Contact Mechanics

1
Contact Mechanics

• Maria Persson Gulda
• Kathleen DiSanto

2
Outline

• 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

3
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

4
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.

5
Derivation Boussinesq and Cerruti Potential
Functions

• Here are the potential functions
• Each satisfy Laplaces equation

6
Special Case Applied Pressure Only

• The potential functions are reduced as follows
• Displacement equations
• By Hookes Law, the stresses are

7
Concentrated Normal Force on an Elastic Half Space

• The displacements are
• The stresses in polar coordinates

8
Concentrated Force Cont.

• Now looking only at the surface, z0
• The displacements in polar coordinates become
• For a general pressure distribution, the
  displacement for any surface point in S, by
  Greens function method, becomes

9
Hertz Pressure

• The Pressure distribution is
• Equation for determining surface displacement
• The Hertz displacement equation

where a is the radius of the contact area
10
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.

11
Examples

• Focus on two examples
• Rigid spherical indenter pushing to deformable
  flat surface.
• Deformable sphere contacting rigid plate.

(2)
(1)
12
Equations to be Used

• where R is the radius of the rigid sphere and
• RS is the radius of the deformable plate

(2) where d is the vertical distance the point
where the load is applied moves and a
is the contact area radius determined by
the equation
(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.
13
Rigid Sphere Contacting Deformable Flat Surface
with Abaqus
Theoretical Contact Radius 11.995 mm Abaqus
Contact Radius 11.6 mm Error 3
14
Deformable Sphere Contacting Rigid Plate with
Abaqus
Theoretical Contact Radius 9.288 mm Abaqus
Contact Radius 8.5 mm Error 6
15
Conclusion

• Contact problems in general are very complicated
  to model numerically and theoretically
• Other factors
• Friction - rough surfaces
• Blunt edges, sharp corners
• Sliding and rolling contact
• Dynamic impact

16
A Special Thank You To

• Dr. Ashkan Vaziri
• Professor James Rice

17
References

• Johnson, K. L. Contact Mechanics, Cambridge
  Cambridge University Press 1985
• Fisher-Cripps, A. C. The Hertzian contact
  surface. J. Materials Science. 199934129-137
• Kogut, L., Etsion, I. Elastic-Plastic Contact
  Analysis of a Sphere and a Rigid Flat. J. of
  Applied Mechanics. 200269657-662
• Johnson, K. L., Greenwood, J. A. An Adhesion Map
  for the contact of elastic Spheres. J. of Colloid
  and Interface Science. 1997192326-333
• Barber, J. R.,Clavarella, M. Contact mechanics.
  Inter. J. of Solids and Structures. 20003729-43

18
Any Questions?