chapter 4 analysis of indentation...
TRANSCRIPT
88
CHAPTER 4
ANALYSIS OF INDENTATION MODEL
An indentation model with a rigid spherical ball and a deformable
flat is considered for the analysis through contact mechanics approach. In this
model a rigid sphere is pressed against a deformable plate (flat) by applying a
concentrated load on the center of the sphere. The analysis of spherical
indentation of an elastic-plastic has been carried out using commercially
available software viz., 'ANSYS' and 'ABAQUS' and presented in this
chapter. 'ANSYS' is used for the analysis of indentation model.
The indentation process is a quasi-static process. For an unloading
process the data has to be restarted from the end of indentation process. For
this simulation the ABAQUS software is having the advantage of restarting
the data than ANSYS. Hence ABAQUS software is made use for the
simulation of loading and unloading process in spherical indentation.
4.1 FINITE ELEMENT MODELING USING ANSYS
4.1.1 Introduction
To study the elastic-plastic deformation of solid homogeneous
materials, a sphere (ball) penetrated into a flat as in an indentation process
(Brinell and Rockwell Hardness testing) was considered. It is assumed that
the material of the ball is considered to be harder than the material of the flat.
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Figure 4.1 Rigid sphere model (RS model)
Figure 4.1 shows that the RS-model (indentation approach). In the
Brinell hardness test a hard ball of diameter ‘D’ is penetrated under a load
‘W’ into the plane surface under test. After removal of the load, the chordal
diameter ‘d’ of the resulting indentation is measured, Brinell hardness HB is
defined as the load W divided by the surface area of the spherical cap formed
by the indentation
22
2WHD 1 1 d / D
B (4.1)
The Meyer hardness HM, is determined by ball indentation in
exactly the same way, but it is defined as the ratio between load applied and
the projected area of the indentation
M 24WH
d(4.2)
4.2 FINITE ELEMENT ANALYSIS FOR INDENTATION
APPROACH CONTACT MODEL
Finite element contact model was created for indentation approach
using ANSYS. In this model sphere is considered as a rigid member and the
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flat is assumed as a deformable member (RS model). The present study aims
to study the effect of tangent modulus for single asperity contact for different
materials, under the loading condition of the RS model. The analysis is
carried out for an elastic-plastic model with different E/Y values. The sphere
size of radius 50 mm and the flat size is 100 mm length and 20 mm thickness
is considered for the analysis.
4.2.1 Method of Simulation
There are two methods to simulate the contact problem. In the first
method force is applied on the rigid body and in the second method is
displacement is applied to the rigid body. In this work the first method is used
for analysing the RS model.
4.2.2 FEA Modeling and Contact Pair Creation
The finite element contact model of a rigid sphere against a
deformable flat is shown in Figure 4.2. Here the spherical ball is considered
as a quarter circle to have the advantage of simulation of axisymmetric
problems. For the contact analysis of the Rigid Sphere model (RS-model),
pair is created between sphere and flat and shown in Figure 4.3.
4.2.3 Mesh Generation
In this analysis element type of plane 82, conta172 and target 169
are used for meshing the model. The deformable body is meshed by finer
elements and the rigid body is meshed by coarser elements to minimize the
computational time and effort. The meshed model is shown in Figure 4.4. The
hemisphere is meshed by rough quad shaped free area mesh and each division
of meshing is 2 mm. The flat plate is meshed by fine quad shaped free area
mesh and each divisions of the meshing is 0.5 mm. The resulting final mesh
consists of 4186 elements in total.
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Figure 4.2 FEA model of rigid sphere and a deformable flat
Figure 4.3 Contact pair of RS model
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Figure 4.4 Finite element meshed model
4.2.4 Boundary Conditions
The Figure 4.5 shows the boundary conditions applied on the RS
model for analysis. The displacement of the nodes lying on the symmetric
axis of the hemisphere and the flat are restricted to move in the radial
direction and allowed to move in the vertical direction. Also the nodal
displacement at the bottom of the flat is restricted in all the degrees of
freedom.
4.2.5 Method of Load Applied
The load is applied for simulation by using pressure on lines
command in the ANSYS software. A load of 20000 N is applied on the top
line of the hemisphere. The maximum number of sub steps given for
computation is 10000 and the minimum number of sub steps is 100 for very
large interference to get accurate results.
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Figure 4.5 Boundary conditions
4.2.6 Material Properties
The material properties are selected based on the Young's modulus
to yield strength ratio. The material properties shown in Table 4.1 were used
for the applications of contact problems such as cylinder over a flat plate,
wheel and rail contact, roller bearings and meshing of gear teeth.
Table 4.1 Material Properties
S.No. Material E ×103
N/mm2Y
N/mm2 E/Y
1 C 45 steel 210 380 552.63
2 Aluminium 70 95 736.84
3 Cast iron 100 130 769.23
4 304 Austenitic steel 120 121 991.74
5 C 15 steel 210 190 1105.26
6 Copper 120 69 1739.13
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4.3 SIMULATION OUTPUT - ANSYS
For this analysis different values of the tangent modulus are taken
to get a fair idea about the effect of the same in different materials i.e.,
500 E/Y 1750. The tangent modulus values are accounted between 0.1E and
0.9E. The following are the output results for deformable flat and a rigid
sphere for the tangent modulus value of 0.5E for different E/Y values upto
991.736 (Close to 1000) and 0.6E for material E/Y value above 1000.
Figure 4.6 Plot of Stress for material E/Y = 552.63 (ET = 0.5E)
Figure 4.6 shows the maximum stress developed in the model for
material E/Y = 552.63 at tangent modulus value of 0.5E. This plot shows that
the hemisphere is penetrated into a deformable half flat. In this model the
maximum stress of 60.123N/mm2 is developed and also observed that the
sink-in is occurred in the contact region near the axis of symmetry.
Figure 4.7 shows the maximum stress developed in the model for
material E/Y = 736.84 at tangent modulus value of 0.5E. This plot shows that
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the hemisphere is penetrated into a deformable half flat. In this model the
maximum stress of 22.374 N/mm2 is developed.
