Engineering, 2010, 2, 205-211
doi:10.4236/eng.2010.24030 Published Online April 2010 (http://www.
Copyright © 2010 SciRes. ENG
A Finite Element Study of Elastic-Plastic Hemispherical
Contact Behavior against a Rigid Flat under Varying
Modulus of Elasticity and Sphere Radius
Prasanta Sahoo*, Biplab Chatterjee
Department of Mechanical Engineering, Jadavpur University, Kolkata, India
Received November 18, 2009; revised January 28, 2010; accepted February 4, 2010
The present study considers a finite element analysis of elastic-plastic axi-symmetric hemispherical contact
for a frictionless deformable sphere pressed by a rigid flat. The material of the sphere is modeled as elastic
perfectly plastic. Analysis is carried out to study the effect of varying modulus of elasticity and sphere radius
in wide range of dimensionless interference until the inception of plasticity as well as in plastic range. Re-
sults are compared with previous elastic-plastic models. It is found that materials with Young’s modulus to
yield strength (E/Y) ratio less than and greater than 300 show strikingly different contact phenomena. The
dependency of E on dimensionless interference at which the plastic region fully covers the surface is ob-
served. However with different radius, finite element study exhibits similar elastic-plastic phenomena.
Keywords: Elastic-plastic Contact, Sphere against Flat, ANSYS, Modulus of Elasticity, Sphere Radius
1. Introduction
Contact is necessary in any engineering application to
transfer force and power hence it is almost an indispen-
sable field of study. The elastic-plastic contact of a
hemisphere with a rigid surface is a fundamental prob-
lem to study the major characteristics like localized de-
formation, the variation in the contact area, the contact
force, displacement and the stress distribution in the de-
formable body. Much interest is devoted in the literature
to the reverse case of indentation loading where a rigid
sphere penetrates an elasto-plastic half space. It is worthy
to emphasize that indentation and hemispherical defor-
mation (this work) are significantly different in the
elasto-plastic and fully plastic regimes. One of the earli-
est models of elastic asperity contact is that of Green-
wood and Williamson [1]. This model uses the solution
of the frictionless contact of an elastic hemisphere and a
rigid flat plane, otherwise known as the Hertz contact
solution [2]. Some works are restricted to pure plastic
deformation of the contacting sphere, based on Abbott
and Firestone [3]. The works on either pure elastic or
pure plastic deformation of the contacting sphere over-
look a wide intermediate range of interest where elas-
tic-plastic contact prevails. An attempt to bridge this gap
was made by Chang et al [4]. Zhao et al [5] used mathe-
matical manipulation to smooth the transition of the con-
tact load and contact area expressions between the elastic
and elastic-plastic deformation regimes. Mathematical
models are replaced using finite element model (FEM)
concepts recently. Kogut and Etsion (KE) [6] used FEM
concept and analyzed the evolution of the elastic-plastic
contact with increasing interference revealing three dis-
tinct stages that ranges from fully elastic, elastic-plastic
up to fully plastic (0 < * 110). The model provides
dimensionless expression for contact area, contact load
and mean contact pressure covering a large range of in-
terference values. They inferred that their analysis is
normalized in such a way that allowed a general solution,
which is independent of specific material and radius of
the sphere. A change in the behavior of the mean contact
pressure was observed in their analysis at *=6, which
marks the elastic limit of the contact area. To generalize
their solution, the numerical solutions were normalized
with respect to their corresponding critical values at
yielding inception, c. The validity of this normalization
was tested by solving the problem for several different
material properties (100 < E/Y < 1000, = 0.3) and
sphere radii (0.1 mm < R < 10 mm). Their results show
that the entire contact zone is plastic when the dimen-
sionless interference ratio (/c = *) is 68 and the rate
of its radial expansion increases substantially. Moreover
when /c = 110, the dimensionless mean contact pres-
sure (p/Y) approaches the value of 2.8. This is identical
to the ratio between the hardness and yield strength
found experimentally for many materials as indicated by
Tabor [7].
