Fatigue, Fracture Mechanics & Durability Design: The Complete Engineering Guide — The 180 MPa Endurance Limit That 627 MPa Steel Actually Delivers, the 1.13 Million Cycles Between a 1 mm and a 10 mm Crack, and the 550 MPa That HIP Gives Back to an As-Built Titanium Part

Every other guide in this series ends at the yield strength. This one begins there, because the overwhelming majority of metal parts that fail in service never exceed yield at all. The arithmetic of durability is exact and it is unforgiving: a machined 30 mm shaft in AISI 1045 cold-drawn (Sut 627 MPa, Sy 531 MPa) has an uncorrected endurance limit of 313 MPa, but after the Marin surface factor ka = 0.818, the size factor kb = 0.862 and a 99% reliability factor ke = 0.814, the number the designer is actually allowed to use is 180 MPa — 29% of tensile and 34% of yield. A 1 mm crack in that shaft growing to 10 mm under a 100 MPa stress range consumes 1.13 million cycles at 3,000 rpm, which is 6.3 hours of running. Fracture mechanics turns inspection into arithmetic: KIC = 50 MPa√m at 300 MPa working stress gives a critical crack of 8.8 mm through-thickness or 7.0 mm at the surface, so the NDT interval writes itself. And manufacturing decides whether any of it is true — as-built laser powder bed Ti-6Al-4V delivers 200–250 MPa at 10⁷ cycles while the same alloy after HIP delivers 550–600 MPa, a 120–175% swing bought entirely with process, not design. This guide works the full stack: Basquin and the Marin factors, Goodman/Gerber/Soderberg, Neuber notches and the q = 1/(1+√a/√r) that turns Kt into Kf, Coffin-Manson strain life, rainflow and Miner, Paris-law integration, Eurocode FAT classes, weld improvement, shot peening and Almen, fretting and corrosion fatigue, and the twenty design rules that separate a part that lasts 10⁷ cycles from one that lasts 10⁵.

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Fatigue, Fracture Mechanics & Durability Design: The Complete Engineering Guide

Every guide in this series has, sooner or later, run into the same wall. The CNC machining guide sized cuts by specific cutting force and stopped at tool deflection; the welding guide sized fillets by throat area; the springs guide sized coils by Wahl-corrected shear; the pressure-vessel guide sized walls by hoop stress against yield. All of them were computing the strength of a part the first time it is loaded. None of them was computing what happens on the ten-millionth time.

That is the gap this guide fills, and it is the widest gap in mechanical design. A part that is strong enough is not the same as a part that lasts. The two questions use different material properties, different safety factors, different inspection regimes and — most expensively — different manufacturing processes. A 627 MPa steel shaft that would pass every static check in this series can break in six hours of running at a stress one-third of its yield strength. Not because the calculation was wrong, but because the calculation was answering a different question.

The numbers in this guide are deliberately concrete, because fatigue is the one branch of machine design where hand-waving kills people. Consider the four that frame everything that follows.

The first is 180 MPa. That is the usable endurance limit of a machined 30 mm shaft in AISI 1045 cold-drawn steel. The textbook starting point is S_e' = 0.5\,S_{ut} = 313 MPa. Apply the Marin corrections for a machined surface (k_a = 0.818), a 30 mm diameter (k_b = 0.862) and 99% reliability (k_e = 0.814) and 313 MPa becomes 180 MPa — 57% of the number a first-pass calculation would have used, and 34% of the material's yield strength.

The second is 1.13 million. That is the number of cycles a 1 mm crack in that shaft takes to grow to 10 mm under a 100 MPa stress range, integrated through the Paris law with the Barsom–Rolfe ferritic steel constants C = 6.9\times10^{-12} and m = 3. At 3,000 rpm it is 6.3 hours. Fatigue life is not mostly initiation; for a part with a 1 mm flaw, it is almost entirely propagation, and propagation is computable.

The third is 8.8 mm. That is the critical crack depth in a 50 MPa√m steel at 300 MPa of working stress — 7.0 mm if the crack is a surface semi-ellipse with Y = 1.12. A crack that size is detectable. A crack that size is also only a factor of nine away from a 1 mm inspection limit, which is nine Paris-law decades of warning if the loads are known. Damage tolerance is the discipline of turning that warning into an inspection interval.

The fourth is 550 MPa. That is what hot isostatic pressing returns to laser powder bed Ti-6Al-4V, which in the as-built condition delivers 200–250 MPa at 10⁷ cycles. Nothing about the alloy changed. The pores closed, the surface remained rough, and the fatigue strength more than doubled — a 120–175% swing purchased entirely with post-processing. No design change, no material change, no geometry change: a process decision worth more than any fillet radius in this guide.

The guide runs in twelve sections:

1. The Physics: Why a Part Breaks Below Yield

1.1 The three-stage failure

Fatigue is the progressive, localised, permanent structural change that occurs in a material subjected to repeated loading, and it produces failure at stresses far below the static strength. The mechanism runs in three stages that are physically distinct and analytically separate.

Stage I — initiation. Under cyclic shear stress, dislocations in favorably oriented surface grains pile up against barriers. Persistent slip bands (PSBs) form — bands of intense, localised cyclic plasticity a few micrometres wide, with dislocation structures distinctly different from the surrounding matrix. At the free surface these bands produce intrusions and extrusions: microscopic ridges and grooves, typically 0.1–1 µm deep, where the slip steps emerge. The intrusion is a micro-notch with a root radius of tens of nanometres, and it is where the crack is born. Stage I cracks grow along crystallographic slip planes, in the direction of maximum shear, and typically extend only one or two grain diameters — 10–50 µm in a fine-grained steel. This stage consumes the majority of life at low stress amplitudes (high-cycle fatigue), and almost none of it at high amplitudes.

Stage II — propagation. Once the crack turns from the slip plane to the plane of maximum tensile stress, growth becomes Mode I opening and the mechanism changes completely. Each cycle blunts the crack tip in tension and resharpens it in compression, leaving a striation — a ridge on the fracture surface whose spacing equals the crack growth increment per cycle, da/dN. Striations are the fingerprint of stage II growth: in a ductile metal at room temperature, one striation is one cycle. They are visible under SEM at spacings from 0.01 µm (near threshold) to several micrometres (near final fracture).

Stage III — final fracture. When the remaining ligament can no longer carry the peak load, the part fails by static overload. On the fracture surface this appears as a rough, dimpled (microvoid coalescence) or cleavage region, distinctly different from the smooth, often discoloured fatigue zone.

The practical consequence is that fatigue life is the sum of initiation life and propagation life, and which term dominates depends on the stress level and the flaw population:

At low stress in a clean, polished laboratory specimen, N_{\text{init}} can be 80–90% of total life. In a real welded structure, a cast component with porosity, or an as-built additive part with 15 µm surface roughness, N_{\text{init}} \approx 0 — the crack already exists as a manufacturing defect, and the entire life is propagation. This is why laboratory S–N curves measured on polished specimens over-predict the life of real hardware so badly, and why the surface and defect terms in §2 and §9 carry more weight than the material's own fatigue ratio.

1.2 The statistics, and why fatigue dominates

Fatigue is not one failure mode among many. It is the failure mode.

The reason is structural: almost every machine component sees a load that varies. A rotating shaft sees a fully reversed bending stress at 1× per revolution — 50 Hz at 3,000 rpm, 1.6×10⁹ cycles in a year of continuous running. A pressure vessel sees a pressurisation cycle per shift. A bridge sees a stress cycle per truck. A turbine blade sees 50–300 Hz depending on blade count and speed. Static design assumes the load is applied once, and almost nothing in service is.

1.3 The historical record is not decoration

Three failures wrote the modern rules.

The de Havilland Comet, 1954. Two Comet 1 aircraft broke up in flight within three months. The investigation traced the failure to a fatigue crack initiating at a rivet hole adjacent to the ADF aerial cutout in the pressurized fuselage skin, propagating until the cabin failed explosively at altitude. A full-scale water-tank test at Farnborough reproduced the failure, and the finding — that stress concentration at a fastener hole plus pressurisation cycling equals crack growth — produced the first serious damage-tolerance thinking in civil aviation, and eventually the fail-safe and safe-life design philosophies.

The Alexander L. Kielland platform, 1980. A semi-submersible accommodation platform capsized in the North Sea after a fatigue crack propagated from a fillet weld around a hydrophone penetrations in a bracing member — a detail with a calculated fatigue life that had been assessed using a conservative but wrong S–N curve, on a joint whose actual weld toe geometry was worse than assumed. 123 died. The legacy is the modern offshore fatigue standard set: DNV-RP-C203, hot-spot stress, and the requirement to fracture-mechanics-assess weld defects rather than assume they are absent.

The Aloha Airlines 737, 1988. A fuselage lap joint failed by multi-site fatigue cracking after 89,000 cycles, tearing open the upper fuselage. The lap joint had been designed as a fail-safe structure with a 0.5 mm crack-growth allowance; the actual disbonding and fretting at the joint changed the load path so that the assumed damage tolerance was not there. The legacy is the widespread damage-tolerance philosophy and the aging-aircraft inspection programmes.

The common thread in all three: the calculated life was optimistic because the assumed initial condition did not match the as-built condition. That is the same failure mode that shows up in a machine shop when a bracket designed for infinite life cracks at the weld toe in three months.

1.4 Reading a fracture surface

A fatigue failure leaves a map, and the map is readable without a microscope for the first pass.

Feature · What it looks like · What it means

Origin · Small flat facet, often at a surface, corner, hole, keyway or weld toe · Where K_t or a defect concentrated stress. Ratchet marks radiating from it = multiple initiation sites

Beach marks · Concentric arcs / clamshells · Crack front positions at load changes (start-stop, overload). Not one per cycle — one per load block

Striations · Parallel micro-ridges, 0.01–5 µm spacing · One per cycle in ductile stage II growth. Spacing = local da/dN

Final fracture zone · Rough, dimpled or cleavage, 45° shear lip · Overload after the ligament was too small. Size of this zone is a crude measure of how far the crack had grown

Discolouration · Brown/black oxide in the fatigue zone · Crack was open to environment for a long time — confirms slow growth

A worked example of the reading: a 40 mm shaft fails with a smooth, oval fatigue zone covering 60% of the cross-section, beach marks at roughly 2 mm intervals, and a 16 mm rough final-fracture zone. The final zone is 40% of the area, so the failure stress was about 40% of the material's ultimate — meaning the shaft was running at roughly 0.4 × 627 ≈ 250 MPa nominal at the moment of fracture. If the design stress was 120 MPa, the crack had grown until the local stress at the reduced section reached the fracture strength. That is the arithmetic to run at the start of every failure investigation, before any metallography.

