finite element simulation of the warm deep drawing process...

Scientia Iranica B (2013) 20(4), 1213{1220

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Finite element simulation of the warm deep drawingprocess in forming a circular cup from magnesium alloysheet

S.A. Ossia and B. Soltani�

Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, P.O. Box 87317-51167, Iran.

Received 6 October 2011; received in revised form 26 February 2013; accepted 21 April 2013

KEYWORDSSheet metal forming;Warm deep drawing;Magnesium alloysheet;Finite elementsimulation.

Abstract. In recent years, due to its low density and high strength, the use of magnesiumalloy in sheet metal forming has been of interest to many researchers. Because of the lowformability of magnesium alloy at room temperature, warm forming has been developed inorder to produce parts of a magnesium alloy sheet. In this paper, the warm deep drawingprocess of a circular cup from a magnesium alloy AZ31B sheet was simulated using the�nite element software ABAQUS/Standard. Simulation results obtained from this studyagree well with experimental results and show that the limit drawing ratio increases withan increase in forming temperature. Maximum LDR was obtained as 2.5 at the formingtemperature of 200�C. Simulation results indicate that heat transfer plays an important rolein increasing formability. By comparing isothermal and non-isothermal warm deep drawing,the heat transfer e�ect and location of sheet failure have been investigated. The e�ects offriction coe�cient and punch speed on the limit drawing ratio were also studied. As a result,increase in the friction coe�cient leads to an increase in the punch force and a decrease inthe limit drawing ratio. Results also show that the maximum friction coe�cient to produceacceptable parts increases with an increase in the forming temperature. In addition, resultsindicate that an increase in punch speed leads to a decrease in the drawing ratio limit.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Lightening of components is of particular importancein di�erent industries in order to reduce weight, whichleads to reduced fuel consumption, energy use andcost. Magnesium alloy is a light alloy that hasreceived much attention in industrial applications inrecent years, due to its low density and high strength.Die casting of magnesium alloys is a popular methodfor manufacturing parts. But, parts which are pro-duced by sheet metal forming processes have good

*. Corresponding author. Tel.: +98 361 5912450;Fax: +98 361 5912424E-mail addresses: [email protected] (S.A. Ossia) [email protected] (B. Soltani)

mechanical properties and surface quality in com-parison with die casting. For these reasons, sheetforming of magnesium alloys has been recommendedfor future development of parts, and because of thepoor formability of magnesium alloy sheeting at roomtemperature, warm forming has been developed inorder to produce parts of the magnesium alloy sheet.The warm forming of magnesium alloy sheet is donefor two reasons. Firstly, warm working leads to theincreased ductility and formability of the sheet, andsecondly, the yield point and forming force decrease atelevated temperatures. Droder [1] studied the warmdeep drawing of magnesium alloy sheets in producingcircular and rectangular parts experimentally, andobtained signi�cant temperature limitations for thisprocess. He performed a uniaxial tensile test at

1214 S.A. Ossia and B. Soltani/Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1213{1220

di�erent temperatures to obtain the stress-strain curveof magnesium alloy sheet. Doege and Droder [2] formedround cups of magnesium AZ31, AZ61 and M1 sheetsby experiment. They showed that these alloys havegood formability in ranges between 200�C and 250�C.In addition, they investigated the in uence of punchvelocity on the limit drawing ratio and reported thatan increase in punch velocity leads to a decrease inthe limit drawing ratio. Yoshihara et al. [3] usedlocalized heating and cooling for increased formabilityof magnesium alloy sheeting to produce a circular cup.They obtained the limit drawing ratio, 5.0, by localheating and cooling using the variable blankholderpressure technique. Yoshihara et al. [4] also investi-gated optimization of magnesium alloy stamping usingthe �nite element method and compared the numericaland experimental results. Palaniswamy et al. [5]studied the formability of magnesium alloy AZ31Bsheet at elevated temperatures using the commercialFE code DEFORM 2D for round cups and DEFORM3D for rectangular pans. They concluded that thelimit drawing ratio increases with an increase in theforming temperature. Neugebauer et al. [6] explainedthe development of forming processes of magnesiumand aluminum alloy sheets at elevated temperatures,and described warm forming and hot stamping thatare being developed to produce light weight materials.El-Morsy and Manabe, [7] and also Huang et al. [8]employed �nite element analysis for simulation of non-isothermal deep drawing. Palumbo et al. [9] conducted�nite element and experimental studies on the warmdeep drawing of magnesium AZ31 alloy sheet. Theyinvestigated the e�ect of heating and punch speedon the limit drawing ratio. Zhang et al. [10] alsosimulated the warm deep drawing of magnesium alloysheets, numerically, and compared the results withexperiments. Chang et al. [11] compared the resultsof experimental and numerical studies on the warmdeep drawing of magnesium alloy AZ31 sheet. Kayaet al. [12] performed, experimentally, the warm deepdrawing of magnesium and aluminum alloy round cupsat elevated temperatures using a servo motor drivenpress.

