Neuber Notch Stress (Local Strain)

Estimate elastic-plastic notch-root stress and strain from nominal load using Neuber's rule and a Ramberg-Osgood curve.

Input(Kt·S)² / E = σ·ε , ε = σ/E + (σ/K′)^(1/n′) → solve σ·ε = Kσ·Kε·(S²/E)

Nominal stress (MPa)or enter ksi — converts to MPaElastic stress-concentration factor KtElastic modulus E (GPa)or enter Msi — converts to GPaCyclic strength coeff. K' (MPa)Cyclic hardening exp. n'

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The engineering

Neuber's rule closes the gap between the elastic Kt from a Peterson chart and what actually happens once the notch root yields. Below yield the local stress is simply Kt·S, but once plastic strain appears the stress rises slower and the strain rises faster — Neuber says the geometric mean of the two concentration factors stays equal to Kt (Kσ·Kε = Kt²). Combined with a Ramberg-Osgood curve it pins down the true σ and ε at the notch root, which is what drives strain-life fatigue.

Sanity check: if σ comes back essentially equal to Kt·S you are still elastic and Neuber added nothing — the interesting cases are when σ drops below Kt·S while ε climbs. Use cyclic (K′, n′) coefficients for fatigue work, not the monotonic curve; a smoother-hardening material shifts more of the concentration into strain.

Where this math comes from

Heinz Neuber, a German engineer working on notch mechanics, published his rule in 1961 in the Journal of Applied Mechanics: for a grooved shaft in shear he proved that the product of the theoretical stress-concentration factor and the strain-concentration factor equals the square of the elastic Kt, even after the notch root goes plastic. It gave designers a way to reach local strains without a full elastic-plastic finite-element run.

The elastic Kt values it consumes trace to R. E. Peterson, whose 1953 Stress Concentration Design Factors — expanded in the 1974 second edition — compiled the photoelastic and analytical charts still cited on every fatigue drawing. Pairing Neuber's rule with Peterson's charts and a Ramberg-Osgood fit became the backbone of the local strain-life fatigue method standardized through SAE.

1943W. Ramberg & W. OsgoodPublish the three-parameter stress-strain curve used here.
1953R. E. PetersonStress Concentration Design Factors compiles elastic Kt charts.
1961Heinz NeuberProves Kσ·Kε = Kt² for notches beyond yield.
1974R. E. PetersonSecond edition expands Kt data for fatigue design.
1988SAE Fatigue Design HandbookCodifies the Neuber local strain-life procedure.

See the full timeline of the math behind every calculator →

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