1. Hertz Contact Stress Theory
UA Optomechanics OPTI 521
Jacob Etter
December 13, 2011

2. Introduction Background Theory and Mathematics Contact Mechanics
Summary and Conclusions

3. Mechanics of Materials
Typical Stress Theory Shows That for An Element Subject to Axial Load
What Happens When Spherical or Cylindrical Surfaces Contact?
Point or Line Contact Results
A  0 so σ  ∞
Images taken from Wikipedia:
Image taken from Contact Mechanics, K.L. Johnson

4. Solids in Contact
In Reality, Under Load the Objects Deform Giving a Contact Area
a << R
A Theory was Required to Predict the Shape of the Contact Area as well as:
Area Growth Under Increasing Load
Stress and Deformation in Both Bodies
Hertz Developed First Analysis of Stress in Two Elastic Solids in Contact

5. Hertzian Contact Stress Theory
1880 Hertz Developed a Theory of the Elastic Deformation of Two Surfaces in Contact
Formulated After Studying Newton’s Interference Fringes Between Two Glass Lenses
Simplifying Assumptions
Surface are Continuous and Non-Conforming
Strains are Small
Surfaces are Frictionless
Solids are Operating in the Elastic Regime
Non-conforming is point or line contact

6. Geometry and Material Properties
Considering the Two Cases Shown Previously
From the Geometry of the Solids we Calculate an Effective Radius,
Similarly, From the Material Properties, An Effective Modulus of Elasticity Can Be Established,

7. Contact Mechanics
The Contact Area of Two Solids Under a Given Load Can Be Shown To Be,
The Maximum Pressure Occurs at the Center and is Shown To Be,
The Compression of the Two Solids Can Also Be Determined As,

8. Contact Stress
The Principle Stresses in the Material Occur in the Orthogonal Planes
σx = σ1, σy = σ2, σz = σ3
At Maximum Pressure the Stress in the Normal Direction is Given By,
Stress in the Orthogonal Directions is Shown to be,
Maximum Shear Stress is,

9. Relation to Optics
In Optics Several Situations Arise Where Contact Stress Must Be Considered
Lens Seats
Kinematic Constraints

10. Sharp Corner Lens Seat
Consider a Spherical Lens in a Sharp Corner Seat
Seating Lens Creates Ring Contact
As a Pre-Load is Applied Stress is Induced in the Glass and the Seat
Glass and Seat Must be Analyzed Under This Load and Any Other Induced Loads

11. Kinematic Constraint
Kinematic Constraint Used for Precision Motion and Alignment
Controls Chosen Degrees of Freedom
Contact Stress Can Degrade Precision

12. Summary
Two Non-Conforming Solids in Contact Can Result is High Stresses
Small Contact Area
Hertz Developed a Theory for Contact Stress
Allows for Prediction of Contact Area, Pressure, Compression and Stress
Contact Stress Becomes a Significant Consideration for Optics
Lens Mounting
Precision Motion and Alignment

13. Questions?