Materials Science · Fatigue, creep and time-dependent damage

Temperature and environmental interactions

Explain how reactions and deformation change one another, compare cycles with elapsed exposure time, and calculate a simple restrained thermal cycle without mistaking it for a complete life prediction.

Useful background: Elasticity, plasticity and tensile strength; Cracks, fracture and toughness. Each worked case explains the quantities it uses.

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After reading, you can

Explain a coupled route from surface-film damage or corrosion pits to cyclic cracking.
Use matched comparisons to distinguish cycle counts, exposure duration and an environmental interaction.
Explain why a hot hold and its controlled variable can change the next part of a cycle.
Derive a uniaxial elastic thermal-stress estimate and identify the processes that would invalidate it.

Cycles and surroundings interact

Corrosion can assist crack initiation and growth during repeated loading. Temperature can alter creep, oxidation and relaxation at the same time. A dry-air fatigue curve cannot simply be transferred to a wet reactive service environment.

Consider a metal surface that develops a corrosion pit before a cyclic crack becomes detectable. Removing material changes the local shape: a pit can concentrate stress and local deformation even if the nominal applied cycle is unchanged. Chemistry has therefore changed the mechanical starting state. Conversely, cyclic surface slip can disrupt a protective film, exposing fresh metal to the solution. Mechanical deformation has changed the chemical boundary condition. This feedback is what an interaction means here.

For a passive metal in a suitable aqueous environment, one proposed route is film rupture at a slip step, dissolution of the freshly exposed metal, and repair of the film. Repair is called repassivation. The interval before another disruption gives reactions and transport time to act. This is one possible route, not a rule that every environment-assisted crack advances only by dissolution; its importance depends on the alloy, film and solution.

Separate the stages when interpreting evidence. A pit can shorten initiation without establishing faster propagation of every existing crack. In a 1999 study of two duplex stainless steels used in papermaking, scratch/repassivation tests and slow-strain-rate tests did not support the proposed frequency or hydrogen explanation for their different service performance; the evidence instead favoured different susceptibility to pitting and pit-associated initiation. A plausible mechanism must survive comparisons that can contradict it.

Define the combined exposure

State hold times, cycling rate, temperature and chemical conditions. A low-frequency test can give reactions more time per cycle. Distinguish time-dependent damage from cycle-dependent damage where possible; coupled mechanisms may prevent a simple independent addition.

For continuous identical cycles at frequency f cycles per second (Hz), a record of N cycles lasts t = N/f seconds. The period is 1/f. If a schedule adds holds, either use a frequency that includes their full duration or add the actual ramp and hold times. Do not count the hold twice or leave it out. The same cycle count can conceal very different time in contact with a reactive fluid.

A useful comparison keeps material state, starting surface or crack, geometry, nominal stress range, stress ratio and endpoint matched. Change the time scale of a stated waveform and record the actual specimen temperature: self-heating can otherwise create another difference. Define the solution as well as its name, including relevant concentration, dissolved gases and replenishment. In hot comparisons, hold temperature fixed across environments before assigning a difference to chemistry.

A reference environment is a defined comparison condition; it is not automatically free of every reaction or time effect. Likewise, nominally equal temperature is not proof of equal local crack-tip chemistry. These limits identify the comparison being made rather than prevent a useful experiment.

Read the same endpoint in cycles and hours

The table is an original hypothetical comparison, not measurements for a named alloy. Imagine matched specimen sets under the same tensile nominal stress cycle, material state, actual temperature and final-separation endpoint. The waveform shape is unchanged while its time scale is changed. Only environment and frequency vary as listed. Treat the displayed values as chosen representative outcomes; no repeat counts or scatter are supplied.

At each frequency, compare the reactive condition with its own reference. Its cycle-count ratio is 80,000/100,000 = 0.80 at 5 Hz and 30,000/100,000 = 0.30 at 0.5 Hz. The supplied pattern is consistent with a larger environmental penalty per cycle in the slower history. It does not establish a population difference or identify the chemical mechanism without repeated observations and supporting evidence.

Now compare only the two reactive rows. The slower test has 37.5% as many cycles to the endpoint, yet lasts 3.75 times as long. Both statements describe the same records. A lower cycle count is not automatically a shorter elapsed life, and a longer elapsed life is not evidence that reactions were harmless. The two normalisations answer different questions.

The four conditions improve on comparing slow reactive loading with fast reference loading alone. They still do not divide total life into independently additive chemical and mechanical parts. Time changes while cycles occur, and the specimen state evolves. Interrupted surface/crack observations and selected additional controls can test what happened in that interval.

