τ
R
ε
,
(5.95)
with
τ
R
=
c
η
/
c
1
the characteristic relaxation time. The force response to a step
ε
(
t
)
=
ε
0
H(
t
) in the strain yields
F
(
t
)
=
F
(
t
)
hom
+
F
(
t
)
part
=
α
e
−
t
/τ
R
+
c
2
ε
0
,
(5.96)
with
an integration constant to be determined from the initial conditions. Deter-
mining the initial condition at
t
α
0 for this problem is not trivial. It is a jump
condition with a discontinuous force
F
and strain
=
0. A way to derive this
jump condition is by using the definition of the time derivative:
ε
at
t
=
F
(
t
+
t
)
−
F
(
t
)
t
F
=
lim
t
→
0
.
(5.97)
Let us take two time points a distance
t
apart, one point on the time axis left of
0
−
and one point on the right side of
t
t
=
0, which we call
t
=
=
0 which we
0
+
. In that case Eq. (
5.95
) can be written as
F
(0)
+
τ
R
F
(0
+
)
call
t
=
F
(0
−
)
(0
+
)
(0
−
)
−
=
c
2
ε
(0)
+
(
c
1
+
c
2
)
τ
R
ε
−
ε
,
(5.98)
t
t
or
t
+
τ
R
(
F
(0
+
)
−
F
(0
−
))
F
(0)
(0
+
)
(0
−
)).
=
c
2
ε
(0)
t
+
(
c
1
+
c
2
)
τ
R
(
ε
−
ε
(5.99)
