OS-V: 0293 Hertzian Contact - Rigid Sphere and Elastic Half Space, S2S Contact with Smoothing

Hertzian contact is demonstrated using OptiStruct for rigid sphere & elastic half space – S2S CONTACT with Smoothing.

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


Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

Benchmark Model

In the model, a 100x100x100 half space (1/4th model) with 2nd order CHEXA elements is used. See Figure 2.

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
Figure 2. Model Details


Analytical Results:

These 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

Table 1.
Pressure/Stress (Magnitude) r/a
Analytical Result OptiStruct (S2S)-Contact Pressure
8482 8545 0
8350 8200 0.2
7700 7930 0.4
6800 6500 0.6
5100 5240 0.8
0 160 1
Figure 3. Analytical Results versus OptiStruct Results


+
The results from FEA are in good agreement with the analytical solution.
Figure 4. Stresses - Z Stress


Figure 5. Contact Pressure