OPTI 521
Optomechanical Engineering
Hertzian Contact Stresses
December 2011
Nicholas LeCain
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OPTI 521
Optomechanical Engineering
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Overview
Hertzian Contact Stresses
Non-Hertzian Contact Stresses
Failure modes
Implications in Opto-Mechanics
Summary
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OPTI 521
Optomechanical Engineering
Overview
• Contact stresses
– Stress developed from
two radii in contact
– Stress
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σ=F/A
Force is constant
Area is infinitely small
Stress approaches infinity
– Deformation occurs until
area is large enough to
reduce stress to below
elastic limit of parts.
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OPTI 521
Optomechanical Engineering
Hertzian Contact Stresses
Ball with no contact force
Deformation Caused by Hertzian
Contact Stresses
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OPTI 521
Optomechanical Engineering
Hertzian Contact Stresses
• Hendrick Hertz first published his
work on contact stresses in 1881.
• Work was based on a few
assumptions.
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Frictionless
Elastic bodies
Isotropic materials
Homogeneous materials
No external shear stress
• Without these assumptions the
equations get out of hand pretty
quickly and an FEA approach to
analysis is required.
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OPTI 521
Optomechanical Engineering
Hertzian Contact Stresses
Spherical Bodies
• Spherical Equations
Radius of
deformed
contact area
Maximum
pressure from
force applied
Note: For a flat surface R would equal
infinity and for a concave surface like a
spherical hole R would be negative
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OPTI 521
Optomechanical Engineering
Hertzian Contact Stresses Spherical
Bodies cont.
• Principle and Shear
Stresses
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OPTI 521
Optomechanical Engineering
•
Hertzian Contact Stresses
Cylindrical Bodies
For Cylindrical contacts instead of a circular contact area an elliptical contact area is
produced. The equations below cover this change.
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OPTI 521
Optomechanical Engineering
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Hertzian Contact Stresses
Cylindrical Bodies cont.
Note: In the cylindrical case the principle stresses are not constant. For more detailed
information on this see Mechanical Engineering Design, Shigley 2004
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OPTI 521
Optomechanical Engineering
Non-Hertzian Contact Stresses
• Applications where the assumptions listed in
the previous slide do not apply fall under NonHertzian contact stresses.
– These applications must be handled with finite
element analysis or with the Smith-Liu equations
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OPTI 521
Optomechanical Engineering
Failure Modes
• Permanent Plastic Deformation of parts
• Fatigue damage
http://www.vibanalysis.co.uk/vibcases/vibch13/vibch13p1.jpg
Fatigue damage on bearing.
Plastic Deformation of Aluminum
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OPTI 521
Optomechanical Engineering
Implications in Opto-Mechanics
• Weight limits for
kinematic mounts
• Design limits of Sharp
Edge lens seats
• Point Contacts for
mirror supports
• Point Contacts of
micrometers
http://assets.newport.com/web900w-EN/images/8566.jpg
http://www.optics.arizona.edu/optomech/Fall11/Notes/27%20
Mounting%20of%20lenses.pdf
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OPTI 521
Optomechanical Engineering
Summary
• Hertzian equations apply to contact stresses
created by the contact of radii.
• Hertzian Equations work well if the stated
above assumptions apply
• If there are exceptions to the assumptions FEA
or more complicated equations must be used.
• In the field of opto-mechanics Hertzian
equations work well for the analysis of
Kinematic mounts and lens seats.
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