106
CHAPTER 4
FINITE ELEMENT ANALYSIS
The finite element method is the one of the most important
developments in numerical analysis. There are many numerical methods for
solving engineering problems, but FEA is the versatile and comprehensive
method for solving complex design problems. During the last three decades,
rubber and rubber-like materials have been simulated by numerical methods,
especially by using Finite Element methods. FEA permits the analysis of
complex structures without the necessity of developing and applying complex
equations.
The Finite Element Analysis (FEA) for an elastomeric component,
with commercial finite element programs such as MARC, was performed in
the late 1970s (Finney & Gupta 1980). The rubber materials are usually
modelled as incompressible hyperelastic materials. At high deformations, the
stress-strain relationship for these materials is nonlinear, and is affected by
dynamic and thermal effects. The verification of the linear analysis is usually
easier than the verification of a non-linear analysis, due to the limited
availability of non-linear analytical solutions.
4.1 NONLINEAR CHARACTERISTICS
Rubbers are capable of recovering substantially in size and shape
after the removal of a load. Large displacements occur or the material behaves
nonlinear, during loading/unloading.
107
There are three major types of non-linearity,
i. Geometric nonlinearity - due to large deformations or snap-
through buckling.
ii. Material non-linearity - due to large strains, plasticity, creep,
or viscoelasticity.
iii. Boundary non-linearity - such as the opening/closing of gaps,
contact surfaces, and follower forces.
The specification of the nonlinear material properties of elastomers
is difficult. Several constitutive theories for large elastic deformations based
on strain energy density functions have been developed for hyperelastic
materials (Gent 2001). These theories, coupled with the FEA, can be used
effectively by design engineers to analyze and design elastomeric products
operating under highly deformed states. The commercial FEA software has its
own ability to take up nonlinear materials for providing more accurate
solutions for the problems.
4.2 MATERIAL MODEL
Rubber materials are characterized by a relatively low elastic
modulus and high bulk modulus. These materials are commonly subjected to
large strains and deformations, and termed as hyperelastic materials. The
hyperelastic materials are capable of experiencing large strains and
deformations. A material is said to be hyperelastic if there exists an elastic
potential W(strain energy density function), that is the scalar function of one
of the strains or deformation tensors, whose derivative with respect to a strain
component determines the corresponding stress component.
108
In nonlinear elasticity, there exist many constitutive models
describing the hyperelastic behavior of rubber like materials (Govindjee &
Simo 1991, Holzapfel & Simo 1996, Simo 1998) and these models are
available in many modern commercial finite element codes. Several
hyperelastic material models are formed from the stress-strain relationship;
e.g., Mooney-Rivlin model, Ogden model, and so on. All the material models
were established from the simple deformation tests, which are necessary for
forming the stain energy density function. It is necessary to have an accurate
knowledge of the material behavior to find out its global characteristics under
distinguished application.
4.2.1 Types of Material Model
The Material models predict large scale material deflection and
deformations. For Incompressible rubber materials, the Mooney-Rivlin,
Arruda-Boyce, and Ogden material models hold well in the analysis. For
Compressible materials, the Blatz-Ko and Hyperfoam models are preferred.
The significance of the material models at different strain rates is given
below.
1. The Mooney-Rivlin model works with incompressible
elastomers with a strain of upto 200%.
2. The Arruda-Boyce model is well suited for rubbers such as
silicon and neoprene, with a strain of upto 300%. This model
provides a good curve fitting, even when the test data are
limited.
3. The Ogden model works for any incompressible material with a
strain of up to 700%. This model gives a better curve fitting
when data from multiple tests are available.
109
4. The Blatz-Ko model works specifically for compressible
polyurethane foam rubbers.
5. The Hyperfoam model can simulate any highly compressible
material, such as a cushion, or a sponge.
4.3 CONSTITUTIVE MODEL FOR RUBBER
A constitutive material law is said to be hyperelastic, if it is defined
by a strain energy function. A hyperelastic material is still an elastic material,
which means, that it returns to its original shape after the forces have been
removed. Hyperelastic material is also called as Cauchy-elastic, which means
that the stress is determined by the current state of deformation, and not the
path or history of Deformation.
