Sheet-Metal Stamping, Plasticity & Mechanical Metallurgy
Hollomon Strain Hardening & Considere Necking Formula
Calculate John Hollomon’s True Flow Stress sigma_true = K * (eps_true ^ n), Armand Considère’s Diffuse Necking True Strain Limit (eps_u = n), Maximum Uniform Engineering Elongation EXP(n) - 1, and Stamping Instability Status.
Why sheet-metal stamping splits right when true plastic strain reaches Hollomon’s exponent n!
Automotive body-in-white stamping engineers choose deep-drawing annealed steels and aluminum alloys with high strain-hardening exponents n because John H. Hollomon’s (1945, AIME) plastic flow law models true stress above yield as sigma_true = K * (eps_true ^ n), where K is the Strength Coefficient (true stress at eps_true = 1.0) and eps_true = LN(1 + e_eng)! By Armand Considère’s (1885) tensile instability criterion, localized thinning (diffuse necking at Ultimate Tensile Strength) begins the instant geometric cross-section shrinkage outpaces work-hardening: d(sigma_true)/d(eps_true) = sigma_true! Differentiating K * eps^n gives n * K * eps^(n - 1) = K * eps^n, proving the remarkable identity that Critical True Uniform Strain at Necking equals n identically (eps_u = n), corresponding to Maximum Uniform Engineering Elongation e_u = EXP(n) - 1!
Verify on an Interstitial-Free Deep-Drawing Steel—Strength_Coefficient K = 600 MPa (B2), Strain_Hardening_Exponent n = 0.25 (B3), and Stamping_Panel_True_Strain eps = 0.0625 (B4 -> 0.0625^0.25 = 0.50 exact!): True Flow Stress is 600 * 0.50 = 300.00 MPa (E2); Considère Necking True Strain Ceiling is 0.2500 (E3); Max Uniform Engineering Elongation is EXP(0.25) - 1 = 0.2840 (28.40% before necking in E4); confirming "Stable Uniform Work-Hardening" (E5)!
| Hollomon-Considère Metric (K=600 MPa, Exponent n=0.25, Panel Strain eps=0.0625) | Formula | Verified Output |
|---|---|---|
| True Plastic Flow Stress (MPa) | K * (eps_true ^ n) | 300.00 MPa (600.00 MPa @ eps=1.0) |
| Considère Necking True Strain | n (From d_sigma / d_eps = sigma) | 0.2500 |
| Max Uniform Eng Elongation e_u | EXP(n) - 1 | 0.2840 (28.40% Engineering Strain) |
| Sheet-Forming Split Gate | Applied_True_Strain <= n | Stable Uniform Work-Hardening |
Hollomon True Flow Stress & Considère Necking Gate Formula
Validate K > 0, physical work-hardening exponent 0 < n < 1, and positive true plastic strain eps_true > 0!
=LET(k_cf, B2, n_ex, B3, ep_t, B4, IF(OR(k_cf <= 0, n_ex <= 0, n_ex >= 1, ep_t <= 0), "Invalid Plasticity Parameter", ROUND(k_cf * (ep_t ^ n_ex), 2)))Post-Considère Stamping Split Check (eps = 0.30) & Unit-Strain K Identity (eps = 1.0)
Stretch panel strain B4 to 0.30 (> n = 0.25): E5 switches to "Necking Instability (eps > n)"! Set B4 = 1.0: Flow Stress E2 equals K = 600.00 MPa identically!
=IF(ISTEXT(E2), E2, IF(B4 > B3, "Necking Instability (eps > n)", "Stable Uniform Work-Hardening"))Try the example
Use the original values or change them to test your understanding.