Figure 4.7 Plot of Stress for material E/Y = 736.84 (ET = 0.5E)
Figure 4.8 Plot of Stress for material E/Y = 769.23 (ET = 0.5E)
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Figure 4.8 shows the maximum stress induced in the model for
material E/Y = 769.23 of tangent modulus value 0.5E. This plot shows that
the hemisphere is penetrated into a deformable flat. In this model the
maximum stress of 26.067 N/mm2 is developed.
Figure 4.9 Plot of Stress for material E/Y = 991.74 (ET = 0.5E)
Figure 4.9 shows the maximum stress induced in the model formaterial E/Y = 991.74 of tangent modulus value 0.5E. This plot shows that
the hemisphere is penetrated into a deformable flat. In this model themaximum stress of 45.974 N/mm2 is developed. Figure 4.10 shows the
maximum stress induced in the model for material E/Y = 1105.26 of tangentmodulus value 0.6E. The maximum stress developed in the model is 56.812
N/mm2.
Figure 4.11 shows the maximum stress induced in the model for
material E/Y = 1739.13 of tangent modulus value 0.6E. The maximum stress
developed in the model is 23.289 N/mm2.
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Figure 4.10 Plot of Stress for material E/Y = 1105.26 (ET = 0.6E)
Figure 4.11 Plot of Stress for material E/Y = 1739.13 (ET = 0.6E)
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Table 4.2 Stress value for material range 500 < E/Y > 1750 of varioustangent modulus value
Tangentmodulus
(ET)N/mm2
Stress in N/mm2
E/Y552.63
E/Y736.84
E/Y769.23
E/Y991.74
E/Y1105.26
E/Y1739.13
0.1E 43.226 2.99 9.496 6.66 41.054 8.0810.2E 43.066 6.956 9.665 14.596 17.352 7.8150.3E 44.301 7.282 11.712 21.07 21.456 19.8270.4E 52.82 16.563 15.347 22.47 20.921 21.9760.5E 60.132 22.374 26.067 45.974 45.082 20.8910.6E 59.932 6.664 24.102 21.172 56.812 23.2890.7E 22.54 5.96 23.411 8.317 11.842 21.2680.8E 15.679 5.992 20.902 9.319 11.895 8.2640.9E 14.196 5.774 7.392 13.926 19.516 6.649
From the Table 4.2, it is clearly shown that the stress induced in thematerial range of 500 < E/Y > 1750 for various tangent modulus value
accounted for analysis between 0.1E and 0.9E. It is clearly shown that, forthese range of materials the maximum stress is lying between the tangent
modulus value of 0.5E and 0.6E. It shows that when the tangent modulus isincrease, the strain hardening effect in the material increases.
Figure A3.1 shows the relation between the tangent modulus and stress
for three different cast iron graded materials (Sjögren 2007). It is observed
that the magnitude of tangent modulus of material increases with the increase
in the magnitude of stress. The comparison has been made between the
simulation results and experimental results. It is concluded that the material
having the effect of tangent modulus between 0.5E and 0.6E.
4.3.1 Comparison of Simulation Output for Different E/Y Values
The comparison of simulation output has been made for differentmaterial E/Y values between 500 and 1750. In the first case the comparison
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has been done for the material between 500 and 1000 (close to E/Y = 991.74)and second the comparison has been done for different material E/Y valuesbetween 1000 and 1750 (Close to E/Y = 1739.13).
Figure 4.12 shows the relationship between the stress induced and
tangent modulus for various materials. The maximum stress induced in thematerial range of 500 < E/Y > 1000 is laying on the tangent modulus of 0.5E.
0.1E 0.2E 0.3E 0.4E 0.5E 0.6E 0.7E 0.8E 0.9E0
10
20
30
40
50
60
Stre
ss (i
n N
/mm
2 )
Tangent modulus (ET) in N/mm2
E/Y = 552.63 E/Y = 736.84 E/Y = 769.23 E/Y = 991.74
Figure 4.12 Stress Vs Tangent modulus for 500< E/Y >1000
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0.1E 0.2E 0.3E 0.4E 0.5E 0.6E 0.7E 0.8E 0.9E
10
20
30
40
50
60
Stre
ss (i
n N
/mm
2 )
Tangent modulus (ET) in N/mm2
E/Y = 1105.26 E/Y = 1739.13
Figure 4.13 Stress Vs Tangent modulus for 1000< E/Y >1750
Figure 4.13 shows the relationship between the stress induced and
tangent modulus for various materials. The maximum stress induced in the
material range of 1000 < E/Y > 1750 is laying on the tangent modulus of
0.6E.
4.4 STUDY OF VARIOUS CONTACT PARAMETERS
The rigid sphere and a deformable flat contact model as shown in
Figure 4.14. The load is applied on the top of the sphere. The sphere is
penetrated in to the flat.
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Figure 4.14 Basic model of rigid sphere and a deformable flat
In this study, the attempt has been made to modify the indentation
depth in the new form by incorporating the tangent modulus. The loading
relationship for the penetration depth is given by the relation
= {9W 2/8D}1/3[ 2{(1 – 2) / (E* + ET) }]2/3 (4.3)
In equation (4.3), W is the applied load, D is the ball diameter, and
the paired material constants , E* and ET are the Poisson’s ratio, Equivalent
young’s modulus and tangent modulus respectively. The E* is given by
2 2* 1 2
1 2
1 11/ EE E
(4.4)
In equation (4.4), 1 and 2 denotes the material properties of ball and
plate respectively.
4.4.1 Estimation of Projected Surface Diameter and Width of
Contact Area
The projected surface diameter (d) of the residual impressed
indentation is shown Figure 4.14. The following relationship gives the
mathematical formula for calculating the diameter
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d = 2 [ (D – )] 1/2 (4.5)
The significant material E/Y value of 991.74 is taken for
observation of various parameters and it is related to the contact behaviour of
the sphere with flat (indentation approach) with incorporating the tangent
modulus is given in the Table 4.3.
Table 4.3 Tangent modulus, Indentation depth and Projected areacontact width
S.No. ET(N/mm2) (mm)
d/2(mm)
1 0.1E 0.095 3.0812 0.2E 0.089 2.9823 0.3E 0.084 2.8974 0.4E 0.080 2.8275 0.5E 0.077 2.7746 0.6E 0.074 2.7197 0.7E 0.071 2.6648 0.8E 0.068 2.6079 0.9E 0.066 2.568
From the Table 4.3, it is clearly known that with increase in tangent
modulus, the indentation depth and contact width are reduced.
