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Journal of Materials Processing Technology 190 (2007) 189–198                                Simulation of springback vari...
190                              P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198          ...
P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198                                           ...
192                                 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198       ...
P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198                                           ...
194                                        P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198...
P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198                                           ...
196                             P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198           ...
P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198                                           ...
198                                     P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198   ...
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Simulation of springback variation in forming of advanced high strength steels

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Simulation of springback variation in forming of advanced high strength steels

  1. 1. Journal of Materials Processing Technology 190 (2007) 189–198 Simulation of springback variation in forming of advanced high strength steels Peng Chen b , Muammer Koc a,∗ ¸ a NSF I/UCR Center for Precision Forming (CPF) and Department of Mechanical Engineering, Virginia Commonwealth University (VCU), Richmond, VA, USA b Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Received in revised form 23 February 2007; accepted 27 February 2007Abstract Variations in the mechanical and dimensional properties of the incoming material, lubrication and other forming process parameters are themain causes of springback variation. Variation of springback prevents the applicability of the springback prediction and compensation techniques.Hence, it leads to amplified variations and problems during assembly of the stamped components, in turn, resulting in quality issues. To predict thevariation of springback and to improve the robustness of the forming process, variation simulation analysis could be adopted in the early designstage. Design of experiment (DOE) and finite element analysis (FEA) approach was used for the variation simulation and analysis of the springbackfor advanced high strength steel (AHSS) parts. To avoid the issues caused by the deterministic FEA simulation, random number generation wasused to introduce uncertainties in DOE. This approach was, then, applied to investigate the effects of variations in material, blank holder force andfriction on the springback variation for an open-channel shaped part made of dual phase (DP) steel. This approach provides a rapid and accurateunderstanding of the influence of the random process variations on the springback variation of the formed part using FEA techniques eliminatingthe need for lengthy and costly physical experiments.© 2007 Elsevier B.V. All rights reserved.Keywords: Springback variation; Variation simulation; Design of experiment; Finite element analysis; Computer experiment; Advanced high strength steel1. Introduction due to challenges in terms of formability, springback, joining and die life issues. During the past two decades, the automotive industry has Springback is an elastically driven change in shape and formbeen in a quest to achieve low-mass vehicles due to government of a part upon unloading after a part is formed. The concernsregulations and consumer demands for light, fuel-efficient and about springback and quality control grow among auto-makerslow-emission vehicles. In addition, requirements for improved as the use of high strength–low weight materials increases [1].safety, performance, and comfort features continue putting Parts made of AHSS demonstrate more springback than partspressure on the automotive manufacturers especially in an envi- made of mild steel do. Moreover, the experience with formingronment where there are ever-increasing global competition of AHSS materials is limited compared to mild steel. Variationand continuous cost reduction demands. As an alternative to of springback prevents the applicability and use of springbackaluminum and magnesium alloys, advanced/ultra high strength prediction and compensation techniques. Hence, it leads tosteels (AHSS) have been proven to be a proper choice of mate- amplified variations and problems during assembly of compo-rial for various body and structural automotive parts meeting the nents, and in turn, results in quality issues. Thus, understanding,required low-mass, affordable cost and increased performance accurate characterization, prediction, control and reduction ofrequirements. However, wide application of high strength steel springback and its variation have become very crucial in termsin many potential auto body and structural parts is still limited of decreasing development times and reducing scrap rate in mass production to achieve cost effective fabrication of AHSS parts. Numerous studies during the last 40 years have attempted to ∗ Corresponding author. Tel.: +1 804 827 7029. determine the controlling factors in springback and find ways to E-mail address: mkoc@vcu.edu (M. Koc). ¸ reduce it. An early study by Baba and Tozawa [2] focused on0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.02.046
