A Study of Elastic Plastic Deformation of Heavily Deformed Spherical Surfaces
by
Saurabh Sunil Wadwalkar
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
December 18, 2009
Keywords: heavy loading, plastic deformation, compression of spheres
Copyright 2009 by Saurabh.S.Wadwalkar
Approved by
Robert L Jackson, Chair, Associate Professor of Mechanical Engineering
Jeffrey Suhling, Quina Distinguished Professor, Mechanical Engineering
Pradeep Lall, Thomas Walter Professor of Mechanical Engineering
Lewis Payton, Associate Research Professor of Mechanical Engineering
ii
Abstract
This work uses a finite element analysis and analytical equations to model elastic-
plastic and fully plastic large deformations of spheres in contact with rigid flat surfaces.
The case considered here is of a deformable sphere compressed by a rigid flat as opposed
to the reverse case of a rigid spherical indenter penetrating a deformable surface. Most
previous work only deals with elastic or elasto-plastic deformation at much smaller
deformations. Based on an extensive literature survey, the work most related to plastic
deformations of spherical surfaces are the papers by Noyan [1] and Chaudhri [2]. Even
existing finite element based models do not explain plastic deformations well. The
current work theoretically explains the initiation and progression of plastic deformations
throughout the sphere.
A model for predicting contact area, pressure and force for plastic deformations
has been proposed based on the FEM simulations and analytical equations derived from
volume conservation. The analytical volume conservation approach is similar to that used
to model the barreling of compressed cylinders. The most important aspect of the model
is the resulting equation relating the average pressure during fully plastic deformation to
the yield strength. The model improves the current state-of-the art by providing equations
relating contact force, area, pressure and interference much further into the fully plastic
regime and for much larger deformations than the previous works. The results have been
compared with existing models and with experimental data. All the results have been
iii
simulated for three different sets of material properties to provide a model that is
applicable to a wide range of materials.
iv
Acknowledgments
I wish to acknowledge my sincere gratitude to my advisor, Dr. Robert L Jackson,
for his great motivation, support and encouragement during the course of this study. I
would like to thank my committee members, Dr. Lewis Payton, Dr.Jeffrey Suhling and
Dr. Pradeep Lall, for their continuous support in this study.
I would like to express deep gratitude and gratefulness to my parents and brother
for their enduring love, immense moral support and encouragement in my life. I wish to
thank all my colleagues and friends at Auburn for their friendship and help.
v
Table of Contents
Abstract????????????????????????????????ii
Acknowledgements???????????????????????????.iv
List of Figures...??????????..??????????????????vi
List of Tables...??????????????????????????..?...viii
Nomenclature??...??????????????????????????...ix
1. Introduction.......................................................................................................???.1
2. Motivation and Objectives??????????????????????.....5
3. Literature Review...........................................................................................................7
4. Finite Element Modeling Methodology......................................................????.21
5. Results and Discussions...............................................................................................31
square4 Finite Element based Model...????????????????.....................33
square4 Comparison with Experimental Results..???????????????...57
square4 Effects of Strain Hardening..????????????????????.63
square4 Effects of Friction across the area of contact?...????????????..69
6. Conclusions..................................................................................................................71
8. Recommendations for Future Work.............................................................................73
Bibliography ......................................................................................................................74
Appendix............................................................................................................................76
vi
List of Figures
Figure 1.1: Stress strain curve for a sphere under compression??????????..3
Figure 3.1: Figure showing the assumption by Jackson and Green????...???...12
Figure 4.1: The two boundary conditions used to model the sphere???????.....22
Figure 4.2: Schematic showing the B.C?s used n the FEM for the two cases????...23
Figure 4.3a: A representation of the FEM mesh and the deformed geometry for the
deformable base case???....????????????????..????......30
Figure 4.3b: A representation of the FEM mesh and the deformed geometry for the
rigid base case????????????????..?????????.............31
Figure 4.4: Distribution of displacements across the sphere?..?????????..32
Figure 4.5: Mesh convergence for maximum displacement across the sphere ????33
Figure 5.1: Stress strain curve for a material exhibiting elastic perfectly plastic
behavior???????????????????????????????.33
Figure 5.2: Mean contact pressure predictions for the deformable base case?...???36
Figure 5.3: Comparison of radius of hemisphere by analytical equation
with FEM results???????????????????????????...37
Figure 5.4: Pressure distribution across contact area??????????????38
Figure 5.5: Nomenclature for the rigid base case???????????????..41
Figure 5.6: Mean contact pressure predictions for the rigid base case.???????.42
Figure 5.7: Effectiveness of constant ????????????????????..43
Figure 5.8: Pressure distribution across contact area??????????????.44
vii
Figure 5.9: Comparison of predictions of JG, MM and current model
with FEM results for the deformable base case for small values of penetration???...48
Figure 5.10: Comparison of predictions of JG, MM and current model with
FEM results for the deformable base case for large values of penetration??????49
Figure 5.11: Comparison of predictions of JG, MM and current model
with FEM results for the rigid base case for small values of penetration??????..51
Figure 5.12: Comparison of predictions of JG, MM and current model
with FEM results for the rigid base case for large values of penetration??????..52
Figure 5.13: Comparison of predictions of contact force according to MM and current
model with FEM results for the deformable base case with small loads??????...55
Figure 5.14: Comparison of predictions of contact force according to JG, MM
and current model with FEM results for the deformable base case with heavy loads?...56
Figure 5.15: Comparison of predictions of contact force according to MM and current
with FEM results for the rigid base case with small loads????????????57
Figure 5.16: Comparison of predictions of contact force according to JG, MM and
current model with FEM results for the rigid base case with large loads??????..58
Figure 5.17: Comparison of experimental and simulation based model for phosphor
bronze material???????????????????????????.?..63
Figure 5.18: Comparison of experimental and simulation based model
for brass material?????????????????????????..??..64
Figure 5.19: Variation of contact pressure with increasing contact radius?????....66
Figure 5.20: Von Mises stress distribution for 4% strain hardening in the sphere...?....68
Figure 5.21: Von Mises stress distribution for 2% strain hardening in the sphere....?...69
Figure 5.22: Von Mises stress distribution for 0% strain hardening in the sphere....?...70
Figure 5.23: Variation of contact force with increasing contact force for increasing
friction across contact surface????.??????????????????...71
Figure 5.24: Variation of contact force with increasing strain hardening??????.73
Figure 5.25: Variation of contact force with increasing friction?????????...74
viii
Figure A.1: Underformed sphere with FEA mesh???????????????.87
List of Tables
Table 4.1: Reaction force results from FEM for mesh convergence????????.30
Table 5.1: Table showing the boundary conditions used for the study???????.32
Table 5.2: Material properties used in (a) Jackson and Green [3]
and (b) the current FEM analysis?????????????????????...41
Table 5.3: Material properties and Microhardness measurements
as given by Chaudhri [2]?????????????????????????56
x
Nomenclature
A individual asperity contact area
nA nominal contact area
JGR
a ?
