Fatigue Testing of Metals: S-N Curve, Endurance Limit and Fatigue Failure Analysis

Most mechanical components fail not under a single overload, but under millions of small, repeated cycles well below their static strength. Fatigue testing quantifies that behaviour through the S-N curve, the endurance limit, and a growing toolkit of mean-stress and notch corrections that let a designer translate a polished laboratory specimen result into a safe working stress for a real part. This article works through how that data is generated, how it is corrected for real-world conditions, and how a fatigue fracture surface is read during failure analysis.

What Is Fatigue Failure?

Fatigue is the progressive, localized structural damage that occurs when a material is subjected to cyclic loading, even when every individual stress peak remains well below the yield strength. Unlike a static overload, which fails a part in a single event once stress exceeds strength, fatigue accumulates damage cycle by cycle until a crack nucleates and grows to a critical size. Fatigue is widely cited across mechanical failure surveys as one of the leading mechanisms behind in-service component failures, which is why S-N data and the corrections covered below are central to mechanical and metallurgical design.

Why Fatigue Differs from Static Failure

A static tensile test loads a specimen once to fracture and reports a single number: Sut. Fatigue instead asks a different question – how many repetitions of a given stress range can the material survive before a crack forms and propagates to failure? Because the answer depends on stress amplitude, mean stress, surface condition, geometry, and environment simultaneously, fatigue strength is never a single fixed property of a material the way Sut or hardness is; it is a function of all of these variables at once.

Stages of Fatigue Failure

Fatigue failure proceeds through three recognizable stages: crack initiation at a stress riser (a surface scratch, inclusion, pore, fillet, or weld toe), slow stable crack propagation driven by the cyclic stress intensity at the crack tip, and final rapid fracture once the remaining cross-section can no longer support the peak load. Each stage leaves a distinct signature on the fracture surface, which is exploited directly during failure analysis later in this article.

The S-N Curve (Stress-Life Approach)

The S-N curve, also called the Wohler curve, is generated by testing a series of nominally identical specimens at different stress amplitudes and recording the number of cycles to failure at each level. The resulting data is plotted with stress amplitude on a linear or log y-axis and cycles to failure on a log x-axis, as shown in Figure 1.

How S-N Data Is Generated

The classical reference test is the rotating-beam fatigue test on an R.R. Moore machine, which subjects a polished, notch-free cylindrical specimen to fully reversed bending (R = -1) at a fixed rotational speed until failure or run-out. Axial and torsional fatigue machines following ASTM-type procedures are also widely used, particularly for components that see axial service loading rather than rotating bending. Because fatigue data is inherently scattered – identical specimens can fail at noticeably different cycle counts at the same stress – results are reported statistically, typically as the stress corresponding to a defined probability of survival rather than a single deterministic number.

Reading the S-N Curve

At high stress amplitudes near Sut, failure occurs in well under 10³ cycles. As stress amplitude decreases, life increases rapidly, and for many wrought steels the curve flattens into a near-horizontal line beyond roughly 10⁶-10⁷ cycles – the endurance limit. Stresses below this level are not expected to produce fatigue failure within the test basis, hence the term “infinite life.” Non-ferrous alloys such as aluminium typically show no such plateau (Figure 1), so designs in these materials are based instead on a fatigue strength quoted at a specific design life.

Low-Cycle vs High-Cycle Fatigue

Below roughly 10³-10⁴ cycles, stresses are high enough to produce measurable plastic strain on every cycle; this low-cycle fatigue (LCF) regime is normally analysed with strain-life (epsilon-N) methods rather than the stress-life approach. High-cycle fatigue (HCF), the focus of this article, covers longer lives where the bulk of the material response stays elastic and the S-N stress-life approach is appropriate.

Endurance Limit and the Marin Equation

The endurance limit measured on a small, polished, notch-free rotating-beam specimen (Se′) is a laboratory idealisation. Real components are larger, have as-machined or as-forged surfaces, see different load types, and require a defined statistical reliability – all of which reduce the usable endurance limit. The Marin equation captures this with a set of multiplicative correction factors:

Se = ka · kb · kc · ke · Se′

ka = surface finish factor
kb = size factor
kc = load type factor
ke = reliability factor
Se′ = unmodified (polished specimen) endurance limit

Unmodified Endurance Limit (Se′)

For wrought steels with Sut below about 1400 MPa, Se′ is commonly approximated as 0.5 x Sut; above that strength level the relationship breaks down and Se′ is typically taken as a fixed value near 700 MPa, since very high-strength steels become increasingly sensitive to inclusions and surface defects rather than following the simple ratio. Wrought iron uses a lower ratio, closer to 0.4 x Sut. These are starting approximations, not substitutes for actual fatigue test data where it is available.

