A Deep Drawing Process by Inverse Finite Element Analysis

Transcription

1 A Deep Drawing Process by Inverse Finite Element Analysis Thaweepat Buranathiti 1,* 1 Division of Materials Technology, School of Energy and Materials, King Mongkut s University of Technology Thonburi (KMUTT), Bangkok, Thailand * Corresponding Author: thaweepat.bur@kmutt.ac.th Abstract Deep drawing process is a means to manufacture a complicated part from sheet metals. To design a deep drawing process, one has to consider many major factors, namely tooling configurations, blank configurations, material properties, and forming conditions. Blank configuration is the focus of this paper. One may simply use simple geometries (rectangular or circles) or approximate the blank configuration from a simple observation of the desired part based on experience. Besides experience-based blank design, this paper presents a systematical means, called Inverse analysis (IA), in determining an optimal blank configuration for a deep drawing process. Due to the complexity of mechanics of deep drawing processes, inverse finite element analysis is needed by having IA incorporate into a finite element scheme. In this paper, IA is explored by using a commercial finite element software package. A number of numerical studies on the effect of blank configurations to the part quality in a deep drawing process were conducted and compared. The quality of the drawing processes from IA and others is numerically tested by using an explicit incremental nonlinear finite element code. The initial blank configuration has shown that it plays an important role in the quality of the product. However, it is observed that if the blank configuration is not greatly deviated from the one from IA, the blank still can result a good product. IA has clearly presented its important role in the systematical design process for deep drawing processes. KEY WORDS: Inverse Analysis, deep drawing process, inverse finite element analysis, blank size estimation Introduction In engineering design processes, trial-and-error method is a traditional means to explore and optimize process conditions in virtually every manufacturing process. However, the current competition forces the design processes to be highly efficient. Therefore, systematical design and process simulations have presented themselves as a vital part in today competitive systems. This paper focuses on a deep drawing process, which is a means in manufacturing of complicated parts from sheet metal used in many industries such as automobile, aerospace, appliance and so on. Deep drawing processes typically involve many complicated physics and boundary conditions. A powerful and widely accepted means for the sheet metal forming process analysis is finite element method or FEM (Chenot and Bay, 1998; Belytschko et al., 2000). For example, U-channel forming (Taylor et al., 1995), wrinkling and tearing prediction (Cao and Boyce, 1997), corner failure (Yao and Cao, 2000), and many more. It is noted that analytical models such as (Wang et al., 1993; Kinsey and Cao, 2000), element free method (Li and Belytschko, 2001) and other techniques are also available for sheet metal forming simulations. For deep drawing processes, the simulation is often computationally expensive and in the design context is also known as forward analysis (Cao et al., 2000). It is typically known that one has to determine the best manufacturing condition that usually requires optimization by repetitive process design for both/either deterministic (Koc et al., 2000; Moshfegh et al., 2000) and/or probabilistic (Chen et al., 2004; Sahai et al., 2004). In optimization of deep drawing processes, the forward analysis leads to an even more expensive design process, which typically requires a large number of forward analyzes to search an optimal value. An alternative design approach for deep drawing processes is inverse analysis (IA). In deep drawing processes, it is well known that design of tooling geometry and conditions plays an important role to the quality and success of the production. It is interesting that the 37

