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3D Triaxial Load Primary Creep (NAFEMS Test 11)

This triaxial load primary creep validation case belongs to solid mechanics. This test case aims to validate the following parameters:

  • Creep material behavior.
  • Standard and reduced integration elements.
  • Automatic time stepping.

The simulation results of SimScale were compared to the analytical results derived from [NAFEMS_R27]\(^1\).


The geometry consists of a cube with edge length \(l\) = 0.1 \(m\).

cube geometry triaxial load primary creep validation
Figure 1: Cube geometry for the present validation project.

The coordinates for the points in the cube geometry are as tabulated below:

Table 1: Cube dimensions in meters.

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Nonlinear static

Mesh and Element Types: The meshes for cases A and B are second-order hexahedral one-element meshes. They were created locally and imported to SimScale. For cases C and D, the standard algorithm was used to generate a second-order tetrahedral mesh.

CaseMesh TypeNumber of NodesElement Type
(A)2nd order hexahedral20Standard
(B)2nd order hexahedral20Reduced integration
(C)2nd order standard235Standard
(D)2nd order standard235Reduced integration
Table 2: Mesh characteristics.

Find below the mesh used for case D. It’s a standard mesh with second-order tetrahedral cells.

cube second order standard mesh
Figure 2: Second-order standard mesh used for cases C and D.

Simulation Setup


  • Steel (linear elastic)
    • \(E\) = 200 \(GPa\)
    • \(\nu\) = 0.3
    • \(\rho\) = 7870 \(kg/m³\)
    • Creep formulation: Time hardening
      • A = 2.6041667e-46 \(1/s\)
      • N = 5
      • M = -0.5

Boundary Conditions:

  • Constraints
    • \(d_x\) = 0 on face ADHE;
    • \(d_y\) = 0 on face ABFE;
    • \(d_z\) = 0 on face ABCD.
  • Surface loads
    • \(t_x\) = 300 \(MPa\) on face BCGF;
    • \(t_y\) = 200 \(MPa\) on face CDHG;
    • \(t_z\) = 100 \(MPa\) on face EFGH.

Advanced Automatic Time Stepping:

For cases A through D, the following advanced automatic time stepping settings were defined under simulation control:

  • Retime event: field change;
  • Target field component: internal variable V1 (accumulated unelastic strain);
  • Threshold value: 0.0001;
  • Time step calculation type: mixed;
  • Field change target value: 0.00008.

Reference Solution

The equations used to solve the problem are derived in [NAFEMS_R27]\(^1\). As SimScale uses SI units, the reference solution was adopted to a time unit of seconds instead of hours.

$$\epsilon_{xx}^c = – \epsilon_{zz}^c = \frac{0.004218}{60} \sqrt{t} \tag{1}$$

$$\epsilon_{eff}^c = \frac{0.004871}{60} \sqrt{t} \tag{2}$$

$$\epsilon_{yy}^c = 0.0 \tag{3}$$

Result Comparison

Find below a comparison between SimScale’s results and the analytical solution presented in [NAFEMS_R27]\(^1\) for the average creep strain \(\epsilon_{xx}^c\) of the cube. The creep time is 3.6e6 \(s\) (equivalent to 1000 hours).

Case[NAFEMS_R27]SimScaleError (%)
Table 3: Comparison of SimScale’s results against an analytical solution.

In Figure 3, we can see how \(\epsilon_{xx}^c\), \(\epsilon_{yy}^c\), and \(\epsilon_{zz}^c\) are evolving for case D.

average creep strain plot
Figure 3: Average creep strain plot for case D.

\(\epsilon_{yy}^c\) and \(\epsilon_{zz}^c\) also show very good agreement with the analytical solution, having an error of 0% and -0.205%, respectively.

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