Contact stresses

Eng.: Mohamed Akram 1

Objects1. Introduction.2. Hertzian theory.3. Hertzian theory assumptions.4. Non Hertzian Contacts.5. Common Engineering Contact Applications .6. Real and Nominal Area of Contact Measurement.7. Experimental Contact Stress Analysis.

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1. Introduction

• contact stress is a description of the stress within mating parts.

• It causes serious problems if not take it into account in some cases.

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1. Introduction

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2. Hertzian theory

• When two curved bodies are brought into contact they initially contact at a single point or along a line.

• With the smallest application of load elastic deformation occurs and contact is made over a finite area.

• A method for determining the size of this region was first described by Heinrich Hertz in 1881.

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3. Hertzian theory assumptions• The strains are small and within the elastic limit.• Each body can be considered as an elastic half space,

i.e., the area of contact is much smaller than the characteristic radius of the body.

• The surfaces are continuous and non-conforming. • The surfaces are frictionless.• The gap (h) between the

undeformed surfaces can be approximated by an expression of the form

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• Cylinders in Contact – Vertical Stress Distribution along Centerline of Contact Area

• The maximum shear and Von Mises stress are reached below the contact area.

• This causes pitting where little pieces of material break out of the surface.

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• Spheres in Contact – Vertical Stress Distribution at Center of Contact Area

• The maximum shear and Von Mises stress are reached below the contact area.

• This causes pitting where little pieces of material break out of the surface.

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4. Non-Hertzian Contacts1. Flat Rigid Planar Punch• A flat-ended punch, of width 2b and of infinite length in the y-

direction, pressed onto an elastic half-space with a force per unit length P .

• The surface of the punch is assumed frictionless .• Contact occurs across the width of the punch, 2b. • The pressure distribution is given by:

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4. Non-Hertzian Contacts

• the deflection can only be presented relative to some datum. The normal deflection uz of the surface, outside the contact region, is given by:

where δ is the normal deflection at an arbitrary datum point.

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4. Non-Hertzian Contacts2. Flat Rigid Axisymmetric Punch• In this case the punch has a circular section of radius a.• It is pressed onto an elastic half-space with a force P. • The surface of the punch is assumed frictionless.• Contact occurs across a circle of radius a and the resulting pressure

distribution is (Timoshenko and Goodier, 1951)

• The penetration ∆ of the punch is given by:

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4. Non-Hertzian Contacts3. Indentation by an Angular Wedge• If a two-dimensional wedge of semi-angle α is pressed onto a

frictionless elastic half-space with a force per unit length, P then contact is made over a rectangular region of semi-width b, such that

• This time the pressure distribution has a singularity at the wedge apex and is given by:

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5. Common Engineering Contact Applications

• Gears• Meshing gear teeth are subjected to bending stresses and contact

stresses. • The appropriate values of R1, R2and Pare then used in the Hertz

relations to determine the geometry of the contact and associated stresses.

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5. Common Engineering Contact Applications

• Ball Bearings• The rolling element is loaded against the conforming grooves in the

inner and outer raceways.• For a radially loaded ball bearing the contact between the ball and

either the inner or outer raceway will be elliptical.• The radii are readily obtained from the ball bearing geometry.• For a bearing containing )z( balls carrying a radial load, F, the

maximum load on the ball, P(located diametrically opposite the loading point) is approximated by :

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5. Common Engineering Contact Applications

• This load and the contact radii can then be used in the expressions for elliptical point contact to determine the contact area and stresses.

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6. Real and Nominal Area of Contact Measurement

• Electrical and Thermal Resistance Measurement of the electrical resistance between contacting surfaces can give information about the true area of contact (Holm, 1967; Bowden and Tabor, 1939).

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6. Real and Nominal Area of Contact Measurement

• Ultrasonic Reflection A wave of ultrasound incident at an interface between two materials will transmit through regions of contact and be reflected back at air gaps. This phenomenon can be used to investigate the true area ofcontact at an interface (Kendall and Tabor, 1971)

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7. Experimental Contact Stress Analysis

• Photoelasticity and Caustics The contacting bodies are modeled in photoelastic material, such as polycarbonate or epoxy resin. For two-dimensional applications a planar model is fabricated and loaded in a polariscope

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ThanksM A