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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 were created with the standard meshing algorithm on the SimScale platform. While a single region refinement is used in the meshes (A) and (C), the meshes in (B) and (D) were created with an additional region refinement around the contact region.

    CaseElement TypeNumber of NodesElement Technology
    (A)1st Order Tetrahedral1559Standard
    (B)1st Order Tetrahedral8229Standard
    (C)2nd Order Tetrahedral34320Reduced Integration
    (D)2nd Order Tetrahedral58973Reduced Integration
    Table 1: The final mesh details for all cases

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

    mesh standard first order tetrahedral elements
    Figure 2: A global refinement is applied in the case A mesh.

    And the mesh (case B), which has a region refinement around the contact region is presented below:

    standard mesh with a local region refinement
    Figure 3: The mesh used for case B, created with an extra local refinement around the contact region

    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] (MPa)SimScale (MPa)Error (%)
    (A)Augmented Lagrange-2798.3-2836.321.359%
    (B)Augmented Lagrange-2798.3-2830.211.140%
    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 (case B), using the Augmented Lagrange contact:

    hertzian contact two spheres cauchy stress σzz penalty contact
    Figure 4: The \(σ_{zz}\) results on the spheres for case B with an Augmented Lagrange contact method

    Last updated: September 24th, 2021