Docs

Hertzian contact between two spheres

Overview

The aim of this test case is to validate the following functions:

  • frictionless penalty contact
  • frictionless augmented lagrange contact

The simulation results of SimScale were compared to the results presented in [SSNV104]. The meshes used in (A) and (C) were created with the automatic-tetrahedralization meshing algorithm on the SimScale platform. The additional meshes in (B) and (D) were locally meshed with refinements.

Import validation project into workspace

Geometry

HertzianContactBetweenTwoSpheres-geometry

Geometry of the sphere

Only one eighth of each of the two spheres (with radius R = 50 mm) is used for the analysis due to the symmetry of the problem.

Analysis type and Domain

Tool Type : Code_Aster

Analysis Type : Static - nonlinear

Mesh and Element types :

Case Mesh type Number of nodes Element type
(A) linear tetrahedral 36779 3D isoparametric
(B) linear tetrahedral 2586 3D isoparametric
(C) quadratic tetrahedral 37905 3D isoparametric
(D) quadratic tetrahedral 14672 3D isoparametric
HertzianContactBetweenTwoSpheres-mesh-a

Mesh used for the SimScale case (A) (similar to mesh (C))

HertzianContactBetweenTwoSpheres-mesh-b

Mesh used for the SimScale case (B) (similar to mesh (D))

Simulation Setup

Material:

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

Constraints:

  • Faces ACD and A’C’D zero x-displacement
  • Faces ABD and A’B’D zero y-displacement
  • Face ABC displacment of 2mm in z-direction
  • Face A’B’C’ displacment of -2mm in z-direction

Physical contact ‘Augmented Lagrange’:

  • Contact smoothing enabled for linear elements and disabled for quadratic elements
  • Frictionless
  • Augmentation coefficient = 100

Physical contact ‘Penalty’:

  • Contact smoothing enabled for linear elements and disabled for quadratic elements
  • Frictionless
  • Penalty coefficient = 10¹⁵

Reference Solution

\[\begin{equation}\label{ref1} \sigma_{ZZ} = \frac{-E}{\pi}\frac{1}{1-{\nu}^2}\sqrt{\frac{2h}{R}} \end{equation}\]\[\begin{equation}\label{ref2} h=2mm-−2mm=4mm \end{equation}\]

The equations used to solve the problem are presented in [SSNV104]. With equation \(\eqref{ref1}\) and equation \(\eqref{ref2}\) the stress \(\sigma_{ZZ}\) at point D results in \(\sigma_{ZZ}\) = −2798.3 MPa.

Results

Comparison of the stress \(\sigma_{ZZ}\) at point D obtained with SimScale with the analytical result of the reference solution [SSNV104_A].

Comparison of the stress \(\sigma_{ZZ}\) in [MPa] at point D
Case Physical Contact [SSNV104_A] SimScale Error (%)
(A) Penalty -2798.3 -2890.8 3.31%
(A) Augmented Lagrange -2798.3 -2893.7 3.41%
(B) Penalty -2798.3 -3141.2 12.25%
(B) Augmented Lagrange -2798.3 -3141.9 12.28%
(C) Penalty -2798.3 -2908.3 3.93%
(D) Penalty -2798.3 -2918.5 4.30%

References

[SSNV104_A](1, 2) Analytical result as stated in [SSNV104]
[SSNV104](1, 2, 3) SSNV104 - Contact de deux sphères