In this paper, the warm deep drawing pro-cess of magnesium alloy AZ31B sheet in a circu-lar die using the implicit �nite element softwareABAQUS/Standard was simulated. In order to val-idate the numerical results, the curves of the punchforce, thickness distribution and the limit drawingratio to the forming temperature were compared withthe experimental study of Droder [1] and the nu-merical investigation of Palaniswamy et al. [5]. Af-ter validation, some parameters in warm deep draw-ing, such as temperature, location of thinning, fric-tion coe�cient and punch speed, have been stud-ied.

2. Numerical simulation

2.1. Governing equationsThe �nite element is very e�ective method for analysisof metal forming processes and for optimizing theirparameters. The �nite element analysis of the warmdeep drawing process is a thermo-mechanical analysisfor mechanical and thermal loads. For the mechanical�eld, the updated Lagrangian formulation is usedbecause of the large deformation of the warm deepdrawing process. The equilibrium statement, writtenas the virtual work equation, and based on the updatedLagrangian formulation, is as follows [13]:Z

tV

t+�tt Sij�t+�t

t "ijdtV =ZtS

t+�tt fSi �uid

tS

+ZtV

t+�tt fBi �uid

tV; (1)

where Sij is the second Piola-Kirchhof stress tensor,"ij is the Green-Lagrange strain tensor, �ui is virtualdisplacement vector, tV is volume, tS is surface at time,t, and fSi and fBi are the components of surface andbody forces.

The full Newton-Raphson iterative method, as anumerical technique, is used to solve the nonlinear equi-librium equations. Because of appearing the Cauchystress after linearization of the equation of motion, andits change under rigid motion, the Jaumann stress rateis also used in the updated Lagrangian formulation.

For the numerical solution of heat transfer in thewarm deep drawing process, the principle of virtualtemperatures is as follows [14]:Z

V��0T k�0dV =

ZV��qBdV +

ZS��SqSdS; (2)

where:

k =

24kx 0 00 ky 00 0 kz

35 ; �0T =h@�@x

@�@y

@�@z

i: (3)

In the heat transfer equation, �� is the virtual displace-ment vector, �0 is the temperature gradient vector, k isthermal conductivity, qB is the rate of heat generationper unit volume, �S is the known surface temperatureon S, and qS is the heat ux input on this surface.

Fully coupled thermo-mechanical analysis is usedto obtain the nodal displacement and temperatures inthis research, because of the inelastic deformation ofthe material and the contact condition in the mechani-cal �eld that should be coupled to the thermal �eld.Therefore, mechanical and thermal solutions a�ecteach other strongly and must be solved simultaneously.An exact implementation of Newton's method is used

S.A. Ossia and B. Soltani/Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1213{1220 1215

to solve the nonlinear coupled system. The coupledequations are as follows [15]:�

Kuu Ku�K�u K��

� ��u��

�=�RuR�

�; (4)

where �u and �� are the incremental displacement andtemperature, Kij are submatrices of the fully coupledJacobian matrix, and Ru and R� are the mechanicaland thermal residual vectors.

