Materials Science · Fatigue, creep and time-dependent damage
Cyclic loading and fatigue initiation
Describe a complete stress cycle, explain how repeated local deformation can initiate a crack, and interpret stress–life observations without turning a stopped test into an infinite-life claim.
Useful background: Elasticity, plasticity and tensile strength; Cracks, fracture and toughness. Each worked case explains the quantities it uses.
Back to the chapterAfter reading, you can
- Calculate mean stress, range, amplitude and stress ratio for a stated constant-amplitude cycle.
- Explain a common metallic fatigue-initiation mechanism and why a monotonic strength comparison is insufficient.
- Distinguish cycles, reversals, crack initiation, final failure and a runout.
- Read a conditioned stress–life dataset, keeping specimen scatter and unobserved outcomes visible.
Repeated loading can accumulate damage
A component can fail after many cycles at stresses below its monotonic tensile strength. Cyclic deformation can initiate cracks at surfaces or defects. The number of cycles, stress range, mean stress, waveform and environment help define the exposure.
Imagine repeatedly pulling a tie that looks almost unchanged after each load is removed. Returning close to its original overall length does not establish that every microscopic region returned to its previous state. A local irreversible change can be small compared with the gauge length, yet matter after many repetitions. Fatigue concerns the evolution under repeated loading; creep concerns time-dependent deformation under sustained loading. An actual history can involve both.
For one common metallic pathway, dislocation slip becomes concentrated in particular bands. Repeated slip at a surface can form small extrusions and intrusions: raised and recessed features. An intrusion can act as an increasingly severe local defect from which a crack develops. This is a mechanism example, not a claim that every fatigue crack begins in a slip band. Inclusions, pores, interfaces and pre-existing damage can provide different starting sites.
A nominal stress below a monotonic yield value does not exclude local cyclic plastic deformation at a defect or within favourably oriented grains. Conversely, a large nominal stress may produce substantial cyclic plasticity across the specimen. Initiation depends on the local history and material state, not only on whether a single gross-section stress crosses one tensile-test endpoint.
Four numbers describe an ideal stress cycle
Use tension-positive axial stress and specify its reference area and location. For a simple nominal uniaxial comparison, σ = F/A₀ uses the original section area A₀. During a constant-amplitude cycle, define its largest and smallest stresses as σ_max and σ_min. The stress range is Δσ = σ_max − σ_min; the amplitude is half that range, σ_a = Δσ/2; the mean is σ_m = (σ_max + σ_min)/2.
These definitions can be checked by reconstructing the extremes: σ_max = σ_m + σ_a and σ_min = σ_m − σ_a. Mean, amplitude and range have stress units. The stress ratio R = σ_min/σ_max is dimensionless and requires a nonzero denominator. The examples here have positive tensile maxima. Fully reversed tension–compression loading has equal positive and negative extremes, giving σ_m = 0 and R = −1.
For an ideal sinusoid, write σ(t) = σ_m + σ_a sin(2πft), where t is time in seconds and f is frequency in cycles per second, or hertz. One cycle is a complete repetition, with period T = 1/f. A reversal is a change from increasing to decreasing stress, or back, at an extremum; it need not switch the stress between tension and compression. Two reversals occur in each complete sinusoidal cycle, including a tension–tension cycle. Saying “a million reversals” and “a million cycles” therefore gives different exposures for that waveform.
The graph compares two ideal histories with the same 200 MPa maximum. One cycles between 20 and 200 MPa; the other between −200 and 200 MPa. Their means, amplitudes and ratios differ. A peak-only comparison discards information even before a fatigue model is chosen.
| σ_min | σ_max | σ_m | Δσ | σ_a | R |
|---|---|---|---|---|---|
| 20 | 200 | 110 | 180 | 90 | 0.10 |
| −200 | 200 | 0 | 400 | 200 | −1 |
| 120 | 200 | 160 | 80 | 40 | 0.60 |
Name the event whose life is being counted
A fatigue life must name its endpoint. Let N_i denote cycles to an explicitly defined initiation observation, such as the first crack detected by a stated method at a stated resolution. Let N_f denote cycles to a specified final event, such as complete fracture. A crack may exist before the instrument resolves it, so “first detected” is not necessarily the first atomic separation.
