Creep-Fatigue Interaction — Life Assessment Methods for High-Temperature Structures
Theory and Physics
Overview — Why Creep and Fatigue Are a Combined Problem
What happens when creep and fatigue occur simultaneously? I understand them individually from textbooks, but I can't visualize a case where both overlap...
Simply put, in equipment operating at high temperatures for long periods—like nuclear reactor piping or steam turbine rotors—creep damage progresses during steady-state operation, and fatigue damage accumulates with each start-up and shutdown. The problem is that these two are not independent.
Not independent, meaning it's not a case of "1+1=2"?
Exactly, it can even be "1+1=3". When voids form at grain boundaries due to creep, fatigue cracks can initiate there and propagate rapidly. Conversely, microcracks formed by fatigue can accelerate creep rupture. Therefore, simply adding up the lifetimes from individual evaluations leads to an unsafe assessment.
Specifically, what kind of components are problematic?
Typical high-temperature components include:
- Nuclear Reactor Piping / Header Tubes: Operation at 550–600°C for over 100,000 hours, with dozens of start-stop cycles per year.
- Steam Turbine Rotors: Thermal fatigue due to thermal gradients on the rotor surface + centrifugal creep during steady operation.
- Gas Turbine Blades: Over 1000°C, centrifugal force + thermal cycles, the most severe conditions.
- High-Temperature Steam Piping in Thermal Power Plant Boilers: Increased fatigue cycles due to DSS (Daily Start-Stop) operation.
I see, they all involve "high temperature" + "cyclic loading". So, how do we quantitatively evaluate this interaction?
Linear Damage Summation Rule (Dc+Df≦D)
The most widely used method is the Linear Damage Summation Rule. It's specified in ASME Section III, Subsection NH and evaluates the sum of the creep damage fraction $D_c$ and the fatigue damage fraction $D_f$:
Here, $D$ is a value on the allowable damage envelope defined for each material. For 304SS, the intersection point is around $(D_c, D_f) = (0.3, 0.3)$, forming a bilinear envelope that is stricter than a simple straight line ($D=1$).
How are $D_c$ and $D_f$ each determined?
Fatigue Damage Fraction $D_f$ is calculated using Miner's rule:
$n_j$ is the actual number of cycles for the $j$-th type of cycle, and $N_{d,j}$ is the design fatigue life under those conditions (read from the S-N curve, divided by a safety factor).
Creep Damage Fraction $D_c$ is determined by the time fraction rule—we'll explain this in detail in the next section.
Is the envelope being stricter (not a straight line $D=1$) related to the earlier "1+1=3" discussion?
Exactly right. When creep and fatigue act alternately, grain boundary voids and intragranular slip synergistically amplify the damage. Therefore, the envelope becomes concave towards the origin. The degree of concavity varies by material, for example:
| Material | Envelope Intersection $(D_c, D_f)$ | Characteristics |
|---|---|---|
| 304SS / 316SS | (0.3, 0.3) | Austenitic, weak grain boundaries |
| 2.25Cr-1Mo Steel | Approx. (0.1, 0.1) | Very strict envelope |
| Alloy 800H | (0.3, 0.3) | Specified in NH |
| Modified 9Cr-1Mo (P91) | (0.1, 0.01) | Recently added, strict constraints |
Time Fraction Rule and Creep Damage
What exactly does the time fraction rule calculate?
It integrates the stress history over the hold time and divides that time by the rupture time. In equation form:
$t_{h,j}$ is the hold time for the $j$-th cycle, and $t_d(\sigma, T)$ is the allowable creep rupture time under that stress $\sigma$ and temperature $T$.
But stress changes during the hold time, right? I've heard that stress relaxation occurs when holding under constant strain...
Good observation! In actual operation, strain-controlled holding is common. This means that strain generated by thermal expansion is constrained and held constant, while stress relaxes over time due to creep. In this case, the stress rate is:
This ordinary differential equation is solved numerically to obtain the stress $\sigma(t)$ at each time step, which is then plugged into $t_d(\sigma, T)$ for integration. Be careful: assuming constant stress at the initial value and ignoring stress relaxation can lead to a significant overestimation of creep damage.
