5. Failure, Fatigue, Fracture and Reliability

Introduction and Context

Mechanical systems rarely fail because of a single abstract “strength” deficiency. Failure is usually the result of an interaction among loading, geometry, material, manufacturing quality, environment, maintenance, and uncertainty. A component may have adequate static strength yet fail after millions of load cycles; a vessel may withstand its design pressure but fracture from a small crack; a bearing may fail through wear rather than yielding. Consequently, advanced mechanical design requires more than calculating nominal stress. It requires identifying credible failure modes, predicting their progression, managing uncertainty, and designing for acceptable reliability.

This chapter develops an integrated approach to failure, fatigue, fracture, time-dependent degradation, and reliability. Static failure theories provide initial design boundaries. Fatigue models address repeated loading. Fracture mechanics explains the significance of cracks and defects. Creep, wear, and corrosion describe important degradation mechanisms. Reliability methods convert uncertain component behaviour into quantitative probabilities, while FMEA and fault-tree analysis connect technical failure modes with system-level decisions.

The central design principle is that safety is not achieved merely by applying a large factor of safety. It is achieved by understanding how failure initiates, how it propagates, how it can be detected, and how its consequences can be controlled.

Learning Objectives

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

1. Distinguish static failure, fatigue failure, fracture, creep, wear, and corrosion mechanisms.

2. Select and apply suitable failure theories for ductile and brittle materials.

3. Construct fatigue-life estimates using stress-life, strain-life, and crack-growth approaches.

4. Evaluate cumulative damage under variable-amplitude loading.

5. Interpret stress-intensity factors, fracture toughness, and critical crack size.

6. Apply basic creep and wear relationships to design decisions.

7. Use reliability distributions and probabilistic design methods.

8. Perform FMEA and fault-tree analysis for mechanical systems.

9. Recognize limitations, ethical responsibilities, and uncertainty in failure prediction.

Failure as a Design Process

Failure is the inability of a component or system to perform its intended function within specified limits. Functional failure may involve complete fracture, excessive deformation, leakage, loss of accuracy, unacceptable vibration, excessive wear, or inability to transmit the required load. The same physical mechanism can therefore have different meanings in different applications. A small gear tooth crack may be tolerable in a low-speed instrument but unacceptable in an aircraft transmission.

Failure analysis begins with four questions: What function must be maintained? What loads and environments are present? By what mechanism could the function be lost? What evidence would reveal the onset of failure? These questions encourage designers to consider both limit states and detectability.

Important terminology includes:

• Failure mode: The physical manner in which a component ceases to meet its function.
• Failure cause: The condition initiating or accelerating the failure mode.
• Failure effect: The local, subsystem, or system consequence.
• Limit state: A boundary separating acceptable from unacceptable performance.
• Factor of safety: The ratio between a limiting capacity and the corresponding design demand.
• Reliability: The probability that an item performs its required function for a stated time and condition.
• Maintainability: The probability that a failed item can be restored within a specified time and procedure.

A deterministic design uses nominal values and prescribed safety factors. A probabilistic design treats loads, dimensions, material properties, and defects as random variables. The latter is essential when scatter is large or failure consequences are severe.

Static Failure Theories

For uniaxial loading, yielding in a ductile material is often estimated by comparing the applied stress with the yield strength. Multiaxial stress states require a failure criterion.

The maximum principal stress theory predicts failure when the largest principal stress reaches the tensile or compressive strength. It is conservative for brittle materials when tensile and compressive strengths differ significantly. Its general condition is

σ1 ≥ S_ut or |σ3| ≥ S_uc,

where σ1 and σ3 are extreme principal stresses and S_ut and S_uc are tensile and compressive strengths.

The maximum shear-stress theory, also called Tresca theory, predicts yielding when the maximum shear stress equals the shear yield strength. For principal stresses,

τ_max = (σ_max − σ_min)/2.

For uniaxial yielding, this gives a yield condition equivalent to a maximum shear stress of approximately S_y/2. Tresca is relatively conservative and useful for ductile metals.

