Finite element analysis of electromagnetic bulging of sheet metals

Transcription

1 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury Finite element analysis of electromagnetic bulging of sheet metals Ali M. Abelhafeez, M. M. Nemat-Alla, M. G. El-Sebaie Abstract Electromagnetic forming is a high velocity forming technique that uses high pulse current to prouce repulsion electromagnetic pressure between a forming coil an the w orkpiece. In the FE moelling of such process two physical moels are involve; electromagnetic moel an mechanical moel in aition to a metho of coupling these moels. Tw o well-known coupling schemes were previously use; strong coupling an loose coupling w hich are either takes long simulation time or gives inaccurate results. Therefore some moifications were mae to the loose coupling scheme to give accurate simulation results in small uration. Material strain harening moels w hich escribe mechanical behaviour of the use material at such high spee forming process are of primary importance to get accurate simulation results. Two harening moels were use in previous researches on this process. But no comparison between them w as mae to conclue the most accurate moel in escribing harening behaviour of the use material. The current investigations introuce a comparison between two harening material moels that use in previous researches. The comparison was mae between results of numerical simulations an experimental results obtaine from literature. The use FE moel is base on moifie loose coupling scheme. Simulation results reveal that rate epenant power law harening moel gives the most accurate results with small average eviation compare w ith experimental ata. It reveals also that moifie loose coupling between mechanical an electromagnetic aspects is an efficient tool for getting accurate simulation results within short time. Inex Terms High spee forming, Electromagnetic forming, bulging of sheet metals, FEM, Strain harening moels, moifie loose coupling metho. 1 INTRODUCTION E LECTROMAGNETIC forming (EM Forming) process epens on generating high intensity transient magnetic fiels by forcing high current to flow through a coil positione very close to the workpiece. When a pulse high current flows through the coil, a transient magnetic fiel is prouce aroun the coil. This changing fiel inuces ey currents in the workpiece oppose in irection with the coil current. The ey currents of the workpiece prouce a magnetic fiel. The magnetic fiels of the coil an the workpiece repel each other, proucing high repulsive pressure. This pressure is consiere as the riving force of eformation. The pulse high current can be generate by charging capacitor banks at high voltage an suenly rain all the charge in the coil. Because EM forming characterizes by high eformation velocity an no-contact between tool an workpiece, the forming limits of metal sheets can be enhance [1] in aition to reucing springback an wrinkling [2]. These characteristics rive the researchers to try to benefit from it; especially in the manufacturing of light weight vehicles boy which is fabricate from Aluminium alloys. Because of Aluminium low fracture strain an high springback; EMF is the ieal forming technique to be use for overcoming these rawbacks an benefiting from its high electrical conuctivity. Ali M. Abelhafeez is currently pursuing master egree program in metal forming at Assiut University, Egypt, PH a.m.abelhafeez@gmail.com M. M. Nemat-Alla, professor of materials science, Assiut University, Egypt. M. G. El-Sebaie, professor of metal forming, Assiut University, Egypt. Several investigations ha been one on EM forming in which few were relate to electromagnetic sheet metals bulging. One of such early investigations was mae by Takatsu et al. [3] in which finite ifference moelling an experimental verification was mae. Takatsu experimental work consiere toay as a benchmark in EM bulging of sheet metals. His experimental ata ha been use by many other previous researchers to verify their numerical simulation. On computing technology avancement, numerical methos began to take increasing role in moelling an simulation of metal forming processes. Fenton an Daehn [4] use a 2D finite ifference coe to simulate EM forming process with a fully electromagnetic mechanical coupling. In orer to valiate their computer coe the results of the experimental work of Takatsu et al. [3] were use as benchmarks. Steinberg work harening material moel was aopte. Although their material moel ignores strain rate effects, the results showe goo agreement with experimental results. This may be attribute to the use of material moel that has much higher initial yiel stress (93 MPa) than the actual value (22 MPa [5]). El-Azab et al. [6] reporte the future nees in moelling EM forming as two major challenges; numerical challenges an material moelling challenges. They claime that numerical solution of the general fully couple problem has not been previously achieve. On the other hans the material challenges are appear in aopting strain rate an temperature epenency through the eformation processes. Only their net effects on the eformation process can be observe in laboratory tests while experimental observation of their effect with time an eformation is ifficult ue to the high spee of eformation an biaxiallity of strains.