Ah, so simply "hold time ÷ rupture time" isn't enough. We need to track stress relaxation.
Exactly. By the way, the creep rate is most commonly expressed by Norton's Law (power law):
$A$ is a material constant, $n$ is the stress exponent (typically 3–8), $Q$ is the activation energy, $R$ is the gas constant, and $T$ is the absolute temperature. Due to the exponential temperature dependence, a mere 50°C difference can change the creep rate by orders of magnitude.
Strain Range Partitioning (SRP)
Are there approaches other than linear damage summation?
There's Strain Range Partitioning (SRP) developed by NASA in the 1970s. It classifies the inelastic strain range into four components:
- $\Delta\varepsilon_{pp}$: Tensile Plastic → Compressive Plastic (pure fatigue)
- $\Delta\varepsilon_{cc}$: Tensile Creep → Compressive Creep
- $\Delta\varepsilon_{pc}$: Tensile Plastic → Compressive Creep
- $\Delta\varepsilon_{cp}$: Tensile Creep → Compressive Plastic (most damaging)
Each component has its own Manson-Coffin type life curve:
Why is $cp$ the most damaging?
When creep progresses on the tensile side, voids form along grain boundaries in the tensile direction. Subsequently, even if rapid plastic deformation occurs on the compressive side, the already opened voids do not fully close. As a result, voids are more likely to grow and coalesce, promoting intergranular fracture. For example, gas turbine blades experience tensile creep due to centrifugal force during operation and compressive plastic deformation during rapid shutdown cooling—a typical $cp$-dominated pattern.
Is SRP used in practice?
It has a track record in the aerospace field (especially NASA-related gas turbine engine design). However, experimentally confirming the four-component strain partitioning is challenging, and the cost of obtaining material data is high. Therefore, linear damage summation (ASME NH) is mainstream in nuclear power, while SRP is used in some NASA-related areas—a kind of division of roles.
Ductility Exhaustion Method and Energy Method
Are there other approaches?
There are several. Let's organize the main ones:
| Method | Basic Concept | Main Application |
|---|---|---|
| Ductility Exhaustion | Defines damage by the consumption rate of creep ductility: $D_c = \int \dot{\varepsilon}_{cr} / \varepsilon_f^* \, dt$ | UK R5 Code (EDF/Nuclear) |
| Energy Method | Evaluates damage by the area of the hysteresis loop (dissipated energy) | Research Level |
| Chaboche Model | Continuum Damage Mechanics (CDM) treats $D$ as an internal variable evolving over time | Advanced Research / Some Commercial Codes |
| ASME NH (Linear Damage Sum) | $D_c + D_f \leq D$ (envelope) | Nuclear / Pressure Vessels (World Standard) |
Is $\varepsilon_f^*$ in the ductility exhaustion method the creep ductility? Is it determined from uniaxial creep tests?
Yes. $\varepsilon_f^*$ is the creep ductility dependent on multiaxiality and temperature, based on the elongation at rupture from uniaxial creep rupture tests. In the UK R5 method, a multiaxiality correction is applied, such as:
This accounts for the effect that higher triaxiality (multiaxial stress state) reduces creep ductility. It's considered more physically reasonable than the time fraction rule and often provides better accuracy for materials like P91 steel (tempered martensitic steel).
History of ASME Code Explicitly Addressing "Creep-Fatigue"
The interaction between creep and fatigue was formally incorporated into design codes starting with the ASME Boiler and Pressure Vessel Code (Section III, Division 1, Subsection NH) in the 1970s. The background was the demonstration that stainless steel piping in Liquid Metal Fast Breeder Reactors (LMFBR) had its life overestimated by conventional fatigue rules alone. Experimental data at the time showed that creep-fatigue tests with hold times at 600°C could have lifetimes less than 1/10th of pure fatigue. This shocking result drove the adoption of the conservative evaluation method using an envelope.
Physical Meaning of Each Term
- Creep Damage Fraction $D_c$: Total amount of creep damage accumulated during hold times. Integrated considering stress relaxation. Corresponds to the physical quantity of grain boundary void nucleation, growth, and coalescence.