  2. 2. 190 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 cthe effect of stretching a sheet by a tensile force, during or after simulation, which is especially designed for the assembly pro-bending, in minimizing springback. Other studies investigated cess (welding) simulation based on the assumption of linearthe role of process variables on springback. Zhang and Lee [3] elasticity.showed the influence of blank holder force, elastic modulus, Generally speaking, Monte-Carlo simulation is effective forstrain hardening exponent, blank thickness and yield strength any kind of non-linear process. However, since large number ofon the magnitude of the final springback strain in a part. Geng FEA simulations is required (more than 100 for reliable results),and Wagoner [4] studied the effects of plastic anisotropy and its it is not practical and too costly to apply it for a complicatedevolution in springback. They developed a constitutive equation case, such as metal forming. Therefore, some researchers [22]for 6022-T4 aluminum alloy using a new anisotropic hardening applied DOE techniques in their FE simulation to effectivelymodel and proved that Barlat’s yield function is more accurate reduce the amount of simulations. However, since the FEA sim-than other yield functions in their case. ulation is deterministic (no random error), replicate observations Some researchers evaluated FE simulation procedures in from running the simulation with the same inputs will be identi-terms of their springback prediction accuracy. Mattiasson et cal. Despite some similarities to physical experiments, the lackal. [5], Wagoner et al. [6], Li et al. [7] and Lee and Yang [8] of random error makes FEA simulations different from physi-found that FEM simulations of springback are much more sen- cal experiments. In the absence of independent random errors,sitive to numerical tolerances than forming simulations are. Li the rationale for least-squares fitting of a response surface is notet al. [9] investigated the effects of element type on the spring- clear [23]. The usual measures of uncertainty derived from least-back simulation. Yuen [10] and Tang [11] found that different squares residuals have no obvious statistical meaning [23,24].unloading scheme will affect the accuracy of the springback According to Welch et al. [25], in the presence of systematicprediction. Similarly, Focellese et al. [12] and Narasimhan and error rather than random error, statistical testing is inappropri-Lovell [13] pointed out that different integration scheme will ate. For deterministic FEA simulation, some statistics, includingalso influence the result of springback simulation. In 1999, F-statistics, have no statistical meaning since they assume thePark et al. [14] and Valente and Traversa [15] attempted to observations include an error term which has mean of zero andlink dynamic explicit simulations of forming operations to static a non-zero standard deviation. Consequently, the use of step-implicit simulations of springback. It was proved that this tech- wise regression for polynomial model fitting is not appropriatenique is very effective for the springback simulation. Li et since it utilizes F-statistic value when adding/removing modelal. [16] explored a variety of issues in the springback simu- parameters [24].lations. They concluded that (1) typical forming simulations In an effort to solve the above problem, McKay et al. [26]are acceptably accurate with 5–9 through-thickness integration introduced Latin hypercube sampling which ensures that eachpoints for shell/beam type elements, whereas springback anal- of the input variables has all portions of its range presented.ysis within 1% numerical error requires up to 51 points, and Sacks et al. [23,27] proposed the design and analysis of com-more typically 15–25 points, depending on R/t, sheet tension puter (DACE) method to model the deterministic output as theand friction coefficient. (2) More contact nodes are necessary realization of a stochastic process, thereby providing a statis-for accurate springback simulations than for forming simulation, tical basis for designing experiments for efficient prediction.approximately one node per 5◦ of turn angle versus 10◦ recom- Kleijnen [28] suggested incorporating substantial random errormended for forming. (3) Three-dimensional shell and non-linear through random number generators. Therefore, it is natural tosolid elements are preferred for springback prediction even for design and analyze such stochastic simulation experiments usinglarge w/t ratios because of the presence of persistent anticlas- standard techniques for physical experiments. Some researcherstic curvature. For R/t > 5.6, shell elements are preferred since (e.g., Giunta et al. [29,30] and Venter et al. [31]), have alsosolid elements are too computation-intensive. For R/t < 5.6, non- employed metamodeling techniques such as RSM for modelinglinear 3D solid elements are required for accurate springback deterministic computer experiments which contain numericalprediction. noise. This numerical noise is used as a surrogate for random Most of the research efforts on springback focused on the error, thus allowing the standard least-squares approach to beaccurate prediction and compensation of springback. The issue applied. However, the assumption of equating numerical noiseof springback variation was seldom concerned. Moreover, none to random error is questionable.of the studies in the area of springback prediction touched In this work, random number generator was used to ensureupon the variation simulation of springback. As lead times the correctness of the regression model. Assuming a Gaussianare shortened and materials of high strength–low weight are distribution, the uncertainty was introduced by random numberused in manufacturing, a fundamental understanding of the generation for controlled factors at different level and uncon-springback variation has become essential for accurate and trolled variables according to their mean and range. By usingrapid design of tooling and processes in the early design stage. this method, the effects of variations in material (mechanicalAs far as the variation simulation is concerned, most avail- properties) and process (blank holder force and friction) onable references are related to the assembly processes [17–21]. the springback variation were investigated for an open-channelIn 1997, Liu and Hu [17] summarized two variation simula- shaped part made of DP steel. The variations in stamping processtion approaches for sheet metal assembly. They are: (1) direct are introduced in Section 2 of the paper. In Section 3, measure-Monte-Carlo simulation, which is popular and good for all kinds ments of springback are defined. Section 4 is the validation of theof random process simulation, and (2) mechanistic variation FE model using existing experimental results in the literature.
