Effects of Temperature Gradient on Structures — Thermal Stress, Deformation Mechanisms, and FEM Coupled Analysis

Good question. Indeed, if it's heated uniformly and can expand freely, the stress is zero. However, when a temperature gradient exists, the story is completely different. The high-temperature part tries to expand significantly, while the low-temperature part "doesn't want to expand that much yet"—this internal constraint generates stress even without external constraints.

Yes, this is the core of the temperature gradient problem. Let's think with a concrete example. Imagine a turbocharger turbine housing. The inner surface on the exhaust gas side is about 800°C, while the cooled outer surface is about 500°C. The inner surface wants to expand greatly, but the outer surface holds it back. As a result, compressive stress occurs on the inner surface, and tensile stress occurs on the outer surface. This stress distribution repeats with each engine start and stop, becoming the cause of thermal fatigue cracking.

Exactly. Stress increases in proportion to the temperature difference $\Delta T$. It is also proportional to Young's modulus $E$ and the coefficient of linear expansion $\alpha$. So aluminum ($\alpha \approx 23 \times 10^{-6}$/K) generates about twice the thermal strain of steel ($\alpha \approx 12 \times 10^{-6}$/K), but its Young's modulus is 1/3 that of steel, so the actual magnitude of thermal stress is determined by the combination of material properties.

First, consider the most basic case—a uniform material under plane stress, completely constrained. For a temperature change $\Delta T$, the free thermal strain is $\varepsilon_{th} = \alpha \Delta T$, but since it's completely constrained and cannot expand, this amount directly generates stress as elastic strain:

Here, $E$ is Young's modulus, $\alpha$ is the coefficient of linear expansion, and $\nu$ is Poisson's ratio. Dividing by $(1-\nu)$ accounts for the effect of biaxial constraint (constraint in two in-plane directions under plane stress). For uniaxial constraint, the denominator is 1; for triaxial constraint (suppression of volume change), it becomes $(1-2\nu)$. The negative sign indicates "temperature increase → compressive stress".

For steel, using $E = 200$ GPa, $\alpha = 12 \times 10^{-6}$ /K, $\nu = 0.3$, the calculation is:

Yes. For a plate of thickness $h$ with a through-thickness temperature distribution $T(y)$, the temperature effect can be separated into a membrane force component and a bending moment component. Taking the midplane as $y=0$:

Intuitively, $N_{th}$ is the "effect of the entire plate expanding uniformly," and $M_{th}$ is the "effect of the plate warping." If the temperature distribution is symmetric about the midplane, then $M_{th} = 0$ (no warping); if it's antisymmetric, then $N_{th} = 0$ (no expansion, only bending).

Exactly. For instance, if the top surface of a bridge deck in direct sunlight is 50°C and the shaded bottom surface is 30°C, assuming a linear gradient $T(y) = 40 + 20y/h$, both $N_{th}$ and $M_{th}$ occur. This thermal gradient bending causes long-span bridges to deform into an arch shape by several millimeters during the day—this is considered as a temperature load in bridge design codes.

Good observation. A bimetallic strip is a structure where two types of metal are bonded together, a device that converts temperature changes into curvature changes by utilizing the difference in their coefficients of linear expansion. There's a classical curvature formula derived by Timoshenko (1925):

Here, $m = t_1/t_2$ (thickness ratio of the two layers), $n = E_1/E_2$ (Young's modulus ratio), $h = t_1 + t_2$ (total thickness). When $m = 1$ (equal thickness) and $n = 1$ (equal stiffness), the curvature is maximized—meaning the combination with the same thickness and stiffness but different coefficients of linear expansion bends the most.

Exactly. In electronic packaging, when a silicon chip ($\alpha \approx 2.6 \times 10^{-6}$/K) is soldered onto an organic substrate ($\alpha \approx 15 \times 10^{-6}$/K), warping occurs upon cooling from the reflow oven due to the same principle as the bimetallic strip. This is a problem directly related to the reliability of solder joints.

Yes. For a flat plate with an in-plane temperature gradient $T(x, y)$, the governing equation of thermoelasticity can be written using the stress function $\phi$ as follows:

The $\nabla^2 T$ on the right-hand side is key. If the temperature distribution is a harmonic function (Laplacian = 0)—meaning under steady-state heat conduction with no internal heat source and a smooth temperature distribution—then the right-hand side becomes zero, and an in-plane temperature gradient alone does not generate internal stress. However, this is only true if the boundaries are free. If constrained at the boundaries, stress will of course occur.

Exactly. In a non-steady (transient) temperature field, $\nabla^2 T = (\rho c_p / k) \, \partial T / \partial t \neq 0$, so internal stress occurs even with free boundaries. Transient states involving rapid temperature changes, like engine startup or jet engine cycles, are the most dangerous—because the maximum temperature gradient occurs at that moment.

Typically, sequential coupling (one-way coupling) is used. The procedure has two steps:

Basically, using the same mesh provides more accurate temperature→displacement mapping. However, in practice, the mesh for thermal analysis (refined for fluid boundary layers) and the mesh for structural analysis (refined for stress concentration areas) are often different. In that case, interpolation mapping (nearest neighbor, projection method, RBF interpolation, etc.) is used to transfer the temperature field. In ANSYS Workbench, if the meshes are identical, they can be linked with a single flag, and even with different meshes, automatic mapping is performed.

Fourier's heat conduction equation is:

$\rho$ is density, $c_p$ is specific heat, $k$ is thermal conductivity, $\dot{q}$ is internal heat generation (e.g., Joule heating). Discretizing with FEM gives:

Good question. The structural FEM equation is $[K]\{u\} = \{F\}$, but the thermal load goes into the force vector $\{F\}$ on the right-hand side. The element-level thermal load vector is:

The decision criteria are the comparison between the thermal time constant of the structure and the speed of temperature change. The thermal time constant is $\tau = \rho c_p L^2 / k$, where $L$ is the characteristic length. If the temperature change is sufficiently slow compared to $\tau$ (i.e., it always nearly follows the steady state), then steady-state is OK. For rapid heating/cooling, transient analysis is essential.

In cases like turbochargers or gas turbines where the transient temperature gradient during startup is maximum, transient analysis is used to pass the temperature distribution at each time step to the structure. The most dangerous moment is often in the transient state, not the steady state, so "Is steady-state sufficient?" should be judged carefully.