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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].


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.

spheres in contact geometry cad model
Figure 1: Geometry of the two spheres in contact

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.

CaseMesh typeNumber of nodesElement type
(A)1st order tetrahedral1826Standard
(B)1st order tetrahedral2586Standard
(C)2nd order tetrahedral107074Standard
(D)2nd order tetrahedral14672Standard
Table 1: The final mesh details for all cases

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

mesh standard first order tetrahedral elements
Figure 2: The mesh used for case A, created with SimScale’s standard meshing algorithm

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

mesh first order tetrahedral elements
Figure 3: The mesh used for case B, created externally with 1st order tetrahedral elements

Simulation Setup


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


  • 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:

$$\sigma_{zz} = −2798.3\ MPa$$


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]\(^1\):

CasePhysical Contact[SSNV104_A]SimScaleError (%)
(A)Augmented Lagrange-2798.3-2884.543.082%
(B)Augmented Lagrange-2798.3-3078.3910.009%
Table 2: The \(σ_{zz}\) results’ comparison for all cases A through D

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

hertzian contact two spheres cauchy stress σzz penalty contact
Figure 4: The \(σ_{zz}\) results on the spheres for case A with a penalty contact

Last updated: May 6th, 2021

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