CHAPTER 4 ANALYSIS OF INDENTATION MODELshodhganga.inflibnet.ac.in/.../10603/17557/9/09_chapter4.pdf4.2.3 Mesh Generation In this analysis element type of plane 82, conta172 and target

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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

90

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.

91

Figure 4.2 FEA model of rigid sphere and a deformable flat

Figure 4.3 Contact pair of RS model

92

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.

93

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

95

the hemisphere is penetrated into a deformable half flat. In this model the

maximum stress of 22.374 N/mm2 is developed.



96

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 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.

97



98

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

99

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

100

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 )


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.

101

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

102

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,

103

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.

104

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


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


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

105

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

106

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.

107

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 )


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 .

108

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.

109

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 )


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

110

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.

111

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)


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.

112

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

114

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

115

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.

116


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.

118

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.

120

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%.

122

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.

123

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


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


spherical indenter. The reaction force in the rigid indenter is 1094 N.

Figure 4.38 Plot for contact pressure in a Aluminium flat


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


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



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


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



reaction force and percentage of indenter movement into the flat. This plot


into a flat for the given indenter displacement of 0.57mm. The percentage of



4.8.3 Simulation Output for Copper

The following are the simulation output of copper material for an


129

Figure 4.45 Plot of Von-Mises Stress in the deformed Copper flat


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 4.47 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


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


Figure 4.50 Equivalent plastic strain plot of Copper flat



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.

132

Figure 4.51 True stress Vs True plastic strain for Copper


material Young's modulus value of 125 x103 N/mm2 and initial yield strength


the nominal stress and strain respectively. The Table 4.11 shows the true

stress and true plastic strain value for Copper.


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



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


135

Figure 4.54 Plot of Von-Mises Stress in the deformed Brass flat


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 Plot of reaction force in the spherical indentation into a

Brass flat

136

Figure 4.56 Plot for contact pressure in a Brass flat


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


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


Figure 4.59 Equivalent plastic strain plot of Brass flat



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


material Young's modulus value of 105 x103 N/mm2 and initial yield strength


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


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%.



reaction force and percentage of indenter movement into a brass flat. This plot


into a flat for the given indenter displacement of 0.325 mm. The percentage of



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


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.


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


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


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



simulation has performed for the indentation depth of 0.25 mm. The pile- up

height is 0.0644 mm.





149





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.

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CHAPTER 4 ANALYSIS OF INDENTATION MODELshodhganga.inflibnet.ac.in/.../10603/17557/9/09_chapter4.pdf4.2.3 Mesh Generation In this analysis element type of plane 82, conta172 and target