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

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 :

Simulation Setup

Material:

• isotropic: E = 20 GPa, ν
  $\nu$

   

  = 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

σZZ=−Eπ11−ν22hR−−−√(1)

h=2mm−−2mm=4mm(2)

The equations used to solve the problem are presented in [SSNV104]. With equation (1)

 and equation (2)

 the stress σZZ

 at point D results in σZZ

 = −2798.3 MPa.

Results

Comparison of the stress σZZ

 at point D obtained with SimScale with the analytical result of the reference solution [SSNV104_A].

References