Creep-Fatigue Interaction
Theory and Physics
Creep-Fatigue Interaction
Professor, what happens when creep and fatigue occur simultaneously?
Under repeated loading at high temperatures (above 350°C for steel), creep damage and fatigue damage accumulate simultaneously. Evaluation considering both is necessary.
ASME NH Linear Damage Rule
Sum of fatigue damage $D_f = n/N_f$ (Coffin-Manson) + creep damage $D_c = t/t_r$ (time fraction). ASME NH specifies the allowable damage using a damage envelope (Creep-fatigue interaction diagram).
Summary
Fundamentals of Creep-Fatigue Interaction
In high-temperature fatigue, "pure fatigue" and "creep" progress simultaneously, and simple linear addition significantly overestimates life. The ASME Boiler & Pressure Vessel Code (Section III, Appendix T) introduced the "Creep-Fatigue Interaction Diagram" in 1974. It established the concept of plotting the fatigue usage fraction Df (= Σni/Nf) on the vertical axis and the creep usage fraction Dc (= Σti/tr) on the horizontal axis, and determining failure when their sum exceeds the design allowable curve.
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 to ignore acceleration". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you pull 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——an iron rod and a rubber band, which stretches more when pulled with the same force? Obviously the rubber. 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 $f_b$ (gravity, etc.) and surface forces $f_s$ (pressure, contact forces, etc.). 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 pitfall here: getting the load direction wrong. Intending "tension" but modeling "compression"——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 vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle——they deliberately absorb vibration 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 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 otherwise specified): 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 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 both mm and m systems. |
Numerical Methods and Implementation
FEM
1. Elastoplastic + creep analysis using Chaboche + Norton (*VISCO)
2. Obtain $\Delta\varepsilon$ from stabilized hysteresis loop → fatigue damage $D_f$
3. Obtain $\dot{\varepsilon}_{cr}$ from high-temperature hold time → creep damage $D_c$
4. Evaluate $D_f + D_c$ using the damage envelope
Summary
Difference Between Time-Fraction Rule and Strain-Fraction Rule
Methods for evaluating creep damage are broadly divided into two. ① Time-Fraction Rule: Quantifies damage by the sum of ratios Σ(ti/tr), where hold time ti is divided by rupture time tr(σ,T). ② Ductility Exhaustion Method: Calculates damage from the ratio of creep strain rate to ductility. A 1998 report by UK AEA Technology showed that for stainless steel 316H, the strain-fraction method gives 30-50% more conservative predictions than the time-fraction method. The time-fraction method is more commonly adopted in nuclear reactor pressure vessel design.
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)
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 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 computational cost is high.
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 total load in small increments rather than all at once. 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: 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
Practical Checklist
Design Example: Gas Turbine First-Stage Rotor Blade
The GE Aviation GE9X (mounted on Boeing 777X) first-stage rotor blade is manufactured from TBC (Thermal Barrier Coating)-coated single-crystal nickel superalloy CMSX-4 and exposed to combustion gas temperatures above 1700°C. Creep-fatigue life analysis of the blade uses a proprietary interaction diagram, evaluating design life by summing takeoff/landing cycles (LCF) and cruise creep. Overhaul intervals are determined in combination with EHM (Engine Health Monitoring) data, typically whichever comes first: about 3000 cycles or 15,000 flight hours.
Analogy: Analysis Flow
The analysis flow is actually very similar to cooking. First, buy the ingredients (prepare CAD model), do the 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, 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 result is 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. 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 the same as "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
Abaqus Creep-Fatigue Analysis Capabilities
Dassault Systèmes Abaqus analyzes creep deformation by combining the CREEP card with time-dependent plasticity (PLASTIC with TIME DEPENDENT). Creep-fatigue damage evaluation can be performed using the Creep-Fatigue Interaction (CFI) module of fe-safe (a fatigue-specific plugin integrated with Abaqus). Ansys Mechanical 2024 R1 added a Creep Fatigue Analysis (CFA) wizard that automatically plots the ASME Section III code interaction diagram. Since SIMcenter Nastran does not support creep, migration to Abaqus/Ansys is common in practice for high-temperature creep problems.
The Three Most Important Questions for Selection
- "What are you solving?": Does it support the physical models/element types needed for creep-fatigue interaction? For example, in fluids, the presence of LES support; in structures, the ability to handle contact/large deformation makes 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 an automatic transmission car (GUI) and a manual transmission car (script).
- "How far will it 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 Technologies
Advanced
Creep-Fatigue Integration via Continuum Damage Mechanics
Continuum Damage Mechanics (CDM) is a framework developed by Lemaitre & Chaboche (1985, École Normale Supérieure de Cachan, France), incorporating a damage variable ω as an internal variable into the constitutive equations. By describing creep damage and fatigue damage with independent damage variables ωc, ωf and solving their evolution equations simultaneously, the interaction can be expressed naturally. Implementation examples of this model in Abaqus User Subroutine (UMAT or CREEP) have been reported for high-temperature fatigue evaluation of Inconel 617 by MPa Forming (2021, Tohoku University).
Troubleshooting
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