Thermal Expansion and Thermal Stress
Thermal Expansion and Thermal Stress: Theoretical Foundations
Thermal Expansion and Thermal Stress
Professor, what are the conditions for stress to occur due to thermal expansion?
Stress is zero if free expansion is possible. Thermal stress occurs only when expansion is constrained:
$\alpha$: Coefficient of linear expansion, $\Delta T$: Temperature change. For the case of full constraint.
Conditions for Thermal Stress Generation
Settings in FEM
Summary
Physical Origin of Coefficient of Thermal Expansion (CTE)
Thermal expansion in solids originates from the asymmetry (anharmonicity) of the interatomic potential. Under the harmonic approximation, thermal expansion would be zero; the Grüneisen constant γ (typically 1~3) represents the degree of this asymmetry. Steel (Fe) has a CTE≈11×10⁻⁶/°C, while aluminum has about 23×10⁻⁶/°C, a roughly twofold difference. When a steel-aluminum joint experiences a 400°C temperature difference, thermal stress can reach around 200 MPa (ΔT×ΔCTE×E≈400×12×10⁻⁶×42GPa). Invar (Fe-36Ni) alloy has an extremely low CTE≈1×10⁻⁶/°C and is used in precision instrument reference gauges and liquefied natural gas (LNG) tank structures.
Computational Methods for Thermal Expansion and Thermal Stress
FEM for Thermal Stress
1. Calculate temperature field (Heat conduction analysis) or directly specify temperature.
2. Calculate thermal stress in structural analysis — Temperature → Thermal strain → Stress.
Thermal strain: $\varepsilon_{th} = \alpha(T - T_{ref})$. $T_{ref}$: Stress-free temperature.
Summary
Thermal Stress Analysis Procedure (Steady-State / Transient)
The standard procedure for thermal stress analysis is a 3-step process: ① Heat conduction analysis (steady-state or transient) to calculate temperature distribution T(x,y,z,t), ② Transfer the temperature field to the structural solver (input temperature-dependent CTE, elastic modulus, yield stress as tables), ③ Calculate thermal strain εth = α(T)×(T−T_ref) at each node, separate it from mechanical strain, and perform mechanical analysis. For transient thermal stress, this must be repeated for all time steps, requiring computational costs tens to hundreds of times higher than steady-state analysis. Note that Ansys Mechanical 2024 R2 has improved memory efficiency for thermal-structural coupled analysis.
Thermal Expansion and Thermal Stress in Practice
Thermal Stress in Practice
Thermal deformation in electronic devices, thermal expansion in piping, engine cylinder blocks, structural response during fire.
Practical Checklist
Thermal Stress Management in Solid Rocket Nozzles
The nozzle throat section of solid rockets (e.g., H3 rocket's SRB-3) is made of C/C composite (carbon fiber reinforced carbon), reaching 3000°C during combustion. The coefficient of thermal expansion is strongly anisotropic: 1×10⁻⁶/°C in the fiber direction and 8×10⁻⁶/°C in the perpendicular direction. 3D FEM is used to analyze the thermal stress generated by the temperature difference between the inner and outer surfaces. In the SRB-3 qualification tests at JAXA's Kakuda Space Center, the maximum principal stress of 1200 MPa predicted by analysis was confirmed to match within ±10% of the strain measurements from optical fiber gauges during combustion tests.
Thermal Expansion and Thermal Stress: Software & Solver Comparison
Tools
Standard support in all FEM solvers. No difference.
Implementation of Coefficient of Thermal Expansion: ECTE vs ICTE Issue
There are two types of coefficients of thermal expansion: secant (ECTE: average from reference temperature) and tangent (ICTE: instantaneous). Confusion between solvers can lead to significant errors. ABAQUS and ANSYS standardly require ICTE input, but MSC Nastran's `MAT1` card requires ECTE (reference temperature 20°C). A paper records an instance where an aircraft engine case design had thermal stress overestimated by up to 40% due to mistakenly using the wrong input format between Nastran and ABAQUS.
Advanced Thermal Expansion and Thermal Stress: Modern Research & Trends
Advanced
Temperature Dependence of CTE and Nonlinear Thermal Stress
Most metals have increasing CTE at higher temperatures (due to the Dulong-Petit law). For Ti-6Al-4V, CTE is 8.6×10⁻⁶/°C at 20°C and 10.8×10⁻⁶/°C at 600°C. Linear calculations assuming constant CTE underestimate stress by 5~15%. Furthermore, for thermo-elastoplastic analysis after yielding, temperature-dependent hardening curves are also needed. In Abaqus/Standard, temperature-dependent CTE can be input in table format in the material card *EXPANSION, and combined with the *PLASTIC card, nonlinear thermal stress analysis is automatically applied.