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MakeCoTech M–K / FLD Explorer v1.3.5

Dedicated Marciniak–Kuczyński forming-limit derivation • FLD-first workflow • von Mises / Hill48 • Swift calibration • strain-path guides • Excel export
Engineering exploration / educational use. Experimental forming-limit data remains the preferred reference for final validation. This tool provides a numerical M–K engineering estimate whose accuracy depends on constitutive inputs, anisotropy, hardening, strain path, localization criterion and imperfection factor f₀.

Case

M–K material / solver

Inputs used by the FLC solverEnter the actual material parameters directly below. The auxiliary Swift-fitting calculator further down the page is intentionally separate and does not update the FLC calculation automatically.
The yield model is used inside the M–K matrix/groove equilibrium solve. Swift hardening remains isotropic in both options.
Direct coefficients are normalized internally by G+H so the RD uniaxial flow stress remains the scalar Swift flow stress. In this in-plane, no-TTS tool only F, G, H and N enter the plane-stress calculation.
α defines material orientation relative to the global major-strain axis x₁. The groove angle φ₀ is independent. r-values require transverse/thickness plastic-strain information; they cannot be obtained from an axial stress–strain curve alone.
Current FLC is up to date.
Swift isotropic hardeningσ̄ = K (ε₀ + ε̄ᵖ)ⁿσᵧ = K ε₀ⁿ   at   ε̄ᵖ = 0K [MPa] scales the flow stress, n controls work hardening, and ε₀ sets the initial yield level. With Hill48, only the yield-surface shape / plastic-flow direction becomes anisotropic; the scalar Swift hardening law remains isotropic.
σᵧ and ε₀ are kept consistent with σᵧ = K ε₀ⁿ. Editing σᵧ updates ε₀; editing K, n or ε₀ updates σᵧ. Set groove step to 1° for the paper's angular resolution.
Model scope. This dedicated FLC builder solves the conventional M–K matrix/groove problem with Swift hardening, evolving groove orientation, compatibility along the groove, normal/tangential traction equilibrium, thickness evolution and a thickness-strain-rate localization criterion. Choose von Mises for isotropic yielding or Hill48 for planar anisotropy calibrated from r-values. Through-thickness shear, anisotropic hardening, texture evolution and non-proportional loading remain outside this version.

Forming Limit Diagram (FLD) — calculated FLC & strain-path guides

Current M–K: all-angle scan1967-style calculation: fixed φ₀=0°, RHS onlyEyckens 2009 AA3003 traditional M–K — digitized Fig. 5
Strain-path guides: uniaxial compression · pure shear · uniaxial tension · plane strain · biaxial tension · equibiaxial tension
FLD zones: safe = below 84% of the current FLC and above the wrinkle indicator · near split = 84–100% of the FLC · split = above the FLC · wrinkle tendency = compression-side indicator below the blue dashed boundary. These shaded zones are visual engineering guides, not additional material criteria.
The larger colored dots are the 100 independently solved radial material rings used by the reduced-order cup model. The cyan dotted reference is an approximate digitization of the traditional M–K curve in Figure 5 of Eyckens et al. for AA3003; it is a like-for-like benchmark only when the AA3003 case is loaded and the yield model is von Mises, matching the paper. The digitization is intended for validation/trend comparison, not as a replacement for the authors’ original numerical dataset. In dense-cloud mode, smaller translucent points and a faint ribbon are interpolated between adjacent solved rings to represent the continuous annular material between them. They make the display visually closer to an FE postprocessor, but they are not additional FE integration points and do not add new physics. Each point moves as punch stroke changes. The blue-outlined dot remains the slider-selected ring. The yellow curve is the current paper-aligned conventional M–K FLC. The dashed violet comparison uses the original 1967 groove-orientation assumption: a thinner groove running along the minor-strain direction, i.e. groove normal n aligned with the major-strain direction (φ₀=0°), and is shown only on the right-hand side. It uses the same current material and the same modern numerical localization detector so the effect of groove-angle treatment can be compared directly; it is not a digitized 1967 published curve.
How to read the FLD. A material point below the selected M–K FLC has not yet reached the modelled localized-necking limit. Reaching/crossing the curve means the M–K localization condition has been met for that proportional strain path. The blue wrinkle region is a separate reduced-order compression indicator and is not part of M–K.
Which M–K criterion is actually used? The yellow current curve uses the Eyckens-type numerical criterion used by this solver: |D₃₃ᵇ| / |D₃₃ᵃ| ≥ Clocwith default Cloc=10, or loss of a solution to the groove equilibrium problem. The violet 1967-style comparison curve fixes φ₀=0° to reproduce the original groove-orientation assumption, but deliberately uses the same modern detector so only the groove-orientation assumption changes.
AUXILIARY VISUALIZATION