?
??
?
? ratio of contact radius to original radius predicted by Jackson and Green
newR
a ?
?
??
?
? ratio of contact radius to original radius predicted by current study
A1, A2 constants based on material properties for deformable base case
A3, A4 constants based on material properties for rigid base case
Ac critical contact area at onset of plastic deformation
AP area of contact during plastic deformation
CEBA area of contact during elastic-plastic range given by CEB model
KEA contact area given by KE model
a radius of area of contact
B material dependent exponent
C critical yield stress coefficient
d separation of mean asperity heights
E modulus of elasticity
E? equivalent modulus of elasticity for the bodies in contact
F contact force
Fc critical contact force
Fp contact force during fully plastic deformation
H contact pressure during fully plastic deformation or hardness
K hardness factor
P contact pressure
CEBP contact force during elastic-plastic range given by CEB model
KEP contact force given by KE model
R1 radius of curvature for deformed hemisphere
R2 radius of curvature for bulged portion of hemisphere
R undeformed radius of the hemisphere
Sy yield strength
V1 initial volume of hemisphere before deformation
V2 volume of deformed geometry
z height of asperity measured from mean of asperity heights
(z) Gaussian distribution
? interference between hemisphere and flat rigid surface
?c critical interference between hemisphere and flat rigid surface
? barreling constant
? Poisson?s ratio
1
CHAPTER 1
INTRODUCTION
Flattening of spherical surfaces in contact with flat rigid surfaces is a problem
which has always received a great deal of attention, especially in regards to bearings,
tribological surfaces, impacting objects, and thermal and electrical contact resistance.
Whenever a sphere is being compressed by a flat surface, it can be classified by different
phases of deformation viz. elastic, elasto plastic and fully plastic deformations. Bodies
undergoing elastic deformation can recover their original shape but if there is plastic
deformation the sphere will get permanently deformed.
To be able to evaluate the behavior of spheres in the elasto plastic and fully
plastic regimes one needs to understand the basic elastic-plastic deformations. The stress
strain relationship for a body under compression is shown in Fig. (1.1). As shown in the
Fig. (1.1), stress increases linearly with strain during small deformations as defined by
Hooke?s law. However, as deformations get larger, the relationship between stress and
strain is no longer linear. Point ?A? is defined as the proportionality limit and upto this
point the curve follows Hooke?s law. The slope of the line till the proportionality limit is
called the Young?s modulus of elasticity. As more and more material starts deforming,
plastic deformations grow and the curve departs from linearity. As permanent
deformations increase, the material becomes saturated with dislocations which prevent
nucleation of new dislocations. This is manifested in the form of increased resistance to
2
deformation and is called work hardening of the material. Materials are many times
purposefully work hardened to increase their strength and resistance against plastic
deformations by techniques such as cold rolling and cold drawing. The rate of work
hardening is defined by the tangent modulus of the stress strain curve beyond the
proportionality limit.
Although the theory of elasticity has been studied in detail and ample material is
available, work related to understanding and explanation of plastic deformation of
spherical surfaces is relatively scarce. Existing models [3], [4], [5], [6] and [7] explain
elastic and elasto plastic deformations and predict the contact pressure and contact area to
much accuracy and rely on the truncation method for explaining fully plastic
deformations. The truncation method proposed by Abbott and Firestone [8] states that
under fully plastic conditions the contact area of an asperity in contact with a flat rigid
can be calculated by truncating the asperity tips as the flat rigid translates an interference,
?. The details of this model will be discussed later in the literature survey chapter. The
limitations in using this approach will be explained in the following sections.
Flattening of a sphere occurs when it is compressed by a rigid surface. As
mentioned above, several studies related to this problem have been published. Some of
them discussed in the literature review (see chapter 3) are Chang et al. [5], Kogut et al.
[9], Jackson and Green [3] and Zhao et al. [10]. These studies discuss elastic and elasto
plastic deformations in detail but lack explanation when the contact area becomes larger.
The present work investigates this flattening problem for spherical surfaces and proposes
a model to better explain the evolution and progression of deformation during elastic,
elasto plastic and fully plastic regimes.
3
Jackson and Green [3] defined a elasto plastic model to address the flattening
problem. They defined a limit for the average contact pressure which is valid upto a/R
=0.41, where a is the contact radius and R is the original radius. According to them above
this value, the deformations become large and the model is not intended for such large
deformations. The current study makes an attempt to extend the Jackson and Green
model and make it applicable to the plastic deformations and validate it till a/R =1.
Initially, the sphere under compressive load has been simulated and analyzed without any
strain hardening and friction. However, some results to understand the effects of strain
hardening and friction have been discussed later. To validate the FEM based model the
results have been compared with existing experimental data [2]. The following section
explains the motivation and objective of the research work. Based upon the research and
comparisons with existing real world data, results will be presented and conclusions will
be made.
4
EtEy
Strain, ?
St
res
s,
?
Et - Tangent modulus
Ey ? Young?s modulus
Figure 1.1: Stress strain curve for a sphere under compression
A
5
CHAPTER 2
MOTIVATION AND OBJECTIVES
Motivation:
Two primary motivating factors for the present investigation on flattening of
spheres by rigid flat surfaces are
1) Better understand evolution and progress of plastic deformation during flattening
of spherical surfaces. Existing models poorly predict contact parameters such as
contact force and area for large deformations.
2) For heavily loaded spherical contact, the effects of strain hardening have scarcely
been documented. It is important to understand this hardening effect in the sphere
under compression.
3) Existing models (see literature survey) do not compare results with real world
experimental data for heavily loaded spherical surfaces. A theoretical description
of compression of spherical surfaces under heavy loading in the fully plastic
regime is not available.
6
Objectives:
Based on these above factors, the chief objectives of the current investigation are
defined as to
1) Provide a comprehensive technical literature review on the flattening problem of
spherical surfaces.
2) Propose a finite element based model which explains and predicts the deformation
behavior of heavily loaded spheres.
3) Present and discuss the effects of strain hardening and friction in the current
study.
7
CHAPTER 3
LITERATURE REVIEW
The problem of compression of spherical surfaces by flat surfaces has
received great attention and a significant amount of work has been published related to
this problem. When a sphere is compressed between flat surfaces, the sphere undergoes
different phases of deformation before complete failure. For low loads, the deformation is
mostly elastic. But as loads get larger, permanent deformation is observed which results
in plastic deformation. Once the sphere load passes a critical value, at a point below the
surface the Von Mises stress exceeds the yield strength and plastic deformation begins.
