Effects of Ellipticity of Contact Area and Poisson’s Ratio on the Yielding
Behavior of Two Contact Solids
Li Po Lin1, Jen Fin Lin2
1
Department of Mechanical Engineering, Southern Taiwan University of Technology
2
Department of Mechanical Engineering, National Cheng Kung University
and 1.79Y ( Y : the yielding stress of material in
Abstract
The present study is presented to study the behavior
simple tension or compression), respectively. The
of two contact solids at yielding and try to establish the
expression of the maximum contact pressure can be
relationships of the maximum contact pressure with the
defined as Pmax  y  KY . Then, there exists a difference
ellipticity ( k ) of a contact area and the Poisson ratio ( )
of about 11.8% between them. They are the limit cases
of a material. The von Mises’ criterion is applied to
of the elliptical contact area formed by the general
determine the depth position of having the maximum
profile of two solids. The dimension analysis reveals
second invariant of deviator stress tensor at yielding,
that the relationship between the load, F , and the
which is found to be on the center of the contact surface,
maximum contact pressure, Pmax , is F  Pmax 3 for the
beneath the contact surface, or occurring both on the
elliptical contact area. Therefore, it may have a
contact surface and beneath the contact surface
maximum difference of up to 40% if the K factor of
simultaneously. The factor of the maximum contact
circular contact area is employed in the calculation of
pressure ( K ) at yielding can be expressed as a function
the critical elliptical contact load, and thus the other
of ellipticity and Poission’s ratio, but there exists a
contact parameters.
discontinuity in the slope of the K  curve when
k
The present study is shown to investigate the
has a sufficiently small value.
behaviors of two contact solids at yielding and try to
Keywords：Microcontact, Ellipticity, Poisson’s ratio,
establish the relationships of the maximum contact
Yielding.
pressure with the ellipticity,
area, the Poisson ratio,
1. Introduction
k , of an elliptical contact
 , and the position of beginning
the yielding. According to the results of Johnson [1],
When two general profile solids come into contact,
Sackfield and Hills [2], the severest stress always occurs
the shape of contact area is an elliptical contact area; it
in the z -axis. The position of this severest stress
k (  b/ a ,
arising along this axis is determined by the Hertz
b and a represent the semiminor axis and semimajor
axis, respectively, b  a , and 0  k  1 ).
pressure distribution using the von Mises’ criterion. The
The study of Johnson [1] showed that the yield
depth in the z -axis, are presented by varying the
begins when material reaches the von Mises’ shear
ellipticity of contact area and the Poisson ratio of
strain-energy criterion. Therefore, the maximum contact
material. The maximum values of the second invariant
pressure of the contact of spheres ( k  1) and parallel
allow us to determine the border of two subregions
cylinders (k  0) , e.g., steel (   0.3 ) were 1.61Y
formed in the
can be characterized by the ellipticity,
values of the second invariant function varying with the
1
k  plot. One of these two subregions
the
plane x  0 and plane y  0 . Therefore, the location of
relationship that is able to start the yielding at the center
yielding is in the z -axis, and the maximum value of
of the contact surface; while the other subregion denotes
J 2 should occur at a certain point in this axis. The
k  relationship, which is the relationship that is
able to begin the yielding in the z -axis beneath the
position of this point
denotes
the
k 
relationship,
which
is
the
Z * can be determined by
obtaining the solution of the following equation
J 2
Z
contact surface. The contact parameters including the
maximum second invariant and the depth position
Z  z/a
0
Z Z *
(3)
corresponding to this maximum second invariant, and
where
the K factor of the maximum contact pressure are
z -coordinate, and Z * denotes the Z parameter where
J 2 has the maximum value at yielding.
thus discussed by varying the ellipticity and Poisson’s
represents
the
dimensionless
The analyses of stresses along the z -axis have been
ratio.
carried out in the studies of Thomas and Hoerch [3].
2. Contact Theory
Then Eq. (2) becomes
J 2  f (k , , Z )Pmax 
2
According to the study of Johnson [1], when two
solids of general shape come into contact, the contours
where
 2k  2
 2  wx1  wx 2  w y1  w y 2
 e 
2

