Materials Today: Proceedings xxx (xxxx) xxx
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Materials Today: Proceedings
journal homepage: www.elsevier.com/locate/matpr
Modeling of multiwall carbon nanotubes reinforced natural rubber for
soft robotic applications – A comprehensive presentation
Elango Natarajan a, C.S. Hassan a, Ang Chun Kit a, M.S. Santhosh b, S. Ramesh c, R. Sasikumar b
a
Faculty of Engineering, UCSI University, Kuala Lumpur, Malaysia
Department of Mechanical Engineering, Selvam College of Technology, Tamilnadu, India
c
Department of Mechanical Engineering, Presidency University, Bangalore, India
b
a r t i c l e
i n f o
Article history:
Received 4 June 2020
Accepted 9 November 2020
Available online xxxx
Keywords:
Nonlinear material model
Multiwall carbon nanotubes
Mullins effect
Quasi-static strain rate
Component design
a b s t r a c t
Soft materials like rubbers, polymers and elastomers are non-linear hyper elastic materials or viscoelastic materials that show the large deformation under loading. One of the nonlinear material model
is to be used in the simulation of soft mechanism. Mooney-Rivlin, Ogden, Polynomial, Arruda-Boyce,
Yeoh, and Neo-Hookean and Extended Tube are some nonlinear hyperelastic material models existing
in the practice. The objexctive of the current research is to present comprehensively the nonlinear material constants of the multiwall nanocarbon (MWCNT) reinforced natural rubber (NR) composites. NR/
MWCNT composites with 5%, 10%, 20% and 30% weight fractions of MWCNTs are prepared through compression molding. The uniaxial tensile test was conducted on dumb bell specimens of ASTM D412-06a
standard. The respective strain–stress data from uni-axial tensile tests are further curve fitted into existing popular nonlinear hyperelastic models. The nonlinear constants of respective model are comprehensively tabulated, which designers can use in the component design and analysis. Finite element contact
analysis of a cylindrical soft finger pressed against a rigid plate is conducted and presented.
Ó 2020 Elsevier Ltd. All rights reserved.
Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials, Manufacturing and Mechanical Engineering for Sustainable Developments-2020.
1. Introduction
Softmaterials like polymers, rubbers, elastomers etc. have ratedependent nonlinear elastic behaviour associated with negligible
residual strain after unloading [1,2] and strain harderning, called
Mullins effect. These kinds of materials are used in soft mechanism
such soft robotic hand, soft actuators etc. The material properties
of these materials can be greatly improved by reinforcement. The
fillers in micro (106) scale or nano scale (109) may greatly
increase the hardness, modulus, wear resistance, tensile strength
etc of the base matrix. The reinforced composites usually will have
lower density than the base matrix, but have high specific strength
and specific modulus. The nature of the base matrix, filler percentage, compatibility of fillers with base matrix, aspect ratio, processing method, dispersion, distribution of fillers in the base matrix,
interfacial structure and morphology are some parameters influencing the improvement of mechanical properties [3]. The composite is called nanocomposite, if nano fillers are used. The
nanofillers will result the greater interface-to-volume ratio of the
resultant composite. The nanofillers are in the form of nanoparticles, nanotubes, nanofibers, nanowires. Carbon nanotubes (CNTs)
is one of the fillers attracted recently the researches for its extremely high modulus [4] and much stronger reinforcing effects in
the polymer matrix than that shown by conventional carbon black
[5,6]. They are available in the form of single wall carbon nanotubes (SWCNT), double wall carbon nanotubes (DWCNT) and
multiwall carbon nanotubes (MWCNT). Blighe et al. [7] and Golshahr [8] used SWCNT and MWCNT, respectively for the reinforcement of silicone rubber Elango et al [9] investigated the effecct of
MWCNT reinforcement in natural rubber and reported that there
is a drastic increase in the material property by incorporating the
MWCNT fillers into the polymer matrix. Prince Jaya Lal et al [10–
12] used nano silica and glass fibers to increase the material properties of the polymer composite. They reported that the increase of
42% in tensile strength and 39.46% in flexural strength was
acheived with 0.75 wt% of nanosilica. Yu et al [13] used NiZn ferrite
nanoparticles to reinforce polypropylene and reported that the filler acts as a nucleating agent to form beta isostatic thermoplastic
https://doi.org/10.1016/j.matpr.2020.11.293
2214-7853/Ó 2020 Elsevier Ltd. All rights reserved.
