ABSTRACT Hertz theory fails when contacting surfaces are expressed by a sum of even polynomials of higher powers than two. An alternative analytical solution implies the knowledge of contact area. In the case of elliptical domains, there are some published proposals for the correlation between pressure distribution and surface normal displacement. This paper identifies the class of high order surfaces which lead to elliptical contact domains and solves a contact between fourth order surfaces

Emanuel N Diaconescu

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DownloadProceedings of WTC2005 
World Tribology Congress III 
September 12–16, 2005, Washington, D.C., USA 
WTC2005-63646 
ELLIPTICAL CONTACT AREA BETWEEN ELASTIC BODIES BOUNDED BY HIGH ORDER 
SURFACES 
 
 
Emanuel N. Diaconescu  
Department of Mechanical Engineering, University of Suceava, emdi@fim.usv.ro 
 
Proceedings of WTC2005 
World Tribology Congress III 
September 12-16, 2005, Washington, D.C., USA 
WTC2005-63646 
ABSTRACT 
Hertz theory fails when contacting surfaces are expressed 
by a sum of even polynomials of higher powers than two.  An 
alternative analytical solution implies the knowledge of contact 
area.  In the case of elliptical domains, there are some published 
proposals for the correlation between pressure distribution and 
surface normal displacement.  This paper identifies the class of 
high order surfaces which lead to elliptical contact domains and 
solves a contact between fourth order surfaces. 
INTRODUCTION 
One of Hertz theory assumptions accepts that bounding 
surfaces are approximated by elliptical paraboloids around 
initial contact point.  This hypothesis leads to an elliptical 
contact domain.  The surfaces described by even monomials of 
higher power than two violate this assumption and Hertz theory 
fails.  Therefore, classical theory of elastic contacts must be 
improved.  The prior guess of contact area shape is a major 
difficulty when finding an analytical solution.  Nevertheless, 
Shtaerman [1], Galin [2] and Gladwell [3] advanced some 
elements of solution for elliptical contact domains. They found 
various correlations between a certain pressure applied over an 
elliptic domain sited in the bounding plane of an elastic half-
space and normal displacement generated in the points of this 
domain.  Shtaerman’s correlation seems to be incorrect and 
Galin’s theorem has a qualitative feature only.  Gladwell’s 
results are based on Legendre function which is difficult to 
apply.  By using the results of Shtaerman and Galin, this author 
advanced previously an explicit correlation between 
generalized Hertz pressure and resulting polynomial surface 
displacement inside an elliptical domain [4].  This is used now 
to identify the class of punch high order surfaces which lead to 
elliptical contact domains, to validate the result by using Hertz 
theory and to analyze the contact between fourth order surfaces. 
CONTACT OF HOMOGENEOUS SURFACES 
 A pressure expressed as a product between typical Hertz 
square root and an even polynomial, applied over an ellipse of 
half-axes a and b, is called generalized Hertz pressure.  This is 
given by following equation: 
 
 
ed From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use∑
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+−−=
1-n
0i
i
2
2
2
2
i2
2
2
2
0 b
y
a
xc  
b
y
a
x1 p)y,x(p , 
 
(1) 
 
where p0 is central pressure and c0=1.  This pressure generates 
following normal displacement in the points of contact domain: 
 
∑ ∑
= =
⎥
⎦
⎤
⎢
⎣
⎡
−η
π
=
n
0i
i
0k
k
k
i
k
i0 I C )1( c  bp 
1)y,x(w , 
 
(2) 
 
where Ik represents following definite integral [4]: 
 
∫
π
π
−
+
+
γ−
γ
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎣
⎡
γ−
⎟
⎠
⎞
⎜
⎝
⎛ γ+γβ
+−−
+
+π
=
2
2
22
1k
22
2
2
2
2
2
1kk
sine1
d
sine1
cos
b
ysin
a
x
b
y
a
x1
)!1k(2
!)!1k2(
I
             (3) 
 
β=b/a being the aspect ratio of the ellipse and e its eccentricity. 
 The punch surface which produces the generalized Hertz 
pressure is found from classical interference condition: 
 
∑ ∑
= =
⎥
⎦
⎤
⎢
⎣
⎡
−η
π
−δ=
n
0i
i
0k
k
k
i
k
i0 I C )1( c  bp 
1)y,x(z , 
 
(4) 
 
in which δ represents central penetration of half-space. 
Equations (1) to (4) define an explicit, bi-univocal 
correspondence between generalized Hertz pressure distribution 
and punch surface.  If pressure distribution is the product 
between typical Hertz square root and an even order 
polynomial of order 2n-2 with unitary free term, the punch 
surface is expressed by an even order polynomial of degree 2n 
with respect to co-ordinates x and y, with no free term. 
 Supplementary, force balance equation holds: 
 