2 configuration of the undeformed blank is often overlooked by many engineers. This paper conducts a comparative study on how the configuration of the blank affects the quality of a deep drawing process. A part configuration is illustrated as a numerical example. The product quality from different setups is consistently evaluated by using an explicit incremental nonlinear finite element method (LS-DYNA). A systematical design approach known as inverse analysis (MSTEP in Dynaform5.2) is presented and conducted to estimate an optimal blank configuration corresponding to the desired part configuration. The analysis results are then analyzed and discussed. Inverse analysis In a deep drawing process, Inverse approach (IA) is a systematical means to offer to estimate important parameters of the forming process. IA is a method that starts from a given desired configuration x of a part and works back to obtain the initial configuration X. In most sheet metal forming processes, a flat blank is X. The problem formulation of IA is set that x, material properties and forming conditions are given but the X of the part and resulting stress-strain states of the deformed part are to be determined. The inverse finite element approach for sheet metal forming has been developed by many researchers presenting in literature: ideal forming theory (Chung and Richmond, 1992a,b), conceptual theory on inverse problems (Chenot et al., 1996), sequential design with ideal forming theory [Chung et al., 1997], initial guess of linear deformation [Lee and Huh, 1998], deformation path iteration method (Park et al. 1999), multi-step with sliding constraint surface (Lee and Cao, 2001), pseudoinverse approach (Guo et al., 2004), an objective function based on forming limit diagram (FLD) (Naceur et al., 2004), a node relocation technique (Lan et al., 2005). For most cases in deep drawing processes, IA is basically a result of the principle of minimization of potential energy. The principle of minimization of potential energy is stated as follows: For conservative systems, of all the kinematically admissible displacement fields, those corresponding to equilibrium extremize the total potential energy. If the extremum condition is a minimum, the equilibrium state is stable. The plastic potential energy Ψ is expressed as the difference between the internal plastic work (W p ) and the external work (W e ) as follows Ψ = W p W e. [1] The external force (W e ) is induced by the friction force and the binder force, and the external force calculated at the final configuration. The minimum of Ψ corresponds to the solution of the stationary value of the first derivation to the design variables U as follows; Ψ R( U) = = 0 U. [2] This system equation is solved by using the Newton-Raphson scheme as follows; R( U) U ( n) { } { } ( n) ΔU = R( U), [3] U ( n+ 1) ( n) = U + α ΔU, [4] where α is a correction factor for numerical stability purpose. 38

3 Numerical examples This paper focuses on an example of deep drawing processes to illustrate the influence of blank configurations to the part quality. A triangular cup desired from the deep drawing process in this paper is presented in Figure 1. The rough dimension of the part is 40 mm depth and 100 mm for each side of the triangular bottom. The fillet is set as 20 mm in radius. The blank is CQ mild steel with the material properties as follows: density ρ of 7850 kg/m 3, Young s modulus E of GPa, Poisson s ratio ν of 0.28, strength coefficient K of MPa, exponent hardening n of 0.226, and the Lankford parameter R00 of 1.45, R45 of 1.10 and R90 of The initial blank thickness t is 1.0 mm. Figure 1: A illustration of the triangular cup as the product in this study. From the given dimension, the tooling dimension is extracted as shown in Figure 2 and used for every case throughout this paper. The tooling consists of a die, a binder and a punch. No drawbead is used. The conditions of the deep drawing process are given as follows: friction coefficient μ of 0.125, and blank holder (binder) force of 20 kn. Binder Punch Die Figure 2: An illustration of the tooling (die, binder and punch) in this study. The forming process is analyzed by using an IA to determine an optimal blank configuration and an explicit incremental nonlinear FEM to analyze the product. Belytschko-Tsay shell element with 5 integration points through the sheet thickness is adopted in this study. The material model of Barlat and Lian (1989) with anisotropic materials under plane stress conditions is adopted. The exponent m in Barlat s yield surface is set as 6.0. An adaptive meshing technique with the maximum of 4 refinement levels is implemented in the FEA model. 39

4 Analysis results A number of blanks in regular geometries (rectangle and circle) shown in Figure 3 are used in the FEA model. The responses of interest from the deep drawing process for the comparison purpose are wrinkle and crack tendency indicated by using strain-based FLD. The ideal shape of the formed part previously presented in Figure 1 is used in IA to estimate an optimal blank configuration. By using the same set of process data, the blank configuration obtained from IA is shown in Figure 3 in comparison with other cases. Figure 3: Initial blank configurations used in this study: (i) blank from IA, (ii) a circle, (iii) a rectangle, (iv) a smaller circle, and (v) a smaller rectangle. For the purpose of this comparative study, the quality of the products from all cases is shown in strain-based FLD in Figures 4-8. Figure 4 presents a case of the blank configuration obtained by using IA. It is observed that there is no significant deformation causing cracks. However, a number of locations (bottom and side walls) indicate insufficient stretch leading to strength problem of the part. The part also has tendency of wrinkling around the die corner, which is not desirable, but it is not critical. Figure 4: The elemental strain plot on a strain-based FLD for the blank obtained by using IA. Figure 5 presents a case of the circular blank. It is observed that there are excessive stress and strain causing cracks, which is critical to the part quality. However, it has no area with insufficient stretching since excessive stretching is presented. 40

