Fatigue & Fracture Analysis — Overview
S-N and ε-N curves, stress concentration factors, mean stress corrections, Miner's rule, fracture mechanics, Paris law, and FEM-based fatigue life prediction tools.
Articles in This Section
1. Why Fatigue Failures Occur
Fatigue is the progressive, localised structural damage that occurs when a material is subjected to cyclic loading — even at stress levels far below the static ultimate tensile strength or yield strength. Roughly 80–90% of all mechanical failures in practice are fatigue-related. The insidious nature of fatigue is that a component can sustain millions of cycles without visible damage, then fracture abruptly without warning when the crack reaches a critical size.
At the microstructural level, cyclic plasticity causes dislocation pile-ups and slip band formation at stress concentrations (notches, holes, welds, surface scratches). These slip bands become persistent slip bands (PSBs) and eventually initiate microscopic cracks. Once initiated, cracks propagate incrementally with each loading cycle — a process captured by fracture mechanics — until the remaining net section can no longer support the load and catastrophic fracture occurs.
Three distinct phases are recognised:
- Crack initiation: Formation of micro-cracks at stress concentrations; dominates total life at high cycle counts (HCF)
- Crack propagation: Stable crack growth under cyclic loading; governed by the stress intensity factor range \(\Delta K\)
- Final fracture: Unstable rapid fracture when \(K_{max}\) reaches fracture toughness \(K_{Ic}\)
2. S-N Curve Approach (Stress-Life, High-Cycle Fatigue)
The Wöhler or S-N (stress-number of cycles) curve is the oldest and most widely used fatigue design tool. It relates the applied alternating stress amplitude \(S_a\) to the number of cycles to failure \(N_f\) for a given material and loading condition.
On a log-log plot, the S-N relationship is often linear for many metals, described by Basquin's equation:
\[ S_a = \sigma'_f \left(2N_f\right)^b \]
where \(\sigma'_f\) is the fatigue strength coefficient (approximately equal to the true fracture stress), and \(b\) is the fatigue strength exponent (Basquin exponent), typically in the range \(-0.05\) to \(-0.12\) for metals.
Many steels exhibit a fatigue limit (endurance limit) \(S_e\) — a stress amplitude below which fatigue failure does not occur for any number of cycles (typically taken at \(10^6\) to \(10^7\) cycles). Aluminium alloys and many non-ferrous metals do not exhibit a true fatigue limit; instead, a fatigue strength at \(10^8\) cycles is used as a design reference. Common rough estimate: \(S_e \approx 0.5 S_{UT}\) for steels with \(S_{UT} \leq 1400\) MPa.
The S-N approach is valid for high-cycle fatigue (HCF) where the bulk of the material remains elastic and the number of cycles exceeds approximately \(10^4\). For low-cycle fatigue dominated by plastic deformation, the strain-life approach is required.
Endurance Limit Modifying Factors
The laboratory-derived endurance limit must be corrected for real-world conditions via the Marin equation:
\[ S_e = k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e \cdot k_f \cdot S'_e \]
where the factors account for surface finish (\(k_a\)), specimen size (\(k_b\)), loading type (\(k_c\)), temperature (\(k_d\)), reliability (\(k_e\)), and miscellaneous effects such as residual stress and environment (\(k_f\)). A polished laboratory specimen can have \(S'_e\) = 350 MPa; the same steel in a hot, corrosive environment with a rough machined surface might yield \(S_e\) ≈ 100 MPa after applying all factors — a factor of 3.5 reduction.
3. ε-N Curve Approach (Strain-Life, Low-Cycle Fatigue)
When cyclic stresses exceed the yield stress — as in thermal cycling, notch roots, or startup/shutdown sequences of power plant components — significant plastic deformation occurs per cycle. The S-N approach breaks down because stress alone cannot characterise the loading when plasticity is involved. The strain-life (ε-N) approach uses total strain amplitude \(\varepsilon_a\) as the controlling parameter.
