⚙️ Materials Science · Engineering
📅 Березень 2026⏱ 12 min🟡 Середній
Material Fatigue: Why Things Break Under Repeated Loading
A paperclip bent once withstands far more force than its weight. Bent back and forth a dozen times, it snaps — at a stress far below what a single push would cause. This is fatigue: the progressive, localised structural damage caused by cyclic loading. It causes 90% of all mechanical failures.
1. Fatigue Mechanism at Micro-Scale
Fatigue failure proceeds in three stages:
- Initiation: Repeated cyclic slip along crystallographic planes (Miller indices) at the surface or at sub-surface defects (inclusions, porosity) forms persistent slip bands (PSBs). At the surface, PSBs create microscopic intrusions and extrusions — providing nucleation sites for cracks. This stage can consume 60–90% of total fatigue life.
- Crack propagation: A micro-crack grows through the material with each cycle. As it grows, the stress intensity factor K at the crack tip increases, accelerating propagation. Stage I: crack follows crystallographic planes. Stage II: crack grows perpendicular to maximum tensile stress. Each cycle leaves a beach mark visible under SEM — a curved line marking the crack front position. In metals, crack growth rate is typically 10⁻⁸–10⁻³ mm/cycle.
- Final fracture: Crack reaches critical size (K ≥ K_IC, fracture toughness). Remaining cross-section can no longer carry the applied load. Rapid fracture. Distinguishable post-fracture: smooth beach-marked area (fatigue zone) + rough granular area (final overload fracture).
De Havilland Comet (1954): The world's first commercial jet airliner suffered three catastrophic fatigue failures. Investigation revealed that the square corners of the windows acted as stress concentrations. The alternating pressurisation cycles (~0.5 bar cabin–atmosphere) grew cracks from rivet holes until catastrophic failure. The accidents led to the modern understanding of aircraft fatigue and rigorous full-scale stress testing now required for certification.
2. S-N Curves (Wöhler Diagrams)
August Wöhler developed systematic fatigue testing in the 1860s on railway axles after several catastrophic failures. His S-N diagram (stress amplitude S vs cycles to failure N on log-log scale) remains the fundamental engineering tool:
S-N relationship (Basquin's equation, high-cycle fatigue):
σ_a = σ_f' · (2N_f)^b
σ_a = stress amplitude (MPa)
σ_f' = fatigue strength coefficient ≈ 1.0–1.1 × UTS
N_f = cycles to failure
b = slope (fatigue strength exponent), typically −0.05 to −0.12
Typical S-N values for steel (AISI 4340, UTS = 1000 MPa):
N = 10³ cycles: σ_a ≈ 800 MPa (high stress, few cycles)
N = 10⁶ cycles: σ_a ≈ 400 MPa
N = 10⁷ cycles: σ_a ≈ 350 MPa (endurance limit for steels)
N > 10⁷ cycles: σ_a ≤ 350 MPa (safe — will never fatigue)
Endurance limit Se (fatigue limit):
Ferrous metals have a true endurance limit (~0.4–0.5 × UTS)
Aluminium alloys do NOT have an endurance limit
→ they will eventually fail at any stress level → design to finite life
3. Stress Concentrations & Notches
Holes, fillets, grooves, and surface defects concentrate stress. The theoretical stress concentration factor K_t amplifies the nominal stress:
K_t = σ_max / σ_nom
Examples (from Peterson's charts):
Circular hole in infinite plate (biaxial stress): K_t = 3 (maximum)
Shaft with shoulder fillet:
r/d = 0.1 (small radius): K_t ≈ 2.5
r/d = 0.4 (generous radius): K_t ≈ 1.5
Fatigue notch factor K_f:
Not all theoretical concentration is effective (local plasticity blunts the tip):
K_f = 1 + q(K_t − 1)
q = notch sensitivity (0 = no effect, 1 = full K_t)
q depends on material and notch radius
For hard steels (brittle): q → 1
For soft metals, small radii: q → 0
Effective endurance limit with notch:
σ_e,notched = σ_e / K_f
4. Crack Propagation: the Paris Law
