4. Fatigue, Fracture and Damage Tolerance

Introduction and engineering context

Mechanical components rarely fail because the applied load exceeds their static strength in a single event. More commonly, failure develops through repeated loading, environmental attack, time-dependent deformation, or the interaction of a pre-existing defect with an applied stress. Fatigue, fracture, creep, corrosion fatigue, and damage tolerance therefore belong to one connected area of mechanical design: the prediction and management of structural degradation.

A component may be adequately sized against yielding and still fail after millions of cycles. A welded joint may contain a small discontinuity that is harmless under static loading but grows progressively under fluctuating stress. A turbine component may retain adequate instantaneous strength while accumulating creep damage at elevated temperature. Safe design consequently requires more than a nominal stress calculation. It requires an understanding of how defects originate, how cracks grow, how material and environment influence growth, and how inspection and maintenance can control risk.

This chapter develops the principal analytical tools used for such decisions. The S–N approach is examined for high-cycle fatigue, while strain-life methods are used for low-cycle fatigue. Mean-stress corrections, cumulative damage, linear elastic fracture mechanics, crack-growth laws, creep, corrosion fatigue, and inspection intervals are then integrated into a damage-tolerance framework.

Learning objectives

After studying this chapter, the reader should be able to:

1. Distinguish high-cycle fatigue, low-cycle fatigue, fracture, creep, and corrosion fatigue.

2. Select and interpret stress-life and strain-life models.

3. Evaluate the influence of mean stress, stress concentration, surface condition, and size.

4. Apply cumulative-damage concepts with appropriate caution.

5. Use stress-intensity and crack-growth relationships to estimate fracture risk and remaining life.

6. Explain the role of fracture toughness, crack closure, threshold behavior, and residual stress.

7. Develop a rational inspection interval using detectable crack size, critical crack size, and crack-growth rate.

8. Recognize limitations, uncertainties, and professional responsibilities in fatigue and damage-tolerance design.

The physical basis of fatigue damage

Fatigue is the progressive, localized deterioration of a material subjected to repeated or fluctuating stress. The nominal stress may remain below the yield strength, yet microscopic plasticity can occur at inclusions, persistent slip bands, weld toes, machining marks, or geometric discontinuities. Repeated slip produces intrusions and extrusions, followed by microcrack initiation. With continued cycling, a crack may propagate through the material until unstable fracture, excessive deformation, or loss of function occurs.

A useful description divides fatigue life into initiation and propagation. In smooth laboratory specimens, initiation may occupy a substantial fraction of total life. In welded, cast, or machined structures containing defects, the initial crack may already be sufficiently large that propagation dominates. This distinction is important: improving surface finish can greatly extend the life of a polished component, whereas inspection and crack-growth control may be more effective for a welded structure.

The stress quantities commonly used are

• stress amplitude, σa = (σmax − σmin)/2;
• mean stress, σm = (σmax + σmin)/2;
• stress range, Δσ = σmax − σmin; and
• stress ratio, R = σmin/σmax.

The same definitions apply to strain. Tensile mean stress generally reduces fatigue life because it promotes crack opening, whereas compressive mean stress often delays crack growth. These effects are not universal or independent of material, surface condition, residual stress, and loading sequence.

High-cycle and low-cycle fatigue

High-cycle fatigue is usually associated with predominantly elastic deformation and a large number of cycles, often above approximately 10⁴–10⁵ cycles. The stress-life or S–N method is the conventional tool. Low-cycle fatigue involves significant cyclic plastic strain, often arising from thermal expansion, start-up and shutdown, forming, or severe transient loading. Its life may range from a few cycles to approximately 10⁴–10⁵ cycles, and strain-life analysis is more appropriate.

The boundary is not a fixed material constant. It depends on the component, loading history, temperature, and definition of failure. A component subjected to a small nominal stress may still experience low-cycle fatigue at a notch because local plasticity is amplified there.

Stress-life analysis: the S–N approach

An S–N curve relates a stress measure, normally alternating stress, to the number of cycles to a specified failure criterion. For many steels in laboratory conditions, the high-cycle portion can be represented by a Basquin-type relation:

σa = σ′f (2Nf)^b,

where σ′f is the fatigue strength coefficient, b is the fatigue strength exponent, and Nf is the number of reversals to failure. Equivalent forms use cycles rather than reversals. The constants must be used consistently with the chosen convention.

