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Elastic Compression of Spheres and Cylinders at Point and Line Contact

Puttock, M.; Thwaite, E.

National Standards Laboratory Technical Paper No. 25; Division of Appled Phyics, National Standards Laboratory, Commonwealth Scientific and Industrial Research Organization (CSIRO), University Grounds, Chippendale, New South Wales, Australia 2008; 64p. 1969.

Elastic compression.

The purpose of this paper is primarily to present in a convenient form the formulae and data for the calculation of the compression effects which occur in the measurement and use of spheres and cylinders in dimensional metrology.

Only Hertzian compression effects are considered in the present paper and these assume that the surfaces in contact are perfectly smooth, that the elastic limits of the materials are homogeneous, and that there are no frictional forces within the contact area. These conditions are closely met with materials and applied forces normally encountered in precise dimensional metrology, and with the surfaces finely lapped.

In the case of surfaces that are not finely lapped the actual compression effects may differ by up to 10% from those calculated using the formulae in this paper. Contributory factors include frictional forces and microstructure variations in the surface leading to variations in elastic modulii. Berndt (1928) has dervived modified formulae to take into account frictional forces arising from non-smooth surfaces and these formulae, in general, lead to compression effects differing from those in this paper by approximately 5%.

It is considered that the formulae given in this paper are sufficiently precise for all practical purposes in precise dimensional metrology.

This paper is in two parts. Part 1 is a series of data sheets giving the appropriate formulae for various specific cases, together with appropriate tables and graphs. Part 2 gives the mathematical derivation of the formulae in a consistent notation and is primarily intended for students with an interest in the subject.

Where the formulae have been partially evaluated for steel the elastic constants used have been those for 1% carbon steel.

Permission to reprint publication on a NIST web site was granted by Commonwealth Scientific and Industrial Research Organization (CSIRO), Australia.

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