  3. 3. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c 191 Fig. 1. Springback variation and its sources.In the following section, variation analysis of an open-channel • Batch-to-batch variation represents the variability among thepart made of advanced high strength steel is presented. The last individual batches, which is mostly caused by material vari-section is discussions and conclusions. ation from batch to batch and the variation introduced by tooling setup.2. Variations in stamping process 3. Measurement of springback for open-channel In this study, the objective is to understand and accurately drawingpredict the variation of springback in an open-channel drawingconsidering the variations of material and process as shown in A schematic view of die, punch, blank and their dimensionsFig. 1. for open-channel drawing, which is used in the analyses for Total variation of springback in the stamping process has this study, is shown in Fig. 3. Fig. 4 shows the formed part afterseveral components. Generally, different variation components springback. Three measurements, namely the springback of wallcan be attributed to different sources [32]. The following are the opening angle (β1 ), the springback of flange angle (β2 ) and side-major categories of variation source (Fig. 2): wall curl radius (ρ) shown in Fig. 5, were used to characterize the total springback considering only the cross-sectional shapes of formed parts obtained before and after the removal of tools. The• Part-to-part variation is also referred to as system-level vari- springback in the direction orthogonal to the cross-section, such ation or inherent variation. It is the amount of variation that as twisting, was not considered since it is negligible is this case. can be expected across consecutive parts produced by the pro- As there is no clear distinction to separate a cross-section curve cess during a given run. It is caused by the random variation for individual measurement of springback angles and sidewall of all the uncontrolled (controllable and uncontrollable) pro- curl, two assumptions deduced from the sample observations cess variables. In the variation simulation of this paper, blank thickness was considered as the uncontrolled variable.• Within batch variation is usually due to the variations of the controlled variables such as BHF, material property and fric- tion. Fig. 3. A schematic view of tools and dimensions for open-channel drawing Fig. 2. Source of variation in a typical forming process. [32].
