Fatigue, Fracture, Wear and Surface Failure

Introduction

Machine components rarely fail because a single static load exceeds the yield strength. More often, failure develops gradually under repeated loading, crack growth, contact stresses, temperature, lubrication failure, or a combination of these conditions. Shafts, gears, springs, bearings, pressure vessels, connecting rods, and welded structures may survive a high one-time load yet fail after millions of smaller load cycles.

This chapter develops methods for evaluating such failures. It connects nominal stress analysis with fatigue data, mean-stress effects, multiaxial loading, cumulative damage, notch sensitivity, fracture mechanics, creep, wear, pitting, and surface-improvement methods. The designer’s objective is not simply to select a material with a high strength; it is to control stresses, defects, environments, surface condition, inspection, and expected service life.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Distinguish fatigue, fracture, creep, wear, and surface-contact failures.
2. Determine stress amplitude, mean stress, and stress ratio for fluctuating loading.
3. Use S–N data and endurance-limit concepts in preliminary fatigue design.
4. Apply mean-stress corrections, notch factors, cumulative-damage rules, and basic multiaxial criteria.
5. Explain crack initiation and propagation using linear elastic fracture mechanics.
6. Identify creep mechanisms and wear modes, including adhesive, abrasive, erosive, fretting, and contact-fatigue wear.
7. Select practical methods for improving fatigue and surface-fatigue life.
8. Diagnose a component failure systematically and recognize the limitations of simplified models.

1. Failure as a Time-Dependent Process

A useful distinction is between strength failure and life failure. Strength failure occurs when the component cannot sustain the present load. Life failure occurs when damage accumulates until the component becomes unsafe, even though the applied stress may remain below the yield strength.

The principal mechanisms considered here are:

• Fatigue: progressive damage caused by repeated or fluctuating stress.
• Fracture: separation caused by crack initiation and crack growth.
• Creep: time-dependent deformation, usually important at elevated temperature.
• Wear: progressive material loss or surface damage caused by relative motion.
• Surface fatigue: repeated contact stress producing pitting, spalling, or cracking.

These mechanisms can interact. A notch may initiate a fatigue crack; corrosion may accelerate crack growth; wear may remove a protective layer; and a surface pit may act as a crack-like stress raiser.

A sound investigation therefore asks five questions: What is the loading history? Where is the highest local stress? What material and surface condition are present? What environment and lubrication apply? How will failure be detected before loss of function?

2. Fluctuating Stress and Fatigue Terminology

For a periodic normal stress with maximum value σ_max and minimum value σ_min, the mean stress, alternating stress, and stress ratio are defined by:

Here σ_m is the mean stress, σ_a is the stress amplitude, and R is dimensionless. Stresses must be expressed in the same units, such as MPa. These definitions apply to a constant-amplitude cycle. For a fully reversed cycle, σ_max = −σ_min, so σ_m = 0 and R = −1. For a repeated cycle in which σ_min = 0, R = 0 and σ_a = σ_m = σ_max ÷ 2.

The nominal stress is calculated from idealized geometry, whereas the local stress includes geometric discontinuities, surface condition, residual stress, and material behavior. Fatigue cracks commonly begin at holes, keyways, threads, shoulders, weld toes, inclusions, machining marks, and corrosion pits.

Worked Example 1: Cycle quantities

A shaft surface experiences σ_max = 180 MPa in tension and σ_min = −60 MPa in compression. Then:

The cycle has a tensile mean stress and a substantial alternating component. The tensile mean stress is generally harmful because it reduces crack-closing during part of the cycle.

3. S–N Curves and Endurance Limits

An S–N curve relates stress amplitude S to the number of cycles to failure N, usually on a logarithmic cycle scale. It is generated by testing several specimens at different stress amplitudes under specified surface, size, temperature, loading, and reliability conditions. The curve is not a universal material property unless those conditions are stated.

For many steels tested in laboratory air under completely reversed bending, the curve approaches an endurance limit. An idealized model is:

where S_e is the corrected endurance strength. “Infinite” means a specified design life, often much greater than the intended service life; it does not guarantee that failure is impossible. Aluminum alloys and many nonferrous alloys generally show a continuously declining S–N curve rather than a sharply defined endurance limit.

