Theoretical model of a metal tube under inversion over circular dies

The inversion of a metal tube over a circular die is an important energy absorption device. Taking into consideration the curvature change during inversion process, this study proposes a two-dimensional theoretical model. Using rigid-perfectly plastic (R-PP) material idealization, its steady inversion load, curling radius after the tube's separation from the die, and the optimization of the die radius are analytically obtained based on energy conservation. Compared with the FE simulations, the current model greatly improves the prediction accuracy of the steady inversion force, with the deviation smaller than 10%. The prediction of the tube radius after inversion agrees well with the FE simulation and experimental data. Out of our expectation, the optimized die radius is found to be different from that of free inversion radius of the same tube. In this study, a simple formula of the optimized die radius is given.