The S–N approach is effective when deformation remains largely elastic and the loading is approximately constant amplitude. Design curves may include statistical scatter and a chosen survival probability. A component curve should account for surface finish, size, reliability, temperature, manufacturing process, residual stress, and stress concentration. Applying a polished-specimen curve directly to a rough, welded, or corroded component is a common and unsafe mistake.

For a notch, the theoretical stress concentration factor Kt describes elastic stress amplification. The fatigue notch factor Kf is generally lower because local plasticity and material sensitivity reduce the full effect of Kt. The fatigue sensitivity factor q is often expressed through Kf = 1 + q(Kt − 1). These relations are empirical approximations; for severe notches, short cracks, or multiaxial loading, local stress–strain analysis is preferable.

Mean-stress corrections

S–N data are often obtained under fully reversed loading, σm = 0, while service loading commonly has a nonzero mean stress. Several empirical corrections are used. The Goodman relation is

σa/Se + σm/Sut = 1,

where Se is the endurance strength at zero mean stress and Sut is ultimate tensile strength. The Soderberg relation replaces Sut with yield strength, producing a more conservative result:

σa/Se + σm/Sy = 1.

The Gerber relation is parabolic and usually less conservative for ductile materials. These equations are design correlations rather than universal laws. They may be unreliable for compressive mean stress, welded details, large plastic strain, and materials with unusual notch or environmental sensitivity. Residual stresses should be treated as part of the local mean-stress state, although their redistribution during cycling may complicate the calculation.

Strain-life analysis and cyclic constitutive behavior

When cyclic plasticity is significant, total strain amplitude is separated into elastic and plastic components. The Coffin–Manson–Basquin equation is commonly written as

Δε/2 = (σ′f/E)(2Nf)^b + ε′f(2Nf)^c,

where Δε/2 is the total strain amplitude, E is Young’s modulus, σ′f and b describe the elastic component, and ε′f and c describe the plastic component. The first term dominates at long lives; the second dominates at short lives.

Strain-life analysis is especially useful for thermal fatigue, pressure-vessel transients, engine components, and notched regions where local strain can be estimated. The Neuber rule is often used to relate nominal elastic stress and strain to local elastic–plastic values at a notch:

σlocal εlocal ≈ (K_t σnom)(K_t εnom),

with suitable modifications for cyclic material behavior. More advanced analyses use finite-element simulations with cyclic plasticity models, such as combined isotropic–kinematic hardening. Cyclic hardening changes the stress response with repeated reversals, while cyclic softening reduces it. Ignoring these effects can lead to incorrect life predictions.

Mean strain and mean stress interact in strain-life analysis. Morrow-type corrections modify the fatigue strength term for mean stress, while Smith–Watson–Topper approaches use a parameter involving maximum stress and strain amplitude, commonly σmax(Δε/2). No single correction is universally best; calibration against representative material data is essential.

Cumulative damage and variable-amplitude loading

Real components experience changing load amplitudes. The simplest cumulative-damage method is Miner’s rule:

D = Σ ni/Ni,

where ni is the number of cycles applied at a particular stress level and Ni is the constant-amplitude life at that level. Failure is conventionally predicted when D reaches unity.

Miner’s rule is attractive because it is simple, but it does not account for load sequence, crack closure, mean-stress evolution, residual stresses, or interaction between damage mechanisms. Overloads may retard crack growth by creating compressive residual stress, whereas some sequences accelerate damage. Consequently, a calculated damage fraction should be treated as an engineering estimate, not a physical law. Variable-amplitude analysis may use rainflow cycle counting to reduce a measured load history into reversals and ranges, followed by mean-stress correction and damage accumulation.

For critical components, fracture-mechanics-based crack-growth calculations are often more informative than a single Miner sum. They can explicitly represent overload retardation, changing geometry, and the transition from detectable crack to unstable fracture.