Selection and peer-review under responsibility of the scientific committee of the International Conference on Materials, Manufacturing and Mechanical Engineering for
Sustainable Developments-2020.
Please cite this article as: E. Natarajan, C.S. Hassan, A. Chun Kit et al., Modeling of multiwall carbon nanotubes reinforced natural rubber for soft robotic
applications – A comprehensive presentation, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.11.293
E. Natarajan, C.S. Hassan, A. Chun Kit et al.
Materials Today: Proceedings xxx (xxxx) xxx
and accelerators were added to NR mixer and the mastication
was continued for about 20 more minutes to open up the filler
aggregates in order to achieve a good NR-filler interface. The masticated nano composite was then placed into the mould and compression moulded at 160 °C for about 20 min for the vulcanization.
The vulcanized filled rubber was then cured for 4 h at 200 °C in a
digital oven. The dumb bell samples of thickness 2 mm as per
ASTM D412-06a standard as shown in Fig. 2 were then cut from
vulcanized rubber composites.
It is obvious that hyper elastic materials result corresponding to
the readjustment of molecular chains. The readjustment of molecular chains is due to various relaxation processes like rearrangement, reorientation, uncoiling. The material do not get sufficient
time for all relaxation processes, if high rate of load is applied,
and thus the material properties will be affected due to incomplete
rearrangement of chains. Considering this factor, the uniaxial tensile tests were conducted at quasi static strain rate of 250 mm/
minute in InstronÒ 3366 universal tensile testing machine. The
samples were marked with equidistant of 33 mm from the center
of the sample and attached to the pneumatic gripper. The tensile
load was gradually increased until the specimen breaks. The elongation of the sample was measured by extensometer attached to
the machine. The recorded strain–stress data was further used to
evaluate the nonlinear material constants as discussed in the next
section.
composite. Though there are many research articles in the reinforcement of polymers, we restrict our literaure as it is not our
focus.
The modelling of the component made of a soft material is a
challenging one for design engineers. Elastomers, polymers and
rubbers are assumed to be isotropic and incompressible materials.
The inelastic response of the materials are neglected and only the
nonlinear elastic response is considered in the modelling. In a finite
strain theory, a scalar strain energy density function (W) defines
the hyperelastic model of the material. The strain energy potential
is represented either as a function of strain invariants W (I1, I2, I3)
or a direct function of the principal stretch ratios W (k1,k2,k3).
Mooney-Rivlin [14], Ogden [15,16], Arruda-Boyce [17], Yeoh [18],
Polynomial [19] and Extended Tube [20,21] are some popular
non-linear hyper elastic models. These models have been used in
the past for modelling elastomers, rubbers, foams, polymers and
biological tissues [9,22–34].
The accuracy of the prediction of the failure of a component
depends on the choice of the material model and the reliability
of input data into the model. It is more important for any design
engineer to appropriately choose the material model that characterizes the behaviour of material. The stability of the material
model and quality of fitting of the experimental strain–stress data
are other factors to be considered in the selection of the model. The
aim of this research is to synthesize the NR/MWCNT composites
with different filler fractions and to evaluate nonlinear material
constants of the composites. The presented material constants
can be used in structural analysis of any component of these
materials.
3. Hyperelastic material models and numerical simulation
Hyperelastic constitutive models are grouped into three; phenomenological models, response type models, and micromechanical models. In phenomenological models, the macro mechanics of
deformation of the material is used to derive the constitutive equation. In response type models, the Invariants and stretch ratios of
hyperelastic strain energy potentials are directly determined from
the experimental data. In micromechanical models, stochastic
kinetics of deforming polymer chains are considered for the modeling. They evaluate the strainenergy potentials based on the
micromechanical deformation of the material.
