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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 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 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 mesh used in cases A and B was created using the standard algorithm within SimScale. The same mesh is used in both cases – the only difference between the runs is the element technology integration. Table 2 shows more details about the cases.

    CaseMesh TypeNumber of NodesElement TypeElement Technology
    (A)Standard2352nd order tetrahedralStandard
    (B)Standard2352nd order tetrahedralReduced integration
    Table 2: Mesh characteristics

    Find below the mesh used for cases A and B. It’s a standard mesh with second-order tetrahedral cells.

    cube second order standard mesh
    Figure 2: Second-order standard mesh used for cases A and B.

    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:

    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 for this primary creep validation case

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

    average creep strain plot
    Figure 3: Average creep strain plot for case B

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

    Last updated: July 21st, 2021