Biomedical Engineering Reference
In-Depth Information
such that the force is given by
F
=
c
η
r
,
(5.63)
which is a purely viscous response. In this case the spring has a constant extension,
and the force response is dominated by the dashpot. This explains why, at a constant
strain rate, the force curve tends towards an asymptote in Fig.
5.12
for sufficiently
large
t
.
(ii) For times
t
>
t
∗
the strain rate is zero. In that case, the force decreases exponen-
tially in time. This is called
relaxation
.If
F
∗
denotes the force
t
t
∗
, the force for
=
t
∗
is given by
t
>
F
∗
e
−
(
t
−
t
∗
)
/τ
.
F
=
(5.64)
The rate of force relaxation is determined by
τ
, which explains why
τ
is called a
t
∗
, the slope of the tangent to the force curve equals
F
t
=
t
∗
=−
relaxation time. At
t
=
F
∗
τ
.
(5.65)
5.3.3
Visco-elastic models based on springs and dashpots: Kelvin-Voigt model
A second example of combined viscous and elastic behaviour is obtained for the
set-up of Fig.
5.10
(b). In this case the total force
F
equals the sum of the forces
due to the elastic spring and the viscous damper:
F
=
c
ε
+
c
η
ε
,
(5.66)
or, alternatively after dividing by
c
η
:
F
c
η
c
c
η
ε
+
ε
.
=
(5.67)
Introducing the
retardation
time:
c
c
,
τ
=
(5.68)
Eq. (
5.67
) may also be written as
F
c
η
1
τ
ε
+
ε
.
=
(5.69)
The set-up according to Fig.
5.10
(b) is known as the Kelvin-Voigt model. In anal-
ogy with the Maxwell model, the solution of this differential equation is given by
t
1
c
η
e
−
(
t
−
ξ
)
/τ
F
(
ξ
)
d
ξ
.
ε
(
t
)
=
(5.70)
0
