The sheet metal forming process has taken an important role in industries because of its various advantages. There have been remarkable advances in sheet metal forming analysis with the increasing need from industries although it is very difficult to determine process parameters before real manufacturing due to very complicated deformation. Deformation during the forming process is effected by many process parameters such as the die geometry, the blank shape, the blank holding force, the bead force, friction and so forth. In spite of its importance on the process parameter determination, the effect of process parameters is yet informed by the experience, the intuition or the time-consuming computer analysis such as incremental finite element methods for small modification during the process design. When the number of process parameters considered becomes larger, it becomes more difficult to determine the optimum values of parameters, which satisfy the design specifications without any problem during the forming process. As the number of parameters considered is increased, it requires more time to choose the optimum parameters by trial and error.
In this paper, a design methodology is proposed to overcome the deficit of the trial-and-error approach in the design stage before manufacturing. A process parameter design system is proposed to determine process parameters directly using the design sensitivity analysis and the optimization scheme. A design sensitivity analysis scheme proposed deals with an elasto-plastic finite element method with explicit time integration and a direct differentiation method. The direct differentiation is concerned with large deformation, the elasto-plastic constitutive relation considering the planar anisotropy, shell elements with reduced integration and complicated contact between the sheet and the dies. The design sensitivities with respect to the process parameter are obtained with the direct analytical differentiation of the equilibrium equation for the finite element analysis. The present result is compared with the result obtained with the finite differentiation method (FDM) in sheet metal forming processes such as a hemi-spherical punch stretching, a cylindrical cup drawing process and a U draw-bending process for the verification of its accuracy and the versatility. The analysis results obtained with the present study show good agreement with the results obtained by FDM. The analysis also provides useful information on the inspection of the failure such as fracture or wrinkling. The abrupt change of the sensitivity of the major strain inspects the initiation of the element necking. The sudden change of sensitivity of the blank holder displacement inspects the initiation of wrinkling. The analysis shows that the sensitivity of state variables provides useful information not only for the strain control but also for the inspection of fracture or wrinkle initiation.
The present algorithm has been implemented the process optimization code and applied to sheet metal forming processes for demonstration of its validity. A new control algorithm of the variable BHF is proposed in order to improve the maximum cup height and to control the strain distribution in the cylindrical cup drawing process. The blank holding force is controlled incrementally in the divided punch stroke interval in order to maximize the cup height and to achieve the desired principal strain distribution of the drawn cup. The objective function is constructed so that the principal strain can follows the desired path within the safe region. The constraint condition is imposed to suppress the sudden lifting of the blank holder, which causes severe wrinkling during the forming process. Other constraints prevent the fracture by excessive stretching, which is prevented by inspecting the principal strain distribution on the forming limit diagram. The numerical experiment confirms that the proposed control algorithm for the variable BHF guarantees the improvement of the cup height and the regularization of the strain level in the weak part of the drawn cup.
The optimization algorithm has been applied to the U draw-bending process in order to reduce the amount of springback and improve the shape accuracy. The bending dominated forming process often suffers difficulties of inaccurate shape after unloading, which is caused by the non-uniformity of the stress distribution along the sheet thickness. The amount of springback can be reduced by increasing the blank holding force. The blank holding force can control the amount of spingback with a limitation due to the excessive stretching in the cup wall. In this paper, a variable blank holding force is proposed in the U draw-bending process in order to reduce the springback amount. The blank holding force is increased in the last 15% of punch stroke interval. In the optimization procedure, the increasing blank holding force is calculated. The objective function is constructed so that the longitudinal stress components in the integration points along the shell thickness have the minimum deviation from the average value. The constraint condition prevents the fracture by excessive stretching, which is predicted by inspecting the principal strain distribution on the forming limit diagram. The other constraint condition suppresses the excessive wall stretching by controlling the effective plastic strain. The obtained optimum blank holding force ensures that the stress distributions have very small inherent deviation caused by bending and unbending in the punch shoulder region and the die shoulder region. The result also shows that higher blank holding force improves the stress deviation by a small amount but excessive stretching occurs during forming. The analysis result after springback simulation confirms that the present optimization algorithm improves that shape accuracy after unloading effectively.
The results demonstrate that the proposed design methodology is applicable to the complicated sheet metal forming analysis and design.