The indentation pressure under elastic, elastic-plastic and fully
plastic conditions may be correlated on a non-dimensional form of pm/Y as a
function of (E* tan /Y) where is the angle of the indenter at the edge of the
contact. With a spherical indenter put tan sin = a/R which varies during
indentation process. Where ‘a’ is width of the contact area (d/2), and ‘R’ is
the radius of the ball (D/2). The material E/Y value of 991.74 is taken for
observation of various parameters such as tangent modulus, indentation depth,
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projected area diameter and ratio of contact width to sphere radius by
incorporating the tangent modulus is given in the Table 4.4
Table 4.4 Tangent modulus, Indentation depth, Projected areadiameter and Ratio of contact width to sphere radius
S.No. ET
(N/mm2) (mm)d
(mm)a/R
1 0.1E 0.095 6.162 0.06162 0.2E 0.089 5.964 0.05963 0.3E 0.084 5.758 0.05764 0.4E 0.080 5.654 0.05655 0.5E 0.077 5.548 0.05556 0.6E 0.074 5.438 0.05447 0.7E 0.071 5.328 0.05338 0.8E 0.068 5.214 0.05219 0.9E 0.066 5.136 0.0514
From the Table 4.4 it is clearly shown that, with an increase in the
tangent modulus, the penetration depth and dimensionless ratio are reduced.
Figure 4.15 shows the diameter of projected surface of the residual
impressed indentation. The tangent modulus of the material increases with the
decrease in projected area diameter. It is conformed that in the smaller contact
area a large amount of load is carried when the body is in contact. The effect
of tangent modulus is greater influence in the contact parameter.
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0.1E 0.2E 0.3E 0.4E 0.5E 0.6E 0.7E 0.8E 0.9E5.0
5.2
5.4
5.6
5.8
6.0
6.2
6.4
Proj
ecte
d ar
ea d
iam
eter
(d) i
n m
m
Tangent modulus (ET) in N/mm2
E/Y = 991.74
Figure 4.15 Projected area diameter Vs Tangent modulus
0.1E 0.2E 0.3E 0.4E 0.5E 0.6E 0.7E 0.8E 0.9E0.050
0.052
0.054
0.056
0.058
0.060
0.062
Dim
ensi
onle
ss ra
tio a
/R
Tangent modulus (ET) in N/mm2
E/Y = 991.74
Figure 4.16 Dimensionless ratio (a/R) Vs Tangent modulus
The plastic strains are, of course, not uniform but, whatever their
quantitative Value, the strain is a function of ratio of width of contact to the
radius of sphere (a/R). Then made a very bold assumption, namely that there
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is a representative strain ( T ) in the specimen which is a power function of
a/R.
Figure 4.16 shows the relationship between the dimensionless ratio
(a/R) to tangent modulus. The tangent modulus increases when the d/D ratio
decreases due to the decrease in projected surface diameter (d). It presents
that when the tangent modulus is increase, the strain hardening effect in the
material increases. The contact size a/R as independent non-dimensional
variable. The value of a/R at the beginning of the finite deformation regime is
independent of the value of the elastic parameters.
4.4.2 Evaluation of Contact area
The estimation of contact area between the two consecutive steps of
penetration is very important for analysis the contact bodies in contact
mechanics. The contact area between the two circles of the sphere and the flat
as shown in Figure 4.17 is expressed by the double integral method in polar
coordinates rdrd , with suitable limits.
Figure 4.17 Contact Area - Circle
The limits are between -2
and2
, scos and 2scos . The area lying inside
the two circles is calculated from the following expression. Where, A1 = inner
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circle, Ao = outer circle, r = position of circle from the centre of contact
between bodies, = Angle at which the circle propagated, dr = radius
between two circles.
Difference in area = rdrd
=23 s
4 (4.6)
where s = size of the circle (Difference in contact width)
The estimation of an area between two circles, the outer circle area
is considered as zero. From this reference the remaining area between the two
consecutive circles are calculated and shown in Table 4.5.
Table 4.5 Difference in area between two consecutive steps of indentation
S.No. ET
(N/mm2)
Projected areacontact
width (d/2) (mm)
Difference in areabetween two circles
(mm2)1 0.1E 3.081 02 0.2E 2.982 0.02313 0.3E 2.897 0.01704 0.4E 2.827 0.01155 0.5E 2.774 0.0066
From the Table 4.5 it is clearly shown that, the increase in the
tangent modulus the difference in area between the two consecutive steps of
indentation is reduced.
Figure 4.18 shows the relation between the difference in area
between two circles of consecutive steps of penetration and tangent modulus.
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It is clearly shown that at 0.3E the material push upto the free surface rapidly
due to straining hardening effect of the material after that the material is very
slow in the free surface.
0.1E 0.2E 0.3E 0.4E 0.5E
0.000
0.005
0.010
0.015
0.020
0.025
Diff
eren
ce in
are
a b
etw
een
two
circ
les
(in m
m2 )
Tangent modulus (ET) in N/mm2
E/Y = 991.74
Figure 4.18 Difference in area between two circles Vs Tangent modulus
4.4.3 Evaluation of Volume of Squeezed Material
The evaluation of volume of material squeezed between the two
contact bodies is very important for contact analysis. The volume of material
between the two circles of the sphere and the flat as shown in Figure 4.19 is
expressed by the triple integral method in Cartesian form 8 dxdydz with
suitable limits. The limits are between zero(0) and s , 0 and 2 2s x . 0 and
2 2s x y .
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Figure 4.19 Volume of squeezed material
The volume lying inside the two circles is calculated from the
following expression
Volume in between two circles = 8 dxdydz
=34 s
3 (4.7)
where s = size of the circle (Difference in contact width)
The estimation of volume between two circles the outer circle
volume is considered as zero. From this reference the volume between the
two consecutive circles are calculated and shown in Table 4.6
From the Table 4.6 it is clearly shown that, the increase in the
tangent modulus the difference in volume between the two consecutive steps
of indentation is reduced.
