Analysis of deep drawing process to predict the forming severity considering inverse finite element and extended strain-based forming limit diagram

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1 Aalysis of deep drawig process to predict the formig severity cosiderig iverse fiite elemet ad exteded strai-based formig limit diagram M. Bosta Shiri a, R. Hashemi b ad A. Assempour c,* a School of biomedical Egieerig, AmirKabir Uiversity of Techology, Tehra, Ira b School of Mechaical Egieerig, Ira Uiversity of Sciece ad Techology, Tehra, Ira c Ceter of Excellece i Desig, Robotics ad Automatio, Departmet of Mechaical Egieerig, Sharif Uiversity of Techology, Tehra, Ira. Article ifo: Received: 23/07/2016 Accepted: 10/01/2018 Olie: 10/04/2018 Keywords: Sheet metal formig, Iverse fiite elemet method, Strai path, Blak shape, Noliear deformatio, Exteded strai-based formig limit diagram. Abstract A ehaced ufoldig iverse fiite elemet method (IFEM) is used together with a exteded strai-based formig limit diagram (EFLD) to develop a fast ad reliable approach to predict the feasibility of the deep drawig process of a part ad determiig where the failure or defects ca occur. I the developed ufoldig IFEM, the meshed part is properly fold out o the flat sheet ad treated as a 2D problem to reduce the computatio time. The large deformatio relatios, oliear material behavior ad frictio coditios i the blak holder zoe are also cosidered to improve the accuracy ad capability of the proposed IFEM. The exteded strai-based formig limit diagram based o the Marciiak ad Kuczyski (M-K) model is computed ad used to predict the oset of eckig durig sheet processig. The EFLD is built based o equivalet plastic strais ad material flow directio at the ed of formig. This ew formig limit diagram is much less strai path depedet tha the covetioal formig limit diagram. Furthermore, the use ad iterpretatio of this ew diagram are easier tha the stress-based formig limit diagram. Fially, two applied examples are preseted to demostrate the capability of the proposed approach. Nomeclature ε Strai compoets matrices σ Stress compoets matrices a Displacemet vector Tractio forces Gree strai tesor compoets T E ij f r Y ( ) I-plae exteral force vector Normal aisotropy factor Yield stress Correspodig author addresses: assem@sharif.edu 1. Itroductio Equivalet stress Equivalet strai Strai ratio Sheet metal formig is oe of the most importat productio methods used i differet idustries such as producig idustrial parts, office ad 39

2 JCARME M. Bosta Shiri, et al. Vol. 8, No. 1 home appliaces, automobile body, airplae parts, etc. [1]. As a whole system, the sheet metal formig cosists of the feasibility study, process plaig, die desig, die maufacturig ad stampig. I order to prevet failures i the trial out process ad reduce desig cost by predictig crackig ad wriklig tedecies, umerical simulatios icludig fiite elemet method (FEM) are applied to check part ad die geometry at early desig stage [2]. But may process parameters, such as die geometry, blak shape, sheet thickess, blak holdig force ad frictio coditio, affect the sheet deformatio ad it is well kow that it is very importat to choose a appropriate value of them to have a successful formig process. The commoly used forward fiite elemet method (FEM), based o the icremetal formulatio, ca cosider these factors ad simulate the process with high accuracy. However, i the forward method, computatios start with the give process parameters that are ukow at the iitial desig stage. Therefore, trial ad error is the ature of the forward method which is very timecosumig. For this reaso, the ecessity of some approaches arises to determie drawig feasibility quickly. While desigig the process of sheet metal formig, the desigers cosider more about how to rapidly calculate the blak shape ad the thickess strai distributio from a give product model [3,4]. Therefore, differet methods have bee developed to estimate the blak shape which has differet accuracies. There have bee several attempts to desig the blak shape ad estimate the strai distributio i a deformed part with deformatio theory of plasticity. It has bee show that these methods have the best accuracy. Majlessi ad Lee [5, 6] showed that usig this theory is reasoable for rapid simulatio i the first stage of desig. They exteded the theory of Levy et al. [7] ad applied it to axisymmetric oe step ad multi-stage problems, obtaiig good results. However, this method caot be applied without cosiderig boudary coditios like frictio ad blak holder force. Therefore, the crash form process caot be aalyzed by this approach. Guo ad Batoz [8, 9] used virtual work theory ad derived a formulatio for field problems as a iverse method to obtai the iitial blak shape ad the thickess distributio i a deformed part. Although their method does ot eed to have iitial boudary coditios, its accuracy reduces the simulatio of parts with vertical walls. The metioed methods cosidered the problem i the 3-D coordiate system ad used oliear strai-stress relatios that have to be solved umerically; as a result, they required high computatio cost ad their covergece depeds o the appropriate selectio of the iitial guess. Based o the work of Liu ad Karima [10], Assempour et al. [11-13] proposed a oe-step iverse fiite elemet method (IFEM), kow as ufoldig techique. Their formulatios are based o the ifiitesimal strai relatios ad the priciple of potetial eergy miimizatio. I their method, the 3D problem is ufolded o the flat sheet ad therefore, treated as 2D oe. Their formulatio eds to a liear system of equatios which ca be easily solved without covergece problems ivolved i oliear methods. This method is very efficiet ad fast i obtaiig the iitial blak shape ad size. Although due to the ature of liear formulatios, the strai values are less accurate compared with the oliear IFEM approaches, they have show that the results accuracy is acceptable, ad it is logical to use their method because it is too fast ad its covergece is guarateed. The calculated strai distributio ca be used i formig limit diagrams (FLD) to determie how close the sheet metal is to tearig whe it is formed to a product shape [14]. The formig limit diagram is used i sheet metal formig aalysis to predict how the sheet metal is close to the eckig poit. The strai-path depedet ature of the formig limit diagram (FLD) causes the method to become ieffective i the aalysis of complex sheet metal formig processes [15, 16]. Recetly, the experimetal ad theoretical results showed that the exteded strai-based formig limit diagram is less sesitive to the strai path effect tha the FLD (e.g., see [17]). Accordig to above reviews, IFEM is a powerful tool i the idustry ad very useful i the early desig stage to predict the iitial blak shape ad decrease the computatio cost by elimiatig trial ad errors for blak shape 40

