Sheet metal forming processes experience very complicated deformations effected by process parameters such as the blank shape, blank holding force, die geometry, sheet thickness, friction, and so on. Although these process parameters influence the deformation mechanism and the quality of deformed parts, the optimum values of process parameters are determined by intuition and experience through trial and error. A more systematic method such as finite element analysis can simulate complicated sheet metal forming processes and provide useful information. However, general finite element analysis is generally carried out with given process parameters and thus requires numerical trial and error with enormous time and cost to determine the optimum values of process parameters. For this reason, some approaches to find directly the process parameters have been developed. However, most design purpose approaches have shortcomings such as geometric restriction, neglecting variation of material properties due to deformation path, and path-independent boundary conditions which causes deformation error.
In order to overcome such problems, a new finite element inverse method is introduced for direct prediction of the blank shapes and strain distributions from desired final shapes. The finite element inverse method in axisymmetric deep drawing enables the determination of process parameters within a small error range in very short computing time before the process design. The finite element inverse method in this papers adopts Hencky's deformation theory, Hill's second yield criterion, and simplified boundary conditions. The (one step) inverse method is, then, extended to the multi-step inverse method in order to reduce the amount of error and to decide the intermediate shape.
The present algorithm has been implemented in a finite element code and applied to drawing of cylindrical and square cup in case of axisymmetric deep drawing processes. Blank shapes and true strain distributions were obtained. And then, the multi-step inverse method was applied to a cylindrical cup, complicated shape such as motor case and cylindrical cup with large limiting drawing ratio. Finally, intermediate shape was decided by using of FLD(forming limit diagram) in safe region. The error induced by the one step inverse analysis was reduced significantly by the multi-step analysis. The process parameters of the above examples were calculated rapidly within a small range of error. Consequently, these examples fully demonstrate that the developed algorithm is a good finite element code for the purpose of process design.