2.2. Finite element modelingA 2D axial-symmetric model is used in the simulationbecause of the axial symmetry of the process. Thegeometry of the sheet and the tools, based on Droder'sstudy [1], are shown in Table 1. Tools were mod-eled as rigid bodies by performing constraints on thedisplacement of all nodes by a reference node. Themechanical properties of the magnesium alloy AZ31Bsheet is shown in Table 2. Figure 1 shows the owstress-strain curve of the magnesium alloy AZ31B sheetat di�erent temperatures used by Palaniswamy et al. [5]in their simulation. The thermal properties of the sheetand the tools are also shown in Table 3. The sheetis modeled as a body with elastic-plastic and thermalproperties. The von Mises yield function is appliedfor the material hardening behavior which is isotropichardening.

Table 1. The geometry of the sheet and the tools [1].

Punch Diameter 100 mmPunch and die corner radius 12 mmInitial punch temperature 25�CSheet thickness 1.3 mm

Table 2. Mechanical properties of magnesium alloyAZ31B sheet [1,2].

Module of elasticity 45 (GPa)Poisson ratio 0.35Flow stress curve Figure 1

Figure 1. Flow stress of magnesium alloy AZ31B sheet atdi�erent temperatures [1,2].

Table 3. Thermal properties of sheet and tools [1,5].

Thermal expansion coe�cient (sheet) 25e-6 (1/�C)Thermal conductivity (sheet) 159 (W/m�C)Heat capacity (sheet) 1.7675 (N/mm2�C)Thermal conductivity (tools) 60.5 (W/m�C)Heat capacity (tools) 3.41 (N/mm2�C)Convection coe�cient 30 (W/m2�C)Interface heat transfer coe�cient 4500 (W/m2�C)

For creating contact between the sheet and thetools, the penalty method, as a contact algorithm withCoulomb friction law, is used. The friction coe�cientis considered 0.1, from the strip draw test done byDroder [1]. In this study, the friction coe�cients forthe punch, the die and the blankholder in contactwith the sheet are the same. According to Table 2,the interface heat transfer coe�cient is considered tobe 4500 W/m2�C for contact regions [5,15]. For theheating of the sheet inside the tool, the sheet shouldbe �xed between the die and the blankholder for afew seconds until the temperature of the sheet reachesthe temperature of the die and the blankholder beforethe drawing process [1,2]. Therefore, for analysisof the warm deep drawing process, the initial tem-peratures of the sheet, the die and the blankholderare considered the forming temperatures. The initialtemperature of the punch is also considered as roomtemperature (25�C). In coupled thermo-mechanicalanalysis, an element should be employed which hasboth displacement and temperature in its degrees offreedom. As a result, in this study, the solid 8-nodeaxisemmetric quadrilateral element, CAX8RT, witha reduced integration technique is used for meshingthe sheet and the tools. The shape function of thiselement for interpolation of displacement is biquadraticand for interpolation of temperature is bilinear. Fig-ure 2 shows the �nite element model of the warm

Figure 2. Finite element model of warm deep drawingprocess.


deep drawing process. The sheet is meshed withsix elements along the thickness, based on optimizedmeshing.

3. Results and discussion

After starting the analysis, the punch, at room tem-perature, would be in contact with the preheatedsheet. So, according to consideration of the interfaceheat transfer and because of high thermal conductivityand the low heating capacity of magnesium alloy, thetemperature of the sheet decreases quickly. Figure 3indicates the temperature variation of the sheet cen-ter with punch displacement for LDR of 2.3 at theforming temperature of 200�C during the drawingprocess.

Initial punch temperature is an important param-eter for improving the formability of a magnesium alloysheet. It should be noted that the temperature of adeformed cup depends on the initial temperature of theworkpiece and tools, the heating of plastic deformation,the heating of friction, and the heat transfer betweenthe workpiece and the tools with the environment.Contacting the punch with the sheet and the thermalinterchange between them leads to an increase in thestrength of the sheet at the bottom and corner of thepunch. Thus, the formability of magnesium alloy sheetin the warm deep drawing process increases.