Figure 4.1 Stress strain curve
Rubber typically undergoes large strains at small loads (low
modulus of elasticity). The stress strain curve of rubber has been presented in
Figure 4.1, on loading and unloading, indicating the stress value at 300%
strain. As the material is nearly incompressible, the Poisson’s ratio is very
close to 0.5, their loading and unloading stress-strain curve is not the same,
110
depending on different influential factors (time, static or dynamic loading,
frequency). Generally, the mechanical behavior of rubber materials is
modeled as hyperelastic. In the present work, the Mooney Rivlin model was
used, in view of the application to analyze the problems involving large
displacement and large deformations.
4.3.1 Mooney Rivlin Model
Material modeling is one of the important parts in the FEA
procedure. The rubber blocks are commonly modeled, using solid elements
with a specific isotropic hyperelastic material model. Even though many
theoretical models were developed to characterize the mechanical behavior of
rubber, one of the most important among them is the Mooney Rivlin model.
This model is extensively used for the stress analysis of rubber components,
and is incorporated in most commercial FEA programs. The Mooney Rivlin
model with two material constants C10 and C01, was considered in the present
analysis.
Rivlin and Sunders developed a hyperelastic material model for
large deformations of rubber (Gent 2001). This material model is assumed to
be incompressible and initially isotropic. The strain energy potential W for a
Mooney-Rivlin material is given as,
W= C10 (I1-3) + C01 (I2-3) + (J-1)2 (5.1)
where C10, C01 and d are material constants, I1, I2 are the invariants of the
elastic strain d=2/K, K is the bulk modulus, and J is the ratio between the
deformed and un-deformed volume.
For hyperelastic materials, simple deformation tests can be used to
determine the Mooney-Rivlin hyperelastic material. This model is extensively
111
used for the stress analysis of rubber components, and is used in the present
study. When applying the finite element analysis for designing rubber
products, the material constants are required as input data. To obtain
sufficiently accurate material constants, combined tests and biaxial tests are
recommended (Roongrote Wangkiet et al 2008). To predict the rubber
behavior based on the Mooney-Rivlin model, the values of C01 and C10 must
be determined. FEA programs can be used to approximate these constants
from the experimental data. The deformation tests are usually carried out in
the laboratory for determining the material behavior. The Mooney-Rivlin
constants obtained from the experimental data were used in the analysis to
determine the deformation behavior of rubber blocks under uniaxial
compression.
4.4 MATERIAL CONSTANTS
It is always recommended to take the data from several modes of
deformation over a wide range of strain values. To obtain the Mooney-Rivlin
coefficients C01 and C10, the deformation tests usually carried out are the
uniaxial tensile test, equal biaxial tensile test, and the volumetric compression
test. To derive the material constants from the test data, the tests were carried
out using a Universal Testing Machine (AGS-2000G, Shimadzu). Different
proportions of CB filled NR/BR blends with other ingredients were mixed in
two roll mills, and the test samples were prepared and tested as per the ASTM
standard.
The different modes of testing were done in the above mentioned
AGS-2000G machine, and the experimental data was extracted for curve
fitting. The obtained data was given as input to the FE software, to extract the
Mooney-Rivlin material constants C01 and C10. Table 4.1 presents the
Mooney-Rivlin material constants for five proportions of CB filled
vulcanizates, tested under various deformation modes in the laboratory and
112
extracted from FE software. These values were used in the FE analysis to
determine the deformation characters of the uniaxially compressed rubber
blocks.
Table 4.1 Mooney-Rivlin material constants and other material properties
Sample No.