  4. 4. 192 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c to construct a circular arc is used. Eq. (1) lists all the equations needed for the calculation of the β1 , β2 and ρ. ox · A0 B0 θ1 = arccos 0 |ox| · A0 B0 ox · A0 B0 θ2 = arccos 0 |ox| · A0 B0 Fig. 4. Open-channel parts after drawing. ox · AB θ1 = arccos |ox| · |AB| AB · ED θ2 = arccos |ED| · |AB| (1) β1 = θ1 − θ1 0 β2 = θ2 − θ2 0 xB + yB − xA − yA − ((yA − yB )/ 2 2 2 2 (yC − yB ))(xC + yC − xB − yB ) 2 2 2 2 xO = 2 xB − xA + (xC − xB )((yA − yB )/(yC − yB )) xA + yA − xB − yB + 2xO (xB − xA ) 2 2 2 2 yO = 2(yA − yB ) ρ= (xA − xO )2 + (yA − yO )2 4. Finite element modeling and validation for the Fig. 5. Illustration of springbacks. open-channel drawing of AHSSare introduced for the springback measurement. Firstly, it is The simulation work for this study is based on the exper-assumed that wall opening angle, flange closing angles and side- imental results of Lee et al. [33]. Information about thewall curl vary independently. Secondly, the sidewall curl could geometry and dimensions of the tooling and blank are pre-be approximated by a piece of circular arc. sented in Fig. 3. The initial dimension of the blank sheet was Fig. 5 also shows the measurements placements (A–E). Two 300 mm (length) × 35 mm (width). Forming was carried out onmeasurements were conducted before springback, namely the x a 150 tonnes double action hydraulic press with a punch speedand y coordinates of A and B, which is denoted as A0 and B0 of 1 mm/s, and the total punch stroke was 70 mm. Blank holderin this work. They are used to compute the wall angle (θ1 ) and0 force (BHF) was 2.5 kN. The blank material used was DP Steelflange angle (θ2 0 ) before springback. After springback, another with the material properties presented in Fig. 6 based on thefive measurements were placed on A–E, which were used in the tensile tests by Lee et al. [34].calculation of the wall angle (θ 1 ), flange angle (θ 2 ) and sidewall Considering the geometric symmetry of the process, onlycurl radius (ρ) after springback. To estimate the sidewall curl half of the blank was simulated. The material was modeled asradius, a curve fitting technique that employs three points (A–C) an elastic–plastic material with isotropic elasticity, using the Fig. 6. Material properties of DP steel [33].
  5. 5. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c 193Table 1Different FEA procedure used in simulation Case 1 2 3 4 5 6 7 8 9Element type Solid Solid Solid Solid Shell Shell Shell Shell ShellContact Soft Soft Soft Hard Soft Soft Hard Hard SoftForming analysis (dynamic) Implicit Implicit Implicit Implicit Implicit Implicit Explicit Explicit ExplicitSpringback analysis (static) Implicit Implicit Implicit Implicit Implicit Implicit Implicit Implicit ImplicitThrough-thickness element number 5 9 21 9 9 21 5 15 9 or integration pointHill anisotropic yield criterion for the plasticity. The coefficients in Table 1) of different element type, different contact condi-of Hill yield criterion (R11 = 1.0, R22 = 1.01951, R33 = 1.00219, tion, different through-thickness element number and differentR12 = 0.992318, R13 = 1.0, R23 = 1.0) were computed from the analysis type were tried in the simulation. The term, soft con-r-values as presented in Fig. 6. The friction coefficient between tact, denotes exponential pressure-overclosure definition for thetools and the sheet blank was assumed to be constant and normal behavior between contacting surfaces.0.1. To determine the appropriate element type, contact con- The comparison of different simulation procedure for theditions, through-thickness element number and analysis type prediction of wall opening angle (θ 1 ), flange angle (θ 2 ) and side-for simulation using ABAQUS, nine combinations (as tabulated wall curl radius (ρ) is shown in Figs. 7–9. It was found that the sidewall curl is very sensitive to the contact condition used in simulation. Since the soft contact tends to soften the contact- ing surface, it actually depresses the sidewall curl, which is not true for advanced high strength steel. Among these combina- tions, case 4 (hard contact), case 7 (hard contact) and case 8 (hard contact) show a good match with the experiment results in all three springback measurements. Hence, hard contact is Table 2 Original experiment design Run order BHF (kN) Friction MaterialFig. 7. Effects of different FEA procedure on the prediction of wall openingangle. 1 13.75 0.15 1.1 2 13.75 0.1 1 3 13.75 0.15 0.9 4 2.5 0.15 1 5 25 0.1 1.1 6 2.5 0.1 1.1 7 25 0.15 1 8 2.5 0.15 1 9 2.5 0.1 1.1 10 13.75 0.05 0.9 11 25 0.05 1 12 2.5 0.1 0.9 13 2.5 0.05 1 14 13.75 0.1 1Fig. 8. Effects of different FEA procedure on the prediction of flange closing 15 13.75 0.1 1angle. 16 13.75 0.05 0.9 17 2.5 0.1 0.9 18 13.75 0.1 1 19 25 0.05 1 20 13.75 0.15 0.9 21 25 0.1 0.9 22 13.75 0.05 1.1 23 13.75 0.1 1 24 25 0.15 1 25 13.75 0.15 1.1 26 25 0.1 0.9 27 13.75 0.05 1.1 28 13.75 0.1 1 29 2.5 0.05 1Fig. 9. Effects of different FEA procedure on the prediction of sidewall curl 30 25 0.1 1.1radius.
