ABSTRACT This work presents a three dimensional (3D) finite element analysis (FEA) of an elastic-plastic hemispherical contact model for two hemispherical bodies sliding across each other with various preset vertical interferences. The boundary conditions, model simplifications, and the normalization scheme are presented. Sample results from this FEA investigation are compared to a semi-analytical solution to validate the methodology

Fellow ASME) Itzhak Green

Member ASME John Moody

CiteSeerX

 
  
 
id1034886671 pdfMachine by Broadgun Software  - a great PDF writer!  - a great PDF creator! - http://www.pdfmachine.com  http://www.broadgun.com 
Downloaded From: httpProceedings of the STLE/ASME International Joint Tribology Conference 
IJTC2007 
October 22-24, 2007, San Diego, California, USA 
          IJTC2007-44301 
AN INVESTIGATION OF THREE DIMENSIONAL ELASTIC-PLASTIC 
HEMISPHERICAL SLIDING CONTACT, PART I:  MODELING AND VALIDAITON 
 
John Moody (Member ASME) and Itzhak Green (Fellow ASME) 
Mechanical Engineering, Georgia Institute of Technology 
Atlanta, GA 30332-0405, USA 
itzhak.green@me.gatech.edu 
Proceedings of ASME/STLE International Joint Tribology Conference 
IJTC2007 
October 22-24, 2007, San Diego, California USA 
IJTC2007-44301 
ABSTRACT 
This work presents a three dimensional (3D) finite element 
analysis (FEA) of an elastic-plastic hemispherical contact 
model for two hemispherical bodies sliding across each other 
with various preset vertical interferences.  The boundary 
conditions, model simplifications, and the normalization 
scheme are presented.  Sample results from this FEA 
investigation are compared to a semi-analytical solution to 
validate the methodology.   
INTRODUCTION 
An investigation of elastic-plastic sliding asperity contact 
using a three dimensional (3D) finite element analysis (FEA) is 
presented in this part. Sliding contact is an important 
phenomenon in both the macro and micro scales. In the macro 
scale, it is important to consider friction, wear, and residual 
deformation that result from contact situations such as in rolling 
element bearings.  In the micro scale, it is known that nominally 
smooth surfaces do indeed have undulations in their surface 
profile and the real area of contact is just a small percentage of 
the nominal area of contact.  These high points, or asperities, 
are known to deform plastically during sliding.  Thus, it is 
important to know how the deformed geometry, residual 
stresses, and surface condition affect the sliding process. This 
information for a pair of asperities provides the kernel of the 
solution for any rough surface described statistically. The model 
presented here has been normalized in order to apply the results 
to both macro and micro scale geometries.   
Though work has been done in the area of surface contact, 
in most cases either simplifying assumptions have ignored 
important phenomena or less than satisfactory results have been 
produced.  There have been many works, mainly based on 
Green [1, 2], which analyzed friction and adhesion of triangular 
shaped contact interfaces, that have analyzed fully plastic 
contact interfaces.  In reality though, the contact junctions 
should realistically be modeled as spherical in shape.   
Historically, work in the area of hemispherical contact has 
been done in a sphere-on-flat configuration.  Faulkner and 
Arnell [3] present the first work that models sphere-on-sphere s://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Usecontact using an FEA approach.  Very few useful results are 
presented in this work and the method resulted in extremely 
long execution times (over 960 hours).  The objective of this 
work is to present a generalized modeling method that can be 
applied to many material combinations.   
 
MODELING METHOD 
A schematic representation of the sliding process is shown 
in Figure 1, lying in the xy plane. Axis x is along the direction 
of sliding, axis y is vertical, and axis z is normal to the said 
plane xy. In this analysis a displacement, Äx is applied to the top 
sphere while the bottom sphere is held stationary.  The 
horizontal displacement imposed to complete sliding increases 
with increasing the vertical interference, ù.  As the vertical 
interference increases, more plastic deformation occurs and 
residual stresses and strains are present upon disengagement.  In 
the model, the top sphere is initially placed just in contact with 
the lower sphere and then slides across the bottom sphere until 
the spheres are no longer in contact.  
 
 
 
Figure 1:  A schematic of the sliding process   
This analysis is done using ABAQUS, a commercial FEA 
software package using linear brick (8-node) elements.  In order 
to take advantage of the symmetry of the problem, each sphere 
is cut in half along the vertical xy plane and no displacement is 
allowed normal to this cut plane.  This can be thought of as a 
roller boundary condition.  Also, an assumption is made, and 
later confirmed, that under the interferences considered here 
there is little to no stress or deformation in areas far from the 
contact region (half-space assumption).  This assumption is 
Top sphere: 
Final position 
 
Top sphere: 
Initial position 
 
  
 
