OS-V: 0291 Hertzian Contact - Elastic Sphere and Rigid Half-space

Hertzian contact is demonstrated using OptiStruct for an elastic sphere and rigid half-space problem.

Hertzian contact (Hertz 1881, 1882) refers to the frictionless contact between two non-conforming bodies.
図 1. Model


Model Files

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Benchmark Model

The properties are:
Material Properties
Value
Young's modulus (E)
210000
Poisson's ratio ( v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaceaadaaakeaacaWG2baaaa@32B4@ )
0.3
R
30
d
0.1 (enforced z-displacement)
The cross-section of the half-space in the ¼ FE model is 100x100.
図 2. Model Details


Analytical Results:

The analytical solutions are provided by Popov, 2010. The derivation of the analytical solution is beyond the technical scope of this document. Some key variables of the analytical solution are as follows:

The contact area is calculated as:
a   =   R d MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadggacaGIGaGaey ypa0JaaOiiamaakaaabaGaamOuaiaadsgaaSqabaaaaa@39B4@
Where,
R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadkfaaaa@356C@
Radius of the sphere.
d MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadsgaaaa@357E@
Depth of indentation.
The applied force is calculated as:
F   =   4 a 3 E ∗ / 3 R MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadAeacaaMc8Uaey ypa0JaaGPaVlaaisdacaWGHbWaaWbaaSqabeaacaaIZaaaaOGaamyr amaaCaaaleqabaGaey4fIOcaaOGaai4laiaaiodacaWGsbaaaa@404B@
Where,
E ∗   =   E / 1   −   v 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadweadaahaaWcbe qaaiabgEHiQaaakiaaykW7cqGH9aqpcaaMc8Uaamyraiaac+cadaqa daqaaiaaigdacaaMc8UaeyOeI0IaaGPaVlaadAhadaahaaWcbeqaai aaikdaaaaakiaawIcacaGLPaaaaaa@4453@
E MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadweaaaa@355F@
Young's modulus.
v MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadAhaaaa@3590@
Poisson's ratio.

The maximum contact pressure/stress is calculated as:

P o   =   3 F / 2 π a 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbwvMCKfMBHbqee0evGueE0jxy aibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0d Xdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vq pWqaaeaabiGaciaacaqabeaabiWacqaaaOqaaiaadcfacaWGVbGaaG PaVlabg2da9iaaykW7caaIZaGaamOraiaac+cadaqadaqaaiaaikda cqaHapaCcaWGHbWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaa aaaa@4290@

Results

表 1.
Pressure/Stress (Magnitude) r/a
Analytical Result OptiStruct (S2S)-Contact Pressure
8482 8525 0
8350 8300 0.2
7700 7675 0.4
6800 6990 0.6
5100 5200 0.8
0 1345 1
図 3. Analytical Results versus OptiStruct Results


The results from FEA are in good agreement with the analytical solution.
図 4. Contact Traction/Normal (Pressure)


図 5. Stresses 2D and 3D - Z Stress


図 6. SPC Forces