Validation Case: Hertzian Contact Between Two Spheres

This validation case belongs to structural dynamics. The aim of this test case is to validate the following parameters at the point of the Hertzian contact between two spheres:

• \(σ_{zz}\) stress using a frictionless penalty contact.
• \(σ_{zz}\) stress using a frictionless augmented Lagrange contact.

The simulation results of SimScale were compared to the results presented in \(^1\).

Geometry

Only one-eighth of each of the two spheres (with a radius of 50 \(mm\) ) is used for the analysis due to the symmetry of the case study.

Analysis Type and Domain

Tool type : Code_Aster

Analysis type : Static nonlinear

Type of contact: Physical

Mesh and element types: The meshes used in (A) and (C) were created with the standard meshing algorithm on the SimScale platform. The additional meshes in (B) and (D) were locally meshed with refinements and uploaded to SimScale.

Case	Mesh type	Number of nodes	Element type
(A)	1st order tetrahedral	1826	Standard
(B)	1st order tetrahedral	2586	Standard
(C)	2nd order tetrahedral	107074	Standard
(D)	2nd order tetrahedral	14672	Standard

Below the 1st order tetrahedral mesh for case A is visualized:

And the mesh that was uploaded for case B is presented below:

Simulation Setup

Material/Solid:

• isotropic:
  • \(E\) = 20 \(GPa\),
  • \(ν\)  = 0.3

Constraints:

• Faces ACD and A’C’D: zero x-displacement
• Faces ABD and A’B’D: zero y-displacement
• Face ABC: displacement of 2 \(mm\) in the z-direction
• Face A’B’C’ displacement of -2 \(mm\) in the z-direction

Physical Contacts:

• Augmented Lagrange:
  • Contact smoothing enabled for linear elements and disabled for quadratic elements
  • Frictionless
  • Augmentation coefficient = 100
• Penalty:
  • Contact smoothing enabled for linear elements and disabled for quadratic elements
  • Frictionless
  • Penalty coefficient = 10\(^{15}\)

Reference Solution

$$\sigma_{zz} = \frac{-E}{\pi}\frac{1}{1-{\nu}^2}\sqrt{\frac{2h}{R}} \tag{1}$$

$$h= 2 mm−(−2 mm ) = 4mm\tag{2} $$

With equation (1) and equation (2) the stress at point D results in:

$$\sigma_{zz} = −2798.3\ MPa$$

Results

Comparison of the stress \(σ_{zz}\) at point D of the Hertzian contact obtained with SimScale with the analytical result of the reference solution [SSNV104_A]\(^1\):

Case	Physical Contact	[SSNV104_A]	SimScale	Error (%)
(A)	Penalty	-2798.3	-2882.1	2.995%
(A)	Augmented Lagrange	-2798.3	-2884.54	3.082%
(B)	Penalty	-2798.3	-3074.14	9.857%
(B)	Augmented Lagrange	-2798.3	-3078.39	10.009%
(C)	Penalty	-2798.3	-2909.3	3.967%
(D)	Penalty	-2798.3	-2909.45	3.972%

It is obvious from the table above that the best results were obtained with SimScale’s 1st order mesh, using the penalty contact:

Last updated: January 26th, 2021