Local bifurcation and instability theory applied to formability analysis

The design and optimization of both sheet metal formed parts and processes are nowadays carried out virtually making use of numerical tools by finite element analysis. Such virtual try-out approach contributes with significant savings in terms of money, time and effort in the design, production and process set-up of deep drawn parts. The analysis of either forming success operation or surface defects, in each of the development phases, is generally performed by means of the material’s forming limit diagram (FLD), since it allows to define a safe region that reduces the probability of: (i) necking; (ii) wrinkling and (iii) large deformation occurrence. However, the FLD represented in the strain space is known to present some disadvantages. To overcome this problem, Ito and Goya proposed a local bifurcation criterion that defines the critical state for a local bifurcation to set in as a function of the stress level to work-hardening rate ratio, leading to a FLD represented in the stress space. This suggests that the FLD obtained is completely objective in the sense that it is completely independent of the strain or stress history paths (Ito et al. 2000). In this work the Ito and Goya model is used to evaluate formability, as well as fracture mode and direction on the deep drawing of a square cup. Since the analysis is performed based on the stress state, it is also possible to determine an instability factor that “measures” the degree of acceleration by current stress for the local bifurcation mode towards fracture. The selected example highlights the potential use of the criterion which, once combined with the finite element analysis, can undeniably improve the mechanical design of forming processes.