1  k 
2
f (k , , Z )    2  2w y1  2w y 2  wz
6  e 
2

  k  2wx1  2wx 2  wz 2
2

 e 

of constant separation are elliptical shapes. If the basic
assumptions and mechanism of contacting surfaces in

Johnson’s work are adopted in the present study, the
onset of the plastic yield of most materials usually occur
when the von Mises’ shear strain-energy criterion
reaches
where
J2
where
J2
*
*
Y2
 k '2 
3
(4)


2










wx1 , w x 2 , w y1 , w y 2 , wz , and e can be
(1)
obtained from ref. [1], and
Pmax denotes the maximum
contact pressure on the contact surface.
represents as the maximum value of the
In Eq. (4), the maximum contact pressure Pmax is
second invariant of the deviator stress tensor ( J 2 ) at
independent of Z , the location where yielding begins
'
when f (k , , Z ) in
yielding; k is the material yield stress in simple shear.
is
The second invariant of the deviator stress tensor can be
Z * represents the dimensionless z -coordinate which
written as
can maximize the f (k , , Z ) value, it can thus be
J2 

1
 1   2 2   2   3 2   3   1 2
6

Eq.
(4)
is
maximum.
If
found by solving the following equation.
(2)
f (k , , Z )
Z
where  1 ,  2 , and  3 are the three principal stresses.
Z Z *
0
(5)
In the study of Sackfield and Hills [2], the stress
Substitution of Z  Z * into Eq. (4) obtains the
distributions formed by the Hertz contact pressure acting
maximum value of J 2 as


J 2*  f k , , Z * Pmax 
on an elliptical contact surface had been developed and
shown that the severest stress state always occurs on
At
such an axis that it is formed by the intersection of
2
the
inception
of
2
yielding,
(6)
Eq.
(6)
gives J 2*  k '2 
Poisson’s ratio (  ), the ellipticity (
k ), and the
dimensionless z-coordinate, Z. Figure 2 shows the J 2
Y2
. Then Eq. (6) can be expressed as
3
J 2*  f (k , , Z * )Pmax  y  k '2 
2
Y2
3
values varying with these three parameters. The solid
(7)
circular points in this figure are shown to note the
Equation (7) can be rewritten as
( Pmax ) y 
1
3 f (k , , Z * )
*
maximum J 2 value ( J 2 ) of each curve. In the case
(8)
Y  K (k , , Z * )Y
k  0.2 , only the curve corresponding
to   0 shows the maximum J 2 value at the contact
of
where
K (k , , Z * ) 
1
area; the other curves in Fig. 2, however, show their
(9)
3 f (k , , Z )
*
maximum value at the position beneath the contact
K (k , , Z * ) denotes the factor of the maximum contact
surface.
pressure arising at yielding. This factor is expressed as a
Figure 3 is applied to predict the position of the
k , and the
yielding if Poisson’s ratio and the ellipticity are
function of the ellipticity of the contact area,
Poisson ratio of a material,
 . The significance of Eq.
available.
(9) is that the required yielding stress of two contacting
Figure 4 shows the variations of the dimensionless
*
depth Z , where the maximum
surfaces at the onset of yielding is no longer a constant.
The factor of the maximum contact pressure,
ellipticity and a Poisson’s ratio. For a Poisson’s
K (k , , Z ) , cannot be solved directly, because there
*
are two unknowns,
J 2* occurs with an
ratio   0.4 , the position of increasing
k and  , in this equation.
J 2* value is
deeper beneath the contact surface as the ellipticity of
Z * curve
However, it can be obtained as a function of one of these
the contact area is reduced. However, the
unknowns if either the Poisson ratio varying in a range
becomes a convex form as Poisson’s ratio is further
of 0 to 0.5 or the ellipticity varying in a range of 0 to 1 is
reduced but is still larger than 0.194. As   0.194 , the
prescribed.
Z * -curves show sharp drops such that the
*
dimensionless depth Z is present at the contact
surface as the k value of a contact area is reduced
If the maximum contact pressures ( Pmax  y ) is
obtained, the critical loads can be expressed as:
Fy 
where Re ,
 3 Re 2
6 E *2
F1 e2 K k , , Z * Y 3
smaller than its critical value. Therefore, the
(10)
Z*
behavior exhibited in the materials with a large value of
Poisson’s ratio is exactly opposite to the behavior
E * , and F1 (e) can be obtained in ref. [1].
demonstrated in the materials with a small Poisson’s
k values.
Consequently, the other contact parameters such as
ratio only when they are evaluated at small
critical interference,  y , the contact area, Ay , and the
For k  0.3 , decreasing Poisson’s ratio of a specimen
average contact pressure, ( Pave ) y can be expressed as
always reduces the
function of the critical load, F y .
value is not lowered to zero; i.e., the maximum J 2
Z * value. Nevertheless, the Z *
*
value ( J 2 ) never occurs on the contact surface. If
k  0.222 , the lowering of Poisson’s ratio reduces the
3. Results and Discussion
*
The second invariant of the deviator stress tensor
non-zero Z value as Poisson’s ratio is still in the range
( J 2 ), as shown in Eq. (2), is expressed as a function of
of relatively high values. However, each of these curves
3
presents a sharp drop to Z *  0 , as Poisson’s ratio is
in these two subregions are quite different. In the
further lowered to the critical value. This critical
subregion 0    0.194 , the K factor presents to be
k value. In
a
general, increasing the ellipticity of the contact area

