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Fatigue and Cyclic Stress Failure

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Stress and Strain Fundamentals→→Fatigue: Cyclic Loading and Failure
fatigue s-n-curve fatigue-limit cycles-to-failure notch-sensitivity

Core Idea

Fatigue is failure under repeated cyclic loading at stresses well below yield strength; failure initiates at surface microstructural features or defects and grows with each cycle. The S-N curve (stress vs. number of cycles to failure) shows that fatigue strength decreases with increasing cycle count, with many metals exhibiting a fatigue limit (threshold stress below which no failure). Stress concentration (notches, surface defects) significantly accelerate fatigue crack initiation.

Explainer

From your prerequisite on stress and strain, you know that steel loaded below its yield strength deforms elastically — the material stores energy and returns to its original shape when the load is removed. This makes the elastic regime appear "safe." But here is the fundamental insight that fatigue reveals: a steel component cycled at 60% of its yield strength — well within the elastic range — can fracture after millions of load reversals. The material is not yielding in any single cycle, yet it is accumulating damage at the microscale. Fatigue failure occurs because cyclic loading progressively opens and extends tiny cracks that would never grow under a single static application of the same stress.

The mechanism begins at a stress concentration — any geometric or microstructural discontinuity where the local stress exceeds the nominal stress by a concentration factor Kₜ. Notches, holes, weld toes, machining marks, and internal inclusions all act as stress concentrators. Even when the bulk of the part remains elastic, the local peak stress at these sites cycles across a range that slowly propagates a crack with each load reversal. This is the initiation phase, which can consume the majority of the total fatigue life. Once a crack of detectable size exists, the propagation phase begins: the crack advances incrementally per cycle according to the Paris law (da/dN ∝ ΔKⁿ, where ΔK is the stress intensity factor range). The part eventually fails by fast fracture when the crack reaches the critical size at which the stress intensity exceeds the fracture toughness.

The S-N curve (Wöhler curve) summarizes the cyclic fatigue behavior of a material by plotting applied stress amplitude S against the number of cycles N to failure. At high stress amplitudes, failure occurs in thousands of cycles (low-cycle fatigue). As stress decreases, the number of cycles to failure increases dramatically — often by orders of magnitude. For steels and titanium alloys, the S-N curve typically flattens at long lives (around 10⁶–10⁷ cycles), defining a fatigue limit: a stress amplitude below which the material can theoretically cycle indefinitely without failure. Aluminum alloys, copper, and most non-ferrous metals exhibit no true fatigue limit — their S-N curves continue declining, so engineers specify a fatigue strength at a defined life (commonly 10⁷ or 10⁸ cycles) as the design allowable.

Surface condition is the single most controllable variable in fatigue design. A polished surface has a higher fatigue strength than a machined surface, which outperforms an as-cast surface, which outperforms a corroded surface — because fatigue cracks almost always initiate at the surface, and surface quality controls the density and depth of potential initiation sites. Shot peening exploits this: bombarding the surface with small steel balls induces compressive residual stresses in the near-surface layer. A fatigue crack cannot open under compression, so the compressive residual must first be overcome before the crack can propagate. This extends fatigue life dramatically — by factors of 2–5× in common engineering metals — which is why aircraft components, springs, and gear teeth are routinely shot-peened after machining.

Practice Questions 5 questions

Prerequisite Chain

Understanding Zero → The Number Zero → Counting to Five → Counting to 10 → Counting to 20 → Counting a Set of Objects Up to 20 → Cardinality: The Last Number Counted → Matching Numerals to Quantities → Subitizing Small Quantities → Addition Within 10 → Number Bonds to 10 → Addition Within 20 → Doubles and Near Doubles → Doubles Facts Within 10 → Near Doubles Facts Within 20 → Mental Math Strategies for Addition → Mental Math: Adding and Subtracting Tens → Addition Within 100 → Repeated Addition as Multiplication → Multiplication as Equal Groups → Multiplication: Arrays → Basic Multiplication Facts (0s, 1s, 2s, 5s, 10s) → Multiplication Facts Within 100 → Division as Equal Sharing → Division as Grouping (Measurement Division) → Division: Grouping (Repeated Subtraction) Model → Division: Fair Sharing Model → Division as Equal Sharing → Division as Grouping → Basic Division Facts → Division Facts Within 100 → Multiplication and Division Fact Families → Relationship Between Multiplication and Division → Division Facts as Inverse of Multiplication → Remainders and Quotients in Division → Division Word Problems → Multi-Step Word Problems → Solving Multi-Step Word Problems → Multiplication Word Problems → Division Word Problems → Introduction to Long Division → Factors and Multiples → Prime and Composite Numbers → Equivalent Fractions → Relating Fractions and Decimals → Decimal Place Value → Integers and the Number Line → Comparing and Ordering Integers → Absolute Value → Adding Integers → Subtracting Integers → Multiplying Integers → Dividing Integers → Unit Rates → Proportions → Percent Concept → Converting Between Fractions, Decimals, and Percents → Operations with Rational Numbers → Two-Step Equations → Solving Multi-Step Equations → Equations with Variables on Both Sides → Angle Pairs: Complementary, Supplementary, and Vertical → Parallel Lines and Transversals → Corresponding Angles → Alternate Interior Angles → Triangle Angle Sum Theorem → Exterior Angle Theorem → Triangle Inequality Theorem → Similar Triangles: AA Similarity → Similar Triangles: SSS and SAS Similarity → Proportions in Similar Triangles → Right Triangle Trigonometry Introduction → Sine, Cosine, and Tangent Ratios → Trigonometric Ratios Review → Radian Measure → Converting Between Degrees and Radians → The Unit Circle → Graphing Sine and Cosine → Graphing Tangent and Reciprocal Trigonometric Functions → Derivatives of Trigonometric Functions → Antiderivatives → Indefinite Integrals → Basic Integration Rules → Riemann Sums → Definite Integral Definition → Double Integrals: Definition and Setup → Iterated Integrals and Fubini's Theorem → Double Integrals over Rectangular Regions → Double Integrals over General Regions → Applications of Double Integrals: Area, Mass, and Moments → Center of Mass → Conservation of Linear Momentum → Elastic Collisions → Inelastic Collisions → Coefficient of Restitution → Collision Analysis and Real-World Applications → Two-Body Collisions in the Center-of-Mass Frame → Reduced Mass and Two-Body Problems → Kinematics in Two Dimensions → Projectile Motion → Circular Motion: Kinematics → Rotational Kinematics → Torque → Moment of Inertia → Rotational Kinetic Energy → The Work-Energy Theorem → Conservation of Mechanical Energy → Work-Energy Principle for Particles → Linear Impulse-Momentum for Particles → Linear Momentum and Impulse in Systems → Conservation of Linear Momentum in Systems → Systems of Particles: Center of Mass and Internal Forces → Rigid Body Kinetics — Force and Acceleration → Angular Impulse and Momentum for Rigid Bodies → Conservation of Angular Momentum → Euler's Equations for Rigid Body Rotation → Gyroscopic Motion, Precession, and Stability → Stability of Equilibrium: Stable, Unstable, and Neutral → Introduction to Statics and Dynamics → Vector Analysis and Components → Scalar and Vector Mechanics → Force Vectors, Components, and Resultants → Stress and Strain Fundamentals → Fatigue and Cyclic Stress Failure

Longest path: 124 steps · 725 total prerequisite topics

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