Biomedical Engineering Reference
In-Depth Information
F
ε
(
c
1
+
c
2
)
ε
0
c
2
ε
0
τ
R
t
t
Figure 5.15
Response of the 3-parameter model to a step in the strain.
Because
F
(0
−
)
(0
−
)
=
0 and
ε
=
0 and the terms with
t
vanish when
t
→
0it
is found that
F
(0
+
)
=
(
c
1
+
c
2
)
ε
0
,
(5.100)
so
α
=
c
1
ε
0
. The solution is shown in Fig.
5.15
:
F
(
t
)
=
ε
0
(
c
2
+
c
1
e
−
t
/τ
R
) .
(5.101)
With Eq. (
5.101
) the step response
G
(
t
) is known. Using the Boltzmann integral
this leads to the general solution of Eq. (
5.95
):
t
c
2
+
c
1
e
−
(
t
−
ξ
)
/τ
R
F
(
t
)
=
ε
(
ξ
)
d
ξ
.
(5.102)
−∞
There are several ways to determine the creep function. We can solve Eq. (
5.95
)
for a step in the force. This can be done by determining a homogeneous and a
particular solution as was done for the relaxation problem. However, it can also be
done by means of Laplace transformation of the differential equation. This leads
to an algebraic equation that can be solved. The result can be transformed back
from the Laplace domain to the time domain.
Instead of again solving the differential equation we can use the relation that
exists between the Creep function and the Relaxation function, Eq. (
5.34
). A
Laplace transformation of G(t) leads to
c
2
s
+
c
1
s
+
1
/τ
R
=
c
2
(
s
+
1
/τ
R
)
+
c
1
s
G
(
s
)
=
.
(5.103)
s
(
s
+
1
/τ
R
)
With Eq. (
5.34
) the Laplace transform of
J
is found:
1
/τ
R
[(
c
1
+
c
2
)
s
+
c
2
/τ
R
]
s
.
s
+
1
J
(
s
)
=
G
(
s
)
=
(5.104)
s
2