Yield locus — von Mises / Hill48

This plot is a visualization. It reflects the yield-model parameters entered in the main M–K material/solver panel above. Edit the actual von Mises / Hill48 inputs there; this graph itself does not define a separate material card.
von Mises referenceHill48 from current anisotropy / RD angle
Plane-stress section in the global σ₁–σ₂ plane with global τ₁₂ = 0. For Hill48, the stress state is rotated into the material RD–TD axes before evaluating the yield function, so changing the RD angle α changes the section seen in global axes. At α = 0°, r₄₅ mainly affects shear/off-axis yielding and therefore has little effect on this zero-global-shear section.
AUXILIARY FITTING CALCULATOR

True plastic curve → Swift fit

Important: this fitting calculator is not connected to the FLC calculation. Use it only to estimate Swift parameters from experimental data. If you want those fitted values in the M–K/FLC calculation, manually enter the resulting Y₀, K, n and ε₀ in the main M–K material/solver inputs above. Young's modulus E is also entered separately in the main input panel.
Enter only the plastic hardening data. The first row should represent the onset of plastic flow: ε̄ᵖ = 0 with non-zero yield stress — do not add a (0,0) point. The values below are an editable example.
Swift fit only. The calculator fits σ̄ = K(ε₀ + ε̄ᵖ)ⁿ to the data shown here. It does not convert engineering data, does not fit the elastic region, and does not automatically send the fitted values to the FLC solver.
experimental / entered pointsSwift fit
0valid plastic points
—Swift K [MPa]
—Swift n
—Swift ε₀
—Swift Y₀ [MPa]
—fit RMSE [MPa]
Editable example data are loaded. Press Fit Swift law to calculate K, n, ε₀ and Y₀.
Data conventionUse true plastic strain and true stress only. Young's modulus and Poisson's ratio are not fitted here; enter them directly in the main FLC material inputs.

How one M–K point becomes the FLC

This animation shows the inner M–K calculation used to create one FLD point. For a chosen matrix strain ratio ρ₂₂, many initial groove orientations φ₀ are tested. The orientation that localizes at the lowest matrix major strain is the critical groove; its matrix strains at localization become one point of the FLC.
Original M–K (1967): physical instability criterion. The original paper describes loss of stability as progressive strain concentration in the thinner groove, while deformation in the adjacent uniform sheet progressively fades; the limiting surrounding-sheet strain is the forming limit ε*. It does not introduce the later universal “10× thickness-strain-rate” threshold used below.
Numerical M–K criterion used by this tool / Eyckens formulation.|D₃₃ᵇ| / |D₃₃ᵃ| ≥ ClocThe Eyckens paper uses Cloc = 10 and a second criterion: localization is also declared if the nonlinear groove equilibrium/compatibility problem no longer has an acceptable solution. The paper notes that the precise large number has little influence because the groove thinning rate rises extremely rapidly near incipient localization. The editable “M–K localization ratio” is Cloc.
Abaqus M–K style criterion — for comparison. Abaqus/Explicit can monitor groove-to-nominal deformation-severity ratios for equivalent plastic deformation, normal deformation, and tangential deformation; the default critical severity factors are 10. Setting all severity factors to zero makes the Abaqus M–K check rely on nonconvergence of the equilibrium/compatibility equations. This tool currently uses the Eyckens thickness-strain-rate detector above, not the full Abaqus three-severity implementation.
Reference basis.
Marciniak & Kuczyński (1967), Limit Strains in the Processes of Stretch-Forming Sheet Metal, International Journal of Mechanical Sciences 9, 609–620, DOI 10.1016/0020-7403(67)90066-5.

Eyckens, Van Bael & Van Houtte (2009), Marciniak–Kuczynski type modelling of the effect of Through-Thickness Shear on the forming limits of sheet metal, International Journal of Plasticity 25, 2249–2268, DOI 10.1016/j.ijplas.2009.02.002.
1 change ρ₂₂ → next strain path2 scan all φ₀3 find first localization4 store (ε₂*, ε₁*)5 repeat → FLCCurrent FLC is up to date.
Left: necking major strain obtained for each tested initial groove orientation φ₀ at the selected ρ₂₂; the minimum is the critical groove. Right: the corresponding limit point is placed at (ε₂, ε₁) = (ρ₂₂ ε₁*, ε₁*). Repeating this outer loop for different ρ₂₂ values constructs the FLC point-by-point.
Original 1967-style versus later full-FLD construction. The original M–K concept assumed an initially thinner, infinitely long groove oriented along the minor-strain direction and used the onset of strain localization / loss of stability to identify the limiting surrounding-sheet strain. Later full-FLD M–K formulations scan possible initial groove orientations and retain the lowest major strain at localization. In the comparison curve above, the original geometry/orientation assumption is isolated by fixing φ₀=0°; the same modern 10×/equilibrium detector is deliberately retained for both curves. Therefore the violet line is a controlled 1967-style calculation, not the historical published 1967 FLC itself.
Reference basis. The conventional M–K formulation and AA3003 validation parameters follow Eyckens, Van Bael & Van Houtte (2009), which revisits the conventional model and uses f₀=0.998, K=184 MPa, ε₀=0.00196 and n=0.224 for AA3003. The historical comparison is labelled “1967-style” because it fixes the groove orientation assumption while retaining the same modern numerical localization detector; it is not a digitized copy of the 1967 curve.