1.5 The endurance limit is not a material constant

One more foundational point, because it drives every calculation in §2. The "endurance limit" of steel — the stress below which a polished laboratory specimen survives 10⁷ cycles — is a property of a specimen, not of a material. It depends on:

The Marin equation in §2 is the bookkeeping system for moving from the polished-specimen number to the number the designer is allowed to use. It is empirical, it is approximate, and it is the best simple tool available — but the error bars are ±20% on the factors and the resulting life prediction can be off by a factor of two to five. That is not a reason to skip it. It is a reason to apply a safety factor to the answer and to test the actual part when the consequence of being wrong is unacceptable.

2. Stress-Life: Basquin, the 10⁷ Convention, and the Marin Chain

2.1 The S–N curve

The stress-life approach, pioneered by August Wöhler in the 1850s–1870s on railway axles, plots the applied stress amplitude S_a against the number of cycles to failure N_f on log–log axes. For most metals in the high-cycle regime the result is a straight line, which is Basquin's equation (1910):

where \sigma_f' is the fatigue strength coefficient (approximately the true fracture stress) and b is the fatigue strength exponent, typically -0.05 to -0.12 for steels and around -0.10 for aluminium alloys. The equivalent form S_a = A N_f^{b} is the same curve with the 2 absorbed into the constant.

For ferrous alloys the curve has a knee: below a certain stress amplitude the specimen survives indefinitely, and the curve becomes horizontal. This is the endurance limit S_e, conventionally defined at 10⁷ cycles (some standards use 5×10⁶ or 10⁸). For non-ferrous alloys — aluminium, copper, magnesium, most titanium alloys — there is no true knee; the curve continues to fall slowly, and the correct design quantity is a fatigue strength at a specified life (e.g. S_{10^7} or S_{5\times10^8}), never an "endurance limit".

The physical origin of the knee in steel is the interstitial carbon: below a critical stress, the strain ageing from carbon/nitrogen atmospheres locks dislocations and prevents the cumulative slip that forms PSBs. Remove the interstitials (very low carbon, or austenitic stainless) and the knee disappears. This is why 304 stainless has no endurance limit in the textbook sense, and why aluminium aircraft structures are designed to a finite life, not infinite life.

2.2 The endurance ratio

The uncorrected endurance limit of a polished, small, rotating-bending specimen correlates with tensile strength:

The cap matters: above about 1400 MPa the endurance ratio falls below 0.5, because the higher-strength steels are also more notch-sensitive and more defect-sensitive. A 2000 MPa maraging steel does not have a 1000 MPa endurance limit in a real part; the surface and defect terms take it down far more aggressively than they do for a 600 MPa steel.

Typical endurance ratios:

Material class · S_e'/S_{ut} · Note

Wrought carbon and alloy steels · 0.40–0.60 · 0.5 is the working default; higher for fine-grained, lower for high-alloy

Cast irons · 0.35–0.45 · Graphite flakes act as internal notches; nodular iron is at the top of the band

Aluminium alloys · 0.25–0.35 · No true limit; ratio quoted at 5×10⁸ cycles

Titanium alloys · 0.45–0.55 · Strongly dependent on surface and microstructure

Copper alloys · 0.25–0.40 · Brass lower than phosphor bronze

Nickel superalloys · 0.35–0.45 · Temperature-dependent; not a room-temperature number

2.3 The Marin equation

The correction chain from the polished specimen to the real part is:

Each factor is a multiplier less than or equal to 1 (except temperature, which can exceed 1 slightly at moderate temperature for steels). The chain is multiplicative because the effects are approximately independent — which is itself an approximation, and one of the reasons the final answer carries ±20–30% uncertainty.

Surface factor k_a. The dominant term for most machined parts. It follows a power law:

with the coefficients tabulated for SI units (S_{ut} in MPa):

Surface finish · a (MPa) · b · k_a at S_{ut}=600 MPa · Typical R_a

Ground · 1.58 · −0.085 · 0.917 · 0.1–0.8 µm

Machined / cold-drawn · 4.51 · −0.265 · 0.828 · 0.8–3.2 µm

Hot-rolled · 57.7 · −0.718 · 0.584 · 3.2–12.5 µm

As-forged · 272 · −0.995 · 0.468 · >12.5 µm

The exponents are the story. For a ground surface the factor barely moves with strength (b = -0.085): a ground 1200 MPa steel retains 0.88 of its specimen limit. For an as-forged surface the factor collapses with strength (b = -0.995): a forged 1200 MPa steel retains only 0.23. The stronger the steel, the more the surface finish matters. This is why high-strength fasteners are thread-rolled and shot-peened rather than cut, and why a "stronger" material can produce a shorter-lived part if the finish is not upgraded with it.

Size factor k_b. For rotating bending of a round bar:

with d in mm. For axial loading, k_b = 1 (no stress gradient); for torsion, use the bending value. The physical basis is statistical: a larger volume of material has a higher probability of containing a critical defect, and a smaller stress gradient means more material sees near-peak stress. At 30 mm, k_b = 0.862; at 100 mm, k_b = 0.729; at 250 mm, k_b = 0.626. A 250 mm shaft retains 63% of the small-specimen limit for size alone, before surface is considered.

Load factor k_c. Rotating-bending data are the reference:

Loading · k_c

Bending · 1.00

Axial · 0.85

Torsion · 0.59

The axial penalty is real and often ignored: a push-pull link at the same nominal stress as a bending beam has 15% less fatigue strength. Torsion is worse, though for combined loading the von Mises equivalent stress plus the torsion factor usually recovers most of the difference.

Temperature factor k_d for steels:

T (°C) · 25 · 50 · 100 · 150 · 200 · 250 · 300 · 350 · 400 · 450 · 500

k_d · 1.000 · 1.010 · 1.020 · 1.025 · 1.020 · 1.000 · 0.975 · 0.943 · 0.900 · 0.843 · 0.768

The slight rise to 150 °C is strain-ageing; the fall above 250 °C is creep and oxidation. Above 400 °C, fatigue is no longer a mechanical problem alone — creep-fatigue interaction takes over and time-at-temperature enters the calculation.

Reliability factor k_e. The S–N curve is a 50% survival line. To design for a higher survival probability:

Reliability · 50% · 90% · 95% · 99% · 99.9% · 99.99%

z_\alpha · 0 · 1.288 · 1.645 · 2.326 · 3.091 · 3.719

k_e · 1.000 · 0.897 · 0.868 · 0.814 · 0.753 · 0.702

Note what this table is really saying: moving from 50% to 99.99% survival costs 30% of the strength. Design for 99% as a default for safety-relevant parts; 90% for benign ones.

Miscellaneous k_f. Corrosion, plating, residual stress, fretting, radiation — all the effects without a neat table. Set k_f = 1 only when there is genuinely nothing.

2.4 Worked example: the 30 mm 1045 shaft

Take a shaft in AISI 1045 cold-drawn, S_{ut} = 627 MPa, S_y = 531 MPa, machined to R_a 1.6 µm, d = 30 mm, rotating bending, room temperature, 99% reliability.

The usable endurance limit is 180 MPa: 57% of the uncorrected value, 34% of yield, 29% of tensile. A designer who used 0.5\,S_{ut} would overestimate the allowable alternating stress by 74%.

Now apply a mean stress. Suppose the shaft carries a steady torque giving \sigma_m = 100 MPa (von Mises) and a rotating bending amplitude \sigma_a = 120 MPa:

The spread between Soderberg and the elliptic criterion is 25% on the safety factor. The choice matters, and §3 explains which one to use.

2.5 The S–N curve as a design tool, and its limits

Three cautions before moving on.

First, the S–N curve is a 50% failure line. Every point on it is the median of a log-normal distribution with a standard deviation of 0.1–0.2 in \log N. A part designed exactly on the curve has a 50% chance of failing before its design life. The reliability factor is how that gets handled in the Marin chain; explicit statistics (§12) is how it gets handled properly.

Second, the curve is for constant-amplitude loading. Real service loads are spectra, not sine waves. Applying a constant-amplitude curve to a spectrum without cycle counting and damage summation is the most common error in industrial fatigue work, and it is usually non-conservative by a factor of 2–10.

Third, the curve is for a smooth specimen of the same material and heat treatment. Weldments, castings, additively manufactured parts, and surface-treated parts do not follow it. They need their own curves, or the appropriate knock-down factors in §8 and §9.

3. Mean Stress: Four Criteria and One Decision

3.1 The mean-stress diagram

A fatigue cycle is characterised by its stress amplitude and mean:

Tensile mean stress reduces fatigue life; compressive mean stress increases it. The mean-stress diagram plots \sigma_a against \sigma_m with four competing boundary lines between the safe and unsafe regions.

Goodman (modified Goodman):

A straight line from (0, S_e) to (S_{ut}, 0). Conservative for ductile metals, simple, and the default in most machine-design texts and in ASME VIII Div 2's elastic route. For brittle materials it is the correct choice, because brittle materials do not yield to redistribute stress.

Gerber:

A parabola through the same two points. It fits the experimental data for ductile steels better than Goodman — which is why Shigley recommends Gerber for ductile materials and Goodman for brittle ones. It is less conservative, sometimes by 20–30% on life at high mean stress.

Soderberg:

A line from (0, S_e) to (S_y, 0). The most conservative of the three because S_y < S_{ut}, and it guarantees no yielding at the notch root under the peak load. Used where yielding itself is unacceptable — precision spindles, instrument flexures, preloaded bearings.

ASME elliptic:

The ASME Boiler and Pressure Vessel Code route for combined loading; less conservative than Goodman by roughly 10–15% in the tension-tension quadrant.

3.2 Which criterion, and when

The choice is a statement about the scatter of the data and the consequence of failure, not about which equation is "true". All four are empirical fits to the same cloud of test points, and the cloud has a standard deviation of 10–20% in stress.

A practical rule used in rotating machinery: use Goodman for the first pass, Gerber for the final weight-optimised design, and Soderberg when the part is a spindle or a preloaded joint. If the four criteria disagree by more than 25% on safety factor, the design is not sensitive to the criterion — it is sensitive to the mean stress, which means the load spectrum is the thing to go and measure.

3.3 Compressive mean stress, and residual stress as a design variable

The mean-stress diagram is usually drawn only in the tensile quadrant, but the compressive side is where the engineering opportunity lives. A compressive mean stress moves the operating point left, away from the failure envelope, and can raise the allowable alternating stress above S_e.