2 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury Correia et al. [7] mae a trial to overcome moelling challenges of EM forming. They use rate epenant power law for escribing strain harening behaviour of Takatsu et al. [3] isk material. They consiere a simple moel of EM bulging process, in which the mechanical an electromagnetic aspects were treate as two inepenent problems. ABAQUS/Explicit FEM software was use to simulate the eformation of the sheet as explaine in Takatsu et al. [3]. The magnetic spatial an temporal pressure istributions were etermine using finite ifference coe which was implemente in ABAQUS user subroutine name VDLOAD to solve magnetic iffusion equations. The obtaine pressure was then use as a loa to the mechanical problem to calculate the eformation. This coupling technique is known by loose coupling. Although they use strain rate epenent power law, their results are not in goo agreement with previous experimental results of Takatsu et al. [3]. This may be ue to the consieration of loose coupling scheme without moifications that gives more accurate simulation results as mentione by Hashimoto et al. [8]. Siiqui et al. [9] enhance electromagnetic part of Correia moel [7] for getting more accurate simulation results. They compare the obtaine simulation results with experimental ata [3] an a goo agreement was notice. They conclue that mesh size effects have negligible influence on results ue to small thickness size of workpiece. Recently, Cui et al. [1] simulate experimental work of Takatsu et al. [3] using ANSYS multi-physics FEM software. Strong coupling scheme was use which accounts for magnetic pressure change with workpiece eformation. Following Fenton an Daehn [4], the simplifie Steinberg Material harening moel was use. The simulation results were in goo agreement with experimental results but simulation process was time consuming. Finally from the previous literature survey it can be conclue that two material harening moels were use to escribe Takatsu [3] isk material behaviour. These two material harening moels are Steinberg moel, an rate epenant power law moel. Previous researchers use ifferent FE simulation moels that utilize these harening moels an goo agreement was obtaine. But right comparison between these harening moels must be with the same FE moel to conclue the most accurate one. In this research, an FE moel was establishe base on moifie loose coupling between electromagnetic an mechanical aspects. This loose coupling strategy was consiere before by many other researchers [7], [11], [12], [13]. The moel was use to compare these harening moels to etermine the most accurate one in escribing mechanical behaviour of the use material at these conitions. 2 PROCESS MODELLING 2.1 Electrical system moel EM sheet metals bulging is a high velocity forming technique in which there is a spiral flat coil positione near to a flat circular blank workpiece. A high repulsion pressure prouce between coil an workpiece when transient high electrical current passes in the coil. Typical process setup is shown in Fig. (1) an imensions are given in table (1). This setup coul be moelle electrically by two mutually couple circuits that shown in Fig. (2). TABLE (1) COIL AND WORKPIECE DIMENSIONS. Coil Workpiece Major raius 32 mm Thickness.5 mm. No. of turns 5 Bulge iameter 8 mm. pitch 5.5 mm Overall iameter 11 mm. Coil/ Workpiece separation istance = 1.6 mm Fig. 1. Typical setup of EM bulging of sheet metals. Fig. 2. Electrical moel of the process. Governing ifferential equations of these magnetically couple circuits accoring to Takatsu et al. [3], is: 1 L1 i1( t ) ( M. i 2( t )) R1. i1( t ) i1( t ) t ; t t c ( L2. i 2( t )) ( M. i1( t )) R2. i 2( t ) t t With initial conitions: i (), i (), [ L i ] t V t t (1)

3 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury Where, L 1, L 2 are inuctance of the coil an workpiece circuit; M is mutual inuctance between the two circuits; R1, R2 are electrical resistances of the coil an workpiece circuit; c capacitance of ischarge circuit;v is initial ischarge voltage of the capacitor banks; i1(t) is ischarging current as function of time for the coil circuit; i2(t) is inuce current function of time for the workpiece circuit. These two simultaneous equations have the mutual inuctance term M which represent the magnetic coupling between the two circuits. The mutual inuctance M an the selfinuctance of the workpiece L 2 change with the workpiece eformation an eformation epens mainly on M an L2. This interepenency makes the mathematical solution of these two equations almost ifficult unless using numerical methos with simplifying assumptions. Assuming that M an L2 are constants an not epen on eformation will simplify this problem. This assumption is part of loose coupling technique use in tying electromagnetic an mechanical aspects. The equivalent circuit moel now coul be obtaine, an the reuce equation for the current will be [8]: 