Kogut and Etsion [8] studied the maximum tangential
load that can be supported by a normally preloaded elas-
tic perfectly plastic spherical contact at the inception of
sliding using an approximate semi analytical solution.
Sliding inception was interpreted as a failure of the con-
tact. They found that when the normal load is less than
the Hertzian critical load the failure occurs on the contact
surface. If the normal load exceeds that critical one, the
failure occurs below the contact surface. Brizmer et al. [9]
analyzed and compared the ductile material yielding in-
ception and the brittle material failure inception for two
different contact conditions between a smooth elastic
sphere and rigid flat and the effect of contact condition
and material properties on the termination of elasticity.
Jackson and Green (JG) [10] extended the KE model to
account for the geometry and the material effects in the
analysis. They used finer mesh than the KE model and
solved for five different yield strengths. Their analysis
showed markedly different behavior for the materials
with different strengths in the transition from elas-
tic-plastic to fully plastic deformations. For calculating
the critical interference, they used material yield strength
(Y) directly in their expressions. Their results show that
the dimensionless mean contact pressure does not reach
2.8 for most of the yield strength values. The JG solution
used E/Y ratio of 952.4, 356.6, 219, 158.1 and 137.74 by
varying the values of ‘Y’ with constant E = 200 GPa.
They also analyzed the interference when plastic defor-
mation first reaches the contacting surface at the far right
end and the contact surface first becomes entirely plastic
for yield strength of 1.619, 1.2653, 0.9115 GPa. Quick-
sall et al. [11] have taken five hypothetical material
properties and studied the error of formulation for KE
and JG models. The first test generated contact area and
contact force data for five hypothetical metals with
Poisson’s ratio, yield strength and elastic modulus typi-
cal of aluminum, bronze, copper, titanium and malleable
cast iron. Non-dimensional interference between *=5
and *=250 were used to generate data in both the elas-
tic-plastic and the plastic regimes except for aluminum at
*=250 and malleable cast iron above *=10. Second set
of tests generated non-dimensional contact area and con-
tact force data for a generic material in which the elastic
modulus and Poison’s ratio were independently varied
with yield strength held constant at 200 MPa. Firstly
Poisson’s ratio was varied between 0.28 and 0.36 with
the elastic modulus held constant at 200 GPa. Then the
elastic modulus was varied between 160 and 240 GPa
with Poisson’s ratio held constant at 0.32. The dimen-
sionless interference was set at *=20, 80 and 250 for
each test iteration. Jackson et al. [12] used the mesh size
same as that of JG model and presented results for a
range of normalized interference *, from 0.571 to 171.
Then the contact force, stress tensor, von mises stresses
and the displacement in both the radial and axial direc-
tions were recorded. After loading conditions had been
simulated; the solution was then restarted and unloaded
completely to simulate the residual stresses and the dis-
placements. In order to measure the effect of the material
properties on the hemispherical deformation, both alu-
minum and steel sphere unloaded from *=135; they
inferred that the deformation of the hemisphere is de-
pendent on the properties of the material and the inter-
Shankar and Mayuram (SM) [13] studied the evolution
of elastic core and the plastic region within the asperity
for different Y/E ratios. They varied the yield strength
from 250 N/mm2 to 2250 N/mm2. E equals to 2.07 x 105
N/mm2. The range of E/Y is from 83.333 to 500. It was
revealed from their analysis that higher dimensionless
interference, * is required for the plastic region to just
touch the contact surface when E/Y > 166.66 and this *
is varied with E/Y ratio. For E/Y< 166.667, this * is
constant and equals to 6. Same trend was observed for
the entire contact surface to be in plastic. For E/Y >
166.66, the value of /c decreases with decrease of E/Y
ratio. When E/Y ratio is less than 166.66, the fully plastic
contact condition exists at a constant interference ratio of
54. They studied the variation of mean contact pressure
(p/Y) as a function of dimensionless interference ratio
and observed that materials having low yield strength (i.e.
when E/Y>300) produces results similar to the KE model
but p/Y ratio never reaches 2.8. Malayalamurti and
Marappan (MM) [14] analyzed the elastic-plastic behav-
ior of a sphere loaded against a rigid flat in two region.