3 JCARME Aalysis of deep drawig... Vol. 8, No. 1 estimatio. Moreover, it ca help to check drawig feasibility if combied with FLD. So the more precise the results, the better the predictio is obtaied. This paper presets a combiatio of two techiques: (1) a iverse fiite elemet aalysis to predict strais i a formed part ad (2) the exteded formig limit diagram which plots formig limits i terms of effective strai versus strai path [18], ad thus provides a formig evaluatio that is ot early as sesitive to strai path as the covetioal formig limit diagram. This combiatio of techologies is very applicable to deep drawig parts sice such parts typically exhibit sigificatly o-liear loadig. I this paper, a ehaced ufoldig IFEM is itroduced. Gree strai relatios ad material hardeig are used for the fast calculatio of strai distributio, ad the exteded strai-based formig limit diagram based o the Marciiak ad Kuczyski (M-K) model [16] is computed ad used for explorig formig severity. The exteded strai-based formig limit diagram is used for ivestigatio of the process [19]. The exteded strai-based formig limit diagram is built based o equivalet plastic strais ad material flow directio at the ed of formig. This ew formig limit diagram is much less strai path depedet tha the covetioal formig limit diagram. Furthermore, the use ad iterpretatio of this ew diagram is easier tha the stress-based formig limit diagram. Fially, two applied examples are simulated by the proposed method ad formig severity, ad their drawig feasibility are ivestigated. 2. Ufoldig iverse fiite elemet formulatios IFEM determies the iitial blak shape as well as strai distributio i a give deformed 3D part (fial part). Assumptios made i this method are as follows: (a) material hardeig law with ormal aisotropy; (b) plae stress coditio; (c) deformatio theory of plasticity ad (d) material with the rigid-plastic model. The formulatios relate the iitial state of the material poits o the flat sheet to their fial state o the give part. I the iverse problem, the geometry of the fial part ad the thickess of the iitial blak sheet, are give while the ukows are the odal coordiates o the flat sheet ad the thickess distributio o the fial part. As the geometry of the 3D part is give, the vertical movemet of material is kow, ad the oly ukow is the displacemets i the horizotal plae. I the ufoldig techique, the geometry of the 3D part is discretized by the membrae elemets, ad all of them are properly ufolded o the flat sheet to covert the 3D problem to 2D oe. I this techique, as it is displayed i Fig. 1, each elemet is rotated about a give axis util its ormal vector becomes parallel to the global z-axis [12]. Fig. 1. Projectio ad ufoldig process for a elemet of fial part. The odal coordiates of projected elemets o the flat sheet are used as iitial guess for the odal positios i the iitial cofiguratio. The coordiates of the iitial odal positio are the improved over the computatios by applyig the priciple of the miimum potetial eergy. The first versio of ufoldig techique was based o the liear fiite elemet formulatios; after that, oliear deformatio relatios have bee implemeted i this techique to improve its accuracy [20]. The icremet of potetial eergy (W ) i a deformed body is: T T W ε σ dv a T ds (1) V S I the drawig process, material poits udergo fiite strais ad therefore the Gree strai tesor is the appropriate form for strai measure: 41

