Fatigue, Fracture, Creep and Durability

Introduction

Many mechanical components fail even though their maximum service stress is below the static yield strength. Shafts, aircraft fittings, pressure vessels, gears, welded structures and turbine components may experience millions of repeated load cycles, contain small manufacturing defects, or operate at elevated temperature. Their reliability depends not only on instantaneous strength but also on time, loading history, crack development, temperature and the environment.

This chapter develops four closely related ideas:

• Fatigue: failure caused by repeated or fluctuating loading.
• Fracture mechanics: prediction of crack initiation, crack growth and unstable fracture.
• Creep: time-dependent deformation and rupture, especially at elevated temperature.
• Durability: the integrated ability of a component to perform safely throughout its intended life.

A useful engineering life assessment normally proceeds from nominal stresses to local stresses, then to crack or damage evolution, and finally to inspection, maintenance or replacement decisions.

Learning Objectives

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

1. Distinguish high-cycle and low-cycle fatigue and use stress-life and strain-life methods.
2. Determine the effects of mean stress, stress concentration and surface condition on fatigue strength.
3. Apply cumulative-damage concepts to variable-amplitude loading.
4. Explain stress-intensity factor, fracture toughness and fatigue-crack-growth laws.
5. Estimate creep strain, rupture life and the significance of stress relaxation.
6. Explain how residual stress, environment, manufacturing and inspection influence durability.
7. Construct a defensible preliminary life-prediction and damage-tolerance strategy.

1. Cyclic Loading and Fatigue Fundamentals

A cyclic stress may be represented by its maximum and minimum values, σ_max and σ_min. The alternating and mean stresses are

where σ_a is the stress amplitude and σ_m is the mean stress. The stress ratio is

For a fully reversed cycle, R = −1 and σ_m = 0. For a repeated tensile cycle, R = 0. These definitions apply to nominal or local stresses, provided the choice is stated clearly.

Fatigue failure generally involves three stages:

1. Initiation: formation of a microscopic crack, often at a surface, inclusion, notch, weld toe or persistent slip band.
2. Propagation: stable growth of the crack under repeated loading.
3. Final fracture: rapid extension when the remaining section can no longer sustain the applied load.

The distinction between initiation and propagation is not absolute. In a highly polished, defect-free specimen initiation may dominate life; in a welded or cast component, a crack-like defect may already exist and propagation may dominate.

High-cycle and low-cycle fatigue

High-cycle fatigue is usually stress-controlled and involves predominantly elastic deformation, often above 10⁴–10⁵ cycles. Low-cycle fatigue is strain-controlled, involves substantial plastic strain and commonly occurs below approximately 10⁴ cycles. The boundary is material- and application-dependent rather than a universal number.

A fatigue test produces an S–N curve, relating stress amplitude to cycles to failure, N. For many steels in laboratory conditions, the curve approaches an endurance limit; aluminium alloys generally do not exhibit a sharply defined endurance limit, so a specified life such as 10⁷ cycles is used instead.

A commonly used Basquin relation for a fully reversed, elastic regime is

Here σ′_f is the fatigue-strength coefficient, b is the fatigue-strength exponent, and N_f is the number of cycles to failure. The factor 2 appears because N_f is cycles while the original formulation is often written in reversals. The relation is empirical and must be calibrated for the material, surface condition, temperature and stress ratio.

Mean-stress effects

A tensile mean stress usually reduces fatigue strength because it promotes crack opening and reduces the compressive part of the cycle. A preliminary correction is the Goodman relation:

where σ_e is the fully reversed endurance strength at the specified life and S_ut is ultimate tensile strength. The relation is a linear approximation and is not a fracture law. The Soderberg relation replaces S_ut with yield strength S_y and is more conservative:

These equations are most appropriate for ductile metals under proportional, uniaxial loading. They should not be applied blindly to notched, welded, multiaxial or corrosive components.

Strain-life method

When plastic strain is important, the Coffin–Manson–Basquin equation is used:

where ε_a is total strain amplitude, E is Young’s modulus, ε′_f is the fatigue-ductility coefficient, and c is the fatigue-ductility exponent. The first term is elastic strain amplitude and the second is plastic strain amplitude. The equation assumes a reasonably stable hysteresis response and is mainly used for smooth specimens or local notch-strain estimates.

