Interactively compute constrained thermal expansion, bimetal bending, and bilayer interface stress based on Timoshenko theory. Useful for solder joint stress analysis and bimetal thermometer design.
The fundamental equation for thermal stress in a material that is fully constrained (cannot expand or contract at all) is derived from Hooke's Law, where the strain from thermal expansion is counteracted by an equal and opposite mechanical strain.
$$ \sigma_{th}= -E \cdot \alpha \cdot \Delta T $$Where:
σth = Thermal stress (Pa)
E = Young's modulus, the material's stiffness (Pa)
α = Coefficient of thermal expansion (1/°C)
ΔT = Temperature change from the stress-free state (T - Tref) (°C)
The negative sign indicates the stress is compressive for a positive ΔT (heating).
For a bimetal strip, Timoshenko's theory provides the curvature. It balances the moments caused by the unequal thermal expansions of the two bonded layers.
$$ \kappa = \frac{6 (1+m)^2 (\alpha_2 - \alpha_1) \Delta T}{t \left[3(1+m)^2 + (1+mn)(m^2 + \frac{1}{mn}) \right]} $$Where:
κ = Curvature (1/m), the inverse of the bend radius.
α1, α2 = CTE of materials 1 & 2.
m = t1/t2, n = E1/E2 (thickness and stiffness ratios).
t = t1 + t2 = total thickness.
This equation shows how curvature depends not just on the CTE mismatch, but critically on the thickness and stiffness ratios you can adjust in the simulator.
Electronics Packaging & Solder Joint Reliability: Microchips, made of silicon, are soldered to circuit boards made of epoxy or metal. Every time you turn your phone or laptop on, it heats up. The large CTE mismatch causes cyclic stress on the tiny solder balls (BGA), leading to fatigue cracks. CAE simulations using this theory predict product lifespan and guide material selection.
Bimetal Thermostats & Circuit Breakers: The classic application. A strip of brass (high CTE) bonded to steel (low CTE) will bend predictably with temperature, making or breaking an electrical contact. This simple, reliable mechanism is used in millions of home thermostats, car turn-signal flashers, and overcurrent protectors.
Aerospace Structures: Aircraft and satellites experience extreme temperature swings from ground operation to high-altitude or space. Long, thin structures like solar panel booms or antenna masts, made of composite materials bonded to metals, can warp or induce high interface stresses if not carefully designed using these principles.
Civil Engineering & Rail Tracks: Concrete roads, bridges, and railway tracks are built with expansion joints to allow for thermal expansion. If these joints fail or are missing, the fully constrained thermal stress calculated by σ = EαΔT can cause concrete to buckle or rails to bend laterally (sun kink), leading to derailments.
A common initial pitfall in this type of analysis is the assumption that "simply choosing materials with a large difference in thermal expansion coefficients will always increase sensitivity." While the bending displacement may indeed be larger, the interfacial stress also skyrockets, increasing the risk of delamination. For instance, combining aluminum with resin can yield significant warpage, but the interfacial shear stress can be several times higher than in a steel-aluminum combination. In practice, the golden rule is to select materials by considering the trade-off between "required displacement" and "allowable stress."
Next, it's easy to forget the limitations of the "small deformation, linear theory" that underlies this simulator. When the temperature difference reaches several hundred degrees Celsius, the actual warpage will deviate from the theoretical value, and the material properties themselves change with temperature. Use this tool strictly as a guideline for initial design; a step involving verification via nonlinear analysis or experimentation is essential for detailed design. For example, in designing a bimetallic thermostat, it's common practice to use this tool to get an approximate curvature first, and then perform verification using finite element analysis that considers contact and large deformations.
Finally, be mindful of interpreting the boundary conditions. The tool's "fully constrained" condition is an ideal case; in reality, few things are perfectly fixed. For instance, the solder joints of a chip component on a substrate often experience lower actual stress than calculated because the substrate itself deforms slightly. Conversely, components built into a high-rigidity enclosure may experience stress close to or even exceeding the calculated value. How you assume the boundary conditions is what determines the accuracy of your analysis.