3.1. Punch forcePunch force is one of the most important parametersin metal forming processes that is equal to the force fordeforming the workpiece plus the force for overcomingthe friction between the tools and the sheet. Figure 4shows the comparison of the punch force curve with thesimulation results of the present work, the experimentalresults of Droder [1] and the numerical analysis ofPalaniswamy et al. [5] for LDR of 2.3 at the formingtemperature of 200�C.

The trend of changing the punch force duringthe process, obtained from FEA (present work and

Figure 3. The temperature variation of sheet center withpunch displacement for LDR of 2.3 at the formingtemperature of 200�C.

Figure 4. Punch force obtained from numericalsimulations and experiment for LDR of 2.3 at the formingtemperature of 200�C.

Palaniswamy et al. [5]), matches with experimentalresults in which maximum punch force occurs in apunch displacement of 20 mm to 40 mm. It is alsoobserved that the punch force curve obtained from thepresent work was closer to experimental investigationthan the result of Palaniswamy et al. [5]. It shouldbe noted that after the peak point, the punch forcedecreases gradually, due to the decrease in the lengthof the sheet under the blankholder.

3.2. Thickness distributionFigures 5 and 6 show the comparison of the thicknessdistribution obtained from FEA with the experimentalresult for LDR of 2.3 at the forming temperatures of200�C and 250�C. The trend of change in thickness (%)obtained from the present work matches completelywith the experimental result and has some improve-ments in comparison with the numerical results ofPalaniswamy et al. [5].

It should be mentioned that maximum thinningoccurs in the length 60 mm to 80 mm of the cup.This shows that the maximum thinning observed inthe cup wall is opposite to cold warm deep drawing,where maximum thinning occurs in the punch radiusregion.

Figure 5. Thickness distribution obtained from numericalsimulations and experiment for LDR of 2.3 at the formingtemperature of 200�C.


Figure 6. Thickness distribution obtained from numericalsimulations and experiment for LDR of 2.3 at the formingtemperature of 250�C.

3.3. The e�ect of forming temperature onlimit drawing ratio

When thinning rate exceeds 25%, it can be concludedthat the sheet is considered to fail [5]. In additionto this, the drawing process has been run at di�erentLDRs and temperatures for investigating thinning andfailure in the sheet. Figure 7 indicates the failedsheets that thinned in the die radius region underdi�erent conditions. Figure 8 displays the dependencyof the limit drawing ratio on the forming temper-ature obtained from simulation, which is comparedwith experimental results. It shows that the limitdrawing ratio increases with an increase in the formingtemperature. Maximum LDR was obtained as 2.5 atthe forming temperature of 200�C, which demonstratesgood agreement between numerical and experimentalresults, whereas Palaniswamy et al. [5] obtained amaximum LDR of 2.4 at the forming temperature of200�C. After the maximum point, the LDR decreaseswith an increase in the forming temperature, becausework hardening decreases with an increase in formingtemperature.

Figure 7. Location of thinning at di�erent temperaturesand LDRs obtained from numerical simulation.

Figure 8. The e�ect of forming temperature on limitdrawing ratio.

3.4. The e�ect of heat transfer on improvingformability of magnesium alloy sheet

The e�ect of heat transfer and temperature distributionon increased formability has been investigated, asuniform temperatures are considered for the tools andthe sheet. Figure 9 shows the comparison betweenisothermal and non-isothermal deep drawing for LDRof 2.5 at the forming temperature of 200�C. Accordingto Figure 9(a), it is concluded that heat transferbetween the tools and the sheet leads to successfuldrawing. However, Figure 9(b) shows that neglectingheat transfer between the tools and the sheet leads tothinning occurring in the punch radius region. Fig-ure 10 also displays the comparison between isothermaland non-isothermal deep drawing for LDR of 2.5 at theforming temperature of 250�C. It is clearly observedthat the punch displacement of non-isothermal drawingis more than in the isothermal one. This is because ofthe fact that heat transfer between the punch and thesheet leads to increase the strength of the sheet in thepunch radius region. So, the thinning happens in thedie radius region. But, in isothermal deep drawing,similar to conventional deep drawing, due to uniformtemperature in the sheet, thinning and failure occur inthe punch radius region.

