HERTZ CONTACT STRESS
THEORY
UA
Optomechanics
OPTI 521
Jacob Etter
December 13, 2011
Introduction
Background
Theory and Mathematics
Contact Mechanics
Summary and Conclusions
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 f rom Wikipedia: http://en.wikipedia.org/wiki/Contact_mechanics
Image taken f rom Contact Mechanics, K.L. Johnson
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
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
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,
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,
Contact Stress
The Principle Stresses in the Material Occur in the Orthogonal Planes
σ
x
=
σ
1
,
σ
y
=
σ
2
,
σ
z
=
σ
3
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,
Relation to Optics
In Optics Several Situations Arise Where Contact Stress Must Be
Considered
Lens Seats
Kinematic Constraints
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
Kinematic Constraint
Kinematic Constraint Used for Precision Motion and Alignment
Controls Chosen Degrees of Freedom
Contact Stress Can Degrade Precision
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
Questions?
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