  6. 6. 194 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 cTable 3 5. Variation simulation of springback and resultsAssumed statistics of variables Mean Range S.D. In this study, we only considered the “part-to-part” andUncontrolled factor “within batch” variations. The variation simulation and anal- Part thickness 1.2 mm 1.18–1.22 mm 0.066667 ysis of the springback of DP steel part are described step by step as follows.Controlled factor BHF level-1 2.5 kN 2.4–2.6 0.033333 BHF level-2 13.75 kN 13.65–13.85 0.033333 Step 1 (Design of experiment). BHF, material property and fric- BHF level-3 25 kN 24.9–25.1 0.033333 tion were chosen as design factors. Box–Behnken RSM design Friction level-1 0.05 0.04–0.06 0.003333 with 2-replicate and 6-center-point was used for this 3-factor and Friction level-2 0.1 0.09–0.11 0.003333 3-level experiment design. The levels of the material property Friction level-3 0.15 0.14–0.16 0.003333 are considered as 110, 100 and 90% of the stress–strain curve Material level-1 90% 88–92% 0.00667 in Fig. 6, which indicates the strength of the material. Table 2 Material level-2 100% 98–102% 0.00667 shows the original experiment table. Material level-3 110% 108–112% 0.00667 Steps 2 and 3Random number generation of controlled and uncontrolled variablesIt is assumed that most random processespreferred. It can be seen that element type and forming anal- conform to a Gaussian distribution. Moreover, irrespective ofysis type do not affect the accuracy of springback prediction the parent distribution of the population, the distribution of themuch. Therefore, to reduce the computation time, hard con- average of random samples taken from the population tends totact, shell element, explicit (dynamic) for forming and implicit be normal as the sample size increases (Central Limit Theo-(static) for springback were used in further simulations. Dif- rem). Therefore, once we know the mean and standard deviationferent through-thickness integration points (5, 9, 15, and 21) of (S.D.) of a random process, we can generate a random numbershell elements were also tried in the simulation, which showed no according to its Gaussian distribution. According to the statisticsmuch influence on the prediction of springback. Therefore, nine chosen in Table 3, the original experiment table was random-through-thickness integration points were used in the further ized as shown in Table 4. Fig. 10 is an illustration of the numbersimulations. randomization.Table 4Randomized (random number generation) experiment table and simulation resultsRun BHF (kN) Friction Material Part thickness β1 (◦ ) β2 (◦ ) ρ (mm) 1 13.649243 0.1405 1.0819 11.8017 16.1242 12.3581 191.5870 2 13.755335 0.0904 0.9814 11.9484 16.5589 12.1747 169.7480 3 13.721143 0.1534 0.8811 11.9619 12.5694 9.4674 301.4050 4 2.3992 0.1512 0.9933 11.9606 18.4420 11.0999 133.4730 5 24.9033 0.0970 1.0923 11.9473 19.8823 13.4837 137.7005 6 2.5053 0.1005 1.0913 12.0388 17.8932 11.0234 138.2284 7 25.0027 0.1466 1.0063 12.0067 17.0396 11.4995 158.0187 8 2.4711 0.1521 0.9910 12.0383 17.5935 10.8937 140.5959 9 2.5291 0.1016 1.0999 12.0638 15.0679 11.1937 140.684110 13.779106 0.0404 0.8937 11.9781 14.1995 11.0920 156.656111 24.9970 0.0536 1.0160 11.9351 17.7311 12.9760 151.331212 2.5072 0.0958 0.8941 12.0725 9.8603 10.0025 178.038813 2.4983 0.0422 0.9933 11.8973 14.9086 9.9908 159.487114 13.757157 0.0995 1.0031 12.0039 16.4520 12.2300 169.079915 13.748318 0.0959 1.0025 12.0272 16.6234 12.2783 166.013216 13.737183 0.0555 0.9081 11.9032 15.3990 11.3933 166.436717 2.4872 0.0970 0.8889 12.0405 14.7347 9.3532 168.844518 