# 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.

## 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 ( $v$ )
- 0.3
- R
- 30
- d
- 0.1

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:

$$a\text{}=\text{}\sqrt{Rd}$$

Where,- $R$
- Radius of the sphere.
- $d$
- Depth of indentation.

The applied force is calculated as:

$$F\text{\hspace{0.17em}}=\text{\hspace{0.17em}}4{a}^{3}{E}^{\ast}/3R$$

Where,
$${E}^{\ast}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}E/\left(1\text{\hspace{0.17em}}-\text{\hspace{0.17em}}{v}^{2}\right)$$

- $E$
- Young's modulus.
- $v$
- Poisson's ratio.

The maximum contact pressure/stress is calculated as:

$$Po\text{\hspace{0.17em}}=\text{\hspace{0.17em}}3F/\left(2\pi {a}^{2}\right)$$

## Results

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 |

The results from FEA are in good agreement with the analytical solution.