This is exactly what shot peening does (§9.2): it imposes a surface layer with −400 to −800 MPa of residual compression, which acts as a negative mean stress on the surface layer where cracks initiate. It is why a peened spring, a peened gear root, and a peened weld toe last two to five times longer than their unpeened counterparts. It is also why the benefit disappears if the part is subsequently ground, machined, or heated above the tempering temperature of the surface layer: the residual stress is a manufactured quantity, and it can be manufactured away.

The same logic runs in reverse for tensile residual stress. A weld toe, a grinding burn, a machined fillet with a torn surface, and an as-built additive surface all carry tensile residual stress, which shifts the operating point right and can consume the entire safety margin without appearing in any nominal stress calculation. Residual stress is not in the free-body diagram, but it is in the part.

3.4 Combined loading

For a shaft under combined bending, torsion and axial load, the standard approach is to compute the von Mises equivalent alternating and mean stresses and apply the chosen criterion to those:

This is conservative when the bending and torsion peaks are out of phase, and slightly non-conservative when they are in phase and the material is notch-sensitive. For a rotating shaft with steady torque, the torsion contributes only to \sigma_m and the bending only to \sigma_a — which is why a shaft can survive an enormous steady torque and fail from a modest rotating bend.

A worked number: a 25 mm shaft under 200 N·m steady torque and 150 N·m alternating bending. Torsional shear \tau_m = 16T/(\pi d^3) = 16 \times 200/(\pi \times 0.025^3) = 65.2 MPa; bending \sigma_a = 32M/(\pi d^3) = 32 \times 150/(\pi \times 1.5625\times10^{-5}) = 97.8 MPa. Then \sigma_m' = \sqrt{3} \times 65.2 = 112.9 MPa and \sigma_a' = 97.8 MPa. With S_e = 180 and S_{ut} = 627: Goodman = 97.8/180 + 112.9/627 = 0.543 + 0.180 = 0.723, n = 1.38. The steady torque alone would never fail this shaft; the rotating bend consumes 54% of the fatigue budget on its own.

4. Notches: K_t, K_f, and the √a That Turns One Into the Other

4.1 Elastic stress concentration

A geometric discontinuity — hole, fillet, keyway, thread root, weld toe, corrosion pit — raises the local elastic stress. The elastic stress concentration factor is:

The classic results are worth memorising because they bound almost every real feature:

Feature · K_t · Note

Circular hole in an infinite plate, uniaxial tension · 3.0 · Independent of hole size — the classic result

Elliptical hole, semi-axes a (⊥ load) and b · 1 + 2a/b · A 10:1 ellipse gives K_t = 21; this is why cracks are sharp

Shoulder fillet, D/d = 2, r/d = 0.05 · 2.1–2.4 · Falls to ~1.6 as r/d \to 0.2

Keyway (end-milled, sharp corners) · 2.0–2.5 · The usual origin of broken shaft ends

Metric thread root, as-cut · 3.0–5.0 · Depends on root radius and pitch; rolled threads are better

Press/shrink fit edge · 2.0–3.0 · Plus fretting (§10.2) — a double penalty

Weld toe, as-welded · 3.0–5.0 · Plus tensile residual stress — the worst combination in this guide

Sharp circumferential groove, r/d = 0.01 · 5–8 · Approaches the crack limit

The elliptical-hole result is the single most important line in the table. It says that the sharpness of a feature, not its size, sets the stress. A 0.1 mm scratch with a 0.001 mm root radius has the same K_t as a 10 mm notch with a 0.1 mm root radius. This is why polishing a shaft raises its fatigue limit, and why a single tool mark in a fillet can start a crack that a much larger but blunter feature never would.

4.2 Notch sensitivity: from K_t to K_f

Real materials do not develop the full elastic K_t at the notch root, because the yielded zone at the root redistributes stress and because the highly stressed volume is tiny. The fatigue notch factor is:

where q is the notch sensitivity, 0 \le q \le 1. Rearranged:

Neuber's empirical relation gives q from the notch root radius r and a material constant \sqrt{a}:

with \sqrt{a} (in √in, S_{ut} in kpsi):

For the 1045 steel of §2 (S_{ut} = 627 MPa = 91 kpsi), \sqrt{a} = 0.0707\ \sqrt{\text{in}}, i.e. a = 0.127 mm. The resulting notch factors for a K_t = 2.5 feature:

Root radius r · q · K_f · Interpretation

0.2 mm · 0.557 · 1.84 · A sharp fillet lets 74% of the elastic concentration through

0.5 mm · 0.665 · 2.00 · Half the benefit of a generous radius is already lost

1.0 mm · 0.737 · 2.11 · Typical machined fillet

2.0 mm · 0.799 · 2.20 ·

4.0 mm · 0.849 · 2.27 · Large radius, q \to 1, K_f \to K_t

Two counter-intuitive results follow.

First, q rises with strength. The \sqrt{a} constant falls as S_{ut} rises, so a high-strength steel has higher notch sensitivity. A 1200 MPa steel with r = 1 mm has q \approx 0.85 versus 0.74 for the 627 MPa steel. The stronger material is more brittle in the fatigue sense: it cannot redistribute the notch stress by local yielding, so it develops more of the elastic concentration. This is the fundamental reason high-strength fasteners, gears and springs are so sensitive to surface damage.

Second, small features are not automatically benign. A 0.2 mm radius reduces q but the feature itself has a higher K_t for the same depth-to-radius ratio, and the two effects partly cancel. The practical conclusion: increase the radius, do not just shrink the feature.

4.3 The short-crack problem and Kitagawa–Takahashi

The K_f approach assumes the notch is large compared with the material's microstructure. When the feature is comparable to the grain size or the intrinsic crack length, the notch-tip stress field is no longer well described by K_t, and the fatigue limit stops rising as the feature shrinks. The Kitagawa–Takahashi diagram captures this: the threshold stress amplitude for a crack of length a is the lower of the plain-fatigue limit and the fracture-mechanics threshold:

El Haddad's correction smooths the transition by adding an intrinsic crack length a_0:

For steel with \Delta K_{th} = 7 MPa√m and \Delta\sigma_e = 180 MPa, a_0 = 0.48 mm. That half-millimetre is the material's intrinsic defect tolerance: cracks shorter than ~0.5 mm behave as if the material had no crack at all, and the plain fatigue limit governs. Cracks longer than ~0.5 mm follow the fracture-mechanics threshold, which falls as 1/\sqrt{a}.

The engineering consequence is the threshold stress table for a surface flaw with Y = 1.12:

Flaw depth a · Threshold stress amplitude \Delta\sigma_{th}

0.1 mm · 353 MPa

0.5 mm · 158 MPa

1.0 mm · 112 MPa

5.0 mm · 50 MPa

10.0 mm · 35 MPa

A 1 mm surface flaw in this steel will not grow at a stress range below 112 MPa. That is the number to compare with the 180 MPa endurance limit: a 1 mm flaw costs 38% of the plain fatigue strength, and a 5 mm flaw costs 72%. Inspection sensitivity and allowable stress are the same calculation seen from two ends.

4.4 Applying K_f in the Marin chain

Strictly, K_f multiplies the nominal stress, not S_e. The two equivalent ways to handle it:

or

Both give the same answer at the elastic level. The second is common in textbook practice and is how the shaft example in §2 would be extended if the shaft had a keyway: K_t = 2.2 at a 0.5 mm corner radius gives q = 0.665, K_f = 1.80, and the allowable alternating stress falls from 180 to 100 MPa. A keyway costs 44% of the fatigue budget — which is why keyed shafts fail at the keyway end, and why keyless interference fits and shrink discs exist.

5. Strain Life: The Low-Cycle Regime Where Stress Is the Wrong Variable

5.1 Why stress-life fails below 10⁴ cycles

Below roughly 10^4 cycles, the applied strain is large enough that the material yields in every cycle, the stress–strain response is a hysteresis loop rather than a line, and the stress amplitude is no longer a good correlating variable — two different materials at the same stress amplitude can have wildly different lives because their ductility differs. The strain-life (ε–N) approach, developed by Coffin and Manson independently in 1953–54, correlates life with strain amplitude instead.

The cyclic stress–strain curve is described by Ramberg–Osgood:

where K' is the cyclic strength coefficient and n' the cyclic strain-hardening exponent (0.10–0.20 for most metals). Note "cyclic": a material that strain-hardens in monotonic tension may cyclically soften, and vice versa. Cyclic softening is common in cold-worked steels and in precipitation-hardened aluminium; cyclic hardening is common in annealed austenitic stainless. The cyclic curve is what matters, not the monotonic one.

5.2 The Coffin–Manson–Basquin relation

Total strain amplitude is the sum of elastic and plastic components:

Four material constants: \sigma_f' (fatigue strength coefficient, ≈ true fracture strength), b (fatigue strength exponent, −0.05 to −0.12), \varepsilon_f' (fatigue ductility coefficient, ≈ true fracture ductility), c (fatigue ductility exponent, −0.5 to −0.7).

The two terms cross at the transition life N_t:

Below N_t, plastic strain dominates and ductility is the useful material property. Above N_t, elastic strain dominates and strength is the useful property. For SAE 1045 with the tabulated constants (\sigma_f' = 948 MPa, b = -0.092, \varepsilon_f' = 0.26, c = -0.445, E = 207 GPa), N_t \approx 46{,}600 cycles — right at the boundary between low-cycle and high-cycle fatigue. This is why the same steel is selected for a crankshaft (LCF, wants toughness) and for a shaft (HCF, wants strength), and why the heat treatment differs.

5.3 Worked strain-life calculation

Using the 1045 constants above:

Total strain amplitude \Delta\varepsilon/2 · Life N_f · Regime

1.0% · 1,330 cycles · Deep LCF — plastic dominates

0.5% · 10,100 cycles · LCF

0.2% · 333,000 cycles · Transition / HCF

0.1% · 23.8 million cycles · HCF — essentially elastic

The sensitivity is brutal: halving the strain amplitude from 0.2% to 0.1% multiplies life by 71. This is the defining characteristic of the LCF regime, and the reason a 10% error in strain measurement in a thermomechanical fatigue test can produce a 3× error in predicted life. It is also the reason LCF testing requires a clip-on extensometer and careful alignment, not a strain gauge on the specimen surface.

5.4 Neuber's notch rule

To apply strain-life to a notched part, the local notch stress and strain must be found. Neuber's rule (1961) states that the elastic stress concentration is the geometric mean of the actual stress and strain concentrations:

For nominal elastic behaviour, e_{nom} = S_{nom}/E, so:

The notch root state is the intersection of this hyperbola with the cyclic stress–strain curve.