4 JCARME M. Bosta Shiri, et al. Vol. 8, No. 1 E 1 u u u u (2) 2 ij i, j j, i k, i k, j It is possible to express the Gree strai tesor as the product of a oliear operator with the odal displacemet of the elemet as follows: e e E Ba (3) where B is the oliear strai operator [21]. B depeds o the derivative of the displacemet field; therefore, it is depedet o the ukow odal displacemets. By differetiatig Eq. (3) ad substitutig it i Eq. (1), the followig relatio is obtaied: abσ a T (4) T T T W dv ds V S A deformable body i a state of mechaical equilibrium state must satisfy the priciple of virtual work as the ecessary ad sufficiet coditio for the stress field. Therefore, the first variatio of the potetial eergy with respect to the ukow odal displacemets must vaish: W T e e e T e e B σ dv TdS B σ dv f 0 e 1 Ve 1 Se a e e e1 V (5) where f correspods to the i-plae exteral forces to create the ukow odal displacemets. The calculatios of the exteral forces are expressed i [21]. After discretizatio of the part ad the itegratio over the all elemets, Eq. (5) ca be expressed as: T e e e B σ t f 0 (6) e1 e1 The stress compoets matrix ca be expressed i terms of strai compoets matrix usig the stress-strai relatioship of the material. Hill s criterio for aisotropic materials uder plae stress coditio has bee used to evaluate this fuctio [22]: f where T 0.5 Y( ) 0 σ Pσ (7) r r r P r Usig this criterio i the flow rule ad adoptig Hecky deformatio theory of plasticity lead to the followig result: σ Pε (8) Equivalet stress is calculated usig Vo Mises criteria ad the equivalet strai is calculated usig Hill s yield criterio [22] as follows: r (1 r) x (1 r) y r 2r x y 0.5 xy Substitutig Eq. (8) i Eq. (6) results i the followig oliear equatio: B e1 T ( ) P 1 Ba e e A t f e1 e 0 1/2 (9) (10) Nodal displacemets ( a ) ca be obtaied by solvig the above system of equatios, usig the Newto-Raphso method. 3. The exteded strai-based formig limit diagram As metioed earlier, the FLD is strai path depedet. So this curve caot be applied to aalyze sheet metal formig process take uder o-liear strai paths. However, a exteded strai-based formig limit curve is preseted, ad this diagram is much less sesitive to strai path chages tha the covetioal FLD. This exteded strai-based FLD is costructed based o effective strais (equivalet strais) at the oset of localized eckig ad material flow directio at the ed of sheet metal formig (Table 1) [16, 17]. I this work, the exteded strai-based formig limit diagram based o the 42

5 JCARME Aalysis of deep drawig... Vol. 8, No. 1 Marciiak ad Kuczyski (M-K) model is computed ad used to predict the oset of eckig durig sheet processig. The curret strai path is defied as the ratio of the icremetal mior strai to the major strai ad is expressed as follows: d 2 d 1 (11) By usig a yield fuctio ad a associated flow rule, the strai ratio ca be related to the stress ratio. 2 1 The exteded strai-based FLD ca be used to specify part quality i the press shop by measurig the pricipal surface strais i areas of cocer. The the strai ratios ad the equivalet strai at the determied locatios ca be calculated (e.g., from Eqs. 10 ad 11). The formig process would be safe if all the measured effective strais are located uder the exteded strai-based FLD. The proposed method ad steps of part aalysis are show i the flowchart of Fig. 2. Criterio Formig limit diagram Formig limit stress diagram (FLSD) Exteded straibased formig limit diagram (EFLD) Table 1. The compariso betwee differet FLD criteria [16]. Advatages Cool aalysis FLD extesively applied to evaluate localized eckig i shee metal formig process because of its straight-forwardess ad suitability of measurig deformatio strais either practically i formig tests or simulatively i fiite elemet simulatio FLSD has the beefit of ot depedig o strai paths FLSD be able to be suitably coverted from covetioal straibased FLD EFLD has the beefit of ot depedig o strai paths The use ad iterpretatio of EFLD is easier tha FLSD Disadvatages Subtle to strai path variatios Limited applicatios i idustrial practice - Fig. 2. Flowchart of usig IFEM together with EFLD to predict drawig feasibility. 43

