A finite element formulation based on non-associated plasticity for sheet metal forming

Abstract

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps. The updated Lagrangian formulation of CBR shell element [Yoon, J.W., Yang, D.Y., Chung, K., 1999. Elasto-plastic finite element method based on incremental deformation theory and continuum based shell elements for planar anisotropic sheet materials. Comp. Meth. Appl. Mech. Eng. 174, 23–56] coupled with the developed constitutive formulations is implemented into the finite element program ADINA 8.1 (2003) via user defined subroutine CUSERG. The results of the cylindrical cup drawing for Al2008-T4 and Al2090-T3 sheet samples are evaluated by comparison with experimental data and predictions of Barlat [Barlat, F. et al., 1997b. Yield function development for aluminum alloy sheets. J. Mech. Phys. Solids 45, 1727–1763] associated model.