Fracture and Fatigue: How Things Actually Fail

A design that keeps every stress below yield, with a factor of safety on top, feels rigorous. It is also the reason a great many structures have failed.

Pressure vessels burst at a third of yield. Axles that ran for a decade snap in normal service. A bridge that carried its design load for forty years falls down under the same load it always carried.

In all of these, the material never yielded anywhere. Something else happened: a crack existed, and a crack changes the problem entirely.

Key idea: Yield strength describes an undamaged part. Real parts contain cracks from casting porosity, weld defects, machining marks, corrosion pits or fatigue, and once a crack is present the governing quantity is not stress but stress and crack size together. Fracture mechanics is the discipline that handles the second case, and it exists because the first case was not enough.

Stress flows through a part like a fluid, and anything it must go around crowds it. For an elliptical hole of half-length $a$ and tip radius $\rho$ in a wide plate, the stress at the tip is

$\sigma_{\max} = \sigma \left(1 + 2\sqrt{\frac{a}{\rho}}\right)$

A circular hole has $a = \rho$, giving a factor of exactly 3, regardless of how large the hole is. Make the ellipse long and sharp and the factor grows without bound; in the limit of a perfectly sharp crack the predicted stress is infinite.

In practice: This is why fillet radii matter more than most other details on a drawing, why a scratch across a glass sheet lets you snap it cleanly, and why an aircraft window is a rounded rectangle rather than a square. The design rule that follows is simple and constantly violated: never leave a sharp internal corner in a loaded part.

An infinite stress at a crack tip is not a usable prediction, so Griffith changed the question in 1920. Instead of asking what the stress is, he asked whether extending the crack releases more energy than it costs.

Growing a crack creates new surface, which costs energy proportional to the crack length. It also relaxes the material either side, releasing stored elastic energy proportional to the length squared. Below a certain size the cost wins and the crack is stable; above it the release wins and the crack runs.

$\sigma_f = \sqrt{\frac{2E\gamma}{\pi a}}$

Fracture stress falls as the inverse square root of crack length. Quadruple the crack and you halve the strength.

Key idea: Fracture is not a stress being exceeded, it is an energy balance tipping over. That reframing is what made brittle failure predictable, and it explains Griffith's glass fibres: strength rose as they thinned because the largest flaw they could contain shrank.

Irwin generalised Griffith in the 1950s into the form engineers use. The severity of a crack is one number, the stress intensity factor:

$K = Y \sigma \sqrt{\pi a}$

where $Y$ is a geometry factor near 1. Every material has a critical value $K_{IC}$, its fracture toughness, and the part fails when $K$ reaches it.

Rearranging gives the number that actually matters in service: the largest crack the part can carry.

$a_c = \frac{1}{\pi}\left(\frac{K_{IC}}{Y\sigma}\right)^2$

Mild steel tolerates a crack you could put a hand through. Glass fails at a flaw sixty microns long, which is why glass strength is really a statement about its surface condition.

This is one of the least intuitive results in engineering, and it has a direct practical consequence: raising the design stress moves the critical flaw size below what routine inspection can find.

Gotcha: A high-strength alloy is only usable if your inspection can reliably detect flaws smaller than its critical crack size. If the smallest crack you can find is 3 mm and the part fails at 2 mm, the material is not fit for that application no matter what its strength is. Specifying the alloy and specifying the inspection regime are the same decision.

Body-centred cubic metals, which includes most structural steel, become dramatically less tough below a transition temperature. Above it a crack blunts by local plastic flow; below it, dislocation motion is too sluggish and the crack propagates by cleaving straight through grains.

The United States built roughly 2,700 Liberty ships between 1941 and 1945, welded rather than riveted so they could be assembled fast. In the cold North Atlantic a large number developed brittle fractures, reported figures vary but run to around a thousand hulls, and a few split completely in two, in some cases while lying at anchor.

Three factors combined: a steel with a transition temperature above North Atlantic sea temperature, sharp corners at hatch openings acting as stress concentrators, and continuous welded construction that gave a crack an uninterrupted path. Riveted ships had cracks too, and the seams stopped them.

In practice: Charpy impact testing across a temperature range is standard for structural steel precisely because of this, and modern specifications state a toughness requirement at the lowest expected service temperature rather than at room temperature.

Fatigue is failure under repeated loading at stresses that are individually harmless. Cyclic stress moves dislocations back and forth at the surface, roughening it into microscopic intrusions, one of which sharpens into a crack that then grows a little on every cycle.

The classic picture is the S-N curve, plotting stress amplitude against cycles to failure, and it splits materials into two camps:

Key idea: Aluminium has no endurance limit. There is no stress low enough to guarantee an aluminium part survives indefinitely, which is why aircraft structures have mandatory life limits measured in flight cycles and steel bridges generally do not. This single material difference shapes an entire regulatory regime.

The de Havilland Comet was the first jet airliner. In January and April 1954 two broke up in flight, and the fleet was grounded.

The investigation, led by Sir Arnold Hall at the Royal Aircraft Establishment at Farnborough, took an intact airframe, G-ALYU, submerged it in a purpose-built water tank, and cycled the cabin pressure to simulate flights. After roughly 3,060 simulated cycles the fuselage tore open. Strain gauges showed stress around cutout corners far above what the design had assumed, and the failure initiated at a rivet hole near a cutout.

Gotcha: The popular version is "square passenger windows". The recovered evidence and the tank test point at a corner of an escape hatch cutout and a rivet hole rather than the passenger windows, and punched rather than drilled rivet holes contributed by leaving tiny cracks. The general lesson, that a sharp corner in a pressurised skin concentrates stress and starts a fatigue crack, is correct; the specific detail is usually misremembered.

The accident aircraft had accumulated 1,290 and 900 pressurised flights, far fewer than anyone expected to matter.

Once a crack exists, its growth per cycle follows the Paris law, an empirical relation that holds across a wide range of materials:

$\frac{da}{dN} = C (\Delta K)^m$

with $\Delta K = Y \Delta\sigma \sqrt{\pi a}$. Integrating gives the number of cycles from a detected flaw to the critical size.

from math import pi, sqrt

C, m, Y = 1e-11, 3.0, 1.0     # typical aluminium alloy constants
ds = 100.0                    # stress range, MPa
a, af, da, N = 0.001, 0.025, 1e-7, 0.0

while a < af:
    dK = Y * ds * sqrt(pi * a)
    N += da / (C * dK ** m)
    a += da

print(f"{N:,.0f} cycles from 1 mm to 25 mm")   # about 909,000

Growing from 1 mm to 5 mm consumes 69 percent of that life. Almost all of a fatigue life is spent while the crack is too small to find easily, and the last few millimetres go quickly.

In practice: Inspection intervals are set from this integral. The interval is chosen so that a crack just below the detection threshold at one inspection cannot reach critical size before the next, which is why the interval depends on the inspection method as much as on the structure.

The field settled on three philosophies, and which one applies is a decision about consequences rather than about materials.

Damage tolerance is the modern default for anything whose failure is unacceptable, and it inverts the usual mindset: you do not assume the structure is sound, you assume it is cracked and prove the crack cannot get dangerous before someone looks again.

Key idea: The historical pattern in every case in this lesson is the same. The mechanism existed long before the failures, the failures made it a design requirement, and the investigation produced the analysis method. Fracture mechanics is largely a record of accidents that were expensive enough to fund the mathematics.