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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 \(m\).

The coordinates for the points in the cube geometry 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**: 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.

Case | Mesh Type | Number of Nodes | Element Type | Element Technology |

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

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

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

**Material**:

- 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.

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*.

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

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

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

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) | 0.133380 | 0.133107 | -0.205 |

(B) | 0.133380 | 0.133107 | -0.205 |

In Figure 3, we can see how \(\epsilon_{xx}^c\), \(\epsilon_{yy}^c\), and \(\epsilon_{zz}^c\) are evolving 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: November 29th, 2023

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