6 JCARME M. Bosta Shiri, et al. Vol. 8, No Results ad discussio The iverse algorithm ad FLD are implemeted i a fiite elemet code ad applied to several examples i sheet metal formig. Amog them, two applied examples, icludig juctio box ad compressor housig, are selected for discussios ad cofirmatios of the method Juctio box example The Juctio box is a square part of 177 mm ad 24 mm sides ad corer radii, respectively. The part is 127 mm deep ad has a flage of 6 mm. The puch ad die profile radii are 5 mm ad the part is draw from a 2 mm sheet of steel. The material properties of this part are as follows: Stress Strai behavior 551 MPa Normal aisotropy factor r 0.8 The miimum rectagular blak size used i idustry is 411 mm 411 mm which its formig is very difficult, ad it is very sesitive to the process parameters [23]. Because of the symmetry coditio, oly a quarter of the part is modeled. The fial part is meshed by 2084 elemets ad aalyzed by the developed iverse method. Figs. 3 ad 4 show the calculated blak shape ad thickess strai distributio o the proposed part, respectively. Fig. 4 shows that strai i puch ad die radius zoes is high, ad rupture or wriklig ca occur i these sectios. To verify the accuracy of the obtaied strai distributio, the forward simulatio is doe with the obtaied blak shape i ABAQUS explicit. Fig. 5 shows the obtaied part ad its thickess strai distributio from forwardig simulatio. It is obvious that the draw part is ear to the desired juctio box ad its flage zoe is similar the desired part. Moreover, it shows that the high strai zoes are the same as predicted by IFEM. Fig. 3. Blak shape calculated by the developed IFEM. Fig. 4. Thickess strai distributio calculated by the developed IFEM. Fig. 5. The obtaied part from drawig simulatio i ABAQUS explicit. 44

7 JCARME Aalysis of deep drawig... Vol. 8, No. 1 The trial ad error step for blak shape ad size determiatio is omitted, ad part displayed i Fig. 5 is obtaied i the first simulatio ru. The thickess strai alog the side of the part calculated by the iverse approach ad forward method is compared i Fig. 5. To show the effect of implemetig large deformatio relatios (Gree strai) ad material oliearity, the part is simulated with liear ufoldig IFEM too, ad the thickess strai distributio alog the side is plotted i Fig. 5. ca be maufactured by deep drawig, but it is difficult. It is observed the strais are too close to the rupture limit lie ad as it was reported i idustry, formig of this part is very severe. So, it is observed that the oliear IFEM together with EFLD has predicted the severity ad feasibility of the drawig. 2 Juctio box example; Exteded FLD; f0=0.99 Equivalet strai Strai ratio IFEM Exteded FLD Fig. 6. Thickess strai alog the juctio box side. It is clear that the results of the iverse method have the same tred as the forward simulatio does, ad they are more precise tha the liear method. The maximum calculated error betwee the ABAQUS results ad the liear ufoldig IFEM ad the oliear ufoldig IFEM is 45% ad 34%, respectively. It should be metioed that because of usig deformatio theory of plasticity, iverse methods are ot highly accurate tools for strai predictio i the part but it is possible to improve them to reduce the error. If the error is high, it caot be used i FLD to predict drawig feasibility. The CPU time used for the oliear ad liear ufoldig IFEM method is 135S ad 60S, respectively. Although the CPU time icreased i the proposed method, it is too faster tha the covetioal iverse ad the forward icremetal methods. Therefore, the predicted strai distributio ca be used i EFLD. Fig. 7 shows the calculated strais i the EFLD for the cosidered part. All strais are below the EFLD ad it ca be cocluded that this part ca be draw. Accordig to deep drawig hadbooks, this part Fig. 7. Strai distributio plotted i the exteded strai-based formig limit diagram (Juctio box example) 4.2. Compressor housig example To show the applicability of the proposed method i predictig the formig feasibility, a compressor housig is studied i this example. The geometry ad material properties of this part are chose as the oes reported i [24]. This part splits i the formig process, ad it is ot possible to form it with this process parameter. Fig. 8 shows the geometry parameters of the housig ad the ruptured part. The material properties of this part are as follows: Stress Strai behavior MPa Normal aisotropyfactor r 0.95 The desired part is modeled ad meshed with 2532 elemets ad aalyzed by IFEM. The the calculated strai distributio is plotted o the exteded strai-based formig limit diagram. Figs. 9 ad 10 show the EFLD plot ad thickess strai distributio o the part, respectively. Regardig Fig. 8, some strai poits are above the EFLD lie which meas that the part will be ruptured i the drawig 45