Worked: K_t = 3, S_{nom} = 150 MPa, 1045 steel. The hyperbola is \sigma\varepsilon = (3 \times 150)^2/207000 = 0.978 MPa. Intersecting with Ramberg–Osgood gives \sigma = 329 MPa, \varepsilon = 0.297\%. Check: K_\sigma = 2.20, K_\varepsilon = 4.10, product = 9.0 = K_t^2. The notch root yields to 329 MPa — below the 531 MPa yield strength, but the local strain of 0.297% is 2× the nominal elastic strain, and the resulting life from the strain-life curve is 1.6 million cycles, versus the 23.8 million the nominal strain of 0.1% would suggest.

That factor of 15 is the notch. It is the entire reason fillet radii, thread forms and weld toe geometry dominate fatigue design.

5.5 Mean stress in strain life

Two corrections are standard.

Morrow subtracts the mean stress from the elastic term only:

Smith–Watson–Topper (SWT) uses the maximum stress as a multiplier, which handles tension-compression cycles with a tensile mean better:

SWT is the default for notched components under tension-compression; Morrow is preferred for fully reversed or compressive-mean loading. Neither handles large compressive mean stress well, because crack closure changes the effective driving force — which is where the fracture-mechanics approach of §7 takes over.

6. Variable Amplitude: Rainflow, Miner, and the Scatter That Eats Your Margin

6.1 Rainflow counting

Service loads are irregular. The standard method for reducing a load history to a set of cycles is rainflow counting, codified in ASTM E1049. The algorithm identifies closed hysteresis loops in the load–time history by treating the history as a sequence of roof peaks down which rain flows; each time the flow is interrupted by a larger peak, a cycle is counted. The output is a histogram of (S_a, S_m) pairs and their counts.

The implementation detail that matters: the counting must be done on the local stress–strain response at the critical location, not on the nominal load. For an elastic structure the two are proportional and the counting can be done on load; for a notch that yields, the local response is path-dependent and the counting has to be done on the reconstructed local history (or on the nominal history with a Neuber correction applied cycle by cycle).

6.2 Miner's rule — and its honest error bar

Palmgren (1924) and Miner (1945) proposed that damage accumulates linearly:

where n_i is the number of applied cycles at stress amplitude S_i and N_i is the life at that amplitude from the S–N curve. Failure is predicted at D = 1.

The rule is used universally because it is simple and because the alternatives require material-specific memory models. It is also wrong in specific, predictable ways:

The practical response is to design to D \le 0.3–0.5 rather than D \le 1 when the spectrum is not well characterised, and to use a damage-tolerance crack-growth calculation (§7) for the critical locations rather than relying on Miner alone.

6.3 The knee, and Haibach's correction

A subtle but important error: Miner applied to the constant-amplitude S–N curve ignores the endurance limit entirely if cycles below the limit are counted as N_i = \infty (no damage) — which is what the S–N curve says. But in a variable-amplitude spectrum, cycles below the constant-amplitude limit do damage, because they follow larger cycles that have already initiated a crack.

Eurocode 3 handles this with a bilinear curve: slope m = 3 up to 5×10⁶ cycles, then slope m = 5 from 5×10⁶ to the cut-off at 10⁸ cycles, below which no damage is counted. The change of slope is empirical and conservative.

Haibach's modification is the continuous version: below the knee, use a slope of 2m - 1 instead of m:

For m = 3 the post-knee slope is 5 — identical to Eurocode's — which is why the two approaches give similar answers for steel. The message is consistent: do not treat the endurance limit as a hard floor when the loading is variable.

6.4 A three-block worked example

Take a spectrum with three blocks, applied to the 1045 shaft of §2 (S_e = 180 MPa, Basquin b = -0.09 fitted so that S_a = 180 MPa at 10^7 cycles):

Block · S_a (MPa) · Cycles n_i · N_i from curve · n_i/N_i

1 · 300 · 20,000 · 3.2\times10^5 · 0.062

2 · 220 · 200,000 · 3.6\times10^6 · 0.056

3 · 150 · 2,000,000 · 6.0\times10^7 · 0.033

Total · · · · 0.151

D = 0.151, so the part is predicted to survive the block roughly 6.6 times — about 14.7 million cycles. Apply a design factor of 3 on damage (i.e. D_{allow} = 0.33) and the allowable repetitions fall to 2.2 blocks. Note that the largest stress block contributes 41% of the damage with only 0.9% of the cycles: damage is dominated by the rare high-amplitude events, which is exactly why a load spectrum truncated at 10⁶ cycles of small loads can be dangerously non-conservative.

7. Fracture Mechanics: Turning Cracks Into Arithmetic

7.1 Griffith and the energy criterion

Griffith (1920) explained why glass fails at stresses far below its theoretical strength: a crack of length 2a in a plate released elastic strain energy faster than it created new surface energy. The critical stress is:

for a central through-crack in an infinite plate, with \gamma_s the surface energy. For glass, \gamma_s \approx 1 J/m² and E = 70 GPa, giving a theoretical strength of ~10 GPa for an atomically sharp flaw and ~70 MPa for a 10 µm crack — the right order for real glass.

Metals are tougher than Griffith predicts because the crack tip yields, dissipating plastic work that is 10³–10⁵ times the surface energy. Irwin (1957) replaced \gamma_s with the plastic work and introduced the stress intensity factor, which is the quantity that made fracture mechanics an engineering tool.

7.2 The stress intensity factor

For a crack of length a under remote stress \sigma:

with Y a geometry factor: 1.0 for a central through-crack in an infinite plate, 1.12 for a surface semi-elliptical crack, 1.12–1.3 for a single edge crack, ~0.7 for a crack at a hole, and tabulated in handbooks (Tada, Paris & Irwin; Rooke & Cartwright) for everything else. Units: MPa√m.

The critical value K_{IC} is the plane-strain fracture toughness, a material property measured to ASTM E399. When K_I reaches K_{IC}, the crack propagates unstably. The link to energy is:

for plane strain, with G_{IC} the critical strain energy release rate.

Typical K_{IC} values (thick section, room temperature, typical heat treatment):

Material · K_{IC} (MPa√m) · Comment

AISI 1020, hot-rolled · 50–90 · Low-carbon steels are tough but not plane-strain valid at small thickness

AISI 4340, tempered 425 °C · ~60 · The classic high-strength steel; drops to ~40 at 200 °C temper

300M · 55–65 · Landing gear steel

17-4PH H900 · 60–90 · Precipitation-hardened stainless

7075-T651 · 24–29 · High strength, low toughness — the trade-off in one line

2024-T351 · 34–40 · Better toughness, lower strength than 7075

Ti-6Al-4V, mill annealed · ~75 · Excellent damage tolerance

AlSi10Mg, LPBF + T6 · 15–25 · Porosity-limited; HIP improves it

Nodular cast iron · 20–30 · Graphite nodules blunt the crack

Alumina (96%) · 3–5 · Ceramics are brittle in the fracture-mechanics sense, not just the colloquial one

The thickness effect is real and often missed: K_{IC} is a plane-strain property, valid only when the specimen is thick enough for the crack-tip plastic zone to be small compared with the thickness. A thin sheet has a higher apparent toughness (plane stress, K_C > K_{IC}) because the plastic zone can extend through the thickness. Using a handbook K_{IC} for a 2 mm sheet is conservative; using a thin-sheet K_C for a 50 mm forging is not.

7.3 Critical crack size

Rearranging K_I = K_{IC}:

Worked: K_{IC} = 50 MPa√m, \sigma = 300 MPa.

Now the sensitivity: at \sigma = 200 MPa, a_c = 19.9 mm (through-crack); at \sigma = 400 MPa, a_c = 4.97 mm. The critical crack size falls with the square of the stress, so a 33% stress increase cuts the tolerable flaw size by 44%. This is why proof testing at 1.1–1.3× working pressure is such an effective damage-tolerance measure: it fails the parts that already contain cracks above the proof-stress critical size, and it leaves a known maximum flaw population in the survivors.

7.4 The Paris law

Subcritical crack growth under cyclic loading follows the Paris law (Paris, Gomez & Anderson, 1961):

with \Delta K = Y\Delta\sigma\sqrt{\pi a}. For ferritic-pearlitic steels, the Barsom–Rolfe constants are C = 6.9\times10^{-12} and m = 3.0 with da/dN in m/cycle and \Delta K in MPa√m. The exponent is the important part: m is 2.5–4 for most metals (3 is the steel default, 3–4 for aluminium, 2–3 for titanium), and it means a doubling of \Delta K multiplies growth rate by 8 at m = 3.

The full growth curve has three regions:

The R-ratio matters throughout: higher mean stress raises da/dN at the same \Delta K and lowers the threshold, because crack closure (Elber, 1970) shields the tip less when the crack is held open. A Paris law measured at R = 0.1 under-predicts growth at R = 0.7 by a factor of 2–5 in the near-threshold region.

7.5 Integrating the Paris law

For a through-crack in an infinite plate (Y = 1):

For m = 3 this reduces to:

Worked: \Delta\sigma = 100 MPa, a_i = 1 mm, a_f = 10 mm, C = 6.9\times10^{-12}, m = 3.

At 3,000 rpm that is 6.3 hours. Three observations:

  1. The answer is dominated by the small-crack end. Going from 1 mm to 2 mm consumes 0.78 million cycles (69% of the total); going from 2 mm to 10 mm consumes the remaining 0.35 million. Most of the life is spent while the crack is invisible.
  2. Raise the stress range by 50% (100 → 150 MPa) and life falls to 3.3\times10^5 cycles — a factor of 3.4, which is 1.5^3 plus the threshold shift. The cube is the whole story.
  3. If the part is 25 mm thick, a_f is capped by the thickness, not by K_{IC}: using a_f = 25 mm gives 1.32\times10^6 cycles. The difference between 10 mm and 25 mm final crack is only 17% of life, because the last part of the growth is fast.

7.6 Damage tolerance and inspection intervals

The damage-tolerance design philosophy inverts the safe-life question. Instead of asking "how long until a crack starts?", it asks "given that a crack of the largest undetectable size exists, how long until it becomes critical, and how often must I inspect?"

The procedure:

  1. Determine the largest flaw that the inspection method can reliably detect — the NDI limit a_{NDI} (for dye penetrant, ~0.5–1 mm; for magnetic particle, ~0.3–0.5 mm; for ultrasonic, 1–3 mm depending on thickness and technique; for radiography, 0.5–2% of thickness).
  2. Calculate the critical size a_c at the design stress.
  3. Integrate Paris from a_{NDI} to a_c to get the inspection interval N_{insp}.
  4. Apply a factor of 2 on the interval (inspect at half the calculated life) to cover scatter in C, m, and the load spectrum.
  5. If the interval is shorter than the practical maintenance interval, change the design: lower the stress, raise the toughness, or improve the inspectability.