The distortion-energy theory, or von Mises theory, predicts yielding when the distortion energy reaches the value at uniaxial yield. The equivalent stress is

σ_e = [((σ1−σ2)^2 +(σ2−σ3)^2 +(σ3−σ1)^2)/2]^{1/2}.

Yielding is predicted when σ_e ≥ S_y. This theory generally correlates well with ductile-metal experiments and is widely used in finite-element design.

The maximum-strain theory compares principal strains with the strain at uniaxial yield. It is less commonly preferred for general metal design because its predictions are sensitive to Poisson’s ratio and may not represent yielding as well as distortion energy.

No criterion is universally correct. Ductile yielding and brittle fracture are different physical events, and a material’s microstructure, temperature, stress state, and loading rate influence the appropriate model. A sound design combines a suitable theory with material data and validation testing.

Fatigue Under Cyclic Loading

Fatigue is progressive damage caused by repeated or fluctuating stress, often at stress levels below the static yield strength. Fatigue life is commonly divided into crack initiation and crack propagation, although the distinction is not always physically sharp. Surface finish, residual stress, notches, size, corrosion, temperature, and manufacturing defects strongly influence life.

A cyclic stress is described by the alternating and mean components:

σ_a = (σ_max − σ_min)/2, σ_m = (σ_max + σ_min)/2.

The stress ratio is R = σ_min/σ_max. Tensile mean stress generally reduces fatigue strength, while compressive mean stress may improve it, although residual stresses and surface damage complicate this interpretation.

The stress-life method is appropriate when the component experiences predominantly elastic deformation and a long-life regime. A common representation of the S–N relationship is Basquin’s equation,

σ_a = σ'_f(2N_f)^b,

where σ'_f and b are material constants and N_f is the number of cycles to failure. Mean-stress corrections such as Goodman, Gerber, or Soderberg modify the allowable alternating stress. The Goodman relation is commonly written as

σ_a/S_e + σ_m/S_ut ≤ 1,

where S_e is the endurance strength and S_ut is ultimate tensile strength. Soderberg uses yield strength instead of ultimate strength and is more conservative.

The strain-life method is preferable when local plasticity occurs, particularly near notches or in low-cycle fatigue. The Coffin–Manson–Basquin relationship is

ε_a = (σ'_f/E)(2N_f)^b + ε'_f(2N_f)^c,

where the first term represents elastic strain amplitude and the second plastic strain amplitude. Notch-plasticity corrections, such as Neuber-type approaches, may be used to estimate local stresses and strains.

The stress-concentration factor, K_t, describes the elastic amplification of nominal stress caused by geometry. The fatigue notch factor, K_f, is often smaller than K_t because of material notch sensitivity. Ignoring this distinction can either overestimate or underestimate fatigue life, depending on the analysis method.

For variable-amplitude loading, the linear Palmgren–Miner rule estimates cumulative damage as

D = Σ(n_i/N_i).

Here n_i is the number of applied cycles at stress level i and N_i is the constant-amplitude life at that level. Failure is conventionally assumed near D = 1. The rule is useful for preliminary design but does not account well for load sequence, overload retardation, corrosion interaction, or changing material condition. Rainflow counting is commonly used to convert a measured load history into fatigue cycles before applying a damage model.

Fracture Mechanics and Crack Growth

Traditional stress analysis treats a component as geometrically continuous. Fracture mechanics recognizes that real components contain cracks, inclusions, pores, machining marks, or weld defects. The central quantity in linear-elastic fracture mechanics is the stress-intensity factor,

K_I = Yσ√(πa),

where Y is a geometry factor, σ is nominal stress, and a is crack size. Mode I loading opens the crack; Modes II and III produce sliding and tearing. Unstable fracture is expected when K_I reaches the fracture toughness K_IC under appropriate plane-strain conditions.

A design check may therefore be written as

Yσ√(πa) < K_IC.

This inequality shows why a component with moderate nominal stress can be unsafe if its crack is sufficiently large. Conversely, a high-strength material may have reduced fracture toughness, creating a trade-off between strength and damage tolerance.

