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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.

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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.

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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.

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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,

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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

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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

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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

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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.

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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

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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

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(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.

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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

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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

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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

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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

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(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

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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

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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

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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

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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

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Figure 4.12 Comparison of linear and lateral strain for the CB filled bonded and non-bonded samples of different aspect ratios

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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

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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

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(a)

(b)

Figure 4.14 Vertical displacements uy of different tread block patterns (a) Square pattern (b) Rectangular pattern