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Table 4.6 Difference in volume of material between two consecutivesteps of indentation
S.No.ET
(N/mm2)
Projected areacontact width
(d/2) (mm)
Volume of materialbetween two circles
(mm3)1 0.1E 3.081 02 0.2E 2.982 0.00413 0.3E 2.897 0.00264 0.4E 2.827 0.00145 0.5E 2.774 0.0006
0.1E 0.2E 0.3E 0.4E 0.5E
0.000
0.001
0.002
0.003
0.004
0.005
Volu
me
of m
ater
ial b
etw
een
two
circ
les
(in m
m3 )
Tangent modulus (ET) in N/mm2
E/Y = 991.74
Figure 4.20 Difference in volume of material between two circles VsTangent modulus
Figure 4.20 shows the relation between the difference in volume
of material between two circles of consecutive steps of penetration and
tangent modulus. It is clearly shown that at 0.3E the volume of the material
which is squeezed out from the contact zone is more, due to the strain
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hardening effect of the material; after that the volume of material is low
down.
4.4.4 Estimation of Angle
The estimation of angle at which the squeezed material comes out
from the contact zone as shown in Figure 4.21 is important for contact
analysis.
Figure 4.21 Angle at which the squeezed material run off
The indentation depth is 'oa' and contact width is 'ob' is shown in
Figure 4.21. Based on these parameters the angle at which the squeezed
material escape from the contact zone is estimated and shown in the
Table 4.7.
From the Table 4.7, it shows that the angle at which the squeezed
material between the contact bodies in the contact region is increased due
to the material pushed towards the free surface by increasing the tangent
modulus.
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Table. 4.7 Angle at which the squeezed material escape
S.No. ET
(N/mm2) (mm)d/2
(mm)Angle
(Degree)1 0.1E 0.095 3.081 88.232 0.2E 0.089 2.982 88.293 0.3E 0.084 2.897 88.344 0.4E 0.080 2.827 88.375 0.5E 0.077 2.774 88.41
0.1E 0.2E 0.3E 0.4E 0.5E
88.22
88.24
88.26
88.28
88.30
88.32
88.34
88.36
88.38
88.40
88.42
Ang
le a
t whi
ch th
e sq
ueez
ed m
ater
ial e
scap
e (D
egre
e)
Tangent modulus (ET) in N/mm2
E/Y = 991.74
Figure 4.22 Angle at which the squeezed material escape Vs Tangentmodulus
Figure 4.22 shows the relation between the angle at which thesqueezed material escape from the contact region and tangent modulus. Theangle is estimated from the reference of line of action of the load applied . Itis clearly shown that the tangent modulus of the material increases as well asthe angle at which the squeezed material comes out from the contact region isalso increases.
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4.5 FINITE ELEMENT MODELING AND ANALYSIS USINGABAQUS
4.5.1 Introduction
To study the elastic-plastic deformation of solid homogeneousmaterials. In this model a rigid sphere is pushed against a deformable plate(flat) by applying a concentrated load on the center of the sphere as shown inFigure 4.23. The sphere (ball) penetrated into a flat like in an indentationprocess (Brinell and Rockwell Hardness testing methods). It is assumed thatthe material of the spherical indenter (ball) is harder than the material of theflat.
Figure 4.23 Indentation approach model (Spherical indentation)
The uniaxial tensile test properties of a material can also determinefrom the spherical indentation technique. The stress-strain curve can bedetermined in different ways from the spherical indentation test. Theindentation parameters are related in three different types of relations such as(i) force and contact radius (ii) force and indentation depth and (iii) meancontact pressure and contact radius. the relation between the load and theindentation diameter follow a power law as given bellow
F = Mdm (4.)
where M and m are the material constants.
113
4.6 METHOD OF SIMULATION IN 'ABAQUS'
There are two methods for simulating the sphere and flat contact
model. They are (i) load control and (ii) displacement control. In this analysis
the displacement control is used for simulating the rigid sphere and a
deformable flat.
4.7 FINITE ELEMENT MODELING
Finite element contact model is created for indentation approach
using 'ABAQUS' is based on the sphere and a flat contact method. In this
model the following assumptions were made for modeling
i) the sphere is a rigid member
ii) flat is a deformable member and
iii) frictionless contact between the indenter and flat.
Figure 4.24 Rigid sphere and a deformable flat contact model -ABAQUS
Figure 4.24 shows the rigid sphere and a deformable flat contact
model generated by using ABAQUS - 6.9. The axisymmetric model is
developed due to the advantage in the analysis procedure. The quarter sphere
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and half of the flat is considered the analysis based on the axisymmetric
property of model. The sphere size of radius is 0.79375 mm (1/16 inch like in
Brinell hardness test) and the flat size is 63 mm length and 10 mm thickness
is considered for modeling (like Specimen size use for Brinell hardness test).
In the modeling procedure the center of the rigid member (sphere) is taken as
a reference point.
4.7.1 Boundary Condition and Loading
Figure 4.25 shows the boundary condition and loading for the
simulation. The nodes lying on the axis of symmetry of the flat displacement
are restricted to move in the radial direction (U1= UR3 = 0). Also the nodes
in the bottom of the flat displacement are restricted to move in the vertical
direction (U2 = 0). In the rigid surface the translations and rotations on a
single node is known as rigid body reference node. In this model the reference
point is assigned on the center of the indenter (sphere). The boundary
conditions applied for this point are restricted to move in radial direction
(U1 = UR3 = 0).
Figure 4.25 Boundary conditions and loading
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The nodes lying on the axis of symmetry of the indenter and flat
displacement is allowed in the vertical direction (U2 0). For loading the
displacement control method was applied. In this method the displacement is
specified as input, which is equal to the penetrated depth of the sphere into a
deformable flat.
4.7.2 Mesh generation
The edges of the flat (plate) are meshed by biased seed edges
method. The finer mesh is generated around the indenter in order to
encompass the region of the higher stress near the contact as shown in
Figure 4.26. The total number of element and nodes generated in the flat is
5000 and 5151 respectively (Table A.3.3). The structured mesh was assigned,
whereas biased mesh control. 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 4.26 Mesh generation - Flat
4.7.3 Material properties
The material properties are selected based on the Young's modulus
and yield strength values. Table 4.8 shows the materials used for the
applications of contact problems and hardness testing.
