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Documentation

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]\(^1\).

Geometry

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 are as tabulated below:

ABCDEFGH
x00.10.1000.10.10
y000.10.1000.10.1
z00000.10.10.10.1
Table 1: Cube dimensions in meters.

Analysis Type and Mesh

Tool Type: Code Aster

Analysis Type: Nonlinear static

Mesh and Element Types: The mesh for cases A through F is a second-order hexahedral one-element mesh. It was created locally and imported to SimScale. For cases G through J, the standard algorithm was used to generate a second-order tetrahedral mesh.

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

CaseMesh TypeNodesCreep FormulationElement Type
(A)2nd order hexahedral20NortonStandard
(B)2nd order hexahedral20NortonReduced integration
(C)2nd order hexahedral20Time hardeningStandard
(D)2nd order hexahedral20Time hardeningReduced integration
(E)2nd order hexahedral20Strain hardeningStandard
(F)2nd order hexahedral20Strain hardeningReduced integration
(G)2nd order standard235NortonStandard
(H)2nd order standard235NortonReduced integration
(I)2nd order standard235Time hardeningStandard
(J)2nd order standard235Time hardeningReduced integration
Table 2: Overview of the mesh, creep formulation, and element technology used for each case.

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

cube second order standard mesh
Figure 2: Second-order standard mesh used for cases G through J in this secondary creep validation.

Simulation Setup

Material:

  • 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 (%)
(A)4.2184.218750.0178
(B)4.2184.218750.0178
(C)4.2184.218750.0178
(D)4.2184.218750.0178
(E)4.2184.218750.0178
(F)4.2184.218750.0178
(G)4.2184.218750.0178
(H)4.2184.218750.0178
(I)4.2184.218750.0178
(J)4.2184.218750.0178
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 E.

\(\epsilon_{yy}^c\) and \(\epsilon_{zz}^c\) also show very good agreement with the analytical solution, having an error of 0% and 0.0178%, respectively.

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