A static strength check asks one question: is the applied stress below the yield strength? Pass that check and a part will not break — under a single, steady load. Yet machine components fail constantly at stresses far below yield. Aircraft fittings, crankshafts, railway axles, bridge welds: they break after months or years of service, under loads they carried without complaint on day one. The reason is fatigue, and explaining it requires a different way of thinking about strength.

This article covers the two pillars of durability analysis — the stress-life approach and fracture mechanics — and works the critical-crack-size calculation that ties them together.

Why this calculation matters

Most real loads are not static. They cycle: a rotating shaft sees alternating bending every revolution, a pressure vessel breathes with every duty cycle, a wing flexes with every gust. Repeated loading does something a single load never does — it grows microscopic flaws into cracks, cycle by cycle, until a crack reaches a size the part can no longer tolerate. Then it breaks suddenly, often with no visible warning.

Because the stress is below yield the whole time, a yield-based check gives a false sense of safety. Fatigue is one of the most common causes of mechanical failure in service. Any rotating, vibrating, or pressure-cycling component needs a durability check, not just a strength check — and that means estimating either how many cycles it survives, or how large a crack it can tolerate.

The two approaches

Stress-life (the S-N curve). Test specimens are cycled at various stress amplitudes and the cycles-to-failure recorded. Plotting stress amplitude against cycles produces the S-N curve: higher stress, fewer cycles. For many steels the curve flattens into an endurance limit — a stress below which the part survives essentially indefinitely. As a rough guide it sits near half the ultimate tensile strength. Real loads also carry a mean stress, and a tensile mean is damaging; the Goodman relation corrects the allowable amplitude for it. Stress-life is the right tool when no crack is assumed and you want a cycle count.

Fracture mechanics. This approach assumes a crack already exists — because in welds, castings, and forgings, one effectively always does. The key quantity is the stress intensity factor K, which measures how severely stress is concentrated at the crack tip:

K = Y * sigma * sqrt(pi * a)

Here sigma is the applied stress, a is the crack length, and Y is a geometry factor near 1. Fast, unstable fracture occurs when K reaches the material's fracture toughness K_IC. Rearranging gives the most useful result in the field — the largest crack a part can tolerate at a given stress:

a_c = (1/pi) * ( K_IC / (Y * sigma) )^2

Between a small initial flaw and that critical size, the crack grows a little each cycle. The Paris law, da/dN = C·(delta-K)^m, describes that growth rate and lets you integrate a remaining fatigue life.

A worked example

A steel plate carries a cyclic tensile stress with a peak of sigma = 200 MPa. The steel has a fracture toughness K_IC = 50 MPa·sqrt(m). Inspection can reliably detect an edge crack; the geometry factor for a single-edge crack is Y = 1.12. What crack size triggers fast fracture?

a_c = (1/pi) * ( K_IC / (Y * sigma) )^2
a_c = (1/pi) * ( 50 / (1.12 x 200) )^2
a_c = (1/pi) * ( 0.223 )^2
a_c = (1/pi) * 0.0498 = 0.0159 m = 15.9 mm

So an edge crack of about 16 mm would cause this plate to fracture suddenly at 200 MPa. That single number drives real decisions. It sets the inspection interval: the technique used must reliably find cracks well below 16 mm, with margin. It sets the retirement rule: a crack approaching that size means the part comes out of service. And combined with the Paris law, the gap between the smallest detectable flaw and a_c becomes a predicted number of safe cycles between inspections. This is the backbone of the damage-tolerant design philosophy used throughout aerospace.

Common mistakes

Trusting a yield check for cyclic loads. Yield strength governs static failure. It says nothing about whether a part survives ten million cycles. Cyclic loading needs its own analysis.

Ignoring stress concentrations. Fatigue cracks start at geometric features — fillets, holes, keyways, weld toes. The nominal stress can be modest while the local stress at a sharp fillet is several times higher. The notch is where the crack begins.

Forgetting mean stress. Two loadings with the same amplitude but different mean stress do not give the same life. A tensile mean shortens it. Use a Goodman-type correction rather than the amplitude alone.

Assuming a flawless part. Welds, castings, and forgings contain defects from the moment they are made. Fracture mechanics starts from that honest assumption; a stress-life analysis that presumes a pristine part can be dangerously optimistic for those processes.

Try the interactive NovaSolver calculator

The relationship between stress, crack size, and toughness is best understood by moving the variables and watching the critical size respond. The fracture mechanics calculator on NovaSolver computes the stress intensity factor and the critical crack size for common crack geometries, so you can see directly how a higher stress shrinks the tolerable flaw.

Related calculators

• Paris law crack growth — to integrate how fast a crack grows and estimate remaining life.
• S-N curve — the stress-life view, including the endurance limit.
• Goodman diagram — the mean-stress correction for cyclic loads with a non-zero average.

The full set is in the fatigue and fracture tools hub.

Closing note

Fatigue and fracture mechanics explain the failures that a strength check never sees coming. The stress-life view answers "how many cycles" for a part assumed sound; fracture mechanics answers "how large a crack" for a part assumed flawed. Together they turn durability from a hope into a calculation: estimate the critical crack size, set inspections well inside it, account for stress concentrations and mean stress, and design so that a crack is found and managed long before it becomes sudden. That is why well-engineered structures last — not because they never crack, but because cracking was planned for.