# Parallelepiped whose Young modulus is function of the temperature¶

## Overview¶

The aim of this test case is to validate the following functions:

- Steady-state thermomechanical analysis
- nonlinear material behavior

The simulation results of SimScale were compared to the results presented in [HPLV100]. Four of the mesh cases were considered; linear and quadratic tetrahedrals, linear and quadratic hexahedrals.

## Geometry¶

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

x [m] | 0 | 20 | 20 | 0 | 0 | 20 | 20 | 0 |

y [m] | 5 | 5 | 5 | 5 | -5 | -5 | -5 | -5 |

z [m] | 5 | 5 | -5 | -5 | 5 | 5 | -5 | -5 |

## Analysis type and Domain¶

**Tool Type** : Calculix, Code_Aster

**Analysis Type** : Steady-state thermomechanical

**Mesh and Element types** :

Case | Mesh type | Number of nodes | Number of 3D elements | Element type | Tool type |
---|---|---|---|---|---|

(A) | linear hexahedrals | 45 | 16 | 3D isoparametric | CalculiX |

(B) | quadratic hexahedrals | 141 | 16 | 3D isoparametric | CalculiX |

(C) | linear tetrahedrals | 44 | 90 | 3D isoparametric | CalculiX |

(D) | quadratic tetrahedrals | 217 | 90 | 3D isoparametric | CalculiX |

(E) | linear hexahedrals | 45 | 16 | 3D isoparametric | Code_Aster |

(F) | quadratic hexahedrals | 141 | 16 | 3D isoparametric | Code_Aster |

(G) | linear tetrahedrals | 44 | 90 | 3D isoparametric | Code_Aster |

(H) | quadratic tetrahedrals | 217 | 90 | 3D isoparametric | Code_Aster |

## Simulation Setup¶

Important

All temperature dependent data is given as a function of *°C*. This also applies for the data in the different simulations on the SimScale platform
(although it says *K*)!

Material:

- isotropic: E = 1000/(800 - T) N/m² where T = -35.5 °C to 75 °C, \(\nu\) = 0.3, \(\rho\) = 2 kg/m³, \(\kappa\) = 1 W/(m °C), \(\alpha\) = 0 1/°C, \(T_{r}\) = 20 °C

The temperature dependent Young’s Modulus was calculated using above mentioned formula. The graph below shows the change with respect to temperature:

Initial Conditions:

- Initial Temperature \(T_{initial}\) = 20°C

Constraints:

- Node
*nO*constrained in x,y and z direction - Node
*nB*constrained in x and z direction - Node
*nC*constrained in x direction

Loads:

- Pressure of -1 Pa on face ADHE and BCFG

Temperature:

- Temperature of 0°C on node nA

Heat flux:

- Surface heat flux of \(q_{s}\) = -2 W/m² on face BCFG
- Surface heat flux of \(q_{s}\) = 2 W/m² on face ADEH
- Surface heat flux of \(q_{s}\) = -3 W/m² on face ABCD
- Surface heat flux of \(q_{s}\) = 3 W/m² on face EFGH
- Surface heat flux of \(q_{s}\) = -4 W/m² on face ABFE
- Surface heat flux of \(q_{s}\) = 4 W/m² on face DCGH

## Results¶

Comparison of temperature on nodes *nO* and *nD* obtained with SimScale with the results presented in [HPLV100].
The Error was calculated with respect to [HPLV100] ¹.

node | Quantity [°C] | [HPLV100] ¹ [°C] | [HPLV100] ² [°C] | Case A [°C] | Error [%] | Case B [°C] | Error [%] | Case C [°C] | Error [%] | Case D [°C] | Error [%] | Case E [°C] | Error [%] | Case F [°C] | Error [%] | Case G [°C] | Error [%] | Case H [°C] | Error [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

nO | T | 40 | 39.99 | 40 | 0 | 40 | 0 | 40 | 0 | 40 | 0 | 40 | 0 | 40 | 0 | 40 | 0 | 40 | 0 |

nD | T | -35 | -35 | -35 | 0 | -35 | 0 | -35 | 0 | -35 | 0 | -35 | 0 | -35 | 0 | -35 | 0 | -35 | 0 |

Comparison of displacements on nodes *nA* and *nD* obtained with SimScale with the results presented in [HPLV100].
The Error was calculated with respect to [HPLV100] ¹.

node | Quantity [m] | [HPLV100] ¹ [m] | [HPLV100] ² [m] | Case A [m] | Error [%] | Case B [m] | Error [%] | Case C [m] | Error [%] | Case D [m] | Error [%] | Case E [m] | Error [%] | Case F [m] | Error [%] | Case G [m] | Error [%] | Case H [m] | Error [%] |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

nA | ux | 15.6000 | 15.5999 | 15.5978 | 0.01 | 15.6000 | 0.00 | 15.5966 | 0.02 | 15.6000 | 0.00 | 15.5988 | 0.01 | 15.6000 | 0.00 | 15.5966 | 0.02 | 15.6000 | 0.00 |

uy | -0.5700 | -0.5701 | -0.3784 | 33.61 | -0.5701 | 0.02 | -0.2205 | 61.31 | -0.5698 | 0.04 | -0.5086 | 10.76 | -0.5701 | 0.02 | -0.2205 | 61.31 | -0.5702 | 0.04 | |

uz | -0.7700 | -0.7700 | -0.5109 | 33.65 | -0.7700 | 0.00 | -0.5292 | 31.28 | -0.7701 | 0.01 | -0.6854 | 10.99 | -0.7700 | 0.00 | -0.5292 | 31.28 | -0.7694 | 0.08 | |

nD | ux | 16.3000 | 16.3000 | 16.0624 | 1.46 | 16.3000 | 0.00 | 16.0057 | 1.81 | 16.2999 | 0.00 | 16.2610 | 0.24 | 16.3000 | 0.00 | 16.0057 | 1.81 | 16.2998 | 0.00 |

uy | -1.7850 | -1.7850 | -1.5831 | 11.31 | -1.7851 | 0.01 | -1.4431 | 19.15 | -1.7848 | 0.01 | -1.7167 | 3.83 | -1.7851 | 0.01 | -1.4431 | 19.15 | -1.7852 | 0.01 | |

uz | -2.0075 | -2.0075 | -1.7310 | 13.78 | -2.0075 | 0.00 | -1.7352 | 13.56 | -2.0076 | 0.00 | -1.9137 | 4.67 | -2.0075 | 0.00 | -1.7352 | 13.56 | -2.0069 | 0.03 |

## References¶

[HPLV100] | (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) HPLV100 - Parallélépipède dont le module d’Young est fonction de la température |

¹ Results by S. ANDRIEUX as mentioned in [HPLV100]: Une solution analytique pour un problème d’élasticité linéaire 3D isotrope avec module d’Young fonction des variables d’espace [V4.90.01]

² Code_Aster results as mentioned in [HPLV100]