The Coffin–Manson–Basquin relationship decomposes total strain into elastic and plastic components:
\[ \varepsilon_a = \underbrace{\frac{\sigma'_f}{E}(2N_f)^b}_{\text{elastic}} + \underbrace{\varepsilon'_f (2N_f)^c}_{\text{plastic}} \]
where \(\varepsilon'_f\) is the fatigue ductility coefficient and \(c\) is the fatigue ductility exponent (typically \(-0.5\) to \(-0.7\)). The transition life \(2N_t\) — where elastic and plastic contributions are equal — defines the boundary between HCF and LCF regimes. For components experiencing LCF (nuclear pressure vessels, aircraft engine discs, solder joints), the strain-life approach must be used.
FEA supports ε-N fatigue by extracting nodal strain amplitudes from nonlinear cyclic analysis steps, then passing these to dedicated fatigue post-processors.
4. Stress Concentration Factors (\(K_t\)) and Notch Sensitivity
A geometric discontinuity — a hole, fillet, notch, or thread root — amplifies the local stress beyond the nominal stress. The theoretical (elastic) stress concentration factor is:
\[ K_t = \frac{\sigma_{\max}}{\sigma_{\text{nom}}} \]
However, the full \(K_t\) is never fully realised in fatigue because of material notch sensitivity. The effective fatigue stress concentration factor \(K_f\) accounts for this:
\[ K_f = 1 + q(K_t - 1) \]
where \(q\) is the notch sensitivity factor (0 = fully insensitive, 1 = fully sensitive). High-strength steels (\(q \approx 0.9\)–1.0) are more notch-sensitive than ductile low-strength steels (\(q \approx 0.5\)–0.7). Neuber's rule and the Peterson method are the standard approaches for estimating \(q\) from material and notch geometry.
In FEA-based fatigue, the FEM directly gives the peak stress at notch roots, effectively incorporating \(K_t\) into the stress field. Care must be taken with mesh convergence at notches — the stress at a notch tip with a zero-radius corner is theoretically unbounded; always use a finite (physical) fillet radius and confirm mesh convergence at the critical location.
5. Mean Stress Corrections: Goodman, Gerber, Soderberg
Real loading rarely has zero mean stress. A tensile mean stress accelerates crack initiation and growth (reduces fatigue life); compressive mean stress retards it (beneficial — the basis of shot peening). Mean stress corrections modify the allowable stress amplitude:
Modified Goodman (most widely used in industry)
\[ \frac{S_a}{S_e} + \frac{S_m}{S_{UT}} = 1 \]
Gerber (less conservative, follows a parabola)
\[ \frac{S_a}{S_e} + \left(\frac{S_m}{S_{UT}}\right)^2 = 1 \]
Soderberg (most conservative — uses yield strength)
\[ \frac{S_a}{S_e} + \frac{S_m}{S_y} = 1 \]
In practice, Modified Goodman is the standard for most structural fatigue work because it is conservative without being excessively so, and it correlates well with steel test data. Gerber is often used for ductile materials where Goodman is overly conservative. Soderberg is used in design codes requiring high safety margins, such as pressure vessel standards.
In a Haigh (constant-life) diagram, these equations appear as lines or curves connecting \((0, S_e)\) on the alternating stress axis to \((S_{UT}, 0)\) or \((S_y, 0)\) on the mean stress axis. Any operating point below the Goodman line has infinite life; above it, finite life requires Miner's rule.
6. Miner's Rule for Variable Amplitude Loading
Real structures rarely experience constant amplitude loading. Road vehicles, aircraft, and wind turbines see complex, variable amplitude load histories. Palmgren–Miner's rule is the standard linear damage accumulation model:
\[ D = \sum_{i=1}^{k} \frac{n_i}{N_i} = 1 \quad \text{(at failure)} \]
where \(n_i\) is the number of applied cycles at stress level \(i\) and \(N_i\) is the number of cycles to failure at that stress level from the S-N curve. The structure is predicted to fail when the damage sum \(D\) reaches 1.0. In practice, failure is often observed at \(D\) between 0.7 and 1.3, reflecting scatter and load interaction effects not captured by the linear model.
Rainflow cycle counting is the standard algorithm for reducing a complex variable amplitude load time history to a set of stress amplitude/mean pairs \((S_{a,i}, S_{m,i})\) with counts \(n_i\). Once the rainflow matrix is generated, each bin is mapped to the S-N curve (with mean stress correction) and damages are summed. This workflow is implemented in all major fatigue post-processors.