Subcritical fatigue crack growth is frequently represented by the Paris law,

da/dN = C(ΔK)^m,

where ΔK is the stress-intensity range and C and m are experimentally determined constants. The Paris regime excludes the near-threshold region and the rapid-growth region close to fracture toughness. More complete models include crack-closure effects, stress ratio, threshold behaviour, and final instability.

Damage-tolerant design specifies inspection intervals such that a crack remains detectable and below the critical size. The analysis requires an initial flaw distribution, crack-growth law, inspection capability, critical crack size, and consequence assessment. Proof testing can reveal some critical defects, but it may also introduce damage and cannot replace a complete inspection and maintenance strategy.

Creep, Wear, and Corrosion

Creep is time-dependent deformation under sustained stress, especially at elevated temperature. It is often divided into primary, secondary, and tertiary stages. Primary creep has a decreasing rate, secondary creep has an approximately steady rate, and tertiary creep accelerates toward rupture. Design must consider both creep strain and creep rupture life.

A simplified steady-state creep relation is the Norton law:

ε̇ = Aσ^n exp(−Q/RT),

where ε̇ is strain rate, A and n are material constants, Q is activation energy, R is the gas constant, and T is absolute temperature. Larson–Miller or similar parameters are used to correlate long-term rupture data. Temperature gradients, thermal cycling, weld zones, and oxidation can significantly alter predictions.

Wear is material loss or surface damage caused by relative motion. Adhesive wear results from local bonding and tearing; abrasive wear involves hard asperities or particles; fatigue wear arises from repeated subsurface stresses; erosive wear is caused by particles or fluid impact. The Archard relationship provides a first estimate:

V = kWL/H,

where V is wear volume, W is normal load, L is sliding distance, H is hardness, and k is a wear coefficient. Lubrication, surface treatment, alignment, contamination, and contact pressure often matter more than nominal hardness alone.

Corrosion is electrochemical or chemical degradation. Uniform corrosion may be predictable through a thickness-loss rate, whereas pitting, stress-corrosion cracking, corrosion fatigue, and galvanic corrosion can be localized and difficult to detect. Material selection, coatings, cathodic protection, drainage, inhibitors, environmental control, and inspection are complementary controls. A fatigue design that ignores corrosion may be unsafe because corrosion pits act as stress raisers and accelerate crack initiation.

Reliability and Probabilistic Design

Reliability analysis accounts for uncertainty in loads, strength, geometry, material properties, and operating conditions. If strength S and load L are random variables, the limit-state function may be expressed as

g(X) = S − L.

Failure occurs when g(X) ≤ 0. The probability of failure is P_f = P[g(X) ≤ 0], and reliability is R = 1 − P_f. The reliability index β measures the distance between the mean operating point and the failure surface in standardized variable space. For normally distributed, independent strength and load,

β = (μ_S − μ_L)/√(σ_S² + σ_L²).

This simple expression is not adequate for all distributions or nonlinear limit states, but it illustrates the effect of variability. Increasing mean strength is not the only solution; reducing load variability or manufacturing scatter may provide greater reliability improvement.

Common life distributions include the exponential distribution, which assumes a constant failure rate, and the Weibull distribution, which can represent infant mortality, random failures, or wear-out depending on its shape parameter β_w. The Weibull reliability function is

R(t) = exp[−(t/η)^β_w],

where η is the characteristic life. A shape parameter below one indicates decreasing failure rate, approximately one indicates constant failure rate, and above one indicates increasing failure rate.

For independent components in series, system reliability is

R_s = ΠR_i,

so a series system is less reliable than any individual component. For independent parallel redundant components, system reliability is

R_s = 1 − Π(1−R_i).

These equations assume independence and perfect switching where relevant; real systems may exhibit common-cause failures and imperfect redundancy.

Reliability-based design is not equivalent to blindly selecting a high safety factor. It requires distributional data, an explicit failure criterion, sensitivity analysis, and decisions about inspection, redundancy, maintenance, and consequences. Monte Carlo simulation, response-surface methods, first-order reliability methods, and Bayesian updating are useful when analytical solutions are difficult.