Marin Modification Factors

The surface finish factor ka is commonly expressed as ka = a x Sut^b, with Sut in MPa and coefficients depending on how the surface was produced:

The size factor kb reduces strength as the loaded cross-section grows, reflecting the greater probability of encountering a critical flaw in a larger volume of material; it applies to bending and torsion but is taken as 1 for axial loading, where the entire cross-section sees the same nominal stress. The load factor kc accounts for the fact that the reference S-N curve is generated in bending: kc is 1.0 for bending, about 0.85 for axial loading, and about 0.59 for torsion. The reliability factor ke shifts the mean-based Se′ down to the value associated with a desired probability of survival:

Materials Without a True Endurance Limit

Aluminium, magnesium, and most non-ferrous alloys do not exhibit a flat endurance-limit plateau; their S-N curves continue to slope downward indefinitely. Design practice for these materials substitutes a fatigue strength reported at a specified long life – commonly 5×10⁸ cycles – in place of Se. Designers working in these alloys should treat any quoted “endurance limit” with caution and confirm the reference cycle count it was measured at.

Mean Stress Effects: Goodman, Gerber, and Soderberg

Rotating-beam data is generated under fully reversed loading, where the mean stress is zero. Real components frequently see a non-zero mean stress superimposed on the cyclic amplitude – a preloaded bolt, a shaft under a steady torque plus a fluctuating bending load, or a pressure vessel cycling above a baseline internal pressure.

Sa = (Smax - Smin) / 2      (stress amplitude)
Sm = (Smax + Smin) / 2      (mean stress)
R  = Smin / Smax            (stress ratio; R = -1 for fully reversed loading)

The Modified Goodman Line

The modified Goodman criterion is the most widely used design line for combining mean and alternating stress, plotted as a straight line between Se on the alternating-stress axis and Sut on the mean-stress axis:

Sa / Se + Sm / Sut = 1 / n

n = factor of safety against fatigue failure

Gerber and Soderberg Criteria

The Gerber criterion replaces the linear Sm/Sut term with a parabolic one, tracking the scatter of real ductile-metal test data more closely and giving a less conservative (higher allowable stress) prediction than Goodman. The Soderberg criterion is the most conservative of the three, using the yield strength Sy in place of Sut, which also guards against first-cycle yielding in addition to fatigue.

Basquin’s Equation and Estimating Finite Fatigue Life

Between the low-cycle region (around 10³ cycles) and the endurance limit (around 10⁶ cycles), the S-N curve for many steels is well approximated by a power-law (Basquin) relationship that plots as a straight line on log-log axes:

Sf = a · N^b

a = (f · Sut)² / Se
b = -(1/3) · log10(f · Sut / Se)

f = fraction of Sut at N = 10³ cycles (commonly taken as ~0.9 for many carbon and alloy steels as a simplifying engineering assumption)

Solving the same expression for N given a known stress amplitude Sf gives the finite-life estimate used in the calculator above: N = (Sf / a)^1/b. This estimate is only valid between the two calibration points – below 10³ cycles the material behaves as a static/low-cycle problem rather than an S-N one, and above Se the model predicts infinite (run-out) life rather than a finite cycle count.

Worked Example

Consider a machined 25 mm steel shaft, Sut = 620 MPa, loaded in bending at 90% reliability with Sa = 180 MPa and Sm = 0. Se′ = 0.5 x 620 = 310 MPa. ka (machined) ≈ 4.51 x 620^-0.265 ≈ 0.83. kb for d = 25 mm in bending ≈ 1.24 x 25^-0.107 ≈ 0.87. kc = 1.0 (bending), ke = 0.897 (90% reliability). Se = 0.83 x 0.87 x 1.0 x 0.897 x 310 ≈ 200 MPa. Since the applied amplitude of 180 MPa is below this corrected Se, infinite life is predicted – try the calculator above with these values, or substitute your own component data, to see the full step-by-step breakdown.

