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    What is Thermal Expansion?

    In a video shot from inside a railway engine, we see the Australian outback from the viewpoint of an engineer as his train slowly chugs along the tracks. Forty seconds into the video, kinks suddenly appear on the railway tracks. Seconds later, we see how the engine room violently shakes as it passes over the spaghetti-shaped railway. Thankfully, the train doesn’t derail and nobody is hurt. The kinks in the tracks are due to thermal expansion. When the rails get too hot under the blazing sun, they expand — leading to compressive stresses that cause buckling. Buckling is an ongoing problem in the railway industry. There are many safety measures in place that prevent buckling tracks. Thankfully, rail accidents due to buckled tracks have become extremely rare. Nevertheless, heat forces trains to reduce speeds which leads to delays, especially on hot summer days.

    buckled railway tracks due to thermal expansion
    Figure 1: Buckled railway tracks due to thermal expansion on an exceptionally hot day. Source: US Department of Transportation

    There are other areas where thermal expansion can cause damage such as construction or pipelines. Figure 2 shows an expansion joint in a bridge to accommodate the expansion of different parts of the bridge.

    bridge expansion joint in austria
    Figure 2: Expansion joint of a bridge over the Danube river in Austria. Source: Wikipedia

    Therefore, thermal expansion must be well-understood in engineering efforts involving heat transfer and high temperatures.

    Time for some thermal expansion physics!

    Changes in temperature lead to changes in the volumes of solids and liquids. To quantify this phenomenon, we use a material and temperature-dependent thermal expansion coefficient γ defined as 

    $$\gamma = \frac{1}{V_0} \left (\frac{\partial V}{\partial T} \right )_p\tag{1}$$

    For small changes in temperature, the change in relative volume \(\frac{\Delta V}{V_0} = \frac{V-V_0}{V_0}\) and the change in temperature \(\Delta T = T – T_0\) are approximately linearly related, such that we can define a mean thermal expansion coefficient:

    $$ \bar \gamma = \frac{\Delta V}{(V_0)(\Delta T)}\tag{2}$$

    In solids, we can simplify the problem to linear expansion where we consider a change in one dimension as opposed to a change in volume. In this case, the change in length is related to the temperature change by a coefficient of linear thermal expansion:

    $$ \alpha = \frac{1}{L_0} \left (\frac{\partial L}{\partial T} \right )_p , \bar \alpha = \frac{\Delta L}{L_0 \Delta T}\tag{3}$$

    The estimation of the mean coefficient of linear thermal expansion works well as long as the coefficient does not change much with the temperature and the change in length is small \(\frac{\Delta L}{L} \text{«} 1\). For isotropic materials, the volumetric thermal expansion is three times the linear coefficient \(\bar \gamma = 3 \bar \alpha\).

    Thermal stress analysis

    Thermal stresses in structures arise when the temperature rises uniformly in solids with physical constraints. A uniform temperature increase in a beam with fixed ends will induce compressive stresses equal to \(\sigma = E \epsilon_T – \alpha E \Delta T\). We derive this by the following superposition of the situations:

    thermal expansion diagram of the free expansion of a cantilever beam
    Figure 3: Superposition of the free expansion of a cantilever beam and a fixed-end beam to calculate compressive stresses due to thermal expansion  

    Non-uniform temperature distribution can also induce thermal stresses which in turn leads to non-uniform thermal expansion within the material. The general thermomechanical case involves a solid with partial mechanical constraints coupled with non-uniform temperature distributions. The temperature distribution, in this case, is described with: 

    $$ \frac{\partial}{\partial x} \left (k \frac{\partial T(x,y,z,t)}{\partial x} \right ) + \frac{\partial}{\partial y} \left (k \frac{\partial T(x,y,z,t)}{\partial y} \right ) + \frac{\partial}{\partial z} \left (k \frac{\partial T(x,y,z,t)}{\partial z} \right ) + Q(x,y,z,t) = \rho c \frac{\partial T(x,y,z,t)}{\partial t}\tag{4}$$