13.791958 0.1035 1.0077 11.9982 16.3857 12.2430 171.579519 24.9737 0.0527 0.9922 11.9569 17.2571 12.6146 156.690920 13.780872 0.1515 0.9056 12.0172 12.6997 9.7055 298.530021 24.9978 0.0977 0.8982 12.0322 15.9928 11.2144 172.023722 13.772126 0.0484 1.1079 11.9815 18.5080 13.6613 122.973323 13.718735 0.0964 0.9919 11.8797 16.5546 12.2819 169.290924 24.9718 0.1483 1.0078 11.9887 17.0759 11.5550 158.116525 13.785849 0.1407 1.1077 12.0855 17.0869 13.9510 178.883326 25.0469 0.1037 0.9091 11.8921 16.4418 11.4263 169.862327 13.768493 0.0428 1.0955 12.0642 18.7483 14.9717 125.456428 13.751131 0.0960 0.9936 12.0129 16.4303 12.1832 169.690129 2.542 0.0480 1.0014 11.9466 15.2362 10.1161 157.662030 24.9880 0.1004 1.0918 12.0695 18.7641 14.3375 139.4873
  7. 7. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c 195 Table 7 Recommended variable level for the minimum variance of β1 , β2 and ρ according to Monte-Carlo simulation Material Friction BHF Min[Var(β1 )] – High Middle Min[Var(β2 )] – High Low Min[Var(ρ)] Middle Middle – equations. For instance, the coefficient of the XMaterial is +1.5867 in Eq. (2), therefore, the bigger the material strength, the larger the springback. Table 5 tabulates the optimal variable level for the minimum of each springback. A more detailed indicationFig. 10. Schematic diagram of the variable randomizations (random number of the relationship between the springbacks and the factors isgeneration). shown in Fig. 11. As in the parameter’s range studied in this work, springback increases with BHF and friction, which agreesTable 5 with the experimental observations [35]. Papeleux and Pon-Recommended variable level for the minimum of β1 , β2 and maximum of ρ thot [35] reported that springback increases with small BHF, Material Friction BHF but decreases as the BHF increases for large force values.Minimum β1 Low Low Low This phenomenon can be explained by the fact that with lowMinimum β2 Low Low Low BHF, the punch induces mostly bending stresses in the mate-Maximum ρ Low Low – rial, but as the blankholder holds the blank more severely, the stresses included by the punching phase become mostly tensile stresses.Step 4 (Simulation). Simulations were run according to Table 4and the simulation results are shown in Table 4 as well. Step 6 (Variation sensitivity analysis). Three methods were used to analyze the effects of the factors on the variation ofStep 5 (Regression analysis). Eqs. (2)–(4) are the regression the springback. Finally, it was found that the springback varia-models of β1 , β2 , and ρ as functions of BHF, material and fric- tion magnitude is too small in this case and not distinguishabletion. The variables in these equations are coded (−1, 0, 1) factors from the system noise.in the DOE. To investigate whether the factors’ effect on eachspringback is significant, analysis of variance (ANOVA) wasused. Factors with a P-value larger than 0.05 were considered as 5.1. Monte-Carlo simulationinsignificant and ignored in the regression model. For example, Monte-Carlo simulation was applied to Eqs. (2)–(4). Accord-the main effect of friction on β1 is negligible: ing to the parameter levels used in the DOE, it is assumedβ1 = 16.5008 + 1.5867XMaterial + 0.7287XBHF that all factors have equal variance in the Monte-Carlo simula- tion, i.e., Var(XMaterial ) = Var(XBHF ) = Var(XFriction ) = 0.32 , with −0.8454XBHF XFriction (2) a zero mean value for each factor (coded factors). Monte-Carlo simulation was run 100 times for each situation (a specific factorβ2 = 12.2319 + 1.3329XMaterial + 0.9646XBHF at a specific level). The corresponding variance of the