# 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.

## Model Files

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

## Benchmark Model

The properties are:
Material Properties
Value
Young's modulus (E)
210000
Poisson's ratio ( $v$ )
0.3
R
30
d
0.1 (enforced z-displacement)
The cross-section of the half-space in the ¼ FE model is 100x100.

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:
Where,
$R$
$d$
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

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