For a component with a notch, nominal stress is not the actual local stress. Elastic stress concentration is described by

For fatigue, the effective fatigue stress-concentration factor is

where q is notch sensitivity, 0 ≤ q ≤ 1. Thus K_f may be less than K_t because local plasticity and material microstructure reduce the full elastic notch effect. The equation is an engineering approximation; q depends on material, notch radius, size and loading mode.

A notch may also produce a beneficial effect if controlled compressive residual stress is introduced, as in shot peening. Conversely, tensile residual stress at a weld toe or machined defect can substantially shorten life.

2. Variable-Amplitude Loading and Cumulative Damage

Real components rarely experience one constant stress amplitude. A spectrum may contain start-up loads, normal operating cycles, overloads, vibration and shutdown transients. The usual procedure is to divide the spectrum into bins and count cycles, commonly using rainflow counting for irregular histories.

Miner’s linear damage rule is

where n_i is the number of applied cycles in stress block i and N_i is the constant-amplitude cycles to failure at that block’s amplitude and mean stress. Failure is often estimated when D ≈ 1.

Miner’s rule is useful because it is simple, but it assumes damage is linear, sequence-independent and unaffected by interaction between stress levels. These assumptions are imperfect. A high tensile overload may retard subsequent crack growth by inducing crack closure, whereas certain sequences accelerate damage. Therefore D = 1 is a design estimate, not a physical universal law.

Worked Example 1: Miner damage

A component experiences 20,000 cycles at a stress level for which the S–N curve gives N₁ = 100,000 cycles, and 50,000 cycles at a second level for which N₂ = 400,000 cycles. The estimated damage is

The remaining linear-damage fraction is 0.675. If the same spectrum repeats, the estimated number of spectrum repetitions to failure is 1/0.325 ≈ 3.08. Because only complete spectra can be applied, failure would be expected during the fourth repetition under the linear rule. This prediction should be modified if inspection evidence or sequence effects are significant.

3. Fracture Mechanics

Strength-of-materials calculations generally treat a component as continuous. Fracture mechanics instead treats a crack as an explicit geometric feature and asks whether it will remain stable, grow, or become unstable.

For a mode-I crack under tensile loading, the stress-intensity factor is commonly written

where σ is the nominal applied stress, a is a characteristic crack size, and Y is a dimensionless geometry factor. The units of K_I are MPa√m when σ is in MPa and a is in metres. The equation assumes linear-elastic material behaviour, a sufficiently sharp crack and a geometry for which the selected Y is valid.

Fracture occurs when the applied K_I reaches the material’s plane-strain fracture toughness K_IC:

K_IC is valid only when specimen thickness, crack size and ligament dimensions satisfy the relevant plane-strain requirements. Otherwise the measured toughness may be a geometry- or thickness-dependent value rather than a valid K_IC.

The fracture safety condition may therefore be expressed as

Solving for the critical crack size gives

This formula is appropriate only when Y is constant or appropriately evaluated at the critical geometry. It does not replace a code-based assessment for complex components.

Fatigue-crack growth

For a crack subjected to cyclic loading, the range of stress intensity is

where ∆σ = σ_max − σ_min. In the Paris regime, crack-growth rate is approximated by

where a is crack length, N is cycles, C and m are experimentally determined constants. The Paris law describes the intermediate stable-growth region. It does not accurately describe the near-threshold region, where crack closure and microstructure matter, or the rapid-growth region near fracture toughness.

For constant geometry and stress range, integration gives a life estimate:

The limits a_i and a_f are the initial and final crack sizes. In practice, Y may vary with crack length and component geometry, so numerical integration is often required.

The effective driving force may be reduced by crack closure. A threshold ∆K_th may exist below which long-crack growth is very slow under specified conditions. Short cracks can grow at values below the long-crack threshold, which is a major limitation in transferring laboratory data directly to components.

Worked Example 2: Critical crack size

A plate has Y = 1.12, nominal tensile stress σ = 150 MPa and fracture toughness K_IC = 50 MPa√m. The critical crack size is

Since 1.12 × 150 = 168 MPa,

Thus a_c ≈ 28 mm for the characteristic crack dimension used in the chosen geometry model. The result is not a permitted operating crack size: a safety factor, inspection uncertainty, residual stress and geometry changes must be considered.

4. Creep and Stress Rupture

Creep is progressive, time-dependent deformation under sustained stress. It becomes important when the homologous temperature, T/T_m, is sufficiently high, where T is absolute temperature and T_m is the material’s absolute melting temperature. A simple engineering guideline is that creep may become significant above roughly 0.3–0.4 T_m for many metals, but the threshold depends strongly on material and service duration.