8 JCARME M. Bosta Shiri, et al. Vol. 8, No. 1 process ad it is impossible to draw it. The elemets whose strais are above the FLD are marked i Fig. 9. It is clear that the predicted rupturig zoe is the same as the experimetal tests show i Fig. 7. It is observed that the method is also predicted rupturig i the cosidered part. Fig. 8. Geometry of Compressor Housig ad the ruptured part, dimesios are i millimeters. Equivalet strai Compressor housig example; Exteded FLD; f0= Strai ratio IFEM Exteded FLD Fig. 9. Strai distributio plotted i the exteded strai-based formig limit diagram (compressor housig example). 5. Coclusios A quick ad reliable method is itroduced to predict the drawig process feasibility. To estimate the strai distributio o the cocept part, a ehaced ufoldig IFEM is developed. The blak shape ad strai distributios are computed based o the potetial eergy miimizatio for the ufolded elemets. To improve the accuracy of the method ad to close to real coditio, the Gree-Lagrage strais is cosidered, ad oliear plastic properties of the material are implemeted i the umerical solutio of the equatio system. As a result, the proposed method has the advatage of low computatio cost of liear ufoldig IFEM ad more precise results. To predict the drawig feasibility of the part, a exteded strai-based FLD is used. This ew FLD is much less strai path depedet tha the covetioal FLD. Fially, two applied examples are studied to show the utility of the method. First, the juctio box is aalyzed as a critical part of formig. The results show that maximum error betwee ABAQUS ad iverse method is 45% for liear ad 34% for oliear IFEM. So, the ufoldig IFEM method is more accurate, ad its result o the exteded FLD is more reliable. The secod example is the compressor housig as a part that is reported caot be deep draw with the desired process parameters. The method predicts both coditios very good. Therefore, the method ca be used i the iitial desig stage to check drawability of the part ad predictig the sectios which are more probable to rupture. Ackowledgmet The authors would like to ackowledge Ira Natioal Sciece Foudatio (INSF) for fiacig this research. Refereces Fig. 10. Thickess strai distributio o the compressor housig example. [1] Y. Liu, X. Peg, Y. Qi, FE simulatio for cocurret desig ad maufacture of automotive sheet-metal parts, Joural of Materials Processig Techology, Vol. 150, No. (1-2), pp , (2004). 46

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10 JCARME M. Bosta Shiri, et al. Vol. 8, No. 1 Mechaical Scieces, Vol. 42, No. 11, pp , (2014). [19] N. Boudeau, J. C. Geli, Postprocessig of fiite elemet results ad predictio of the localized eckig i sheet metal formig, Joural of Materials Processig Techology, Vol. 60, No. (1-4), pp , (1996). [20] M. Bosta shiri, A. Assempour, Some improvemets o the ufoldig iverse fiite elemet method for simulatio of deep drawig process, The Iteratioal Joural of Advaced Maufacturig Techology, Vol. 72, No. 1, pp , (2014). [21] A. Khoei, Computatioal Plasticity i Powder Formig Processes, 1 st Editio, Elsevier, (2005). [22] R. A. Hill, Theory of the yieldig ad plastic flow of aisotropic metals, Proc. R SocLod A., Vol. 193, pp , (1948). [23] A. Assempoor, M. Karima, Iteractio of Part Geometry ad Material Properties with Formig Severity ad Toolig Desig for Box-Shaped Stampigs, SAE. Iteratioal Cogress Detroit, Michiga, Feb. pp , (1992). [24] A. Assempoor, Simulatio of formig severity ad blak developmet i geeral box-shaped parts, Iteratioal coferece o egieerig applicatios of mechaics, Tehra, Ira, Jue, pp , (1992). How to cite this paper: M. Bosta Shiri, R. Hashemi, A. Assempour, Aalysis of deep drawig process to predict the formig severity cosiderig iverse fiite elemet ad exteded strai-based formig limit diagram Joural of Computatioal ad Applied Research i Mechaical Egieerig, Vol. 8, No. 1, pp , (2018). DOI: /JTE URL: 48