C01
N/mm2
C10
N/mm2
Density kg/m3
Youngsmodulus E
N/mm2
1 0.02527 0.4943 930 1.898
2 0.02085 0.5331 1010 2.050
3 0.05419 0.8503 1054 2.870
4 0.05049 1.0517 1089 3.620
5 0.09803 1.5936 1134 4.450
4.5 3D FE MODEL
3D solid models of rubber blocks with different aspect ratios were
created and analyzed using ANSYS package to simulate the compression
deformation and to study the stress-strain characteristics. 3D (3Dimensional)
models of cylindrical geometry with different aspect ratios were modeled with
platens, for analyzing the deformation behavior of bonded samples, and have
been compared with the experimental results. Further, an analysis of the
rubber blocks similar to tyre tread blocks has been carried out as their
performance provides a better idea for the optimization of the tyre design. As
the analysis of the rubber block having different shapes similar to tyre tread
blocks has high scope in tyre design, its analysis is made necessary. The
influence of the groove void ratio in the tyre tread design can be optimized,
by analyzing the performance of a single tread block under compression and
113
shear loading. The 3D model of rubber blocks of different shapes similar to
tyre tread blocks was modeled and analyzed in the MARC software.
4.6 ELEMENT CHARACTERISTICS
Solid 185, CONTA174 and TARGET 170 elements are used for
meshing the 3D solid model of the rubber blocks, and to define the contact
conditions. Solid 185 is a structural solid used for modeling 3D solid
structures. It is defined by 8 nodes having three degree of freedom at each
node. These elements have plasticity, hyperelasticity, stress stiffening, creep,
and large deflection capabilities. It also has a mixed formulation capability,
for simulating deformations of nearly incompressible elastoplastic materials
and fully incompressible hyperelastic materials. CONTA174 is a 3D, 8-node,
higher order quadrilateral element used for defining the contact on a 3D solid.
TARGET 170 is used to represent various 3D target surfaces for the
associated contact element CONTA174. The contact elements themselves
overlay the solid elements describing the boundary of the deformable body,
and they are potentially in contact with the target surface.
4.7 FEA OF RUBBER BLOCKS
The Contact problems are highly nonlinear, and require significant
computer resources to solve them. Surface-to-surface contact elements can be
used to model rigid-flexible between surfaces in the present analysis.
CONTA174 is a 3D, 8-node, higher order quadrilateral element, used for
defining the contact on a 3-D solid. Since the contact surface is attached to
areas or volumes that are meshed with solid elements, ANSYS automatically
determines the outward normal needed for contact calculations. Figure 4.2
illustrates the FEA procedure adopted to analyze the deformation behavior of
rubber blocks.
114
Figure 4.2 FEA procedure
4.7.1 Modeling and Meshing of Rubber Blocks
The 3D geometry of cylindrical models with different aspect
(a/h 0.5 to 1) ratios was modeled along with platens. Bonded block models
were also modeled with specified dimensions to analyze the deformation
behavior. Figure 4.3 (a) and (b) represents the 3D models of non-bonded and
bonded rubber blocks. The solid 185hex element was used to mesh the rubber
block and the platens. The contacts have been defined between the rubber
block and platens. TARGET 170 element was used to represent the 3D target
Create model geometry
Define target and contact surface
Mesh the model
Material model and assignment of material properties
Apply necessary Boundary conditions and loads
Analysis and result
Element selection
115
surfaces for the associated contact element CONTA174. The Mooney Rivlin
constants were assigned to the hyperelastic rubber material and compression
platen with linear properties of steel. The material property of the platen
material was assigned as E=2.1x105 N/mm2, µ=0.3 and = 7800 kg/m3. The
respective material properties were assigned to the models, and an analysis
was conducted. Figure 4.4 (a) and (b) presents the meshed 3D models of non-
bonded and bonded rubber blocks.
(a)
(b)
Figure 4.3 3D models of cylindrical samples (a) Non-bonded sample (b) Bonded sample
116
(a)
(b) Figure 4.4 Meshed models (a) Non-bonded sample (b) Bonded sample
4.7.2 Boundary Conditions and Loads
In this step, the constraint and loading conditions on the nodes of the 3D solid are imposed. The general Boundary conditions adopted were fixing the cylindrical rubber blocks between the platens, and compress them uniaxially. The top and bottom surfaces are in friction contact with the steel platens and the friction value has been chosen as 1. The bottom platen is fixed, and compressive force was applied over the top surface of the platen. Different compressive loads (165 N to 825 N) were applied on the top platen, similar to the experiment conducted using the imaging tool, and simulation was performed. Different compressive forces were applied over the top platen, and the rubber blocks were strained. A similar analysis was performed for the cylindrical rubber blocks of various aspect ratios with their respective material properties. Thus, the simulation was carried out, using FE software to estimate the linear and lateral dimension variation.
117
4.7.3 FEA Output
Different compressive loads have been applied similar to the
experimental tests, and simulation was done. The large displacement option
has been selected, and the analysis was performed. FE analysis was carried
out with prudence, and the obtained results are discussed in the succeeding
sections in detail.
Figure 4.5 clearly illustrates the un-deformed and deformed
configurations of the non-bonded rubber block under uniaxial compressive
force. ux, uy and uz are the displacements along the principal axes x, y and z
respectively. The vertical displacement uy represents the linear deformation,
whereas ux and uz represent the lateral deformation. Deformation being
uniform in x and z axes, only the displacement values ux along x-axis has
been listed. Table 4.2 presents the FE results for the linear and lateral
deformations of CB filled cylindrical vulcanizates of different aspect ratios.
The deformation values presented in the Table 4.2 have been obtained from
the FE analysis for the material constants C01 =0.05049 and C10=1.0517 for
the CB filled samples.
Figure 4.5 Un-deformed and deformed configuration of non-bonded rubber block
118
Figure 4.6 presents the compressive deformation of the non-bonded
rubber block with its vertical displacement uy for cylindrical samples with the
aspect ratio (a/h) 1. A similar analysis was performed on the bonded rubber
blocks of various shape factors, and the results are presented herein, for
understanding the deformation behavior of NR/BR blended cylindrical
vulcanizates. Figures 4.7 a, b and 4.8 present the vertical, horizontal
displacements uy , ux and Usum of the deformed bonded rubber block. Table 4.3
presents the FE results of the linear and lateral deformations of unfilled
cylindrical vulcanizates with different aspect ratios. The deformation values
mentioned in Table 4.3 have been obtained from the FE analysis for the
Mooney Rivlin material constants C01 =0.02527and C10=0.4943 for the
unfilled samples, with and without bonding plates. In the unfilled samples,
many of the lower aspect ratios (a/h < 0.7) models are not solved, due to the
instability of the material under higher compressive forces. The element
distortion was found to be very high in these cases, and also they were unstable for
all the compressive force values which cannot be solved further.
Figure 4.6 Deformed non-bonded rubber block
119
Table 4.2 FE results of the linear and lateral deformations of CB filled cylindrical vulcanizates with different aspect ratios
Aspect ratio(a/h)
Non-bonded Samples Bonded samples Compressive
load (N)
Verticaldeformation
uy(mm)
Horizontal deformation
ux(mm)
Verticaldeformation
uy(mm)
Horizontal deformation
ux(mm)
165 1.274 0.376 1.378 0.385330 2.114 0.668 2.320 0.686
0.5 495 2.760 0.911 3.061 0.938660 3.867 1.282 3.676 1.160825 4.526 1.482 4.205 1.357165 1.056 0.325 0.919 0.323330 1.841 0.592 1.587 0.589
0.6 495 2.487 0.823 2.128 0.818660 3.040 1.028 2.586 1.022825 3.523 1.207 2.987 1.195165 0.812 0.278 0.764 0.281330 1.447 0.516 1.361 0.521
0.7 495 1.988 0.723 1.869 0.733660 2.461 0.917 2.314 0.925825 2.881 1.101 2.710 1.091165 0.633 0.240 0.580 0.240330 1.143 0.452 1.051 0.452
0.8 495 1.591 0.643 1.458 0.643660 1.991 0.819 1.821 0.817825 2.351 0.983 2.151 0.978165 0.503 0.209 0.442 0.207330 0.914 0.398 0.804 0.393
0.9 495 1.284 0.571 1.128 0.564660 1.618 0.731 1.421 0.723825 1.924 0.883 1.690 0.871165 0.403 0.183 0.338 0.181330 0.733 0.350 0.615 0.345
1 495 1.035 0.505 0.868 0.498660 1.313 0.651 1.100 0.642825 1.570 0.788 1.314 0.777
120
Table 4.3 FE results of the linear and lateral deformations of unfilled cylindrical vulcanizates with different aspect ratios
Aspect ratio(a/h)