In a history where initiation is observed and the crack subsequently grows to final failure, the recorded intervals can be written N_f = N_i + N_g, where N_g is the subsequent cycle count. The partition depends on the detection criterion and specimen. It is not a fixed fraction for all materials. A component with an existing crack can have little or no initiation interval under the chosen definition.
Surface state matters because a local feature changes the loading of a possible initiation site. A sharp machining mark, a pore near a surface, residual stress or a reactive environment may change that history. Comparing polished laboratory specimens with rough, joined or corroding components therefore requires more evidence than matching their material name. A treatment that helps one initiation mechanism need not improve every crack-growth or service-life criterion.
A curve belongs to its method
An S–N curve relates a selected stress measure to cycles under specified conditions. Some systems show an apparent long-life plateau; others do not. A specimen that survives a stopped test is a runout, not proof of infinite life.
Read both axes and the endpoint definition. Here S means nominal stress amplitude σ_a, and N means cycles to complete fracture of a specimen. Hold stress ratio, waveform, frequency, temperature, environment, geometry and surface preparation fixed within the comparison. A curve using stress range would have different vertical numbers even for the same cycles; a curve for initiation would describe a different event.
The table below is an invented small dataset for explaining the record, not measured fatigue evidence or a design curve. At one amplitude, two specimens can fail at different cycle counts. The lower-amplitude row shows survival only up to the stopping count. Neither a smooth line through a few points nor the smallest tested amplitude establishes a guaranteed threshold for a population.
Life axes are often logarithmic because counts can span several factors of ten. On such an axis, 10⁴ to 10⁵ and 10⁵ to 10⁶ have equal spacing: each is a tenfold increase. Equal spacing does not mean an added fixed number of cycles. A failure marker is an observed endpoint; a runout marker points beyond the observation window instead of supplying an invented failure time.
| Amplitude σ_a (MPa) | Specimen 1 | Specimen 2 |
|---|---|---|
| 250 | Fractured at 9,000 cycles | Fractured at 11,000 cycles |
| 200 | Fractured at 70,000 cycles | Fractured at 95,000 cycles |
| 150 | Fractured at 600,000 cycles | Fractured at 800,000 cycles |
| 125 | Unfractured when stopped at 2,000,000 | Unfractured when stopped at 2,000,000 |
Keep exposure, inference and mechanism separate
A runout is a right-censored observation: the event count lies beyond the observed count, while its exact value is unknown. It is useful information and should be retained. Treating it as an exact failure invents an event; discarding it throws away observed survival. Estimating a life distribution or a confidence bound needs an appropriate analysis and more information than these two specimens supply.
Frequency converts cycle count into elapsed time only for a stated schedule. At constant f, N = ft. At the same N, changing f changes elapsed exposure and can change heating, relaxation or environmental interaction. Equal cycle counts and equal maxima do not automatically establish equivalent damage. The elementary arithmetic is not a rule that fatigue depends only on count.
Load blocks, irregular waveforms, multiaxial stress and growing cracks require additional descriptions. The next fatigue-growth lesson develops those questions. This lesson supplies a complete cycle description and a way to read evidence; it does not assign a component lifetime from a single monotonic strength, one runout or the illustrated dataset.
Worked examples
The same maximum load can hide a different cycle
A nominal axial specimen has original area A₀ = 50 mm². Its tensile force cycles between 1000 and 10,000 N. Find the stress descriptors. Compare a second cycle between −10,000 and 10,000 N using the same area.
- For the first cycle, σ_min = 1000/50 = 20 N/mm² = 20 MPa and σ_max = 10,000/50 = 200 MPa.
- Its range is 200 − 20 = 180 MPa; its amplitude is 90 MPa; its mean is (200 + 20)/2 = 110 MPa; R = 20/200 = 0.10. Check 110 ± 90 gives 200 and 20.
- The second cycle has σ_min = −200 MPa and σ_max = 200 MPa. Its range is 400 MPa, amplitude 200 MPa, mean zero and R = −1.
- The maxima agree, but the second cycle traverses a larger stress interval and changes sign. These calculations identify different exposures. Predicting their lives would require suitable fatigue evidence; no life ratio follows just by dividing their amplitudes.
The first cycle has mean 110 MPa and amplitude 90 MPa; the second has mean zero and amplitude 200 MPa. A shared 200 MPa maximum does not make them the same cycle.
Read a failure and a stopped test differently
Use the illustrative S–N table. What does the 200 MPa row show? What can be reported for the 125 MPa row, and does it establish that cracks never initiated?
