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.
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.
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
Table 1: The final mesh details for all cases
Below the 1st order tetrahedral mesh for case A is visualized:
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:
Figure 3: The mesh used for case B, created externally with 1st order tetrahedral elements
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
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%
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:
Figure 4: The \(σ_{zz}\) results on the spheres for case A with a penalty contact
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