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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.
65
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
The analysis is carried out for material E/Y = 200 with yield
strength value of this ratio. The various parameters such as equivalent plastic
75
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.
Figure 3.9 Equivalent plastic strain plot of material E/Y = 200
Figure 3.9 shows the scalar plastic strain developed in the model.
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 3.10 Reaction force plot of material E/Y = 200
Figure 3.11 Displacement of nodes plot for material E/Y = 200
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Figure 3.12 Contact pressure plot of material E/Y = 200
Figure 3.13 Displacement of node number 62 simulation output plot ofmaterial E/Y = 200
Figure 3.13 shows the simulation output plot of displacement of
node 62 for material E/Y = 200. This plot gives the relationship between the
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.
3.2.4.3 Analysis of Material E/Y = 250
The analysis is carried out for material E/Y = 250 with yield
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
following are the results obtained from the simulation.
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.
79
Figure 3.14 Equivalent plastic strain plot of material E/Y = 250
Figure 3.15 Reaction force plot of material E/Y = 250
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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Figure 3.16 Displacement of nodes plot for material E/Y = 250
Figure 3.17 Contact pressure plot of material E/Y = 250
Figure 3.18 shows the simulation output plot of displacement of
node 62 for material E/Y = 250. This plot gives the relationship between the
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.
82
3.2.4.4 Analysis of Material E/Y = 300
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
following are the results obtained from the simulation.
Figure 3.19 Equivalent plastic strain plot of material E/Y = 300
Figure 3.19 shows the scalar plastic strain developed in the model.
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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Figure 3.20 Reaction force plot of material E/Y = 300
Figure 3.21 Displacement of nodes plot for material E/Y = 300
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
Figure 3.22 shows the simulation output plot of displacement of
node 62 for material E/Y = 300. This plot gives the relationship between the
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.