A commonly used preliminary relation for a corrected endurance strength is:

S'_e is the laboratory specimen endurance strength. The factors represent, as applicable, surface finish k_a, size k_b, load type k_c, temperature k_d, reliability k_e, and miscellaneous effects k_f. These factors are empirical and must be taken from an appropriate design standard or material source. They should not be transferred between materials or test conditions without justification.

An S–N design is principally a stress-life method. It is efficient for high-cycle fatigue, where nominal elastic behavior is expected. It is less reliable for low-cycle fatigue, severe plastic strain, weld defects, short cracks, corrosion fatigue, and components with large manufacturing variability.

4. Mean-Stress Corrections

S–N data are often reported for fully reversed loading, while machine components frequently experience nonzero mean stress. A common linear correction is the Goodman relation:

S_e is the endurance strength for the relevant fully reversed condition, S_ut is ultimate tensile strength, and n is the fatigue factor of safety. The relation is conservative for many ductile materials but is an approximation, not a law of fracture mechanics.

The Gerber parabola is:

It is often less conservative in the tensile mean-stress region. The Soderberg relation uses yield strength S_y instead of ultimate strength:

Soderberg is usually more conservative because yielding is a limiting condition. These relations are most appropriate for ductile metals under proportional, nominally elastic loading. Compressive mean stress may improve fatigue resistance, but blindly extrapolating a correction into large compressive stresses is unsafe. Local yielding, residual stress relaxation, surface effects, and crack closure must be considered.

Worked Example 2: Goodman assessment

A steel component has S_e = 240 MPa, S_ut = 720 MPa, σ_a = 120 MPa, and σ_m = 180 MPa. Using Goodman:

The Goodman fatigue factor of safety is:

This is a preliminary result. The designer must still check yielding, local notch stress, surface finish, size, temperature, residual stress, and required reliability.

5. Notches, Stress Concentration, and Surface Condition

A geometric discontinuity raises local stress. The theoretical elastic stress-concentration factor is:

For fatigue, the relevant factor is the fatigue stress-concentration factor K_f:

where q is notch sensitivity, 0 ≤ q ≤ 1. A sharp notch in a highly notch-sensitive material has q close to 1; a blunt notch or notch-insensitive material has a smaller q. Some empirical expressions estimate q from a material length constant a and notch radius ρ, but such expressions are material- and unit-dependent and should only be used with validated data.

The alternating and mean stress concentration effects need not be identical. In elastic design, K_f is often applied to alternating stress, while a separate treatment is used for mean stress. If the notch causes local yielding, elastic concentration factors can overpredict or misrepresent the actual stabilized stress; elastic–plastic methods are then required.

Important fatigue modifiers include:

• surface roughness and machining direction;
• component size and stressed volume;
• residual tensile or compressive stress;
• decarburization, inclusions, and heat-treatment defects;
• corrosion, fretting, and temperature;
• manufacturing scatter and inspection quality.

A generous fillet, polished surface, correct thread relief, rolled thread, and avoidance of abrupt section changes are often more effective than simply increasing nominal diameter.

6. Cumulative Damage and Variable-Amplitude Loading

Real components experience changing loads. The linear Palmgren–Miner rule estimates cumulative damage as:

n_i is the number of applied cycles in stress block i, and N_i is the number of cycles to failure if block i acted alone at that stress. Failure is conventionally predicted when D ≈ 1.

Worked Example 3: Miner damage

A component experiences 20,000 cycles at a stress level for which N₁ = 100,000 cycles, and 60,000 cycles at another level for which N₂ = 300,000 cycles. Thus:

The estimated remaining damage capacity is 0.60, subject to the assumptions and uncertainty of the rule.

Miner’s rule is convenient but limited. It neglects load-sequence effects, crack retardation or acceleration, mean-stress changes, overload plasticity, environmental changes, and scatter. For critical components, rainflow cycle counting should be used to convert measured irregular histories into fatigue cycles, followed by an appropriate damage model. Fracture-mechanics crack-growth integration may be preferable when an initial flaw and inspection plan are known.

