# 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.

## Geometry¶

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 |

## 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¶

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].

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% |