Tribo Presentation

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

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