hertz contact stress theory ua optomechanics opti 521 jacob etter december 13, 2011
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HERTZ CONTACT STRESS THEORYUA 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 from Wikipedia: http://en.wikipedia.org/wiki/Contact_mechanics
Image taken from 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 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,
𝜎𝑥 = 𝜎𝑦 = −𝑝𝑚𝑎𝑥 ൞൦1− 𝑧𝑎tan−1൮ 1ฬ𝑧 𝑎Τ ฬ
൲൪ሺ1+𝜈ሻ− 12൬1+𝑧 𝑎Τ 2൰ൢ
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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