7. Multiaxial Fatigue

Machine components often experience bending, torsion, axial force, and contact loading simultaneously. The stress state may be represented by normal stresses σ_x, σ_y, σ_z and shear stresses τ_xy, τ_yz, τ_zx. A simple scalar equivalent-stress approach is useful for some proportional loading cases.

For static or slowly varying ductile yielding, the von Mises equivalent stress is:

σ_1, σ_2, and σ_3 are principal stresses. For fatigue, however, the amplitude and mean components of the stress tensor must be separated. Criteria such as maximum shear strain, critical-plane methods, and equivalent-stress methods may be used depending on material and loading history.

For a circular shaft under bending moment M and torque T, the nominal surface stresses are:

where d is shaft diameter. These equations assume a solid circular shaft, elastic behavior, and stresses evaluated away from discontinuities. If bending and torsion are in phase and proportional, a von Mises-type alternating combination may be written:

A corresponding mean combination may be used cautiously. Nonproportional loading, phase differences, axial stress, local plasticity, and critical-plane crack growth can make this simplification inaccurate.

8. Fracture Mechanics

Fatigue design based only on nominal stress may miss the effect of an existing crack, weld flaw, inclusion, or corrosion pit. Linear elastic fracture mechanics describes the intensity of the near-tip stress field through the stress-intensity factor:

K is the stress-intensity factor, Y is a dimensionless geometry factor, σ is the applied nominal stress, and a is a crack-size parameter. Units of K are commonly MPa√m when σ is in MPa and a is in metres. The equation assumes predominantly linear elastic behavior, a crack sufficiently large compared with microstructural dimensions, and a geometry for which the selected Y is valid.

For Mode I opening, unstable fracture is expected when:

K_IC is plane-strain fracture toughness. The plane-strain condition requires sufficient thickness and constraint; otherwise a geometry-dependent fracture toughness may be appropriate. The plastic-zone size must also be small relative to crack and specimen dimensions for linear elastic assumptions to be valid.

Fatigue crack growth is often correlated by the Paris law:

where da ÷ dN is crack growth per cycle, C and m are experimentally determined constants, and ∆K = K_max − K_min. The Paris region excludes the near-threshold region and the rapid-growth region near fracture toughness. Mean stress, crack closure, environment, load sequence, and crack geometry affect the constants and applicability.

A damage-tolerant design may specify an initial detectable crack size, inspect at intervals, estimate crack growth, and require that the crack remain below a critical size. This approach complements, rather than replaces, good stress design and manufacturing control.

9. Creep and Elevated-Temperature Damage

Creep is progressive deformation under sustained stress. It becomes important when the homologous temperature, T_abs ÷ T_m, is sufficiently high, where T_abs is absolute temperature and T_m is the material melting temperature in kelvin. The threshold depends on material, stress, time, and required accuracy.

A commonly used empirical steady-state relation is Norton’s law:

ε̇_s is steady-state creep strain rate, A is a temperature-dependent material constant, σ is applied stress, and n is the stress exponent. For a wider temperature range, an Arrhenius factor may be included, but constants must be obtained experimentally for the material and temperature range.

The three conventional creep stages are primary creep, in which the rate decreases; secondary creep, with an approximately steady rate; and tertiary creep, in which damage and necking accelerate deformation. Creep rupture, oxidation, thermal fatigue, and creep–fatigue interaction may control high-temperature component life. Time–temperature parameters can correlate data, but extrapolation beyond the tested range is hazardous.

Design responses include reducing stress, selecting creep-resistant alloys, controlling temperature, using thermal barriers or cooling, allowing expansion, and specifying inspection for distortion and cracking.

10. Wear and Surface Failure

Wear is not a single mechanism. Correct diagnosis is essential because a remedy for abrasive wear may be ineffective or harmful against fatigue pitting.

Adhesive wear

Local asperities weld together and then break during sliding. It is promoted by high load, poor lubrication, incompatible materials, and inadequate hardness. Control methods include lubrication, surface hardening, compatible material pairs, and reduced contact stress.

Abrasive wear

A hard asperity or particle cuts or ploughs a softer surface. Two-body abrasion involves a hard counterface; three-body abrasion involves loose particles. Filtration, seals, hardness, and wear-resistant coatings are important controls.