Original chosen comparison under matched nominal loading, temperature and endpoint. Values are teaching inputs; the hours are calculated from N/f. No scatter, allowable or service-life prediction is supplied.
Environment	f (Hz)	Cycles to endpoint N	Elapsed time (h)
Reference	5.0	100,000	5.56
Reference	0.5	100,000	55.56
Reactive	5.0	80,000	4.44
Reactive	0.5	30,000	16.67

Original chosen outcomes: reference at 5 Hz has 100,000 cycles and 5.56 h; reference at 0.5 Hz has 100,000 cycles and 55.56 h; reactive at 5 Hz has 80,000 cycles and 4.44 h; reactive at 0.5 Hz has 30,000 cycles and 16.67 h. Solid bars denote 5 Hz, hatched bars 0.5 Hz. The slower reactive condition has fewer cycles but more hours than its fast counterpart. These teaching inputs have no supplied scatter or design meaning.

A hold changes what the next excursion meets

At a temperature where creep is appreciable, a sustained tensile stress can continue to deform the material during a hold. In contrast, a hold at fixed total strain can show stress relaxation as elastic strain is exchanged for time-dependent strain. The imposed variable matters: a nominal hold label does not specify a constant stress, constant strain or constant crack-driving force.

Reactions can also act during the hold. For example, an oxide layer may grow while a hot surface is under one part of its deformation history. If the layer cannot accommodate the following excursion, it can crack and expose the substrate or provide sites for cracking. Deformation changes the film; film growth or damage changes the next deformation response. Simply adding a dry fatigue life and a separate creep life does not represent that feedback.

A study of 316 stainless steel in air at 816 °C examined several creep–fatigue waveforms and exposure durations. Fractography and metallography showed oxidation and carbide precipitation in long-exposure specimens. Its proposed mechanisms included an oxide formed during compressive creep cracking on a later tensile excursion. This alloy- and history-specific evidence motivates examining both waveform and time, rather than adopting a universal rule that a slower cycle is always worse.

That study changed creep stress levels to change exposure times and used particular inelastic strain histories; it was not a clean isolation of frequency alone. Its fitted life relations are not transferred into the teaching table. Nor does the presence of an oxide establish that it always accelerates damage: whether a film protects, breaks or alters contact needs evidence for the actual system.

Temperature cycling can generate a stress cycle

A uniform temperature change can create stress through restraint, even without a separately prescribed force cycle. Start with a homogeneous slender bar that is stress-free at T₀. Use small uniaxial elastic strain, constant Young’s modulus E and constant linear expansion coefficient α over the chosen interval. Temperature is uniform throughout the bar, lateral expansion is free, and buckling, yielding and creep are excluded.

Write total axial strain as ε = σ/E + αΔT, with tensile stress positive and ΔT = T − T₀. The first term is mechanical elastic strain; the second is the strain the material would develop on free heating. If its ends can move freely, σ = 0 and the bar length changes. If ideal rigid supports hold its original length, ε = 0, so σ = −EαΔT. For positive α, heating produces compression and cooling below T₀ produces tension. Support reactions supply the axial forces.

In this model a heating-and-return cycle retraces the same straight σ-versus-ΔT line. It has no hysteresis or accumulated damage by itself. It supplies a possible cyclic stress input; whether that causes fatigue needs material and local-response evidence. The ideal straight line cannot also describe stress relaxation during a long hot hold, plastic flow or a temperature gradient. Those processes require a different model.

Expansion mismatch between bonded materials can create internal stresses even when a whole assembly is not fixed. Its force balance and compatibility differ from this single fully restrained bar. Use the linked thermal-expansion lesson for that extension rather than assigning −EαΔT to every part of a real joint.

For the original uniaxial linear-elastic model, E = 100 GPa and α = 12 × 10^−6/K, the fixed-length bar stress falls from 0 MPa at ΔT = 0 K to −120 MPa at 100 K. The axially free bar remains at 0 MPa. The heating-and-return path retraces each line; this is not a hysteresis curve or damage prediction. Constant properties, uniform temperature, lateral freedom and no creep, yielding or buckling are stipulated.

Choose a comparison that can distinguish causes

Suppose pits are suspected of shortening initiation. First compare otherwise matched smooth and pre-pitted specimens in the same reference condition, measuring pit geometry and observing where cracks start. This tests a geometric starting-state contribution, although pre-pitting may also change residual stress or near-surface material; those differences must be considered. Then compare matched pre-pitted sets in reference and reactive environments. Continued differences may involve ongoing reactions, but a final endpoint alone still cannot say whether initiation or propagation changed. Interrupted crack-size measurements help separate those stages.

