Tribo Presentation
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Transcript of Tribo Presentation
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8/10/2019 Tribo Presentation
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HERTZ CONTACT
STRESS THEORY
NG HAN PEI
KGY 130024
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Introduction
Background
Theory and Mathematics
Contact Mechanics
Summary and Conclusions
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Mechanics of Materials
Typical Stress TheoryShows That for AnElement Subject to AxialLoad
What Happens When
Spherical or CylindricalSurfaces Contact? Point or Line Contact Results
A0 so
Images taken from Wikipedia: http://en.wikipedia.org/wiki/Contact_mechanics
Image taken from Contact Mechanics, K.L. Johnson
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Solids in Contact
In Reality, Under Load the Objects
Deform Giving a Contact Area
a
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Hertzian Contact Stress Theory
1880 Hertz Developed a Theory of the Elastic
Deformation of Two Surfaces in Contact
Formulated After Studying Newtons InterferenceFringes 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
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Contact between a sphere and a
half-space
An elastic sphere of radius R indents an elastic half-
space to depth d
thus creates a contact area of radius,
Similarly, From the Material Properties,
An Effective Modulus of Elasticity Can
Be Established,are the elastic moduli
are the Poisson's
ratios associated
with each body.
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Contact between a sphere and
a half-space
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,
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Contact between two spheres
For contact between two spheres of radii R_1 and R_2,
the area of contact is a circle of radius a. The equations
are the same as for a sphere in contact with a half plane
except that the effective radius R is defined as
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Contact between two crossed
cylinders of equal radius R
This is equivalent to contact between a sphere of radius
R and a plane
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Contact Stress
The Principle Stresses in the Material Occur in the Orthogonal
Planes
x = 1, y= 2, z= 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,
= = 1 tan1 1
1 + 1
2 1 + 2
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Relation to Optics
In Optics Several Situations Arise Where Contact Stress
Must Be Considered
Lens Seats
Kinematic Constraints
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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
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Kinematic Constraint
Kinematic Constraint Used for Precision Motion and
Alignment
Controls Chosen Degrees of Freedom
Contact Stress Can Degrade Precision
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Summary
Two Non-Conforming Solids in Contact CanResult is High Stresses
Small Contact Area
Hertz Developed a Theory for Contact StressAllows for Prediction of Contact Area, Pressure,
Compression and Stress
Contact Stress Becomes a Significant
Consideration for Optics Lens Mounting
Precision Motion and Alignment
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Thanks