lowers the critical Poisson’s ratio. For steel material,
the increasing rate of K is elevated by increasing the
  0.3 , the Z value for k  0 is found at 0.705;
*
and the Z value for k  1 is found at 0.48. These

two results are identical to the results shown in the
K (0, , Z * )  1.201  2.741  2.624 2
studies of Johnson [1] and Sackfield and Hills [2].
increasing the
Poisson’s ratio is dependent upon the
*
polynomial
as
a
function
of
as: K (0, , Z )  1  1.594  8.445 ; therefore,
*
value
of
2
the
specimen.
In
the
subregion
0.194    0.5 , the K factor is expressed as:
Since the yielding strength Y is a material property,

.
Therefore,
value gradually lowers the increasing
rate of K due to the rise in Poisson’s ratio. There
it is a constant value for a material. The K factor as
exists a discontinuity in the
Eq. (9) shown, is linearly proportional to the maximum
  0.194 when the ellipticity k  0 . This can be
contact pressure at yielding. Therefore, the magnitude of
attributed to the shift of the yielding point from the
the K factor can be taken as the yielding resistance of
contact surface to be somewhere beneath the contact
a material.
surface. It can be noted that the occurrence of the
discontinuity in the K -curve slope exists only in the
Figure 5 is presented to be convenient for the
investigation of the K -factor values varying with
Poisson’s ratio

K -curve slope at
cases
with
their
0  k  0.222 .
as well as with two extreme values
k
value
The
values
in
the
range
of
k
and

of
k . In the case of k  1 (the circular contact area),
corresponding to these continuities in the K -curve
the maximum-contact-pressure factors varying in a
slope are shown just on the border of these two
range of 0    0.5 can be expressed by a polynomial
subregions. The critical Poisson’s ratio is dependent
as:
upon the ellipticity value, as it is always lowered by
of
K (1, , Z * )  1.2997  0.9084  0.4853 2 .
As
the
results of K shown in Fig. 5 illustrate, all these curves
k value; however, the discontinuity
property is absent from the K -curve if k and  are
are present to be a non-linear form of Poisson’s ratio. In
shown in subregion B of Fig. 3. For steel materials,
the CEB model [4], the maximum contact pressure is
Poisson’s ratio is about a value of 
increasing the
( Pmax  y  0.454  0.41 H ). Errors are thus generated as
 0.3 , the K
factor for k  0 is 1.79 and, k  1 is 1.613.
Therefore, within the range of 0  k  1 , the K
compared with the results shown in Fig. 5. Elevating
factor
Poisson’s ratio of a material thus increases the value

