Biomedical Engineering Reference
In-Depth Information
ε
(
t
)
ε
(
t
)
=
J
(
t
)
ε
0
t
=
0
t
Figure 5.7
Typical example of the strain response after a unit-step in the force.
F
ε
ε
1
(
t
)
F
1
F
0
ε
0
(
t
)
0
t
1
0
t
1
t
t
Figure 5.8
Superposition of responses for a linear visco-elastic material.
Proportionality means, that increasing the force with some factor
F
0
leads to a
proportional increase in the strain:
F
(
t
)
=
H(
t
)
F
0
→
ε
(
t
)
=
J
(
t
)
F
0
.
(5.19)
Superposition implies, that applying a load step
F
0
at
t
=
0, with response:
ε
0
(
t
)
=
J
(
t
)
F
0
,
(5.20)
followed by a load step
F
1
at
t
=
t
1
with individual response:
ε
1
(
t
)
=
J
(
t
−
t
1
)
F
1
,
(5.21)
leads to a total response, which is a summation of the two:
ε
(
t
)
=
ε
0
(
t
)
+
ε
1
(
t
)
=
J
(
t
)
F
0
+
J
(
t
−
t
1
)
F
1
.
(5.22)
This is graphically shown in Fig.
5.8
.
Both principles can be used to derive a more general constitutive equation for
linear visco-elastic materials. Assume, we have an arbitrary excitation as sketched
in Fig.
5.9
. This excitation can be considered to be built up by an infinite number
of small steps in the force.
