chapter 3 analysis of flattening approach...

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CHAPTER 3

ANALYSIS OF FLATTENING APPROACH MODEL

In this chapter, the rigid flat and a deformable spherical ball model

(flattening model) is considered for the study through contact mechanics

approach. This study has been focused on the material characteristic and

comparison of various existing contact models for analysing the contact

characteristics.

3.1 FINITE ELEMENT MODELING USING ABAQUS

3.1.1 Introduction

The finite element method is employed to study the elastic-plastic

deformation of solid homogeneous materials. Finite Element Analysis (FEA)

software package 'ABAQUS' version- 6.9 is used in the simulation process.

The brief overview of 'ABAQUS' software and analysis procedure is

presented in this chapter. The main objective of this analysis is to study the

behaviour of the model in an elastic and elastic-plastic region.

3.1.2 About 'ABAQUS' Software

ABAQUS software is a powerful tool for engineering simulation

programme based on the finite element method. The linear analysis module is

used to analyse the most complex non-linear problems.

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In 'ABAQUS' software there are two main analysis modules such

as ABAQUS - standard/explicit. It consist of three stages: (i) pre-processing

(ii) simulation and (iii) post processing.

An 'ABAQUS' working interference is called as ABAQUS - CAE.

This includes all the options for generating models, to submit the job for

analysis and review the results. ABAQUS - CAE is used for present analysis

to pre-processor of different stages of the model creation starting from

creation of parts, material properties, assembly, steps, interaction properties,

load, mesh, contact interactions, job creation and submission from the

respective module and post processor to execute the results using

visualization module.

3.1.3 Nonlinear analysis in ABAQUS

Nonlinear analysis is carried out for two purposes, one is for

geometry nonlinearity and the other one is for material nonlinearity. The

Newton-Raphson method is used for nonlinear analysis in the ABAQUS

software. In the nonlinear analysis problem ABAQUS split the analysis into a

number of load increments and resolve the approximate equilibrium

configuration at the end of each load increments and several iterations takes

place to find the solution for the given load conditions.

3.1.3.1 Geometry nonlinearity

Geometry nonlinearity is described by the magnitude of the

displacement affects the response of the structure. In an indentation process

the rigid indenter penetrated into the base material (deformed body) the large

deformation occurs below and around the indenter.

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3.1.3.2 Material nonlinearity

The linear relationship at low strain values is given by Hook's law.

The stress is directly proportional to strain which is true for most of the

metals. The geometry nonlinearity is described, for higher strain value the

material yields at a particular point where the relationship becomes nonlinear

and irreversible.

3.1.4 Material Characteristics and Unit System of the Models

The material characteristic of model is very important in all type of

engineering simulation tools. The model behaviour is depending on the

material properties. In ABAQUS software package the material properties are

defined in such a way that mechanical properties, tangential behaviour, elastic

properties, plastic properties, mass properties etc. The two models are

considered for this study as described below.

3.1.4.1 Linear elastic model

The deformation is very small, the elastic deformation is observed

in all the materials. In an isotropic linear elastic model, the deformation is

proportional to the applied load. For an uniaxial tension state the stress-strain

relationship can be expressed as

E = / (3.1)

where E is Young's modulus, is uni-axial stress and is uni-axial strain.

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3.1.4.2 Power law work hardening model

For an elasto-plastic model the idealised single power law function

is in the form of

= K n (3.2)

where is the true stress, is logarithmic strain, n is strain hardening

exponent and K is the strength coefficient.

The external force applied on it will undergo plastic deformation

beyond elastic limit. Most of the engineering materials such as metallic and

alloys are obeying the power work hardening model approximately which is a

material constitutive relation. The modified uniaxial stress-strain curve of a

stress free material can be expressed as

= E for YE

(3.3)

= K n for YE

(3.4)

where Y = Yield strength, E = Young's modulus and K =nEY

Yis the work

hardening rate. In the equation (3.4) if the value n is zero, then the equation

reduces to an elastic-perfectly plastic material. The elasto-plastic properties of

a power law material is completely characterized by four independent

parameters, i.e., Young's modulus E, work hardening exponent n, yield

strength Y and Poisson's ratio .

