Thermo-mechanical cycle fatigue (TMF)
Theory and Physics
What is TMF?
Professor, is TMF the same as thermal fatigue?
TMF (Thermo-Mechanical Fatigue) is a problem where temperature and mechanical loads cycle simultaneously. Simple thermal fatigue involves only temperature cycling, but TMF includes simultaneous fluctuations in pressure or centrifugal forces. Engine cylinder heads and turbine blades are typical examples.
IP vs. OP
Is there a big difference in lifespan between IP and OP?
It can differ by several times to ten times. OP often has a shorter lifespan (due to accelerated oxidation).
Summary
Definition and Types of TMF (Thermo-Mechanical Fatigue)
Thermo-Mechanical Fatigue (TMF) is fatigue where temperature cycles and mechanical strain cycles act simultaneously. The failure mechanisms differ between "in-phase TMF (IP-TMF)," where temperature and strain change in phase, and "out-of-phase TMF (OP-TMF)," where they are out of phase. IP-TMF involves coupling of oxidation and fatigue at high temperature and high strain, dominated by intergranular fracture, while OP-TMF involves crack initiation from oxide film cracking during low-temperature tensile strain. For high-temperature structures like gas turbine blades, OP-TMF often governs lifespan, and the formulation organized by Nissley (1995, P&W) is still referenced today.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration." Haven't you 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, 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 the 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 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 ending up with "compression"—it 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, right? Because the vibration 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 vibration energy to improve ride comfort. 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 heterogeneity.
- 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 or 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 inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the 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 TMF
1. Apply temperature cycle + mechanical load cycle simultaneously
2. Use Chaboche model for elastoplasticity + Creep (*VISCO)
3. Obtain stabilized hysteresis loop
4. TMF life prediction (Coffin-Manson + creep damage + oxidation damage)
Summary
TMF Testing Procedure (ISO 12111)
The international standard for TMF testing, ISO 12111 (established 2011), specifies simultaneous control testing where a round bar specimen is induction heated while strain is applied by a mechanical tensile testing machine. The temperature range is the material's service temperature (e.g., 200–950°C for nickel-based superalloys), and the mechanical strain range is typically 0.5–2.0%. Heating/cooling rates are standard at 5–10°C/sec, with one cycle taking about 5–20 minutes, and total test duration ranging from several days to weeks. Equipment costs are around 50–100 million yen per unit, with MTS Systems or Instron high-temperature testing machines being mainstream. In Japan, NIMS (Tsukuba), Toshiba ESS (Yokohama), and Tohoku University (Sendai) possess such facilities.
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 Mid-side 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 the tangent stiffness matrix every iteration. Achieves quadratic convergence within the convergence radius but has high computational cost.
Modified Newton-Raphson Method
Updates the 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 track 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 improves accuracy 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 judgment should be based on total cost-effectiveness.
Practical Guide
TMF in Practice
Engine cylinder heads, exhaust manifolds, turbine blades. Creep-fatigue evaluation per ASME NH.
Practical Checklist
Turbocharger Housing Life Prediction
Automotive turbocharger turbine housings (made of SiMo cast iron) undergo cycles of 20–900°C with each engine start-stop. Each cycle generates 0.1–0.3% mechanical strain, with a typical design target lifespan of 100,000–200,000 cycles (equivalent to 15–20 vehicle years). Continental AG (Germany) established a TMF analysis flow linking Nastran→Abaqus, adopting a method that evaluates IP/OP-TMF damage as independent variables and then sums them using Miner's rule. They identified hotspots with over 40% damage via analysis and achieved double the lifespan through shape optimization (increasing fillet R) (2019, SAE Paper 2019-01-0281).
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 the mesh quality is poor, the results will be a mess no matter how excellent the solver is.
Common Pitfalls for Beginners
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 will be far from reality. Verify 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 constraints is actually the most critical step in the entire analysis.
Software Comparison
Tools
Major Software Supporting TMF Analysis
Comparison of TMF-dedicated analysis software: Abaqus/Standard offers high flexibility for nonlinear creep-fatigue coupling models, used by Rolls-Royce and MTU Aero for engine blade design. Ansys Mechanical (with nCode DesignLife integration) automates TMF post-processing compliant with ISO 12111, offering high practical efficiency. fe-safe (under DS) can automatically distinguish IP/OP-TMF and display damage contours. SIMcenter Nastran excels at linear-nonlinear coupling but has limitations for advanced TMF. MATDAT's MATERIAL PROPERTY database contains numerous TMF material constants for nickel superalloys and is used in design standards.
Three Most Important Questions for Selection
- "What are you solving?": Does it support the necessary physical models/element types for thermo-mechanical cycle fatigue (TMF)? For example, presence of LES support for fluids, contact/large deformation capability for structures can make a difference.
- "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 automatic (GUI) and manual (script) transmission cars.
- "How far will you expand?": Choosing with future expansion in mind—scaling up analysis (HPC support), deployment to other departments, integration with other tools—leads to long-term cost reduction.
Advanced Technologies
TMF Frontiers
Discovery of Thermo-Mechanical Fatigue: History of Jet Engine Development
Thermo-Mechanical Fatigue (TMF) was recognized as an independent failure mode during the development of Pratt & Whitney's JT9D engine in the 1960s. To understand the phenomenon where turbine blades fractured in under 1,000 cycles under thermo-mechanical loads of 700–1,050°C per flight cycle, In-phase/Out-of-phase TMF testing machines were developed. It was found that TMF life for IN738LC alloy was only 1/4 of its isothermal fatigue life.
Troubleshooting
TMF Troubles
Related Topics
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