Fatigue, Fracture and Creep Design

Introduction

Machine components rarely fail because of a single static load. Shafts, gears, connecting rods, springs, pressure vessels and aircraft structures commonly experience loads that vary with time. Repeated loading can cause fatigue failure at stresses substantially below the yield strength. A component may also fail because a small crack grows progressively, or because prolonged exposure to high temperature produces creep deformation and rupture.

Fatigue, fracture and creep design therefore require more than comparing one calculated stress with one yield strength. The designer must identify the stress history, material condition, surface quality, size, temperature, manufacturing defects, residual stresses and required service life. This chapter develops methods for finite-life and infinite-life fatigue design, mean-stress correction, cumulative damage, crack-growth assessment and high-temperature design.

Learning Objectives

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

1. Distinguish fluctuating, alternating and mean stresses.
2. Interpret S–N curves and determine finite- and infinite-life strength.
3. Apply endurance-limit modifiers and mean-stress criteria.
4. Design components using Goodman, Soderberg and Gerber relations.
5. Estimate cumulative fatigue damage using Miner’s rule.
6. Explain crack initiation, stress concentration and linear-elastic fracture mechanics.
7. Use the Paris crack-growth law within its range of validity.
8. Describe creep mechanisms, creep curves and rupture design.
9. Select appropriate safety factors and recognize limitations of simplified models.

1. Nature of Fatigue Failure

Fatigue is progressive, localized structural damage produced by cyclic or fluctuating stress. It usually occurs in three stages:

• Crack initiation: microscopic damage develops at persistent slip bands, inclusions, surface marks, keyways or weld toes.
• Stable crack propagation: a crack grows incrementally under repeated loading.
• Final fracture: the remaining uncracked section can no longer carry the applied load.

A fatigue fracture commonly has a relatively smooth region containing beach marks or striations and a rough final-fracture region. Unlike ductile overload failure, fatigue may occur with little visible plastic deformation.

A useful classification is:

For a uniaxial stress cycle,

where σ_m is mean stress, σ_a is alternating stress amplitude, and σ_max and σ_min are maximum and minimum stresses. The stress range is

The stress ratio is

These equations apply to a regular, uniaxial stress cycle. For multiaxial loading, the designer must define an equivalent stress or use a multiaxial fatigue criterion.

The load factor, often called the amplitude ratio, is

when σ_m is nonzero. A positive tensile mean stress generally reduces fatigue strength, whereas a compressive mean stress often improves it, although excessive compression can introduce buckling or contact problems.

2. Stress Concentration and Notch Effects

Geometric discontinuities such as holes, shoulders, threads and keyways amplify local elastic stress. The theoretical stress-concentration factor is

It is obtained from elasticity solutions, charts or finite-element analysis and does not itself include material notch sensitivity.

Fatigue behavior is represented more realistically by the fatigue stress-concentration factor,

where q is the notch-sensitivity factor, 0 ≤ q ≤ 1. Thus K_f approaches 1 for a notch-insensitive material and K_t for a highly notch-sensitive material. The relation is empirical and should be used with compatible material, notch-radius and size data.

For a component under nominal alternating and mean stresses, a first estimate is

and, for a sharp notch under elastic conditions,

Residual stresses, surface treatment and local plasticity can make these approximations inaccurate. Notches are especially important because fatigue cracks often initiate at the point of maximum local tensile stress.

3. S–N Behavior and Endurance Strength

An S–N curve relates stress amplitude S to the number of cycles to failure N under specified conditions. The conditions must include material, surface finish, specimen size, load type, stress ratio, temperature and failure criterion. Consequently, an S–N curve is not a universal material constant.

For many steels tested in air at moderate temperatures, the curve becomes approximately horizontal beyond a large number of cycles. The corresponding stress is called the endurance limit S_e. Nonferrous alloys such as aluminum commonly do not exhibit a sharp endurance limit; their fatigue strength must be specified at a selected life, such as 10⁷ cycles.

A common finite-life representation is the Basquin relation:

where S_a is alternating stress amplitude, N is cycles to failure, A is a fitted coefficient and b is a negative exponent. The equation is empirical and applies only over the data range used for fitting. Taking logarithms gives

which permits a straight-line fit on log–log coordinates.