5 Figure 6 presents a case of the rectangular blank. It is observed that there are excessive stress and strain causing cracks. Also, there is no area with insufficient stretching. Figure 7 presents a case of the smaller circular blank compared to Figure 5. It is observed that no significant deformation causing cracks. The bottom of the part also does not indicate insufficient stretch. Only small area of side wall indicates insufficient stretch. The part also has tendency of wrinkling around the die corner like before. Figure 5: The elemental strain plot on a strain-based FLD for the circular blank. Figure 6: The elemental strain plot on a strain-based FLD for the rectangular blank. Figure 7: The elemental strain plot on a strain-based FLD for the smaller circular blank. 41

6 Figure 8 presents a case of the smaller rectangular blank compared to Figure 6. It is observed that there is no significant deformation causing cracks and there are insufficient stretch problems. The part also has tendency of wrinkling around the die corner. Figure 8: The elemental strain plot on a strain-based FLD for the smaller rectangular blank. Discussions and concluding remarks The illustrative examples in this paper have shown that the difference of blank configurations significantly affects the quality of the part produced by a deep drawing process. A systematical means for designing an optimal blank configuration in a deep drawing process is inverse analysis (IA). IA only needs the desired final configuration x* and limited data of forming conditions in the model to obtain an approximation of the initial configuration X and an approximated stress-strain state at the final configuration. However, a blank configuration other than the one obtained from IA can still provide a good quality of the part if the blank configuration does not greatly deviate from it. Based on FLD, it appears that the smaller circular blank in Figure 7 results a better part than the one from IA due to insufficient stretching at the bottom. It is also an industrial practice that initial blank configurations should be smooth and have little trimming cost. It seems arguable that a further advancement of inverse methods seems to have small effect to the quality in the forming process design but, on the other hand, it definitely helps design processes that directly need the accurate prediction of stress and strain during the optimization search. It is noted that the algorithm of IA in general does not take insufficient stretching problems into account as the objective of the IA model is to minimize the potential energy. Based on FLD plot of the blank configuration from IA shows some area at the bottom is not stretched enough. It should be noted that the nature of the inverse problem is ill-posed. A simple analogy is that the combination of 1 and 4 is 5. Inversely, the inverse problem is to determine what is the combination of x and y to be 5 [x +y = 5 x=?, y=?]. It can be observed that this problem has multiple solutions like virtually all optimization problems have. Therefore, the way to formulate the model including constraints and initial guess values are very important to achieve the optimal solution. As seen in literature, some advances in IA came from the search constraints. It is worth noting that the final manufacturing conditions from IA should be eventually verified by an incremental nonlinear finite element method before a physical tryout can be taken. In addition, the computational cost of IA is typically a small fraction of that of an incremental nonlinear FEA. However, the prediction accuracy of stress and strain from IA is still less than that of incremental FEA. The inverse analysis (IA) clearly offers itself as an important part of the design methodology for deep drawing processes. 42