1 L i ( t ) R i ( t ) i ( t ) t V c c c c c t cc With initial conitions: Where, Lc is total inuctance of the system; Rc is total electrical resistance of the system; Cc is total capacitance of the system; ic(t) is the current passing in the coil; V is initial ischarge voltage of the capacitor banks. The solution of Eq. (2) is given by [14]: V _ t i ( t ) e sin( t ) (3) L Where, 1 L C c i (), [ L i ] V t c c c t Rc ( ) 2L c c c 2 Values of these electrical parameters are given by table (2). TABLE (2) PROCESS ELECTRICAL PARAMETERS. Total inuctance L c 2.86 µh Total Capacitance C c 4 µf Charging voltage V 6 kv Total Resistance R c 28.5 mω It can be notice that the electric current function has an exponentially ecaye sinusoial wave form as shown in Fig. (3). t (2) Fig. 3. Variation of electric current i(t) with time Magnetic pressure moel Magnetic pressure generate from magnetic fiel of the coil current is the riving force of workpiece eformation. Its istribution over the workpiece is not uniform an it is varying with time an with workpiece eformation. Thus magnetic pressure is epenent on workpiece eformation an eformation is epenent on applie magnetic pressure. This interepenency makes perfect etermination of this pressure epens mainly on the electromagnetic-mechanical aspects coupling scheme use. Assuming magnetic pressure to be inepenent on workpiece eformation will simplify the analysis with small relative errors in results. Therefore the general magnetic pressure function P(r,t) can be consiere as a function of blank raius multiplie by a pressure function of time, i.e. P( r, t ) f ( r) p( t ) The temporal variation of magnetic pressure was expresse before [8] as: B ( t ) B ( t ) pt () iff Where, B(t) is magnetic flux ensity between workpiece an coil; Biff (t) is iffuse flux ensity; an µ is magnetic permeability of the air. Magnetic flux ensity; B(t); is given by [15], B ( t ) K i ( t ) Where; K is constant epens on workpiece an coil geometry an skin epth. The iffuse magnetic flux can be neglecte for nonmagnetic materials like Aluminium [16], thus; Bt () pt () 2 2 Consequently from (3), (6) an (7) the final form of temporal pressure behaviour is 1 2 V 2 _ 2t 2 p( t ) (8) K ( ) e sin ( t ) 2 L c (4) (5) (6) (7)

4 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury Such variation of temporal behaviour of the magnetic pressure is shown in Fig. (4). The spatial istribution at a given time can be etermine by stuying the magnetic flux ensity istribution over the isk raius. This istribution can be foun by numerically solving Maxwell equations of electromagnetism for specific coil-workpiece geometry. Such numerical solution is beyon the scope of this research paper. Previous calculate ata [3] of spatial pressure istribution at time of maximum pressure value was use; after normalizing; to etermine spatial istribution function. Normalize ata was obtaine by iviing all pressure values with maximum p(t) value an is shown in fig. (5). simulation package, an implicit ynamic finite element coe, which is use in sheet eformation analysis. The ynamic equilibrium equation is given by (9), an the Newmark time integration metho is use to solve it [17]. M u C u K u F (9) Where M represents the mass matrix (Kg), C is the amping matrix (Kg/s), K is the stiffness matrix (Kg/s 2 ), F is loa vector (N). Accoring to Takatsu experimental work [3], workpiece mechanical an electrical properties are liste in table (3). TABLE (3) WORKPIECE MATERIAL PROPERTIES Density 275 Kg/m 3 Young s Moulus 8.7 GPa Poisson s ratio.33 Yiel stress 22 MPa. Electric conuctivity MS/m The workpiece is meshe with quarilateral finitemembrane-strain element with reuce integration points on its surface an 9 integration points through its thickness. Total number of elements is 4 an the meshe workpiece is shown in fig. (6). Fig. 4. Temporal behaviour of magnetic pressure. Fig. 6.Non-eforme workpiece mesh. During eformation, workpiece outer perimeter is consiere to be fixe. There are two harening moels that are extensively use in moelling JIS A15-O material which use in Takatsu et al. [3] experimental work. Simplifie Steinberg moel [4], [1] as given by (1), an rate epenent power law [7], [9] as given by (11). 93(1 125 ).1 (1) Fig. 5. Pressure spatial istribution over workpiece raius. Now magnetic pressure function has been ientifie an etermine. This function is entere to the mechanical FE moel as surface pressure on the bottom surface of the blank with spatial an temporal variation specifie before. 2.3 Mechanical FE moel The magnetic pressure calculate by (4) is use as bounary conition to the structure moel an entere to an FEM (11) Where is the effective stress in MPa; is the effective strain. These harening moels are shown in fig. (7). To ecie the most accurate moel in escribing harening behaviour of this material; An FE simulation was built which is base on a moifie loose coupling scheme. Next section escribes this coupling scheme.

