An Enhanced Hybrid Springback Compensation Approach: Springback Path – Displacement Adjustment Method For Complex Shaped Products of Sheet Metal Forming

Springback compensation is critical in sheet metal forming. Advanced techniques have been adopted in the design stage of various sheet metal forming processes, e.g. stamping, some of which are for complex shaped products. However, the currently available numerical approaches are not always sufficiently accurate and reliable. To improve the accuracy of springback compensation, an enhanced hybrid springback compensation method named Springback Path – Displacement Adjustment (SP-DA) method has been developed in this study based on the well-known conventional displacement adjustment (DA) method. Its effectiveness is demonstrated using FEM analysis of low, medium and high strength steels adopted in automobile industry, in which a symmetrical model owning geometry complexity similar to an auto body panel was established. The results show this new enhanced SP-DA method is able to significantly improve the accuracy of springback compensation comparing to conventional displacement adjustment technique. were conducted. Low, medium and high strength steels adopted in automobile industry were considered. The results obtained using SP-DA method were compared with those using DA method. The results show the new SP-DA method is able to significantly improve the accuracy of springback compensation in sheeting metal forming of complex shaped product, on which the influence of high strength of the materials and high springback is minimized.


Introduction
Cracking, wrinkling and springback are the most common defects in sheet metal forming, e.g. stamping while springback is the hardest to handle [1][2][3]. Springback is a process in which the internal stress in a part is released after removing the constraint of the die or trimming addendum surface [4][5]. It may lead to undesirable geometries and rejected products [6][7]. Currently the issue of springback may be resolved in the stage of pre-production or postproduction. The pre-production solution is to alter tool geometry manually based on the difference between the actual and target dimensions of the fabricated parts [8]. Theoretically this method may completely eliminate springback, but it could incur high cost and much time to achieve an accurate profile for compensation. The post-production solution is to adjust process parameters, e.g. increasing blank holder force or restrain force of drawbead so both the inner and outer sides of a sheet metal forming product are subjected to tensile stress [9]. This may partly reduce springback while increase the risk of cracking. Pre-production solution is preferred when it is possible. Springback needs to be determined using numerical methods before a compensation may be conducted at the stages of design and construction of a stamping die [10], as illustrated in Fig. 1. Springback compensation mesh is acquired through an iterative algorithm integrating the numerical simulation and analysis of springback at the stage of design [11], based on which the die may be manufactured. The deviation between an actual stamping product and its design needs to be checked. If further springback compensation is necessary, the actual stamping product may be scanned for creating a modified mesh to further modify the die [12][13]. This study focuses on springback compensation in the stage of design.  Figure 1 The process of springback compensation in stamping Geometric features need to be considered when determining the exact position of each node in the springback compensation mesh. For example, as shown in Fig. 2, node i locates in a flat section in the design model. Its corresponding node in the springback compensation mesh may be located at 1 i or 2 i , which has little influence on the construction of the new surface. On the other hand, for node j locating at the edge of an arc, the position of its corresponding node j  in the springback compensation mesh needs to be determined cautiously for ensuring the geometrical feature is transferred to the new surface properly. Two methods have been developed to construct the surface for springback compensation. One is force descriptor method (FDM) proposed by Karafillis and Boyce [14], which is a stress reverse iteration compensation method. Reverse springback can be obtained by artificially reversing the stress in inner and outer layers of a sheet. It may be hard to achieve convergence 3 in the iterative calculation in the case of large springback and asymmetric parts. In order to improve the condition of convergence, Wu [15][16] introduced a factor determined by the deviation of the key dimension of the part to multiply the stress. Anagnostou [17] proposed a modified FDM by introducing a compensation coefficient based on the distance between the positions of a node before and after springback. However, these cannot completely resolve the issue of convergence.
The other method is geometric compensation based on the displacement of a node, which has been widely adopted. The distance between the positions of a node before and after springback along the moving direction of the tool is set as the value of springback compensation. Using displacement adjustment (DA) method [18], a node is moved to the spring compensation mesh along the direction opposite to the moving direction of the tool, which has relatively fast convergence. DA method is effective for most areas in a part except for side wall because the compensation direction on side wall is perpendicular to the moving direction of the tool. Weiher [19] proposed a modified DA method, in which the distance between the position of a node before and after springback is taken as springback value and compensation is carried out along the direction simply opposite to the direction of springback. The method has good convergence, however, it may incur large errors when the springback is large or the product has a complex shape. Yang [20] proposed another modified DA method by introducing an angle compensation coefficient. It has high accuracy and efficiency for simple shaped parts but it is not an easy task to determine the angle coefficient everywhere in a complex shaped part.
Considering the significant influence of the complexity of a sheet metal forming product on spring compensation, there is still a large room for further research. The aim of this study is to develop an enhanced hybrid springback compensation approach, i.e. Springback Path -Displacement Adjustment (SP-DA) method for acquiring accurate springback compensation mesh taking into account all geometric features. The springback compensation mesh is used to construct the surface of tools iteratively till the target may be achieved in numerical simulation. The results obtained using this new enhanced approach are compared with those obtained using Weiher's method.

Methodology
It is assumed that the moving path of a node during springback and compensation are always similar. Springback compensation does not change die design parameters, binder surface, addendum surface, complexity of the surface and geometrical features. A symmetrical model owning a geometrical complexity that is similar to that of auto body panel was established and finite element method (FEM) simulation was carried to acquire springback at every node by employing commercial package LS-DYNA. The thickness of the blank is 1mm. Three steels ST14F, BH300 and DP500 were adopted, which represent low, medium and high strength steel respectively adopted in automobile industry for observing the effect of the strength on springback. The nominated chemical compositions and mechanical properties of these steels are shown in Table 1 and Table 2 respectively. The stress-strain relationship is described by = (1) where is true stress, is true strain, K is hardening coefficient, n is hardening exponent.