springback was recorded in Table 6. Table 7 summarizes the optimal variable −0.3929XFriction − 0.7297XBHF − 0.5529XBHF XFriction 2 level for the minimum variance of each springback according to (3) Table 6.r = 169.234 − 8.384XMaterial + 18.482XMaterial XFriction (4) 5.2. Sensitivity analysis The effect of each factor on each springback could be deter- According to Eqs. (2)–(4), the variance of β1 , β2 and ρ aremined by the sign of the corresponding coefficient in the above expressed as Eqs. (5)–(7) via linearized sensitivity analysis.Table 6Springback variation Material Friction BHF Low Middle High Low Middle High Low Middle HighVar(β1 ) 0.0667 0.0667 0.0667 0.5349 0.324 0.2563 0.3501 0.2571 0.3207Var(β2 ) 0.1146 0.1146 0.1146 0.4798 0.3276 0.2367 0.1866 0.1926 0.2656Var(ρ) 37.419 0 37.419 73.712 7.1786 10.414 11.813 11.813 11.813
  8. 8. 196 P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c Fig. 11. Response surface plots of (a) β1 , (b) β2 and (c) ρ. Var(r) = [−27.3 − 19.35XFriction + 27.194XMaterial ]2Var(β1 ) = [1.8861 − 1.6648XMaterial ]2 Var(XMaterial ) ×Var(XMaterial ) + [−54.444XBHF ]2 Var(XBHF ) +[1.028 − 0.8454XFriction ]2 Var(XBHF ) +[22.745 − 19.35XMaterial + 19.82XFriction ]2 +[0.8454XBHF ]2 Var(XFriction ) (5) ×Var(XFriction ) (7)Var(β2 ) = [1.3329 + 0.2899XBHF ]2 Var(XMaterial ) The variance of the response is determined by the variance +[0.9646 − 1.4594XBHF + 0.2899XMaterial ]2 of each factor and the sensitivity coefficient (the quantity in the square parentheses). To minimize the variance of the springback, ×Var(XBHF ) + [−0.3929 − 0.5528XBHF ]2 the most efficient way is to minimize the sensitivity coeffi- ×Var(XFriction ) (6) cients in the equation. Table 8 tabulates the optimal variable
  9. 9. P. Chen, M. Ko¸ / Journal of Materials Processing Technology 190 (2007) 189–198 c 197Table 8 (Tables 7 and 8) actually do not have any meaning becauseRecommended variable level for the minimum variance of β1 , β2 and ρ according the springback variations are totally random and uncontrollableto sensitivity analysis in this case. In other words, conclusions from both methods Material Friction BHF are neither correct nor wrong. Since the system-level noisesMin[Var(β1 )] High High Middle were introduced by random number generation (Table 3) in ourMin[Var(β2 )] High Middle High computer experiment, we can solve the problem by reducingMin[Var(ρ)] High Low Middle the standard deviations used in the random number generation. However, this kind of adjustment would not be easy in reality, since the tuning of the system-level noise is usually impossibleTable 9Extracted data (β1 ) used in MINITAB for Taguchi analysis in most cases.BHF (kN) Friction Material S.D. (β1 ) Mean (β1 ) 6. Conclusions13.75 0.15 1.1 0.68073 16.605613.75 0.1 1 0.09154 16.5008 The effects of BHF, material and friction on springback13.75 0.15 0.9 0.09214 12.6346 and springback variation of DP steel channel have been ana- 2.5 0.15 1 0.59998 18.017825 0.1 1.1 0.79069 19.3232 lyzed parametrically using the FEA and DOE with random 2.5 0.1 1.1 1.99779 16.4806 number generation (computer experiment). On the basis of the25 0.15 1 0.02567 17.0578 quantitative and qualitative analysis made herein, the following13.75 0.05 0.9 0.84817 14.7993 conclusions could be drawn.25 0.05 1 0.33517 17.4941 The sidewall curl is very sensitive to the contact condition in 2.5 0.1 0.9 3.44672 12.2975 2.5 0.05 1 0.23165 15.0724 the simulation; hard contact is preferred for high strength steel.25 0.1 0.9 0.31749 16.2173 Springback variation in this case is not distinguishable from13.75 0.05 1.1 0.16992 18.6282 