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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 \(m\).

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

A | B | C | D | E | F | G | H | |

x | 0 | 0.1 | 0.1 | 0 | 0 | 0.1 | 0.1 | 0 |

y | 0 | 0 | 0.1 | 0.1 | 0 | 0 | 0.1 | 0.1 |

z | 0 | 0 | 0 | 0 | 0.1 | 0.1 | 0.1 | 0.1 |

**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:

Case | Mesh Type | Nodes | Element Type | Creep Formulation | Element Technology |

(A) | Standard | 235 | 2nd order tetrahedral | Norton | Standard |

(B) | Standard | 235 | 2nd order tetrahedral | Norton | Reduced integration |

(C) | Standard | 235 | 2nd order tetrahedral | Time hardening | Standard |

(D) | Standard | 235 | 2nd order tetrahedral | Time hardening | Reduced integration |

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

**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

- Norton:

**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.

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*.

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

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

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

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] | SimScale | Error (%) |

(A) | 4.218 | 4.21875 | 0.0178 |

(B) | 4.218 | 4.21875 | 0.0178 |

(C) | 4.218 | 4.21875 | 0.0178 |

(D) | 4.218 | 4.21875 | 0.0178 |

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

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