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Table 4.8 Material Properties
S.No. Material E ×103
N/mm2Y
N/mm2
1 Steel 210.83 2002 Aluminium 70 1303 Copper 125 1254 Brass 105 135
4.8 LOADING CONDITION
The finite element simulation has been performed by using the
condition of frictionless contact between the indenter and the flat for spherical
indentation approach. The indentation process is assumed to be quasi-static
approach, in which no time effect is considered. Hence ABAQUS - Standard
method is used for indentation approach. The elastic-plastic material models
is analyzed under loading condition of spherical indenter of radius
0.79375 mm. The objective of the analysis under this loading condition is to
determine the indentation diameter for different materials and various contact
parameters like contact pressure, Von-Mises stress, strain, equivalent plastic
strain and reaction force in the indenter.
4.8.1 Simulation output for Steel
The following are the simulation output of steel material for an
indentation depth of 0.18 mm in ABAQUS - Standard.
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Figure 4.27 Plot of Von-Mises Stress in the deformed Steel flat
Figure 4.27 shows the Von-Mises stress developed in the deformed
flat. The maximum stress is developed in the contact region between indenter
and the flat. The minimum stress is away from the contact region. The
maximum and minimum stresses are 480 N/mm2 and 0.1444 N/mm2.
Figure 4.28 Plot of reaction force in the spherical indentation into aSteel flat
Figure 4.28 shows the reaction force developed in the rigid
spherical indenter. In the displacement control method, for the applied
displacement the Reaction force (RF) on the indenter is the summation of
force over the contact zone along the penetration direction. The reaction force
in the indenter is 904.9 N.
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Figure 4.29 Plot for contact pressure in a Steel flat
Figure 4.29 shows the contact pressure between indenter and the
contact surface nodes of the deformed flat. The maximum contact pressure
1196 N/mm2 is in-between the rigid indenter and the deformed flat surface.
The minimum contact pressure is zero at the top surface nodes of the flat.
Figure 4.30 Plot for strain in X-direction - Steel flat
Figure 4.30 shows the strain developed in the model. The plot
shows the deformed shape of the loaded flat. The minimum strain is
developed near the edge of the contact between the indenter and the flat. It is
shown in a circle in the plot. The maximum strain is developed under the
indenter surface in the flat.
119
Figure 4.31 Plot for displacement of nodes in Y-direction - Steel flat
Figure 4.31 shows that displacement of nodes in the loaded flat.
The minimum displacement of the nodes under the indenter and the
maximum at the edge of the contact. The displacement of nodes under the tip
of the indenter is approximately equal to the displacement of the indenter.
Figure 4.32 Equivalent plastic strain plot of Steel flat
Figure 4.32 shows the scalar plastic strain developed in the model.
PEEQ is an integrated measure of plastic strain. The plot shows the deformed
shape of the loaded flat. The maximum plastic strain is developed in the flat
under the indenter. The zero plastic strain in outside the contact zone.
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Figure 4.33 True stress Vs True plastic strain for Steel
Figure 4.33 shows the true stress and true plastic strain for the
material Young's modulus value of 210x103 N/mm2 and initial yield strength
value of 200 MPa. The true stress and true plastic strain are calculated from
the nominal stress and strain respectively. The Table 4.9 shows the true stress
and true plastic strain value of steel.
Table 4.9 True stress and True plastic strain - Steel
S. No True stress(MPa)
True plasticstrain
1 200.2 02 246 0.023743 294 0.047844 374 0.094365 437 0.13886 480 0.1814
121
Figure 4.34 Plot for displacement of rigid indenter into a Steel flat
Figure 4.34 shows the simulation output plot of displacement of
spherical indenter reference point. This plot gives the relationship between the
displacement of indenter in vertical direction to penetrated into a steel flat and
percentage of indenter movement into the flat. This plot is an evidence for the
spherical indenter is completely (100%) penetrated into a steel flat for the
given indenter displacement of 0.18 mm. It is clearly shown that, the indenter
is gradually (linear) penetrated into a flat when the percentage of the indenter
movement is reached upto 100%.
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Figure 4.35 Plot of reaction force Vs percentage of indenter movementfor a Steel flat
Figure 4.35 shows the simulation output plot of reaction force in
the indenter reference point. This plot gives the relationship between the
reaction force and percentage of indenter movement into a steel flat. This plot
also an evidence for the spherical indenter is completely (100%) penetrated
into a flat for the given indenter displacement of 0.18 mm. The percentage of
indenter movement is increases the reaction force is gradually increased upto
1077 N for 100% movement.
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4.8.2 Simulation Output for Aluminium
The following are the simulation output of aluminium material for
an indentation depth of 0.57 mm in ABAQUS - Standard.
Figure 4.36 Plot of Von-Mises Stress in the deformed Aluminium flat
Figure 4.36 shows the Von-Mises stress developed in the deformed
flat. The maximum stress developed under the contact region between
indenter and the flat. The minimum stress is away from the contact region.
The maximum and minimum stresses are 237.5 N/mm2 and 0.2362 N/mm2.
Figure 4.37 Plot of reaction force in the spherical indentation into aAluminium flat
[[
124
Figure 4.37 shows the reaction force developed in the rigid
spherical indenter. The reaction force in the rigid indenter is 1094 N.
Figure 4.38 Plot for contact pressure in a Aluminium flat
Figure 4.38 shows the contact pressure between indenter and the
contact surface nodes of the deformed flat. The maximum contact pressure
720 N/mm2 is in-between the rigid indenter and the deformed flat surface. The
minimum contact pressure is zero at the top surface nodes of the flat.
Figure 4.39 Plot for strain in X-direction - Aluminium flat
Figure 4.39 shows the strain developed in the model. The plot
shows the deformed shape of the loaded flat. The minimum strain 0.3480 is
developed near the edge of the contact between the indenter and the flat. It is
125
shown in a circle in the plot. The maximum strain 1.879 is developed under
the indenter surface in the flat.