FMEA and Fault-Tree Analysis

Failure Modes and Effects Analysis (FMEA) is a bottom-up method. For each component, the team identifies functions, failure modes, causes, local effects, system effects, existing controls, and recommended actions. Severity, occurrence, and detectability are often scored to prioritize action. A numerical risk-priority number can support discussion, but it should not replace engineering judgment: different combinations of severity, occurrence, and detectability can produce the same score.

Fault-tree analysis is a top-down method. It begins with a defined top event, such as loss of braking, and decomposes it using AND and OR logic into contributing events. OR gates represent alternative paths to failure; AND gates represent combinations that must occur together. For independent basic events, probabilities can be combined using Boolean relationships, but common-cause dependence must be explicitly considered.

FMEA is especially useful during design reviews and maintenance planning, whereas fault trees are valuable for system-level hazards and quantitative probability assessment. Used together, they reveal both component-level weaknesses and system-level combinations.

Integrated Applied Case: Fatigue and Reliability of a Pump Shaft

Background

Consider an illustrative high-pressure process pump driven by an electric motor. The shaft transmits torque through a keyed coupling and rotates continuously, while hydraulic forces produce bending. The pump is expected to operate for many years with planned periodic inspection.

Situation and Problem

A preliminary design based on static von Mises stress shows a safety factor of 2.2 relative to yield. However, field measurements indicate torque fluctuations during start-up and pressure pulsations during operation. A keyway creates a local stress concentration, and the pump handles a mildly corrosive liquid. The design team must determine whether static adequacy implies acceptable service life.

Analysis

The shaft is modelled using combined bending and torsion. Nominal alternating bending stress and mean torsional stress are calculated from the measured load history. The keyway is included through an appropriate fatigue concentration factor. A modified Goodman diagram is then used with corrected endurance strength accounting for surface finish, size, reliability, and corrosion effects.

Because start-up produces local plasticity at the keyway, a strain-life check is also performed. The resulting life is compared with a variable-amplitude estimate using rainflow cycle counting and Miner damage. Inspection records from similar shafts suggest that small machining marks are credible initial flaws. A fracture-mechanics calculation determines the crack size at which K_I reaches the material’s fracture toughness. The required inspection interval is selected so that predicted crack growth remains below that size with margin.

Finally, an FMEA identifies keyway cracking, bearing seizure, coupling misalignment, and corrosion pitting as principal failure modes. A fault tree for “loss of pump function” shows that shaft fracture is only one branch; seal failure and motor protection trips are additional paths. The analysis also identifies common-cause contributors such as poor alignment and inadequate lubrication.

Approach and Solution

The redesign increases shaft diameter locally, improves the keyway geometry, specifies a controlled surface finish, and applies a corrosion-resistant treatment compatible with the pumped fluid. Start-up control reduces torque transients. Condition monitoring is added through vibration and motor-current measurements. Periodic inspection focuses on the keyway and shoulder fillets, while maintenance procedures verify alignment and lubrication.

Outcome

The redesigned shaft satisfies the static, fatigue, crack-growth, and reliability targets under the specified assumptions. More importantly, the design no longer depends on a single calculation. It combines improved geometry, controlled operation, inspection, monitoring, and maintenance.

Learning Points

Static strength is only one limit state. Fatigue data must reflect the actual mean stress, surface condition, environment, and load spectrum. Crack tolerance and inspection capability are part of the design, and reliability improvement often comes from reducing variability and preventing common-cause failures rather than merely increasing nominal strength.

Challenges, Limitations, and Professional Responsibilities

Failure predictions are uncertain because material data are scattered, operating loads may be poorly characterized, and models are often calibrated under laboratory conditions. S–N data may not transfer directly from polished specimens to welded or corroded components. Miner’s rule may misrepresent load-sequence effects. Paris-law constants are sensitive to environment and stress ratio. Weibull parameters can be unstable when few failures are observed.

Common mistakes include using yield criteria for fatigue, ignoring residual stresses, treating theoretical stress concentration as the complete fatigue correction, assuming statistical independence without evidence, using a high reliability target without considering inspection capability, and interpreting a risk-priority number as an absolute risk measure. Another serious error is neglecting human and organizational causes such as incorrect assembly, inadequate training, poor documentation, or deferred maintenance.

