Acta Mechanica

, Volume 229, Issue 6, pp 2495–2519 | Cite as

Calibration of a complete homogeneous polynomial yield function of six degrees for modeling orthotropic steel sheets

  • Wei Tong
Original Paper


For better modeling plane-stress anisotropic plasticity of steel sheets, a direct calibration method is proposed and detailed for establishing a positive and convex sixth-order homogeneous polynomial yield function with up to sixteen independent material constants. The calibration method incorporates parameter identification, convexity testing, and if needed, an adjustment of an initially calibrated but non-convex yield function toward a convex one. Some advantages of the calibration method include (i) a systematic solution of only linear equations for the sixteen material constants of a steel sheet with various degrees of planar anisotropy, (ii) a practical numerical implementation of the necessary and sufficient conditions for convexity certification of the calibrated or adjusted yield function, and (iii) an incremental procedure using a parameterized version of the initially calibrated and non-convex yield function that can always lead to an approximate sixth-order yield function with guaranteed convexity. Results of applying the proposed calibration method to successfully obtain convex sixth-order yield functions are presented for three steel sheets with experimental measurement inputs from various types and numbers per type of uniaxial and biaxial tension tests.

List of symbols


The orthotropic material symmetry axes corresponding to the rolling (RD), transverse (TD), and normal (ND) directions of a thin sheet metal

\(\sigma _x\), \(\sigma _y\), \(\tau _{xy}\)

Three in-plane Cartesian (two normal and one shear) components of an applied Cauchy stress \(\pmb {\sigma }\) in the orthotropic coordinate system of the sheet metal

\(\varPhi _2\), \(A_1\),..., \(A_4\)

Hill’s 1948 quadratic anisotropic yield function [9] in plane stress and its four material constants

\(\varPhi _4\), \(A_1\),..., \(A_9\)

Gotoh’s 1977 fourth-order anisotropic yield function [6] in Cartesian stress components (\(\sigma _x\), \(\sigma _y\), \(\tau _{xy}\)) and its nine material constants

\(\varPhi _6\), \(A_1\),..., \(A_{16}\)

The sixth-order homogeneous polynomial anisotropic yield function in Cartesian stress components (\(\sigma _x\), \(\sigma _y\), \(\tau _{xy}\)) and its sixteen material constants

\(\sigma _1\), \(\sigma _2\), \(\theta \)

The so-called intrinsic variables of an applied plane stress \(\pmb {\sigma }\) according to Hill [12, 13], namely, the in-plane principal stresses (\(\sigma _1,\sigma _2\)) and the loading orientation angle \(\theta \) between \(\sigma _1 (\ge \sigma _2)\) and the rolling direction of the sheet metal

\(\sigma _{\theta }\), \(r_{\theta }\), \(\sigma _b\), \(r_b\)

Yield stresses and plastic strain ratios under uniaxial tension (\(\sigma _1=\sigma _\theta > 0,\sigma _2=0\)) at the loading orientation angle \(\theta \); and yield stress and plastic strain ratio under equal biaxial tension (\(\sigma _1=\sigma _2=\sigma _b>0\))

\(\sigma _{p\theta }\), \(\sigma _{s\theta }\)

Yield stresses under near plane-strain tension (\(\sigma _1=2\sigma _2=\sigma _{p\theta }>0\)) and under pure shear stress (\(\sigma _1=-\sigma _2=\sigma _{s\theta }>0\)) at the loading orientation angle \(\theta \)

\(\phi _{6}\), \(F(\theta ), G(\theta ),H(\theta ), N(\theta )\)

The sixth-order yield function recast in intrinsic variables in a compact form of seven homogeneous principal stress terms and its four in-plane anisotropic functions. \(F_0\),...,\(F_6\), and so forth are the 25 nonzero Fourier cosine series coefficients of those four functions

\(\varPsi _{6A}\), \(\varPsi _{6B}\), \(\varPsi _{6C}\)

Three sub-determinants or leading principal minors of the Hessian matrix of the sixth-order yield function \(\varPhi _6\) in Cartesian stress components (\(\sigma _x\), \(\sigma _y\), \(\tau _{xy}\))

\(\psi _{6A}\), \(\psi _{6B}\), \(\psi _{6C}\)

Three sub-determinants \(\varPsi _{6A}\), \(\varPsi _{6B}\), \(\varPsi _{6C}\) of the Hessian matrix of the sixth-order yield function \(\varPhi _6\) recast in intrinsic variables (\(\sigma _1\), \(\sigma _2\), \(\theta \))

\(\rho \), \(\omega \)

The polar coordinates for the two principal stresses \(\sigma _1\) and \(\sigma _2\)


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Lyle School of EngineeringSouthern Methodist UniversityDallasUSA

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