Fracture mechanics: from defects to instability

Fracture mechanics treats a crack as a geometrical feature that intensifies stress near its tip. In linear elastic fracture mechanics, the mode-I stress-intensity factor is commonly written

K_I = Y σ√(πa),

where Y is a geometry factor, σ is the applied nominal stress, and a is a characteristic crack dimension. A crack becomes unstable when K_I reaches the relevant fracture toughness, K_IC, under plane-strain conditions. If constraint is lower, the measured critical value may be K_C rather than a valid plane-strain K_IC.

The critical crack size can be estimated by rearranging the relation:

a_c = (1/π)[K_IC/(Yσ)]².

This equation is valuable for damage-tolerance decisions, but only if crack geometry, loading mode, thickness, constraint, material condition, and environment are represented realistically. Plane stress, mixed-mode loading, plasticity, and ductile tearing may require elastic–plastic parameters such as J-integral or crack-tip-opening displacement.

The crack-driving force is not determined solely by nominal stress. Residual stress, thermal stress, contact loading, bolt preload, and pressure transients may contribute to K. For fatigue crack growth, the principal range is ΔK = Kmax − Kmin. A crack-growth threshold ΔKth may exist under specified conditions, but it is not a guaranteed safe limit because closure, environment, frequency, and load ratio influence the measured threshold.

Fatigue crack-growth laws

The Paris law describes the intermediate stable-growth regime:

da/dN = C(ΔK)^m,

where a is crack length, N is cycle count, and C and m are material/environment parameters. Integrating the law between an initial crack size ai and a final size af gives an estimate of propagation life. The law is not reliable near the threshold region, where growth rates become very small, or near instability, where growth accelerates sharply. A more complete description may use a Forman or NASGRO-type relation that includes stress ratio, threshold behavior, and fracture toughness.

Crack-growth analysis distinguishes three stages: near-threshold growth, approximately Paris-regime growth, and rapid acceleration toward fracture. Inspection planning must consider the entire path from the smallest reliably detectable flaw to the critical size, not merely the final fracture condition.

Damage tolerance and inspection intervals

Damage-tolerant design assumes that defects may exist and demonstrates that they will be detected or remain harmless before reaching a critical condition. The basic logic is:

1. Define credible initial flaws based on manufacturing quality, service history, and inspection capability.

2. Determine the critical crack size using fracture mechanics and the applicable failure criterion.

3. Calculate crack-growth life under the actual or conservatively enveloped load spectrum.

4. Establish a detectable crack size using the selected non-destructive examination method.

5. Set an inspection interval that provides adequate margin between detectable and critical sizes.

6. Specify repair, replacement, or continued-monitoring actions if an indication is found.

If crack growth from ad to ac requires Nprop cycles, an inspection interval should be shorter than Nprop by a justified safety factor or by a regulatory requirement. The interval must also account for inspection reliability, access, human factors, uncertainty in crack location, and the possibility of rapid growth after an overload or environmental change. A nominal interval based only on average crack-growth data is inadequate for a safety-critical structure.

Residual-strength assessment determines whether a component containing a detected crack can continue operating. It compares the applied K or J with fracture resistance and considers stable tearing, leak-before-break, plastic collapse, and the consequences of failure. The most appropriate strategy depends on whether failure is detectable, contained, gradual, or catastrophic.

Creep and creep–fatigue interaction

Creep is time-dependent deformation under sustained stress, usually important at elevated temperature. It involves primary creep with decreasing rate, secondary creep with an approximately steady rate, and tertiary creep with accelerating deformation leading to rupture. A common empirical relation for steady-state creep is Norton’s law:

ε̇ = Aσ^n exp(−Q/RT),

where ε̇ is creep strain rate, A and n are material constants, Q is activation energy, R is the gas constant, and T is absolute temperature.

Creep rupture is often correlated using time–temperature parameters such as Larson–Miller methods, but extrapolation beyond the experimental range must be treated cautiously. In components subject to repeated start-up and shutdown, fatigue damage and creep damage interact. Tensile strain reversals can initiate cracks, while hold periods at high temperature allow creep deformation and grain-boundary damage. A linear interaction diagram may be used as an initial screening tool, but detailed life assessment requires appropriate constitutive models and material data.