The strain invariants I1, I2, I3 are used in Mooney-Rivlin, Polynomial, Arruda-Boyce and Yeoh models to represent the strain energy
density function W. The principal stretch ratios k1, k2, k3 are used in
Ogden model to represent strain energy density function. All these
models are available in commerical finite element (FE) software.
The substitution of experimentally measured data into the strain
energy function of the respective model will result the material
constants.
The general form of Mooney-Rivlin model is
2. Sample preparation and experimentation
MWCNT, NANOCYLÒ NC7000TM series, was supplied by Nanocyl,
Belgium. RMA 1 natural rubber was supplied by Aerospace Engineers Pvt. Ltd. RMA 1 rubber is pure in composition, light in colour,
acid, salt and alkaline resistant material that make it to be used in
many industrial applications including tyres. MWCNTs are thin
tube shaped materials produced by the supplier through the Catalytic Chemical Vapour Deposition (CCVD) process. The average
diameter, length and density of the tube are 9.5 nm, 100 mm and
1.3 g/cm3 respectively. 5%, 10%, 20% and 30% of MWCNTs were
considered in the preparation. Fig. 1 shows the TEM image of
MWCNT used and Table 1 shows the weight fractions of the materials used in synthesing the nano composites. RMA 1 and ingredients mentioned in Table 1 were initially mixed together in two roll
mill for about 20 min. The predetermined quantity of MWCNTs
i
j
k
W ðI1 ; I2 ; I3 Þ ¼ R1
i;j;k¼0 C i;j;k ðI 1 3Þ ðI2 3Þ ðI3 1Þ
ð1Þ
where, Ci,j,k are material constants. The shear modulus and bulk
modulus of the material can be estimated from Mooney-Rivlin constants as
Go ¼ 2ðC 10 þ C 01 Þ
Ko ¼
2
d
ð2Þ
ð3Þ
The parameter d allows for the inclusion of compressibility, it is
assumed to be zero for incompressible materials. If uniaxial tensile
test data are used for modeling, the parameter d can be estimated
as;
d ¼ 1=500Go
ð4Þ
The poison’s ratio and linear elastic modulus of the material can
be estimated as;
Fig. 1. TEM image of NC7000TM MWCNT – scale: 100 nm (from the supplier).
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E. Natarajan, C.S. Hassan, A. Chun Kit et al.
Table 1
Materials used and their weight fractions.
Description
1
2
3
4
RMA 1 natural rubber (47 Shore A)
NC7000TM MWCNT (weight fraction)
Length = 1.5 mm, diameter = 9.5 nm, density = 1.3 g/cc.
95
5
90
10
80
20
70
30
Other ingredients and accelerators
4020 – 0.0016%, TDQ – 0.0015, Zinc oxide – 0.005%, Stearic acid – 0.002%, TMT – 0.0026%, CBS – 0.001%, Sulphur (accelerator) – 0.001%
v¼
3K 0 2G0
6K 0 þ 2G0
ð5Þ
Eo ¼ 6ðC 10 þ C 01 Þ
ð6Þ
The general form of Ogden model is
h
i
W ðk1 ; k2 ; k3 Þ ¼ Rni¼1 lai k1 ai þ k2 ai þ k3 ai 3
Fig. 2. Dumb bell specimen as per ASTM D412-06a standard.
i
Fig. 3. Curve fit of NR/5% MWCNT data into a) Mooney-Rivlin b) Ogden c) Neo-Hookean d) Yeoh e) Polynomial f) Arruda-Boyce models.
3
ð7Þ
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Materials Today: Proceedings xxx (xxxx) xxx
Eq. (1) can also be redefined in terms of first two invariants as,
3
ð8Þ
I2 ¼ ka12 þ ka22 þ ka32 3
ð9Þ
a1
a1
a1
I1 ¼ k1 þ k2 þ k3
This is model is generalized form of Rivlin model.
The extended-tube model is a physics-based polymer model,
that uses the strain-energy potential.
