An extended M-K model under surface traction with adjustable gradient distribution

The forming limit of sheets usually corresponds to the strain accumulation before localized necking and is closely related to the loading state. In the viscous pressure forming (VPF) method, the surface traction generated by the force-transmitting medium with high-viscosity fluid-flow characteristics on the sheet exhibits the obvious gradient distribution near the necking region. However, existing necking instability theories have not factored in the surface traction gradient distribution (STGD) and can't accurately explain the inherent mechanism of the VPF method with this loading state to improve the forming limit. To deal with the issue, STGD-related ‘stress balance’ equations are established and an extended M-K model with STGD response under 3D stress state (STGD-3D-MK model) is proposed, based on the analysis of the adjustable STGD generated in the VPF method during the necking development. The proposed model is used to explore the influence and mechanism of the STGD on the forming limit, as well as to provide suggestions on how to achieve the full positive effect of tangential surface traction. This research demonstrates that the forming limit is sensitive to the STGD, which affects the ability of surface traction to improve the forming limit. The STGD changes the stress path of the groove, affecting the development of necking. In addition, the critical STGD that can increase the limit strain is also provided and its magnitude is related to the strain path. Finally, the VPF-based experimental results are provided to support the effectiveness of the analysis of the STGD-3D-MK model. This work provides a new perspective on the loading method to improve the forming limit.