Two Efficient Algorithms of Plastic Integration for Sheet Forming Modeling

A fast method called the “inverse approach” for sheet forming modeling is based on the assumptions of the proportional loading and simplified tool actions. To improve the stress estimation, the pseudo-inverse approach was recently developed: some realistic intermediate configurations are geometrically determined to consider the deformation paths; two new efficient algorithms of plastic integration are proposed to consider the loading history. In the direct scalar algorithm (DSA), an elastic unloading-reloading factor $γ$ is introduced to deal with the bending-unbending effects; the equation in unknown stress vectors is transformed into a scalar equation using the notion of the equivalent stress, thus the plastic multiplier $Δλ$ can be directly obtained without iterative resolution scheme. In the $γ$-return mapping algorithm, the equivalent plastic strain increment estimated by DSA is taken as the initial solution in Simo’s return mapping algorithm, leading to a stable, efficient, and accurate plastic integration scheme. The numerical experience has shown that these two algorithms give a considerable reduction of CPU time in the plastic integration.