Figure 4.40 Plot for displacement of nodes in Y-direction - Aluminiumflat
Figure 4.40 shows that displacement of nodes in the loaded flat.
The maximum displacement of the nodes under the indenter and the minimum
at the edge of the contact. The displacement of nodes under the tip of the
indenter is approximately equal to the displacement of the indenter.
Figure 4.41 Equivalent plastic strain plot of Aluminium flat
Figure 4.41 shows the scalar plastic strain developed in the model.
PEEQ is an integrated measure of plastic strain. The plot shows the deformed
shape of the loaded flat. The maximum plastic strain 2.084 is developed in the
126
flat under the indenter. The plastic strain is zero in the outside of the contact
zone.
Figure 4.42 True stress Vs True plastic strain for Aluminium
Figure 4.42 shows the true stress and true plastic strain for thematerial Young's modulus value of 70 x103 N/mm2 and initial yield strengthvalue of 130 MPa. The true stress and true plastic strain are calculated fromthe nominal stress and strain respectively. The Table 4.10 shows the truestress and true plastic strain value for Aluminium.
Table 4.10 True stress and True plastic strain - Aluminium
S. No True stress(MPa)
True plasticstrain
1 130 02 147 0.04693 187 0.09234 207 0.13605 222 0.17886 237.5 0.2196
127
Figure 4.43 Plot for displacement of rigid indenter into a Aluminium flat
Figure 4.43 shows the simulation output plot of displacement of
spherical indenter reference point. This plot gives the relationship between the
displacement of indenter in vertical direction to penetrated into a Aluminium
flat and percentage of indenter movement into the flat. This plot is an
evidence for the spherical indenter is completely (100%) penetrated into a
aluminium flat for the given indenter displacement of 0.57mm. It is clearly
shown that, the indenter is gradually (linear) penetrated into a flat when the
percentage of the indenter movement is reached upto 100%.
128
Figure 4.44 Plot of reaction force Vs percentage of indenter movementfor Aluminium flat
Figure 4.44 shows the simulation output plot of reaction force in
the indenter reference point. This plot gives the relationship between the
reaction force and percentage of indenter movement into the flat. This plot
also an evidence for the spherical indenter is completely (100%) penetrated
into a flat for the given indenter displacement of 0.57mm. The percentage of
indenter movement is increases the reaction force is gradually increased upto
1094 N for 100% movement.
4.8.3 Simulation Output for Copper
The following are the simulation output of copper material for an
indentation depth of 0.45 mm in ABAQUS - Standard.
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Figure 4.45 Plot of Von-Mises Stress in the deformed Copper flat
Figure 4.45 shows the Von-Mises stress developed in the deformed
flat. The maximum stress developed under the contact region between
indenter and the copper flat. The minimum stress is away from the contact
region. The maximum and minimum stresses are 275 N/mm2 and
0.1949 N/mm2.
Figure 4.46 shows the reaction force developed in the rigid
spherical indenter. The reaction force in the rigid indenter is 1013 N.
Figure 4.47 shows the contact pressure between indenter and the contact
surface nodes of the deformed Copper flat. The maximum contact pressure
863.6 N/mm2 is in-between the rigid indenter and the deformed flat surface.
The minimum contact pressure is zero at the top surface nodes of the flat.
Figure 4.46 Plot of reaction force in the spherical indentation into aCopper flat
130
Figure 4.47 Plot for contact pressure in a Copper flat
Figure 4.48 Plot for strain in X-direction - Copper flat
Figure 4.48 shows the strain developed in the model. The plot
shows the deformed shape of the loaded Copper flat. The minimum strain
0.253 is developed near the edge of the contact between the indenter and the
flat. It is shown in a circle in the plot. The maximum strain 1.426 is developed
under the indenter surface in the copper flat.
131
Figure 4.49 Plot for displacement of nodes in Y-direction - Copper flat
Figure 4.49 shows that displacement of nodes in the loaded Copper
flat. The maximum displacement of the nodes under the indenter and the
minimum at the edge of the contact. The displacement of nodes under the tip
of the indenter is approximately equal to the displacement of the indenter.
Figure 4.50 Equivalent plastic strain plot of Copper flat
Figure 4.50 shows the scalar plastic strain developed in the model.
PEEQ is an integrated measure of plastic strain. The plot shows the deformed
shape of the loaded flat. The maximum plastic strain 1.554 is developed in the
copper flat under the indenter. The plastic strain is zero in the outside of the
contact zone.
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Figure 4.51 True stress Vs True plastic strain for Copper
Figure 4.51 shows the true stress and true plastic strain for the
material Young's modulus value of 125 x103 N/mm2 and initial yield strength
value of 125 MPa. The true stress and true plastic strain are calculated from
the nominal stress and strain respectively. The Table 4.11 shows the true
stress and true plastic strain value for Copper.
Figure 4.52 shows the simulation output plot of displacement of
reference point of a spherical indenter. This plot gives the relationship
between the displacement of indenter in vertical direction to penetrated into a
copper flat and percentage of indenter movement into the flat.
133
Table 4.11 True stress and True plastic strain - Copper
S. No. True stress(MPa)
True plasticstrain
1 125 02 135 0.04693 175 0.09864 220 0.14825 250 0.1986 275 0.2478
Figure 4.52 Plot for displacement of rigid indenter into a Copper flat
This plot is an evidence for the spherical indenter is completely
(100%) penetrated into a copper flat for the given indenter displacement of
0.45 mm. It is clearly shown that, the indenter is gradually (linear) penetrated
into a flat when the percentage of the indenter movement is reached upto
100%.
134
Figure 4.53 Plot of reaction force Vs percentage of indenter movementfor Copper flat
Figure 4.53 shows the simulation output plot of reaction force in
the indenter reference point. This plot gives the relationship between the
reaction force and percentage of indenter movement into a copper flat. This
plot also an evidence for the spherical indenter is completely (100%)
penetrated into a flat for the given indenter displacement of 0.45 mm. The
percentage of indenter movement is increases the reaction force is gradually
increased upto 1013 N for 100% movement.
4.8.4 Simulation Output for Brass
The following are the simulation output of brass material for an
indentation depth of 0.325 mm in ABAQUS - Standard.