expressed
in
a
linear
function
of

as:
K k ,0.3, Z
is
*
expressed
  1.794  0.552k  0.5669k
2
 0.1928k
as:
3
k  0 (the contact of two
parallel cylinders), the variations of K factor in a
range of 0    0.5 can be divided into subregions
with   0.194 as the critical Poisson’s ratio. In either
of these two subregions, the K factor shows the
( J 2 ) is generally expressed as a function of Poisson’s
ratio ( ) of a material and the ellipticity (k ) of the
characteristic of increasing by increasing Poisson’s ratio
contact area. The position of the maximum J 2 value
of the specimen. However, the increasing rates of K
( J 2 ) at yielding is more apt to begin at the center of the
of K . As to the case of
4. Conclusions
1. The second invariant of the stress deviator tensor
*
4
contact surface if both the ellipticity
ratio

k and Poisson’s
Analysis, Vol. 18, pp.101-105, 1983.
of a material is sufficient small. Conversely,
4. Chang, W. R., Etsion, I., and Bogy, D. B., An
yielding begins beneath the contact surface as well as in
Elastic-Plastic Model for the Contact of Rough
k and  are sufficient large. The
Surfaces, ASME J. of Tribol, Vol. 109, pp. 257-263,
the z -axis if both
border of these two subregions shows that an increase in
1987.
the ellipticity of a contact area can lower the critical
Poisson’s ratio of yielding.
2. For a Poisson’s ratio   0.4 , the depth position
Z * of increasing J 2 * value is deepened beneath the
contact surface as the ellipticity of the contact area is
reduced. For a Poisson’s ratio in a range of 0.4 and
0.194, the Z *  k curve is presented to be a convex
form. As the material Poisson’s ratio is less than 0.194,
the depth corresponding to the J 2
*
Fig. 1 The contact of two general profile solids.
parameter is
sharply shifted to the center of the contact area, as the
ellipticity of the contact area is reduced smaller than its
critical value.
3. The K factor of the maximum contact pressure
at yielding ( Pmax  y ) is expressed as a function of
and
.
k
For   0.194 , the K value varies with the
Fig. 2 Variations of the dimensionless second invariant
ellipticity, presenting monotonic drops as the ellipticity
is increased. If
of the deviator tensor with the dimensionless
  0.194 , the K curve exhibites a
Z.
convex form such that its peak value is not present at
k  0 , but is dependent upon the  value. There
exists a discontinuity in the slope of the K  curve
when the ellipticity of the contact area has a sufficiently
small value.
5. References
1. Kragel’skii, I.V., and Mikhin, N.M., Handbook of
Friction Units of Machines, ASME Press, New York,
Fig. 3 Two subregions of the
1988.
2. Johnson, K. L., Contact Mechanics, Cambridge
University Press, Cambridge, 1985.
3. Sackfield, A., and Hills, D.A., Some Useful Results in
the Classical Hertz Contact Problem, J. of Strain
5
k - diagram.
可解決已發表研究中橢圓形接觸面積，但仍使用圓形
接觸面積最大接觸壓力因子計算之問題。
關鍵字：微接觸，橢圓比，卜桑比，降伏。
Fig. 4 Variations of the dimensionless depth
the ellipticity, k , and Poisson’s ratio,
Z * with
.
Fig. 5 Variations of the factor of the maximum contact
pressure with Poisson’s ratio,
.
接觸面積橢圓比與卜桑比對二接觸固
體降伏行為之效應
林黎柏 1 林仁輝 2
1
南台科技大學機械系副教授
2
國立成功大學機械系教授
摘要
本研究乃欲建立在降伏狀態下最高接觸壓力
( Pmax ) y 與橢圓比( k )、材料卜桑比（ ）及位置（ Z ）
*
之 關 係 。 固 體 內 部 是 否 達 到 降 伏 ， 以 von Mises
criterion 來判斷。結果顯示，當發生降伏時，其位
置可能在表面上、表面下，或在接觸表面上及表面下
同時發生，視 k 與 之值而定。最大接觸壓力可表為
( Pmax ) y  KY （ K ：最大接觸壓力因子， Y ：降伏強
度），而 K 則為橢圓比， k ，卜桑比， ，之函數。
6