Biomedical Engineering Reference
In-Depth Information
F
(
d
d
ξ
)
Δ
ξ
Δ
F
≈
ξ
ξ
Δ
ξ
+
t
Figure 5.9
An arbitrary force history in a creep test.
The increase
F
of the force
F
between time steps
t
=
ξ
and
t
=
ξ
+
ξ
is equal
to
dF
d
ξ
ξ
=
F
(
F
≈
ξ
)
ξ
.
(5.23)
The response at time
t
as a result of this step at time
ξ
is given by
=
F
(
ε
ξ
ξ
J
(
t
−
ξ
(
t
)
)
) .
(5.24)
The time-dependent force
F
(
t
) as visualized in Fig.
5.9
can be considered as a
composition of sequential small steps. By using the superposition principle we
are allowed to add the responses on all these steps in the force (for each
ξ
).
This will lead to the following integral expression, with all intervals
ξ
taken
as infinitesimally small:
t
)
F
(
ε
(
t
)
=
J
(
t
−
ξ
ξ
)
d
ξ
.
(5.25)
ξ
=−∞
This integral was derived first by Boltzmann in 1876.
In the creep experiment the load is prescribed and the resulting strain is mea-
sured. Often, the experimental set-up is designed to prescribe the strain and to
measure the associated, required force. If the strain is applied as a step, this is
called a relaxation experiment, because after a certain initial increase the force
will gradually decrease in time. The same strategy as used to derive Eq. (
5.25
) can
be pursued for an imposed strain history, leading to
t
F
(
t
)
=
G
(
t
−
ξ
)
ε
(
ξ
)
d
ξ
,
(5.26)
ξ
=−∞
with
G
(
t
) the relaxation function.