7. Fracture Mechanics: Stress Intensity Factor and Crack Growth
Linear Elastic Fracture Mechanics (LEFM)
For a crack of half-length \(a\) in an infinite plate under remote tensile stress \(\sigma\), the Mode I stress intensity factor is:
\[ K_I = \sigma \sqrt{\pi a} \]
More generally, a geometry correction factor \(F\) accounts for finite plate width, crack shape, and loading mode:
\[ K_I = F \cdot \sigma \sqrt{\pi a} \]
Fracture occurs when \(K_I\) reaches the material's fracture toughness \(K_{Ic}\) — a material property measured per ASTM E399. Typical values: structural steel 50–100 MPa√m, aluminium alloys 20–45 MPa√m, ceramics 1–5 MPa√m. The critical crack size at fracture is:
\[ a_c = \frac{1}{\pi} \left(\frac{K_{Ic}}{F \sigma}\right)^2 \]
This is the basis of damage tolerance design: if you know the initial flaw size (from NDT), you can compute when the crack reaches \(a_c\) and set inspection intervals accordingly.
Paris Law for Crack Growth
In the stable crack growth regime, the crack growth rate per cycle relates to the stress intensity factor range \(\Delta K = K_{max} - K_{min}\):
\[ \frac{da}{dN} = C \left(\Delta K\right)^m \]
where \(C\) and \(m\) are material constants determined from experimental crack growth tests (ASTM E647). For structural steels, \(m \approx 3\)–4; for aluminium, \(m \approx 3\)–5. The total crack growth life from initial crack \(a_i\) to critical crack \(a_c\) is obtained by integration:
\[ N_\text{grow} = \int_{a_i}^{a_c} \frac{da}{C(\Delta K)^m} \]
Below the fatigue crack growth threshold \(\Delta K_{th}\), cracks do not propagate. Above fracture toughness \(K_{Ic}\), propagation is unstable. The Paris law is valid only in the intermediate "Paris regime" between these limits.
8. FEM-Based Fatigue Tools
Dedicated fatigue post-processors integrate directly with FEA results to automate the fatigue life calculation workflow:
| Tool | Developer | FEA Interfaces | Key features |
|---|---|---|---|
| nCode DesignLife | HBK (Hottinger Baldwin Messtechnik) | Abaqus, ANSYS, Nastran, LS-DYNA | S-N, ε-N, multiaxial, weld fatigue, rainflow, test data import |
| FEMFAT | Magna International / ECS | Abaqus, ANSYS, Nastran, PERMAS | Strong in automotive NVH fatigue; local stress approach; weld module |
| fe-safe | Simulia (Dassault Systèmes) | Abaqus (native), ANSYS, Nastran | Tight Abaqus integration; Verity weld method; multiaxial critical plane |
| MSC Fatigue | MSC Software | Nastran (native), Adams | Closely integrated with MSC Nastran SOL 101/103; MBD fatigue workflow |
| ANSYS nCode (embedded) | ANSYS / HBK | ANSYS Mechanical native | Streamlined S-N fatigue within ANSYS Workbench environment |
The typical workflow: (1) run linear elastic FEA to obtain nodal stress tensors for each load case; (2) define the load time history or load spectrum; (3) import FEA results and load history into the fatigue tool; (4) specify material S-N data and modifying factors; (5) run rainflow counting + damage summation; (6) output contour plots of safety factor and cycle life. This entire process is automated and can be run parametrically for design optimisation.
9. Weld Fatigue: IIW Guidelines and Hot-Spot Stress Method
Welds are the most common fatigue failure location in engineering structures. Their complex geometry (weld toe, weld root, undercut) creates high stress concentrations, and the residual tensile stresses from welding effectively shift the mean stress to a high tensile value regardless of the applied loading.
The IIW (International Institute of Welding) Fatigue Design Recommendations are the internationally recognised standard for weld fatigue assessment. They define a system of FAT (fatigue class) curves — S-N curves parameterised by the structural stress range \(\Delta\sigma\) at \(2 \times 10^6\) cycles. FAT 71 means the S-N curve crosses 71 MPa at \(2 \times 10^6\) cycles; FAT 160 is for high quality full penetration butt welds in pristine condition.
