A stationary predictor corrector method for the simulation of elastic-plastic bending of axially moving plates with non-material finite elements

Abstract

Current simulation models for continuous sheet metal forming usually rely on transient simulation strategies that yield the stationary forming state after a large number of time steps. This transient progression of states involves a gradual downstream transport of internal plastic variables. Since the intermediate states have little practical relevance, this established strategy is both numerically inefficient and inconvenient for conducting parameter studies with respect to the desired stationary state. The here presented method overcomes these disadvantages by removing the time dependence altogether. In particular, a novel stationary predictor corrector algorithm is developed for the iterative solution of the problem of elastic-plastic bending of axially moving plates in an established finite element framework. A mixed kinematic parametrisation in the spirit of arbitrary Lagrangian Eulerian methods (ALE) is used for the finite element discretisation of a thin rectangular metal sheet that is modelled as an unshearable Kirchhoff-Love plate. Out-of-plane distributed, self-equilibrated loadings are imposed at spatially fixed lines to mimic the continuous, bending dominant roll forming process. Higher load magnitudes induce plastic deformations, which need to be transported in downstream direction through the non-material finite element mesh. A previously developed structural plasticity model is employed to formulate the corresponding constitutive laws directly in terms of plate curvature strains and stress resultants. The iterative solution of the axially moving plate bending problem is achieved by repeated application of elastic predictor and plastic corrector steps. Contrary to standard return-mapping schemes typically employed by transient algorithms, the plastic corrector phase is modified to additionally account for the advection of plastic variables in downstream direction. The condition of stationary operation is imposed directly such that the change of the plastic variables for a given material point is solely determined by convection. A spatial finite difference scheme is applied to solve the corresponding stationary advection problem along the streamlines of material particles, which are in alignment with the integration points of the regular finite element mesh. Clamped and free boundary conditions are imposed at the upstream and downstream boundaries of the open control domain, respectively. At steady state operation, plastic deformations arise in close proximity to the distributed external loading and persist in downstream direction. Conventional transient time-stepping simulations, conducted for the sake of reference, are clearly outperformed by the proposed stationary algorithm in terms of numerical efficiency.