For a component, the corrected endurance limit is often estimated by

where S′_e is the laboratory rotating-beam endurance limit and the factors represent surface finish, size, loading, temperature, reliability and miscellaneous effects, respectively. The exact factor definitions vary among design standards. These modifiers must not be mixed indiscriminately between sources.

For steels, a preliminary estimate sometimes used for polished specimens is S′_e ≈ 0.5 S_ut, where S_ut is ultimate tensile strength, subject to material and standard limitations. This is only an estimate; actual test data are preferred.

Worked Example 1: Corrected Endurance Limit

A steel has a polished rotating-beam endurance limit S′_e = 300 MPa. The selected factors are k_a = 0.85, k_b = 0.90, k_c = 1.0, k_d = 0.95, k_e = 0.897 and k_f = 1.0. Then

The product of the factors is approximately 0.653, so

This value represents an estimated component endurance limit, not a guarantee of infinite life. It is valid only for the assumed loading, environment, reliability and surface condition.

4. Mean-Stress Correction and Fatigue Criteria

Because a tensile mean stress accelerates crack opening and growth, fatigue strength depends on both σ_a and σ_m. Common design criteria are expressed in terms of alternating stress, mean stress, ultimate strength S_ut, yield strength S_y and endurance strength S_e.

Goodman criterion

This linear relation is widely used for machine design. It is generally conservative relative to the Gerber parabola in the tensile mean-stress region.

Soderberg criterion

Because yield strength is lower than ultimate strength, the Soderberg relation is more conservative. It is useful where yielding must be strongly avoided.

Gerber criterion

The Gerber relation often gives a better empirical fit for ductile metals under tensile mean stress, but it is less conservative and should not be extrapolated carelessly.

Here n is the fatigue factor of safety. These equations assume proportional loading, a known stress cycle, appropriate endurance data and a ductile material model. They do not independently account for corrosion, fretting, residual stress, creep, weld defects or complex multiaxial histories.

For completely reversed loading, σ_m = 0, and the relations reduce to σ_a ≤ S_e/n. For a tensile static load with σ_a = 0, the Goodman relation becomes σ_m ≤ S_ut/n, whereas the Soderberg relation becomes σ_m ≤ S_y/n.

Worked Example 2: Goodman Design Check

A shaft experiences nominal stresses σ_max = 180 MPa and σ_min = −60 MPa. The corrected endurance limit is S_e = 200 MPa and the ultimate strength is S_ut = 600 MPa. Determine the Goodman factor of safety.

First,

The Goodman usage ratio is

Since U = 1/n,

The design passes a Goodman requirement of n = 1.25, but the margin may be inadequate if stress estimates, surface condition or loading spectrum are uncertain.

5. Finite-Life and Infinite-Life Design

An infinite-life design requires the operating stress to remain below the corrected endurance strength after mean-stress correction and all relevant concentration factors are applied. “Infinite” means a specified engineering target, commonly 10⁶–10⁷ cycles or the life required by the application; it does not mean mathematically unlimited life.

Finite-life design is appropriate when a component is to be replaced after a known number of cycles, when the material lacks a true endurance limit, or when weight reduction justifies controlled damage. A design workflow is:

1. Establish the load spectrum and required life.
2. Convert loads to local stresses, including stress concentration and fatigue notch effects.
3. Separate each cycle into mean and alternating components.
4. Correct material data for surface, size, temperature, reliability and environment.
5. Apply a mean-stress relation or a strain-life method where plasticity is significant.
6. Determine the allowable life and compare it with the requirement.
7. Check static yielding, buckling, wear, bearing stresses and fracture.

The stress-life approach is most suitable for high-cycle fatigue, where nominal plastic strain is small. Low-cycle fatigue requires strain-life analysis, commonly represented by the Coffin–Manson–Basquin equation:

where ε_a is total strain amplitude, σ′_f is fatigue strength coefficient, E is Young’s modulus, N_f is reversals to failure expressed as cycles in the equation convention, b is fatigue-strength exponent, ε′_f is fatigue-ductility coefficient and c is fatigue-ductility exponent. The notation and definition of N_f must be kept consistent with the test data.