In ABAQUS the plastic properties are defined from the power law

hardening material model. The true stress- strain data are prepared from the

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equations (3.3) and (3.4). The plastic strain ( p) is calculated by using the

following equation

pYE

(3.5)

3.1.5 Unit System in ABAQUS

In ABAQUS software package does not indicate any unit system.

The user could enter the input values in a consistent unit system throughout

the problem.

3.1.6 Method of Applying Load

In ABAQUS there are two methods for simulating the Sphere and a

flat contact model. They are (i) load control and (ii) displacement control. The

load control method is used for analysing the flattening approach model. The

displacement control method is used for analysing an indentation approach

model.

3.1.6.1 Load control method

In this method a concentrated force is applied to the top surface of

the sphere. The total load is applied in an incremental steps as the load

applied is large to avoid the non-linearity in the finite element analysis. The

entire load increments are divided such that the total time for the step 1.

Hence the load-displacement data is obtained.

3.1.6.2 Displacement control method

In this method the displacement is specified as input, which is equal

to the penetrated depth of the sphere into a deformable flat. For the applied

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displacement the reaction force is calculated which is equal to the applied

force over the contact zone along the penetrated direction. Hence the force

displacement data is obtained.

3.1.7 Contact Interaction for Contact Approach Problems

In ABAQUS the account of the contact interaction is very

important to formulate the contact. The two components are available in the

interaction between contacting surfaces: one normal to the contact surface and

another tangential to the contact surface. The normal component referred as

contact pressure and the tangential component referred as relative motion of

the surfaces involving friction. From the reference of ABAQUS user manual

the rigid surface is defined as a 'master' surface and the deformable body

contact surface is defined as a 'slave' surface.

3.2 FINITE ELEMENT ANALYSIS FOR FLATTENING

APPROACH CONTACT MODEL

Finite element contact model is created for flattening approach

using ABAQUS is based on the sphere and a flat contact method. In this

model the assumptions has been made for sphere and flat (specimen) such as

the sphere is a deformable member and the flat is a rigid member. The factors

such as material data and contact constraint are regularized in the ABAQUS -

Explicit compared with ABAQUS - standard. It is concluded that an explicit

method is suitable for analysing such type of contact problems in ABAQUS.

The analysis is carried out in ABAQUS - explicit for an elastic-plastic model

with different E/Y values to optimizing the analysis. The sphere size of radius

31.5 mm and the flat size is 200 mm length and 20 mm thickness is

considered for finite element analysis.

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3.2.1 Analysis of Elastic-Plastic Model Using ABAQUS - Explicit

In ABAQUS the contact problems are simulated in ABAQUS -

Explicit mode. The Figure 3.1 shows the basic contact model of sphere and a

flat has generated in the ABAQUS. The axisymmetric model is developed due

to the advantage in the analysis procedure. The quarter sphere and half of the

plate is considered for the analysis based on the axisymmetric property of

model. In this model the sphere is a deformable member and a flat is a rigid

member.

Figure 3.1 Basic contact model of sphere and a flat

3.2.2 Mesh Generation

The edges of the quarter sphere are meshed by biased seed edges

method. The fine area of the mesh near the tip of the hemisphere is varied in

order to encompass the region of the higher stress near the contact as shown

in Figure 3.2. The total number of element and nodes generated in the sphere

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is 5779 and 5920 respectively (Table A3.1). The element type of CAX4R type

was used for all the simulations in which the letter or number indicates the

type of element which is of Continuum type, Axisymmetric in nature has 4-

nodes bilinear and Reduced integration with hour glass respectively.