Engineering decisions must be traceable. Assumptions, data sources, uncertainty ranges, inspection limits, and acceptance criteria should be documented. Safety-critical designs require applicable codes, standards, certification procedures, independent review, and configuration control. Deliberately accepting an undocumented defect or suppressing adverse test results is professionally unacceptable because failure consequences may affect workers, the public, and the environment.

Digital Tools and Emerging Developments

Finite-element analysis is widely used to obtain local stresses and strains, but its results require mesh-convergence checks, realistic boundary conditions, and validation. Digital twins and condition-monitoring systems combine sensor data with degradation models to estimate remaining useful life. Machine-learning methods can detect unusual vibration or acoustic-emission patterns, but they require representative training data and careful control of false alarms and missed detections.

Probabilistic finite-element methods, Bayesian updating, and physics-informed machine learning are increasingly used to combine inspection evidence with prior material and reliability models. Additive manufacturing creates new opportunities but introduces process-dependent defects, anisotropy, residual stress, and qualification challenges. Advanced materials and surface engineering may improve fatigue, wear, and corrosion performance, although long-term environmental data are often limited.

Best-Practice Design Workflow

A defensible failure-prevention workflow is to:

1. Define the required function, life, environment, and consequences of failure.

2. Identify credible failure modes and limit states.

3. Characterize loads, transients, cycles, defects, and uncertainty.

4. Select failure theories and degradation models appropriate to the material and regime.

5. Evaluate local stresses, fatigue life, fracture tolerance, and time-dependent damage.

6. Convert variability into reliability or probability-of-failure measures.

7. Use FMEA and fault-tree analysis to examine component and system interactions.

8. Design controls through geometry, material, process, monitoring, inspection, and maintenance.

9. Validate important assumptions through testing, field data, and independent review.

10. Update the reliability assessment as operating and inspection evidence becomes available.

Chapter Summary

Mechanical failure is a system-level engineering issue involving loads, materials, geometry, environment, manufacturing, operation, and maintenance. Static failure theories provide criteria for yielding or brittle fracture, while fatigue methods address damage under repeated loading. Stress-life methods suit predominantly elastic high-cycle conditions; strain-life methods address local plasticity and low-cycle fatigue. Miner’s rule provides a preliminary cumulative-damage estimate for variable loading.

Fracture mechanics evaluates cracks through stress-intensity factors, fracture toughness, and crack-growth laws. Creep, wear, and corrosion introduce time-, motion-, and environment-dependent degradation. Reliability methods represent uncertain strength, load, and life using limit states and distributions such as Weibull. FMEA and fault-tree analysis connect physical failure mechanisms with system consequences and risk controls. Effective design combines analytical models with inspection, monitoring, maintenance, standards, validation, and professional judgment.

Key Terms

Failure mode: The physical manner in which a component loses its intended function.

Limit state: A boundary between acceptable and unacceptable performance.

Von Mises stress: An equivalent stress used primarily to predict yielding in ductile materials.

Tresca criterion: A yielding theory based on maximum shear stress.

Fatigue: Progressive damage caused by repeated or fluctuating loading.

Stress-life method: Fatigue method relating stress amplitude to cycles to failure.

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

Mean stress: The average of maximum and minimum stress in a cycle.

Miner’s rule: A linear cumulative-damage rule for variable-amplitude fatigue.

Stress-intensity factor: A parameter describing the crack-tip stress field.

Fracture toughness: A material’s resistance to unstable crack extension.

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

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

Wear: Progressive surface damage or material loss caused by contact and motion.

Corrosion fatigue: Fatigue degradation accelerated by a corrosive environment.

Reliability: Probability of performing a required function for a specified time.

Weibull distribution: A flexible life distribution used to represent changing failure rates.

FMEA: A bottom-up analysis of failure modes, causes, effects, and controls.

Fault-tree analysis: A top-down logical analysis of combinations leading to a system failure.

Common-cause failure: Failure of multiple elements due to one shared cause.

