Fracture Mechanics & Fatigue Notch Analysis
Molski-Glinka ESED Notch Strain vs Neuber Formula
Calculate Elastic Pseudo-Notch Stress K_t * S_nom, Molski-Glinka Equivalent Strain-Energy Density (ESED) Elastic-Plastic Notch Strain, Neuber Hyperbola Strain, and Plane-Stress/Plane-Strain Ratio.
Bound thick-plate plane-strain notch plasticity between Molski-Glinka ESED and Neuber’s Rule!
When nominal far-field stress S_nom stays safely elastic (S_nom < sigma_y) yet theoretical elastic peak stress S_e = K_t * S_nom exceeds yield strength sigma_y at a fillet or bolt hole, Neuber’s Rule (sigma * epsilon = S_e^2 / E) often overpredicts notch-root plastic strain in thick, constrained components because it equates total + complementary energy! K. Molski and G. Glinka (1981, Materials Science and Engineering) proposed the Equivalent Strain-Energy Density (ESED) criterion: surrounded by a vast elastic bulk, the localized plastic zone absorbs the exact same strain-energy density W_p as hypothetical elastic material W_e = S_e^2 / (2 * E)! For an elastic-perfectly-plastic notch root (sigma_notch = sigma_y once S_e > sigma_y), equating W_p = sigma_y^2 / (2*E) + sigma_y * (epsilon_Glinka - sigma_y / E) to S_e^2 / (2*E) yields the clean closed-form Glinka Notch Strain epsilon_Glinka = (sigma_y / (2 * E)) * ((S_e / sigma_y)^2 + 1), compared against Neuber Notch Strain epsilon_Neuber = (sigma_y / E) * (S_e / sigma_y)^2!
Verify on Far_Field_Nominal_Stress S_nom = 200 MPa (B2), Elastic_Concentration_Factor K_t = 3.0 (B3 -> Elastic Pseudo-Stress S_e = 600.00 MPa in E2), Yield_Strength sigma_y = 300 MPa (B4 -> overstress ratio r = 600 / 300 = 2.0), and Youngs_Modulus E = 200 GPa (200,000 MPa, B5 -> yield strain = 0.0015): Molski-Glinka Plane-Strain Notch Strain is 0.0015 * (2^2 + 1) / 2 = 0.003750 (E3); Neuber Plane-Stress Notch Strain is 0.0015 * 2^2 = 0.006000 (E4); giving a 1.60x Neuber/Glinka conservatism spread (E5)!
| Notch Plasticity Metric (S_nom=200 MPa, Kt=3.0, sigma_y=300 MPa, E=200 GPa) | Formula | Verified Output |
|---|---|---|
| Elastic Pseudo-Stress S_e (MPa) | K_t * S_nom | 600.00 MPa (r = S_e / sigma_y = 2.0) |
| Molski-Glinka ESED Notch Strain | eps_y * ((S_e / sigma_y)^2 + 1) / 2 | 0.003750 (0.001200 @ S_nom=80) |
| Neuber Hyperbola Notch Strain | eps_y * (S_e / sigma_y)^2 | 0.006000 (0.001200 @ S_nom=80) |
| Neuber / Glinka Strain Ratio | eps_Neuber / eps_Glinka | 1.60x (1.00x Elastic @ S_nom=80) |
Molski-Glinka ESED Localized Notch-Root Strain Formula
Guard S_nom >= sigma_y ("Gross Section Yield (S_nom >= sigma_y)"—both Glinka and Neuber require elastic net-section containment!) and return purely elastic S_e / E whenever S_e <= sigma_y!
=LET(s_nm, B2, k_tc, B3, sg_y, B4, md_e, B5, IF(OR(s_nm <= 0, k_tc < 1, sg_y <= 0, md_e <= 0), "Invalid Notch Input", IF(s_nm >= sg_y, "Gross Section Yield (S_nom >= sigma_y)", LET(s_el, k_tc * s_nm, e_mp, md_e * 1000, IF(s_el <= sg_y, ROUND(s_el / e_mp, 6), ROUND((sg_y / (2 * e_mp)) * (((s_el / sg_y) ^ 2) + 1), 6))))))Elastic Convergence Check (S_e <= sigma_y)
Lower S_nom to 80 MPa (S_e = 240 MPa <= 300 MPa): both Glinka (E3) and Neuber (E4) evaluate to 0.001200 and their ratio E5 equals 1.00!
=IF(ISTEXT(E3), E3, ROUND(E4 / E3, 2))Try the example
Use the original values or change them to test your understanding.