∫∫ =
A
Qdxdy)y,x(p , 
 
(5) 
 
where Q is normal load. 1 Copyright © 2005 by ASME 
: http://www.asme.org/about-asme/terms-of-use
Downl A contact problem on elliptical area possesses n+1 
unknowns: central pressure p0, normal approach δ, ellipse half-
axes and coefficients of pressure distribution.  Equation (5) 
yields p0 and remaining unknowns satisfy Eq.(4).  As z(0,0)=0, 
δ results imposing the free term in right side member of Eq.(4) 
be zero. There are n-1 unknown coefficients in pressure 
equation, including those equal to zero, and n+1 unknowns in 
the problem.  Surface equation has s=0.5(n2+3n) coefficients. 
From these, 0.5(n2+n-2) are zero and n+1 nonzero.  Only n+1 
surface coefficients are useful to solve the problem (2 nonzero 
and n-1 equal to zero), whereas 0.5(n2-n) zero coefficients are 
redundant.  The problem is determined and the domain is 
elliptical only if n=1, which represents a Hertz contact.  
Usually, the problem is undetermined and contact domain is 
non-elliptical. To force contact area be elliptical, surface 
polynomial must contain only n+1 independent coefficients, 
from which only two are different of zero.  The remaining n-1 
nonzero coefficients must comply with right side member of 
Eq.(4).  Only homogeneous polynomial surfaces satisfying this 
restriction yield an elliptical contact area. 
HERTZ POINT CONTACT 
 In Hertz point contacts n=1 and punch equation is: 
 
22 y Bx A)y,x(z += . (6) 
 
According to Eq.(1), the pressure takes the known Hertz form: 
 
  
b
y
a
x1 p)y,x(p 2
2
2
2
0 −−= . 
 
(7) 
 
After some calculations, Eq.(4) takes following form: 
 
⎥
⎦
⎤
⎢
⎣
⎡
−−−η=−−δ )DK(
b
y
D
a
xKbp ByAx
2
2
2
2
0
22 , 
 
(8) 
 
K and E being complete elliptic integrals of eccentricity e, and 
[ ] 2eEKD −= .  The identification of coefficients in Eq.(8) 
and force balance Eq.(5) yield all known Hertz formulae. 
 
HOMOGENEOUS FOURTH ORDER SURFACES 
 Following forth order surface equation: 
 
222244 y Ex Dyx Cy Bx A)y,x(z ++++=  (9) 
 
is homogeneous if D=E=0.  As n=2 s=5 contact area is elliptical 
if Eq.(9) contains only 2 independent nonzero coefficients, let 
them be A and B.  The coefficient C cannot be imposed priory, 
its value resulting from Eq.(4) by coefficients identification. 
After performing implied calculations, pressure distribution is: 
 
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++−−=
2
2
2
2
2
2
2
2
0 b
y
a
x  21 
b
y
a
x1p)y,x(p , 
 
(10) 
 
whereas contact elements are: 
 
5
a A 12
Q  5 na η= , ( ) ( )[ ]5 22
4a
E e1 2K e2
e 
1n +−+
π
= ; (11) 
 
5
b B 12
Q  5 nb η= , ( ) ( )[ ] E e2 12K e32
e
n 5 222
4b
−−β−
π
β
= ;(12)  
oaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Us5
2
3
p
0
Q B A
54
125 
n
p
ηπ
= , 
ba
p nn
1n = ; 
 
(13) 
 
5 44Q A
648
625 
2
n
η=δ δ , 
an 
K3n
π
=δ . 
 
(14) 
 
The eccentricity of contact ellipse yields from equation: 
 
( )
)e(E)e21(2)e(K)2e5e3(
)e(E)e1(2)e(K)e2(
e1
B
A
224
22
22
−−+−
+−+
−= . 
 
(15) 
 
Coefficient C of surface equation must have following value: 
 
[ ]K )e1(2E )e2( 
bae
bp  3
C 22
224
0 −−−
η
= .  
 
(16) 
 
 The pressure distribution given by Eq.(10) is shown in 
dimensionless coordinates in Fig.1, for an aspect ratio 5.0=β . 
 
 
 
Figure 1. Pressure distribution between fourth order surfaces 
CONCLUSIONS 
A punch bounded by a surface described by a high order 
homogeneous equation, pressed against an elastic half-space, 
generates an elliptical contact domain if it contains only two 
independent non-zero coefficients.  The remaining coefficients 
result from identification of coefficients in Eq. (4). 
Hertz equations for point contacts yield as a particular case 
of advanced formulae, thus validating the theory advanced. 
Homogeneous fourth order polynomial surfaces lead to an 
elliptical contact area if they possess two independent 
coefficients and the third is given by Eq.(16).  Contact elements 
are found by aid of Eqs. (11) to (15). 
REFERENCES 
[1] Shtaerman, I., 1949, “Contact Problems in the Theory of 
Elasticity”, (in Russian), Gostehizdat, Moscow, Chaps. III, IV. 
[2] Galin, L.A., 1953, “Contact Problems in the Theory of 
Elasticity”, (in Russian), Gostehizdat, Moscow, Chap. II. 
[3] Gladwell, G.M.L., 1978, “Polynomial Solutions for an 
Ellipse on an Anisotropic Elastic Half-Space,” J. Mech. Appl. 
Math., XXXI, Part. 2, pp. 251-260. 
[4] Diaconescu, E.N., 2003, “Hertz Theory Revisited,” Private 
communication, Opening Plenary Session of 10th Romanian 
Tribology Conference, Galatzi, 14 pp. 
pressure
x
y
2 Copyright © 2005 by ASME 
e: http://www.asme.org/about-asme/terms-of-use


WTC2005-63646 ELLIPTICAL CONTACT AREA BETWEEN ELASTIC BODIES BOUNDED BY HIGH ORDER SURFACES

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WTC2005-63646 ELLIPTICAL CONTACT AREA BETWEEN ELASTIC BODIES BOUNDED BY HIGH ORDER SURFACES

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