IEEE Robotics & Automation Magazine - September 2022 - 82

Table 1. The strain energy function and the Drucker stability for the hyperelastic constitutive models fitted
on the experimental stress-strain data. As indicated in the Drucker stability column, the results of the neoHookean
model and the first-order Ogden law are stable for all strains presented in the experimental data.
Constitutive Models Strain Energy Function (W)
Neo-Hookean
C () ()2
10 I 3
r1-+ - 1
D1
1 J
Mooney-Rivlin CC I01 2
Ogden
Yeoh
i==1 Dk//1 () ()2k
aaa
N
1
a
n
i
i
N
mmm 3++- +-1
k
ii i
123
J
CC CII I33 3 3
10 1201 () () () () () ()6
4
-+ -+ -+ -+ -+ -
rr r
2
30 1
D1
1 J
1
D2
1 J
1
4
D3
1 J
1
10 () () ()2
D1
I 33 1 J
r1-+ -+ - 1
Drucker Stability
Stable for all strains
Stable for m<75 %
First order: Stable for all
strains, Second order:
Stable for
m<58 %
Stable for m<89 %
where z is the displacement vector of the point p. Under
compressible assumption J 1! where
J = de (),Ft
`
- 1
chy-Green tensor (B) can be defined:
BF ·· ,
==J
FB BF Frr= rr
TT
3
2
where
with the three invariants (I1, I2, J) defined as:
()
Itr
Itrtr
J
11
2
2
2
1
(( ))
2
1
== ++
=+ =+ +
== =
BB
de BF
B mm m ,
22
2
2
[( )( )]
tdet()
3
2
mm mm mm
mmm
1
2
2
2
123
,
ii1
2
2
3
2
1
2
3
2
,
(3)
vm W CJ mi IJ mi
2mi
ii
where λi are the principal stretches and are related to principal
strains (e) by the relation
r
r
IJ I
IJ I
1
2
=
=
-
-
3
2
3
4
me .=+ Under the
assumption of compressibility, the first two invariants are
written as follows:
1
2
,
,
(4)
Under the assumption that the material is isotropic, the
strain energy function can be written as a function of the
Table 2. The norm of residuals and the RMSE for
the fitting of the first-order Ogden law and the
neo-Hookean model on the DBPM-F5000-r0.5
hyperelastic material. The measure unit is in
MPa. The values of the neo-Hookean law for both
parameters are slightly lower, so that means this
behavior better approximates the material.
Hyperelastic
Model
Norm of
Residuals RMSE Material Parameters
Ogden N = 1 0.256
Neo-Hookean 0.248
0.019 na 1 972
D .1144
11
1
==
=
0218
0.018 CD0 109
..
1
..
10==1150
== -+
+2
2
mi
D1
J
2
()
1 mi
2
J
3
2
3
5
1
2mi
Because ==mm the principal Cauchy
stresses (6) can be expressed as
JJ( / ),ii123
mmm 22
vi
== m -+ -
-
m W 2CJ 8 2
i
J
2
2
mi
10
3
5
i
I
3
1 B
D1
J
2 ().
1
(7)
The nominal stresses (N), on the other hand, can be
derived from the Cauchy stresses (v), thanks to the following
relationship:
NF .J1 v=
-
(8)
Now that a relation is formed between the nominal stress
and the material parameters C10 and D1 [(7) and (8)], these
parameters can be fitted using a Levemberg Marquadt algorithm.
This is a least square method algorithm, which is
governed by three parameters: the norm of residuals, the
plot of residuals and the root mean-square error (RMSE).
The first parameter, the norm of residuals, is a parameter
for the " goodness " of the fitting. The closer to zero, the
higher the quality of the fitting. The plot of residuals is a
measure of the error of the fitting residuals around the
middle value that is selected to be zero by definition. Lastly,
82 * IEEE ROBOTICS & AUTOMATION MAGAZINE * SEPTEMBER 2022
2
3
2
(2)
deformation gradient can be split into two contributions; a
volumetric J 3
the
j and an isochoric part Fr^h. The left Cauthree
invariants of left Cauchy-Green deformation tensor,
as the equations listed in Table 1. In what follows, as an
example, the strain energy function is derived for a compressible,
isotropic, and homogeneous material for the neoHookean
model. The methodology is identical for all
constitutive laws presented in Table 1. The neo-Hookean
strain energy function (W) is given by:
WC () () .
=- +r
10
1
I
3
1
D1
J
1
2
(5)
For a compressible, isotropic, and homogeneous material,
the equation for the principal Cauchy stresses is written as:
10 ; -- 2E
2J
.
(6)
IEEE Robotics & Automation Magazine - September 2022

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