135
Figure 4.54 Plot of Von-Mises Stress in the deformed Brass flat
Figure 4.54 shows the Von-Mises stress developed in the deformed
Brass flat. The maximum stress developed under the contact region between
indenter and the brass flat. The minimum stress is away from the contact
region. The maximum and minimum stresses are 350 N/mm2 and
0.1934 N/mm2.
Figure 4.55 shows the reaction force developed in the rigid
spherical indenter. The reaction force in the rigid indenter is 1070 N.
Figure 4.55 Plot of reaction force in the spherical indentation into a
Brass flat
136
Figure 4.56 Plot for contact pressure in a Brass flat
Figure 4.56 shows the contact pressure between indenter and the
contact surface nodes of the deformed Brass flat. The maximum contact
pressure 1191 N/mm2 is in-between the rigid indenter and the deformed flat
surface. The minimum contact pressure is zero at the top surface nodes of the
flat.
Figure 4.57 Plot for strain in X-direction - Brass flat
Figure 4.57 shows the strain developed in the model. The plot
shows the deformed shape of the loaded Brass flat. The minimum strain
0.1581 is developed near the edge of the contact between the indenter and the
flat. It is shown in a circle in the plot. The maximum strain 0.9546 is
developed under the indenter surface in the Brass flat.
137
Figure 4.58 Plot for displacement of nodes in Y-direction - Brass flat
Figure 4.58 shows that displacement of nodes in the loaded Brass
flat. The maximum displacement of the nodes under the indenter and the
minimum at the edge of the contact. The displacement of nodes under the tip
of the indenter is approximately equal to the displacement of the indenter.
Figure 4.59 Equivalent plastic strain plot of Brass flat
Figure 4.59 shows the scalar plastic strain developed in the model.
PEEQ is an integrated measure of plastic strain. The plot shows the deformed
shape of the loaded brass flat. The maximum plastic strain 1.029 is developed
in the copper flat under the indenter. The plastic strain is zero in the outside of
the contact zone.
138
Figure 4.60 True stress Vs True plastic strain for Brass
Figure 4.60 shows the true stress and true plastic strain for the
material Young's modulus value of 105 x103 N/mm2 and initial yield strength
value of 135 MPa. The true stress and true plastic strain are calculated from
the nominal stress and strain respectively. Table 4.12 shows the true stress
and true plastic strain value for Brass.
Table 4.12 True stress and True plastic strain - Brass
S. No. True stress(MPa)
True plasticstrain
1 135 02 180 0.043 230 0.0754 255 0.15 310 0.156 350 0.2
139
Figure 4.61 Plot for displacement of rigid indenter into a Brass flat
Figure 4.61 shows the simulation output plot of displacement of
reference point of a spherical indenter. This plot gives the relationship
between the displacement of indenter in vertical direction to penetrated into a
brass flat and percentage of indenter movement into the flat. This plot is an
evidence for the spherical indenter is completely (100%) penetrated into a
brass flat for the given indenter displacement of 0.325 mm. It is clearly shown
that, the indenter is gradually (linear) penetrated into a flat when the
percentage of the indenter movement is reached upto 100%.
Figure 4.62 shows the simulation output plot of reaction force in
the indenter reference point. This plot gives the relationship between the
reaction force and percentage of indenter movement into a brass flat. This plot
also an evidence for the spherical indenter is completely (100%) penetrated
into a flat for the given indenter displacement of 0.325 mm. The percentage of
indenter movement is increases the reaction force is gradually increased upto
1070 N for 100% movement.
140
Figure 4.62 Plot of reaction force Vs percentage of indenter movementfor Brass flat
4.8.5 Parameters Measured from the Simulation - ABAQUS
The following are the results were obtained from the simulation of
spherical indentation under loading condition for the different materials at
different indentation depth.
From the Table 4.13, it is clearly shown that in the spherical
indentation process the reaction force in the rigid indenter increases as well as
the indentation diameter is also increased.
141
Table 4.13 Simulation output parameters - ABAQUS
S. No. Material Reaction Force(RF2) N
Indentation diameter(d) mm
1 Steel 904.9 0.982 Aluminium 1094 1.393 Copper 1013 1.224 Brass 1070 1.07
4.9 UNLOADING CONDITION (SPRINGBACK ANALYSIS)
The finite element simulation has been performed under unloadingof the loaded spherical indenter into a flat. The penetrated indenter is returnback to its initial position from the end of the indentation process is known asSpringback analysis. In the springback simulation in which the materialrecovers its elastic deformation after the indenter is unloaded. This analysis isperformed from the developed model in 'ABAQUS' for the loading conditionby using the commands restart, copy model, edit attributes and so on. Themain objective of this analysis is to determine the parameters which have notbeen studied in the experimental work like residual stress and strain.
4.9.1 Stress Distribution Simulation Output for Steel (Unloading)
The following are the simulation result of stress distribution forunloading the spherical indenter to bring back to its initial position. Themodel which is used in the loading condition of spherical indenter the samemodel is used and restart the results which is obtained in the loading step.
Figure 4.63 shows the Von-Mises stress developed in the deformedsteel flat in the loading condition. The maximum stress developed under thecontact region between indenter and the flat. The minimum stress is outsidefrom the contact region. The maximum and minimum stresses are 480 N/mm2
and 0.1444 N/mm2 before unloading the indenter.
142
Figure 4.63 Von-Mises stress plot of Steel flat before unloading
Figure 4.64 shows the Von-Mises stress distribution in the
deformed steel flat after unloading. It is observed that the stress is released
under the indenter and migrate into the left edge of the deformed flat. The
value of maximum stress 480 N/mm2 is same which is obtained in the loading
step. But it is migrate to other location. The minimum stress outside the
contact region is released from the flat after unloading.
Figure 4.64 Von-Mises stress plot of Steel flat after unloading
It is important to analysis in this point, to study the behaviour of the
displacement of the particular node in the left edge of the deformed flat. The
further study has been carried out in such a node (Node number 5051). The
143
study has focused on the displacement of the particular node in the loading
and unloading steps. The displacement of the indenter is same as that of in the
loading case of steel material.
