Biomedical Engineering Reference
In-Depth Information
In the case of a
constant
force
F
, the strain response is given by
F
c
(1
e
−
t
/τ
) .
ε
(
t
)
=
−
(5.71)
This phenomenon of an increasing strain with a constant force (up to a maximum
of
F
/
c
) is called
creep
.For
t
τ
:
Ft
c
η
ε
≈
,
(5.72)
corresponding to a viscous response, while for
t
τ
:
F
c
ε
≈
,
(5.73)
reflecting a purely elastic response.
5.4
Harmonic excitation of visco-elastic materials
5.4.1
The Storage and the Loss Modulus
In this section some methods will be described that can be used to calculate the
response of a visco-elastic material for different excitations. The section is aimed
at closed form solutions for the governing equations. Fourier and Laplace trans-
forms and complex function theory are used. First, the methods will be outlined
in a general context. After that, an example of a standard linear model will be
discussed.
In the previous section first-order differential equations appeared to describe
the behaviour of simple visco-elastic models, however, in general, a linear visco-
elastic model is characterized by either a higher-order differential equation (or a
set of first-order differential equations):
p
M
d
M
F
dt
M
q
N
d
N
p
1
dF
q
1
d
dt
+···+
ε
dt
N
,
p
0
F
+
dt
+···+
=
q
0
ε
+
(5.74)
or an integral equation:
t
)
F
(
ε
(
t
)
=
J
(
t
−
ξ
ξ
)
d
ξ
(5.75)
0
t
F
(
t
)
=
G
(
t
−
ξ
)
ε
(
ξ
)
d
ξ
.
(5.76)
0
Both types of formulation can be derived from each other.
When a model is used consisting of a number of springs and dashpots, the creep
and relaxation functions can be expressed by a series of exponential functions. It is