Äx 
ù 
Sliding direction 1 Copyright © 2007 by ASME 
: http://www.asme.org/about-asme/terms-of-use
Downlreasonable if one considers the fact that the contact half-width 
is much smaller than the radius of the sphere and as such, the 
stress distribution near the contact region is unaffected by the 
conditions far away from it.  To take advantage of this each 
sphere is cut in half in the horizontal plane and once again a 
roller boundary condition is imposed on the top surface of the 
top sphere while vertical displacement is completely 
constrained on the bottom surface of the bottom sphere. The 
end result is a quarter-sphere model for each spherical body.   
This analysis considers steel-on-steel and aluminum-on-
copper contact.  The material is modeled as elastic-perfectly 
plastic, but in order to help convergence, a 2% strain hardening 
based on the Youngs modulus was used on the higher 
interference cases.  This small amount of strain hardening has 
been shown to not significantly affect the results yet drastically 
improves upon execution time [4].   
In the elastic domain and up to the onset of plasticity, the 
Hertzian solution [5] is used to obtain critical values of load, 
contact half-width, and strain energy [6].  As explained by 
Jackson et al [7], hardness is not implemented as a unique 
material property as it varies with the deformation itself as well 
as with other material properties such as yield strength, 
Poissons ratio, and the elastic modulus. Instead, the critical 
vertical interference, ùc, as derived by Green [6] for 
hemispherical contact, is employed. This quantity is derived by 
using the distortion energy yield criterion at the site of 
maximum von Mises stress by comparing the stress value with 
the yield strength, Sy. The critical values of force, Pc, contact 
area, Ac, and interference, ùc, are: 
 
 
 
2 233 2
22
( ) R
  ;   ; 
6 ' 2 '2 '
yy y
c c c
CS RCS CS
P A R
E EE


 
    
 
                  (1) 
 2 3 2
2 2
((1 2 2 (1 ) 2( )(1 ) )1
2 (1 )
e
o
ArcCot
p
      

     


        (2) 
Where: 
 
1 2
1 1 1
R R R
                                                                           (3)  
2 2
1 2
1 2
1 11
'E E E
  
                                                               (4) 
0.38167 0.33136c                               (5) 
21.30075 0.87825 0.54373C                                 (6) 
1 2min( , )y y yCS CS CS                                                       (7) 
The value of C is obtained from elasticity considerations, 
and the critical parameters are obtained at the point of yielding 
onset. The maximum elastic energy that can possibly be stored 
(up to the point of yielding onset) is used to normalize the net 
energy loss due to plastic deformation after sliding, and is given 
by Green [6] as: 
 
5 3
460 '
y
c
CS R
U
E

                                                                   (8)  
oaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use 
 
Table 1 presents the material properties used in this 
analysis for steel-on-steel and aluminum-on-copper sliding 
contact.  The critical values are calculated for both contact pairs 
and presented in Table 2.  Since all the quantities are 
subsequently being normalized by the critical parameters in Eq. 
(1), the ensuing results apply for any geometry scale (and for as 
long as continuum prevails); therefore, the radii for the 
asperities in the FE model are subjectively (and conveniently) 
chosen to be R1 = R2 = 1 m. It should be noted that the critical 
parameters are used solely for the purpose of normalization of 
the results, whether frictional or frictionless sliding. 
 
Table 1: Material properties for the two spheres 
Property Steel Aluminum Copper 
E 200 GPa 68.0 GPa 130 GPa 
Sy 911.5 MPa 310 MPa 331 MPa 
õ 0.32 0.326 0.33 
 
Table 2: Critical values of parameters at the onset of plasticity 
for sliding between two spherical asperities 
1. Aluminum is yielding first  
REPRESENTATIVE RESULTS AND VALIDATION 
In order to validate the FEA method, the results are 
compared to a semi-analytical numerical method (SAM).  For 
more information on the specifics of the methodology used here 
see Boucly et al [8].  Briefly, this semi-analytical method uses 
contact pressure on the surface that can be thought of as the 
summation of concentrated normal loads over the area of 
contact.  Each of these concentrated loads has a corresponding 
influence on the displacement throughout the body. This 
influence is quantified using influence coefficients, which are 
actually the discretized form of Green's functions.  The semi-
analytical method takes advantage of this by using the 
superposition principle to sum at each location in the region of 
interest the displacements due to the contact pressure.  Once 
this information is gathered the stresses, strains, and 
deformations can be calculated based on the material properties 
from the compatibility and equilibrium relations.  An iterative 
process is used to incorporate the residual deformations present 
from a previous load step.   
Figures 2 and 3 present a comparison of the horizontal and 
vertical reaction forces for the different vertical interferences 
for steel-on-steel frictionless sliding contact, respectively.  As 
shown in the figures, the results are nearly identical for the 
smaller interference cases.  As shown in Figure 2, with 
Parameter Steel-on-Steel Al-on-Cu1 
CSy 1.493 GPa 509.9 MPa 
ùc 0.2214 mm 0.1261 mm 
Pc 346.1 kN 67.32 kN 
Ac 347.8  mm
2 198 mm2 
Uc 30.65 J 3.395 J 2 Copyright © 2007 by ASME 
: http://www.asme.org/about-asme/terms-of-use
 
