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
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