A typical creep curve has three stages:

1. Primary creep: decreasing creep rate because of work hardening.
2. Secondary creep: approximately constant minimum creep rate, often used for design correlations.
3. Tertiary creep: accelerating deformation due to necking, cavitation, damage and section loss, ending in rupture.

A power-law representation of steady-state creep is the Norton relation:

where ε̇_s is steady-state creep strain rate, A is a material constant, σ is stress, n is the stress exponent, Q is activation energy, R is the universal gas constant and T is absolute temperature. The constants are valid only over the temperature and stress range used to determine them.

For rupture-life correlation, a Larson–Miller parameter is often used:

where T is in kelvin, t_r is rupture time in hours and C_LM is an empirical constant. Data collapse using this parameter is approximate and material-specific; it should not be extrapolated far beyond the test range.

Creep-fatigue interaction occurs when a component experiences repeated thermal or mechanical cycles at high temperature. Tensile hold periods can cause creep damage, while cyclic plasticity causes fatigue damage. A simple interaction diagram may use

where D_f is fatigue damage and D_c is creep damage. This linear interaction is only a preliminary screening model. Constitutive models, hold-time tests and applicable design codes are required for critical components.

Stress relaxation

Under imposed total strain, stress may decrease with time as creep strain accumulates. This is stress relaxation. It is important in bolted joints, seals, springs, thermal fittings and prestressed structures. A joint designed using initial preload alone may lose clamping force at elevated temperature, even if total deformation appears small.

5. Residual Stress, Environment and Manufacturing

Residual stress is self-equilibrating stress remaining after manufacture or prior loading. Sources include welding, casting, machining, grinding, heat treatment, cold working and assembly interference.

• Tensile residual stress assists crack opening and generally reduces fatigue and fracture resistance.
• Compressive residual stress can delay surface crack initiation and growth.
• Residual stress may redistribute during yielding, creep, thermal cycling or crack growth.

Fatigue in a corrosive environment is called corrosion fatigue. Hydrogen embrittlement can reduce ductility and accelerate cracking in susceptible alloys. Fretting fatigue occurs when small-amplitude relative motion damages contacting surfaces. Elevated temperature may cause oxidation, creep, diffusion and environmental attack simultaneously.

Durability therefore depends on a chain of conditions:

Design geometry → material and processing → residual stress → service spectrum → environment → inspection and maintenance

A strong design avoids abrupt section changes, specifies suitable surface finish and heat treatment, controls weld quality, manages residual stress and defines inspection acceptance criteria.

6. A Damage-Tolerance Life-Prediction Procedure

A practical assessment should use the following sequence:

1. Define the mission: loads, temperatures, frequencies, environment, start-stop events and intended life.
2. Characterize material condition: strength, fatigue data, fracture toughness, crack-growth data, creep properties and variability.
3. Calculate nominal stresses: include axial, bending, torsion, pressure and thermal stresses.
4. Identify local effects: notches, holes, weld toes, threads, inclusions and contact regions.
5. Select the life method: S–N for elastic high-cycle fatigue, strain-life for local plasticity, or crack-growth analysis when defects are credible.
6. Establish initial flaw assumptions: based on manufacturing capability, nondestructive examination and applicable standards.
7. Propagate damage: use a stress-life damage rule, crack-growth integration or a creep-fatigue model.
8. Check unstable fracture: compare K_I with the appropriate toughness.
9. Define inspection intervals: the interval must be short enough to detect a crack before it reaches the allowable size.
10. Validate: compare predictions with component tests, field data and appropriate statistical confidence limits.

Inspection reliability matters. If an inspection method cannot reliably detect the assumed initial crack size, the analysis is nonconservative. Probability of detection, false calls, accessibility, calibration and operator competence must be considered.

7. Illustrative Case: Fatigue Assessment of a Notched Rotating Shaft

Background

Consider a steel rotating shaft with a shoulder fillet. The shaft transmits torque and bending moment, producing a repeated bending stress at the fillet. The nominal alternating bending stress is 120 MPa, and the nominal mean stress is 40 MPa. The local fatigue concentration factor is K_f = 1.6. The material has an ultimate tensile strength of 600 MPa and a fully reversed endurance strength at the required life of 250 MPa.

