Constitutive Equations Based on Non-associated Flow Rule for the Analysis of Forming of Anisotropic Sheet Metals

Boxun Wu, Koichi Ito, Naomichi Mori, Tetsuo Oya, Tom Taylor, Jun Yanagimoto

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)


In this study, an anisotropic constitutive model based on the non-associated flow rule was developed for anisotropic sheet metals. This model was defined in the quadratic form of the Hill’s anisotropic function under a general three-dimensional stress condition. The anisotropic parameters for the yield function were identified using the directional planar yield stresses, bulge yield stress and shear yield stress, while those for the plastic potential function were identified using the directional r-values. A full expression related to the non-associated flow rule was applied and the model was implemented into the finite element code ABAQUS. A static-implicit analysis and the solid element were applied. Capability of the developed model for predicting the anisotropic behavior of sheet metal was investigated by considering two different sheet metal forming processes: cylindrical cup drawing of AA2090-T3, A6061P-T6 and SPCE; and hole expansion forming test of A6016-O. Cup heights and through-thickness strain distributions obtained from the simulations were compared with the experimental data. Results demonstrate that the developed material model considering 3D condition can improve accuracy of predicting the anisotropic behaviors. Furthermore, the simple formulations are efficient and user-friendly for computational analyses and solving the common industrial sheet metal forming problems.

Original languageEnglish
Pages (from-to)465-480
Number of pages16
JournalInternational Journal of Precision Engineering and Manufacturing - Green Technology
Issue number2
Publication statusPublished - 2020 Mar 1


  • Anisotropic material
  • Constitutive behavior
  • Finite element
  • Sheet metal forming

ASJC Scopus subject areas

  • Renewable Energy, Sustainability and the Environment
  • General Materials Science
  • Mechanical Engineering
  • Industrial and Manufacturing Engineering
  • Management of Technology and Innovation


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