Algorithm
As shown in Fig  To calculate the two unit tangent vectors ( , , )  translation, which can be determined using equation (2).

FEM simulation of forming process
A model of stamping part owning sufficient geometry complexity was carefully designed as shown in Fig. 6 (a). Only a quarter needs to be considered in the FEM simulation considering the symmetry, as shown in Fig. 6   The profiles of the blank before and after stamping are shown in Fig. 8(a). The profile of the die in the simulation is constructed according to that of the part as shown in Fig. 8(b). Explicit FEM was carried out using symmetrical boundary condition as shown in Fig. 8(c). The 7 die moves down while the punch at bottom is fixed. A fixed pad is set to be 3 mm away from the punch for restraining the overarching of the middle part of the blank during stamping. The die, punch and pad are defined as rigid body. Both the tools and blank are meshed and the mesh size is 3 mm. The blank adopts BT element and there are 5 Gauss integral points along its thickness direction. There are 42558 shell elements, including 41979 quadrangle elements and 579 triangular elements. The speed of the die is 2 m/s. The gap between the punch and die is 1 mm at the end of the simulation.

FEM analysis on springback
Implicit FEM was adopted in the analysis on springback after the forming tools are removed. The forces at all nodes at the end of the simulation of stamping are applied in the beginning of this analysis as boundary condition. The forces were released in ten steps and 10% of total force was released in each step to acquire ten positions of each node during springback for constructing the moving path of the node. Full integration element is adopted in the blank. In order to define springback, 66 nodes on bottom plane were selected as reference, i.e. they were constrained in three direction of x, y and z, as shown in Fig. 9.

Comparison of springback using SP-DA and DA methods
FEM analysis results of springback after 100% unloading are shown in Fig. 10. The displacements at locations A, B, C and D after springback are listed in Table 3.   Table 4 shows the displacement at location B that is farthest from the reference points after each step of unloading for further investigation. The values after steps 1 to 9 are much smaller than that in the tenth step, i.e. the first nine position points are very much close. This is because the forces at all nodes at the end of stamping are much higher than the internal stress that causes springback due to the existence of blank hold force etc. In order to acquire evenly distributed position points for generating the springback path by interpolation, a specific elastic modulus was assigned in each step from step 1 to step 8. This method does not change the springback path. Table 5 shows the specific elastic modulus assigned and the displacements at locations A, B, C and D in steps 1 -8 in the case of steel ST14F. The results at location B are illustrated in Fig. 11.  , i = 0, 1,…, n, a cubic parameter accumulated chord length spline interpolation method described in Equation (3) is applied to determine the accumulated chord length of each point.
where sk is accumulated chord length of position point, lk is chord length between two adjacent points, as shown in Fig.12. The relationship between s and x, y and z at each point is shown in Table 6, which can be used for obtaining  The second derivative of two endpoints are specified as zero so equation (6) (j=0, 1, …, n) can be obtained from equation (5) and equation (6).
1 , The similar procedure was used for ( ) and ( ). Then the curve of springback path may be created.
Adopting the method described in section 2.1, the tangent vectors of two endpoints on the meshes before and after spingback can be calculated, followed by the determination the spring compensation path, then the construction of spring compensation mesh.
The results obtained using SP-DA method are compared to those obtained using DA method, as shown in Figs. 13 -15 for ST14F, BH300 and DP500 respectively. The lengths of the four curves are listed in Table 7. represents the lengths in CAD model. f V and s V represent the lengths before and after springback respectively. VDA represents the length after springback using DA method and DA SP V  represents the length after springback using SP-DA method. defined in Equation (8) represents the difference between the length of a curve before springback and that after springback obtained using DA method while − represents the difference using SP-DA method. Fig. 16 shows the difference in DA method increase with the increase of springback. On the other hand, the effect of springback on the difference in SP-DA method is very small, which means SP-DA method is more accurate and reliable. SP-DA method converges in a stable way and is in particular effective when dealing with large springback.

Figure 16
Comparison of the lengths of the four curves of three steels at locations A, B, C and D

Verification
It is necessary to verify the assumption made in the beginning, i.e. the moving path of a node during springback is always similar to the moving path of the node in springback compensation. The springback after stamping obtained in FEM analysis using the model of the 14 die modified based on SP-DA compensation and that based on the original design were compared. The results of steel BH300 shown in Fig. 17 Fig. 17 (b). is the springback path curve obtained in the original design model and Vsp_1 is the tangent vector at point B. The above was done at all four locations A, B, C and D and the results are shown in Table 8. It can be seen that both and ∆ are very small, which means the two paths are highly similar, i.e. the assumption that the moving path of a node during springback is always similar to the moving path of the node in springback compensation can be verified.

Conclusion
This research proposed an enhanced hybrid springback compensation method named Springback Path -Displacement Adjustment (SP-DA) method. A model of stamping part owning sufficient geometry complexity was carefully designed. FEM simulation of a stamping process and analysis on springback were conducted. Low, medium and high strength steels adopted in automobile industry were considered. The results obtained using SP-DA method were compared with those using DA method. The results show the new SP-DA method is able to significantly improve the accuracy of springback compensation in sheeting metal forming of complex shaped product, on which the influence of high strength of the materials and high springback is minimized.