5 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury [3] experimental ata. It is clear that the use strain harening moels have a great effect on the simulation results. Obviously, there is a moerate agreement between the experimental results an the rate epenent harening moel simulation results. In aition to a large eviation between the other harening moel simulation results an Takatsu results are clearly appeare. Fig. 7. Effective stress-strain iagrams for use harening moels (ashe lines represent rate epenent moel an soli line represents Steinberg moel). Fig. 8. Moifie pressure temporal behaviour. 2.4 Moifie loose coupling scheme Loose coupling epens on fining the magnetic pressure spatial an temporal istribution inepenent of workpiece eformation. This is of course untrue an always gives overestimate simulation results. A moification of this coupling scheme to get more accurate results is mentione by Hashimoto et al. [8]. They reporte that only the first wave of current affects eformation of workpiece. Thus the moifie loose coupling scheme consiers only the first wave of current to etermine magnetic pressure temporal behaviour. On the other han, the spatial pressure istribution function f(r) was specifie before from electromagnetic FE simulation. Therefore the total loa to be applie on the workpiece surface is P(r,t) = p(t) moifie. f(r), where f(r) is as given by Fig. (5), an p(t) moifie is given by (12) an shown in fig. (8) K i ( t ), t pt moifie 2, t (12) Where, is the perioic time of current wave i(t) an equals 67 µs. 3 FE SIMULATION AND RESULTS ANALYSIS Two runs of the simulation were one each with one of the harening moels previously liste. The final eforme profile of the workpiece for each harening moel is presente in fig. (9) in conjunction with, Takatsu Fig. 9. Final profile of eforme workpiece w ith using various harening moels. The relative errors between simulation results an experimental work are calculate over the whole eforme workpiece an presente in Fig. (1). The maximum relative error was about 9 % for Steinberg moel while it was about 45 % rate epenent harening moel. It is obvious that the most accurate strain harening moel is rate epenent power law which has a relative error with average value over isk raius of 3 %. This result is logic since the strain rate in this process has very large values an can t be neglecte. The effective plastic strain istribution on the final eforme isk is shown in Fig. (11). It is clear that a maximum value of effective plastic strain of.45 was achieve at blank

6 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury centre for rate epenent strain harening moel. For the other strain harening moel; the maximum effective plastic strain value was achieve at blank ege which may be not true. Fig. 12. Effective strain rate variation w ith time for element at blank centre. Fig. 1.Relative absolute error between FEM simulation an experimental ata. Fig. 13. Effective strain rate variation w ith time for element at 2 mm from centre. Fig. 11. Effective plastic strain istribution. Effective plastic strain rate for elements at blank centre an at raial istance of 2 mm are plotte against time as shown in Fig. (12) an (13). It can be notice that the maximum strain rate over the whole blank attaine at centre. The strain rate at blank centre has its maximum value near en of eformation on contrast with strain rate at raial istance 2 mm. This coul be explaine by the earlier movement of blank outer perimeter than inner perimeters. 4 CONCLUSIONS In the current investigations a simple an accurate finite element moel for the EM forming of sheet metals was introuce. Two ifferent strain harening material moels were consiere in simulation an the obtaine results were compare with publishe experimental ata. From the simulation results an comparison between them the following points can be conclue: 1- The introuce simulation moel gives simulation results in goo agreement with experimental ata when use with rate epenent power law harening moel. This is logic since high strain rates were achieve in this process an thus strain rate effects can t be neglecte. 2- The final eforme blank profiles in t have any change in shape for any use strain harening moel but the changes are only in imensions of the final eforme blank. This means that the final profile epens mainly on the pressure spatial an temporal istribution. 3- For rate epenent power law moel; maximum values of strains an strain rates are achieve at blank centre. This means that failure is possible to occur at centre of the blank.

