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    3D Triaxial Load Secondary Creep (NAFEMS Test 6(a))

    This triaxial load secondary 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 an 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 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: For this validation case, a single second-order tetrahedral mesh was created in SimScale with the standard algorithm.

    The secondary creep formulation and element technology vary from case to case. The table below contains an overview of the configurations:

    CaseMesh TypeNodesElement TypeCreep FormulationElement Technology
    (A)Standard2352nd order tetrahedralNortonStandard
    (B)Standard2352nd order tetrahedralNortonReduced integration
    (C)Standard2352nd order tetrahedralTime hardeningStandard
    (D)Standard2352nd order tetrahedralTime hardeningReduced integration
    Table 2: Overview of the mesh, creep formulation, and element technology used for each case.

    Find below the second-order standard mesh used in this validation case:

    cube second order standard mesh
    Figure 2: This second-order mesh consists of tetrahedral elements.

    Simulation Setup


    • Steel (linear elastic)
      • \(E\) = 200 \(GPa\)
      • \(\nu\) = 0.3
      • \(\rho\) = 7870 \(kg/m³\)
      • Three creep formulations are used. Find here the parameters for each of them.
        • Norton:
          • A = 8.6805556e-48 \(1/s\)
          • N = 5
        • Time hardening:
          • A = 8.6805556e-48 \(1/s\)
          • N = 5
          • M = 0
        • Strain hardening:
          • A = 8.6805556e-48 \(1/s\)
          • N = 5
          • M = 0

    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 all cases, 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.

    $$\epsilon_{xx}^c = – \epsilon_{zz}^c = \frac{0.004218}{3600} t \tag{1}$$

    $$\epsilon_{eff}^c = \frac{0.004871}{3600} t \tag{2}$$

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

    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 the analytical solution for this secondary creep validation case.

    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 0.0178%, respectively.

    Last updated: July 21st, 2021