7 References Barlat, F., and Lian, J Plastic behavior and stretchability of sheet metals. Part I: A yield function for orthotropic sheets under plane stress conditions. International Journal of Plasticity 5: Belytschko, T., Liu, W.K., and Moran, B Nonlinear Finite Elements for Continua and Structures. New York: John Wiley & Sons. Buranathiti, T., Cao, J., Chen, W., Baghdasaryan, L., and Xia, Z.C Approaches for model validation: methodology and illustration on a sheet metal flanging process. ASME Journal of Manufacturing Science and Engineering 128: Cao, J., and Boyce, M.C A predictive tool for delaying wrinkling and tearing failures in sheet metal forming. Journal of Engineering Materials and Technology 119: Cao, J., Li, S., Xia, Z.C., and Tang, S.C Analysis of an axisymmetric deep drawn part forming using less forming steps. Journal of Materials Processing Technology 117: Chen, W., Baghdasaryan, L., Buranathiti, T., and Cao, J Model validation via uncertainty propagation and data transformations. AIAA Journal 42: Chenot, J.-L., and Bay, F An overview of numerical modeling techniques. Journal of Materials Processing Technology 80-81:8-15. Chenot, J.-L., Massoni, E., and Fourment, L Inverse problems in finite element simulation of metal forming processes. Engineering Computations 13: Chung, K., and Richmond, O. 1992a. Ideal forming--i. Homogeneous deformation with minimum plastic work. International Journal of Mechanical Sciences 34: Chung, K., and Richmond, O. 1992b. Ideal forming--ii. Sheet forming with optimum deformation. International Journal of Mechanical Sciences 34: Chung, K., Barlat, F., Brem, J.C., Lege, D.J., and Richmond, O Blank shape design for a planar anisotropic sheet based on ideal forming design theory and FEM analysis. International Journal of Mechanical Sciences 39: eta/dynaform: User s Manual, Version 5.2. Kinsey, B., and Cao, J An analytical model for tailor welded blank forming. Journal of Manufacturing Science and Engineering 125: Koc, M., Allen, T., Jiratheranat, S., and Altan, T The use of FEA and design of experiments to establish design guidelines for simple hydroformed parts. International Journal of Machine Tools & Manufacture 40: Lan, J., Dong, X., and Li, Z Inverse finite element approach and its application in sheet metal forming. Journal of Materials Processing Technology 170: Lee, C., and Cao, J Shell element formulation of multi-step inverse analysis for axisymmetric deep drawing process. International Journal for Numerical Methods in Engineering 50: Lee, C.H., and Huh, H Blank design and strain estimates for sheet metal forming processes by a finite element inverse approach with initial guess of linear deformation. Journal of Materials Processing Technology 82: Li, G., and Belytschko, T Element-free Galerkin method for contact problems in metal forming analysis. Engineering Computations 18: LS-DYNA: User s Manual, Version 970. Moshfegh, R., Li, X., and Nilsson, L Gradient-based refinement indicators in adaptive finite element analysis with special reference to sheet metal forming. Engineering Computations 17:

8 Naceur, H., Delameziere, A., Batoz, J.L., Guo, Y.Q., and Knopf-Lenoir, C Some improvements on the optimum process design in deep drawing using the inverse approach. Journal of Materials Processing Technology 146: Park, S.H., Yoon, J.W., Yang, D.Y., and Kim, Y.H Optimum blank design in sheet metal forming by the deformation path iteration method. International Journal of Mechanical Sciences 41: Sahai, A., Schramm, U., Buranathiti, T., Chen, W., and Cao, J Sequential optimization and reliability assessment method for metal forming processes. Paper read at the 8th International Conference on Numerical Methods in Industrial Forming Processes NUMIFORM 2004, at Columbus. Taylor, L., Cao, J., Karafillis, A.P., and Boyce, M.C Numerical simulations of sheet metal forming. Journal of Materials Processing Technology 50: Wang, C., Kinzel, G., and Altan, T Mathematical modeling of plane-strain bending of sheet and plate. Journal of Materials Processing Technology 39: Yao, H., and Cao, J Predicting the corner failure depths in the deep drawing of 3D panels using simplified numerical and analytical model. ASME Journal of Manufacturing Science and Engineering 123: Acknowledgements The authors would like to express their gratitude to KMUTT for providing partial financial support and Engineering Technology Associates (ETA) for providing the commercial software package. 44