Problem

Estimate whether a Goodman-based fatigue check is acceptable for the local bending stress. This is an illustrative preliminary calculation, not a complete shaft design.

Analysis

The local alternating and mean stresses are estimated as

Applying the Goodman interaction index,

A value below 1.0 indicates that the preliminary linear Goodman criterion is satisfied. The margin by this criterion is 1 − 0.875 = 0.125, or 12.5% of the normalized interaction capacity.

Approach and solution

The design should next check combined bending and torsion using an appropriate multiaxial fatigue criterion, confirm the notch radius and surface condition, assess mean-stress relaxation if local yielding occurs, and examine the fillet for machining marks. If the shaft is safety-critical, a crack-growth assessment should be performed using a credible initial flaw and inspection capability.

Shot peening may improve resistance by introducing compressive surface stress, but its intensity, coverage and retained benefit must be verified. Increasing the fillet radius, reducing surface roughness and lowering the stress spectrum may provide more robust improvements than relying solely on a larger nominal strength.

Outcome and learning points

The preliminary check passes, but the relatively small margin shows why a simple nominal-stress calculation can be misleading. Fatigue design is a system problem involving geometry, material, manufacturing, loading history and inspection. A pass against one criterion is not proof of safe life under all service conditions.

8. Limitations, Standards and Professional Practice

Fatigue and fracture predictions contain uncertainty from material scatter, load variability, geometry factors, residual stress, environmental effects and inspection error. Safety factors should reflect consequence of failure and quality of evidence, not merely tradition.

Engineering decisions should follow applicable standards and regulations. Examples include pressure-vessel, aerospace, railway, automotive, welding and structural codes. These documents specify allowable stresses, fatigue curves, flaw-assessment procedures, inspection requirements and qualification tests. A textbook equation cannot override a governing code.

Ethically, engineers must report uncertainty, avoid suppressing unfavorable test results, maintain traceable material and inspection records, and communicate remaining life honestly. Digital monitoring and AI can assist with load-spectrum classification, anomaly detection and sensor fusion, but they do not eliminate the need for validated physics, qualified personnel and human accountability. Models trained on one operating population may fail when temperature, material batch, maintenance history or load spectrum changes.

Contemporary durability practice increasingly combines:

• digital twins linked to measured loads and temperatures;
• structural-health-monitoring sensors for strain, vibration and acoustic emission;
• probabilistic fracture mechanics and Bayesian updating;
• additive-manufacturing defect characterization;
• physics-informed machine learning for crack-growth forecasting; and
• risk-based inspection that prioritizes consequence and probability of failure.

These tools are most reliable when used to supplement, not replace, conservative mechanics-based assessment and physical validation.

Chapter Summary

Fatigue is a time-dependent failure process caused by cyclic stress or strain. Stress-life S–N methods are suitable for predominantly elastic high-cycle conditions, while strain-life methods account for local plasticity and are useful in low-cycle fatigue. Mean stress, notches, surface condition, residual stress and environment strongly influence fatigue life. Miner’s rule provides a convenient but approximate method for variable-amplitude loading.

Fracture mechanics explicitly represents cracks. The stress-intensity factor K_I = Yσ√(πa) measures the crack-tip driving force, and unstable fracture is approached when it reaches the applicable fracture toughness. Paris’ law estimates stable fatigue-crack growth but has limited validity near threshold and rapid fracture.

Creep is time-dependent deformation at elevated temperature. Primary, secondary and tertiary stages describe its evolution; Norton’s law estimates steady-state strain rate, while Larson–Miller methods correlate rupture data. Creep-fatigue interaction, stress relaxation and environmental attack must be considered in hot components.

Durability is achieved through integrated design, material selection, manufacturing control, load characterization, residual-stress management, inspection and maintenance. Reliable life prediction requires explicit assumptions, validated data, suitable standards and honest treatment of uncertainty.

Key Terms

Fatigue: Failure caused by repeated or fluctuating loading.

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.

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

Strain-life method: Fatigue method relating strain amplitude to cycles to failure.

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

Fatigue notch factor: Effective concentration factor for fatigue, including notch sensitivity.

Miner’s rule: Linear cumulative-damage approximation for variable loading.

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

Fracture toughness: Material resistance to unstable crack extension under specified conditions.

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

Crack closure: Reduction in effective crack-driving force because crack faces contact during part of a cycle.