    The solution of the above equation is a necessary step preceding the thermal stress analysis of the solid figure shown in Figure 4. The first step in a thermal stress finite element analysis is the solution of the heat flow equation to calculate the temperature field. Subsequently, the temperature field is used as an equivalent to a body force to carry out the stress analysis.

    diagram of a solid with temperature field distribution
    Figure 4: Solid with temperature field distribution T, thermal fluxes q, heat source Q, and mechanical forces P

    Figure 4 shows a solid with a temperature field distribution T(x, y, z, t) produced by thermal fluxes \(q_1, q_2, q_3, … \) and a heat source or sink represented by Q. The solid is also subject to mechanical forces \(P_1, P_2, P_3, P_4, … \). If we discretize the solid into finite elements, the strain of the elements is described by:

    $$\begin{Bmatrix}\epsilon\end{Bmatrix} = \begin{Bmatrix}\epsilon _M\end{Bmatrix} + \begin{Bmatrix}\epsilon _T\end{Bmatrix}\tag{5}$$

    \(\begin{Bmatrix}\epsilon _M\end{Bmatrix}\) is the strain induced due to mechanical forces and stresses:

    $$\begin{Bmatrix}\epsilon _M\end{Bmatrix} = \begin{bmatrix}D\end{bmatrix}^{-1}\begin{Bmatrix}\sigma\end{Bmatrix}\tag{6}$$

    \(\begin{bmatrix}D\end{bmatrix}^{-1}\) is the inverse of the materials stiffness matrix. \(\begin{Bmatrix}\epsilon _T\end{Bmatrix}\) is the strain induced by the temperature change due to heat fluxes and heat sources:

    \(\begin{Bmatrix}\epsilon _T\end{Bmatrix} =
    \begin{pmatrix}
    \epsilon _{T,xx} \\
    \epsilon _{T,yy} \\
    \epsilon _{T,zz} \\
    \end{pmatrix} =
    \begin{pmatrix}
    \alpha \Delta T \\
    \alpha \Delta T \\
    \alpha \Delta T \\
    \end{pmatrix}\tag{7} \)

    After a bit of math (detailed in the sources at the end of this page), we arrive at the well-known element equation:

    $$\begin{bmatrix}K_e\end{bmatrix} \begin{Bmatrix}\Phi\end{Bmatrix} = \begin{Bmatrix}f_M\end{Bmatrix} + \begin{Bmatrix}f_T\end{Bmatrix} \tag{8}$$

    In this manner, thermal expansion is incorporated into the finite element method.

    Thermomechanical Analysis in SimScale

    The Thermomechanical module in SimScale couples the effects of structural loads and thermal expansion. It enables you to virtually test and predict the behavior of structures and hence solve complex structural engineering problems subjected to static and thermal loading conditions. The FEA simulation platform uses scalable numerical methods that can calculate mathematical expressions that would otherwise be very challenging due to complex loading, geometries, or material properties.

    You can try the Thermomechanical module on SimScale by creating an account and following this tutorial on a simulation of an engine piston. In the figure below, you see the results from the tutorial. On the left, is the temperature field, while in the middle we see the stress distribution. We can see how the stresses correlate with the temperature gradients, especially on the top of the piston head. On the right, is the deformation (20x scale). These simulations are very easy to set up and any user can run several such simulations in parallel to quickly generate useful results.

    With our cloud-native simulation platform, we empower every engineer to innovate faster by making high-fidelity engineering simulation technically and economically truly accessible at any scale.

    Happy simulating!

    References:

    [1] Logan, Daryl, A First Course in the Finite Element Method, Chapter 15: Thermal Stress, Cengage Learning, 2012.

    [2] Hsu, Tai-Ran, The Finite Element Method in Thermomechanics, Chapter 3: Thermoelastic-plastic Stress Analysis, Allen and Unwin, 1986.

    [3] Hsu, Tai-Ran, Lecture Notes on Thermal Stress Analysis of Solid Structures Using Finite Element Method, Introduction to Finite Element Method, San Jose University, 2017.

    Last updated: December 30th, 2022

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