Erosive wear

Particles or droplets carried by a fluid strike a surface. Velocity, impact angle, particle size, and material toughness influence the rate. Flow redesign, protective liners, and erosion-resistant materials may be required.

Fretting wear

Small-amplitude oscillatory motion at a nominal contact produces debris, oxidation, and sometimes fretting fatigue. It is common in fitted joints, splines, bolted assemblies, and bearing seats. Increasing contact pressure appropriately, eliminating relative motion, improving surface treatment, and controlling contamination can help.

Corrosive and tribochemical wear

Chemical reaction weakens a surface and removes reaction products during motion. Material selection, coatings, inhibitors, sealing, and environmental control are necessary.

A basic adhesive-wear estimate is given by Archard’s relation:

V is wear volume, W is normal load, L is sliding distance, H is characteristic hardness of the softer material, and k is a dimensionless wear coefficient determined experimentally. The relation assumes a defined wear regime and is not a universal law. It does not directly predict transitions between mild and severe wear, nor does it represent abrasive particles, lubrication changes, or surface fatigue without appropriate modification.

11. Contact Stress, Pitting, and Surface Fatigue

Gears, rolling bearings, cams, and wheels transmit load through small contact areas. Hertzian contact theory estimates subsurface stresses for idealized elastic bodies. Repeated rolling or rolling–sliding contact can initiate subsurface or near-surface cracks, leading to micropitting, pitting, spalling, or destructive scuffing.

• Micropitting consists of very small gray areas and shallow material loss, often associated with roughness, lubricant film insufficiency, and high local stress.
• Pitting produces discrete cavities caused by fatigue cracks and material removal.
• Spalling is larger-scale detachment of material from a fatigued surface.
• Scuffing is severe adhesive damage associated with lubricant-film breakdown and high temperature.

The ratio of lubricant film thickness to composite surface roughness is often used to assess film separation. Since the value depends on lubricant viscosity, speed, load, temperature, roughness, and geometry, it should be calculated using the relevant bearing or gear standard rather than a generic value.

Surface-fatigue life is improved by correct geometry, alignment, clean lubricant, adequate film thickness, suitable hardness and case depth, controlled roughness, and avoidance of overloads and contamination.

12. Fatigue-Life Improvement

Fatigue improvement should follow the hierarchy of reducing the damaging stress, improving the material and surface, and controlling service conditions.

1. Geometric improvement: increase fillet radius, remove sharp grooves, improve thread run-outs, align load paths, and reduce eccentricity.
2. Surface improvement: polishing, superfinishing, roller burnishing, deep rolling, shot peening, and laser peening can introduce compressive residual stress and reduce crack initiation.
3. Thermal and chemical treatment: carburizing, nitriding, induction hardening, and carbonitriding create hard, wear-resistant cases. Case depth and transition properties must match the stress field.
4. Manufacturing control: avoid grinding burns, laps, seams, decarburization, weld undercut, and poor surface finish. Use nondestructive inspection where appropriate.
5. Environmental control: prevent corrosion, control temperature, filter lubricants, and avoid fretting conditions.
6. Load management: reduce shock, resonance, overload, and unnecessary speed variation.

Compressive residual stress is beneficial only within the relevant depth and stability range. It may relax under high temperature or plastic deformation. Shot peening can increase roughness and may damage a thin or brittle surface if poorly controlled. Every treatment requires process specifications and verification.

13. Applied Case: Illustrative Gearbox Shaft Failure

Background

A steel gearbox shaft in an industrial conveyor operates continuously. It carries a fluctuating bending load from a helical gear and a steady transmitted torque. After eight months, a crack is found near the keyway.

Problem

The original design used nominal bending stress below the corrected endurance strength. The shaft nevertheless failed earlier than expected.

Analysis

Inspection shows a sharp keyway corner, longitudinal machining marks, and fretting corrosion beneath the gear hub. The measured load history includes start-up torque peaks. The nominal stress calculation did not include the keyway fatigue concentration, residual tensile stress from machining, or the transient start-up cycles. Fractography shows a relatively smooth beach-marked region followed by a rough final-fracture region, consistent with fatigue initiation and subsequent overload.

