Biomedical Engineering Reference
In-Depth Information
Therefore, by having a suitable strain energy function, the derivation of
ψ
with respect
to the invariants can be obtained and the stresses (here
σ
11
) can be found. For example,
consider a strain energy function of the following general form [10]:
i
=
0
j
=
C
ij
I
1
−
3
i
I
2
−
3
j
ψ
=
(6.12)
0
which can be used for a wide range of applications. For a stress-free condition in the
reference configuration (undeformed state),
ψ
should be equal to zero. Since in the ref-
erence configuration,
λ
1
=
λ
2
=
λ
2
= 1, and from Equation 6.8 ,
I
1
=
I
2
= 3, therefore,
I
1
and
I
2
are subtracted by 3. In addition, for the same reason, the constant
C
00
(when
i
=
j
= 0) must be zero.
In the above equation when
i
=0,
j
=1and
i
=1,
j
= 0, the two-term Mooney - Rivlin
equation is obtained, that is,
ψ
=
C
10
(
I
1
−
3). By substitution of partial
derivation of
ψ
with respect to
I
1
, that is,
∂ψ
/
∂I
1
=
C
10
and
I
2
, that is,
∂ψ
/
∂I
2
=
C
01
into
Equation 6.11, we find that:
σ
11
=
2
λ
2
3) +
C
01
(
I
2
−
−
λ
−
1
C
10
+
C
01
λ
−
1
(6.13)
The constants
C
01
and
C
10
in Equation 6.13 can be obtained from the stress - strain
relationship of the material. To develop a more accurate model, more terms can be used.
For instance, the three-term Mooney - Rivlin model for incompressible materials, which
was used for the present study, is in the form:
ψ
=
C
10
I
1
−
3
+
C
01
I
2
−
3
+
C
11
I
1
−
3
I
2
−
3
(6.14)
In the uniaxial stress - strain compression test, a cubic specimen of elastomeric material
with dimensions 17
×
17
×
19 mm was cut out of a sheet and placed between two plates
designed for compression tests. Then, with a displacement rate of 2 mm s
-1
, the sample
was compressed until the force reached to the machine's force limit (40 N) and the test
was terminated. The compression tests were undertaken using an MCR (modular compact
rheometer) 500 from Physica, Anton Paar. In order to reach to a stable and repeatable
condition, before recording the test results, three compression tests were done. The output
results of the corresponding software were in the form of force and displacement, which
were converted to the engineering strain and stress using the area and the initial thickness
of the specimens. The dependency of the results on the strain rate was examined by
repeating the test at different strain rates. For the selected rubber-like material, negligible
difference between results was observed. Three samples were tested under the above-
mentioned conditions and the results showed that the maximum error was less than 4%
among the samples. The numerical values were averaged and the result was plotted, as
shown in Figure 6.1. The shape of the curve is similar to those of compression tests done
on abdominal organs [1], which illustrates the similarity of the hyperelastic behavior of
elastomeric material to that of real tissue.
In order to find the constants of the model, using the least squares fitting procedure,
the averaged curve was used. The experimental values of engineering stress - strain were
then compared with the curve obtained from the optimization procedure for the three-term
Mooney - Rivlin model, as illustrated in Figure 6.1.