Worked: a_{NDI} = 1 mm (dye penetrant, conservative), a_c = 8.8 mm, \Delta\sigma = 100 MPa → N_{insp} = 1.1\times10^6 cycles; inspect at 550,000 cycles. At 3,000 rpm continuous, that is 3.1 hours of running — which tells you immediately that this component cannot be run continuously without either a much lower stress or a much better inspection method. Damage tolerance is not a paper exercise; it produces numbers that either fit the maintenance schedule or do not.

7.7 Leak-before-break

For pressure vessels and piping, the preferred failure mode is leakage rather than rupture: a through-wall crack that leaks is detectable and repairable, while a critical crack is an explosion. Leak-before-break requires that the critical crack length through the wall be greater than the wall thickness, so the crack penetrates and leaks before it becomes unstable:

This is achievable in thin-wall, high-toughness vessels (austenitic stainless piping in nuclear plants is the classic case) and difficult in thick-wall, high-strength vessels. The design consequence: choose toughness over strength when the consequence of rupture is unacceptable, because a_c \propto (K_{IC}/\sigma)^2 and doubling K_{IC} quadruples the tolerable flaw.

7.8 When linear elastic fracture mechanics fails

LEFM assumes the plastic zone at the crack tip is small compared with the crack and the remaining ligament. When it is not — low-strength high-toughness steels, thin sections, or loading near general yield — the appropriate tools are:

For a machine shop, the practical implication is simpler: if the section is thin and the steel is tough, LEFM is conservative; if the section is thick and the steel is strong, LEFM is the right tool and the toughness number matters more than the yield strength.

8. Welded Joints: The Worst Fatigue Detail in Any Structure

8.1 Why welds are different

A welded joint is not a material with a fatigue strength; it is a detail with a fatigue strength. Three things change at a weld, and all three are bad:

  1. Geometry. The weld toe is a notch with a root radius of 0.1–0.5 mm as-welded, giving K_t of 3–5. Undercut, spatter, stop-start craters and toe angles of 120–160° make it worse.
  2. Residual stress. The weld metal and heat-affected zone (HAZ) cool under restraint, producing tensile residual stress approaching yield at the toe. A weld toe is therefore already at R \approx 0.5–0.8 before any service load is applied, which puts it in the worst part of the mean-stress diagram.
  3. Defects. Porosity, slag, lack of fusion, lack of penetration and micro-cracks are built in. The largest of these acts as a pre-existing crack, so N_{init} \approx 0.

The consequence: welded joints have no true endurance limit. The apparent knee at 5×10⁶ cycles is a convention; variable-amplitude loading below the knee still grows cracks. This is why every structural fatigue code treats welds as finite-life details, not infinite-life ones.

8.2 Eurocode detail categories

EN 1993-1-9 defines a family of S–N curves, each labelled by its fatigue strength at 2×10⁶ cycles in N/mm²:

Detail category · Example detail

160 · Base metal, rolled, away from welds

140 · Base metal with edges rolled or ground

125 · Butt weld, ground flush, with transition

112 · Butt weld, machined flush

90 · Butt weld, as-welded, full penetration

80 · Transverse butt weld, as-welded

71 · Transverse attachment, fillet, toe failure

56 · Cruciform joint, fillet, toe failure

45 · Fillet weld, throat failure

36 · Fillet weld, root failure (load-carrying)

The curves share a common shape:

with \Delta\sigma_D = 0.737\,\Delta\sigma_C (the constant-amplitude fatigue limit at 5×10⁶) and a cut-off at 10⁸ cycles of \Delta\sigma_L = 0.549\,\Delta\sigma_D \approx 0.405\,\Delta\sigma_C.

Worked: a FAT 80 detail at a 100 MPa stress range survives 2\times10^6 \times (80/100)^3 = 1.02\times10^6 cycles. The same 100 MPa on base metal (FAT 160) gives 8.2\times10^6 cycles — a factor of eight, for the same steel, the same load, and a weld that passes every visual inspection. The weld costs more fatigue life than changing from mild steel to 700 MPa quenched-and-tempered steel gains.

At FAT 36 (fillet root), 100 MPa gives only 9.3\times10^4 cycles. This is why load-carrying fillet welds are avoided in fatigue-critical structures and why full-penetration butt welds are specified wherever the load path allows.

8.3 Hot-spot stress and the structural stress method

Nominal stress is ill-defined at a weld because the stress field is dominated by the local geometry. The hot-spot stress method extrapolates the stress from two or three reference points outside the notch region to the weld toe, capturing the structural (membrane + bending) stress without the notch singularity. IIW defines the extrapolation distances (0.4t and 1.0t for the two-point method) and assigns FAT classes to the hot-spot stress rather than the nominal.

The alternative is the structural stress approach (Dong, Battelle), which uses through-thickness stress linearisation and a master S–N curve that is claimed to be independent of joint type. Both methods exist because the nominal-stress approach requires a detail classification for every geometry, and real structures contain geometries that the classification tables do not cover.

For a machine shop building a bracket or a frame, the practical version is simpler: treat the weld toe as a FAT 71–80 detail unless the weld is ground flush and fully penetrated, in which case FAT 90–112 applies. Do not use the base-metal curve at a weld, ever.

8.4 Weld improvement methods

Four methods reliably buy back one to three detail categories:

TIG dressing. Re-melting the toe with a tungsten arc eliminates the toe notch and produces a smooth transition with a radius of 2–4 mm. Typical improvement: one to two FAT classes (a factor of 1.3–1.6 on stress range at 2×10⁶). Requires clean, dry plate and a skilled operator; the benefit is lost if the toe is re-worked afterwards.

Burr grinding. Grinding the toe with a burr to a depth of 0.5–1 mm and a radius of 2–4 mm removes the notch and any small toe defects. Improvement: one to two FAT classes, similar to TIG dressing, with less heat input and more tolerance for site work.

High-frequency mechanical impact (HFMI). A hardened pin oscillating at 200–300 Hz hammers the toe, imposing a deep compressive residual layer (0.5–1.5 mm) and a smooth, work-hardened groove. Improvement: two to three FAT classes (factor 1.5–2.5), and the only method with a codified procedure (IIW recommendations, 2016) that accounts for steel grade. This is the current best practice for heavy fabricated structures.

Shot peening. Similar mechanism, shallower layer, less consistent at a toe because the geometry shields the peening media. Improvement: 20–60% on life, highly dependent on coverage and intensity.

All four share a failure mode: the improvement is a residual-stress effect, and it is erased by any subsequent operation that yields the surface in tension — proof loading beyond yield, straightening, or thermal exposure above the tempering temperature. A peened weld that is subsequently flame-straightened is an unpeened weld with extra steps.

8.5 Distortion, misalignment and the secondary bending penalty

A 1 mm axial misalignment in a 10 mm plate produces a secondary bending moment. The resulting stress increase at the weld toe is often 50–100%, and the fatigue life penalty is the cube of that: a factor of 3–8. Misalignment is the single most under-appreciated weld fatigue variable, and it is controllable by fixture design, not by weld procedure. A well-fitted joint with a modest weld beats a perfect weld on a badly aligned joint every time.

The same logic applies to angular distortion: a 2° angular misalignment in a butt joint can double the local bending stress at the toe. Fabrication tolerances are fatigue variables, and the inspection that matters is the one before welding.

9. Manufacturing: Where the Fatigue Strength Is Actually Decided

9.1 Surface integrity

The surface is where fatigue cracks start, in 80–90% of cases. Three properties of the surface matter: roughness, residual stress, and microstructure (including white layers, overtempered zones, and micro-cracks).

Roughness. The k_a table in §2 is the design-level treatment. At the process level, the numbers are:

Process · Typical R_a (µm) · k_a at 600 MPa (from §2) · Fatigue penalty vs ground

Polished · 0.05–0.2 · 0.92–0.95 · Baseline

Ground · 0.1–0.8 · 0.917 · 0–5%

Fine turning · 0.4–1.6 · 0.87–0.90 · 5–10%

Rough turning / milling · 1.6–6.3 · 0.80–0.85 · 10–15%

Drilling · 1.6–6.3 · 0.80–0.85 · 15% plus hole K_t

Broaching / reaming · 0.4–1.6 · 0.87–0.90 · 5–10%

Hot rolling · 3.2–12.5 · 0.584 · 35–40%

As-forged · >12.5 · 0.468 · 50%

As-built LPBF · 7–20 · 0.45–0.55 (effective) · 50–70%

The last row is not a Marin factor in the textbook sense — the roughness is a population of notches, and the effective k_a depends on the local surface angle, powder adhesion and layer thickness. It is nevertheless the right order of magnitude, and it is why §9.3 exists.

Residual stress. Machining does not just cut metal; it deforms it. Turning with a worn tool, or with a large nose radius and high feed, can leave tensile residual stress of 300–600 MPa in the surface layer. Grinding is worse: a grinding burn — visible as temper colours, detectable by nital etching — produces an overtempered zone with tensile residual stress and, in hardened steel, a re-hardened white layer that is brittle and crack-prone. A burned 60 HRC bearing race can lose 30–50% of its fatigue life and still pass a dimensional inspection.

The remedy is process control: sharp tools, correct coolant, controlled depth of cut, and a finishing pass that removes the damaged layer. For critical surfaces, specify the surface integrity requirement — "no temper colours, no white layer, residual stress compressive or below +100 MPa" — and inspect for it. A surface finish specification that names only R_a does not control residual stress, and it is the residual stress that often decides the life.

9.2 Shot peening and Almen control

Shot peening hammers the surface with spherical media (steel, ceramic or glass, 0.1–1.0 mm) at 50–100 m/s. Each impact creates a small plastic indentation, and the surrounding elastic material springs back into compression. The result is a surface layer, 0.1–0.5 mm deep, with −400 to −800 MPa of compressive residual stress and work-hardened microstructure.

The process is controlled by Almen intensity — a measure of the arc height of a standard Almen strip (A, C or N) peened under the same conditions. An 8A intensity produces a compressive layer of roughly 0.2–0.3 mm in steel; a 12C produces 0.4–0.6 mm. Coverage is specified as a percentage of the surface fully indented; 100% coverage is the point at which the surface is fully dimpled, and 200% is a common specification for critical parts (peening for twice the time needed for 100%).

