Optimization of initial blank shape predicted based on inverse finite element method

Abstract

For stamping of sheet metals and converting them to specific product shapes without failure, the initial blanks should be correctly designed. Therefore, initial blank design is a critical step in stamping design procedure. In the present paper for calculating the total deformation gradient and its relation to each step's deformation gradient tensor $(F)$, a modified kinematics formulation will be introduced. This formulation has been used in connection with the ideal forming theory for predicting the initial blank shape of the specified products with defined thickness. In the ideal forming theory, each material element is prescribed to deform in a minimum plastic work path and ideal process is obtained when the deformations are most evenly distributed in the final products. The latter has been assumed for developing an inverse finite element method (FEM) code to predict the blank shape and size in one step, which has been applied for rectangular cup. Even the results show the capability of the new algorithm in designing the initial blank shape for stamping products but the predicted blank shapes are oversized. For overcoming such problem, some kind of optimization must be applied. By considering all the conditions, forward FEM has been selected for optimization of blank shape and size. The accuracy of optimized blank shape and size has been evaluated using experimental work. For strain measurements, over the surface of blanks square grid have been printed. The results of experimental work confirmed the applied procedure for defining the shape and size of rectangular blank. The examination of thickness strain using printed grids show the superiority of optimized blanks.

Introduction

In order to perform a successful sheet metal forming operation and avoid shape error, tearing and wrinkling defects, process and material variables such as tools geometry, blank holding force, friction, blank shape, sheet thickness and material properties, etc. should be optimized. One of the main parameters, which must be defined at the beginning of any sheet metal forming processes, is the initial blank shape and size. Using an optimum blank not only leads to material saving but also minimizes defects. The initial blank shape and size can be determined by experimental trial and error or numerical methods. Since the first method usually requires a significant number of iterations before achieving the final shape it is time consuming and due to the wasting of large amount of sheet material during the trial work it is expensive as well. In order to overcome these problems, several numerical methods have been developed in recent decades. These numerical techniques can be categorized in slip line [1], [2], geometrical mapping [3] and finite element-based methods [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Because of inherent capability of finite element techniques, the last group has received most attention and used in connection or combination with other concepts. Batoz et al. proposed an inverse finite element method (FEM) to obtain the initial blank shape and the thickness strain distribution by assuming membrane stress and total deformation theory of plasticity [4], [5]. Lee et al. by using the same procedure and considering boundary condition tried to find the initial blank shape in one and multi-steps [6], [7], [8]. Shi et al. used elements with bending capability in connection with one step procedure and tried to predict the initial blank shape [9]. Yang et al. used the same assumptions and tried to apply the technique without considering the exact shape of tools [10]. Guo et al. extended mentioned method for considering the effect of constraint imposed by draw bead for predicting the initial blank shape of complicated products [11]. Kim et al. using FEM proposed roll back method in order to predict the optimum blank shape [12]. Ku et al. used backward tracing technique for designing the initial blank [13]. After introduction of ideal forming theory by Chung et al., it has been applied as another way for prediction of final product and initial blank shape using various optimization methods such as the volume addition/subtraction technique [14], [15], [16], [17], [18]. Shim used combination of FEM and the sensitivity method for prediction of initial blank shape [19].

The applied methods have their own characteristics and among them finite element-based methods seem to be the best choice. They can be categorized into forward and inverse finite element techniques. The forward approach requires suitable initial guess and experienced user. In the case of complex shapes, initial guess becomes very important and wrong estimation can lead to the extensive computational trial and error. When initial shape of the blank and the exact die contour are unknown, analyzing it in forward manner becomes very difficult and time consuming. Therefore, the most attractive procedure is inverse techniques. The inverse technique can result in suitable initial blank shape if the applied method has been based on sound theory, which considers the path dependence of plastic deformation. In ideal forming theory the path dependency of plastic deformation is considered and assumes that the elements deform in the minimum plastic work path. In the previous work, it has been shown that it is possible to improve the calculations’ accuracy in designing the blank shape by considering the deformation gradient in each discretized step of the forming process using inverse finite element [20]. Even using mentioned modification contributes to attain more satisfactory results but the predicted blank is oversized. For predicting the final blank shape and size, exact effect of boundary condition must be considered which if not applicable, adds a lot of complications to the calculations and extended calculation time or some kind of optimization should be used. For optimization of blank size various methods can be used. One of the optimization methods that besides optimization of blank size assures the successfulness of stamping operation is forward simulation of the predicted blank.

In the following, after brief discussion about the modified kinematical formulation and governing energy equation, the newly developed inverse finite element code will be introduced. Then the predicted blank shapes and sizes for rectangular shape without flange using mentioned code and optimized by forward simulation will be also presented and compared with experimental results.