Biomedical Engineering Reference
In-Depth Information
equations
are
material
independent
and
thus
often
referred
to
as
universal
equations.
3.2.5.1 One-Dimensional Case
Analogue to the (one dimensional) outlines of
Sects. 3.2.3.1
and
3.2.4.1
, in the
following, it is demonstrated how a simple material model such as that for elastic
material behaviour can be derived from balance equations. Here, the (local) bal-
ance of mechanical energy (in the one-dimensional case) is used
w
¼
re
:
ð
3
:
108
Þ
In (
3.108
), the time rate of change of the strain energy w (on the left-hand side)
and the stress power re (with the strain rate e on the right-hand side) are balanced.
The strain energy in (
3.108
) is still unknown and an appropriate formulation,
therefore, must be made. If, in the simplest case, the strain energy w (in the
following referred to as strain energy function or sef for short) is formulated in a
first step as a function of the current strain e(t) and, in a second step as a quadratic
function e (in (
3.109
)
2
c represents a material coefficient)
w
ðÞ¼
ce
2
w
¼
w
ðÞ¼
w e
ð½
and more precise
ð
3
:
109
Þ
differentiation of (
3.109
)
1
with respect to time applying the chain rule and equating
with (
3.108
) leads to
e
¼
0
ð
3
:
110
Þ
w
d
dt
f
w
½
e
ð
t
Þg¼
dw
de
dt
dw
dw
de
r
ð
3
:
108
Þ
de
e
¼
re
or
de
For arbitrary strain rates e it can be deduced from (
3.110
)
2
r
¼
dw
de
ð
3
:
111
Þ
which leads to the decisive result that, based on the power-energy equation, the
stress can be derived by differentiation of w with respect to the strain! In this case,
w is referred to as the ''potential for the stress''. Furthermore, employing (
3.109
)
2
-
(
3.111
) the sought material model reads
r
¼
d
de
ð
ce
2
Þ¼
2ce
¼
f
ð
e
Þ
ð
3
:
112
Þ
which can be identified as the well-known H
OOKE
model using 2c: = E.
Conclusion: At given structure of the sef, the corresponding material model can
be derived by differentiation with respect to strain, following the previously out-
lined steps. It is understood that in the above example the formulation (
3.109
)
2
has
been chosen accordingly, with respect to the (one-dimensional) H
OOKE
model. In
case of three-dimensional non-linear (hyper-) elasticity, for example, the relations