Biomedical Engineering Reference
In-Depth Information
x
B
=
0
+
vt
,
(5.5)
with
0
the initial length of the dashpot. The actual length
at time
t
is given by:
=
x
B
−
x
A
=
0
+
vt
.
(5.6)
Hence, the elongational rate is given by
1
d
dt
=
v
0
+
vt
.
D
=
(5.7)
This shows, that if one end is moved with a constant velocity, the elongational rate
decays with increasing time
t
. Maintaining a constant elongational rate is possible
if the velocity of point B is adjusted as a function of time. Indeed, a constant
elongational rate
D
implies that the length
must satisfy
d
dt
=
D
,
1
(5.8)
subject to the initial condition
=
0
at
t
=
0, while
D
is constant. Since
d
ln (
)
1
d
dt
=
=
D
,
(5.9)
dt
the solution of Eq. (
5.8
) is given by
=
0
e
Dt
.
(5.10)
This means that to maintain a constant elongational rate, point B has to be
displaced exponentially in time, which is rather difficult to achieve in practice
(moreover, try to imagine how the force can be measured during this type of
experiment!).
5.2.1
Small stretches: linearization
If
u
B
and
u
A
denote the end point displacements, introduce:
=
u
B
−
u
A
.
(5.11)
The stretch
λ
may be expressed as
λ
=
0
+
0
.
(5.12)
ε
Introducing the strain
as
ε
=
0
,
(5.13)
the stretch is written as
λ
=
1
+
ε
.
(5.14)