the system-level noise. Therefore, it is uncontrollable in this case. In order to reduce springback variation, the standard devi- ations used for variable randomization has to be decreased;level for the minimum springback variation suggested by Eqs. virtually, it means that a system-level adjustment of the press(5)–(7), which does not agree with Table 7. This discrepancy has to be performed to reduce the part-to-part variation of thewas explained by the third method. equipment. On the other hand, if the springback variation is large and uncontrollable, then the springback compensation technique5.3. Taguchi approach has to be chosen with it in mind. A methodology for the variation simulation of springback Taguchi analysis was used to analyze the springback vari- was developed, which provides a rapid understanding of theation. MINITAB, a statistical software, was used to analyze influence of the random process variations on the springbackthe existing experiment results (Table 4). MINITAB can auto- variation of the formed part using FEA techniques eliminatingmatically extract data (standard deviation and mean) from the the need for lengthy and costly physical experiments.available experimental observations. For example, Table 9 is theextracted data of β1 used in MINITAB for Taguchi analysis. In ReferencesMINITAB, the main effects of each design factor on the standarddeviations of the response are obtained via regression analysis, [1] W.D. Carden, L.M. Geng, D.K. Matlock, R.H. Wagoner, Measurement ofand the significance of these effects were tested via analysis springback, Int. J. Mech. Sci. 44 (2002) (2002) 79–101. [2] A. Baba, Y. Tozawa, Effects of tensile force in stretch-forming process onof variation (ANOVA) and F-tests. P-values (P) were used to the springback, Bull. JSME 7 (1964) 835–843.determine which of the effects in the model are statistically sig- [3] Z.T. Zhang, D. Lee, Effects of process variables and material propertiesnificant, which are compared with a -level of 0.05. As shown on the springback behavior of 2D-draw bending parts, in: Automotivein Table 10, none of the effects are significant, which indicates Stamping Technology, SAE, 1995, pp. 11–18.that the springback variation in this case is not distinguishable [4] L.M. Geng, R.H. Wagoner, Role of plastic anisotropy and its evolution on springback, Int. J. Mech. Sci. 44 (1) (2002) 123–148.from the system-level noise. In other words, the springback [5] K. Mattiasson, A. Strange, P. Thilderkvist, A. Samuelsson, Simulation ofvariation is not controllable in this case. Therefore, the conclu- springback in sheet metal forming, in: 5th International Conference onsions based on Monte-Carlo simulation and sensitivity analysis Numerical Methods in Industrial Forming Process, New York, 1995, pp. 115–124. [6] R.H. Wagoner, W.P. WDCarden, D.K. Carden, Matlock, Springback afterTable 10 drawing and bending of metal sheets, vol. 1, in: Proceedings of the IPMMF-tests for the standard deviation of each springback ’97—Intelligent Processing and Manufacturing of Materials, 1997, pp. P-value 1–10. [7] K.P. Li, L.M. Geng, R.H. Wagoner, Simulation of springback with the S.D. (β1 ) S.D. (β2 ) S.D. (ρ) draw/bend test, IPMM ’99, IEEE, Vancouver, BC, Canada, 1999, ISBNMaterial 0.283 0.088 0.853 0-7803-5489-3, p. 1.BHF 0.098 0.648 0.747 [8] S.W. Lee, D.Y. Yang, An assessment of numerical parameters influenc-Friction 0.635 0.495 0.542 ing springback in explicit finite element analysis of sheet metal forming process, J. Mater. Process. Technol. 80–81 (1998) 60–67.
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