OS-V: 0292 Hertzian Contact - Rigid Sphere and Elastic Half Space, 2D Axisymmetric, S2S Contact

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

Since the 2D axisymmetric analysis is computationally less expensive compared to 3D analysis, a bigger half space size is used for the 2D axisymmetric analysis. In the model, CQAXI and CTAXI are the elements. Use PAXI and PCONT as card image for the property. See FIGURE for model details.

Model Files

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

rigid_sph_elas_hsph_axi.fem

Benchmark Model

Since the 2D axisymmetric analysis is computationally less expensive compared to 3D analysis, a bigger half space size is used for the 2D axisymmetric analysis. In the model, CQAXI and CTAXI are the elements. Use PAXI and PCONT as card image for the property. See Figure 2.

The properties are:

Material Properties: Value
Young's modulus (E): 210000
Poisson's ratio ( $v$ ): 0.3
R: 30
d: 0.1

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:(1)

a = \sqrt{R d}

Where,

$R$: Radius of the sphere.
$d$: Depth of indentation.

The applied force is calculated as:(2)

F = 4 a^{3} E^{*} / 3 R

Where,

E^{*} = E / (1 - v^{2})

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

The maximum contact pressure/stress is calculated as:(3)

P o = 3 F / (2 π a^{2})

Results

Table 1.
Pressure/Stress (Magnitude)		r/a
Analytical Result	OptiStruct (S2S)-Contact Pressure
8482	8434	0
8350	8272	0.2
7700	7696	0.4
6800	6960	0.6
5100	5110	0.8
0	943	1

The results are in good agreement with the analytical solution, although contact smoothing is not supported in 2D analysis.