The general form of Extended tube model is
W¼
m and k of Ogden model are shear modulus and limiting network stretch respectively. The linear elastic modulus of the material can be estimated from Ogden constants as:
Eo ¼ 2G0 ð1 þ v Þ
¼
ð10Þ
The general form of Yeoh model is
W ¼R
n
i¼1 C i ðI1
3Þ
i
ð11Þ
n
I;J¼0
#
2Ge
2
þ
In
1
d
ð
ð
I
3
Þ
Þ
þ 2
1
b
1 d2 ðI1 3Þ
3
X
1
2
kb
i 1 þ ðJ 1Þ
d
There are 5 material constants involved in this strain energy
function. Gc represents crosslinked network modulus, Ge represents constraint network modulus, b is empirical parameter
ð0 6 b 6 1Þ; d, and d are extensibility and incompressibility parameters respectively. b is physically derived for a given polymer network as a function of the amount of solvent, solution fraction,
network defects and filler.
The model may also be chosen based on the elongation of the
material. In general, Neo-Hookean model is used for the material
that elongates upto 30%. Mooney-Rivlin is used when the elongation is in the range of 30–200%. Arruda-Boyce model, Polynomial
model and Yeoh model are suitable for elongation upto 300%.
Ogden model is better model for the material having elongation
upto 700%. The uniaxial stress–strain data or volumetric data of
the material are measured from uni-axial or bi-axial test or shear
test as per ASTM standard. The user can use any one of the test data
or multiple test data for the modeling. The experimental data are
curve fitted into respective hyperelastic material model to obtain
the material constants. Mooney model presents two parameters,
three parameters, 5 parameters, and 9 parameters material con-
ð12Þ
where n is number of chain segments, kB is Boltzmann constant, h is
temperature in Kelvin, N is number of chains in network of crosslinked polymer. This model is based on the statistical mechanics
of material with cubic representative volume element containing
eight chain along the diagonal direction.
The polynomial hyperelastic model is formulated in terms of
the two strain invariants I1 and I2 of the Cauchy-Green deformation
tensor. The strain energy function is
W ðI1 ; I2 Þ ¼ R C i;j ðI1 3Þi ðI2 3Þj
"
1 d2 ðI1 3Þ
i¼1
where, Ci are material constants, 2C1 is the initial shear modulus of
the material. When n = 1, Yeoh model reduces to Neo-Hookean
model.
The general form of Arruda-Boyce model is
pffiffiffi
pffiffiffi
sinhb
W ¼ NkB h n bkchain nln
b
Gc
2
ð13Þ
Fig. 4. Curve fit of NR/10% MWCNT data into a) Mooney-Rivlin b) Ogden c) Neo-Hookean d) Yeoh e) Polynomial f) Arruda-Boyce models.
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E. Natarajan, C.S. Hassan, A. Chun Kit et al.
Fig. 5. Curve fit of NR/20% MWCNT data into a) Mooney-Rivlin b) Ogden c) Neo-Hookean d) Yeoh e) Polynomial f) Arruda-Boyce models.
norm/residual values is considered as the better curve fit of the
model. It helps the user to assess the result whether to accept
the results. The another better way to assess the curve fit is Visual
fit mode in which the user can visually see the curve fit of experimental data and decide whether the results are acceptable. If the
results are not acceptable, the user can choose the higher order,
or increase the iterations and repeat the curve fitting.
Figs. 3–6 depict curve fit of experimentally measured data into
various models. The quality of the curve fit was visually noticed in
all the models in all cases of the composites, which regarded the
accuracy of the results. Moreover the error norm/residual values
obtained from all simulations was less than 50 which confirms
the quality of the curve fit of the models alternatively. The nonlinear material constants of NR composites of different weight fractions are presented in Tables 2 and 3. The design engineers can
use one of the models of his own choice in their analysis.
stants. Ogden model presents the constants with 1st order, 2nd
order and 3rd order. It adopts Levenberg-Marquardi nonlinear least
squares optimization algorithm to determine material constants of
Ogden’s stress deformation function. In general, the material property is more accurately represented by higher order models. Finite
element analysis (FEA) software such as ANSYS, Marc, COMSOL,
ABACUS provide the curve fit modules and options to curve fit
the strain–stress data to derive the material constants of different
nonlinear models. The curve fitting process is based upon the
regression analysis using least sqaure method. The material constants can be found from the experimental strain–stress data and
constitutive equation for the principal true stress r11 under uniaxial. The quality of the fitting is mostly assessed by comparing visually the curves obtained with hyperelastic models to the
experimental data.