- At amplitude 200 MPa, the two stated specimens fractured at 70,000 and 95,000 cycles. The observations differ by 25,000 cycles; 95,000/70,000 ≈ 1.36. They do not provide one exact life shared by every specimen.
- At 125 MPa, both specimens remained unfractured when stopped at 2,000,000 cycles. Retain each as a runout at its stopping count, with N_f greater than the observed count under the stated endpoint convention.
- Do not enter 2,000,000 as the exact failure life. Those failure events were not observed, and a population survival probability or an infinite-life limit has not been established.
- The endpoint is fracture, not first detectable crack. Unfractured specimens may contain initiation sites or growing cracks; the table gives no crack-inspection record. The appropriate conclusion is bounded survival of the specified endpoint.
The failure row contains two actual illustrative endpoints with scatter. The stopped row contains two right-censored fracture lives, not exact failures, guaranteed future survival or proof of no initiation.
Count complete repetitions before interpreting time
An ideal sinusoidal test runs at 8 Hz for 45 minutes. How many cycles and reversals occur? If a different constant-frequency test stops unfractured at one million cycles at 5 Hz, how long was its cyclic exposure?
- Convert 45 minutes to 2700 s. With 8 cycles/s, N = 8 × 2700 = 21,600 complete cycles.
- This waveform has two reversals per cycle, so the reversal count is 43,200. The factor two belongs to the stated cycle definition.
- For the second test, t = N/f = 1,000,000/5 = 200,000 s, about 55.56 h of continuous cycling. Interruptions would require a separate elapsed-time record.
- The specimen is a runout at one million cycles. The time conversion does not turn that observation into a failure life or permit transfer to a different frequency or environment.
21,600 cycles and 43,200 reversals in the first test; about 55.56 hours of continuous cycling in the second, with its failure life still unobserved.
Try the reasoning
A cycle has σ_max = 150 MPa and σ_min = −50 MPa. Find its mean, range, amplitude and R. A worksheet calls the 200 MPa range its amplitude; explain the factor-of-two error.
Show the answer and reasoning
σ_m = 50 MPa, Δσ = 200 MPa, σ_a = 100 MPa and R = −1/3.
The amplitude is measured from the mean to either extreme, whereas the range spans both extremes. Reconstructing 50 ± 100 gives 150 and −50, which checks both definitions.
Two nominally identical specimens have the same 200 MPa stress maximum. One cycles between 20 and 200 MPa, the other between 120 and 200 MPa. Compare mean and amplitude, and explain why neither the maximum nor one polished-surface result fixes the life of a rough component.
Show the answer and reasoning
The means are 110 and 160 MPa, and the amplitudes are 90 and 40 MPa. Their stress histories differ despite the common maximum; the component also has a different possible initiation-site condition.
Changing the minimum changes both mean and range. Surface geometry and state can change local cyclic deformation. A valid life comparison requires evidence matching those conditions, not a peak-only ranking.
A specified inspection first detects a crack at 80,000 cycles; complete fracture occurs at 120,000. Find the subsequent growth interval. Would an instrument with poorer crack resolution necessarily report the same initiation count?
Show the answer and reasoning
The subsequent interval is 40,000 cycles. A poorer detection method can report a different first-detected count, even for the same physical history.
For these defined observations, N_g = N_f − N_i. An initiation record includes its detection threshold; it does not automatically mark the first microscopic separation.
A test stops unfractured at 2,000,000 cycles. A learner plots it as a failure at exactly that count and averages it with two observed failures. Identify the altered information and the missing evidence for a population claim.
Show the answer and reasoning
The learner replaced a lower bound by an unobserved exact event. Population inference needs retained censoring, the sample/test design and a suitable life-distribution analysis; one stopped specimen does not supply a guaranteed lifetime.
Observed survival beyond the stopping count is information, but its unknown future endpoint cannot be silently inserted as a measured failure. The illustration also does not establish a universal probability distribution or endurance limit.
What to carry forward
- A fatigue exposure needs a complete stress-history description, not only a maximum.
- Range and amplitude differ by two; cycles and reversals need a stated waveform convention.
- Local repeated deformation can initiate a crack while an average or monotonic comparison misses the relevant history.
- Name the endpoint, preserve specimen scatter and keep a runout as observed survival with an unknown future event count.
References
Teaching resources and research supporting this lesson’s explanations and examples.