Fatigue improvements:

Component · Improvement · Mechanism

Coil springs · 2–5× life · Compressive layer at the wire surface, where the torsion crack starts

Gear roots · 30–100% · Compressive layer at the fillet, plus reduced notch sensitivity

Weld toes · 20–60% · Compressive layer at the toe

Fastener threads · 2–3× · Rolled threads plus peening; cut threads are 2–3× worse than rolled to start with

Turbine blades · 50–200% · Compressive layer at the dovetail root

The limits: peening cannot help a part that fails from an internal defect; it does not help compressive-mean loading; and the benefit is lost above the tempering temperature of the material (roughly 200–250 °C for a 4340 temper, higher for a maraging steel). Shot peening a part that will run at 400 °C is buying a residual stress that will relax away in the first hundred hours.

Laser peening produces a deeper layer (1–2 mm), higher magnitude (−1 to −1.5 GPa), and no surface roughening — but at 5–20× the cost. It is used on turbine engine fan blades and on weld toes in nuclear piping where the extra depth is worth the price.

Cold expansion of fastener holes: a tapered mandrel pulls a split sleeve through the hole, expanding it by 2–4% and leaving a compressive ring around the bore. Fatigue life improvement is 3–5×, and the method is standard in aircraft wing skins. The hole must be reamed first, and the expansion must be measured, not assumed.

9.3 Additive manufacturing: the fatigue problem in full

This is the section that matters most for a platform connecting AM capacity to customers, because AM fatigue is where the gap between the datasheet and the part is widest.

The four process-specific fatigue variables:

  1. Surface roughness. As-built LPBF surfaces have R_a 7–20 µm; EBM 15–40 µm. The roughness depends on build angle: a 90° (vertical) surface is smoother than a 45° down-skin, which is rougher than a horizontal top surface. Down-skins have partially fused powder particles attached, staircase effects on inclined surfaces, and dross on overhangs. These are not cosmetic; valleys 50 µm deep act as crack initiators, and the effect is strongest at low stress amplitudes where initiation is otherwise difficult.
  2. Porosity and lack-of-fusion. Keyhole pores are roughly spherical and relatively benign; lack-of-fusion (LoF) defects are irregular, sharp and aligned with the build layers, and are the worst fatigue initiators. LoF occurs when the laser energy density is insufficient to fully melt the previous layer, and it is a process-control failure, not a random event. Typical as-built porosity is 0.1–1%; HIP reduces it below 0.01% but does not remove gas porosity trapped inside the metal.
  3. Residual stress. The rapid melt-solidify cycle produces tensile residual stress at the surface of up to ~800 MPa in Ti alloys and 200–400 MPa in steels, balanced by compression in the bulk. This causes distortion, cracking and delamination, and it shifts the surface operating point to the wrong side of the mean-stress diagram. Stress relief (2 h at 650 °C for Ti64, 2 h at 900 °C for stainless) is mandatory for load-bearing parts.
  4. Microstructure and anisotropy. The columnar grain structure grows along the build direction, so fatigue properties differ between vertical and horizontal specimens, and between specimens built at different locations on the plate. A single "Ti-6Al-4V" datasheet number is an average over a distribution that can span 100 MPa.

The measured consequence, for Ti-6Al-4V:

Condition · Fatigue strength at 10⁷ cycles · Note

Wrought, polished · 500–600 MPa · Baseline

EBM as-built · 200–250 MPa · Surface and porosity dominated

EBM stress-relieved · 200–250 MPa · Stress relief alone does not fix surface or porosity

EBM + HIP · 550–600 MPa · Porosity closed; surface still rough

LPBF as-built · >50% below wrought · Roughness and LoF

LPBF + machined to S_a 0.5 µm · Highest of the set · Surface is the controlling variable

LPBF + HIP + machined + peened · Approaches wrought · The full post-processing chain

The key insight is the ordering: HIP fixes porosity, machining fixes roughness, and neither substitutes for the other. A HIPed part with an as-built surface still fails from surface initiation; a machined part with internal LoF still fails from a subsurface defect. The post-processing chain has to address both.

Process-specific standards: ASTM F2924 (PBF Ti-6Al-4V), ASTM F3001 (EBM Ti-6Al-4V), ASTM F3301 (post-processing of AM Ti alloys), and ISO/ASTM 52900 for terminology. For fatigue-critical AM parts, the drawing should specify: build orientation, HIP cycle, surface finish on fatigue-critical surfaces, and the NDT method for internal defects — and the supplier should be able to show coupons built in the same orientation and post-processed the same way.

For a customer sourcing an AM part on FabFlow, the practical ask is: "Which surfaces are fatigue-critical, and what is the as-built roughness and post-processing on those surfaces?" A quote that specifies "Ti-6Al-4V, HIP" without a surface finish requirement is a quote for a part with 200–250 MPa fatigue strength, not 550.

9.4 Casting and forging

Castings carry porosity, shrinkage, hot tears and inclusions. The fatigue strength of a casting is dominated by the largest defect in the highly stressed volume, which is why casting factors (1.0–2.0 on allowable stress) exist in pressure-vessel codes and why radiography of the critical section is specified. A well-made investment casting can approach wrought fatigue strength; a poorly fed sand casting cannot.

Forgings carry grain flow, which is beneficial when the flow lines follow the part contour and detrimental when a machining operation cuts across them. A crankshaft forged and then machined has grain flow following the webs; the same shape machined from plate has grain flow running straight through the fillets, and fatigue life drops by 30–50%. This is why "forged" is a fatigue specification, not just a strength specification.

9.5 A process-selection table

Process · Typical fatigue debit vs polished wrought · Dominant cause · Best remedy

Wrought + machined + polished · 1.0 (baseline) · — · —

Wrought + turned · 0.85–0.95 · Surface roughness · Finish pass, peen

Wrought + welded · 0.3–0.5 · Toe notch + residual stress · TIG dress, HFMI

Casting · 0.4–0.7 · Porosity · HIP, radiography, casting factor

Forging (flow lines intact) · 0.9–1.1 · — · —

Forging (machined across flow) · 0.5–0.7 · Grain flow · Reorient, redesign

LPBF + HIP + machined · 0.8–0.95 · Residual porosity · Full post-processing chain

LPBF as-built · 0.3–0.5 · Roughness + LoF · Machine critical surfaces, HIP

EBM + HIP · 0.85–1.0 · Surface roughness · Machine or peen

10. Environment: Corrosion Fatigue, Fretting and Hydrogen

10.1 Corrosion fatigue

A corrosive environment removes the endurance limit entirely. The S–N curve continues to fall, and at 10⁸ cycles the fatigue strength can be 20–40% of the air value. The mechanism is straightforward: the corrosion pit is a notch (K_t 2–5), the crack tip dissolves, and the repassivation film is ruptured by each cycle, exposing fresh metal.

The engineering consequences:

The pit-to-crack transition is the same Kitagawa–Takahashi arithmetic as §4.3: a 0.2 mm pit in steel with \Delta K_{th} = 7 MPa√m has a threshold stress range of about 160 MPa; grow the pit to 0.5 mm and the threshold falls to 100 MPa. Corrosion is a slow, continuous way of making the flaw bigger.

10.2 Fretting fatigue

Fretting occurs when two surfaces in contact experience small relative slip — typically 5–100 µm — under a normal load. The slip destroys the oxide film, produces wear debris, and initiates cracks at the contact edge. Fretting fatigue strength can be 50–80% below the plain fatigue strength, and the cracks start at stresses that would be harmless without the contact.

Where it bites:

The countermeasures are, in order of effectiveness:

  1. Eliminate slip — increase the interference or preload so the contact stays in partial slip or stick. A shrink fit with 0.03–0.05 mm interference per 50 mm diameter is a fatigue measure, not just a torque-transmission measure.
  2. Shot peen the contact edge — compressive residual stress suppresses crack initiation in the fretting zone.
  3. Use a sacrificial coating or insert — MoS₂, PTFE, or a soft metal shim, which accommodates the slip in a low-friction layer.
  4. Move the contact away from the stress concentration — a relief groove or a change in the fit geometry so the contact edge is not at the peak stress.
  5. Reduce the normal load if the joint design allows.

Fretting is the reason a shaft that passes a plain fatigue calculation can still fail at a bearing seat, and it is why the fit edge deserves the same design attention as a fillet.

10.3 Hydrogen embrittlement

High-strength steels absorb hydrogen during plating, pickling, welding, or cathodic protection, and the hydrogen reduces the cohesive strength at the crack tip. The result is subcritical crack growth at K_I well below K_{IC}, often with a delay between loading and failure — the classic "broke overnight after plating" failure.

The rules:

The fatigue connection: hydrogen embrittlement accelerates crack growth and lowers the threshold, so a plated high-strength fastener that passes a static proof test can fail in service under cyclic load at a fraction of its rated strength. If a part is both high-strength and plated, it needs a bake and a fatigue assessment, not just a tensile certificate.

11. Twenty Design Rules, Each With a Number

  1. Assume S_e = 0.5\,S_{ut} only as a starting point. Apply the full Marin chain; the answer is typically 50–60% of the starting point.
  2. Design to 99% reliability for safety-relevant parts. k_e = 0.814, not 1.0.
  3. Never use a nominal stress at a notch. Multiply by K_f, and compute K_f from q, not K_t.
  4. Fillet radius is a fatigue variable. Going from r = 0.2 to r = 2 mm at K_t = 2.5 moves K_f from 1.84 to 2.20 — but the larger radius also lowers K_t itself, so the net gain is larger than the q table suggests. Specify the radius on the drawing, not just the profile.
  5. Keep the stress range below 112 MPa if a 1 mm flaw is possible. That is the Kitagawa threshold for steel at \Delta K_{th} = 7 MPa√m.
  6. For welded joints, use FAT 71–80 unless the toe is improved. Base-metal curves do not apply at a weld.
  7. Specify HFMI or TIG dressing for fatigue-critical welds. It buys 1.5–2.5× life for 5–15% of the fabrication cost.
  8. Control misalignment to 1 mm in 10 mm. A 1 mm offset can cost a factor of 3–8 in life.
  9. Do not design an infinite-life part in aluminium. There is no endurance limit; design to a finite life and inspect.
  10. Use D \le 0.3–0.5, not D \le 1, for variable-amplitude loading. Miner's rule has a 0.3–3 scatter band.
  11. Count cycles with rainflow, not with a peak count. ASTM E1049.
  12. For LCF, use strain, not stress. Below ~10⁴ cycles the stress amplitude is the wrong variable.
  13. At a notch in LCF, use Neuber. K_t^2 = K_\sigma K_\varepsilon; the local strain can be 2–5× the nominal.
  14. Specify surface integrity, not just R_a. Residual stress and white layer matter more than roughness on hardened steels.
  15. Shot peen, then do not touch the surface. Any subsequent grinding, straightening, or heating above temper removes the benefit.
  16. HIP and machine additively manufactured fatigue parts. HIP alone gives 550 MPa; as-built gives 250 MPa; both are needed for wrought-equivalent performance.
  17. For pressure vessels, check leak-before-break. a_c > t at design stress; if not, raise toughness or lower stress.
  18. Set inspection intervals from Paris integration at half the calculated life. N_{insp} = 0.5\,N_p(a_{NDI} \to a_c).
  19. Bake high-strength plated steel. 190–230 °C for 1–4 h within 1 h of plating (ASTM B850).
  20. Test the actual part when failure is unacceptable. A coupon S–N curve is not a component test; the difference is the surface, the residual stress, and the load path.