The fully plastic regime is defined as when the entire contact area is deforming
plastically.
Most previous spherical contact models consider the elastic or elastic-plastic
case, but do not study the fully plastic regime in depth [5], [9] and [3]. Chaudhri et al. [2]
and Noyan at al. [1] have conducted experimental analysis of spherical contact in the
fully plastic regime, but no theoretical studies appear to have been conducted for the
flattening case. There is a great deal of work which also considers indentation in the
elastic, elasto-plastic and fully plastic regime [3], [11], [7] and [12]. Indentation means
that the sphere is rigid and the flat surface deforms. However, the current work is
concerned only with flattening rather than indentation. In flattening, the flat surface is
8
rigid and the sphere is deformable. Whenever a metal sphere is compressed between flat
rigid surfaces, it undergoes elastic, elasto plastic and fully plastic deformation. In this
work we refer to this case as flattening. Jackson and Kogut [11] also compared these two
cases and showed how their behaviors are very different. This configuration is relevant in
many other areas such as forging and anisotropic conductive films [13], [14], [15] and
[16]. The following sections are aimed to give a summary of the literature that is
available related to flattening of spherical surfaces.
Experimental work ? Plastic compression of spheres
Two of the most noteworthy previous works on spherical flattening in the fully
plastic regime are the experiments conducted by Chaudhri et al. [2] and Noyan [1].
Chaudhri et al. [2] conducted experiments to characterize the behavior of spheres made of
different materials and with different prior treatments (work hardened or annealed). The
different materials used for the experiments were phosphor bronze (92% Cu, 8% Sn),
brass (60% Cu and 40% Zn) and aluminum. The experimental setup used by Chaudhri et
al. [2] included two different flat surfaces for compression at high and low loads. For low
loads, the spheres were compressed between sapphire plates backed by a glass plate
which rested on a strong metal support. A calibrated graticule in the viewing microscope
measured the diameter of contact area. For high loads (plastic deformation) the sphere
was compressed between polished plane tool-steel platens at a cross head speed of 5 mm
min-1 in a J.J modeled T5000 testing machine. The diameter of the area of contact was
measured by an optical microscope after unloading. A detailed discussion of the
evolution and progress of deformation is presented. The effects of lubrication on fully
9
plastic contact have also been studied. The current work uses this experimental data to
compare to and validate the FEM results.
In Chaudhri et al. [2] the spheres, both undeformed and compressed were sectioned and
measured for microhardness across the diameter of the sections using a Leitz miniload
hardness tester. Care was taken so that the indentations do not interfere with each other.
The hardness measurements for the as-received phosphor bronze before and after loading
revealed that there was hardly any hardening left in the material. Thus, these can be
treated as elastic perfectly plastic materials in the experiments. The current work models
the sphere as elastic perfectly plastic initially and will use the experimental results to
validate the simulation results.
The experimental work by Noyan [1] focused mostly on compression of solid
spheres of various materials between parallel platens. During the experiments, the
variation in contact area and area of the central plane of symmetry with plastic
deformation was monitored. They defined two normalized parameters which are
independent of size and material of the sphere. This indicated that according to them,
plastic deformation of spheres was controlled by geometry. They also mapped the
microhardness throughout the sphere and predicted the distribution of deformation. Some
of the conclusions of these experiments are:
square4 An increase in contact area is a function of the depth of penetration of the flat
rigid surface
square4 The hardness distribution in the deformed spheres is symmetric across the central
plane
10
square4 As the compressive strain increases, the plastic deformation progresses deeper
into the sphere.
The current work will confirm these findings with analysis of finite element modeling
results for spheres without any strain hardening. But an attempt to understand strain
hardening effects during compression will also be made later.
Fully plastic contact models
Tabor [17] studied the contact between a sphere and a flat surface under
compression loads. He showed from slip-line theory that the hardness of a perfectly
plastic spherical indentation should be about
ySH ?= 8.2 (1)
The hardness, H, here is defined as the average pressure during fully plastic contact or
indentation. These formulations are empirical and have been defined for a variety of
materials like aluminum, copper and mild steel. According to Tabor [17], the relationship
between mean pressure and the yield stress changes from ySP ?= 1.1 to ySH ?= 8.2
during the transition from elasto plastic to fully plastic deformations. Notice, the pressure
is addressed by P and H. For fully plastic deformations, the mean contact pressure
(hardness) is also denoted by H. These relationships are derived from frictionless
compression experiments.
11
Although often attributed to the earlier work by Abbot and Firestone [8] but
probably actually derived by Greenwood and Tripp) [18], a model for contact area of a
fully plastic spherical contact was created by simply truncating the sphere geometry with
the flat surface. Then the contact area can be approximately calculated by truncating the
sphere tip as it translates an interference,?, without deformation into the flat surface. For
a hemisphere, this approximated fully plastic contact area is be given by
?piRAP 2= (2)
The contact force is then just Eq. (2) multiplied by the contact pressure which in this case
is the hardness, since the contact is assumed to be fully plastic and is given by,
HRFp ?pi2= (3)
Eq. (3) has been proven erroneous by many works [3], [7] and [11] for both the
indentation and flattening cases. This is because it is overly simplistic and neglects the
actual elastic-plastic contact mechanics that take place during contact. The major
criticism in addition to this is that it does not conserve the volume of the material
deforming plastically.
Ishlinsky [31] also proved analytically by using the Harr-Karman criterion of
plasticity that it is possible to determine the mean pressure for spherical contact.
According to Islinsky [31] the value of the constant in the relation between the mean
12
contact pressure and the yield strength is 2.84. This value of the constant was confirmed
by Johnson [32] who mention a value of 2.84 for the constant in their work. Whereas,
Ashby [33] reported that the value of constant in their study was found to be 3.3. This
discrepancy in the value of this constant was studied in detail and a range of 2.8 to 3.3
was observed to be reported in various texts referred to in the current study.
Elastic and Elasto plastic models
Chang et al. [5] developed a plastic contact model (CEB) that supplemented with
the GW model explained later. The GW model is an elastic contact model. CEB model
used the volume conservation principle similar to the current study to approximate a
elastic-plastic contact. Assumptions of the CEB model are,
square4 A fixed relationship between yield strength and hardness, H = 2.84Sy.
square4 The hemisphere behaves elastically below the critical interference ?c and fully
plastically above it.
square4 Deformation is localized near the hemisphere?s tip.
According to the CEB model, the contact area and force for ?/?c >1, that is the elastic-
plastic range are given by
( )cCEB RA ??pi ?= 2 (4)
( )KHRP cCEB )2 ??pi ?= (5)
13
where,
REKHc
2
'2 ??
