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> > Validation Case: Bimetallic Strip Under Thermal Load

# Problem Statement

Bimetallic strips are subject to bending due to heat and can be used to convert change in temperature to mechanical displacement finding application in thermostats for example. The associated physics coupling thermal and mechanical domains are amenable to analytical solution making them ideal for validation of the coupled thermal and mechanical solvers.

The resulting comparison of the simulation results against the analytical calculations validates the use of the following conditions for thermomechanical static analysis in Solve:

• Thermomechanical Solver
• Restraints

## Geometry

The goemetry is composed of two metal strips of equal size bonded to each other across their major surfaces. Each strip being 20~\text{mm} long, 0.1~\text{mm} thick and 0.3~\text{mm} wide.

PartMaterialLengthWidthThickness
Part 1Iron20~\text{mm}0.3~\text{mm}0.1~\text{mm}
Part 2Cast Iron20~\text{mm}0.3~\text{mm}0.1~\text{mm}

## Materials:

The materials assigned to the parts are taken directly from OnScale’s library of materials.

PartMaterialYoung’s ModulusThermal ExpansionThermal Conductivity
Part 1Cast Iron200~\text{GPa}1\times 10^{-5}~\text{K}^{-1}60~\text{Wm}^{-1}\text{K}^{-1}
Part 2Iron200~\text{GPa}2\times 10^{-5}~\text{K}^{-1}60~\text{Wm}^{-1}\text{K}^{-1}

The poisson’s ratio for both materials was set to zero so that the strain response was only in the direction of the applied stress, giving a more appropriate comparison to the 2D case of the analytical solution.

## Boundary Conditions A full displacement restraint was applied to the surface at one of the lengthwise ends of the bimetallic strip while a second restraint constraining motion in the direction perpendicular to the lengthwise axis and parallel to the plane interface between the two materials was applied to the opposite end. Since this was a coupled thermomechanical study, additionally, a temperature constraint of 100~^\circ\text{C} was applied to the exposed major face of part 2. The remaining surfaces defaulted to zero heat flux adiabatic condition with no explicitly applied loads or constraints.

## Meshing

OnScale automatically generates 3D second order tetrahedral meshes at 3 tiered densitites. The meshing statistics are:

Mesh TierNumber of VerticesNumber of CellsDegrees of Freedom
Fine11,07445,250295,204
Medium4,39214,066105,900
Coarse3,2509,75476,340

# Analytical Solution

The equations describing the deflection of the bimetallic strip presented here are found in :

d_x = l\cdot\left(T-T_\text{0}\right)\frac{\gamma_a -\gamma_b}{2} d_z = \frac{l^2}{2}\cdot\frac{6\left(\gamma_b -\gamma_a\right)\cdot\left(T-T_\text{0}\right)\cdot\left(t_a+t_b\right)}{t_b^2\cdot K_1} \sigma_{xx} = \frac{\left(\gamma_b -\gamma_a\right)\cdot\left(T-T_\text{0}\right)\cdot E_b}{K_1}\left(3\cdot\frac{t_a}{t_b}+ 2 - \frac{E_a}{E_b}\left(\frac{t_a}{t_b}\right)^3\right)

where l is the length of the bimetallic strip, t_a and t_b are the thicknesses of the top and bottom strips respectively, E_a and E_b are the Young’s moduli of the top and bottom strips, \gamma_a and \gamma_b are the coefficients of thermal expansion respectively. The temperature of the strips is T and T_\text{0} is the temperature at which the strip is not bent, which for this simulation is 20~^{\circ}\text{C}, \sigma_{xx} is the stress component at the base of the strip and K_1 is evaluated from:

K_1 = 4 + 6\cdot\frac{t_a}{t_b} + 4\left(\frac{t_a}{t_b}\right)^2 + \frac{E_a}{E_b}\left(\frac{t_a}{t_b}\right)^3 + \frac{E_b}{E_a}\cdot\frac{t_b}{t_a}

The respective symbols for the parameters for the analytical solution are in tbl. 1.

Table 1: Symbol indexing for different parts of the strip.
PartThicknessYoung’s ModulusThermal expansion
Part 1t_aE_a\gamma_a
Part 2t_bE_b\gamma_b

## Results

The values for d_x and d_z are taken from the middle of the free end of the strip, at the contact point between the top and bottom plate.