Corrosion fatigue and environmentally assisted cracking

Corrosion fatigue occurs when a corrosive environment combines with cyclic stress. The environment may remove protective films, accelerate crack-tip reactions, promote hydrogen uptake, or prevent the formation of beneficial crack-closure effects. Consequently, fatigue strength may be reduced and a conventional endurance limit may disappear.

Important controls include material selection, coatings, cathodic protection where appropriate, drainage, stress reduction, corrosion-resistant design, environmental monitoring, and inspection. Shot peening can introduce beneficial compressive residual stress, but its effectiveness may decline if the surface is damaged or if corrosion removes the treated layer. Hydrogen embrittlement and stress-corrosion cracking are related but distinct mechanisms; they require separate diagnosis because prevention and allowable stress limits differ.

An illustrative case: inspection planning for a pressure-containing vessel

Background

Consider an illustrative high-pressure cylindrical vessel made from a quenched-and-tempered steel. The vessel experiences repeated pressure cycles, with occasional pressure excursions. A longitudinal weld contains a planar flaw that is not detected during initial inspection but is assumed, for design purposes, to have an initial depth ai. The weld region has a geometry factor Y that changes moderately as the crack grows.

Situation and analysis

A nominal stress-life calculation suggests that the pressure-cycle stress is below the material’s corrected endurance strength. However, the weld detail is not a polished specimen and contains residual tensile stress. A damage-tolerance assessment is therefore selected instead of relying solely on an S–N curve.

First, the critical crack depth ac is estimated from K_IC = Yσ√(πa), using the maximum credible pressure and the appropriate fracture toughness for the weld heat-affected zone. Second, the pressure history is converted into stress-intensity ranges. A crack-growth relation of the form da/dN = C(ΔK)^m is integrated from ai to a detectable size ad and then from ad toward ac. The analysis includes a conservative stress ratio and considers whether the occasional pressure excursion produces crack-growth retardation or, alternatively, an immediate increase in crack-driving force.

Strategy and solution

The vessel operator adopts a two-level control plan. Volumetric examination is scheduled at an interval shorter than the predicted propagation life from ad to ac, with additional inspection after any pressure excursion beyond the design envelope. The inspection procedure is qualified for the weld geometry and includes sizing uncertainty. If a crack exceeds the acceptance limit, the vessel is removed from service for weld repair or replacement. Operating pressure is also restricted temporarily if an indication is detected but remains below the critical condition.

Outcome

The example illustrates why a low nominal stress does not by itself establish safety. The critical decision variables are the initial flaw assumption, fracture toughness, crack-growth rate, inspection reliability, and consequence of failure. The resulting plan is not a promise that cracks will never occur; it is a controlled demonstration that credible cracks will be found or remain noncritical before failure.

Learning points

• Welded structures should not be represented by smooth-specimen endurance data without suitable corrections.
• Critical crack size and detectable crack size are different quantities.
• Inspection interval is a life-management decision involving uncertainty, not merely a calendar choice.
• Pressure excursions and environmental changes require reassessment.

Limitations, risks, and common errors

Fatigue and fracture calculations are sensitive to the quality of input data. Common errors include confusing stress amplitude with stress range, mixing cycles and reversals in strain-life equations, applying Goodman corrections outside their validated range, ignoring residual stress, using inappropriate specimen data, and treating Miner’s rule as exact.

Further risks arise from assuming that an inspection method detects every flaw, neglecting crack orientation, using a two-dimensional geometry factor for a three-dimensional crack, and extrapolating Paris-law data into threshold or near-fracture regions. Statistical scatter is substantial, particularly in crack-growth rates and weld performance. Conservative design should address model uncertainty, parameter uncertainty, load uncertainty, and inspection uncertainty separately rather than hiding all uncertainty inside one arbitrary safety factor.

Professional, regulatory, and digital considerations

Safety-critical fatigue assessment should be traceable: loads, material condition, geometry, assumptions, software version, validation evidence, and acceptance criteria should be recorded. Engineers must distinguish code compliance from actual fitness for service. Applicable pressure-vessel, aerospace, railway, offshore, or power-generation standards may prescribe inspection methods, allowable flaws, toughness tests, and reassessment procedures.

