This validation case belongs to thermomechanics, with the case of a bimetallic strip under thermal load. The aim of this test case is to validate the following parameters:

- Thermomechanical solver
- Multiple materials and bonded contact

The simulation results of SimScale were compared to theoretical computations derived from [Roark’s]\(^1\).

The geometry used for the case is as follows:

It represents a strip with length \(l\) of 10 \(m\), width \(w\) of 1 \(m\) and thickness \(t\) of 0.1 \(m\), composed of two strips each with thickness \(t_a , t_b\) of 0.05 \(m\). Nodes N1 and N3 are located at mid-thickness and node N2 is located at the bottom surface as shown in figure 1.

**Tool Type**: Code_Aster

**Analysis Type**: Thermomechanical steady state with static inertia effect

**Mesh and Element Types**:

Mesh Type | Number of Nodes | Element Type |
---|---|---|

2nd order hexahedral | 3652 | Standard |

The hexahedral mesh was computed locally and uploaded into the simulation project.

**Material**:

- Top Strip:
- Elastic Modulus \(E_a = \) 200 \(GPa\)
- Poison’s ratio \(\nu_a = \) 0
- Density \( \rho_a = \) 7870 \( kg/m^3 \)
- Thermal conductivity \( \kappa_a = \) 60 \( W/(mK) \)
- Expansion ratio \( \gamma_a = \) 1e-5 \( 1/K \)
- Reference temperature 300 \(K\)
- Specific heat \(C_{pa} = \) 480 \( J/(kgK) \)

- Bottom Strip:
- Elastic Modulus \(E_b = \) 200 \(GPa\)
- Poison’s ratio \(\nu_b = \) 0
- Density \( \rho_b = \) 7870 \( kg/m^3 \)
- Thermal conductivity \( \kappa_b = \) 60 \( W/(mK) \)
- Expansion ratio \( \gamma_b = \) 2e-5 \( 1/K \)
- Reference temperature 300 \(K\)
- Specific heat \(C_{pb} = \) 480 \( J/(kgK) \)

**Boundary Conditions**:

- Constraints:
- N1 restrained with \( d_x = d_y = d_z = \) 0 \(m\)
- N2 restrained with \( d_x = d_y = \) 0 \(m\)
- N3 restrained with \( d_y = \) 0 \(m\)
- Fixed temperature \(T = \) 400 \(K\) applied on top and bottom surfaces

- Contacts:
- Bonded contact between the strips

The reference solution is of the analytical type, as presented in [Roark’s]\(^1\). It is given in terms of the displacements of the free end of the strip and the stress at the bottom surface:

$$ d_x = l (T – T_0) \frac{\gamma_a + \gamma_b}{2} $$

$$ d_z = \frac{ 3 l^2 (\gamma_b – \gamma_a) (T – T_0)(t_a + t_b) }{ t_b^2 K_1} $$

$$ \sigma_{bottom} = \frac{ (\gamma_b – \gamma_a)(T – T_0) E_b }{ K_1 } \Big[ 3 \frac{t_a}{t_b} + 2 – \frac{E_a}{E_b} \Big( \frac{t_a}{t_b} \Big)^3 \Big] $$

$$ K_1 = 4 + 6 \frac{t_a}{t_b} + 4 \Big( \frac{t_a}{t_b} \Big)^2 + \frac{E_a}{E_b} \Big( \frac{t_a}{t_b} \Big)^3 + \frac{E_b}{E_a} \frac{t_b}{t_a} $$

The computed solutions are:

\(d_x= 0.015\ m\)

\(d_z = 0.75\ m\)

\(\sigma_{bottom} = 50\ MPa\)

A comparison of displacements at point N3 and stress \(\sigma_{XX}\) at point N2 with theoretical solution is presented below:

POINT | FIELD | COMPUTED | REF | ERROR |
---|---|---|---|---|

N3 | DX \([m]\) | 0.015 | 0.015 | 0.00 % |

N3 | DZ \([m]\) | 0.7479 | 0.75 | -0.28 % |

N2 | SIXX \([MPa]\) | 48.7631 | 50 | -2.47 % |

Illustration of the deformed shape and stress distribution on the bimetallic strip below:

Advanced Tutorial: Thermomechanical Analysis of an Engine Piston

References

- (2011)”Roark’s Formulas For Stress And Strain, Eighth Edition”, W. C. Young, R. G. Budynas, A. M. Sadegh

Note

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Last updated: May 19th, 2021

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