A systematic approach of the energy method is proposed for analysis of axisymmetric and three-dimensional sheet metal forming. In the proposed formulation, the total deforming region is divided into several regions by the geometric characteristics and the contact boundary condition. The velocity field is assumed in each region and is constructed over the global deforming region satisfying the conditions of the kinematically admissible velocity field. Then, the solution is obtained by optimizing the total energy with respect to some given parameters assumed in the velocity field and the geometric profile.
For axisymmetric sheet metal forming, the total deforming region consists of several sections of four types defined by simple geometry. In order to verify the effectiveness of the proposed method, hemispherical punch streching is analyzed considering the frictional effect at the contact region as well as axisymmetric cup drawing by considering the bending effect. The validity of the present method is confirmed by comparing the computed results with the reported finite element solutions as well as with the corresponding experimental results.
For analysis of three dimensional sheet metal forming, a formulation is introduced in which a continuous velocity field is appropriately constructed and the hydrostatic bulging with elliptic diaphragm is analyzed as a simple example for three-dimensional sheet metal forming. The computed results are compared with the existing experimental results and both results are shown to be in good agreement.
In order to analyze general sheet metal forming by the proposed energy method, an approach is proposed in which the geometric shapes of subregions are constructed by sweeping the section curves defined at the boundaries of each region and the velocity field is constructed by sweeping the velocity boundary functions at each region. Hydrostatic bulging with a rectangular diaphragm and deep drawing by elliptic and clover type punches are analyzed. The computed results are then compared with the corresponding experiments or the finite element solutions and it is found that the theoretical predictions are in excellent agreements with them.
It is, therefore, shown that the present approach of the energy method is effective in analyzing axisymmetric and three-dimensional sheet metal forming and is further applicable to more complicated industrial problems.