 
Downincreasing preset interference, the semi-analytical results 
diverge from the FEA results once the spheres have passed the 
point of vertical alignment.  The vertical reaction force curves, 
as shown in Figure 3, are also nearly identical for all the 
interference cases presented. The discrepancy between the FEA 
and SAM results at larger interferences may be due to the 
different implementation of material hardening: FEA uses a 
bilinear hardening (giving a larger force response in the loading 
phase), while SAM uses an elastic-perfectly plastic material.  
This comparison of results suggests that both the FEA and SAM 
adequately model the sliding contact phenomenon. 
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
x/R
F
x/
P
c
2ùc ABAQUS
2ùc SAC
6ùc ABAQUS
6ùc SAC
9ùc ABAQUS
9ùc SAC
12ùc ABAQUS
12ùc SAC
15ùc ABAQUS
15ùc SAC
 
Figure 2:  A comparison of the SAM and FEA results for 
the tugging force. 
 
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
x/R
F
y/
P
c
2ùc ABAQUS
2ùc SAC
4ùc ABAQUS
4ùc SAC
6ùc ABAQUS
6ùc SAC
9ùc ABAQUS
9ùc SAC
12ùc ABAQUS
12ùc SAC
15ùc ABAQUS
15ùc SAC
 
Figure 3:  A comparison of the SAM and FEA results for 
the normal reaction force. 
 
 
CONCLUSIONS 
The main conclusions can be summarizes as followed: 
1. A FEA modeling method is presented to analyze 3D 
hemispherical sliding contact.  
2. A normalization scheme based on the distortion energy 
criterion is presented in order to apply the results to any 
scale.    
 
loaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Us3. Representative results are presented and validated 
against a semi-analytical technique.  The FEA and SAM 
results agree rather well for all cases examined of 
frictionless sliding.  Typically the SAM code runs in less 
than 10 percent of the FEA execution time. 
ACKNOWLEDGMENTS 
This research is supported in part through the Department 
of Defense Multidisciplinary Research Program of the 
University Research Initiative as Office of Naval Research 
Grant N00014-04-1-0601, entitled "Friction & Wear under Very 
High Electromagnetic Stress."  Dr. P. Peter Schmidt serves as 
Program Officer.  Information conveyed in this manuscript does 
not necessarily reflect the position or policy of the Government, 
and no official endorsement should be inferred. Thanks are 
afforded to Mr. Boucly and Prof. Nelias for furnishing the semi-
analytical computer program and adjusting it to our needs. 
REFERENCES 
1. Green, A.P., The plastic yielding of metal junctions 
due to combined shear and pressure. Journal of 
Mechanics and Physical Solids, 1954. 2: p. 197-211. 
2. Green, A.P., Friction between unlubricated metals: a 
theoretical analysis of the junction medel. Proceedings 
of the Royal Society, 1955: p. 191-204. 
3. Faulkner, A. and R.D. Arnell, The development of a 
finite element model to simulate the sliding interaction 
between two, three-dimensional, elastoplastic, 
hemispherical asperities. Wear, 2000. 242(1-2): p. 
114. 
4. Etsion, I., Y. Kadin, and Y. Kligerman, Unloading of 
an elastic-plastic spherical contact. International 
Journal of Solids and Structures, 2005. 42: p. 3716-
3729. 
5. Johnson, K.L., Contact Mechanics. 1985: Cambridge 
University Press. 
6.         Green, I., Poisson Ratio Effects and Critical Values in 
Spherical and Cylindrical Hertzian Contacts. 
International Journal of Applied Mechanics, 2005. 
10(3): p. 451-462. 
7. Jackson, R., I. Chusoipin, and I. Green, A Finite 
Element Study of the Residual Stress and Deformation 
in Hemispherical Contacts. ASME, Journal of 
Tribology, 2005. 127(3): p. 484. 
8. Boucly, V., D. Nelias, and I. Green, Modeling of the 
Rolling and Sliding Contact Between Two Asperities. 
ASME, Journal of Tribology, 2005, 129 (2): p. 235-
245. 3 Copyright © 2007 by ASME 
e: http://www.asme.org/about-asme/terms-of-use


IJTC2007-44301 AN INVESTIGATION OF THREE DIMENSIONAL ELASTIC-PLASTIC HEMISPHERICAL SLIDING CONTACT, PART I: MODELING AND VALIDAITON

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=B7650FBDB8E342EA6BDD9D1981EB21D3?doi=10.1.1.1077.6933&rep=rep1&type=pdf

IJTC2007-44301 AN INVESTIGATION OF THREE DIMENSIONAL ELASTIC-PLASTIC HEMISPHERICAL SLIDING CONTACT, PART I: MODELING AND VALIDAITON

Abstract

Similar works

Full text

Available Versions

CiteSeerX