Digital tools provide substantial value. Finite-element models estimate local stress and strain; automated rainflow algorithms process measured load histories; probabilistic fracture mechanics represents distributions of flaw size and crack-growth rate; and structural-health-monitoring systems can combine strain, vibration, acoustic-emission, or guided-wave data. Machine-learning methods may assist anomaly detection, but they do not replace validated mechanics, qualified inspection, or engineering judgment. Data drift, false negatives, and lack of explainability are serious concerns when automated systems influence maintenance decisions.

Current developments and best practice

Modern practice is moving toward probabilistic damage tolerance, digital twins, risk-based inspection, and integrated life management. Probabilistic methods estimate the probability of failure rather than relying only on a single conservative crack size. Digital twins update a structural model using inspection and operational data, although their reliability depends on sensor quality and model calibration.

A robust engineering workflow combines the following principles:

• characterize the actual load spectrum and environment;
• use material and weld data representative of the component;
• select S–N, strain-life, or fracture-mechanics methods according to the governing mechanism;
• include mean stress, residual stress, temperature, and environmental effects;
• validate assumptions with inspection and service data;
• define detectable and critical flaw sizes independently;
• include human and equipment reliability in inspection planning; and
• reassess the component after overloads, repairs, design changes, or unexpected indications.

Chapter Summary

Fatigue is a progressive failure process caused by repeated or fluctuating loading, while fracture mechanics describes the influence of cracks and defects on structural integrity. High-cycle fatigue is commonly evaluated using S–N curves, whereas low-cycle fatigue requires strain-life relations that include elastic and plastic strain. Mean stress, notches, surface condition, residual stress, size, temperature, and environment strongly affect life.

Miner’s rule offers a simple estimate for variable-amplitude damage but neglects load-sequence effects and should not be treated as exact. Linear elastic fracture mechanics relates crack size and applied stress to stress intensity, with fracture occurring when the crack-driving force reaches material resistance. Paris-type laws estimate stable fatigue crack growth, but threshold and near-fracture regions require more complete models. Damage-tolerant design links assumed initial flaws, detectable flaws, critical flaws, crack-growth life, and inspection reliability. Creep, creep–fatigue interaction, corrosion fatigue, and environmentally assisted cracking require specialized treatment. Safe professional practice depends on validated data, traceable assumptions, appropriate standards, qualified inspection, and recognition of uncertainty.

Key Terms

Fatigue: Progressive damage caused by repeated or fluctuating loading.

High-cycle fatigue: Fatigue dominated by elastic deformation and generally long life.

Low-cycle fatigue: Fatigue involving significant cyclic plastic strain.

S–N curve: Relationship between stress amplitude and cycles to a defined failure criterion.

Strain-life method: Fatigue method relating elastic and plastic strain amplitudes to life.

Stress amplitude: Half the difference between maximum and minimum stress.

Mean stress: Average of maximum and minimum stress in a cycle.

Stress ratio: Ratio of minimum stress to maximum stress.

Miner’s rule: Linear cumulative-damage approximation based on fractions of constant-amplitude life.

Stress concentration factor: Ratio of local elastic peak stress to nominal stress.

Stress-intensity factor: Crack-tip parameter representing the intensity of the elastic stress field.

Fracture toughness: Resistance of a material to unstable crack extension under specified conditions.

Critical crack size: Crack dimension at which the applied crack-driving force reaches failure resistance.

Paris law: Power-law relation between fatigue crack-growth rate and stress-intensity range.

Crack closure: Partial closure of a fatigue crack during part of a loading cycle.

Damage tolerance: Design philosophy assuming flaws exist and controlling their growth or consequences.

Creep: Time-dependent deformation under sustained stress, usually at elevated temperature.

Corrosion fatigue: Fatigue damage accelerated by a corrosive environment.

Rainflow counting: Algorithm for converting variable-amplitude histories into fatigue cycles.

Risk-based inspection: Inspection planning based on probability and consequence of failure.