7 International Journal of Scientific & Engineering Research Volume 3, Issue 2, Febraury REFERENCES [1] Seth, M., Vohnout V. J., Daehn G. S., "Formability Of Steel Sheet In High Velocity Impact", J. mater. Process. Tech., 168, pp. 39-4, (25). [2] Pamanabhan, M., "Wrinkling An Springback In Electromagnetic Sheet Metal Forming An Electromagnetic Ring Compression", Master thesis, The Ohio State University, (1997). [3] Takatsu, N., Kato, M., Sato, K., Tobe, T., "High Spee Forming Of Metal Sheets By Electromagnetic Force", J.S.M.E., 31(1), p. 142, (1988). [4] Fenton, G. K., Daehn, G.S., "Moeling Of Electromagnetically Forme Sheet Metal", J. mater. Process. Tech., 75, pp. 6-16, (1998). [5] Kono, K., Suzuki, H., "Research On The Accuracy Of Sheare Proucts By Different Working Principles In Precision Shearing", J. mater. Process. Tech., 56, pp. 7-77, (1996). [6] El-Azab, A., Garnich, M., Kapoor, A., "Moeling Of The Electromagnetic Forming Of Sheet Metals: State-Of-The-Art An Future Nees", J Mater. Process. Tech., 142, pp , (23). [7] Correia, J. P. M., Siiqui, M.A., Ahzi, S., Belouettar, S., Davies, R., " A Simple Moel To Simulate Electromagnetic Sheet Free Bulging Process.", Int. J. Mech. Sci., 5, pp , (28). [8] Hashimoto, Y., Hieki, H., Miki, S., Hieaki, N., "Local Deformation An Buckling Of A Cylinrical Al Tube Uner Magnetic Impulsive Pressure", J Mater. Process. Tech., 85, pp , (1999). [9] Siiqui, M. A., Correia, J. P. M., Ahzi, S., Belouettar, S., "A Numerical Moel To Simulate Electromagnetic Sheet Metal Forming Process.", Int. J. Mater. Form., 1, pp , (28). [1] Cui, X., Mo, J., Xiao, S., Du, E., Zhao, J., "Numerical Simulation Of Electromagnetic Sheet Bulging Base On FEM", Int. J. Av. Manuf. Tech., pp. 1-8, (211). [11] Imbert, J. M., "Increase Formability an the Effects of the Tool/Sheet Interaction in Electromagnetic Forming of Aluminum Alloy Sheet", M.Sc., University of Waterloo, (25). [12] Oliveira, D. A., "Electromagnetic Forming of Aluminum Alloy Sheet: Experiment an Moel.", M.Sc., University of Waterloo, (22). [13] Pérez, I., Aranguren, I, González, B, Eguia, I, "Electromagnetic Forming: A New Coupling Metho", Int. J. Mater. Form., 2, pp , (29). [14] Xu, W., Fang, H., Xu, W., "Analysis Of The Variation Regularity Of The Parameters Of The Discharge Circuit With The Distance Between Workpiece An Inuctor For Electromagnetic Forming Processes", J. mater. Process. Tech., 23, pp , (28). [15] Zhang, H., Murata, M., Suzuki, H., "Effects Of Various Working Conitions On Tube Bulging By Electromagnetic Forming", Journal of Materials Processing Technology, 48, pp , (1995). [16] Kleiner, M., Beerwal, C., Homberg, W., "Analysis of Process Parameters an Forming Mechanisms within the Electromagnetic Forming Process", Annals of the CIRP, 54, pp , (25). [17] Yu, H. P., Li, C.F., Deng, J.H., "Sequential Coupling Simulation For Electromagnetic Mechanical Tube Compression By Finite Element Analysis", J. Mater. Process. Tech., 29, pp , (29).