Thermal Expansion and Thermal Stress
Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. So here's a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common pitfall here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades away. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would keep swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic inhomogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic Material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-Static Assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-Applicable Cases: For large deformation/large rotation problems, geometric nonlinearity is required. For nonlinear material behavior like plasticity and creep, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system. |
Numerical Methods and Implementation
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 (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.
Linear Elements (1st-Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated by reduced integration or B-bar method).
Quadratic Elements (with Midside Nodes)
Capable of representing curved deformation. Stress accuracy improves significantly, but degrees of freedom increase by about 2~3 times. Recommended when stress evaluation is critical.
Full Integration vs Reduced Integration
Full Integration: Risk of over-constraint (locking). Reduced Integration: Risk of hourglass modes (zero-energy modes). Choose appropriately for the situation.
Adaptive Mesh
Automatic refinement based on error indicators (e.g., ZZ estimator). Efficiently improves accuracy in stress concentration areas. Includes h-method (element subdivision) and p-method (order increase).
Newton-Raphson Method
Standard method for nonlinear analysis. Updates tangent stiffness matrix every iteration. Provides quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using the initial value or every few iterations. Cost per iteration is low, but convergence speed is linear.
Convergence Criteria
Force residual norm: $||R|| / ||F_{ext}|| < \epsilon$ (typically $\epsilon = 10^{-3}$ to $10^{-6}$). Displacement increment norm: $||\Delta u|| / ||u|| < \epsilon$. Energy norm: $\Delta u \cdot R < \epsilon$.
Load Increment Method
Applies the full load not all at once, but in small increments. The arc-length method (Riks method) can trace beyond extremum points on the load-displacement curve.
Analogy: Direct Method vs. Iterative Method
The direct method is like "solving simultaneous equations accurately with pen and paper"—reliable but takes too long for large-scale problems. The iterative method is like "repeatedly guessing to approach the correct answer"—starts with a rough answer but improves accuracy with each iteration. It's the same principle as looking up a word in a dictionary: opening to an estimated page and adjusting forward/backward (iterative method) is more efficient than searching sequentially from the first page (direct method).
Relationship Between Mesh Order and Accuracy
1st-order elements are like "approximating a curve with a ruler"—represented by straight line segments, so accuracy is limited. 2nd-order elements are like "flexible curves"—can represent curved changes, dramatically improving accuracy even with the same mesh density. However, computational cost per element increases, so judgment should be based on total cost-effectiveness.
Practical Guide
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.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, you buy the ingredients (prepare the CAD model), do the prep work (mesh generation), put it on the heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—which step in cooking is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Pitfalls Beginners Often Fall Into
Are you checking mesh convergence? Do you think "the calculation ran = the results are correct"? This is actually the most common trap for CAE beginners. The solver will always return "some answer" for the given mesh. But if the mesh is too coarse, that answer can be far from reality. Confirm that results stabilize across at least three levels of mesh density—neglecting this leads to the dangerous assumption that "the computer gave the answer, so it must be correct".
Thinking About Boundary Conditions
Setting boundary conditions is like "writing the problem statement" for an exam. If the problem statement is wrong? No matter how accurately you calculate, the answer will be wrong. "Is this surface really fully fixed?" "Is this load really uniformly distributed?"—Correctly modeling the real-world constraint conditions is often the most critical step in the entire analysis.
Software 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.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models and element types required for thermal expansion and thermal stress? For example, in fluids, the presence of LES support; in structures, the ability to handle contact and large deformations can differ.
- "Who will use it?": For beginner teams, tools with rich GUIs are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between an automatic transmission car (GUI) and a manual transmission car (script).
- "How far will you expand?": Selection considering future expansion of analysis scale (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
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.
Troubleshooting
Troubleshooting
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