??
?
?= pi? and ?41.0454.0 +=K
The limitations of the CEB model are that it assumes the fixed relationship between
hardness and yield strength. This assumption was proved incorrect by Jackson and Green
[3]. Also, the model has discontinuity at ?c.
Kogut and Etsion [9] performed an FEM analysis for the flattening case of sphere
in contact with a flat rigid surface. Their work gives a very detail stress distribution study
in the contact area. The contact force and area are defined for ranges of ?/?c.
For 1? ?/?c ?6,
= 425.1/03.1 ccKE PP ?? (6)
= 136.1/93.0 ccKE AA ?? (7)
For 6? ?/?c ?110,
= 263.1/40.1 ccKE PP ?? (8)
= 146.1/94.0 ccKE AA ?? (9)
The KE model similar to the CEB model assumes a fixed relationship between hardness
and yield strength H = 2.84Sy. Also, the model equations are discontinuous at ?/?c =1
and 6. KE model only describes deformations till ?/?c=110. Beyond this point the
truncation model [8] is assumed to define the fully plastic deformations.
14
Zhao et al. [10] worked on a elasto-plastic asperity microcontact model for rough
surfaces in contact. The model incorporates the transitional regime from elastic to fully
plastic deformations. The model like the CEB [5] and KE [9] models assumes truncation
model for fully plastic deformations. Nuri [19] reported rough surface contact parameters
by experimentally measuring them. Jackson and Green [4] compared their predictions of
contact radius to show that the bulk material below the asperities would undergo extreme
deformation. [20] gave analytical approximations for modeling rough surface contacts.
Jackson and Green [3] find that Eq. (3) can overpredict the contact force. They propose a
FEM based elasto-plastic contact model for the contact between a deformable sphere and
a flat rigid surface. Their work finds that the hardness or the fully plastic average contact
pressure actually varies with the deforming geometry of the sphere. Chaudri et al. [2] also
confirmed this experimentally. When the contact pressure is plotted against the contact
radius, a limit appears to emerge for the average pressure during fully plastic contact.
According to Jackson and Green [3], as a/R increases, the limiting average pressure to
yield strength ratio must change from Tabor?s [17] predicted value of approximately 2.84
to a theoretical value of 1 when a=R. This has been depicted in the Fig. (3.1).
15
Figure 3.1: Figure showing the assumption by Jackson and Green.
As the interference increases the contact geometry changes and when the contact radius
a=R, the geometry is similar a cylinder in contact with a flat rigid surface. For the case of
the cylinder in compression, the value of P/Sy is theoretically equal to 1.
By fitting a function to their FEM results, Jackson and Green [3] provide the following
formula:
?
?
?
?
?
?
?
?
???
?
???
?
????????=
? 7.0
82.0exp184.2 RaSP
y
(10)
+=
=
0
84.2
R
a
S
P
y
1
1
=
?
R
a
S
P
y
10
184.2
<<
>>
R
a
S
P
y
16
Note that in Eq. (10) P is used instead of H as the symbol for the average pressure during
elasto-plastic contact. This is used here to emphasize that the P predicted by Eq. (10)
varies with the deformation of the sphere, whereas the traditional value of H does not.
The FEM based model also provides predictions of the contact force during elastic-plastic
contact as,
ccyccc CS
P
F
F
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
?
?
???
?
???
???+
???
?
???
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
?
?
???
?
???
??= 9
5
2
3
12
5
25
1exp14
4
1exp (11)
where,
( )?736.0exp295.1=C
The ratio of pressure to the yield strength in Eq. (11) is calculated using Eq. (10).
Quicksall et.al [21] also verified these results for a wider range of material properties by
varying E ( Young?s modulus) and Sy (Yield Strength).
In another study, Jackson and Green [4] related the contact radius of the sphere to
the interference ?, of the flat rigid surface into the sphere. They worked on predictions of
the contact radius during elastic and elastic- plastic contact. Jackson and Green [4]
predicted the ratio of contact radius to the original radius as
2/
9.1
B
cRR
a
???
?
???
?=
?
??
(12)
where,
17
???
?
???
?
???
?
???
?=
'23exp14.0 E
SB y
Due to the limited range of FEM cases used to find Eq. (12), these models are only valid
for normalized contact radii of 0 0.41
Jackson and Green [3] provided the FEM data used to build the model in their
work. Modifying Eq. (12) involves using both the JG data and the FEM results from
current study. This is accomplished by fitting an equation to the FEM results of Jackson
and Green [3] and the new FEM data acquired in this work that tracks surface
deformation farther in the fully plastic regime. The equation provides a relationship
between the contact radius and penetration. For the deformable base case, the resulting fit
equations differ from all the FEM results by an average of 5% and are given as,
???
?
???
??
???
?
???
?+?
?
??
?
?=?
?
??
?
?
ccJGnew
AARaRa ???? 2
2
1 (24)
where
JGR
a ?
?
??
?
? is given by Eq. (5) and
148.3
1 0826.0 ???
?
???
??=
E
SA y
;
545.1
2 3805.0 ???
?
???
??=
E
SA y
49
The normalized contact radius (a/R) predicted by Eq. (24) is compared Jackson
and Green [4] and FEM results in Fig (5.9) and (5.10). The results have been divided into
2 figures for small and large deformations. Small deformations are defined till a/R =0.41
and large deformations extend till a/R=1. Note that the material properties of both sets
viz. JG model and current study, of data are slightly different (see Table 5.2).
Table 5.2: Material properties used in (a) Jackson and Green [3] and (b) the current FEM
analysis.
Material Yield strength, (Sy) GPa Equivalent modulus of elasticity (E') Gpa
1a 0.9115 228.2
1b 1 228.2
2a 0.5608 228.2
2b 0.5 228.2
3a 0.21 228.2
3b 0.2 228.2
Mayuram and Megalingam [23] also built a elasto plastic spherical contact model,
essentially studying the same problem. They provide a set of equations to predict the
contact area and contact force during elastic, elasto-plastic and fully plastic deformations.
These equations are valid till ?/?c = ? and are given previously in this work. In order to
compare the contact area given by the MM model, the equations for contact area are
converted to give contact radius by substituting critical area equation as shown here for
one of the penetration ranges defined by Eq. (14) resulting in,
50
2R
A
R
a c
pi= (25)
As shown in Figs. (5.9) and (5.10), the JG model (Eq. 12) and MM model (Eq.
25) model compare well for very small values of a/R but as the contact radius increases,
the predictions progressively depart from the FEM results.
