Contact Mechanics
Maria Persson Gulda
Kathleen DiSanto
Outline
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What is Contact Mechanics?
The two different kind of contacts.
Boussinesq and Cerruti Potential Functions
The specific case of an Applied Normal Force
Hertz Equations- Derivation, Assumptions
Rigid Sphere Contacting a Deformable Plate
Deformable Sphere Contacting a Rigid Plate
What is Contact Mechanics?
“[The theory of contact mechanics] is
concerned with the stresses and
deformation which arise when the
surfaces of two solid bodies are
brought into contact.”
Professor Johnsson
Two kinds of contact

Conforming contacts


The two surfaces fit exactly or closely
together without deformation
Non-conforming contact

The surfaces, or one of the two
surfaces, deforms when there is a
contact area in between them.
Derivation: Boussinesq and Cerruti
Potential Functions
Here are the potential functions:

F1 
 q ,   d  d
x
S
G1 
 q ,   d  d
y
S
H1 
QuickTime™ and a
decompressor
are needed to see this picture.
S


    x     y   z
2

2
1
2 2

 p,   d  d
F
F  1   qx  ,  ln   z  d  d
z
S
G
G  1   qy  ,  ln   z  d  d
z
S
H
H  1   p ,  ln   z  d  d
z
S
  z  ln   z  

2

 0
Each satisfy Laplace’s equation:
Special Case: Applied Pressure Only


The potential functions are reduced as follows:

By Hooke’s Law, the stresses are:
F  F1  G  G1  0
H
H 1
1
1  1  H   p, ln(   z)  d  d


  p ,  d  d
z
z z

S
S
 Displacement
 equations:
1 

 
ux  
1 2  1  z 
4 G 
x

x 
1 
 
uz 
21    z 
4G 
z 
1 

 
uy  
1 2  1  z 
4 G 
y
y 


1  
 2
 21 
x 
 z 2  (1 2 ) 2 
2
2  z
x
x 
1 
 21
 2 
 xy   (1 2 )
2

2 
xy
xy 
1  
 2
 21 
y 
 z 2  (1 2 ) 2 
2
2  z
y
y 
1  2
 yz   z
2 yz
1 
 2 
z 
  z 2 
2 z
z 
1  2
 zx   z
2 xz
Concentrated Normal Force on an Elastic
Half Space
1 

2
p, d  d  P


1
2 2
S
H P
H1
 H  P  ln(   z)  

z
z 



QuickTime™ and a
decompressor
are needed to see this picture.
  x  y  z
2
The displacements are:
P xz
x 
ux 
  1 2 


4G 
   z
P z 2 21  
P yz
y 
 3 

uy 
 3  1 2 
 uz 
4 G 
 
4G 
  z


The stresses in polar coordinates:
 1
P 
z  3zr 2 
r 
1 2  2  2  5 
2 
r 
r
  
3P z 3
z  
2  5
 1
P
z
z 
   1 2  2  2  3 
2
r  
r
3P rz 2
 rz  
2  5
Concentrated Force Cont.
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
Now looking only at the surface, z=0
The displacements in polar coordinates become:
1   P
1 2  P

ur z 0  
uz z 0 
4 G r
2G r
For a general pressure distribution, the displacement for
any surface point in S, by Green’s function method,
becomes:

1  2
uz z 0 
E
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decompressor
are needed to see this picture.


 ps,  ds  d
S
Hertz Pressure

The Pressure distribution is:
po a 2  r 2 
1/ 2
p(r) 




a
Equation for determining surface displacement:
1  2
uz z 0 
E

where a is the radius of the contact area
 ps,  ds  d
S
The Hertz displacement equation:
1  2 po
uz z 0 
2a 2  r 2 ,r  a

E 4a
Hertz Theory of Elastic Contact

Assumptions:

The radii of curvature of the contacting bodies
are large compared with the radius of the
circle of contact.

The dimensions of each body are large
compared to the radius of the circle of
contact.

The contacting bodies are in frictionless
contact.

The surfaces in contact are continuous and
nonconforming.
Examples

Focus on two examples:
1.
2.
Rigid spherical indenter pushing to deformable flat
surface.
Deformable sphere contacting rigid plate.
(2)
(1)
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Equations to be Used
(1) – where R’ is the radius of the rigid sphere and
RS is the radius of the deformable plate
(2) – where δ is the vertical distance the point where the load is
applied moves and a is the contact area radius determined
1/ 3
by the equation: a  3 P  R

4  E*

(3) – h is the original distance between a point on the rigid sphere
and the deformable plate before load application.

(4) – These are the equations of
displacement derived
previously
(5) – This states that the translation of the point of load
application equals the surface displacement of the
plate and sphere plus the original distance
between the surfaces.
Rigid Sphere Contacting Deformable Flat
Contact Radius: 11.995 mm
Surface with Abaqus Theoretical
Abaqus Contact Radius: 11.6 mm
Error: 3%
Deformable Sphere Contacting Rigid
Theoretical Contact Radius: 9.288 mm
Plate with Abaqus
Abaqus Contact Radius: 8.5 mm
Error: 6%
Conclusion
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Contact problems in general are very
complicated to model numerically and
theoretically
Other factors
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Friction - rough surfaces
Blunt edges, sharp corners
Sliding and rolling contact
Dynamic impact
A Special Thank You To:
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Dr. Ashkan Vaziri
Professor James Rice
References
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Johnson, K. L. Contact Mechanics, Cambridge:
Cambridge University Press; 1985
Fisher-Cripps, A. C. The Hertzian contact surface. J.
Materials Science. 1999;34:129-137
Kogut, L., Etsion, I. Elastic-Plastic Contact Analysis
of a Sphere and a Rigid Flat. J. of Applied
Mechanics. 2002;69:657-662
Johnson, K. L., Greenwood, J. A. An Adhesion Map
for the contact of elastic Spheres. J. of Colloid and
Interface Science. 1997;192:326-333
Barber, J. R.,Clavarella, M. Contact mechanics. Inter.
J. of Solids and Structures. 2000;37:29-43
Any Questions?