Figure 3.2 Mesh generation - sphere

3.2.3 Boundary Conditions and Loading

As shown in Figure 3.3, the rigid flat is completely restricted in all

the directions at the reference point of the flat (U1 = U2 = U3 = UR1 = UR2 =

UR3 = 0). A radial constraint is applied to the symmetric axis. The load

control method is used for simulation of an elastic model. The pressure load

2000 N/mm2 is uniformly distributed at the top surface of the sphere.

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Figure 3.3 Boundary conditions

3.2.4 FE analysis of flattening model with different material

properties

An analysis has been carried out for elastic-plastic flattening

approach contact model with different materials in terms of Young's modulus

to yield strength ratio (E/Y values). Considering these aspects, the analysis is

performed to study the behaviours of the single asperity contact model with

different material properties and development of plastic region in the

deformed sphere. The contact problem and elastic-plastic material property

made an analysis as a nonlinear and difficulty to converge the solution in

ABAQUS/Explicit. To overcome the problem of converging the solution the

load control method is applied for loading the sphere with loading step time

as one. The sphere is loaded by a uniform distributed pressure of 2000 N/mm2

at the top surface of the quarter sphere. The different E/Y values of 100, 200,

250 and 300 are considered. The material properties are considered for

analysis E = 200GPa and = 0.32.

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3.2.4.1 Analysis of material E/Y = 100

The analysis is carried out for material E/Y = 100 with yield

strength value of this ratio. The various parameters such as equivalent plastic

strain, reaction force in rigid flat, displacement of nodes, contact pressure and

displacement of node number 62 (maximum displaced node) are observed in

the analysis for study the behaviour of elastic-plastic contact model. The

following are the results obtained from the simulation.

Figure 3.4 Equivalent plastic strain plot of material E/Y = 100

Figure 3.4 shows the scalar plastic strain developed in the model.

PEEQ is an integrated measure of plastic strain. The plot shows the deformed

and undeformed shape of the loaded sphere. For the proportional yield

strength of E/Y = 100 material the maximum plastic strain is approximately

20%. The maximum plastic strain is developed in the sphere near the contact

region at the edge of the contact. The shape of the deformed sphere shows

that the buckling has occurred at the bottom of the sphere. Figure 3.5 shows

the reaction force (RF2) developed in the model normal to the contact surface.

In the flattening approach problem the rigid body is a flat and the deformed

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body is a sphere. The reaction force is developed in the rigid flat to support

the loaded sphere.

Figure 3.5 Reaction force plot of material E/Y = 100

Figure 3.6 shows that displacement of nodes in the loaded sphere.

The minimum displacement of the nodes at near the axis of symmetric and

the maximum at the top right edge of the sphere. This maximum displacement

occurred away from the contact region.

Figure 3.7 shows the contact pressure at the surface nodes of the

deformed sphere. The maximum and minimum contact pressure are lying in

between the flat and sphere and at the surface nodes in the right edge of the

sphere respectively.

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Figure 3.6 Displacement of nodes plot for material E/Y = 100

Figure 3.7 Contact pressure plot of material E/Y = 100

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Figure 3.8 Displacement of node 62 simulation output plot of materialE/Y = 100

Figure 3.8 shows the simulation output plot of displacement of

node 62 for material E/Y = 100. This plot gives the relationship between the

displacement and applied load. The horizontal axis indicates the percentage of

load applied to the model. This plot is an evidence for the sphere which is

fully loaded within the stipulated time. When the time increases the sphere is

gradually loaded upto the maximum value and also the particular node is

gradually displaced from its initial position to the maximum displacement of

5.705 mm.

3.2.4.2 Analysis of Material E/Y = 200


strength value of this ratio. The various parameters such as equivalent plastic

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strain, reaction force in rigid flat, displacement of nodes, contact pressure and

displacement of node number 62 are observed in the analysis for study the

behaviour of elastic-plastic contact model. The following are the results

obtained from the simulation.



The plot shows the deformed shape of the loaded sphere. For the proportional

yield strength of E/Y = 200 material the maximum plastic strain is

approximately 51%. The maximum plastic strain is developed in the sphere

near the contact region at the edge of the contact and the top of the right edge

of the sphere.

Figures 3.10 to 3.12 show the simulation outputs of material E/Y =200 for different parameters reaction force, displacement at nodes and contactpressure at surface nodes respectively. It is observed that, the E/Y valueincreases the contact area between the sphere and a flat is also increased. Thedeformation is maximum at the edge of the contact and at the free end of thesphere.

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Figure 3.13 Displacement of node number 62 simulation output plot ofmaterial E/Y = 200



displacement and applied load. The horizontal axis (time) indicates the

increment of total load applied to the model. This plot is an evidence for the

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sphere fully loaded within the stipulated time. When the time increases the

sphere is gradually loaded upto the maximum value and also the particular

node is gradually displaced from its initial position to the maximum

displacement of 5.141 mm.