It appears that Eq. (25) is limited to smaller deformations (see Fig 5.9) and as the
deformations get larger, the model does not seem to agree with the FEM results (see
Fig.5.10). The current study (Eq. 24) does not compare as well as the JG model Eq. (12)
to the FEM data at lower interferences (see Fig. (5.9), but for large interferences Eq. (24)
compares much better with the current FEM results. The percentage difference between
the current and JG model for small interferences is a maximum of 5.18% and a minimum
of 1%. The trend seems to be the same for all the three material properties used for the
study.
51
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 1 10 100 1000
?/?c
a/R
FEM mtl. 1a
FEM mtl. 2a
FEM mtl. 3a
Eq. (12)
Eq. (24)
Eq. (25)
Mtl. 3a
Mtl.1a
Mtl.2a
Figure 5.9: Comparison of predictions of Eq. (12), (24) and (25) with FEM results for the
deformable base case for small values of penetration.
52
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
100 1000 10000 100000?/?
c
a/R
FEM mtl. 1b
FEM mtl. 2b
FEM mtl. 3b
Eq.(2)
Eq.(12)
Eq. (24)
Eq. (25)
Mtl. 3b
Mtl. 1b
Mtl. 2b
Figure 5.10: Comparison of predictions of Eq. (12), (24) and (25) with FEM results for
the deformable base case for large values of penetration.
53
For the rigid base case (case 2), equations describing the ratio between the contact
radius and spherical radius are also fit to the FEM data of Jackson and Green [3] and the
current work using the same form given in Eq. (24). The resulting fit equations for the
rigid base case differ from the FEM data by an average of 2.5% and are given as
???
?
???
??
???
?
???
???
?
??
?
?=?
?
??
?
?
ccJGnew
AARaRa ???? 4
2
3 (26)
where,
605.5
1583933 ??
?
?
???
?=
E
SA y
;
8939.0
4 0034.0 ???
?
???
?=
E
SA y
Figs. (5.11) and (5.12) show the predictions of the current model for the rigid base
case (Eq. 26) compared with the FEM, JG and MM models. Again, for clarity the results
are presented for two different penetration levels in Figs. 5.11 (for small interferences)
and Fig. 5.12 (for large interferences). The results show trends similar to the deformable
base case, that, for small deformations, the Jackson and Green [4] model (Eq. (12))
compares the best with the FEM results. This is expected since the JG model is meant for
only elasto plastic deformations. But the most interesting observation is that the model
almost replicates the FEM data for small deformations. The MM model [24] results
deviate significantly from the FEM at fairly small values of interference. From Fig.
(5.12), it is very evident that the current model (Eq. 26) predictions are the most accurate
when compared to the FEM results when large deformations occur.
54
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.1 1 10 100 1000
?/?c
a/R
FEM mtl. 1a
FEM mtl. 2a
FEM mtl. 3a
Eq. (12)
Eq. (25)
Eq. (26)
Mtl.1a
Mtl.3a
Mtl.2a
Figure 5.11: Comparison of predictions of Eq. (12), (25) and (26) with FEM results for
the rigid base case for small values of penetration.
55
0
0.2
0.4
0.6
0.8
1
1.2
1.4
100 1000 10000 100000?/?c
a/R
FEM mtl. 1b
FEM mtl. 2b
FEM mtl. 3b
Eq. (2)
Eq. (12)
Eq. (25)
Eq.(26)
Mtl.1b
Mtl.3bMtl.2b
Figure 5.12: Comparison of predictions of Eq. (12), (25) and (26) with FEM results for
the rigid base case for large values of penetration.
56
5.1.c Contact force
Jackson and Green [3] provide an equation for the contact force during elasto
plastic deformations of a sphere valid up to a/R=0.41 (see Eq. (11)). The current work
aims to provide an extended model that is capable of producing accurate predictions of
the contact force by modifying Eq. (11) and extending it into the fully plastic deformation
range. As deformations get larger, the first term in Eq. (11) approaches zero and the
second term involving the contact pressure becomes dominant in predicting the contact
force. The current study proposes that the contact pressure should be multiplied by the
contact area for accurate predictions of the contact force, resulting in the following
modified equation:
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
?
?
???
?
???
????
?
??
?
?+
???
?
???
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
?
?
?
???
?
???
??= 9
52
22
3
12
5
25
1exp1
4
1exp
cnewcccc R
aR
F
P
F
F
?
?pi
?
?
?
? (27)
Eq. (11) does not contain a term which can address the increasing contact area during
plastic deformations. Also, the value of P in Eq. (11) is calculated using Eq. (10) which
has already been proven to give inaccurate predictions for fully plastic contact pressure
with large deformations.
The new equation proposed in Eq. (27) uses the new contact area calculated from
Eq. (24) and (26) .The contact pressure P, is calculated by modifying Eq. (13) for contact
pressure predictions.
57
First the predictions of Eq. (27) for case 1 (the deformable base) have been shown
in Figs. (5.13) and Fig. (5.14). Fig. (5.13) shows the comparison in the range of the FEM
data provided by Jackson and Green (small deformations up to a/R =0.41) and Fig. (5.14)
shows the comparison in the range of the new FEM data (large deformations up to
a/R =1) from the current work. Comparisons of the current model with FEM results in
Figs (5.13) and (5.14) reveal that the current model for contact force compares very well
for both small and large interferences. For small interferences the current model
predictions and Eq. (11) are almost indiscernible and so the predictions of Eq. (11) are
not shown. As the deformations get larger, as shown in Fig. (5.14), the differences
between the FEM results, the current model, Eq. (11), and Eq. (15) become very
profound. In fact, the differences between the FEM results, the current model and the
MM model (Eq. (15)) appears to sometimes be several orders of magnitude. Overall the
current model performs well and better than the other models at predicting the contact
force over the considered range of interferences. It is also interesting to note that the new
equations capture some interesting trends in the force-interference curves shown in Fig.
5.14 for the rigid base case. There is a slight?s? shape to the FEM results that the new
model equations also successfully capture.
* The current model is a combination of Equations depending upon the case considered.
For the Deformable base case, the current model is a combination of Eq. (13), (24) and
(27). And for the rigid base case it is a combination of Eq. (13), (26) and (27).