Figure 9. The e�ect of heat transfer on improvingformability: (a) Non-isothermal deep drawing; and (b)isothermal deep drawing.


Figure 10. The e�ect of heat transfer on improvingformability: (a) Non-isothermal deep drawing; and (b)isothermal deep drawing.

3.5. The e�ect of friction coe�cient on limitdrawing ratio at di�erent temperatures

Lubrication between tools and the sheet is one of themost signi�cant parameters in the deep drawing pro-cess and has great in uence on the forming force. Thehigh friction coe�cient causes an increase in the frictionforce and the punch force, which leads to thinning inthe sheet. Figure 11 shows the e�ect of the frictioncoe�cient on the limit drawing ratio at the formingtemperatures of 150�C and 200�C. It shows that themaximum friction coe�cient for successful drawingdecreases with an increase in the limit drawing ratio. Inother words, if LDR increases, the lubrication rate forproducing acceptable parts should be increased. It isalso observed that with an increase in temperature from150�C to 200�C, the maximum friction coe�cient forsuccessful drawing increases. So, it can be concludedthat an increase in the forming temperature leads to adecrease in lubrication rate between the tools and thesheet.

3.6. The e�ect of punch speed on limitdrawing ratio

Punch speed is an important factor, which a�ectsheat transfer between the tools and the sheet. Themaximum cup height with low punch speed is higherthan maximum cup height with high punch speed.The reason is that the heat transfer is more e�ectiveat low punch speed than at high punch speed [7].In this study, the punch speed was considered to be

Figure 11. The e�ect of friction coe�cient on limitdrawing ratio at di�erent temperatures.

5 mm/s at all temperatures and LDRs, based on theexperimental work of Dorder [1]. For analyzing thee�ect of punch speed on fracture and thinning, thepunch speed was increased gradually until thinningoccurred in the sheet.

Figure 12 shows the deformed sheet at di�erentpunch speeds and LDRs at the forming temperature of200�C. The sheet is thinned in the die radius region.Figure 13 displays the curve of the limit drawing ratio-punch speed for LDR of 2.3, 2.4 and 2.5 at the formingtemperature of 200�C.

It shows that the maximum punch speed foracceptable drawing decreases with an increase in thelimit drawing ratio. This is because the necessary timefor heat transfer between the tools and the sheet shouldbe increased, with an increase in the limit drawingratio, which leads to a decrease in punch speed.

Figure 12. Location of thinning at di�erent punch speedsand LDRs obtained from numerical simulation.

Figure 13. The e�ect of punch speed on limit drawingratio.


4. Conclusions

In this study, the warm deep drawing of a magnesiumalloy AZ31B sheet in a circular die was simulated in theimplicit �nite element software, ABAQUS/Standard,using elastic-plastic fully coupled thermo-mechanicalanalysis. The obtained results were compared withthe experimental work of Droder [1] and the numericalinvestigation of Palaniswamy et al. [5] in order tovalidate the FE analysis, and then further parametersof forming were studied. The conclusions are as follows:

1. The forming force at the forming temperature of200�C and di�erent LDRs has been investigatedand it is concluded that punch force increaseswith an increase in the limit drawing ratio at con-stant temperature. The trend of the punch force-displacement curve obtained from the numericalsimulation for LDR of 2.3 at the forming tempera-ture of 200�C matches well with the experimentalwork of Droder [1], and had some improvementscompared to the numerical study of Palaniswamyet al. [5]. It must be noted that maximum punchforce occurs in a punch displacement of 20 mm to40 mm in all investigations.

2. The trend of the thickness distribution obtainedfrom the present work for LDR of 2.3 at the formingtemperatures of 200�C and 250�C has good agree-ment with the experimental work of Droder [1] andhad some improvements compared to the numericalinvestigation of Palaniswamy et al. [5]. It should bealso noticed that maximum thinning occurs in thelength 60 mm to 80 mm of the cup. This could bedue to the fact that temperature distribution fromthe sheet center to the ange region increased andthus the strength of the sheet decreased. So, it canbe concluded that maximum thinning occurs in thecup wall.