A sustained-load control in the same environment and temperature can ask whether a crack advances without cycling at that load. A no-load exposure control followed by reference-environment cycling can ask whether prior exposure left a damaging surface state. Neither control reproduces cyclic film rupture, and subtracting their damage from cyclic damage requires an additional model. A null result also has a duration and detection limit.

Record the histories that the mechanism needs: actual temperature and stress or strain during holds, time in the fluid, surface/pit state and crack size where measurable. Choose repeats and measurement resolution before claiming a difference exceeds scatter. This evidence can reject an oversimplified explanation while leaving more than one coupled mechanism possible. The aim is a warranted inference, not a universal correction factor for every service environment.

Worked examples

Fewer cycles can still mean more exposure time

A slower cyclic test in a reactive fluid gives fewer cycles to failure. Does that prove frequency alone changes intrinsic fatigue resistance? Use the four chosen outcomes above to calculate elapsed times and compare the two reactive tests.

For each continuously cycling test, t = N/f. The fast reference lasts 100,000/5 = 20,000 s ≈ 5.56 h. The slow reference lasts 100,000/0.5 = 200,000 s ≈ 55.56 h. The frequency change increases time per cycle tenfold.
The fast reactive test lasts 80,000/5 = 16,000 s ≈ 4.44 h. The slow reactive test lasts 30,000/0.5 = 60,000 s ≈ 16.67 h. Each time was divided by 3600 s/h; no hold is added to this continuous schedule.
Within the reactive condition, N_slow/N_fast = 30,000/80,000 = 0.375, whereas t_slow/t_fast = 60,000/16,000 = 3.75. Thus the slower record ends after fewer cycles but after more elapsed exposure.
The reference records have equal cycle counts, while the reactive/reference ratio changes from 0.80 to 0.30. This chosen pattern points toward a frequency–environment interaction per cycle. It does not prove one intrinsic frequency law, establish significance without scatter, or show whether pits, crack propagation or another process controlled the endpoint.

No. The slower reactive record has 37.5% as many cycles but 3.75 times the exposure time. The matched reference comparison strengthens an interaction interpretation; it does not uniquely identify a mechanism or population life.

Equal cycle counts can carry unequal hot exposure

Two chosen high-temperature schedules each apply 2000 cycles. The ramps and non-hold parts occupy 2.0 s per cycle in both. Schedule B adds an 8.0 s hold at peak total tensile strain to every cycle; schedule A has no added hold. Compare durations. During a B hold, measured stress falls. Does a falling hold stress, or return to the same stress at later cycle peaks, demonstrate zero hold damage?

Schedule A lasts 2000 × 2.0 = 4000 s ≈ 1.11 h. Its full frequency is 1/2 = 0.50 Hz.
Schedule B lasts 2000 × (2.0 + 8.0) = 20,000 s ≈ 5.56 h. Added hold exposure is 16,000 s ≈ 4.44 h, and the full-cycle frequency is 1/10 = 0.10 Hz. Do not label B 0.50 Hz and silently omit the holds.
B gives five times the total hot exposure with the same count of cycles. Its strain-controlled hold can relax stress as time-dependent strain develops. Falling stress is not proof that deformation, oxidation or local damage stopped.
A returning or repeating nominal stress value is also not a measurement that surface chemistry, pits, cavities or crack length returned to their earlier states. The example supplies no damage rate or life law, so no numerical damage multiplier follows from the fivefold duration.

A lasts about 1.11 h and B 5.56 h, including 4.44 h of added holds. Stress relaxation identifies a changing mechanical response under strain control; it does not quantify or eliminate hold damage.

Derive the load created by a heating cycle

Use the ideal restrained-bar model with chosen E = 100 GPa and α = 12 × 10^−6 K^−1. Starting stress-free at T₀, heat uniformly to T₀ + 100 K and return to T₀. Find peak stress, stress range, mean stress and amplitude. A second identical bar of length 200 mm is axially free: find its peak elongation and stress.

