Numerical determination of forming limit diagrams of orthotropic sheets using the `3G' theory of plasticity A. Haddad, P. Vacher, R. Arrieux * Laboratory for Applied Mechanics, ESIA-University of Savoie (CESALP), B.P. 806, 74016 Annecy, France Abstract This paper deals with the application of the `3G' theory of plasticity to the numerical determination of forming limit diagrams and forming-limit stress curves. It describes the straining of materials by means of sliding in planes at 458 to the principal stress direction. This theory of plasticity combines three elements: (i) ¯ow relations between the principal slidings and the principal stresses; (ii) a yield surface; and (iii) a hardening law, which is a relationship between a limit shear stress and the three glidings. Here it is applied to Marciniak and Kuczinsky's model which assumes a heterogeneous material represented by a strip of lesser thickness. When displacements are imposed to the edges of the model, straining occurs and little by little it concentrates into the heterogeneous region, thus simulating the occurrence of necking and its growth. This method allows the determination of the forming-limit strain and stress diagrams, and the study of in¯uence of the material behaviour parameters on the level and the shape of these curves. This work provides a tool to detect the onset of necking at the time of the numerical simulation of deep drawing operations, using FEM software where the 3G behaviour law is still implemented. # 1999 Elsevier Science S.A. All rights reserved. Keywords: Forming limit diagrams; Necking deep drawing; Plasticity; Anisotropy 1. Introduction According to the importance of the quantity of converted materials, the deep drawing of thin steel sheets is one of the most important manufacturing processes. A lot of research corresponding to this ®eld of activity has been carried out. It concerns three main directions: (i) the behaviour laws of the material; (ii) The numerical simulation of drawing opera- tions by means of the ®nite-element method; and (iii) the prediction of plastic instability. Presently, Hill's quadratic model [1] is generally accepted to describe the behaviour of thin steel sheets, and it is used in most calculation codes. It is also used for the determination of a very ef®cient criterion to predict the onset of necking at the time of the numerical simulation of the forming opera- tion: i.e. the forming-limit stress diagram. This criterion can be determined in two ways [2]: (i) by means of a step-by-step plastic calculation along the experimental strain paths of laboratory tests; and (ii) in an analytical way, using Marci- niak's two-zone model [3]. Recently, the Centre de Recherche Metallurgique (CRM) in Lie Áge proposed a new model to describe the behaviour of thin orthotropic sheets [4]. This model allows the description of the material anisotropy in a more general way than the classical Hill's theory. It assumes that the plastic deforma- tion is due to glidings in planes at 458 to the principal stress directions. Moreover, the hardening law is not a single curve between the equivalent stress and the equivalent strain, but a relationship between the threshold shear stress and the three glidings. This model was implemented in a calculation software [5]. In a recent work the present authors proposed a method to determine the forming limit stress diagram [6] using this model, in a step-by-step plastic calculation along the strain paths of blanks strained up to the point of plastic instability on an experimental device. Here is proposed the determination of the FLD using Marciniak and Kuczinsky's model with two zones, in which are introduced the `3G' behaviour law. 2. Behaviour law According to physical considerations about the disloca- tion displacements in the most strained plans, this model assumes that the total straining d" is the superposing of shear Journal of Materials Processing Technology 92±93 (1999) 419±423 *Corresponding author. Tel.: +33-450-66-6099; fax: +33-450-66-6020 E-mail address: arrieux@univ-savoi.fr (R. Arrieux) 0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved. PII:S0924-0136(99)00170-3