Required field
Required field
Required field

# Parallelepiped whose Young Modulus is Function of the Temperature

## Overview

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

• 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.

Import validation project into workspace

## Geometry

ABCDEFGH
x [m]020200020200
y [m]5555-5-5-5-5
z [m]55-5-555-5-5

## Analysis type and Domain

Tool Type : Calculix, Code_Aster

Mesh and Element types :

CaseMesh typeNumber of nodesNumber of 3D elementsElement typeTool type
(A)linear hexahedrals45163D isoparametricCalculiX
(C)linear tetrahedrals44903D isoparametricCalculiX
(E)linear hexahedrals45163D isoparametricCode_Aster
(G)linear tetrahedrals44903D isoparametricCode_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, ν

$\nu$

= 0.3, ρ

$\rho$

= 2 kg/m³, κ

$\kappa$

= 1 W/(m °C), α

$\alpha$

= 0 1/°C, Tr

${T}_{r}$

= 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 Tinitial

${T}_{initial}$

= 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

• Pressure of -1 Pa on face ADHE and BCFG

Temperature:

• Temperature of 0°C on node nA

Heat flux:

• Surface heat flux of qs

${q}_{s}$

= -2 W/m² on face BCFG

• Surface heat flux of qs

${q}_{s}$

= 2 W/m² on face ADEH

• Surface heat flux of qs

${q}_{s}$

= -3 W/m² on face ABCD

• Surface heat flux of qs

${q}_{s}$

= 3 W/m² on face EFGH

• Surface heat flux of qs

${q}_{s}$

= -4 W/m² on face ABFE

• Surface heat flux of qs

${q}_{s}$

= 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]
nodeQuantity [°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 [%]
nOT4039.99400400400400400400400400
nDT-35-35-350-350-350-350-350-350-350-350

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]
nodeQuantity [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 [%]
nAux15.600015.599915.59780.0115.60000.0015.59660.0215.60000.0015.59880.0115.60000.0015.59660.0215.60000.00
uy-0.5700-0.5701-0.378433.61-0.57010.02-0.220561.31-0.56980.04-0.508610.76-0.57010.02-0.220561.31-0.57020.04
uz-0.7700-0.7700-0.510933.65-0.77000.00-0.529231.28-0.77010.01-0.685410.99-0.77000.00-0.529231.28-0.76940.08
nDux16.300016.300016.06241.4616.30000.0016.00571.8116.29990.0016.26100.2416.30000.0016.00571.8116.29980.00
uy-1.7850-1.7850-1.583111.31-1.78510.01-1.443119.15-1.78480.01-1.71673.83-1.78510.01-1.443119.15-1.78520.01
uz-2.0075-2.0075-1.731013.78-2.00750.00-1.735213.56-2.00760.00-1.91374.67-2.00750.00-1.735213.56-2.00690.03

## 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]

Last updated: January 8th, 2021