For material E/Y = 200, the proportional yield strength value the

plastic strain also occurred at the top right edge of the loaded sphere. It shows

that the plastic strain has been developed outside the contact region if the

sphere is loaded uniformly. It is observed that if the E/Y value increases the

behaviour of the material is distorted within the contact region and also

outside of the contact. So it is required to study the behaviour of material by

increasing the E/Y ratio beyond the value 200 and hence an attempt has been

made.



strength value of this ratio. Similarly the various parameters such as

equivalent plastic strain, reaction force in rigid flat, displacement of nodes,

contact pressure and displacement of node number 62 are observed in the

analysis for study the behaviour of elastic-plastic contact model. The


Figure 3.14 shows the scalar plastic strain developed in the model.The plot shows the deformed and undeformed shape of the loaded sphere. Forthe proportional yield strength of E/Y = 250 material the maximum plasticstrain is developed in the top of the right edge of the sphere is shown in a boxprovided in the plot. The minimum plastic strain is within the contact region.The material is trying to tear out in this edge.

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From Figure 3.15 to 3.17 shows the simulation outputs of material

E/Y = 250 for different parameter such as reaction force, displacement at

nodes and contact pressure at surface nodes respectively. It is observed that

for further increases in the E/Y value the maximum deformation is slowly

move to the free end i.e., away from the contact region and the material trying

to tear at this end.

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displacement and percentage of load applied. This plot is an evidence for the

sphere fully loaded within the stipulated time. When the time increases the

sphere is gradually loaded upto the maximum value and also the particular

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node is gradually displaced from its initial position to the maximum

displacement of 5.272 mm.

Figure 3.18 Displacement of node number 62 simulation output plot of

material E/Y = 250

For material E/Y = 250, the proportional yield strength value

the plastic strain occurred at the top right edge of the loaded sphere. It shows

that the plastic strain has developed outside the contact region if the sphere is

loaded uniformly. It is observed that if the E/Y value increases the behaviour

of the material is distorted outside of the contact region. It shows a different

behaviour of material and trying to tear out in the edge. So an attempt has

been made to study the behaviour of material by increasing the E/Y ratio

beyond the value 250.

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The analysis is carried out for material E/Y = 300 with the

proportional yield strength value. Similarly the various parameters such as

equivalent plastic strain, reaction force in rigid flat, displacement of nodes,

contact pressure and displacement of node number 62 are observed in the

analysis for study the behaviour of elastic-plastic contact model. The




The plot shows the deformed shape of the loaded sphere. For the proportional

yield strength of E/Y = 300 the maximum plastic strain is developed at the top

of the right edge of the sphere. The minimum plastic strain within the contact

region. The material is trying to tear out in this edge.

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From Figure 3.20 and 3.21 shows the simulation outputs of material

E/Y = 300 for different parameter such as reaction force and displacement at

nodes respectively. It is observed that, the contact area between the deformed

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sphere and the rigid flat is decreased and the node displacement is minimum

in the sphere at the contact region. The sphere is completely tear out at the

edge.

Figure 3.22 Displacement of node number 62 simulation output plot ofmaterial E/Y = 300



displacement and percentage of load applied. This plot is an evidence for the

fact of sphere not completely loaded within the stipulated time. When the

time increases the sphere is gradually loaded upto 82.5% of applied load and

also the particular node is gradually displaced from its initial position to the

maximum displacement of 3.716 mm.

The simulation is turn out at 82.5% of applied load. So that the

reaction force is zero and contact pressure cannot be simulated. For the same

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load and simulation procedure this material E/Y = 300 shows different

behaviour in the elastic-plastic model than the other materials E/Y value of

100, 200 and 250.

3.3 COMPARISON OF VARIOUS PARAMETERS OF

DIFFERENT E/Y VALUES

The comparison has been made for different parameters are

simulated from ABAQUS/Explicit.

