Deformation Mechanism-Based Analysis of Blank Holder Force Paths in Hydromechanical Deep Drawing of Parabolic Cups

In hydromechanical deep drawing (HMD) of parabolic cups made of SPCC cold-rolled low-carbon steel grade JIS G 3141, localized thinning at the acute apex is a persistent failure that conventional “black-box” optimization methods cannot physically explain. This study develops a mechanism-informed methodology that integrates physical experiments with a validated finite element (FE) model. Interrupted drawing experiments then divided the HMD process into four transient stages (pre-bulge, thin-out, draw-in, and re-strike) and identified the thin-out stage, at a punch stroke of 73 mm, as the critical failure window. An LS-DYNA FE model was calibrated with these measurements, and forming limit diagram (FLD) analysis was used to evaluate the strain paths. A linear blank holder force (BHF) profile that keeps the force low during the thin-out stage shifted the average strain state from a uniaxial tensile mode (strain ratio β = −0.50) toward a drawing-dominated mode ( β = −0.69), increasing material draw-in by 59.1% and reducing the maximum thinning from 30 to 25%. To avoid the flange wrinkling caused by an excessive BHF reduction, a bi-linear BHF profile was proposed to lower the force during the thin-out stage and increase it during the draw-in and re-strike stages; FE simulation predicts a further reduction in maximum thinning to 23.74% without flange wrinkling. As the four-stage deformation sequence and the thin-out-driven failure mechanism arise from the part’s acute curvature, this procedure of stage identification, FLD-based diagnosis, and stage-dependent BHF design may be transferable to other severely curved axisymmetric HMD components.