The free thermal strain at the hot endpoint is αΔT = 12 × 10^−6 × 100 = 0.0012. A temperature difference of 100 K also equals a difference of 100 °C; no absolute temperature is inserted into αΔT.
The restrained bar has ε = 0, so σ = −100,000 MPa × 0.0012 = −120 MPa at the hot endpoint. At T₀ it returns to 0 MPa. The negative sign means compression; supports exert the required reactions.
For that 0 to −120 to 0 MPa cycle, σ_max = 0 and σ_min = −120 MPa. The range is 120 MPa, the mean is −60 MPa and the amplitude is 60 MPa. Heating and cooling between these endpoints is one full cycle, not two independent cycles.
The free bar has ΔL = αΔT L₀ = 0.0012 × 200 mm = 0.240 mm and σ = 0 in this uniform uniaxial model. Equal temperature histories therefore need not mean equal stress histories.
Both results assume the stipulated elastic/no-creep/no-buckling response throughout. No fatigue life or hot-hold relaxation follows from that reversible calculation.

The restrained endpoint is −120 MPa; range 120 MPa, mean −60 MPa and amplitude 60 MPa. The free bar elongates 0.240 mm with zero axial stress under the stated model.

Try the reasoning

Why is reporting only 300 cycles insufficient when one schedule takes 2 s per cycle and another adds an 18 s hold to each of those cycles? Calculate both durations and state what the ratio does not establish.
Show the answer and reasoning
The durations are 600 s (10 min) and 6000 s (100 min), a factor of ten. The added holds total 5400 s (90 min).
Time-dependent deformation or reactions may continue during holds. Ten times the duration is not automatically ten times the damage; the controlled variable and evolving state also matter.
In two chosen observation blocks, the same defined crack dimension grows by 0.060 mm over 20,000 cycles. One block runs continuously at 2 Hz and the other at 0.2 Hz. Calculate mean growth per cycle and per second. Does the slower block show ten times more growth per cycle?
Show the answer and reasoning
Both mean rates are 3.0 × 10^−9 m/cycle. The durations are 10,000 and 100,000 s, giving 6.0 × 10^−9 and 6.0 × 10^−10 m/s respectively. No: the per-cycle means match.
The slower per-second rate here is ten times lower because the same supplied advance occupies ten times as long. Changing normalisation does not demonstrate a changed mechanism; these are chosen finite block averages, not constant instantaneous rates.
A report compares reactive-fluid cycling at 90 °C with reference-gas cycling at 20 °C and attributes every life difference to the fluid. Propose two additional comparison conditions and explain what they resolve.
Show the answer and reasoning
Add reference-gas cycling at 90 °C and reactive-fluid cycling at 20 °C, with the same loading, starting state and endpoint and verified specimen temperatures.
The resulting temperature-by-environment comparison permits a fluid contrast at each temperature and a temperature contrast in each environment. Replicates and physical observations are still needed; the original two tests confounded both changes.
Pre-pitted specimens reach a visible-crack endpoint earlier than smooth specimens in the same reference environment. Does this prove reactive fluid increases the growth rate of an already existing long crack? Give a more direct next comparison.
Show the answer and reasoning
No. The result concerns a changed starting surface and a visible-crack endpoint. Compare matched known cracks under reference and reactive conditions, measuring crack extension against cycles and time at matched loading and temperature.
A pit can affect the initiation path. Propagation from a known crack is a different question; preserve the definition of crack size and track its changing driving force when comparing growth.
Use the thermal worked model but reduce the heating excursion to 50 K before returning to T₀. Find restrained endpoint stress and amplitude. Would allowing uniform free expansion give that same stress cycle?
Show the answer and reasoning
Restrained endpoint stress is −60 MPa and the 0 to −60 to 0 MPa cycle has amplitude 30 MPa. Uniform free expansion gives zero axial stress in the stated model.
The elastic stress is proportional to the temperature change only with the same restraint and model assumptions. A temperature history alone does not specify a stress history.
At fixed peak total strain, stress relaxes from one measured value to a smaller one during a hot hold. At the next cycle peak the original value is reached again. A learner concludes all hold damage was reversed. Identify the missing inference and a useful observation.
Show the answer and reasoning
A repeated nominal stress value does not establish recovery of crack length, pit geometry, oxide state or microscopic damage. Interrupted crack/surface measurements, with the actual stress–strain and exposure histories, can test whether the material state changed.
Stress is one measured response, not a complete state description. Relaxation and reaction can change what the next excursion encounters even when selected endpoint readings repeat.

What to carry forward

Chemistry can change geometry and local deformation; deformation can expose new reactive surface.
Report the cycle schedule and elapsed exposure together, with actual temperature and a defined environment.
Matched comparisons can support an interaction without uniquely identifying its mechanism or statistical significance.
A hot hold can change stress, strain, films and the material state before the next excursion.
Thermal stress follows restraint and compatibility; a reversible elastic estimate is not an accumulated-damage or life law.

References

Teaching resources and research supporting this lesson’s explanations and examples.