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Verification Problems
This manual presents solved verification models including NAFEMS problems.
NAFEMS Nonlinear Static Analysis
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.

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Verification Problems
This manual presents solved verification models including NAFEMS problems.
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  - 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.
  - OS-V: 0292 Hertzian Contact - Rigid Sphere and Elastic Half Space, 2D Axisymmetric, S2S Contact
    Hertzian contact is demonstrated using OptiStruct for rigid sphere and elastic half-space –2D axisymmetric, S2S CONTACT.
  - 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.
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- Elements
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- Aeroelastic Analysis
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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.

rigid_sph_elas_hsph.fem

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

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 = \sqrt{R d}

Where,

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

The applied force is calculated as:

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:

P o = 3 F / (2 π a^{2})

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