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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.

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Geometry

ParallelepipedWhoseYoungModulusIsFunctionOfTheTemperature-geometry

Geometry of the 3D box

  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
ParallelepipedWhoseYoungModulusIsFunctionOfTheTemperature-mesh-caseA

Mesh used for the SimScale Case A

ParallelepipedWhoseYoungModulusIsFunctionOfTheTemperature-mesh-caseC

Mesh used for the SimScale Case C

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:

ParallelepipedWhoseYoungModulusIsFunctionOfTheTemperature-youngsmodulus

Young’s modulus change over 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.

Comparison of the temperature on nodes nO and nD [°C]
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.

Comparison of the displacements on nodes nA and nD [m]
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]