Figure 4.65 Displacement of node 5051 in Y-direction in loading step -Steel flat
Figure 4.65 shows the simulation output plot of displacement of
node 5051 in the loading step. This plot gives the relationship between the
displacement of node in the direction of indenter penetration and percentage
of indenter movement into a steel flat. It shows that for the complete
movement of indenter the node displaced at a distance of 0.1842 mm.
Figure 4.66 shows the simulation output plot of displacement of
node 5051 in the unloading step. This plot gives the relationship between the
displacement of node in the direction opposite to the indenter penetration and
percentage of indenter movement in unloading step of steel flat. It shows that
for 10% of unloading the particular node is displaced from 0.1842 mm to
0.18125 mm and for further unloading the node displacement is constant.
144
Figure 4.66 Displacement of node 5051 in Y-direction in unloading step -steel flat
Figure 4.67 Displacement of node 5051 in Y-direction in loading andunloading steps - steel flat
145
Figure 4.67 shows the simulation output plot of displacement of
node 5051 in loading and unloading step. It shows that at the end of the
loading step the node reached the maximum displacement of 0.1842 mm. It
has been slightly greater than the applied displacement in the indenter
(0.18 mm). At the starting of the unloading step the node displaced at a
distance of 0.00295 mm in the upward direction and then no further
displacement in the node. It is clear that the indentation depth is increased by
0.00125 mm than the applied once.
4.9.2 Strain Distribution Simulation Output for Steel Flat
(Unloading)
The following are the simulation result of strain distribution for
unloading the spherical indenter to bring back to its initial position.
Figure 4.68 Strain plot of Steel flat before unloading
Figure 4.68 shows the strain developed in the model before
unloading the indenter. The plot shows the deformed shape of the loaded steel
flat. The minimum strain 0.09164 is developed near the edge of the contact
between the indenter and the flat. The maximum strain 0.5666 is developed
under the indenter surface in the steel flat.
146
Figure 4.69 Strain plot of Steel flat after unloading
Figure 4.69 shows the strain developed in the model after
unloading the indenter. The plot shows the deformed shape of the unloaded
steel flat. The strain rate at an edge of the contact between the indenter and
the flat is reduced by 0.005 after unloading . The strain rate under the indenter
surface in the steel flat is reduced by 0.0061
4.10 FINITE ELEMENT ANALYSIS FOR PILE-UP CONDITION
IN SPHERICAL INDENTATION
The finite element simulation has been performed to study the pile-
up condition for different elastic-plastic material model for same indentation
depth of 0.25 mm in a spherical indentation process. The sphere radius of
0.79375 mm and flat size is 63 × 10 mm is used for simulation. The main
objective of this study is to determine the pile-up height of the deformed
material and classify the pile-up condition based on the material properties.
4.10.1 Material Properties for Analysing the Pile-Up Condition
The different materials are selected for analysis based on the
Young's modulus to initial yield strength ratio. The following are the
materials shown in Table 4.14 used for the applications of contact problems.
147
Table 4.14 Material Properties
S.No. Material E/Y1 Steel 10502 Aluminium 5383 Copper 10004 Brass 778
4.10.2 Analysis of Pile-Up Condition for Different Materials
The following are the simulation results for different materials to
analysing the pile-up condition. For all the materials the elastic-plastic
indentation is performed based on the assumptions (i) the material is not
subjected to body force (ii) the material is isotropic and homogeneous and
(iii) the system is isothermal.
Figure 4.70 Plot of Pile-Up condition for material E/Y = 1050
Figure 4.70 shows the pile-up of material E/Y = 1050. The
simulation has performed for the indentation depth of 0.25 mm. The pile-up
height (h) is equal to 0.0481 mm.
148
Figure 4.71 Plot of Pile-Up condition for material E/Y = 538
Figure 4.71 shows the pile-up of material E/Y = 538. The
simulation has performed for the indentation depth of 0.25 mm. The pile- up
height is 0.0644 mm.
Figure 4.72 Plot of Pile-Up condition for material E/Y = 1000
Figure 4.72 shows the pile-up of material E/Y = 1000. The
simulation has performed for the indentation depth of 0.25 mm. The pile- up
height is 0.0564 mm.
149
Figure 4.73 Plot of Pile-Up condition for material E/Y = 778
Figure 4.73 shows the pile-up of material E/Y = 778. The
simulation has performed for the indentation depth of 0.25 mm. The pile- up
height is 0.0456 mm.
4.10.3 Parameters measured from the simulation of pile-up condition -
ABAQUS
The following are the results were obtained from the simulation of
spherical indentation under loading condition for the different materials at
same indentation depth of 0.25 mm.
Table 4.15 Simulation output parameters - ABAQUS
S. No. Material E/YValue
Pile-up height(h) in mm FEA
1 Steel 1050 0.04812 Aluminium 538 0.06443 Copper 1000 0.05644 Brass 778 0.049
150
From the Table 4.15, it is clearly shown that in the spherical
indentation process the maximum pile-up height has occurred in the material
E/Y = 538. The material pile-up has developed in the material having low
strain hardening. And also the height of the pile-up is dependent on the
indenter penetrated depth.
4.11 CHAPTER SUMMARY
The indentation approach contact model has been investigated by
considering the tangent modulus of the material in the frictionless rigid
indenter penetrated into an deformable half-space in the axisymmetric
condition in 'ANSYS'. The effect of the tangent modulus has been studied and
the contact parameters such as an area between the two consecutive steps of
indentation, volume of squeezed material and angle at which the squeezed
material run off were evaluated based on the effect of tangent modulus. The
tangent modulus increases when the d/D ratio decreases due to the decrease in
projected surface diameter (d). When the tangent modulus is increase, the
strain hardening effect in the material increases.
The simulation has been performed in the 'ABAQUS' software
for the spherical indentation model. The ABAQUS software is used for
simulation having the advantage of restarting the data for unloading process.
The analysis was carried out for the different materials at different indentation
depth under loading and unloading conditions. The reaction force in the rigid
indenter and indentation diameter are estimated in the loading condition. The
springback (unloading) analysis has been performed, the elastic property of
the material has recovered and the stress and strain distribution are
investigated in the indentation area. The material pile-up height is analysed
for different material by applying the equal indentation depth.