The design team calculates σ_m and σ_a from the measured bending cycle, applies an appropriate fatigue concentration factor, and checks the mean-stress effect using a Goodman diagram. The start-up history is converted into cycles and included in a preliminary Miner calculation. A crack-tolerance check uses K = Yσ√(πa) with the inspection-detectable crack size.

Solution

The keyway is redesigned with a larger radius and improved relief. The gear hub fit and surface condition are revised to reduce fretting. The shaft is manufactured with controlled finish and inspected for grinding damage. A modest increase in diameter reduces bending and torsional stress because shaft stress varies with 1 ÷ d³ for the stated solid-shaft equations. The surface is locally treated to produce suitable compressive residual stress, and start-up control reduces torque peaks.

Outcome

The revised design meets the required fatigue factor of safety under normal operation and has a defined inspection interval for crack detection. The result is not attributed to material strength alone; it follows from integrating loading, geometry, manufacturing, environment, and inspection.

Learning Points

• Nominal stress can conceal a severe local fatigue condition.
• Service transients may dominate cumulative damage.
• Fretting and surface defects can initiate cracks at stresses below the nominal endurance limit.
• A credible solution combines redesign, process control, monitoring, and validation testing.

14. Limitations, Standards, and Professional Practice

Fatigue data contain scatter. A design value must identify material heat treatment, specimen orientation, surface finish, size, loading mode, temperature, environment, reliability, and statistical basis. Safety factors should reflect consequence of failure, uncertainty, inspectability, and load knowledge—not habit alone.

Relevant design work should use recognized standards and validated databases, such as standards for fatigue testing, gear rating, bearing life, weld quality, fracture assessment, and nondestructive examination. The exact standard depends on the industry and jurisdiction. Engineers must maintain traceability of material certificates, heat-treatment records, inspection results, calibration, and design assumptions.

Computer-aided finite-element analysis is valuable for locating stress concentrations and contact regions, but mesh refinement, boundary conditions, nonlinear contact, residual stress, and material models require engineering judgment. AI and machine-learning systems can assist with anomaly detection, predictive maintenance, image-based fracture classification, and load-history processing. They should not replace validated mechanics, calibrated sensors, qualified inspection personnel, or accountable engineering approval. Training-data bias and false negatives are serious safety risks.

Chapter Summary

Fatigue is progressive damage caused by cyclic stress and is commonly evaluated using stress amplitude, mean stress, stress ratio, and S–N data. Endurance strength must be corrected for surface, size, loading, temperature, reliability, and other conditions. Goodman, Gerber, and Soderberg relations provide approximate mean-stress corrections with different conservatism.

Notches, keyways, threads, surface roughness, residual stress, corrosion, and fretting strongly affect fatigue life. Variable-amplitude loading can be estimated with Miner’s rule, while rainflow counting and fracture-mechanics methods provide more realistic analysis for complex histories and detectable cracks. Multiaxial fatigue requires consideration of normal and shear stress amplitudes, loading phase, and critical planes.

Fracture mechanics uses the stress-intensity factor K = Yσ√(πa), fracture toughness, and crack-growth laws to assess crack tolerance. Creep controls deformation and rupture at elevated temperature. Wear includes adhesive, abrasive, erosive, fretting, corrosive, and surface-fatigue mechanisms. Archard’s relation offers only a regime-specific estimate of wear volume. Pitting, spalling, micropitting, and scuffing are controlled through contact-stress reduction, alignment, hardness, lubricant quality, cleanliness, and surface engineering.

Reliable design integrates mechanics, manufacturing, environment, inspection, standards, and consequence-based safety decisions.

Key Terms

Fatigue: Progressive damage 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: Minimum stress divided by maximum stress.

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

Endurance strength: Stress associated with a specified long fatigue life.

Goodman relation: Linear mean-stress fatigue criterion using ultimate strength.

Notch sensitivity: Material response to a geometric stress raiser.

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

Miner’s rule: Linear cumulative-damage approximation.

Multiaxial fatigue: Fatigue under combined normal and shear stresses.