58
1
10
100
1000
10000
1 10 100 1000?/?
c
F/F
c
FEM mtl. 1a
FEM mtl. 2a
FEM mtl. 3a
Eq. (15)
Eq. (27)
Figure 5.13: Comparison of predictions of contact force according to Eq. (15) and (27)
with FEM results for the deformable base case with small loads
59
100
1000
10000
100000
1000000
100 1000 10000 100000?/?
c
F/F
c
FEM mtl. 1b
FEM mtl. 2b
FEM mtl. 3b
Eq. (1)
Eq. (11)
Eq. (15)
Eq. (27) Mtl. 2b
Mtl. 3b
Mtl. 1b
Figure 5.14: Comparison of predictions of contact force according to Eq. (11), (15) and
(27) with FEM results for the deformable base case with heavy loads
60
Next, the current model of contact force for the rigid base case using Eqns. (13,
26 and 27) is compared to the FEM data and the previous models given by JG (Eq. (11))
and MM (Eq. (15)) as shown in Figs. (5.15) and (5.16). It is expected that the JG model
(Eq. (11)) works better for the rigid base because they have studied asperity contact
which is similar to the rigid base case. As seen from the comparison with the FEM results
in Fig. (5.15) and (5.16), this is especially true when deformations get larger (a/R
approaches 1). The observed trends for deformable and rigid base case in figs. (5.15) and
(5.16) are significantly different.
The plots also show that the model by Megalingam and Mayuram [23] (Eq. (15)),
significantly underpredicts the FEM results for large deformations. As expected, the new
equations based on barreling and volume conservation agree well with the FEM results.
61
1
10
100
1000
10000
1 10 100 1000?/?
c
F/F
c
FEM mtl. 1a
FEM mtl. 2a
FEM mtl. 3a
Eq. (15)
Eq. (27)
Figure 5.15: Comparison of predictions of contact force according to Eq. (15) and (27)
with FEM results for the rigid base case with small loads
62
100
1000
10000
100000
1000000
100 1000 10000 100000?/?
c
F/F
c
FEM mtl. 1b
FEM mtl. 2b
FEM mtl. 3b
Eq. (1)
Eq. (11)
Eq. (15)
Eq.(27)
Mtl. 2bMtl.3b
Mtl.
Figure 5.16: Comparison of predictions of contact force according to Eq. (11), (15) and
(27) with FEM results for the rigid base case with large loads
63
5.1.d Comparisons with Existing Experimental Measurements
In order to validate the current model predictions for contact pressure and contact
radius, the results were compared with experimental data measured by Chaudhri et al. [2].
They reported experimental results for the compression of metal spheres of different
material properties (phosphor bronze, aluminum and brass) between two smooth parallel
platens. The deforming geometry of the spheres in this experiment can be correlated to
the deformable base case (case 1) in the current work.. The spheres used in the
experiment were brass, aluminum and phosphor bronze all with diameters of 3.175mm.
The current work compares the new model results with the results for the brass and
phosphor bronze spheres given by Chaudhri [2] since the resulting a/R ratios for these
tests are in the range of the current study compared to aluminum. The phosphor bronze
spheres were work hardened in an attempt to cause there behavior to be like an elastic-
perfectly plastic material when compressed under heavy loads. This was done to allow
for comparison with existing models which mostly assume the material to behave elastic-
perfectly plastically. The current model has also been modeled intially as elastic-perfectly
plastic in the FEM simulations.
Measurements of Vickers hardness were provided by [2] before and after
compression. A Leitz microhardness testing machine with an accuracy of ? 4% and a
load of 50 gms-force (0.49 N) was used. This information has not been mentioned in their
work, but a thorough literature survey of the equipment used in the experiments was done
to find the accuracy of the results.
According to the hardness measurements following data has been given by
Chaudhri [2].
64
Table 5.3: Material properties and Microhardness measurements as given by
Chaudhri et al. [2].
Hardness (GPa)
Material
Poisson?s
Ratio, ?
Elastic modulus,
E (GPa)
Before
compression
After
compression
Phosphor bronze 0.35 115 2.72 ? 0.06 2.68 ? 0.06
Brass 0.37 120 1.8 ? 0.08 2.22 ? 0.08
Using these values of hardness and standard values of elastic modulus for brass
and phosphor bronze, a comparison is made between the predictions of the current model
for the deformable base case and the experimental results [2]. In dimensional form the
current model for the deformable base case is given as,
y
new
SRRRaP ??
?
?
???
?
?
?
?
?
?
?
???
?
???
? ?
?
??
?
???=
1
cos192.084.2 pi (28)
where, R1 is calculated using Eq. (10), (a/R)new is calculated from Eq. (24), and R is the
original radius of the sphere.
The value of the yield strength is not explicitly provided by Chaudhri et al. [2].
Instead it has to be calculated using the Vicker?s hardness measurements given in table
5.3. Vicker?s hardness measurements conducted by [17], [31] and [33] revealed that the
value of the constant c, is between 2.8 to 3.3. These results have been found for
frictionless compression experiments (similar to the current study). The current work
65
finds a value which H/Sy = 3.15 provides a trend closest to the experimental results given
by Chaudhri [2] for phosphor bronze and brass. The results have been presented in Figs.
(5.17) and (5.18). The data compares the predictions of the current model for three
different values of the constant ,c (2.8, 3.15 and 3.3).
The experimental data given by Chaudhri et al. [2] was extracted using
DataThief?. The predictions of the current model (Eq. 28) are compared with this
extracted data and are shown in Figs. (5.17) and (5.18). The trends of the experiments
and model are qualitatively and quantitatively very similar. The average error between
the model and the measurements is about 9% for the brass results and 7% for the
phosphor bronze. Considering that in reality there is some hardening and friction
occurring in the tests that is not considered by the model, this shows surprisingly good
agreement between them.
The R-squared value to determine the co-efficient of correlation between the
experimental data and the current model has also been calculated. For the case where c is
3.15, the R-squared value for the correlation was found to be 0.9851 and 0.9707 for
phosphor bronze and brass respectively. The R-squared value shows how closely the
trends of the results being compared are related to each other. The maximum R-squared
value is unity. As mentioned previously, the phosphor bronze spheres were work
hardened to achieve newly elastic perfectly plastic behavior. For Phosphor bronze, the
percentage error between the experimental the current model is observed to be lower and
the R-squared number is higher compared to brass. This is because the current model is
simulated as elastic perfectly plastic sphere and phosphor bronze shows a behavior
closest to this (see table. 5.3).
66
This suggests that the new model presented in the work can be used effectively to
predict the behavior of heavily deformed spheres, especially when a material has little
strain hardening. Strain hardening and friction which add complexity to the problem have
not been considered in this section but has been discussed later. The study by Chaudhri
[2] also mentions the possibility of a barreling mechanism for predicting the deforming
geometry of spheres during compression. The current study has studied this possibility in
detail and confirms these possibilities in the previous sections.
Figure 5.17: Comparison of experimental and simulation based model for phosphor
bronze material
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
a/R
P
(G
Pa
)
Experimental data
H/Sy = 2.84
H/Sy = 3.15
H/Sy = 3.3
67
Figure 5.18: Comparison of experimental and simulation based model for brass material
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
a/R
P
(G
Pa
)
Experimental data
H/Sy = 2.84
H/Sy = 3.15
H/Sy = 3.3
68
5.2 Effects of strain hardening
Strain hardening also known as work hardening occurs when metals undergo
plastic deformation. The material essentially gains resistance to permanent deformations.
