Engineering & Science
Neuber's Rule: Local Elastic-Plastic Notch Stress & Strain Concentration
Calculate the Neuber hyperbola product sigma * epsilon = (Kt * S)^2 / E (MPa) at geometric stress raisers and solve local notch-root total strain under elastic-perfectly-plastic yielding.
Stop overpredicting notch-root stress once local yielding redistributes load!
At a bolt hole, fillet, or keyway with elastic stress concentration factor Kt = 2.5 under nominal net-section stress S = 240 MPa, naive linear elasticity predicts Kt * S = 600 MPa—exceeding the alloy's yield strength sigma_y = 450 MPa! Once the notch root yields locally, stress concentration K_sigma drops below Kt while local plastic strain concentration K_epsilon rises ABOVE Kt!
Heinz Neuber (1961) discovered that the geometric mean of plastic stress and strain concentration remains equal to Kt: K_sigma * K_epsilon = Kt^2! For nominal elastic net sections (e = S / E), this defines the Neuber hyperbola product sigma * epsilon = ((Kt * S)^2) / E! For an elastic-perfectly-plastic (EPP) material where yielded notch stress caps at MIN(Kt * S, sigma_y), local notch total strain follows directly in closed form: epsilon_notch = IF(Kt * S <= sigma_y, (Kt * S) / E, ((Kt * S)^2) / (E * sigma_y))! At Kt = 2.5, S = 240 MPa, E = 200,000 MPa, sigma_y = 450 MPa, Neuber's product is 1.800 MPa and local notch strain is 1.800 / 450 = 0.00400 (0.400% vs naive elastic 0.300%)!
| Nominal Load S (Kt = 2.5, E = 200 GPa, sy = 450 MPa) | Pseudo-Elastic Kt * S | Neuber Product (Kt*S)^2 / E | EPP Notch-Root Strain epsilon |
|---|---|---|---|
| Yielded Notch (S = 240 MPa) | 600 MPa (> 450 MPa Yield!) | 1.800 MPa | 0.00400 (Plastic amplification!) |
| Purely Elastic Notch (S = 120 MPa) | 300 MPa (<= 450 MPa Elastic) | 0.450 MPa | 0.00150 (= 300 / 200,000) |
| Invalid Kt < 1 or E <= 0 | Guarded Input Error | Non-physical geometry | "Invalid Notch Parameters" |
Neuber Hyperbola Energy Product sigma * epsilon = (Kt * S)^2 / E (MPa)
Computes the invariant stress-strain product at the notch root for intersection with Ramberg-Osgood or EPP curves.
=IF(OR(B2 < 1, B3 < 0, B4 <= 0, B5 <= 0), "Invalid Notch Parameters", ROUND(((B2 * B3) ^ 2) / B4, 3))Local Notch-Root Total Strain (Elastic-Perfectly-Plastic Closed Form)
Returns (Kt * S) / E in the elastic regime and ((Kt * S)^2) / (E * sigma_y) once Kt * S exceeds sigma_y.
=IF(OR(B2 < 1, B3 < 0, B4 <= 0, B5 <= 0), "Invalid Notch Parameters", ROUND(IF((B2 * B3) <= B5, (B2 * B3) / B4, ((B2 * B3) ^ 2) / (B4 * B5)), 5))Try the example
Use the original values or change them to test your understanding.