Biomedical Engineering Reference
In-Depth Information
For sufficiently small strain levels, i.e.
|
ε
|
1, and using the notation
ε
=
d
ε/
dt
it can be written:
1
λ
d
dt
=
1
D
=
+
ε
ε
≈
−
ε
ε
≈
ε
(1
)
.
(5.15)
1
|
ε
|
Consequently, if
1 then
1
d
dt
≈
ε
=
1
0
d
dt
,
D
=
(5.16)
such that Eq. (
5.1
) reduces to
F
=
c
η
ε
.
(5.17)
Remark
In the literature, the symbol
is frequently used to denote the elonga-
tional rate for large filament stretches instead of
D
.
ε
5.3
Linear visco-elastic behaviour
5.3.1
Continuous and discrete time models
Biological tissues usually demonstrate a combined viscous-elastic behaviour as
described in the introduction. In the present section we assume geometrically
and physically linear behaviour of the material. This means that the theory leads
to linear relations, expressing the force in the deformation(-rate), and that the
constitutive description satisfies two conditions:
•
superposition
The response on combined loading histories can be described as the
summation of the responses on the individual loading histories.
•
proportionality
When the strain is multiplied by some factor the force is multiplied
by the same factor (in fact proportionality is a consequence of superposition).
To study the effect of these conditions a unit-step function for the force is
introduced, defined as H(
t
) (Heaviside function):
0f
t
<
0
1f
t
≥
0
.
H(
t
)
=
(5.18)
=
Assume that a unit-step in the force
F
(
t
)
H(
t
) is applied to a linear visco-elastic
material. The response
(
t
) might have an evolution as given in Fig.
5.7
. This
response denoted by
J
(
t
) is called the
creep compliance
or
creep function
.
ε