Factors That Reduce Fatigue Strength in Real Components

Notches and Stress Concentration

Fillets, keyways, holes, threads, and weld toes all act as geometric stress risers. The theoretical stress concentration factor Kt depends only on geometry, while the fatigue stress concentration factor Kf accounts for the material’s notch sensitivity q (0 ≤ q ≤ 1): Kf = 1 + q(Kt – 1). Brittle, high-strength materials tend toward q ≈ 1 (Kf ≈ Kt), while more ductile materials can show q well below 1, partially blunting the geometric effect under cyclic loading.

Surface Condition and Residual Stress

Because fatigue cracks almost always initiate at the surface, surface condition has an outsized effect on fatigue life. Decarburization, grinding burn, and plating-induced hydrogen pickup (see hydrogen-induced cracking) all reduce fatigue strength, while compressive-residual-stress treatments such as shot peening and surface rolling improve it. Quenching and tempering and other heat-treatment routes also change Sut and therefore the baseline Se′ used throughout this analysis.

Corrosion Fatigue and Environment

A corrosive environment combined with cyclic stress can eliminate the endurance-limit plateau entirely, since pits act as continuously multiplying stress risers as they deepen (see pitting corrosion and general corrosion mechanisms). Corrosion-fatigue life is therefore frequency- and environment-dependent in a way that pure mechanical fatigue is not, and laboratory S-N data generated in air can significantly overstate service life in aggressive media.

Temperature Effects

At elevated service temperature, fatigue strength interacts with creep, and conventional S-N data generated at room temperature no longer applies directly; thermal cycling near weld heat-affected zones introduces additional residual-stress and microstructural complications that should be evaluated with dedicated creep-fatigue data rather than the Marin/Goodman approach covered here.

Fatigue Failure Analysis: Reading the Fracture Surface

Crack Initiation Sites

Initiation almost always occurs at the surface, at the location of the highest local stress combined with the lowest local strength – a machining mark, an inclusion, a fillet radius, a keyway corner, or a weld toe. Multiple simultaneous initiation sites are common in heavily stressed or notch-sensitive components and produce multiple beach-mark origins converging across the fracture face, often separated by ratchet marks.

Propagation Zone: Beach Marks and Striations

As the crack advances cycle by cycle, it leaves behind concentric beach marks visible to the naked eye (often associated with changes in loading or shutdown periods) and, at much finer scale under SEM, individual striations corresponding to single load cycles. Both features curve with their concave side facing the initiation site, which is the single most reliable visual clue for tracing a fatigue fracture back to its origin.

Final Fracture Zone

Once the crack has reduced the load-bearing cross-section enough that the remaining ligament can no longer support the peak applied load, the part fails rapidly by overload, producing a comparatively rough, often fibrous or granular texture quite different from the smoother striated region. The relative size of the final fracture zone compared with the total cross-section gives a rough indication of how overstressed the component was relative to its fatigue design margin – a small final fracture zone suggests a long, slow crack growth under modest overstress, while a large one suggests the part was operating close to or above its intended capacity.

Common Failure Patterns

Rotating shafts commonly fail at fillets, keyways, or press-fit boundaries with a characteristic curved or “fir-tree” beach-mark pattern. Welded structures frequently initiate fatigue cracks at the weld toe in the heat-affected zone, where geometric discontinuity, residual stress, and microstructural changes combine; coarse martensite or untempered regions near a weld toe can further lower local notch sensitivity tolerance. Bolted joints often show fatigue at the first engaged thread root, where stress concentration is highest.

Industrial Significance and Standard Test Methods

Fatigue design governs the service life of rotating machinery shafts, vehicle suspension and powertrain components, aircraft structures, pressure vessels subject to cyclic pressurisation, and welded steel structures under wind or traffic loading. Reliable S-N and life-prediction data underpins inspection intervals, retirement-for-cause decisions, and safety factors across all of these applications.

Many of the variables discussed above – hardness-related strength estimates, microstructural condition after heat treatment, and toughness behaviour under impact loading – link directly back to other mechanical testing methods. See the related guides on hardness testing methods and Charpy impact testing for complementary mechanical property data, and the calculators hub for related engineering tools.

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