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Oct 29, 2013 (3 years and 9 months ago)

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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
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?