The experimentally measured strain–stress data of NR/
MWCNTs composite of different filler fractions were applied to
Mooney-Rivlin (3 parameters), Ogden (3rd order), Polynomial,
Yeoh, and Arruda-Boyce models through curve fitting. ANSYS provides options to curve fit nonlinear hyperleastic models and viscoelastic models. Three control parameters of the nonlinear
regression model are number of iterations, residual tolerance and
coefficient change tolerance. The solution stops when both the
residual tolerance and the coefficient change tolerance of the erron
norm are met, or the number of iterations is met. For the current
analyses, the number of iterations was set to 1000 and normalized
least square fit option was chosen.
Visual fit and the error norm/residual values are two factors
through which the acceptability of the results are determined.
When the curve fit plot is printed on the screen, the error norm/
residual values is also printed on the GUI window. The smaller error
4. Case study
The purpose of this case study is to show the application of nonlinear model in FE analysis. In the past, Elango et al [26] developed
a contact model for power grasping using soft finger. They
attempted with Silicone, Viton, Neoprene elastomeric materials.
The power grasping is a type of grasping in which many contact
points will be used for grasping the object. The numerical study
is required for grasp manipulations before authors take up the path
planning. Fig. 7 shows the contact model of the cylindrical soft finger pressed against a rigid body as power grasping, in which the
deformation of the finger (d0) and contact width (w) are measured
against the normal force (F).
5
E. Natarajan, C.S. Hassan, A. Chun Kit et al.
Materials Today: Proceedings xxx (xxxx) xxx
Fig. 6. Curve fit of NR/30% MWCNT data into a) Mooney-Rivlin b) Ogden c) Neo-Hookean d) Yeoh e) Polynomial f) Arruda-Boyce models.
Table 2
Nonlinear constants of Mooney-Rivlin, Neo-Hookean, Yeoh and Arruda-Boyce models.
Filler %
5%
10%
20%
30%
Mooney-Rivlin (3 parameters) model
Neo-Hookean model
Yeoh model
C10
C01
C11
m
C10
C20
m
Arrda-Boyce model
k
1.096958
1.416277
0.864849
2.11687
1.03862
0.357
2.487001
7.965075
0.03872
0.04437
0.088338
0.58387
0.813225
2.221848
4.61169
7.043666
0.345177
1.140026
2.53247
3.583066
0.002238
0.00506
0.02545
0.0095
0.706919
2.221848
4.61169
7.043666
5.45516
81,681,725
67,985,149
23,248,482
Table 3
Nonlinear constants of Ogden and polynomial models.
Filler %
5%
10%
20%
30%
Ogden (3rd order) model
Polynomial model
m1
a1
m2
a2
m3
a3
C10
C01
0.171536
0.794482
0.623713
0.170897
2.358751
1.919898
3.118536
4.701839
0.171536
0.794482
20.41128
21.32771
2.358751
1.919898
0.251752
0.390028
0.171536
0.794482
20.52394
21.36175
2.358751
1.919898
0.249966
0.404207
0.554358
1.074612
1.692903
2.83703
0.3798
0.069335
1.370455
1.390453
b) When the load is applied over the top, the deformation of
the soft finger is uniform through out the length, and hence,
2D FE model was rendered and analysis was done
accordingly.
c) Due to the symmetry of the problem, only quarter of the
cylindrical finger was modelled and plane strain condition
was applied for the analysis.
d) The rigid plate was considered to be made of mild steel with
modulus of elasticity E = 2.05 105 N/mm2 and poison’s
ratio t = 0.3.