12. Testing, Standards, and the Ledger

12.1 The test methods that produce the curves

Property · Standard · Specimen / method · Notes

High-cycle S–N (force control) · ASTM E466, ISO 1099 · Smooth or notched axial specimen, R controlled, 1–100 Hz · Run-outs at 10⁷; 12–20 specimens for a curve

Low-cycle fatigue (strain control) · ASTM E606 · Extensometer-mounted specimen, 0.1–5 Hz · Total strain amplitude 0.2–2%; 10–30 specimens

Rotating bending · ISO 1143 · Cantilever or four-point rotating specimen · The classic Wöhler test; cheap, but a stress gradient

Fatigue crack growth rate · ASTM E647 · CT or M(T) specimen, load shedding · da/dN vs \Delta K, threshold by decreasing \Delta K

Plane-strain toughness · ASTM E399 · CT or SENB, thickness > 2.5(K_{IC}/S_y)^2 · Valid only if the plastic zone is small

Elastic-plastic toughness · ASTM E1820 · J-integral / CTOD · For ductile steels and thin sections

Cycle counting · ASTM E1049 · Rainflow / reservoir · The data-reduction standard

S–N data analysis · ASTM E739 · Linear regression on log–log · Confidence bands, not just a line

Statistical endurance limit · ISO 12107 · Staircase (Dixon–Mood), 15–20 specimens · Step size 5–10% of the estimated limit

Hydrogen embrittlement · ASTM F519, F1624 · Notched tensile, incremental step load · For plated high-strength steel

The staircase method deserves a note because it is the cheapest way to get a defensible endurance limit. Test the first specimen at an estimated stress; if it survives 10⁷ cycles, raise the stress by one step; if it fails, lower it. After 15–20 specimens, the mean and standard deviation of the endurance limit come from the counts of failures at each level. It is more informative per rupee than a full S–N curve when the question is "what is the limit?" rather than "what is the curve?".

12.2 The design codes

Code · Scope · Fatigue approach

EN 1993-1-9 · Steel structures · Detail categories FAT 36–160, bilinear S–N

BS 7608 · Steel structures (UK) · Detail classes, similar to Eurocode

IIW recommendations · Welded joints, global · Hot-spot stress, FAT classes, improvement factors

DNV-RP-C203 · Offshore structures · S–N curves in seawater with CP, thickness correction

ASME VIII Div 2, Part 5 · Pressure vessels · Elastic stress analysis, fatigue curves, K_e factor

ASME III · Nuclear components · Design fatigue curves with environmental factors

API 579 / BS 7910 · Fitness-for-service · Failure assessment diagrams, LEFM/EPFM

ASTM F2924 / F3001 / F3301 · AM Ti-6Al-4V · Material and post-processing requirements

IS 800 · Indian steel structures · No fatigue design curve; Indian practice adopts EN 1993-1-9 or BS 7608

The Indian gap is worth stating plainly: IS 800:2007 does not contain a fatigue design procedure. Indian structural designers working on cranes, bridges and machine frames either adopt Eurocode 3 or BS 7608, or rely on OEM-specific rules. For Indian Railways rolling stock, RDSO specifications govern axle and bogie fatigue; for automotive, the OEM's own durability standard (often derived from a global platform) governs. The absence of a national fatigue code is one reason Indian fabrication shops often under-specify fatigue-critical details — not because the engineering is unknown, but because the drawing never asked for it.

12.3 The ledger

Fatigue work is expensive relative to static testing, and the costs are worth budgeting explicitly. Indicative Indian market figures for 2025–26:

Test · Machine time · Indicative cost

HCF S–N, per specimen · 2–8 h · ₹6,000–15,000

Full S–N curve, 12–15 specimens · 40–80 h · ₹1.2–3.0 lakh

LCF, per specimen (E606) · 4–12 h · ₹15,000–35,000

FCGR, per specimen (E647) · 8–20 h · ₹25,000–60,000

K_{IC}, per specimen (E399) · 2–6 h · ₹20,000–45,000

Staircase endurance limit, 15–20 specimens · 30–60 h · ₹1.0–2.5 lakh

Machine-hour rates at Indian labs run roughly ₹2,000–5,000/h for servo-hydraulic frames, higher for environmental chambers, high-temperature rigs, or multiaxial setups. Institutions with servo-hydraulic fatigue capability include CMTI (Bengaluru), ARAI (Pune), NAL (Bengaluru), BHEL R&D (Hyderabad), and several IIT materials/testing laboratories; commercial labs in Pune, Chennai and Bengaluru also offer ASTM E466/E606 testing.

Compare that with the cost of getting it wrong. A single in-service fatigue failure of a ₹40,000 shaft in a ₹2 crore machine can cost ₹5–20 lakh in downtime, replacement and consequential damage — 5–20× the cost of the entire test programme, before any safety or liability cost. The economics of fatigue testing are not close.

12.4 A practical programme for a small manufacturer

For a shop that cannot afford a full characterisation, the minimum defensible programme is:

  1. One static tensile test of the actual material and heat treatment, to get S_{ut} and S_y for the Marin calculation.
  2. One staircase endurance limit on a polished specimen, 15 specimens, to calibrate the material's fatigue ratio (is it 0.45 or 0.55?).
  3. Three or four component tests at the design stress, to catch the effects the coupons miss — surface, residual stress, fit, assembly load.
  4. One overload proof test at 1.2–1.5× design load, to screen out parts with critical flaws.

That is roughly ₹2–4 lakh and two to three weeks, and it converts a design based on textbook factors into a design based on the actual material and process. For a product that will sell in thousands of units, it is the cheapest reliability insurance available.

Frequently Asked Questions

What is the difference between fatigue strength and endurance limit?

The endurance limit is the stress amplitude below which a ferrous specimen survives indefinitely (conventionally 10⁷ cycles) — it is a horizontal asymptote on the S–N curve. Fatigue strength is the stress amplitude that produces a specified finite life, e.g. S_{10^5} or S_{10^7}. Aluminium, copper and most titanium alloys have no true endurance limit; their S–N curves keep falling, so only a fatigue strength at a stated life is meaningful. Using "endurance limit" for aluminium is a category error that leads to infinite-life designs that fail.

Why did my part fail at 30% of yield strength?

Because it was loaded cyclically, and fatigue does not care about yield. The local stress at a notch can be 2–3× the nominal stress, the surface residual stress can add another 300–600 MPa in tension, and the material's usable endurance limit after the Marin corrections is typically 25–35% of yield. A 30%-of-yield nominal stress with a K_f of 2 and a machined surface is already at or beyond the fatigue limit. The fracture surface will show you which: beach marks and a smooth fatigue zone mean cyclic loading; a rough dimpled zone with no fatigue zone means overload.

How do I calculate fatigue life for a welded joint?

Do not use the base-metal S–N curve. Classify the detail to EN 1993-1-9 or IIW (FAT 36–160), then use N = 2\times10^6(\Delta\sigma_C/\Delta\sigma)^3 up to 5×10⁶ cycles and the m = 5 branch beyond. For variable amplitude, apply Miner with a D limit of 0.3–0.5. If the joint is fatigue-critical, specify HFMI or TIG dressing and use the improved FAT class — the improvement is codified in the IIW recommendations and is worth 1.5–2.5× life.

What is the Paris law and what are C and m?

The Paris law is da/dN = C(\Delta K)^m, the empirical relationship between crack growth per cycle and stress intensity range. For ferritic-pearlitic steels the Barsom–Rolfe constants are C = 6.9\times10^{-12} and m = 3.0 with da/dN in m/cycle and \Delta K in MPa√m. The exponent m is 2.5–4 for most metals; the constant C is unit-sensitive and must always be quoted with its units. Integrate the law from the initial flaw to the critical size to get propagation life, and remember that the answer is dominated by the small-crack end of the integration.

How big a crack is dangerous?

Critical crack size is a_c = (1/\pi)(K_{IC}/(Y\sigma))^2. For K_{IC} = 50 MPa√m at 300 MPa and Y = 1, a_c = 8.8 mm; at 400 MPa it falls to 5.0 mm. The dangerous size therefore depends on the stress, which is why proof testing and stress reduction are both crack-size interventions. A crack that is harmless at 150 MPa can be critical at 300 MPa, and the square in the equation means a doubling of stress quarters the tolerable flaw.

Does shot peening really increase fatigue life?

Yes, by 20–100% on components that fail from the surface, and by 2–5× on springs and gear roots where the crack initiates at a surface tensile hot spot. The mechanism is a compressive residual layer of −400 to −800 MPa, 0.1–0.5 mm deep, which acts as a negative mean stress. The benefit is erased by grinding, straightening, or service above the tempering temperature, and it does nothing for a part that fails from an internal defect. Specify Almen intensity and coverage, not just "shot peen".

Why is additive manufacturing fatigue strength so low?

Because the as-built surface is a population of 50 µm-deep notches, the process leaves 0.1–1% porosity including sharp lack-of-fusion defects, and the residual stress is tensile at the surface (up to ~800 MPa in Ti alloys). As-built LPBF Ti-6Al-4V delivers 200–250 MPa at 10⁷ cycles; HIP closes the porosity and raises that to 550–600 MPa, but the surface still initiates cracks. Machining the fatigue-critical surfaces to S_a 0.5 µm is what closes the remaining gap to wrought.

How often should I inspect a fatigue-critical part?

Set the interval from Paris-law integration: N_{insp} = 0.5\,N_p(a_{NDI} \to a_c), where a_{NDI} is the largest flaw your inspection method reliably finds and a_c is the critical size at the design stress. Halve the calculated interval to cover scatter in C, m and the load spectrum. If the resulting interval is impractically short, the design needs to change — lower stress, higher toughness, or a better inspection method — not the interval.

What is the difference between K_t and K_f?

K_t is the elastic stress concentration factor, a purely geometric quantity. K_f is the fatigue notch factor, which accounts for the fact that the material yields and redistributes stress at the notch root: K_f = 1 + q(K_t - 1), where q = 1/(1 + \sqrt{a}/\sqrt{r}) is Neuber's notch sensitivity. K_f \le K_t always, and the gap widens for blunt notches in low-strength materials. Use K_f for fatigue, K_t for static stress and for LCF where the notch root yields.

