Biomedical Engineering Reference
In-Depth Information
ωε
0
e
i
ω
t
∞
−∞
G
(
s
)e
−
i
ω
s
ds
ωε
0
G
∗
(
)e
i
ω
t
.
F
(
t
)
=
i
=
i
ω
(5.90)
In this equation
G
∗
(
) can be recognized as the Fourier transform of
G
(
t
) (see
Appendix
5.5
). Apparently the force has the same form as the strain, only the
force has a complex amplitude. If we define the
Complex Modulus
E
∗
(
ω
ω
)as
E
∗
(
ω
)
=
i
ω
G
∗
(
ω
)
=
E
1
(
ω
)
+
iE
2
(
ω
) ,
(5.91)
substitution of (
5.91
) into (
5.90
) gives the real part of
F
(
t
):
F
(
t
)
=
ε
0
E
1
cos(
ω
t
)
−
ε
0
E
2
sin(
ω
t
) .
(5.92)
It is clear again that
E
1
and
E
2
are the Storage and Loss modulus. The above
expression specifies the form of the force output in the time domain. We can also
directly derive the relation between
E
(
ω
) and
G
(
ω
) by using a Fourier transform
of Eq. (
5.89
):
F
∗
(
G
∗
(
ωε
∗
(
E
∗
(
ε
∗
(
ω
)
=
ω
)
i
ω
)
=
ω
)
ω
) .
(5.93)
The Complex Modulus is similar to the transfer function in system theory. The
Storage Modulus is the real part of the transfer function, the Loss Modulus is the
imaginary part.
When experiments are performed to characterize visco-elastic, biological mate-
rials, the results are often presented in the form of either the Storage and the Loss
Modulus as a function of the excitation frequency, or by using the absolute value of
the Complex Modulus, in combination with the phase shift between input (strain)
and output (force), as a function of the frequency. In the case of
linear
visco-
elastic behaviour these properties give a good representation of the material (this
has to be tested first). As a second step often a model is proposed, based on a com-
bination of springs and dashpots, to 'fit' on the given moduli. If this is possible, the
material behaviour can be described with a limited number of material parameters
(the properties of the springs and dashpots) and all possible selections of proper-
ties to describe the material under consideration can be derived from each other.
This will be demonstrated in the next subsection for a particular, but frequently
used model, the standard linear model.
5.4.3
The standard linear model
The standard linear model can be represented with one dashpot and two springs,
as shown in Fig.
5.14
. The upper part is composed of a linear spring, the lower
part shows a Maxwell element.