3. The obtained results show that the limit drawingratio increases with an increase in forming temper-ature. Maximum LDR was obtained as 2.5 at theforming temperature of 200�C, which demonstratesgood agreement between the numerical analysisof the present work and the experimental result,whereas Palaniswamy et al. [5] obtained a max-imum LDR of 2.4 at the forming temperatureof 200�C. After the maximum point, the LDRdecreased with an increase in the forming temper-ature.

4. The reasons behind the improvements of thesimulation results of the present work beingnon-comparable to the numerical investigation ofPalaniswamy et al. [5] are:

- The sheet was meshed with six elements alongthe thickness in the present work whereas the

sheet was meshed with four elements in thenumerical study of Palaniswamy et al. [5].

- Exact implementation of Newton's method (fullycoupled analysis) was used in the present work,while Palaniswamy et al. [5] probably used an ap-proximate implementation of Newton's method(weak coupled analysis) in their investigation.

5. The heat transfer e�ect and location of sheet failurehave been studied by comparing isothermal andnon-isothermal warm deep drawing. In isothermaldeep drawing, as in conventional deep drawing,thinning occurs in the punch radius region due touniform temperature in the sheet. But, in non-isothermal deep drawing, heat transfer betweenthe punch and the sheet leads to an increase inthe strength of the sheet in the punch radiusregion and delays the thinning to that of the dieradius. Therefore, the punch displacement of non-isothermal drawing is more than that of isothermaldrawing.

6. The friction coe�cient, as an important parameterin the deep drawing process, was investigated. Re-sults show that an increase in the friction coe�cientleads to an increase in the punch force and adecrease in the limit drawing ratio. In addition,results show that the maximum friction coe�cientto produce parts without defect increases with anincrease in forming temperature.

7. Punch speed, as an e�ective factor in the deepdrawing process, was also investigated. Resultsshow that the maximum punch speed required toproduce acceptable parts decreased with an increasein the limit drawing ratio.

References

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2. Doege, E. and Droder, K. \Metal formability of mag-nesium wrought alloys-formability and process technol-ogy", Journal of Material Processing Technology, 115,pp. 14-19 (2001).

3. Yoshihara, S., Nishimura, H., Yamamoto, H. andManabe, K. \Formability enhancement in magnesiumalloy stamping using a local heating and coolingtechnique circular cup deep drawing process", Journalof Material Processing Technology, 142, pp. 609-613(2003).

4. Yoshihara, S., MacDonald, B.J., Nishimura, H., Ya-mamoto, H. and Manabe, K. \Optimisation of mag-nesium alloy stamping with local heating and coolingusing the �nite element method", Journal of MaterialProcessing Technology, pp. 153-154, 319-322 (2004).

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at elevated temperatures", Journal of Material Pro-cessing Technology, 146, pp. 52-60 (2004).

6. Neugebauer, R., Altan, T., Geiger, M., Kleiner, M. andSterzing, A. \Sheet Metal Forming at Elevated Tem-peratures", CIRP Annals - Manufacturing Technology,55(2), pp. 793-816 (2006).

7. El-Morsy, A-W. and Manabe, K-I. \Finite elementanalysis of magnesium AZ31 alloy sheet in warmdeep drawing process considering heat transfer e�ect",Materials Letters, 60, pp. 1866-1870 (2006).

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Biographies

Seyed Amir Ossia was born in Babol, Mazandaran,Iran, in 1983. He received his BS degree in MechanicalEngineering from the University of Mazandaran, Iran,in 2007, and his MS degree in Mechanical Engineeringfrom the University of Kashan, Iran, in 2009. Hisresearch interests include metal forming, �nite elementmethod and automotive engineering.

Behzad Soltani was born in Abadan, Khouzestan,Iran, in 1960. He received his BS and MS degreesin Mechanical Engineering from Isfahan Universityof Technology, Iran, in 1988 and 1991, respectively,and his PhD degree in Mechanical Engineering fromChalmers University of Technology, Sweden, in 1996.He is currently Assistant Professor in the Departmentof Mechanical Engineering at the University of Kashan.His research interests include continuum mechanics,nonlinear �nite element and mesh free methods.

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