Creep: Time-dependent deformation under sustained stress.

Stress relaxation: Reduction of stress under approximately constant total strain.

Creep rupture: Failure after time-dependent deformation at elevated temperature.

Residual stress: Self-equilibrating stress remaining after manufacture or prior loading.

Damage tolerance: Design philosophy that permits specified defects while preventing catastrophic failure.

Probability of detection: Likelihood that an inspection method detects a flaw of a specified size.

Durability: Ability of a component to perform safely for its intended service life.

Review Questions

Short Answer Questions

1. Define stress amplitude, mean stress and stress ratio for a cyclic load.
2. Distinguish high-cycle fatigue from low-cycle fatigue.
3. What physical meaning is assigned to the stress-intensity factor K_I?
4. Describe the three stages of a typical creep curve.
5. Why can tensile residual stress reduce fatigue life?

Long Answer Questions

1. Explain the development and engineering use of an S–N curve, including its limitations.
2. Derive the critical crack-size expression from the condition K_I = K_IC and discuss its assumptions.
3. Compare the stress-life, strain-life and crack-growth approaches to life prediction.
4. Explain Miner’s rule and discuss why actual damage may depend on load sequence.
5. Describe a complete damage-tolerance program for a safety-critical component.

Analytical / Application Questions

1. A component experiences 10,000 cycles at a level with N₁ = 50,000 cycles to failure and 30,000 cycles at a level with N₂ = 200,000 cycles. Calculate Miner damage and estimate the number of repeated blocks to failure.
2. A plate has Y = 1.0, applied stress σ = 100 MPa and K_IC = 40 MPa√m. Calculate the critical crack size using the linear-elastic expression and state why the result must be treated as a preliminary estimate.
3. A rotating shaft passes a Goodman check but has poor surface finish and tensile weld residual stress nearby. Recommend at least four additional engineering actions before approving it for service, and justify each action.

Multiple Choice Questions with Answers

Question 1

For a cyclic stress with σ_max = 120 MPa and σ_min = −40 MPa, what is the stress amplitude?

A. 40 MPa

B. 80 MPa

C. บ0 MPa

D. 160 MPa

Correct Answer: B. 80 MPa

Question 2

Which method is most appropriate when local plastic strain is significant?

A. S–N stress-life method only

B. Strain-life method

C. Goodman relation only

D. Elastic beam theory only

Correct Answer: B. Strain-life method

Question 3

In the Goodman relation, the mean stress is commonly normalized by which material property?

A. Young’s modulus

B. Fracture toughness

C. Ultimate tensile strength

D. Poisson’s ratio

Correct Answer: C. Ultimate tensile strength

Question 4

If n₁ = 20,000 cycles and N₁ = 100,000 cycles, what is the Miner damage contribution?

A. 0.02

B. 0.20

C. 5.0

D. 80,000

Correct Answer: B. 0.20

Question 5

Which expression represents the mode-I stress-intensity factor for a simple crack geometry?

A. K_I = Yσ√(πa)

B. K_I = σ/a

C. K_I = Eε²

D. K_I = ρgV

Correct Answer: A. K_I = Yσ√(πa)

Question 6

Unstable fracture is approached when the applied stress-intensity factor reaches the applicable value of:

A. Endurance limit

B. Yield strain

C. Fracture toughness

D. Creep exponent

Correct Answer: C. Fracture toughness

Question 7

Which creep stage generally has an approximately constant strain rate?

A. Primary

B. Secondary

C. Tertiary

D. Elastic recovery

Correct Answer: B. Secondary

Question 8

What is the usual effect of compressive residual stress at a free surface?

A. It generally delays surface crack initiation and growth

B. It always causes immediate creep rupture

C. It eliminates all stress concentrations

D. It increases tensile crack opening

Correct Answer: A. It generally delays surface crack initiation and growth

Question 9

The Paris law primarily describes which region of fatigue-crack growth?

A. Stable intermediate-growth region

B. Only crack nucleation before a crack exists

C. Only instantaneous overload fracture

D. Elastic deformation without cracking

Correct Answer: A. Stable intermediate-growth region

Question 10

Why is inspection reliability important in damage-tolerance design?

A. It determines whether assumed flaws can actually be detected

B. It eliminates the need for fracture toughness data

C. It guarantees Miner damage equals one

D. It prevents all residual stress

Correct Answer: A. It determines whether assumed flaws can actually be detected