6. Variable-Amplitude Loading and Cumulative Damage

Real components experience several stress levels. If n_i cycles are applied at stress level i and the constant-amplitude life at that level is N_i, Miner’s linear damage rule estimates failure when

The quantity D is cumulative damage. In design, a value below 1 is required, often with additional margin. Miner’s rule assumes damage is linear, sequence effects are negligible and each N_i is obtained for the relevant stress ratio and environment. These assumptions can be poor when overloads cause crack retardation, when corrosion changes during service or when low-frequency high-amplitude cycles dominate.

Worked Example 3: Miner Damage

A component experiences 20,000 cycles at a stress level for which N₁ = 100,000 cycles and 100,000 cycles at another level for which N₂ = 1,000,000 cycles. The estimated damage is

If this block repeats unchanged, the predicted number of blocks to failure is approximately 1/D = 3.33 blocks. Since service occurs in whole blocks, failure would be expected during the fourth block under the model. This result is an estimate, not a precise failure prediction.

For measured random histories, cycle-counting methods such as rainflow counting convert the history into equivalent stress cycles. Mean-stress correction and S–N data are then applied to each counted cycle.

7. Fracture Mechanics and Crack Growth

Traditional fatigue design assumes that a component is initially crack-free. Fracture mechanics instead treats a crack as an explicit defect and evaluates whether it will remain stable or grow.

For a crack of characteristic size a under nominal tensile stress σ, the mode-I stress-intensity factor is

where Y is a dimensionless geometry factor. Linear-elastic fracture mechanics is applicable when the plastic zone near the crack tip is small compared with crack size, ligament dimensions and specimen thickness.

Fast fracture is avoided when

where K_IC is plane-strain fracture toughness. For thin sections or significant plasticity, other toughness measures and elastic–plastic fracture mechanics may be necessary.

Under cyclic loading,

The Paris law estimates stable crack-growth rate:

where da/dN is crack extension per cycle, C and m are material/environment constants, and ΔK is stress-intensity range. The Paris law generally applies to the intermediate, approximately linear region of a da/dN versus ΔK plot. It does not accurately describe threshold behavior at very low ΔK or rapid growth near fracture toughness.

For constant-amplitude loading and constant Y, integration gives

where a_i is initial crack size and a_f is the critical or inspection-limit size. In real geometries Y may vary with crack length, requiring numerical integration.

Crack-growth design supports damage-tolerant engineering: specify an allowable initial flaw, inspect at defined intervals, calculate the detectable crack size, and ensure that a crack cannot grow from detectable to critical between inspections. This approach is central to aircraft, pressure-vessel and power-plant safety.

8. Creep and High-Temperature Failure

Creep is time-dependent permanent deformation under sustained stress, generally important above approximately 0.3–0.4 times the absolute melting temperature for many metals. The exact threshold depends on alloy, stress, microstructure and environment.

A typical creep curve contains:

1. Primary creep: decreasing creep rate because of strain hardening.
2. Secondary creep: approximately constant minimum creep rate.
3. Tertiary creep: accelerating rate caused by necking, void formation, grain-boundary cracking or microstructural degradation.

A common empirical relation for steady-state creep is Norton’s law:

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 in kelvin. The equation is valid only over the temperature and stress range used to determine its constants.

Creep rupture data are often presented as stress versus rupture time at fixed temperature. The Larson–Miller parameter is one correlation used for extrapolation:

where P is the Larson–Miller parameter, T is absolute temperature, t_r is rupture time in hours and C is an empirical constant. Extrapolation beyond the tested range can be highly uncertain and must be supported by appropriate standards and data.

High-temperature design must consider:

• creep deformation and dimensional tolerances;
• creep rupture and stress relaxation;
• oxidation, corrosion and carburization;
• thermal gradients and thermal fatigue;
• material changes such as grain growth or precipitation coarsening;
• interaction between fatigue and creep.

A component may satisfy an elastic fatigue criterion at room temperature yet fail rapidly at operating temperature. Creep–fatigue interaction is especially important in turbines, boilers, exhaust manifolds and nuclear components. Hold periods at tensile strain can cause stress relaxation and damage by creep cavitation, while cycling causes fatigue cracks.

