Thermo-Mechanical Design

No, evaluating thermal stress is just one small aspect. What's more important is the degradation of dimensional accuracy due to thermal deformation. For example, the stage of a semiconductor lithography system requires 0.1nm accuracy, but a temperature fluctuation of just 0.01°C can cause deformation of several nm.

Exactly. That's why "thermo-mechanical design" includes everything from the selection of special materials like Invar (Fe-Ni low-expansion alloy, CTE ≈ 1.2×10⁻⁶/K), optimal placement of cooling channels, to the design of constraint conditions. It's not just about calculating $\sigma = E \alpha \Delta T$ and being done.

These five fields are typical:

You can see how scary it is with numbers. For example, if the temperature of a 1m long steel piece (CTE ≈ 12×10⁻⁶/K) rises by 10°C:

0.12mm—it's not a problem for ordinary mechanical design, but it's fatal for machine tools with μm-level machining accuracy. Moreover, because the temperature distribution is not uniform, you get "warping" and "twisting," not just simple expansion. This is the true fear of thermal deformation.

Exactly. That's why at the design stage, you need to think from both sides: "how to minimize the non-uniformity of temperature distribution" and "how to withstand the remaining non-uniformity." The former is cooling design, the latter is material selection and constraint design.

Yes. For example, on an electronic substrate, a silicon chip (CTE ≈ 2.6×10⁻⁶/K) is soldered onto an FR-4 substrate (CTE ≈ 14×10⁻⁶/K). During temperature cycles, the CTE difference directly becomes interfacial shear stress.

Here $\gamma$ is the shear strain in the solder, $L$ is the half-width of the chip, $h_{solder}$ is the solder height. The larger the CTE difference, the larger the chip, and the thinner the solder, the greater the shear strain. This is the root cause of BGA solder cracks.

Sharp observation. And the same problem exists in turbine blades. TBC (Thermal Barrier Coating, ceramic-based, CTE ≈ 10×10⁻⁶/K) is sprayed onto a Ni superalloy, but the CTE of the Ni superalloy is about 13×10⁻⁶/K. This difference causes TBC delamination during repeated start-stop cycles.

First, the heat conduction equation (transient):

Here $\rho$ is density, $c_p$ is specific heat, $k$ is thermal conductivity, $Q$ is internal heat generation. And the structural equilibrium equation:

The key to coupling is the constitutive law. Elastic strain is found by subtracting thermal strain from total strain:

That's the basics. However, when material properties like elastic modulus $E(T)$ or yield stress $\sigma_y(T)$ are temperature-dependent, the structural behavior depends on temperature, so even "one-way coupling" accurately reflects temperature effects. Furthermore, when structural deformation changes contact conditions and alters thermal resistance (e.g., contact thermal conductance in bolted joints), bidirectional coupling is needed.

That's a major pitfall. There are many cases where transient stress is greater than steady-state stress. Take a steam turbine rotor as an example. During startup, when hot steam hits the outer surface, the outside heats up quickly but the inside is still cold. At this time, a large temperature gradient occurs with compression on the outside and tension on the inside.

When steady-state is reached, the temperature gradient relaxes and stress decreases. In other words, the transient conditions during startup and shutdown are the most severe. In practice, steam turbine manufacturers always perform transient analysis for each startup pattern (cold start / warm start / hot start) to determine the allowable heating rate (°C/min).

That's why in thermo-mechanical design, transient analysis of the entire lifecycle (startup → steady-state → shutdown → restart) is essential. This is the point that is fundamentally different from simple "thermal stress calculation."

There are three main approaches:

A typical example is forging or friction stir welding (FSW). Heat generation from plastic deformation $Q_{plastic} = \eta \cdot \sigma : \dot{\varepsilon}^p$ ($\eta$ is the thermal conversion coefficient of plastic work, typically 0.9) raises temperature, which changes material properties, which changes deformation behavior—a complete feedback loop. Abaqus's `*Coupled Temperature-Displacement` is a representative step that handles this strong coupling.

The structural finite element equation is:

Here, the thermal load vector $\{\mathbf{F}_{th}\}$ is obtained from the thermal strain of each element:

$\mathbf{B}$ is the strain-displacement matrix, $\mathbf{D}$ is the elasticity matrix. In other words, through the transformation "temperature difference → thermal strain → equivalent nodal forces," the thermal effect enters the right-hand side of the structural equation in the same form as mechanical loads. This makes implementation easy because it can be solved with the same solver.

Using the backward Euler method for FEM discretization of the heat conduction equation:

$[\mathbf{C}]$ is the heat capacity matrix, $[\mathbf{K}_T]$ is the thermal conductivity matrix. Backward Euler (implicit method) is unconditionally stable, so you can use a large time step, but a sufficiently small time step is needed to capture transient temperature peaks. Guidelines:

This is a really important point in practice. Thermal analysis may converge with a coarse mesh, but structural analysis needs a fine mesh at stress concentration areas. When meshes are non-matching, temperature data mapping (transfer) is needed: