熱衝撃解析
Theory and Physics
Thermal Shock
Professor, what is thermal shock?
Thermal stress generated by rapid temperature changes. Typical in ceramics, glass, refractories. When the surface is rapidly cooled, tensile stress occurs on the surface → cracking.
Thermal Shock Resistance
Kingsley's R factor: $R = \sigma_f(1-\nu)/(E\alpha)$. Higher R indicates better thermal shock resistance.
Summary
Fracture Mechanics of Thermal Shock: Thermal Shock Coefficient R
Thermal Shock is instantaneous thermal stress caused by rapid temperature changes, a primary failure mechanism causing cracks in ceramics. Hasselman (1969, GE R&D Center) defined thermal shock resistance as R=σf(1−ν)/(Eα) (first thermal shock resistance, critical temperature difference for crack initiation). High-toughness ceramic ZrO₂ (PSZ) has an R about 3 times that of Al₂O₃, one reason it's used for insulation/TBC coatings. Glass (soda-lime) has an R of about 80°C, and quenching experiments have demonstrated a 50% failure probability when dropped into 100°C water.
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 so acceleration is negligible". It 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. Now a question—an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. 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"—different concepts.
- External Force Term (Load Term): Body force $f_b$ (gravity, etc.) and surface force $f_s$ (pressure, contact force, etc.). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression"—sounds like a joke, but it actually happens when coordinate systems rotate 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. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—intentionally absorbing vibration energy for a smoother ride. What if damping were zero? Buildings would keep shaking 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, stress-strain relationship is linear
- Isotropic material (unless specified otherwise): Material properties are independent of direction (anisotropic materials require separate tensor definition)
- Quasi-static assumption (for static analysis): Ignores inertial/damping forces, considers only 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 extension is needed
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load/elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Note unit inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note distinction between engineering strain and logarithmic strain (for large deformation) |
| 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 Shock
1. Transient Heat Conduction Analysis — Temperature distribution changing over time
2. Structural Analysis at Each Time Step — Temperature distribution → thermal stress
3. Identify Time of Maximum Stress — When temperature gradient is maximum
Summary
Procedure for Transient Thermal Stress Analysis
Transient analysis for thermal shock follows the flow: ① Set boundary conditions for instantaneous cooling/heating (surface heat transfer coefficient h value is dominant), ② Time-integrate the transient heat conduction equation using an implicit method (Crank-Nicolson method), ③ Calculate thermal strain + elastic stress from the temperature distribution at each time step. For ceramics, low thermal conductivity creates temperature differences of several hundred °C between surface and interior, causing rapid stress changes (tension/compression) on the surface. This process can be automated with Ansys Transient Thermal → Static Structural coupled analysis, with a recommended time step of 1/10 or less of the thermal diffusion time (L²/α).
Linear Elements (1st Order Elements)
Linear interpolation between nodes. Low computational cost but low stress accuracy. Beware of shear locking (mitigated with reduced integration or B-bar method).
Quadratic Elements (with Mid-side Nodes)
Can represent 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 mode (zero-energy mode). Choose appropriately.
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 each iteration. Quadratic convergence within convergence radius, but high computational cost.
Modified Newton-Raphson Method
Updates tangent stiffness matrix using initial value or every few iterations. Cost per iteration is low, but convergence 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 total load incrementally in small steps rather than all at once. The arc-length method (Riks method) can trace beyond limit 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 accuracy improves with each iteration. It's the same principle as looking up a word in a dictionary: it's more efficient to estimate where to open it and adjust forward/backward (iterative method) than to search 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 a "flexible curve"—can represent curved changes, dramatically improving accuracy even at the same mesh density. However, computational cost per element increases, so judge based on total cost-effectiveness.
