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Validation Case: Hyperelastic Equibiaxial Tension Test

This validation case belongs to an hyperelastic equibiaxial tension test in solid mechanics. The aim of this test case is to validate the following parameters:

  • Hyperelasticity under equibiaxial tension

The simulation results of SimScale were compared to the results derived from [Treloar]\(^1\).

Geometry

The geometry used for the case is as follows:

cube geometrical model
Figure 1: Geometry model for the cube

The cube has an edge length of 1 \(m\).

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Static Non Linear

Mesh and Element Types:

The mesh consists of one eight nodes hexahedral element, computed locally and uploaded to the platform.

CaseMesh TypeNumber of
Nodes
Element Type
ALL1st order Hexahedral 8Standard
Table 1: Finite element model for each case
one element hexahedral mesh for equibiaxial validation case, testing hyperelastic equibiaxial tension
Figure 2: One hexahedral element mesh for the cube

Simulation Setup

Material:

Hyperelasticity material, with the following parameters for each type of law:

  • Mooney-Rivlin:
    • \(C_{10} = \) 1.751 \(Pa\)
    • \(C_{01} = \) 4.625e-2 \(Pa\)
    • \(D_{1} = \) 1e-6 \(1/Pa\)
  • Neo Hooke:
    • \(C_{10} = \) 2.398 \(Pa\)
    • \(D_{1} = \) 1e-6 \(1/Pa\)
  • Signorini:
    • \(C_{10} = \) 5.365 \(Pa\)
    • \(C_{01} = \) -2.477 \(Pa\)
    • \(C_{20} = \) 6.337e-1 \(Pa\)
    • \(D_{1} = \) 1e-6 \(1/Pa\)

Boundary Conditions:

  • Constraints:
    • Face ABFE with zero x-displacement
    • Face AEHD with zero y-displacement
    • Face ABCD with zero z-displacement
    • Face DCGH with imposed 4 \(m\) x-displacement
    • Face BFGC with imposed 4 \(m\) y-displacement

Reference Solution

The reference solution is of the experimental type and was extracted from [Treloar]\(^1\). It corresponds with the nominal stress-strain material curve. The values were extracted using WebPlotDigitizer.

Result Comparison

Comparison of the nominal stress-strain curves computed from reaction force on the faces with zero displacement, versus the reference data is shown in Figure 3:

nominal stress strain result curves comparison for equibiaxial validation case
Figure 3: Nominal stress-strain curves comparison with Treloar \(^1\)

Following are the contours of the final deformed shape and the von Mises stress from the Money-Rivlin case results, where the hyperelastic equibiaxial tension load effect can be appreciated:

deformed shape color plot for equibiaxial validation case
Figure 4: Final deformed shape of the cube, with color representing deformation magnitude for the Money-Rivlin case

References

  • L. R. G. Treloar, “Stress-strain data for vulcanised rubber under various types of deformation”, Trans. Faraday Soc., 40:59–70, 1944.

Note

If you still encounter problems validating you simulation, then please post the issue on our forum or contact us.

Last updated: July 15th, 2020

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