One is when E/Y > 300, another when E/Y < 300. They
observed that when E/Y<300, the fully plastic average
contact pressure or hardness is not constant. However,
the hardness and interference at the inception of fully
plastic condition are constant for materials with E/Y>300.
The present work aims to study the qualitative and quan-
titative contact characteristics of single asperity contact
using commercial FEM software ANSYS. Analysis is
carried out to study the effect of varying modulus of
elasticity and sphere radius in wide range of dimen-
sionless interference until the inception of plasticity as
well as in plastic range.
2. Finite Element Model and Procedure
To improve upon the efficiency of computation, an
axi-symmetric 2-D model is used. The present study
utilizes the commercial program ANSYS. The hemi-
sphere is modeled by a quarter of a circle, due to its axi-
symmetry. A line models the rigid flat. The model re-
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Copyright © 2010 SciRes. ENG
fines the element mesh near the region of contact to al-
low the hemisphere’s curvature to be captured and accu-
rately simulated during deformation. The model uses
quadrilateral, four node elements to mesh the hemisphere.
The resulting ANSYS mesh is presented in Figure 1.
The nodes on the axis of symmetry are fixed in radial
direction. Likewise the nodes on the bottom of the
hemisphere are fixed in both axial and radial direction.
This boundary condition is valid for the modeling of as-
perity contacts for two reasons: first, the asperities are
connected to a much larger bulk material at the base;
second, since the high stress region occurs near the con-
tact, the boundary condition at the base of the hemi-
sphere will not greatly affect the solution because of
Saint Venant’s principle. The bilinear isotropic harden-
ing (BISO) option in the ANSYS program is chosen to
account the elastic-plastic material response for the sin-
gle asperity model. The rate independent plasticity algo-
rithm incorporates the von Mises criterion. Tangent
modulus is assumed as zero for validating the results
with other elastic perfectly plastic models. The KE
model uses a maximum of 2944 nodes and JG model
uses a constant of 11,101 elements for their analysis.
However the present work uses a maximum of 11069
elements for the radius of 1 mm. The mesh density is
iteratively increased until the contact force and contact
area differed by less than 1% between iterations. In addi-
tion to mesh convergence, the model also compares well
with the Hertz elastic solution at interferences below the
critical interference. The contact force of the model dif-
fers from the Hertzian solution by no more than 2%
while the contact radius differs by less than 10%. The
smaller error in the contact force is due to overall force
balance enforced by ANSYS. But, the contact radius is
obtained from a discrete mesh which has a finite resolu-
tion. Apart from this, the magnitude of the contact ele-
ment stiffness also has some effect on such contact radii,
although not on the overall force balance. This work uses
Lagrangian multiplier method. The tolerance of current
work is set to 1% of the element width.
There are two ways to simulate the contact problem.
The first applies a force to the rigid body and then com-
putes the resulting displacement. The second applies a
displacement and then computes the resulting contact
force. In both methods, the displacement, stress, and
strain can be determined as well as the contact pressure.
In this work the latter approach is used. This method is
used because the resulting solution converges more rap-
idly than the former. In this work contact parameters are
normalized using the JG model’s expression for critical
interference (c), critical load (Pc) and critical contact
area (Ac) to form dimensionless parameters. The critical
interference (c) is expressed as [10]
c = (CY/2E/)2R (1)
Where C=1.295 exp(0.736). The Poisson’s ratio, ,
to be used here is that of the material which yields first.
The critical load, Pc, is then may be calculated as [10]
Pc = (4/3)(R/E/) 2.(CY/2) 3 (2)
Figure 1. Finite element mesh of a sphere generated by ANSYS.
