IEEE Robotics & Automation Magazine - September 2022 - 81

TA instruments, with switchable tensile and compression
clamps. For the tensile test, rectangular samples of 30.0 ×
3.8 × 2.3 mm3 are made, while for the compression test,
cylindrical specimens are constructed with a diameter of
8.10 mm and a thickness of 3.20 mm. Both the uniaxial
compression and tension tests are performed using a strain
ramp of 1% per second. The tensile test is performed until
fracture to derive the ultimate strain and stress at fracture.
Both the tensile and compression engineering stress-strain
curves are plotted in the graph in Figure 3(a). The samples
could withstand strains of more than 100%, typical for soft
robotics applications. In addition, a Poisson ratio (ν) of 0.44
was determined. To do so, five measurements are taken at a
series of discrete strains (5%, 10%, 25%, 50%) and this for
five samples. Via linear regression on the collection of longitudinal
and related lateral strain measurements, the Poisson
ratio was determined.
Fitting Constitutive Laws on the
Hyperelastic Behavior
In the literature, a number of constitutive laws exist that
can be fitted onto the hyperelastic behavior presented in Figure
3(a), including neo-Hookean, Mooney-Rivlin, Ogden,
and Yeoh constitutive models. In the following discussion,
the material is assumed to be compressible, isotropic, and
homogeneous. The characteristic strain energy functions
(W) for the earlier mentioned constitutive laws are presented
in Table 1 for compressible, isotropic, and homogeneous
hyperelastic materials. In these functions, I1, I2, and J are
the three invariants of the left Cauchy-Green deformation
tensor. The functions are characterized by the compressibility
factor Di and the material constants Cik, μi, and αi. In
what follows, we present the methodology for fitting these
constitutive on the experimental data of the nominal stress
(N1), nominal strains 1f^h [Figure 3(a)], and Poisson ratio
of 0.44.
First of all, a material sample with a volume of R3
X!
is considered, which is subjected to a uniaxial mechanical
test. The effect of its compressibility is expressed through
the Poisson ratio. Consequently, the transversals strains, 2f
and
through the formula:
ff of==The
three components ,,
23 1
.
ff and 3f are also called prin12
cipal
strains. As written in [18], considering a generic point
,
p !X the deformation gradient F is defined as
FF ,
==d z
xp J 3
1
r
(1)
f3 , can be written as a function of the nominal strain
f 1 and the Poisson ratio (ν),
Stress-Strain Curve
Stress-Strain Curve Fitting
0.1
0.2
0.3
0.4
0.5
-0.2
-0.1
Raw Data
50
Strain (%)
(a)
100 150
0.1
0.2
0.3
0.4
0.5
-0.2
-0.1
0.01
0.02
0.03
0.04
0.05
Raw Data
Mooney-Rivlin
Ogden N = 1
Ogden N = 2
Neo-Hookean
Yeoh
050100 150
Strain (%)
(b)
-0.05
-0.04
-0.03
-0.02
-0.01
Residuals
Zero Value
Neo-Hookean
Ogden N = 1
050100
Strain (%)
(c)
Figure 3. (a) Experimental data of uniaxial tension/compression test. (b) Fitting of the strain-stress curve of DBPM-F5000-r0.5. Each
fitted constitutive law in the plot has a similar trend, but the Drucker stability have to be checked, to understand if the result of the
fitting gives a stable or unstable solution, for all of the strains of the material. (c) Plot of residuals of DPBM-F5000-r0.5 of the remaining
hyperelastic models that respect Drucker stability. The neo-Hookean (blue) and the first-order Ogden (green) laws have quite the same
variation around the zero middle value (red). Anyway, the trend of neo-Hookean law is slightly tighter than the Ogden one.
SEPTEMBER 2022 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
81
Stress (MPa)
Stress (MPa)
Residuals (MPa)
IEEE Robotics & Automation Magazine - September 2022

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