Influence of the Plastic Properties on the Loss of Stability and Limit Strains of Sheet Metals Under Biaxial Tension

In their pioneering papers [1-3], Marciniak and Kuzinski introduced for the first time a theory and explanations for the loss of stability of sheet metals subjected to biaxial tension. It was hypothesized that an initial inhomogeneity in the sheet develops into a localized neck (or groove) in the direction normal to the larger principal stress. The initial homogeneity is taken to be a local variation in the sheet thickness. A detailed explanation of the mechanism of deformation and analytical treatment was provided for isotropic materials governed by the von Mises yield criterion and the special version of Hill’s orthotropic criterion proposed by Hill [4] for the case of materials with planar anisotropy (r-values considered to be the same in any in-plane orientation, the common value R being different from unity) (see [1]). This analysis and the ensuing strain-based method for establishing forming limit curves (FLC) have been the object of intense research for the last five decades. In particular, during the last decade there have been intense efforts towards enhancing the accuracy and robustness of experimental methods used for establishing and measuring the limit strains (e.g. development of new tests and procedures for determining the FLC) and gaining improved understanding of the influence of various material or process parameters on the limit strains. From a theoretical standpoint, crystal plasticity models have been recently used in conjunction with FLC as and efforts have been devoted to formulation of stress-based limit curves (e.g. see [5]). 

In this paper, we revisit the analysis of Marciniak and Kuzinski [2] of limit strains in sheet forming stretching for the case of materials displaying planar anisotropy, characterized by a unique parameter R. First, it is put into evidence deficiencies in Hill’s [4] planar yield function and a correction is provided. Next, using the improved formulation the analysis of the deepening of the groove and formation of the localized neck is provided. The improvements brought about by the corrections in the yielding and plastic dissipation formulation are discussed and pertinent examples are provided.

References:

[1] Marciniak, Z. and Kuczyński, K., 1967. Limit strains in the processes of stretch-forming sheet metal. International journal of mechanical sciences, 9(9), pp.609-620. 

[2] Marciniak Z. Analysis of necking preceding fracture of sheet metal under tension, La Metallurgica Italiana (1968) No 8., 701-709. 

[3] Marciniak Z. Deformation limite lors de la traction des tôles aux propriétés viscoplastiques. Formage et traitement des metaux (1970), pp. 23-28.

[4] Hill R (1950) The mathematical theory of plasticity. Oxford University Press. 

[5] Banabic D. et al (2010) Sheet metal forming processes. Springer, Berlin Heidelberg.