- Fatigue Damage Fraction $D_f$: Total damage from cyclic loading (start-stop cycles). Accumulated using Miner's rule. Corresponds to intragranular slip band formation and surface crack propagation.
- Envelope $D$: Allowable limit reflecting the synergistic effect of creep-fatigue interaction. Determined experimentally for each material. Located inside the straight line ($D=1$), indicating the severity of the interaction.
- $n$ (Stress Exponent) in Norton's Law: Represents the stress dependence of steady-state creep. $n=1$ indicates diffusion creep (Nabarro-Herring), $n \geq 3$ indicates dislocation creep. Typical for practical metals is $n=3\sim8$.
Assumptions and Application Limits
- The linear damage summation rule does not consider "sequence effects"—actual life differs between creep-first→fatigue and fatigue-first→creep.
- Under multiaxial stress states, results vary depending on whether Mises equivalent stress or maximum principal stress is used—the NH code primarily uses Mises equivalent stress.
- For non-uniform temperatures (with thermal gradients), differences arise between local evaluation at the hottest point and volume-averaged evaluation.
- For very long durations (over 100,000 hours), creep data relies on extrapolation, making validation of extrapolation methods like the Larson-Miller parameter essential.
- Applicable temperature range: Creep typically becomes significant above about 0.4Tm (40% of the melting point) of the material.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Stress $\sigma$ | Pa (MPa) | The unit of $A$ in Norton's law depends on the unit of $\sigma$. Often based on MPa. |
| Creep Rate $\dot{\varepsilon}_{cr}$ | 1/s (1/h) | Sometimes given as 1/h in literature; careful conversion needed. |
| Activation Energy $Q$ | J/mol (kJ/mol) | $R=8.314$ J/(mol·K). $Q$ is often given in kJ/mol. |
| Rupture Time $t_d$ | h (hours) | ASME NH data is in hours. |
| Creep Ductility $\varepsilon_f^*$ | Dimensionless (mm/mm) | Be careful of confusion between % notation and decimal notation. |
Numerical Methods and Implementation
FEM-Based Creep-Fatigue Analysis Flow
So, what are the steps when actually performing creep-fatigue evaluation with CAE?
Roughly, there are four steps:
- Thermal Analysis: Perform transient thermal analysis for start-up → steady-state → shutdown. Obtain temperature history $T(\mathbf{x}, t)$.
- Structural Analysis: Perform elastoplastic creep analysis using the temperature history as a load. Obtain stress and strain history.
- Damage Parameter Calculation: Calculate $D_f$ from the strain range and $D_c$ from the stress history during hold times.
- Envelope Check: Verify that $(D_c, D_f)$ falls within the ASME NH envelope.
In practice, a consideration of "elastic follow-up" may be inserted between steps 2 and 3. In areas with low constraint, creep strain concentrates due to elastic follow-up, which can cause major problems if overlooked.
What is added to a typical elastoplastic analysis for elastoplastic creep analysis?
Strain decomposition is key. Total strain is divided as follows:
Elastic $\varepsilon_{el}$, plastic $\varepsilon_{pl}$ (rate-independent yielding), creep $\varepsilon_{cr}$ (time-dependent deformation), and thermal strain $\varepsilon_{th}$. The FEM solver calculates the increment for each component at each integration point and updates the stress. An implicit method (backward Euler) is stable for integrating creep increments.
Implementation of Constitutive Laws
Are there creep laws other than Norton's law? How do solvers choose?
Summarizing the main creep constitutive laws:
| Constitutive Law | Equation | Characteristics |
|---|---|---|
| Norton's Law (Steady-State Creep) | $\dot{\varepsilon}_{cr} = A\sigma^n$ | Simplest. Only steady state. Secondary creep region. |
| Time Hardening Law | $\dot{\varepsilon}_{cr} = A\sigma^n \cdot m \cdot t^{m-1}$ | Can represent primary creep. $t$ is time. |
| Strain Hardening Law | $\dot{\varepsilon}_{cr} = (A\sigma^n)^{1/m} \cdot m \cdot \varepsilon_{cr}^{(m-1)/m}$ | Robust to stress variations. Widely used in practice. |
| Theta Projection Method | $\varepsilon = \theta_1(1-e^{-\theta_2 t}) + \the... |
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