The physical phenomena described across this series—from anisotropic plastic deformation and springback to necking and fracture—are predicted, optimized, and controlled in modern press shops using advanced Finite Element Analysis (FEA). Before physical tooling is cut, virtual stamping simulations model the nonlinear material response, boundary contact mechanics, and thermal-mechanical interactions. Accurate numerical predictions depend on selecting appropriate yield criteria, strain-hardening laws, and material damage models calibrated to experimental sheet test data.
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+———————+ +———————–+ +———————–+
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| Uniaxial / Biaxial | –> | Constitutive Material | –> | FEA Solver Engine |
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| Tension Test Data | | Model Calibration | | (LS-DYNA / AutoForm) |
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+———————+ +———————–+ +———————–+
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|
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+———————+ +———————–+ v
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| Die Geometry / Process| <– | Virtual Stamping | <– +———————–+
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| Compensation Adjust | | Anomaly Diagnostics | | Springback, Wrinkling |
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+———————+ +———————–+ | & Splitting Predictions|
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+———————–+
Advanced Yield Criteria for Anisotropic Sheet Metals
Classical isotropic yield criteria—such as von Mises or Tresca—assume that material yield strength is identical in all directions. However, cold-rolled sheet metals exhibit crystallographic texture that causes anisotropic yielding. To accurately capture direction-dependent plastic flow, specialized anisotropic yield functions have been developed.
Hill’s Quadratic Anisotropic Criterion (Hill ’48)
Hill’s 1948 anisotropic yield function extends the von Mises criterion by adding weighting parameters ($F, G, H, N$) aligned with the sheet’s orthotropic symmetry axes (rolling direction, transverse direction, and normal direction):
$$F(sigma_{yy} – sigma_{zz})^2 + G(sigma_{zz} – sigma_{xx})^2 + H(sigma_{xx} – sigma_{yy})^2 + 2Ltau_{yz}^2 + 2Mtau_{zx}^2 + 2Ntau_{xy}^2 = 1$$
While Hill ’48 effectively captures basic planar anisotropy in interstitial-free steels, it fails to accurately predict the plastic response of aluminium alloys due to the “anomalous behavior” where the yield strength in balanced biaxial tension deviates significantly from uniaxial values.
Barlat Non-Quadratic Yield Functions (Yld2000-2d, Yld2004-18p)
To overcome the limitations of quadratic criteria in aluminium and advanced high-strength steels, Barlat and co-workers developed non-quadratic yield functions using linear transformations on the Cauchy stress tensor. The general form of Barlat’s yield function incorporates an exponent $a$ related to the material’s crystal structure ($a = 6$ for $BCC$ steels, $a = 8$ for $FCC$ aluminium alloys):
$$Phi = vert{}S’_1 – S’_2vert{}^a + vert{}2S”_2 + S”_1vert{}^a + vert{}2S’_1 + S”_2vert{}^a = 2bar{sigma}^a$$
Where $S’$ and $S”$ are stress tensors modified by linear transformation matrices calibrated using experimental yield stresses and Lankford $r$-values along multiple orientations ($0^circ, 45^circ, 90^circ$, and balanced biaxial state). These non-quadratic functions accurately capture asymmetric yield surfaces, enabling precise simulation of thinning and earing in complex deep-drawn parts.
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Yield Surface Comparison
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Von Mises (Isotropic Circular/Elliptic)
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– – – Hill ’48 (Anisotropic Quadratic)
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—– Barlat Yld2000 (Non-Quadratic Aluminium Fit)
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Sigma_2
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^
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. – | – .
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/ |
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/ * * | * *
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| | |
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|–+—–+—–+–|—-> Sigma_1
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| | |
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* * | * * /
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| /
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‘ – | – ‘
Constitutive Hardening Models and Springback Prediction
When a stamped component is removed from the die cavity, the release of internal elastic stresses causes the part to unbend and distort—a phenomenon known as springback. Predicting springback requires constitutive models that accurately describe material behavior during complex loading, unloading, and reverse-bending strain paths.
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Bauschinger Effect During Reverse Loading
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True Stress (sigma) ^
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| Forward Plastic Loading
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| /
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| /
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+———+ (Unloading Path)
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/| /
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/ | /
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/ | +———– (Yielding occurs earlier in reverse)
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/ | /
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Reverse Loading –+ | /
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+—————————–> True Strain (epsilon)
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Isotropic vs. Kinematic Hardening: Pure isotropic hardening assumes that the yield surface expands uniformly in all directions during plastic deformation. However, when sheet metal undergoes reversed bending over a die radius, it exhibits the Bauschinger Effect—the yield strength in reverse loading is significantly lower than in forward loading.
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Yoshida-Uemori Kinematic Hardening Model: To capture the Bauschinger effect and transient work-hardening degradation, advanced FEA simulations utilize the Yoshida-Uemori model. This framework employs a two-surface system (a yield surface sliding inside a bounding surface) to track the evolution of back-stress tensors. Incorporating kinematic hardening reduces springback prediction errors from $30%$ down to within $5%$ of physical stamped parts.
Damage Mechanics and Forming Limit Prediction
Beyond predicting geometric shape, stamping simulations evaluate local material damage to prevent splits and micro-void coalescing during drawing.
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Continuum Damage Mechanics (CDM): Models such as the Lemaitre damage model track internal scalar damage variables ($D$), where $0 le D < 1$. As plastic strain accumulates, $D$ increases, softening the local elastic modulus ($E_{eff} = E(1-D)$) until macroscopic cracking occurs.
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GISSMO (Generalized Incremental Stress-State Dependent Damage Model): Integrated extensively in LS-DYNA, GISSMO uses triaxiality-dependent fracture strain curves to calculate cumulative damage along arbitrary, non-linear strain paths:
$$Delta D = frac{a}{epsilon_f(eta)} D^{left(1 – frac{1}{a}right)} Delta epsilon_p$$
Where $eta = frac{sigma_m}{bar{sigma}}$ is the stress triaxiality and $epsilon_f(eta)$ is the fracture strain envelope. This allows engineers to predict ductile fracture under shear, uniaxial tension, and equal-biaxial stretching conditions simultaneously.
Aluminium Sheets Kigali Rwanda