9. Applied Case: Fatigue Design of a Stepped Shaft

Background

Consider an illustrative steel transmission shaft with a shoulder fillet. The shaft is subjected to rotating bending and steady torque. The design life is at least 10⁷ bending cycles. The shaft has an ultimate strength of 700 MPa, yield strength of 500 MPa and an estimated corrected endurance limit of 210 MPa.

Problem

At the critical shoulder, the calculated nominal bending stress amplitude is 110 MPa and the nominal torsional stress is 80 MPa. The fatigue concentration factor for bending is K_f = 1.6. The torsional concentration factor is 1.3. Determine whether a simplified equivalent-stress Goodman check gives a satisfactory design factor of safety of 1.5.

Analysis and approach

For a rotating shaft, bending stress is approximately fully reversed, so its mean component is zero. The local bending amplitude is

The local torsional stress amplitude depends on whether torque is steady or fluctuating. Here torque is stated to be steady, so it produces a mean shear stress:

For a preliminary distortion-energy combination, use

and

With σ_m = 0 and τ_a = 0,

Applying Goodman with the required factor n = 1.5 gives the allowable usage condition

The left side is

whereas 1/n = 0.667. The design fails substantially.

Solution options

The engineer could increase the shoulder fillet radius, reduce the nominal stresses by increasing shaft diameter, improve surface finish, use compressive residual stress such as shot peening, select a higher-strength fatigue-resistant material, reduce torque, or redesign the load path. The fillet change must be checked for manufacturability and for its effect on bearing and assembly requirements.

Outcome and learning points

The example demonstrates that a shaft can fail an infinite-life criterion even when nominal stresses appear moderate. Concentration factors and steady torsion create a damaging mean equivalent stress. A final design should use a validated multiaxial fatigue criterion, check yielding and fracture, and verify the result through testing or an appropriate standard.

10. Design Practice, Standards and Limitations

Good fatigue and creep design begins with accurate service data. Designers should record load spectra, start–stop events, vibration, temperature cycles, corrosion exposure and maintenance procedures. Material certificates and heat-treatment records are important because inclusions, hardness and residual stresses strongly affect fatigue performance.

Finite-element analysis is useful for locating hot spots, but mesh refinement does not automatically produce a valid fatigue life. Singular stresses at idealized sharp corners require a notch or fracture-mechanics interpretation. Experimental correlation is essential for novel geometries.

Reliability design should distinguish variability in material strength, manufacturing defects, load uncertainty and model error. A single factor of safety may conceal these different uncertainties. Safety-critical parts require traceable requirements, independent review, inspection plans and compliance with relevant standards such as fatigue, pressure-vessel, aerospace or power-generation codes.

Ethically, a designer must not report a calculated life as a guarantee when the material data or operating spectrum are uncertain. Inspection intervals, retirement criteria and foreseeable misuse are part of responsible design. Digital monitoring and machine-learning models can assist with load identification and anomaly detection, but they require representative training data, physical validation and human engineering judgment. They should not replace fracture-critical inspection or established safety margins without evidence.

Chapter Summary

Fatigue is progressive damage under cyclic loading and is governed by mean stress, alternating stress, stress concentration, surface condition, size, temperature, environment and material microstructure. Stress amplitude and mean stress are obtained from the maximum and minimum values of a cycle. S–N curves support high-cycle fatigue design, while finite-life relations are used when a material has no practical endurance limit or when a specified replacement life is acceptable.

Goodman, Soderberg and Gerber relations provide different mean-stress corrections. Soderberg is generally the most conservative because it uses yield strength. Miner’s rule estimates cumulative damage under variable-amplitude loading but neglects sequence effects. Fracture mechanics treats cracks explicitly through stress intensity, fracture toughness and crack-growth laws such as the Paris relation. Creep is time-dependent deformation at elevated temperature; Norton’s law and rupture correlations are useful only within validated ranges. Safe professional design combines calculations with material data, inspection, testing, standards and appropriate uncertainty management.

Key Terms

Fatigue: Progressive damage caused by repeated or fluctuating stress.

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.

Endurance limit: Stress level associated with a specified very long fatigue life.