Hot-Spot Stress Method
The hot-spot stress (also called structural stress or geometric stress) is the stress at the weld toe extrapolated from the nominal stress field, excluding the sharp local notch effect of the weld toe itself. In FEA, it is calculated by linear extrapolation from integration points at 0.4t and 1.0t (or 0.5t and 1.5t) from the weld toe, where \(t\) is the plate thickness. This approach is mesh-insensitive compared to using raw peak FEA stress at the weld toe (which diverges with mesh refinement).
Verity / Battelle Structural Stress Method
The Verity method (implemented in fe-safe) computes a mesh-insensitive structural stress using nodal forces rather than nodal stresses, making it formally mesh-independent and well-suited for shell element weld models. It is increasingly adopted in automotive and heavy machinery fatigue design as an alternative to the hot-spot method.
Concept Q&A — Student & Professor
Professor, I've always thought that as long as I keep stress below yield, the part won't fail. But fatigue failures happen below yield, right? How does that work?
Exactly right — and it's what makes fatigue so treacherous. Static yield is a one-shot event: if stress exceeds yield, you get permanent deformation, which is usually visible. But fatigue is cumulative and invisible. Even at 60% of yield stress, microscopic slip bands form at stress concentrators — scratches, corners, inclusions — over thousands of cycles. These bands slowly develop into cracks that propagate a tiny increment per cycle. The component looks perfectly fine until the crack reaches a critical size and it fractures suddenly. The Aloha Airlines Boeing 737 fuselage failure in 1988 is a textbook example: stress levels were well within static allowables, but millions of pressurisation cycles drove crack growth that wasn't caught in time.
So the S-N curve tells us how many cycles a material survives at a given stress. But I've seen some curves that show a flat region at the bottom — the "endurance limit." Does that mean below a certain stress, the part lasts forever?
For many steels in laboratory air conditions, yes — there appears to be a true endurance limit. Below roughly 40–50% of UTS, microscopic cracks never initiate or arrest before propagating. This was the basis of classical machine design for decades. The important caveats: aluminium alloys do NOT have a true endurance limit — their S-N curve keeps declining even at \(10^8\) cycles, which is why aircraft structures use a finite design life. Also, in corrosive environments even steels lose their endurance limit — seawater drops it by 50–70%. And under very high cycle fatigue (VHCF, beyond \(10^9\) cycles), some steels with inclusions fail below what was thought to be the endurance limit, due to internal crack initiation from subsurface inclusions.
What about the ε-N approach? When do I need to use that instead of S-N?
The rule of thumb is: if the expected life is below about \(10^4\)–\(10^5\) cycles, you're in the low-cycle fatigue regime where plastic strain per cycle is significant. S-N assumes elastic behaviour — once you're cycling into plasticity, the S-N approach under-predicts damage because it doesn't account for the plastic strain amplitude. Classic examples: nuclear pressure vessel startup/shutdown cycles (maybe a few thousand over 40 years), turbine disc thermal cycling, solder joints in electronics under thermal expansion cycling. For all of these, ε-N (Coffin–Manson) is the right framework. In FEA this requires a proper elastoplastic cyclic material model and extraction of local strain amplitudes at the critical location.
I've seen "Kt" mentioned a lot. My FEA gives me the actual peak stress at a notch. Do I still need to apply Kt on top of the FEA stress?
No — and this is a common source of double-counting. If your FEA mesh is fine enough to capture the stress concentration at the notch, then the peak nodal stress already incorporates the Kt effect. You should NOT multiply it by Kt again. The Kt charts in handbooks (Peterson's) are for analytical calculations using the nominal (average) section stress. In FEA you directly compute the local stress field, so notch concentration is already embedded in your result — as long as the mesh is converged at the critical location. What you DO still need to apply is the notch sensitivity correction (converting Kt to Kf) if you're using the nominal stress S-N approach. In a local stress approach (critical plane, hot-spot), Kf is not applied separately.