Practical Guide
Practical Checklist
Thermal Shock Evaluation for Nuclear Reactor Emergency Core Cooling
During activation of a nuclear reactor's Emergency Core Cooling System (ECCS), low-temperature cooling water (~20°C) is rapidly injected into the high-temperature reactor pressure vessel (~320°C). This 300°C temperature difference thermal shock (Pressurized Thermal Shock, PTS) generates transient tensile stress up to 400 MPa on the vessel wall. US NRC Regulatory Guide 1.99Rev.2 requires fracture toughness evaluation considering irradiation embrittlement (RTNDT transition temperature). Evaluation combining Westinghouse's HEATH analysis code with 3D-FEM has become an international standard. In Japan, Toshiba Energy Systems & Solutions conducts equivalent evaluations.
Analogy for Analysis Flow
The analysis flow is actually very similar to cooking. First, buy ingredients (prepare CAD model), do prep work (mesh generation), apply heat (solver execution), and finally plate it (visualization in post-processing). Here's an important question—in cooking, which step is most prone to failure? Actually, it's the "prep work". If mesh quality is poor, results will be a mess no matter how good the solver is.
Common Pitfalls for Beginners
Are you checking mesh convergence? Do you think "calculation ran = 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 is far from reality. Confirm that results stabilize across at least three mesh densities—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 real-world constraint conditions is often the most critical step in the entire analysis.
Software Comparison
Tools
All FEM solvers support thermal-structural coupling. No difference.
Solver-Specific Approach Comparison for Thermal Shock Analysis
Methods for thermal shock analysis differ significantly by solver. ABAQUS/Explicit can track contact/fracture using explicit methods and was adopted for delamination analysis of GE's gas turbine thermal barrier coatings (TBC). ANSYS tracks crack growth with ADPCM (Adaptive Thermo-Mechanical Coupling) mesh. MSC Nastran has provided a Thermo-Mechanical Fatigue module specialized for 1,000+ cycles since 2019.
Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for thermal shock analysis? For example, presence of LES support for fluids, contact/large deformation capability for structures creates differences.
- "Who will use it?": For beginner teams, tools with rich GUI are suitable; for experienced users, flexible script-driven tools are better. Similar to the difference between automatic (GUI) and manual (script) transmission cars.
- "How far will you expand?": Selection considering future analysis scale expansion (HPC support), deployment to other departments, and integration with other tools leads to long-term cost reduction.
Advanced Technology
Advanced
Fracture Mechanics of Thermal Shock: Ceramic Quenching Experiments
Thermal shock fracture in ceramics was quantified by Kingery's 1955 alumina specimen quenching experiment. Quenching from 900°C into water initiates cracking at ΔT ≥ 200°C, and the critical temperature difference can be predicted by the product of Biot number and fracture toughness KIC. In modern solar panel manufacturing, thermal shock during quenching is analyzed with ABAQUS/Explicit, and cooling rate designs achieving 95% survival rate for Si3N4 substrates with KIC=2.0 MPa√m have been commercialized.
Troubleshooting
Troubles
Accuracy Issue in Setting Heat Transfer Coefficient (h value)
The surface heat transfer coefficient h value setting greatly influences thermal shock analysis results. In quenching experiments (e.g., dropping high-temperature ceramic into water), boiling heat transfer occurs, with h = 10,000–50,000 W/m²K in nucleate boiling region and h = 200–400 W/m²K in film boiling region—a difference of over 100 times. Assuming a constant h value can cause maximum thermal stress to deviate by over 50%, as shown in a 2015 report by the Japan Fine Ceramics Center (JFCC, Nagoya). Identifying h by combining experimental IR thermometer data with inverse heat conduction analysis is key to improving accuracy.
When You Think "The Analysis Doesn't Match"
- First, take a deep breath—Panicking and randomly changing settings makes the problem more complex
- Create a minimal reproducible case—Reproduce the thermal shock analysis problem in its simplest form. "Subtractive debugging" is most efficient
- Change one thing and re-run—Changing multiple things simultaneously makes it unclear what worked. The principle of "control experiment" same as in science
- Return to physics—If results are non-physical like "objects floating against gravity", suspect fundamental input data errors
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