Evaluation of Forming Limit by the 3 Dimensional Local Bifurcation Theory

Abstract

A theoretical prediction and evaluation method for the sheet metal formability is developed on the basis of the three-dimensional local bifurcation theory previously proposed by authors. The forming limit diagram represented on the plane defined by the ratio of stress component to work-hardening rate is perfectly independent of plastic strain history . The upper and the lower limit of the sheet formability are indicated by the 3D critical line and the Storen -Rice's critical line on this plane, respectively. In order to verify the above mentioned behavior of the proposed forming limit diagram, the experimental research is also conducted. From the standpoint of the mechanical instability theory, a new concept called instability factor is introduced. It represents a degree of acceleration by current stress for developing the local bifurcation mode toward a fracture. The instability factor provides a method to evaluate a forming allowance which is useful to appropriate identification for a forming limit and to optimize the forming condition. The proposed criterion provides not only the moment to initiate the necking but also the local bifurcation mode vector and the direction of necking line.

Local bifurcation and instability theory applied to formability analysis

“…In those cases the localized mode vector m is consistent with cleavage action, which may be regarded as the most dangerous situation for fracture. In fact, the results suggest that thickness reduction, caused by local straining, occurs only for positive stress ratios [12].…”

“…For 1≤λ f <2 the status is warning and, finally, for 2≤λ f <3 the status is danger. This instability factor allows estimating if the forming process can continue without risk of fracture [12]. Figure 3 presents the interpretation of the instability factor in the FLD on the σ i /h plane.…”

“…Thus, materials with lower ductility present lower values for K C . Since for each material different limit curves are predicted, according with its mechanical properties, the value for the material parameter K C can be determined adjusting the limit strain obtained experimentally with the one predicted by S-R theory, for a specific strain path, such as uniaxial tension or equibiaxial strain path [12]. Figure 2 highlights that the influence of this material parameter is higher for the 3D limit curve.…”

“…Thus, it suggests that the forming limit diagram represented in the s/h space is completely objective. Both the Stören-Rice mode (S-R) limit and the 3D mode limit can be represented in this space [1,12]. The local necking initiates at the S-R limit and develops towards the 3D mode limit with the increase of the stress state.…”

“…Furthermore, the development of local bifurcation is accelerated by the absolute value of the functional I[m, n, s] [8]. Thus, the degree of acceleration by the current stress state for local bifurcation mode towards fractures can be evaluated using an instability factor, defined as [12] I ¼ I½m; n; s ½ Á W d ðm; n; sÞ SR ¼ hH…”