Validation Case: Thick Hollow Thermoelastic Cylinder

This validation case belongs to thermomechanics, with the case of a thick hollow thermoelastic cylinder under temperature gradient conditions. The aim of this test case is to validate the following parameters:

• Thermomechanical solver

The simulation results of SimScale were compared to the analytical results presented in [HPLA100]\(^1\).

Geometry

The geometry used for the case is as follows:

It represents a quarter section of a hollow cylinder with an internal radius of 19.5 \(m\) and an external radius of 20.5 \(m\), length of 10.0 \(m\). Face ABCD coincides with the XZ plane, face EFGH with the YZ plane and face CHGD with the XY plane.

Analysis Type and Mesh

Tool Type: Code_Aster

Analysis Type: Thermomechanical, Linear, Steady State.

Mesh and Element Types:

Case	Mesh Type	Number of Nodes	Element Type
A, B	1st order tetrahedral	22105	Standard
C, D	2nd order tetrahedral	39674	Standard

The meshes were computed using SimScale’s standard mesh algorithm and automatic sizing.

Simulation Setup

Material:

• Young’s module \( E\) = 20 \( Pa \)
• Poisson coefficient \( \nu\) = 0.3
• Density \( \rho\) = 8000 \( kg/m^3 \)
• Expansion coefficient \( \alpha\) = 1e-5 \( 1/°C \)
• Reference temperature \( T_0\) = 0 \(°C\)

Boundary Conditions:

• Displacements
  • Fixed value DY = DZ = 0 on face ABCD
  • Fixed value DX = DZ = 0 on face EFGH
  • Fixed value DZ = 0 on faces CHGD and AFEB
• Temperatures (Cases A, C) – Bending deformation
  • Fixed temperature value of \(T\) = -0.5 \(°C\) on the internal face (BEHC)
  • Fixed temperature value of \(T\) = 0.5 \(°C\) on the external face (AGDF)
• Temperatures (Cases B, D) – Radial expansion
  • Fixed temperature value of \(T\) = 0.1 \(°C\) on the internal face (BEHC)
  • Fixed temperature value of \(T\) = 0.1 \(°C\) on the external face (AGDF)

Reference Solution

The reference solution for the thermoelastic cylinder is of the analytical type, as presented in [HPLA100]\(^1\). It is given in terms of the longitudinal stress onthe internal and external faces:

Cases A and C, internal face: \( \sigma_{ZZ}\) = 1.4321427 \( MPa \)

Cases A and C, external face: \( \sigma_{ZZ}\) = -1.4250001 \( MPa \)

Cases B and D, internal face: \( \sigma_{ZZ}\) = -0.2 \( MPa \)

Cases B and D, external face: \( \sigma_{ZZ} \) = -0.2 \( MPa \)

Result Comparison

Comparison of the computed stresses with the reference solution is presented, at the representative points:

CASE	POINT	SIMSCALE \( \sigma_{ZZ} \) \([Pa]\)	REFERENCE \( \sigma_{ZZ} \) \([Pa]\)	ERROR
A	C	1.1593Ee6	1.4321e6	19.1 %
	D	-1.1943e6	-1.4250e6	16.2 %
B	C	-2.0010e5	-2.0e5	0.0 %
	D	-1.9993e5	-2.0e5	0.0 %
C	C	1.4342e6	1.4321e6	-0.1 %
	D	-1.4228e6	-1.4250e6	0.2 %
D	C	-2.0000e5	-2.0e5	0.0 %
	D	-2.0000e5	-2.0e5	0.0 %

It is found that general results agree with the reference solution. As usual, the first order tetrahedral mesh gives bad accuracy on the bending case, although for the pure expansion it gives good results. It is expected that a finer mesh will improve the accuracy of the solution.

Illustration of the stress distribution for the bending case of the thermoelastic cylinder is shown below:

Related Tutorials

Advanced Tutorial: Thermomechanical Analysis of an Engine Piston

Last updated: September 30th, 2020