This happens because the material gets saturated with dislocations which prevent
formation of any more dislocations. The current study has to this point not considered
strain hardening in the sphere under compression. However in reality there will be some
strain hardening in almost all metallic materials.
Therefore an attempt was made to model the sphere under compression while
including strain hardening effects. The current work compares its simulation results with
the maps of microhardness distribution provided by Noyan [1] at various levels of
penetration and draws some interesting observations and conclusions. Noyan [1]
conducted experiments by compressing spheres and measuring the hardness across the
sphere and mapped these microhardness values throughout the depth of the sphere for
4%, 11%, 20% and 58% compressive macroscopic strain. Compressive strains are the
difference between the initial (undeformed) and final compressed height of the sphere.
The sphere is divided into zones depending on the hardness measured. This gives an
exact idea of the birth and progression of stress distribution in the sphere. He also
concluded that the area of contact and the area at the centre of the sphere are independent
of material if sphere and the size.
In the finite element analysis, strain hardening is introduced into the spheres by
varying the tangent modulus of the material. The current work considers tangent modulus
of 1%, 2% and 4% hardening of the material. Discussions with other scholars in this field
69
who have conducted experiments to study strain hardening revealed that hardening
reaches values upto 4% during compression tests. The compressive strain considered by
Noyan [1] reach a maximum value of 58%. The current study considers compressive
strains upto 50%. Hence, the Von Mises stress distribution for the various levels of strain
hardening in the current study can be compared with almost all of the microhardness
plots given by Noyan [1]. The FEM predictions of contact pressure at 0%, 1%, 2% and
4% tangent moduli strain hardening are given in Fig. (5.19).
Figure 5.19: Variation of contact pressure with increasing contact radius
0
1
2
3
4
5
6
7
8
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
a/R
P/
Sy
0% Strain Hardening
1% Strain Hardening
2% Strain Hardening
4% Strain Hardening
70
The inclusion of hardening appears to neglect the geometric effects on hardness during
spherical flattening (see [7],[2]). Essentially, hardening counteracts the trend of Eq. (13)
causing the H/Sy to increase instead of decreasing as a/R increases. This may be why this
phenomenon had not been experimentally recognized until Chaudhri et al. [2].
In order to study the mapping of the stress distribution inside the sphere, the Von
Mises stress distribution at each load step is mapped and shown in Figs. (5.20) and (5.21)
and (5.22). The figures show Von Mises stresses for one material with three different
levels of strain hardening (4%, 2% and 0%). The trends suggest that the maximum Von
Mises stress zone for 4% and 2% strain hardening cases in Figs. (5.20) and (5.21)
migrates from just below the contact surface to the center of the sphere. These results
have been compared with the experimental work presented by Noyan [1]. The
comparisons reveal similar trends in both the studies. The zone indicating maximum Von
Mises stress levels represented by the red zone in Figs. (5.20) and (5.21) can be compared
to the ?C? zone in the experimental work by Noyan [1]. In this experimental work by
Noyan [1], the sphere is divide into zones based on microhardness measurements and ?C?
zone is the hardest zone. As the compressive strain increases the ?C? zone increases in
size and migrates to the center of the sphere similar to the current observations in Figs.
(5.20) and (5.21).
71
Figure. 5.20: Von Mises stress distribution for 4% strain hardening in the sphere.
72
Figure 5.21: Von Mises stress distribution for 2% strain hardening in the sphere.
73
Figure. 5.22: Von Mises stress distribution for 0% strain hardening in the sphere.
74
5.3 The Effect of Friction
When a sphere is in contact with a flat rigid surface, in reality there will be some
friction across the contact area. The study has not considered the effects of friction on the
contact pressure and area until now. In this section an attempt is made to understand the
effects of friction on the contact pressure and area. Friction is introduced in the contact
area in the finite element modeling code by varying the coefficient of friction across the
contact area.
Figure 5.23: Variation of contact force with increasing contact radius
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
a/R
F/F
c
Coeff friction 0
Coeff friction 0.1
Coeff friction 0.2
Coeff friction 0.3
Coeff friction 1
75
The results in Fig. (5.23) show that for increasing coefficients of friction across the
contact area, the variation in contact force is not much for light loading however as the
deformations get larger in Fig. (5.22), friction plays an important role in predicting the
contact force. Comparisons with strain hardening results suggest that friction does not
have a large effect relative to strain hardening. Also, increasing coefficients of friction
affects the predictions in Fig. (5.23) only after a/R = 0.5. This suggests that friction does
not play a major role until a/R becomes larger than 0.5.
It is important to understand whether strain hardening or friction has a greater
effect on the contact force predictions. Figs. (5.24) and (5.25) show the effect of strain
hardening and friction on the rigid base case predictions. It can be seen from the
comparison that strain hardening starts affecting the predictions from small values
contact radius (a/R=0.2) and friction plays a minor role compared to strain hardening and
only affects the predictions for values of contact radius above 0.5. Hence friction plays a
secondary role to strain hardening in predictions of contact force.
76
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
3.0E+04
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
a/R
F/F
c
0% strain hrd
4% strain hrd
Figure 5.24: Effect of increasing strain hardening across the contact area on predictions
of contact force
77
0.E+00
1.E+03
2.E+03
3.E+03
4.E+03
5.E+03
6.E+03
7.E+03
8.E+03
9.E+03
1.E+04
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
a/R
F/F
c
Coeff fric 0
Coeff fric 1
Figure 5.25: Effect of increasing friction across the contact area on predictions of contact
force
78
CHAPTER 6
CONCLUSIONS
This work presents an FEM based model to predict the behavior of spherical
surfaces being heavily compressed by flat rigid surfaces. In the initial part of the study,
the material has been modeled as elastic-perfectly plastic to exclude strain hardening
effects and friction across the contact area. Later the effects of strain hardening have also
been studied. The model works well for both elasto-plastic deformations and for fully
plastic deformations. Therefore a new model has been provided which can model a
significantly larger range of deformation for spherical contacts. The model is based on
the results of a FEM simulation of heavily loaded spherical contacts. The equations have
been formulated using volume conservation theory and barreling theory for the
compression of cylinders. Probably, the most important finding is that the effect of
bulging or barreling must be considered in calculating a/R. The results show that when
deformations are small, that the Jackson Green model may actually provide slightly more
accurate results, but as deformations get larger the current model produces more accurate
results in comparison to the FEM results.
