Springback Evaluation with Several Phenomenological Yield Criteria

In the last decades, several orthotropic phenomenological yield criteria were proposed to accurately describe the anisotropic behaviour of the rolled metallic sheets that are widely used in the production of sheet metal formed parts. In this work, the authors evoke the implementation of several yield criteria in the implicit finite element code DD3IMP, namely the Hill 1948 [1], the Cazacu & Barlat 2001 [2] and the isotropic Drucker’s yield criterion mixed with a linear transformation [2]. The Numisheet’2002 benchmark “Unconstrained Cylindrical Bending” was selected to evaluate the influence of the yield criteria on the springback evaluation. A 6111-T4 aluminium alloy was used. Its mechanical characterization was performed by Alcoa [3]. The results show that only the Cazacu and Barlat 2001 yield criterion fits very accurately both the uniaxial tensile yield stresses and the r-values. All the other yield criteria fits rather well the experimental r-values. Concerning the influence of the yield criteria on springback, only a minor influence was found. However, the results obtained from the Cazacu and Barlat 2001 yield criterion are clearly the closest to the experimental ones [4]. Introduction At the present time, the reliability of finite element method simulations is still a dilemma, which easily explains the reason of the continuous efforts and developments that have been done by all the scientific community in the last decades. In sheet metal forming, numerous works have been published concerning the improvement of both the behaviour laws and the initial yield locus description. In fact, for computer simulation of sheet metal forming processes, a quantitative description of plastic anisotropy by the yield locus of the material is required. The accuracy of the numerical results is obviously correlated with the accuracy of this description. Several phenomenological yield criteria are implemented in the implicit finite element code DD3IMP (contraction of ‘Deep Drawing 3-D IMPlicit code’) [5]. DD3IMP is a 3-D elastoplastic finite element code with an updated Lagrangian formulation, following a full implicit time integration scheme, large elastoplastic strains and rotations, with several isotropic and anisotropic constitutive models (8 yield criteria and 7 isotropic/kinematic hardening laws). The Coulomb’s law models the frictional contact problem, which is treated with an augmented Lagrangian approach. This code has a 3-D finite element library with different types of elements and integration schemes. The main goals of this work are to compare the fitting accuracy of the proposed yield criteria to the experimental results, and to present a comparative study of the influence of those yield criteria in the evaluation of the springback effects. Numisheet’2002 benchmark “Unconstrained Cylindrical Bending” [4] was select to evaluate the influence of the yield criteria on springback. A 6111-T4 aluminium alloy was used. The mechanical characterization was performed by Alcoa [3]. A new methodology (the minimization of a functional) to the identification of the anisotropy parameters of the yield criteria is also followed and presented. Materials Science Forum Online: 2004-05-15 ISSN: 1662-9752, Vols. 455-456, pp 732-736 doi:10.4028/www.scientific.net/MSF.455-456.732 © 2004 Trans Tech Publications Ltd, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications Ltd, www.scientific.net. (Semanticscholar.org-11/03/20,15:41:33) Title of Publication (to be inserted by the publisher) 733 The Implemented Phenomenological Yield Criteria. A yield surface is generally described by an implicit equation of the form: 0 Y σ = − = F , (1) where Y is the flow stress, which evolution is given by a hardening law, and σ is the equivalent stress that “measures” the stress state. Y can also be seen as the size of the yield surface and therefore their evolution law is also the evolution law of the yield surface. To describe σ , and furthermore the anisotropic plastic response of the rolled metal sheets, several yield criteria proposed in the bibliography were implemented. Hill 1948 yield criterion [1]. Basically, the yield criterion proposed by Hill is a generalization of the von Mises’s distortional energy criterion to orthotropy. The well known and widely used quadratic Hill 1948 yield criterion is given by the expression ( ) ( ) ( ) 1 2 2 2 2 2 2 2 2 2 2 yy zz zz xx xx yy yz zx xy F G H L M N   = − + − + − + + +     σ σ σ σ σ σ σ τ τ τ , (2) where σ is the tensile equivalent stress (a measure of the yield surface), F , G , H , L M and N are the anisotropy parameters of the yield criterion, and ... xx xy σ τ are the stress components of the deviatoric Cauchy stress tensor. Alternatively, if the kinematic hardening is taken into account, the deviatoric stress tensor ( ′ σ ) should be replaced by the effective stress tensor ( ′ − σ X ), where X is the backstress tensor. CB2001 yield criterion [2]. The CB2001 yield criterion is a generalization of the Drucker’s isotropic yield criterion to orthotropy. To describe yielding of orthotropic materials, a generalization of the second and third invariants of the deviatoric stress tensor were developed. These generalized stress invariants were then used to extend the Drucker’s isotropic yield criterion to orthotropy: ( ) ( ) { } 1 6 3 2 0 0 2 3 27 J c J   = −     σ , (3) where 0 2 J and 0 3 J are the second and third generalized invariants of the effective stress tensor: ( ) ( ) ( ) 2 2 2 0 2 2 2 3 1 2 2 4 5 6 6 6 6 xx yy yy zz xx zz xy xz yz a a a J a a a σ σ σ σ σ σ σ σ σ = − + − + − + + + , (4)