Thermal Stress Analysis
Sequential Coupling, Thermal Fatigue & Electronics

Sequential coupling — also called one-way coupling — is the standard approach for the vast majority of thermal stress problems. Step 1: run a thermal analysis (steady-state or transient) using your heating/cooling boundary conditions. The output is a nodal temperature field. Step 2: read that temperature field into a structural analysis as a predefined "initial condition" or applied load at each node. The structural solver then converts nodal temperatures into thermal strains and computes the resulting stresses. The two analyses use the same mesh (or a mapped mesh if they differ). This is efficient because the thermal conductivity doesn't depend on the deformation — the two problems are weakly coupled and one-way transfer is accurate.

Full coupling is needed when the deformation meaningfully changes the thermal response — or vice versa. The main cases: (1) Thermoelastic damping in precision resonators and MEMS devices — vibration causes tiny temperature oscillations that dissipate energy; (2) Bulk heating due to plasticity — in metal forming or high-speed machining, plastic work converts to heat (usually ~90%) and that heat changes the material's yield stress; (3) Frictional contact problems where heat generation from sliding depends on contact pressure, which depends on thermal expansion. For most engineering components — electronics, turbine discs, exhaust manifolds — sequential coupling is perfectly adequate and much cheaper.

In a 3D isotropic material, the thermal strain acts identically to an initial strain — a uniform volumetric expansion. The thermal strain vector is:

The first three entries are the normal strain components (equal in all directions for isotropic expansion), and the last three are the shear strains (zero — thermal expansion doesn't shear). The mechanical (elastic) strain that generates stress is total strain minus thermal strain: \(\{\varepsilon_\text{mech}\} = \{\varepsilon_\text{total}\} - \{\varepsilon_\text{th}\}\). In the global stiffness equation, thermal loading appears as an equivalent nodal force vector \(\{F_\text{th}\} = \int [B]^T [D]\{\varepsilon_\text{th}\}\,dV\), which gets added to the right-hand side alongside mechanical loads.

The reference temperature \(T_\text{ref}\) (also called the stress-free temperature) is the temperature at which the structure is assumed to have zero thermal stress — the state at which it was manufactured or assembled with no thermal loading. Every temperature in your analysis is measured as \(\Delta T = T - T_\text{ref}\). If you set \(T_\text{ref}\) incorrectly, your thermal stress will be wrong by a constant offset. A common mistake: setting \(T_\text{ref} = 0°C\) for a structure that was stress-relieved at 200°C after welding — you'd overestimate residual thermal stress dramatically. For electronics solder joints, \(T_\text{ref}\) is typically the solder reflow temperature (~220°C for SAC305) — the joint is stress-free at that temperature and then cools down to room temperature, generating the initial thermal stress.

Thermal fatigue is real and it's nasty — but it's quite different from conventional high-cycle vibration fatigue. Each thermal cycle (ΔT heating and cooling) generates a plastic strain range Δϵ_p at stress concentrations. Since the temperature swings are large but infrequent (power-on/power-off cycles for electronics, cold start cycles for an engine manifold), you're in the low-cycle fatigue regime — typically 10³ to 10⁵ cycles to failure. The governing relationship is the Coffin-Manson law:

where \(\varepsilon_f'\) is the fatigue ductility coefficient and \(c\) is the fatigue ductility exponent (typically -0.5 to -0.7). The FEM workflow: run a nonlinear structural analysis over one thermal cycle with temperature-dependent plasticity, extract the stabilised plastic strain range, then use Coffin-Manson or IPC-9701 (for solder joints) to predict cycles to failure.