Non-bonded Samples Bonded samples Compressive
load (N)
Verticaldeformation
uy(mm)
Horizontal deformation
ux(mm)
Verticaldeformation
uy(mm)
Horizontal deformation
ux(mm) 165 2.119 0.699 2.222 0.710330 - - 3.448 1.193
0.5 495 - - 4.362 1.583660 - - - -825 - - - -165 1.736 0.619 1.589 0.619330 2.819 1.071 2.553 1.068
0.6 495 3.636 1.444 - -660 - - - -825 - - - -165 1.356 0.542 1.304 0.546330 2.278 0.959 2.188 0.965
0.7 495 2.999 1.312 2.878 1.313660 3.647 1.611 3.467 1.623825 - - - -165 1.068 0.475 1.011 0.476330 1.844 0.858 1.745 0.857
0.8 495 2.471 1.187 2.331 1.180660 2.999 1.479 2.827 1.464825 3.449 1.883 3.282 1.738165 0.856 0.418 0.786 0.415330 1.505 0.766 1.381 0.760
0.9 495 2.045 1.07 1.873 1.059660 2.508 1.342 2.294 1.325825 2.916 1.591 2.671 1.572165 0.689 0.368 0.614 0.364330 1.228 0.683 1.091 0.675
1 495 1.686 0.96 1.495 0.949660 2.086 1.212 1.847 1.196825 2.443 1.443 2.160 1.421
121
(a)
(b) Figure 4.7 Bonded rubber block with (a) Vertical deformation uy
Horizontal deformation ux
Figure 4.8 Bonded rubber block total deformation usum
122
4.8 FEA RESULTS AND DISCUSSION
The influence of the aspect ratio on the compressive loading of the
NR/BR blended rubber samples of different geometries was analyzed under
uniaxial compression. The variation in the lateral dimension for the associated
linear deformation was analyzed under uniaxial compressive force, using FEA
software.
Figure 4.9 depicts the maximum bulge radius, Rmax, as a function of
Compressive load for uniaxially Compressed CB unfilled and filled
cylindrical rubber blocks of different aspect ratios. In the CB unfilled rubber
block samples of low aspect ratio, the deformation behavior was nonlinear.
As the rubber distortion is high at larger compressive force, the convergence
of the solution is difficult to achieve. Many of the values in Table 4.3 have
been left blank for a/h 0.5 to 0.7, as the solution has not converged due to
high distortion in the elements. Owing to better dimensional stability and
material property, the CB filled samples for all aspect ratios has been solved,
and the results also converged. As the application of the gum compound is
limited, more significance has been given to analyze the CB filled
vulcanizates, and the results are presented.
The maximum bulge radius values obtained for the applied
compressive load, showed linear variation for all the aspect ratios. For the CB
filled samples, the increase in the maximum bulge radius at the mid plane has
been progressive, and showed steady bulging for all compressive loads and
aspect ratios. The ultimate aim of the FE analysis conducted on the rubber
cylinders of different aspect ratios using ANSYS software is to evaluate the
123
deformed height and the increase in the lateral dimension at the mid plane, to
check the consistency of the developed image tool results.
Figure 4.9 Rmax as a function of Compressive force for CB filled non-bonded cylindrical rubber blocks of different aspect ratios
A similar analysis has been conducted on bonded vulcanizates,
and the deformation characters are presented in Tables 4.2 and 4.3.
Compressive force was applied over bonded rubber blocks of different aspect
ratios, and their deformation characteristics were evaluated. When the bonded
rubber blocks of different aspect ratios were compressed the variation in the
maximum bulge radius has increased progressively, similar to the non-bonded
samples. Since the rubber is softer than the bonding plate, a majority of the
deformation has occurred in the rubber part only. Almost a negligible increase
in the bulge radius has been observed in the bonded and non bonded cylinders
of different aspect ratios, under the same magnitude of compressive force.
The rubber blocks bonded to the rigid steel plates were analyzed for
their behavior, under a constant loading area. Moreover, the deformation was
taken by the hyperelastic rubber blocks for all the compressive loadings, and
124
the deformation of the steel plate was found to be almost zero. The analysis
result showed approximately the same values of increase in the maximum
bulge radius, for the bonded and non bonded samples, with and without the
bonding plate. It was inferred from the FE analysis that the major deformation
has been taken over only by the rubber materials, in both the bonded and non-
bonded samples. The net effect of the applied compressive load on the bonded
samples with a constant loading area was transferred to the rubber part which
made it to deform. But, the obtained maximum bulge radius remains almost
the same in both the bonded and non-boned samples for the same magnitude
of applied compressive loads with negligible variations.
Figure 4.10 depicts the linear strain as a function of compressive
force for the CB filled bonded cylindrical rubber blocks of different aspect
ratios. During the analysis, the lower aspect ratio samples had shown higher
value for linear strains in both the CB filled and unfilled samples. Their strain
behavior was also found to be nonlinear, which makes the analysis of the
shape factor effect on the slender rubber blocks more significant. Thus, an
extensive knowledge in acquiring the deformation characteristics of long
slender rubber blocks is necessary for evaluating the nonlinear behavior in
many applications.
Figure 4.11 depicts the lateral strain as a function of Compressive
load for the CB filled non-bonded cylindrical rubber blocks of different aspect
ratios. The slender cylindrical blocks of lower aspect ratios showed
comparatively uneven lateral dimension variation, under uniaxial compression.
The FEA software output for linear deformation and its variation in lateral
dimensions of the CB filled bonded and non-bonded cylindrical rubber blocks
of different aspect ratios is presented in Figure 4.12. The comparison of the
125
dimensionless linear and lateral strain under applied compressive force for the
same aspect ratios of rubber blocks bonded with steel plates, and directly
under the compression platen, showed approximately the same variation in
the lateral dimension (bulge radius) at the mid plane.
Figure 4.10 Linear Strain as a function of compressive force for the CB filled bonded cylindrical rubber blocks of different aspect ratios
Figure 4.11 Lateral Strain as a function of compressive force for the CB filled non-bonded cylindrical rubber blocks of different aspect ratios
126
Figure 4.12 Comparison of linear and lateral strain for the CB filled bonded and non-bonded samples of different aspect ratios
127
4.8.1 FEA Analysis of Tread Block Geometries
An analysis was performed on the rubber blocks of different
geometries in the MARC software. The 3D model of the rubber blocks of
different geometries, similar to tyre tread blocks was modeled and analyzed in
the MARC software. All the geometries are modeled identical to the tyre
tread block patterns, having square, rectangular and other shapes. The present
studies have their own significance in analyzing the tread block geometries
for optimizing the tread design. The tread block samples of different shapes
analyzed in this study, would include better information on the deformation
behavior under compressive forces. Figure 4.13 (a) and (b) show the rubber
block with load and boundary condition, and their analysis result for vertical
displacements uy.
Similar to the analysis performed on the cylindrical blocks, an
analysis has been carried out for rubber blocks of different geometries. All the
parameters, such as material constants, material model and boundary
conditions are set the same as in the previously conducted cylindrical
sample’s analysis, except the block geometries. The boundary condition and
load step was assigned identical, and analyzed for their deformation behavior.
Figure 4.14 (a) and (b) present the vertical displacement uy of the deformed
square and rectangular geometry, under the normal compressive force. The
deformation of the rubber blocks of different geometry with unique properties
has been analyzed and the variation was inferred in the displacement and
induced stresses. The linear and lateral dimension variations for the square
and rectangle have been distinguished from the cylindrical geometry under
the same magnitude of compressive force. Thus, the analysis conducted on
various tread block geometries would be helpful, for a clear understanding of
128
their deformation behavior under uniaxial compressive force, to optimize the
tyre tread design.
(a)
(b)
Figure 4.13 (a) Rubber block imposed with load and boundary conditions (b) Vertical displacements uy
129
(a)
(b)
Figure 4.14 Vertical displacements uy of different tread block patterns (a) Square pattern (b) Rectangular pattern