Fracture toughness: Resistance to unstable crack extension.

Stress-intensity factor: Measure of crack-tip loading severity.

Paris law: Empirical relation for fatigue crack-growth rate.

Creep: Time-dependent deformation under sustained stress.

Adhesive wear: Wear caused by local bonding and separation of sliding surfaces.

Abrasive wear: Material removal by hard asperities or particles.

Fretting: Damage caused by small-amplitude oscillatory motion.

Pitting: Local surface cavities caused by contact fatigue.

Shot peening: Surface treatment that induces beneficial compressive residual stress.

Review Questions

Short Answer Questions

1. Define stress amplitude, mean stress, and stress ratio for a fluctuating cycle.
2. Why is an endurance limit not automatically a guarantee of infinite service life?
3. Distinguish theoretical stress concentration factor K_t from fatigue factor K_f.
4. State two assumptions and two limitations of Miner’s rule.
5. What physical meaning does the stress-intensity factor K have?

Long Answer Questions

1. Explain the construction, use, and limitations of an S–N curve for machine-component design.
2. Compare Goodman, Gerber, and Soderberg mean-stress criteria and discuss when each may be appropriate.
3. Explain how geometry, surface finish, residual stress, manufacturing, and environment influence fatigue life.
4. Describe the principal mechanisms of wear and recommend a control strategy for each.
5. Explain the stages of creep and discuss design responses for high-temperature components.

Analytical / Application Questions

1. A component has σ_max = 240 MPa and σ_min = −80 MPa. Calculate σ_m, σ_a, and R. Interpret the effect of the mean stress.
2. A steel part has S_e = 260 MPa, S_ut = 780 MPa, σ_a = 130 MPa, and σ_m = 195 MPa. Use the Goodman relation to calculate the fatigue factor of safety and state the assumptions behind the result.
3. A shaft has a measured variable-amplitude history and a visible surface pit. Propose a failure-analysis plan integrating stress concentration, rainflow counting, fracture mechanics, inspection, and surface improvement.

Multiple Choice Questions with Answers

Question 1

For σ_max = 180 MPa and σ_min = −60 MPa, what is the stress amplitude?

A. 60 MPa

B. 120 MPa

C. 180 MPa

D. 240 MPa

Correct Answer: B. 120 MPa

Question 2

A fully reversed stress cycle has which stress ratio R?

A. −1

B. 0

C. 0.5

D. 1

Correct Answer: A. −1

Question 3

Which mean-stress relation uses yield strength and is generally the most conservative of the three listed criteria?

A. Gerber

B. Goodman

C. Soderberg

D. Paris

Correct Answer: C. Soderberg

Question 4

If n₁ = 20,000 and N₁ = 100,000, what damage fraction is assigned to that stress block in Miner’s rule?

A. 0.02

B. 0.20

C. 0.50

D. 5.00

Correct Answer: B. 0.20

Question 5

In K = Yσ√(πa), which variable represents crack size?

A. Y

B. σ

C. π

D. a

Correct Answer: D. a

Question 6

The Paris law primarily describes which phenomenon?

A. Steady-state creep deformation

B. Fatigue crack growth

C. Elastic contact deformation

D. Abrasive particle impact

Correct Answer: B. Fatigue crack growth

Question 7

Which wear mechanism is most closely associated with small-amplitude oscillatory motion?

A. Fretting wear

B. Erosive wear

C. Three-body abrasion

D. Scuffing only

Correct Answer: A. Fretting wear

Question 8

In Archard’s relation, increasing the hardness H while other variables remain constant generally causes predicted wear volume to:

A. Increase

B. Decrease

C. Remain exactly unchanged

D. Become negative

Correct Answer: B. Decrease

Question 9

Which treatment is intended to introduce compressive residual stress at a component surface?

A. Shot peening

B. Decarburization

C. Grinding burn

D. Overheating

Correct Answer: A. Shot peening

Question 10

Which description best characterizes secondary creep?

A. A decreasing creep rate

B. An approximately steady creep rate

C. A rapidly accelerating crack-free rate

D. Instantaneous elastic strain only

Correct Answer: B. An approximately steady creep rate