The results of the finite element based model have also been verified with
experimental data for different materials like brass and phosphor bronze provided by
Chaudhri et al. [2]. The FEM based model compares surprisingly well with these
79
previous results, and without the use of any additional fitting parameters. There is some
difference in the results because there is some hardening and friction occurring in the
experimental measurements. The current study confirms the suggestions by Chaudhri [2]
that barreling of cylinders has similarities in behavior to large deformations in spherical
contact. Also, through various literature sources referred during current research, the
constant in the relationship between contact pressure and yield strength (see Eq. (29))
seems have values ranging from 2.84 to 3.3. The current study found out that for
phosphor bronze and brass the value that best replicated the experimental results is 3.15.
This work also studies the effect of strain hardening in spherical contact with
severe deformation. Results of Von Mises stress distribution have been compared with
Noyan [1] and similar patterns of hardening are found. Also, friction will play a
secondary role in predictions of contact area and pressure. Some preliminary results have
been shown to understand the variation of contact force with increasing contact area for
various values of coefficient of friction. In the future the authors would like to further
investigate these additional effects.
80
CHAPTER 7
RECOMMENDATIONS FOR FUTURE WORK
The current study presents the results of a FEM model and proposes a closed form
equation for predicting contact pressure and area for spheres compressed under heavy
loading. The results have been verified with real world experimental data and are in good
agreement. The comparisons reveal that the results are in better agreement with phosphor
bronze material than brass since this material (phosphor bronze) was work hardened
before compression to have elastic perfectly plastic materials. In contrast, the brass
spheres were not work hardened. In the simulations too, the spheres were modeled as
elastic perfectly plastic. Hence the current model is more applicable to this case than
actual spherical deformation in real applications.
An attempt was made to understand the effects of strain hardening and friction
during compression of spheres. Preliminary results have been shown in the sections
above. More work is needed to fully describe these effects. Fig. (5.19) reveals that
friction will play a crucial role in predictions of contact force and pressure as the contact
radius increases.
81
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83
APPENDIX
Finite element analysis Code
1. No Strain hardening considered ?
/PREP7
CYL4,0,0,1,0,0,90
!*
ET,1,PLANE82
!*
KEYOPT,1,3,1
KEYOPT,1,5,0
KEYOPT,1,6,0
!*
!*
MPTEMP,,,,,,,,
MPTEMP,1,0
MPDATA,EX,1,,200E9
MPDATA,PRXY,1,,0.3
TBDE,BISO,1,,,
TB,BISO,1,1,2,
TBTEMP,0
TBDATA,,,2E9,,,,
APLOT
-------------------------------------------MESHING---------------------------------------------------
SMRT,1
MSHAPE,0,2D
MSHKEY,0
!*
CM,_Y,AREA
ASEL, , , , 1
CM,_Y1,AREA
CHKMSH,'AREA'
CMSEL,S,_Y
!*
AMESH,_Y1
84
!*
CMDELE,_Y
CMDELE,_Y1
CMDELE,_Y2
!*
K,5,2,1,0,
LSTR, 2, 5
!*
/REPLOT
!*
/COM,
------------------------------CONTACT PAIR CREATION - START ---------------------------
CM,_NODECM,NODE
CM,_ELEMCM,ELEM
CM,_KPCM,KP
CM,_LINECM,LINE
CM,_AREACM,AREA
CM,_VOLUCM,VOLU
/GSAV,CWZ,GSAV,,TEMP
MP,MU,1,0
MAT,1
MP,EMIS,1,7.88860905221E-031
R,3
REAL,3
ET,2,169
ET,3,172
R,3,,,1.0,0.1,0,
RMORE,,,1.0E20,0.0,1.0,
RMORE,0.0,0,1.0,,1.0,0.5
RMORE,0,1.0,1.0,0.0,,1.0
RMORE,10.0
KEYOPT,3,3,0
KEYOPT,3,4,2
KEYOPT,3,5,0
KEYOPT,3,7,0
KEYOPT,3,8,0
KEYOPT,3,9,0
KEYOPT,3,10,1
KEYOPT,3,11,0
KEYOPT,3,12,2
KEYOPT,3,2,3
KEYOPT,2,2,0
KEYOPT,2,3,0
85
--------------------------------GENERATE THE TARGET SURFACE ------------------------
LSEL,S,,,4
CM,_TARGET,LINE
TYPE,2
LATT,-1,3,2,-1
TYPE,2
LMESH,ALL
-----------------------------GENERATE THE CONTACT SURFACE -------------------------
LSEL,S,,,1
CM,_CONTACT,LINE
TYPE,3
NSLL,S,1
ESLN,S,0
ESURF
*SET,_REALID,3
ALLSEL
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
LSEL,S,REAL,,3
/PSYMB,ESYS,1
/PNUM,TYPE,1
/NUM,1
EPLOT
ESEL,ALL
ESEL,S,TYPE,,2
ESEL,A,TYPE,,3
ESEL,R,REAL,,3
LSEL,S,REAL,,3
CMSEL,A,_NODECM
CMDEL,_NODECM
CMSEL,A,_ELEMCM
CMDEL,_ELEMCM
CMSEL,S,_KPCM
CMDEL,_KPCM
CMSEL,S,_LINECM
CMDEL,_LINECM
CMSEL,S,_AREACM
CMDEL,_AREACM
CMSEL,S,_VOLUCM
CMDEL,_VOLUCM
86
/GRES,CWZ,GSAV
CMDEL,_TARGET
CMDEL,_CONTACT
/COM,
---------------------------------CONTACT PAIR CREATION - END --------------------------
/MREP,EPLOT
/REPLOT
FINISH
/SOL
FINISH
/PREP7
FINISH
/SOL
FINISH
/PREP7
FINISH
/SOL
FINISH
/PREP7
FINISH
/SOL
-------------------------------------BOUNDARY CONDITIONS -------------------------------
FLST,2,79,1,ORDE,4
FITEM,2,1
FITEM,2,3
FITEM,2,204
FITEM,2,-280
!*
/GO
D,P51X, , , , , ,UY, , , , ,
FLST,2,79,1,ORDE,4
FITEM,2,2
FITEM,2,-3
FITEM,2,127
FITEM,2,-203
!*
/GO
D,P51X, ,0, , , ,UX, , , , ,
FLST,2,1,4,ORDE,1
FITEM,2,4
!*
/GO
DL,P51X, ,UY,-0.5
87
--------------------------------------ANALYSIS TYPE----------------------------------------------
ANTYPE,0
NLGEOM,1
DELTIM,0.1,0,0
AUTOTS,0
TIME,100
OUTRES,ALL,-10
/STATUS,SOLU
88
Figure A.1: Undeformed sphere with FEA mesh