Mean stress corrections — I've heard Goodman, Gerber, Soderberg. Which one should I actually use in practice?
Modified Goodman for most structural engineering work — it's the industry default for a reason. It's linear, easy to apply, and matches steel test data reasonably well. It's somewhat conservative at high mean stress (compared to Gerber which is more accurate for ductile materials) but errs on the safe side. Soderberg is used when your design code demands it — it replaces UTS with yield strength in the denominator, so it's more conservative than Goodman, sometimes excessively so for high-ductility steels. Gerber gives better correlation for very ductile materials but is non-conservative on the unconservative side for brittle or high-strength materials. For most automotive and aerospace S-N calculations, start with Goodman unless your material data or design standard specifies otherwise.
For variable amplitude loading, Miner's rule says sum the damage fractions. But I've also heard it's unreliable. Should I worry about that?
Miner's rule is the industry workhorse because it's simple and gives reasonably conservative results for most fatigue loading if you apply appropriate safety factors. It's unreliable in two specific situations: first, when load sequence matters — for example, a single large overload can significantly retard crack growth in the following cycles because of crack tip residual compressive stresses; Miner's rule is completely blind to this. Second, it's non-conservative in service environments where cyclic frequency affects crack growth (corrosion-fatigue, fretting). In practice, the scatter in fatigue data itself is often 5–10× in life, so a factor of 2 on the Miner's sum (design to D = 0.5 instead of D = 1.0) is a reasonable safety allowance for most ground vehicle or general machinery applications. Aerospace is more demanding — detailed spectrum analysis with crack growth tracking replaces pure Miner's rule for primary structure.
Can you explain the Paris law in plain terms? The equation looks straightforward but I'm not sure what it's physically describing.
The Paris law says: the amount a crack grows per cycle depends on the stress intensity factor range ΔK raised to a power m. ΔK = Kmax − Kmin measures the range of the "crack driving force" — how much the stress field at the crack tip intensifies and relaxes each cycle. When ΔK is small (thin crack in low-stress region), growth per cycle is tiny — maybe nanometres. As the crack grows, ΔK increases (because K scales with √a), so growth rate accelerates. When K_max hits fracture toughness K_Ic, the crack runs unstably. The power m being 3–4 for steel means that doubling ΔK gives you 8–16× faster crack growth — this explains why crack growth seems slow forever and then suddenly catastrophic. Paris law is the basis for all damage tolerance calculations: inspect at intervals short enough that a crack cannot grow from the detectable size to critical size between inspections.
We use fe-safe at work and I often see it comparing Goodman-corrected S-N results to a "critical plane" option. Which should I use?
For uniaxial or nearly uniaxial loading — a simple bar in tension-compression — the S-N with Goodman is perfectly adequate and faster. Critical plane methods (Brown–Miller, Fatemi–Socie) become essential when the loading is multiaxial with varying principal stress directions. Think of a crankshaft or a wheel hub: bending and torsion act simultaneously and the angle of maximum stress rotates with the loading cycle. In those cases, a uniaxial S-N analysis on the equivalent von Mises stress can significantly over- or underestimate life depending on the phase relationship. Critical plane finds the specific plane on which fatigue damage is maximum, which correctly captures the damage mechanism for crack initiation under complex stress states. Use Goodman for early design screening; switch to critical plane when loading is confirmed as multiaxial.
What about weld fatigue? I've heard that welds are always the weak link and the IIW method is the standard, but I've never actually applied it.
IIW recommendations are the go-to for structural weld fatigue — bridges, cranes, offshore structures, pressure vessels, railway carriages. The key idea is that the weld geometry (toe angle, undercut, lack of penetration) makes the local stress concentration highly variable and hard to model accurately in FEA. Instead, IIW gives you FAT class S-N curves calibrated to specific joint types and quality levels, using the "structural stress" or "nominal stress" at the weld rather than the raw FEA peak. You identify your joint type from the IIW catalogue (there are around 80 standard joint types), pick the FAT class, and compare your computed stress range to the FAT S-N curve. For shell model welds in nCode or fe-safe, the Verity method does this automatically using nodal forces — much more mesh-insensitive than extracting raw weld toe stresses.
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