Paul Paris (1963) discovered a power-law relationship between crack growth rate and stress intensity factor range ΔK:
Paris Law:
da/dN = C · (ΔK)^m
a = crack half-length (m)
N = number of cycles
ΔK = K_max − K_min = Δσ · Y · √(πa) (stress intensity factor range)
Y = geometry correction factor
C, m = material constants (empirically determined)
For many structural steels: C ≈ 10⁻¹², m ≈ 3 (ΔK in MPa√m, da/dN in m/cycle)
For aluminium alloys: C ≈ 10⁻¹¹, m ≈ 3-4
Crack growth regions:
Region I (ΔK < ΔK_th): no crack growth (threshold, ~3-5 MPa√m for steel)
Region II: Paris law (stable growth)
Region III (K > K_IC): rapid fracture
Integrating Paris law to find fatigue life:
N_f = ∫[a₀ to a_c] da / [C·(ΔK)^m]
a₀ = initial crack size (NDT detection limit, ~0.5-2 mm)
a_c = critical crack size = (1/π)(K_IC / (Y·σ_max))²
5. Cumulative Damage: Miner's Rule
Real components experience variable amplitude loading — not uniform sinusoidal cycles. Miner's rule (linear damage accumulation, 1945):
Miner's Rule:
D = Σᵢ (nᵢ / N_fᵢ)
nᵢ = number of cycles at stress level σᵢ
Nfᵢ = life at stress level σᵢ (from S-N curve)
D = cumulative damage
Failure when D = 1 (often conservative; experimental failure at D = 0.7–2)
Rainflow cycle counting (for irregular load histories):
Reduces a complex time history to a set of stress ranges and means.
Identified by the flowing-rain algorithm (Matsuishi & Endo 1968).
Standard ISO 4600 / ASTM E1049.
Example (aircraft wing):
Take-off/landing (high amplitude, low cycle): n₁/N_f1 = 0.15
Gust loads (medium amplitude, 10⁵ cycles): n₂/N_f2 = 0.45
Vibration (low amplitude, 10⁷ cycles): n₃/N_f3 = 0.30
Total D = 0.90 → expected remaining life 10% of design life used
6. Mean Stress Effects
Fatigue life depends not just on stress amplitude but also on the mean (static) stress. A tensile mean stress reduces fatigue life; compressive mean stress improves it:
Goodman line (conservative, safe side):
σ_a / Se + σ_m / UTS = 1 (failure boundary)
Gerber parabola (mean of experimental scatter):
σ_a / Se + (σ_m / UTS)² = 1
Modified Goodman for design:
σ_a / Se + σ_m / σ_y = 1 (yield stress instead of UTS)
R-ratio (stress ratio):
R = σ_min / σ_max
R = −1: fully reversed (no mean, highest fatigue damage for given amplitude)
R = 0: pulsating tension (common in bolted joints)
R = 0.1-0.5: typical operational range for aircraft structures
Shot peening, cold working of holes, and prestressed bolts all introduce compressive residual stresses at the surface (σ_m→ negative), significantly improving fatigue life. Turbine blades are laser shock peened to depths of 1-2 mm to prevent compressor fatigue cracking.
7. Design Against Fatigue
- Generous radii at stress concentrations: Increasing fillet radius at a shoulder from r/d = 0.05 to r/d = 0.2 can cut K_f in half — doubling fatigue life for the same nominal stress.
- Surface finish: Polished surfaces improve life significantly. Surface roughness corrections S_b: mirror polish → 0.9 × Se, machined → 0.7 × Se, hot-rolled → 0.4 × Se, corroded → 0.1 × Se.
- Safe-life vs damage-tolerance: Aviation traditionally used safe-life design — retire components before expected crack initiation. Modern practice uses damage-tolerance: assume cracks exist (from initial defects, manufacturing flaws), calculate crack growth rate, and schedule inspections before critical size is reached.
- Non-destructive testing (NDT): Ultrasonic, eddy current, magnetic particle, and X-ray inspection find cracks down to ~0.5 mm. Setting inspection intervals requires knowing crack growth rates and critical crack sizes via fracture mechanics.
- Structural Health Monitoring (SHM): Embedded piezoelectric actuators and sensors, fibre Bragg gratings, and acoustic emission monitoring can detect crack growth in real-time on bridges, aircraft, and wind turbines, enabling condition-based maintenance.