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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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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

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:

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

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

Kinematic Constraint

Kinematic Constraint Used for Precision Motion and Alignment

Controls Chosen Degrees of Freedom

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?