Similarly, the critical contact area is calculated as [10]
Ac = 3(CYR/2E/) 2 (3)
In these expressions, E/ is defined as
1/E/ = (1-1
2)/E2 (4)
Where E1, E2, and 1, 2, are Young’s moduli and Pois-
son’s ratios of the two materials, respectively. In case of
the rigid flat, E2. These critical values predict analyti-
cally the onset of plasticity. These values are, therefore,
used to normalize the results. The dimensionless parame-
ters are as follows: * = /c, P* = P/Pc, A* = A/Ac.
3. Numerical Results and Discussion
The results of the finite element model are presented
for a variety of interferences. The materials selected
cover steel, grey cast iron and aluminum used in engi-
neering applications. While yield strength and Poisson’s
ratio are held constant, four different values of elastic
modulus (E) are chosen. These are 70, 80, 103, 200 GPa.
Figure 2 represents the results of the mean contact
pressure to yield strength ratio (p/Y) as a function of
dimensionless interference ratio (/c) obtained from the
finite element analysis. Here the study is made with con-
stant yield strength (Y) and Poison’s ratio (). The values
of which are taken as 0.21 GPa and 0.32 respectively.
With the variation of elastic modulus different E/Y ratios
are obtained. Thus the E/Y ratios are 333.31, 380.95,
490.476 and 952.4 for the elastic modulus of 70, 80, 103,
200 GPa respectively. The range of /c is from 100 to
above 500. It is revealed from Figure 2 that when
E/Y>300; the trend obtained is similar to the JG [10]
model with small changes in magnitude. JG inferred that
a material with higher Y has less mean contact pressure
in plastic range. Here material with higher E has higher
Figure 2. Dimensionless mean contact pressure versus di-
mensionless interference for E/Y > 300.
Figure 3. Dimensionless contact load as a function of di-
mensionless interference for E/Y > 300.
Figure 4. Dimensionless contact area as a function of di-
mensionless interference for E/Y > 300.
Figure 5. Dimensionless mean contact pressure as a func-
tion of dimensionless interference for E/Y < 300.
Copyright © 2010 SciRes. ENG
mean contact pressure as the dimensionless interference
increases in the plastic range. MM [14] concluded the
material independent behavior for E/Y > 300. They had-
taken the average results of four materials. Moreover
they modeled the material of the sphere as an elastic lin-
ear hardening material. A 1% linear hardening was se-
lected for their work. The present work modeled the ma-
terial of the sphere as an elastic perfectly plastic material
and the results entirely deviate from their result as mean
contact pressure to yield strength ratio (p/Y) never
reaches 2.8. Figure 3 shows the variation of dimen-
sionless contact load as a function of dimensionless in-
terference, when E/Y is greater than 300. The result in-
dicates the similarity with Quicksall et al. [11] findings.
With the increased modulus of elasticity (E), the value of
dimensionless load increases in the plastic range.
Dimensionless contact area as a function of * (di-
mensionless interference) is plotted in Figure 4. The
values for dimensionless contact area continue to in-
crease with interference even past fully plastic condition.
For the contact area, all the models follow the same gen-
eral trend, but they differ only in magnitude. The result
shows that the dimensionless contact area is independent
of E when E/Y > 300.
Study is also made for the materials of E/Y < 300. For
this case Y is taken as 1.619 GPa, with constant Pois-
son’s ratio of 0.32.With four E values of 70, 80, 103 and
200 GPa the corresponding E/Y ratios are 48.178, 54.972,
71.0315 and 137.739 respectively. These values are well
below 300. The behaviors of these materials in elastic as
well as in elastic-plastic ranges are also analyzed. The
average contact pressure to yield strength ratio is plotted
in Figure 5 as a function of dimensionless interference
(*). It is clearly evident from the plot that dimensionless
mean contact pressure does not reach the value of 2.8
even if for steel with E = 200 GPa. There is a decrease in
mean contact pressure after reaching its peak value,
though the peak value is dependent on E. As the value of
E increases, peak mean contact pressure also increases.