Due to the symmetry of the geometry, only a quarter of the finger was considered in the current nonlinear contact analysis. The
FE model was developed in ANSYS 18.0 with the following
considerations;
a) The solid cylinder of diameter 17.8 mm, made of NR/20%
MWCNT material was considered for the current analysis.
The material is a homogeneous, incompressible, nonlinear
hyper elastic, isotropic material. Hence, the nonlinear material model, Ogden constants were used as a material model.
6
Materials Today: Proceedings xxx (xxxx) xxx
E. Natarajan, C.S. Hassan, A. Chun Kit et al.
has plasticity, hyper elasticity, stress stiffening, large deflection,
and large strain capabilities. Fig. 8 shows the mesh model of soft
cylindrical finger with 2155 nodes. The nonlinear material model
with Ogden constants were selected and m1 = 0.623713,
a1 = 3.118536, m2 = 20.41128, a2 = 0.251752, m3 = 20.52394,
a3 = 0.249966 were input to the model.
The mixed formulation was also applied in the simulation. The
soft-to-hard contact pair was created between the cylinder and the
plate. The constrains in x-direction were appropriately imposed
along the axis of symmetry. The coupled elements on the top of
the soft cylinder are expected to have uniform pressure. All directions of the rigid target surface were fixed. The load of 100 N was
applied over the soft cylinder which is normal to the rigid target
surface. The FE model was simulated after imposing appropriate
boundary conditions stated above. The simulation was computed
in 2 s in the Intel(R) Core(TM) I7-7500U CPU @ 2.70 GHz
2.90 GHz with 16 GB RAM. The deformation of the soft finger
was measured at the reference node associated with the top
surface.
Fig. 9 shows the deformation and Von-Mises stress of the cylinder at 100 N of normal load. The maximum deformation noticed on
the finger is 3.591 mm. The maximum Von-Mises stress noticed on
the cylinder is 10.6633 N/mm2. The ultimate tensile strength of the
NR/20% MWCNT is 14.735 MPa [9]. Since the nominal stress (r0) is
less than the strength (Su) of the material, the design is regarded to
be safe.
Fig. 7. Contact model of a soft finger pressed against the rigid object [26].
5. Conclusions
The aim of this paper is to provide the nonlinear material constants of NR/MWCNT nano composites comprehensively to the
readers and designers, which they can use in the component
design and analysis. MWCNTs, filler weight fraction of 5%, 10%,
20% and 30% were mixed with RMA 1 NR matrix in two roll mill
thoroughly. The masticated NR/MWCNT composites were compression moulded from which dumb bell specimens were then prepared. Uniaxial tensile tests were conducted as per ASTM ASTM
D412-06 a standard and uniaxial stress–strain data of each sample
was recorded. The experimental data were then applied into
Mooney-Rivlin, Ogden, Neo-Hookean, Yeoh, Polynomial and
Aruda-Boyce models and the respective nonlinear material constants were evaluated and tabulated for the component design
and analysis. A case study-FE analysis was conducted on a cylindrical soft finger when it is pressed against a rigid mild steel plate. The
simulation results show that the induced nominal stress from
100 N is within the strength of the material, which regarded the
safe component design.
Fig. 8. Mesh model – NR/20% MWCNT cylinder pressed against a mild steel rigid
plate.
e) The non-linear with large deformation condition was
applied, as the material of the soft finger is a hyperelastic
material.
The accuracy of any FE analysis depends on the mesh elements
and material model used in the analysis. Hence, they were chosen
very carefully in order to get the converged results from the analysis. The non-linear hyper elastic quadrilateral elements were used
to perform FE meshing. The four noded 2D Plane elements chosen
Fig. 9. (a) Deformation plot (b) Von-Mises stress plot.
7
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Materials Today: Proceedings xxx (xxxx) xxx
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CRediT authorship contribution statement
Elango Natarajan: Conceptualization, Methodology, Writing review & editing. C.S. Hassan: Software, Data curation. Ang Chun
Kit: Writing - original draft. M.S. Santhosh: Visualization, Investigation. S. Ramesh: Supervision. R. Sasikumar: Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
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