Biomedical Engineering Reference
In-Depth Information
f
(
t
)
step
t
f
(
t
)
block
t
f
(
t
)
ramp
t
Figure 5.13
Some 'simple' functions that can be used as loading histories and for which closed form solutions
exist for linear visco-elastic models.
F
(
t
)
=
ε
0
E
1
cos(
ω
t
)
−
ε
0
E
2
sin(
ω
t
) ,
(5.80)
with:
E
1
=
(
F
0
/ε
0
) cos(
δ
) the
Storage Modulus
and
E
2
=
(
F
0
/ε
0
) sin(
δ
) the
Loss Modulus
. These names will become clear after considering the amount of
energy (per unit length of the sample considered) dissipated during one single
loading cycle. The necessary amount of work for such a cycle is
2
π/ω
W
=
F
ε
dt
0
2
π/ω
=−
[
ε
0
E
1
cos(
ω
t
)
−
ε
0
E
2
sin(
ω
t
) ]
ε
0
ω
sin(
ω
t
)
dt
0
2
=
πε
0
E
2
.
(5.81)
It is clear that part of the work is dissipated as heat. This part, given by Eq. (
5.81
)is
determined by
E
2
, the Loss Modulus. During loading the
E
1
related part of
F
also
contributes to the stored work, however this energy is released during unloading.
That is why
E
1
is called the Storage Modulus.
5.4.2
The Complex Modulus
In literature on visco-elasticity the
Complex Modulus
is often used, which is
related to the Storage and Loss Modulus. To identify this relation a more formal
way to study harmonic excitation is pursued. In the case of a harmonic signal the
Boltzmann integral for the relaxation function can be written as