There is a similarity of the trend with the findings of MM
[14]. But MM inferred that peak value would reach after
the dimensionless interference of 90 only. Present work
observed that the peak value of mean contact pressure in
all the cases occurred well below the dimensionless in-
terference of 90. It can be seen from Figure 5, the loca-
tion at which the mean pressure attain its peak value and
its magnitude is entirely dependent on modulus of elas-
ticity (E) of the material.
Figure 6 represents the plot of dimensionless load (P*)
versus *. It is clear from the figure that for different
value of E up to the inception of fully plastic contact the
trend is linear with marginal change in magnitude. This
is similar to the prediction of KE [6]. This trend is not
observed in higher interference ratio. Figure 7 represents
the plot of dimensionless contact area with * when E/Y
< 300. The FEM results from this study indicate that
dimensionless contact area decreases slightly with in-
creased elastic modulus of elasticity. Same trend was
observed by Quicksall et al. [11].
When interference increases, the elastic core com-
pletely disappears and the fully plastic region reaches the
contact surface as shown in Figure 8, thus the transition
to the fully plastic state occurs. KE model predicted this
transition limited to specific dimensionless ratio of 68.
JG [10] and SM [13] predicted that this ratio at which
this transition occurs is not constant. In the present work
it is observed that this ratio is dependent on E. As the
value of E increases the dimensionless ratio at which the
transition occurs is also increasing. There is no indication
of constant value for the present four E/Y ratio though
SM also observed a constant value of 54 when E/Y <
166.66. Figure 9 is the plot of * versus E/Y where it is
clear that in the present domain of work SM overesti-
mates the interferences marginally. The present work
also studied the values of dimensionless contact load,
Figure 6. Dimensionless contact load versus dimension- less
interference for E/Y < 300.
Figure 7. Dimensionless contact area versus dimension-less
interference for E/Y < 300.
Copyright © 2010 SciRes. ENG
Figure 8. Inception of fully plastic contact.
Figure 9. End of elastic-plastic region based on the disap-
pearance of elastic core.
contact area and mean contact pressure with different
sphere radii, the study is made with radii of 1, 2, and 3
mm. No significant deviation is observed. Thus it can be
inferred that earlier mentioned dimensionless contact
characteristics are independent of the sphere radius.
4. Conclusions
The present work considers 2D axi-symmetric finite
element model of an elastic perfectly plastic hemisphere
in contact with a rigid flat surface. A comparison is also
made with other existing models. The material is mod-
eled as elastic perfectly plastic, and yielding occurs ac-
cording to the Von Mises criterion. It has been clearly
shown that the dimensionless mean contact pressure ratio
is not constant at 2.8 in the plastic range but depends on
modulus of elasticity whether E/Y is greater or lesser
than 300. But the peak value (i.e. maximum mean con-
tact pressure ratio), the dimensionless interference at
which this value occurs, the trend of variation are all
dependent on E/Y ratio. Dimensionless contact area,
contact load also depends on modulus of elasticity. But
dimensionless contact load, contact area and mean con-
tact pressure is independent of the radius of sphere. The
development of the elastic core and the transition to the
plastic region within the surface for the wide range of
interference is studied using Von Mises stress. It is ob-
served that the dimensionless interference ratio at the
instance when the plastic region fully covers the surface
is dependent on the values of modulus of elasticity when
Copyright © 2010 SciRes. ENG
other properties are kept constant.
5. Nomenclature
Symbol Meaning Unit
P Contact load N
R Radius of the sphere mm
p Pressure GPa
Interference m
A Contact area
E Modulus of Elasticity GPa
Y Yield strength of sphere GPa
Poison’s ratio
5.1 Subscripts
c = critical values
5.2 Superscripts
/ = equivalent
* = dimensionless
6. References
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