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Murakami √area calculator

The fatigue limit of a part containing a defect, from two numbers: hardness and defect size. σw = C·(HV + 120) / (√area)1/6. It is a threshold for crack initiation, not a life — and the sixth root is the part everyone underestimates.

Inputs

Hardness of the matrix, and the √area of the largest defect in the highly stressed volume.

Typical √area by route
Or convert from surface roughness

√area ≈ 2.97·Rz, and Rz ≈ 4·Ra on a machined surface.

σw

Defect-initiation threshold, fully reversed (R = −1)

—
—
Intrinsic limit (defect-free)—
Defect costs you—
If it were internal / surface—
Halving √area would gain—

How it is calculated

This is the expression METALLAI uses inside its own fatigue chain, written out so you can check it.

σw = C · (HV + 120) / (√area)1/6 [MPa, √area in µm] C = 1.43 surface defect C = 1.56 internal defect intrinsic (defect-free reference) ≈ 0.5 × UTS, with UTS ≈ 3 × HV

Murakami's result was that the projected area of the defect normal to the maximum principal stress predicts fatigue strength far better than the defect's shape does. A pore, an inclusion and a machining groove of the same √area behave about the same. That is what makes one number enough.

The sixth root is the whole story

Because √area enters to the power −1/6, defect size is a weak lever:

Reduce √area byFatigue limit rises by
2×+12 %
4×+26 %
10×+47 %
64×+100 % (double)

So cleanliness programmes give real but modest returns, and it is usually cheaper to raise hardness or lower the stress than to chase the last inclusion. The same arithmetic run backwards is more alarming: a defect ten times larger than you assumed costs you about a third of the fatigue limit, which is why the largest defect in the stressed volume — not the average — is the one to measure.

Where the defect size comes from

Measure it if you can: fractography on a failed part, or CT on a casting. Failing that, these are the values METALLAI assumes by route.

Route√area (µm)Route√area (µm)
Forging10Squeeze casting30
Cold rolling10Low-pressure die60
Extrusion12Permanent mould80
Hot rolling15Investment casting100
High-pressure die100Die casting120
Gravity casting150Sand casting250
This is a threshold, not a life. It estimates the stress amplitude below which a crack will not initiate from a defect of that size. Above it, the relation says nothing about how many cycles the part lasts — that needs an S–N or fracture-mechanics calculation. Reading a Murakami limit as a design life is a common and serious mistake. It also assumes a small defect in a hard matrix, fully reversed loading, and no mean stress; a positive mean stress requires a Goodman or Haigh correction on top.

Why the defect almost always governs

Below a certain defect size the intrinsic, defect-free limit takes over — the Kitagawa–Takahashi picture. But that crossover happens at a very small size:

HardnessCrossover √area
HV 20012.6 µm
HV 3005.7 µm
HV 4303.3 µm
HV 7001.9 µm

In other words, for any defect you can actually see, and for every casting route in the table above, the defect governs. And the harder the steel, the smaller the defect that is enough to take control — which is why high-strength steels are so much more defect-sensitive, and why raising hardness without also improving cleanliness often buys less fatigue performance than expected.

Questions

What is the Murakami √area method?
An empirical relation predicting the fatigue limit of a steel containing a small defect from just hardness and the square root of the defect's projected area: σw = C(HV + 120) / (√area)1/6, MPa with √area in µm. C is 1.43 at the surface, 1.56 internally.
What is the difference between a surface and an internal defect?
Only the constant: 1.43 versus 1.56. So the same defect is about 9 % less damaging internally, because the surface removes the constraint on one side and gives the crack a free path. In practice the surface defect is the one that governs — which is why surface finish and shot peening matter as much as they do.
How much does shrinking a defect improve the fatigue limit?
Far less than people expect, because size enters as a sixth root. Halving √area buys about 12 %. Doubling the fatigue limit would need a 64-fold reduction. Cleanliness pays, but modestly — raising hardness or lowering the stress is usually cheaper.
Does the Murakami limit give a fatigue life?
No — it is a threshold, not a life. It gives the amplitude below which a crack should not initiate from that defect. Above it, you need an S–N or fracture-mechanics calculation. Treating it as a design life is a common and serious misreading.
What defect size should I use?
Measure it — fractography on a failed part, or CT on a casting — and use the largest defect in the highly stressed volume, not the average. Failing that, use the route table above, or convert from surface roughness: √area ≈ 2.97·Rz, with Rz ≈ 4·Ra on a machined surface.

σw is one link in the chain

METALLAI runs the whole fatigue chain from a composition and a route: an ideal Basquin baseline, porosity and surface knockdowns, this Murakami check, a Goodman mean-stress correction, and a P10 design life — with the terms printed as a decomposition that sums to the answer.

Run the full fatigue chain — free Hardenability calculator