Previous guides in this series: Industrial Lubrication & Tribology · Abrasive Machining & Precision Grinding · Springs & Spring Design · Industrial Wastewater Treatment · O-Rings, Gaskets & Mechanical Seals · Machine Safety & Guarding · PCB Fabrication & Bare Board Manufacturing · Rolling-Element Bearings · Gears & Gearboxes · CNC Machining · Welding Processes · Metal Additive Manufacturing.

References

[1] R. G. Budynas and J. K. Nisbett, Shigley's Mechanical Engineering Design, 11th ed., McGraw-Hill, 2020 — Chapter 6, "Fatigue Failure Resulting from Variable Loading": the Marin equation S_e = k_a k_b k_c k_d k_e k_f S_e', the surface-factor coefficients (a = 1.58/4.51/57.7/272, b = -0.085/-0.265/-0.718/-0.995), the size-factor equations, the load and temperature factors, the reliability factor k_e = 1 - 0.08z_\alpha, and the Goodman/Gerber/Soderberg/ASME-elliptic criteria. [2] J. A. Bannantine, J. J. Comer and J. L. Handrock, Fundamentals of Metal Fatigue Analysis, Prentice Hall, 1990 — strain-life constants for SAE 1045 (\sigma_f' = 948 MPa, b = -0.092, \varepsilon_f' = 0.26, c = -0.445, K' = 1197 MPa, n' = 0.196), Neuber notch analysis, rainflow counting, and the transition-life calculation. [3] R. I. Stephens, A. Fatemi, R. R. Stephens and H. O. Fuchs, Metal Fatigue in Engineering, 2nd ed., Wiley, 2000 — the three-stage fatigue mechanism, persistent slip bands, and the statistics of S–N data. [4] ASM International, ASM Handbook, Vol. 19: Fatigue and Fracture, 1996 — the standard reference for fatigue mechanisms, fractography (beach marks, striations, ratchet marks), and the 80–90% service-failure attribution. [5] ASCE Committee on Fatigue and Fracture Reliability, "Fatigue reliability: introduction," Journal of the Structural Division, 108(1), 1982 — the 80–90% figure for metallic structures; and the 1983 Battelle Columbus Laboratories/NBS study of 230 service failures (141, or 61%, fatigue-related). [6] ASTM E466-21, Standard Practice for Conducting Force Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials; ASTM E606/E606M-21, Strain-Controlled Fatigue Testing; ASTM E647-23, Measurement of Fatigue Crack Growth Rates; ASTM E399-22, Plane-Strain Fracture Toughness of Metallic Materials; ASTM E1820-23, Fracture Toughness Measurements; ASTM E739-23, Statistical Analysis of Linear or Linearized Stress-Life and Strain-Life Fatigue Data; ASTM E1049-85(2017), Cycle Counting in Fatigue Analysis; ASTM E468-18, Presentation of Constant Amplitude Fatigue Test Results. [7] ISO 1099:2017, Metallic materials — Fatigue testing — Axial force-controlled method; ISO 1143:2021, Rotating bar bending fatigue testing; ISO 12107:2012, Statistical planning and analysis of fatigue data — the staircase (Dixon–Mood) method. [8] EN 1993-1-9:2005, Eurocode 3: Design of steel structures — Part 1-9: Fatigue — detail categories FAT 36–160, the m = 3/m = 5 bilinear curve, the constant-amplitude fatigue limit at 5×10⁶ cycles, and the 10⁸-cycle cut-off. [9] A. Hobbacher (ed.), Recommendations for Fatigue Design of Welded Joints and Components, IIW Document IIW-2259-15, 2nd ed., Springer, 2016 — hot-spot stress, FAT classes, and the HFMI/TIG-dressing improvement factors. [10] DNV-RP-C203, Fatigue Design of Offshore Steel Structures, 2021 — seawater S–N curves with cathodic protection, thickness correction, and the hot-spot stress method. [11] BS 7608:2014+A1:2015, Guide to fatigue design and assessment of steel products. [12] ASME Boiler and Pressure Vessel Code, Section VIII, Division 2, Part 5, Design by Analysis Requirements — elastic fatigue analysis, the K_e plasticity factor, and the fatigue curves. [13] P. C. Paris, M. P. Gomez and W. E. Anderson, "A rational analytic theory of fatigue," The Trend in Engineering, 13, 1961, pp. 9–14; P. C. Paris and F. Erdogan, "A critical analysis of crack propagation laws," Journal of Basic Engineering, 85(4), 1963, pp. 528–533 — the Paris law. [14] J. M. Barsom and S. T. Rolfe, Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics, 3rd ed., ASTM, 1999 — the ferritic-pearlitic steel constants C = 6.9\times10^{-12}, m = 3.0 (m/cycle, MPa√m), and the R-ratio dependence of the threshold. [15] W. Elber, "The significance of fatigue crack closure," in Damage Tolerance in Aircraft Structures, ASTM STP 486, 1971, pp. 230–242 — plasticity-induced crack closure and the R-ratio effect. [16] H. Neuber, "Theory of stress concentration for shear-strained prismatical bodies with arbitrary nonlinear stress-strain law," Journal of Applied Mechanics, 28(4), 1961, pp. 544–550 — the notch rule K_t^2 = K_\sigma K_\varepsilon and the \sqrt{a} material constant. [17] L. F. Coffin, "A study of the effects of cyclic thermal stresses on a ductile metal," Transactions of the ASME, 76, 1954, pp. 931–950; S. S. Manson, "Behavior of materials under conditions of thermal stress," NACA TN-2933, 1953 — the Coffin–Manson plastic-strain life relation. [18] A. A. Griffith, "The phenomena of rupture and flow in solids," Philosophical Transactions of the Royal Society A, 221, 1921, pp. 163–198; G. R. Irwin, "Analysis of stresses and strains near the end of a crack traversing a plate," Journal of Applied Mechanics, 24, 1957, pp. 361–364; J. R. Rice, "A path independent integral and the approximate analysis of strain concentration by notches and cracks," Journal of Applied Mechanics, 35, 1968, pp. 379–386 — Griffith, the stress intensity factor, and the J-integral. [19] H. Kitagawa and S. Takahashi, "Application of fracture mechanics to very high cycle fatigue," Proceedings of the 2nd International Conference on Mechanical Behavior of Materials, 1976, pp. 627–631; M. H. El Haddad, K. N. Smith and T. H. Topper, "Fatigue crack propagation of short cracks," Journal of Engineering Materials and Technology, 101, 1979, pp. 42–46 — the Kitagawa–Takahashi diagram and the intrinsic crack length a_0. [20] N. E. Dowling, Mechanical Behavior of Materials, 5th ed., Pearson, 2019 — fatigue testing methods, strain-life analysis, and the statistics of fatigue data. [21] NIST, "Fatigue Properties of a Titanium Alloy (Ti-6Al-4V) Fabricated Via Electron Beam Melting (EBM): Effects of Internal Defects and Residual Stress," NIST Publication 919957 — as-built and stress-relieved EBM Ti-6Al-4V at 200–250 MPa at 10⁷ cycles, HIPed at 550–600 MPa. [22] "Mapping the Tray of Electron Beam Melting of Ti-6Al-4V: Properties and Microstructure," Materials, 2019 (PMC6539714) — HIPed fatigue limit ~550 MPa versus as-built ≤300 MPa, and the role of voids and lack-of-fusion defects. [23] "Roughness Reduction" (PBF surface integrity review), ScienceDirect Topics — as-built R_a 7–20 µm for LPBF and 15–40 µm for SEBM, and the >50% fatigue debit of as-built surfaces versus wrought Ti-6Al-4V. [24] "Pandora's Box — Influence of Contour Parameters on Roughness and Subsurface Residual Stresses in Laser Powder Bed Fusion of Ti-6Al-4V," Materials, 2020 (PMC7436019) — as-built S_a 13–30 µm, S_z 138–264 µm, and surface tensile residual stress up to ~800 MPa. [25] M. S. Hamidi Nasab et al., "Effect of surface post-treatments on the fatigue behaviour of LPBF AlSi10MG," and related PBF post-processing studies — vibro-finishing, sand blasting and machining compared; machined S_a 0.5 µm gives the highest fatigue limit. [26] ASTM F2924-14(2021), Standard Specification for Additive Manufacturing Titanium-6 Aluminum-4 Vanadium with Powder Bed Fusion; ASTM F3001-14(2021), EBM Ti-6Al-4V; ASTM F3301-18a, Standard for Post Processing Methods for Additive Manufacturing of Ti-6Al-4V; ISO/ASTM 52900:2021, Additive manufacturing — General principles — Fundamentals and vocabulary. [27] ASTM B850-98(2019), Standard Guide for Post-Coating Treatments of Steel for Reduction of the Risk of Hydrogen Embrittlement; ASTM F519-18, Mechanical Hydrogen Embrittlement Evaluation of Plating/Coating Processes and Service Environments; ASTM F1624-12(2018), Measurement of Hydrogen Embrittlement Threshold in Steel by the Incremental Step Loading Technique. [28] API 579-1/ASME FFS-1, Fitness-For-Service, 2021; BS 7910:2019, Guide to methods for assessing the acceptability of flaws in metallic structures — failure assessment diagrams and the elastic-plastic fracture assessment routes. [29] Indian practice: IS 800:2007, General Construction in Steel — Code of Practice (no fatigue design curve); RDSO specifications for rolling-stock axles and bogies; and the fatigue testing capabilities of CMTI Bengaluru, ARAI Pune, NAL Bengaluru and BHEL R&D Hyderabad. [30] SAE J1099, Technical Report on Low Cycle Fatigue Properties of Ferrous and Non-Ferrous Materials, 2018 — the strain-life constant tabulation used in §5.

Note on sources: the governing equations are Basquin for stress life, Coffin–Manson–Basquin for strain life, Neuber for notches, Miner for damage and Paris for crack growth; the specifications are ASTM E466/E606/E647/E399/E1820, ISO 1099/1143/12107, EN 1993-1-9, IIW, DNV-RP-C203, BS 7608 and ASME VIII Div 2. Where the literature differs — Miner sums between 0.3 and 3, K_{IC} varying by 20% between heats, Paris constants that shift with R-ratio — this guide prints the working number and the standard behind it, because a fatigue calculation is a prediction with a factor of two attached, and the only defence against that is to state the assumption, test the real part, and inspect at an interval derived from the arithmetic rather than the calendar.

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