Review Questions

Short Answer Questions

1. Distinguish between a failure mode, failure cause, and failure effect.

2. Why is the von Mises criterion commonly used for ductile metals?

3. Define stress ratio and mean stress in a fatigue cycle.

4. What physical meaning does fracture toughness have?

5. State the significance of the Weibull shape parameter.

Long Answer Questions

1. Explain the assumptions, uses, and limitations of the stress-life and strain-life fatigue methods.

2. Derive the role of stress intensity in determining critical crack size and discuss damage-tolerant design.

3. Explain creep, wear, and corrosion mechanisms and identify suitable design controls for each.

4. Compare FMEA and fault-tree analysis in terms of direction of reasoning, outputs, and applications.

5. Discuss how probabilistic design differs from the use of a conventional deterministic factor of safety.

Analytical / Application Questions

1. A shaft has a high static safety factor but operates under torque fluctuations, contains a keyway, and is exposed to corrosion. Develop an analysis plan to evaluate its fatigue reliability.

2. A two-component series system has reliabilities of 0.98 and 0.97 over one mission. Calculate system reliability and explain why improving only one component may have limited benefit.

3. A measured load history contains several stress ranges and occasional overloads. Explain how rainflow counting, cumulative damage, crack-growth analysis, and inspection planning could be combined to estimate safe service life.

22. Multiple Choice Questions with Answers

Question 1

Which failure theory is generally most appropriate for predicting yielding in a ductile metal under multiaxial stress?

A. Maximum principal stress theory

B. Distortion-energy theory

C. Griffith brittle-fracture theory

D. Archard wear law

Correct Answer: B. Distortion-energy theory

Question 2

A component has maximum and minimum cyclic stresses of 120 MPa and -40 MPa. What is its stress amplitude?

A. 40 MPa

B. 80 MPa

C. 120 MPa

D. 160 MPa

Correct Answer: B. 80 MPa

Question 3

What is the principal limitation of applying Miner’s rule to variable-amplitude fatigue?

A. It cannot use S–N data

B. It assumes linear damage and generally neglects load-sequence effects

C. It applies only to brittle materials

D. It requires fracture toughness data for every cycle

Correct Answer: B. It assumes linear damage and generally neglects load-sequence effects

Question 4

According to linear-elastic fracture mechanics, unstable crack growth is expected when:

A. Nominal stress reaches zero

B. The stress intensity factor reaches fracture toughness

C. The component reaches its endurance limit

D. Wear volume becomes constant

Correct Answer: B. The stress intensity factor reaches fracture toughness

Question 5

In the Paris crack-growth relation, da/dN represents:

A. Crack-growth rate per loading cycle

B. Applied mean stress

C. Critical crack size

D. Fracture toughness

Correct Answer: A. Crack-growth rate per loading cycle

Question 6

A Weibull shape parameter greater than one generally indicates:

A. A decreasing failure rate

B. A constant failure rate

C. An increasing failure rate associated with wear-out

D. Zero probability of failure

Correct Answer: C. An increasing failure rate associated with wear-out

Question 7

For two independent components connected in series, the system reliability is:

A. The sum of component reliabilities

B. The product of component reliabilities

C. One minus the product of reliabilities

D. Always greater than both component reliabilities

Correct Answer: B. The product of component reliabilities

Question 8

Which statement best distinguishes FMEA from fault-tree analysis?

A. FMEA is bottom-up, whereas fault-tree analysis is top-down

B. FMEA applies only to electrical systems

C. Fault-tree analysis cannot represent logical combinations

D. FMEA requires no identification of causes

Correct Answer: A. FMEA is bottom-up, whereas fault-tree analysis is top-down

Question 9

A corrosive pit is dangerous in a fatigue-loaded shaft primarily because it:

A. Eliminates mean stress

B. Acts as a local stress raiser and may initiate a crack

C. Increases fracture toughness

D. Prevents crack propagation

Correct Answer: B. Acts as a local stress raiser and may initiate a crack

Question 10

Which design action most directly supports damage-tolerant design?

A. Assuming that no manufacturing defects exist

B. Selecting an inspection interval based on predicted crack growth and critical crack size

C. Using only the static yield strength

D. Replacing all probabilistic data with a single nominal value

Correct Answer: B. Selecting an inspection interval based on predicted crack growth and critical crack size