Review Questions

Short Answer Questions

1. Distinguish high-cycle fatigue from low-cycle fatigue.

2. Define stress amplitude, mean stress, and stress ratio.

3. State the principal assumptions of the Paris crack-growth law.

4. What is meant by critical crack size?

5. Why can corrosion fatigue eliminate an apparent endurance limit?

Long Answer Questions

1. Explain the S–N method, including the effects of mean stress, surface finish, size, and stress concentration.

2. Derive and interpret the Coffin–Manson–Basquin strain-life relation.

3. Explain how linear elastic fracture mechanics is used to assess a cracked component.

4. Discuss the advantages and limitations of Miner’s rule for variable-amplitude loading.

5. Describe a complete damage-tolerance procedure for establishing inspection intervals.

Analytical / Application Questions

1. A component is subjected to a tensile mean stress in addition to alternating stress. Explain how Goodman and Soderberg corrections would alter the allowable alternating stress and why their predictions differ.

2. A detected crack is larger than the assumed initial flaw but smaller than the calculated critical size. Identify the additional information required to decide whether the component may remain in service.

3. A turbine component experiences cyclic start-up loading and long high-temperature holds. Propose an assessment strategy that accounts for fatigue, creep, creep–fatigue interaction, inspection, and uncertainty.

22. Multiple Choice Questions with Answers

Question 1

A component experiences significant cyclic plastic strain at a notch during thermal transients. Which method is most appropriate for primary life estimation?

A. Stress-life S–N analysis only

B. Strain-life analysis

C. Static yield-strength comparison only

D. Miner’s rule without strain data

Correct Answer: B. Strain-life analysis

Question 2

In the relation K_I = Yσ√(πa), increasing crack size while holding stress constant generally causes K_I to:

A. Decrease linearly

B. Remain unchanged

C. Increase with the square root of crack size

D. Increase with the square of crack size

Correct Answer: C. Increase with the square root of crack size

Question 3

Which statement best describes Miner’s rule?

A. It explicitly models crack-tip plasticity and closure

B. It sums fractions of constant-amplitude fatigue life

C. It predicts fracture toughness from tensile strength

D. It applies only to creep rupture

Correct Answer: B. It sums fractions of constant-amplitude fatigue life

Question 4

Compared with the Goodman relation, the Soderberg relation is generally more conservative because it uses:

A. Ultimate tensile strength instead of yield strength

B. Yield strength instead of ultimate tensile strength

C. Fracture toughness instead of yield strength

D. Endurance strength instead of alternating stress

Correct Answer: B. Yield strength instead of ultimate tensile strength

Question 5

A crack-growth calculation based on the Paris law is least reliable when the crack is:

A. In the intermediate stable-growth regime

B. Subjected to a known constant stress-intensity range

C. Near the threshold or approaching unstable fracture

D. Growing under repeated tensile loading

Correct Answer: C. Near the threshold or approaching unstable fracture

Question 6

The principal purpose of comparing detectable crack size with critical crack size is to:

A. Determine elastic modulus

B. Establish whether inspection can provide adequate warning before failure

C. Calculate the material’s ultimate tensile strength

D. Eliminate the need for fracture toughness data

Correct Answer: B. Establish whether inspection can provide adequate warning before failure

Question 7

Which loading feature generally tends to reduce fatigue life in a ductile component?

A. A beneficial compressive residual stress

B. A tensile mean stress

C. A polished surface

D. A lower stress range

Correct Answer: B. A tensile mean stress

Question 8

For a high-temperature component subjected to sustained stress and repeated start-up cycles, the most appropriate assessment recognizes:

A. Only static yielding

B. Only high-cycle fatigue

C. Creep–fatigue interaction

D. Only corrosion fatigue

Correct Answer: C. Creep–fatigue interaction

Question 9

Rainflow counting is used primarily to:

A. Measure fracture toughness

B. Convert a variable load history into fatigue cycles

C. Calculate Young’s modulus

D. Determine crack geometry factors experimentally

Correct Answer: B. Convert a variable load history into fatigue cycles

Question 10

A damage-tolerant inspection interval should primarily be based on the predicted time for a crack to grow:

A. From zero size to any size

B. From detectable size to critical size, with suitable margins

C. Only during the first loading cycle

D. From yield stress to ultimate stress

Correct Answer: B. From detectable size to critical size, with suitable margins