Finite-life design: Design for a prescribed, limited number of cycles.

Infinite-life design: Design below a specified endurance-strength criterion.

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

Fatigue notch factor: Effective stress concentration factor for fatigue.

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

Soderberg criterion: Conservative linear relation using yield strength.

Miner’s rule: Linear cumulative-damage hypothesis.

Fracture toughness: Resistance to unstable crack extension.

Stress-intensity factor: Crack-tip parameter describing elastic stress severity.

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

Creep: Time-dependent permanent deformation under sustained load.

Creep rupture: Failure after prolonged high-temperature loading.

Norton’s law: Power-law relation for steady-state creep rate.

Review Questions

Short Answer Questions

1. Define mean stress and alternating stress for a regular stress cycle.
2. Why does a tensile mean stress usually reduce fatigue life?
3. Distinguish theoretical stress concentration from fatigue stress concentration.
4. What are the three stages of a typical creep curve?
5. State the range of validity of the Paris crack-growth law.

Long Answer Questions

1. Explain the construction, interpretation and limitations of an S–N curve.
2. Compare Goodman, Soderberg and Gerber fatigue criteria, including their assumptions and relative conservatism.
3. Describe a complete engineering workflow for infinite-life shaft design.
4. Explain how fracture mechanics changes the approach to fatigue design.
5. Discuss creep deformation, creep rupture and creep–fatigue interaction in high-temperature components.

Analytical / Application Questions

1. A component has σ_max = 240 MPa, σ_min = −40 MPa, S_e = 180 MPa and S_ut = 600 MPa. Calculate σ_m, σ_a and the Goodman factor of safety.
2. A variable-amplitude history applies 30,000 cycles where N₁ = 120,000 and 200,000 cycles where N₂ = 2,000,000. Calculate Miner damage and estimate the number of identical blocks to failure.
3. A crack grows according to the Paris law in a plate with a geometry factor that changes with crack length. Explain why numerical integration is required and identify the information needed to determine inspection intervals.

Multiple Choice Questions with Answers

Question 1

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

A. 60 MPa

B. 120 MPa

C. 180 MPa

D. 240 MPa

Correct Answer: B. 120 MPa

Question 2

Which mean-stress criterion uses yield strength in its linear denominator?

A. Goodman

B. Gerber

C. Soderberg

D. Paris

Correct Answer: C. Soderberg

Question 3

If n_i = 20,000 cycles and N_i = 100,000 cycles, the Miner damage contribution is:

A. 0.02

B. 0.20

C. 5.0

D. 80,000

Correct Answer: B. 0.20

Question 4

The theoretical stress-concentration factor is defined as:

A. Nominal stress divided by yield strength

B. Local peak elastic stress divided by nominal stress

C. Ultimate strength divided by endurance strength

D. Alternating stress divided by mean stress

Correct Answer: B. Local peak elastic stress divided by nominal stress

Question 5

In the Paris law, da/dN represents:

A. Stress range per cycle

B. Crack extension per cycle

C. Mean stress per cycle

D. Cycles per unit stress

Correct Answer: B. Crack extension per cycle

Question 6

Which creep stage generally has an approximately constant creep rate?

A. Primary

B. Secondary

C. Tertiary

D. Fracture initiation

Correct Answer: B. Secondary

Question 7

For a completely reversed stress cycle, the mean stress is:

A. Equal to the maximum stress

B. Equal to the minimum stress

C. Zero

D. Infinite

Correct Answer: C. Zero

Question 8

The fracture condition in linear-elastic fracture mechanics is approached when:

A. K_I approaches K_IC

B. S_e approaches zero only

C. R approaches one

D. D approaches zero

Correct Answer: A. K_I approaches K_IC

Question 9

Norton’s creep equation includes which temperature dependence?

A. A logarithmic dependence only

B. An exponential term involving activation energy and absolute temperature

C. No temperature dependence

D. A term involving stress ratio R only

Correct Answer: B. An exponential term involving activation energy and absolute temperature

Question 10

A Goodman usage ratio of 0.70 corresponds to a Goodman factor of safety of approximately:

A. 0.70

B. 1.00

C. 1.43

D. 2.70

Correct Answer: C. 1.43