Table 3.1 Comparison of various parameters of different E/Y values

S.No. Parameters Units

Young's modulus to Yield strength ratio (E/Y)values

100 200 250 300

1 Equivalent plasticstrain (PEEQ) - 2.055 5.175 1.10 × 102 8.466 × 101

2 Reaction Force(RF2)

N 5.7 × 106 5.039 × 106 4.155 × 106 0

3 Spatial nodesdisplacement (U2)

mm 7.222 9.634 9.806 5.661

4 Contact pressure(CPRESS) N/mm2 4.877 × 103 4.913 × 103 7.923 × 103 -

5 Displacement ofnode number 62 mm 5.7 5.141 5.272 3.716

6 Percentage ofload applied - 100 100 100 82.5

From the Table 3.1, it is clearly shown that the material E/Y = 300

shows different results and the percentage of load applied for simulation is

82.5. It shows that this material is not completely loaded even though its turn

out from the simulation. This range of material is independent of material

properties in the elastic-plastic range of analysis. The yield strength of the

material decreases the steep increases in the PEEQ and CPRESS values for

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the materials E/Y = 250 and 300. The material slowly get tear out at an edge

of the sphere for E/Y = 250 and completely tear out for E/Y = 300.

From the Table 3.1, it can be seen that the equivalent plastic strain

is increased upto the E/Y = 250. The further increases in the E/Y ratio the

scalar plastic strain decreases.

From the Table 3.1, it is clearly shown that the E/Y ratio increases

the reaction force (RF2) normal to the contact surfaces developed in the rigid

flat is decreased due to the increases of equivalent plastic strain upto

E/Y = 250. After that the reaction force is zero for material E/Y = 300 due to

the failure of sphere.

From the Table 3.1, it can be seen that the E/Y ratio increases the

spatial nodes displacement (U2) in the deformed sphere is increased due to the

increases of equivalent plastic strain if the value is upto E/Y = 250. After that

the displacement is decreasing for material E/Y = 300 due to an incomplete

loading of sphere.

From the Table 3.1, it is clearly shown that the E/Y ratio increases

the contact pressure between the deformed sphere and rigid flat is also

increased due to an increases in the spatial nodes displacement upto E/Y =

250. After that it is zero for material E/Y = 300 due to the failure of sphere.

From Table 3.1, it can been seen that the E/Y ratio increases the

displacement of node 62. It has been slightly decreased and again increased

for further increases in the E/Y ratio. But for material E/Y=300 the

displacement is very much decreased due to an incomplete loading of sphere.

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From the Table 3.1, clearly shows that the E/Y ratio increases the

percentage of load applied for simulation is constant upto E/Y = 250. But for

material E/Y = 300 it is decreased due to an incomplete loading of sphere. For

material E/Y = 300, the loading is done 82.5% only. This is due to the

material behaviour in the elastic-plastic analysis and the simulation is turn out

from this rage of material characteristics in the elastic-plastic analysis.

From Equation (2.16) the critical interference is calculated as , c =

0.32 mm for the E/Y value of the material (Steel) = 250, E = 200 x103

N/mm2, = 0.32, interference = 9.806 mm. Therefore the dimensionless

interference ratio value, c = 30.643. This dimensionless interference ratio

value is lying between the initial surface yield and initial fully plastic region

(Table A3.2). It is concluded that the simulation has been carried out within

the elastic-plastic region.

3.4 CHAPTER SUMMARY

The analysis of a deformable sphere and a rigid flat (flattening

model) has been made in ABAQUS software. The different elastic-plastic

materials are considered for analysis based on E/Y ratio. The sphere has

loaded by a uniform pressure load at the top surface of the quarter sphere due

to the advantage of the axisymmetric of the model. The simulation results

shows that the maximum plastic strain occurs in the deformed sphere at the

near of the contact region for lower E/Y value of material and move away

from the contact region to the free end of the sphere if the E/Y value

increases. The simulation is turn out for the material E/Y = 300. It shows that

for an elastic-plastic analysis is not material dependent for the material E/Y =

300. The critical value of E/Y has been identified as E/Y = 250. If the E/Y

value is less than 250, it shows the different behaviour compared with E/Y =

300 in both simulation and analytical measurement.

chapter 3 analysis of flattening approach...

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