991 Investigation into a new incremental forming process using an ajustable punch set for the manufacture of a oubly curve sheet metal S J Yoon an D Y Yang* Department of Mechanical Engineering, Korea Avance Institute of Science an Technology, Taejon, Korea Abstract: For exible manufacture of a oubly curve sheet metal, a new sheet metal forming process is propose that uses a movable punch set compose of two pairs of lower support punches an one upper centre punch. The process is characterize by exibility compare with the existing conventional processes because it oes not require any complex-shape ie an the loss of material is minimize by unconstraine forming. The process involves very little thickness strain by applying the principle of bening eformation instea of stretching. In the preliminary nite element metho (FEM) simulation it is foun that the curvature of the forme sheet metal is etermine by the istance between the supporting punches working in pairs an the forming epth of the centre punch. In the experiments using sheets of aluminium, a metho with ege forming is propose for proucing sheet metal in the esire shape. The equation using process variables such as the istance between supporting punches in pairs an the forming epth of the centre punch is propose for the preiction of curvature of the forme shape. The equation is revise by the experimental results an the FEM simulation results containing springback information. It is then shown that the raius of curvature for the forme sheet metal can be preicte by the revise equation unless a very large amount of springback occurs. Keywors: incremental sheet metal forming, new forming process, ieless forming, unconstraine forming, oubly curve sheet NOTATION a b L R c R s R w R x half-istance between a pair of support punches along the x axis (mm) half-istance between a pair of support punches along the y axis (mm) forming istance between a centre punch point an the next centre punch point ˆ 5 mm length of a sie of a square workpiece ˆ 100 mm raius of curvature of the centre punch hea (mm) raius of curvature of the support punch hea (mm) raius of curvature of the eforme workpiece representing R x an R y (mm) raius of curvature erive from the point ata on the x axis of the eforme workpiece (mm) The MS was receive on 29 August 2000 an was accepte after revision for publication on 5 February 2001. *Corresponing author: Department of Mechanical Engineering, Korea Avance Institute of Science an Technology, Science Town, Taejon 305-701, Korea. R y raius of curvature erive from the point ata on the y axis of the eforme workpiece (mm) t w workpiece thickness ˆ 2 mm z forming epth of the centre punch (mm) z 0 forming epth of the centre punch for ege forming ˆ 2 3 z (mm) 1 INTRODUCTION In orer to evelop new proucts in the inustry, it is important to reuce the lea time an the evelopment cost. As a way of meeting these requirements, incremental forming processes have been use as competitive means for rapi prouct evelopment. Especially in manufacturing sheet metal prototypes, some new trial manufacturing processes capable of eforming general sheet metal into an arbitrary shape have been introuce. Incremental sheet metal forming has taken its place as an evective trial manufacturing metho. It goes way back to the conventional spinning process B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
992 S J YOON AND D Y YANG [1, 2] by which only axisymmetric parts can be mae. Recently, many other processes that can manufacture non-axisymmetric parts as well as axisymmetric parts have been evelope [3± 10]. The incremental sheet metal forming process enables a esire shape to be manufacture by incremental eformation in a local contacte region. A hemispherical hea tool moves over the arbitrary surface which is a slice part of the whole geometry, locally but continuously stretching the workpiece. Accoringly, complicate proucts can be fabricate with a simply shape punch an non-eicate ie. Powell an Anrew [3] evelope an incremental forming process for forming ange sheet metal parts without eicate tooling by repeate localize eformation. The complete formation of a shrink ange over a former by a groove roller, moving aroun the ange in a series of passes, is analyse by the nite element metho. Matsubara et al. [4] esigne an evelope a new process of sheet metal forming that enable a sheet metal blank to be forme into various shapes through repeate local stamping between a simple upper punch an a simple lower ie. Various conical an pyramial parts were successfully forme with a numerical control (NC) milling machine by Matsubara [5]. Research into the forming limit for the incremental sheet metal forming process has been carrie out by Iseki an Kumon [6]. It was shown that the forming limit curve is consierably larger than that base on theories of plastic instability. Kitazawa et al. [7] have propose the inverse metho in which a stretchexpane blank is inverte upsie own an forme into a nal shape with sharp corners. An attempt has been mae by Shima et al. [8] to evelop a exible spin-forming system, where spin forming is performe without a manrel. It was shown that the thickness variation is in uence by the fee rate of the rollers. Cho [9] evelope an automate incremental sheet metal forming machine controlle by computer an introuce the multiple pass forming metho base on linear blening in orer to improve the formability. Kim an Yang [10] propose the ouble-pass forming metho base on the sine law of the conventional shear spinning process in orer to preict the nal thickness strain, which improve the formability of the incremental sheet metal forming process. The intermeiate shape is etermine from the preicte thickness strain so as to istribute the strain uniformly, an then the sheet metal is eforme by the ouble-pass forming metho unergoing the calculate intermeiate shape passes. Most incremental sheet metal processes have some common characteristics, so that a ie is necessary even though it is non-eicate, a ange with consierable loss of material is neee to hol the blank an the workpiece is mainly forme by continuous local stretching. For manufacturing a generally oubly curve sheet metal, the previously propose methos have such isavantages. An evective incremental sheet metal forming process is neee. The new process shoul be more exible than other conventional processes without using any eicate ie. The loss of material shoul be minimize by unconstraine forming, an it shoul bring about very little thickness strain by applying the principle of bening eformation instea of stretching. In this paper, a new forming process is propose an evelope by carrying out experiments an simulation. Simulation is introuce to verify the new process. Experiments are also implemente in orer to establish the forming metho of the new process for manufacturing a oubly curve sheet metal. Finally, a relationship of experimental parameters an the raius of curvature of the forme sheet is propose to control the eforme shape more evectively. Fig. 1 Punch Set Support punch Punch Set Center punch Schematic iagram of the new incremental forming process using a punch set Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 993 2 DESCRIPTION OF THE NEW FORMING PROCESS 2.1 Characteristics of the new forming process The principle of the new process is illustrate in Fig. 1. The punch set consisting of one centre punch an two pairs of support punches plays a key role in this process. By using the punch set as forming tools, the new process enables the manufacture of a oubly curve sheet metal without any eicate ie by continuous bening eformation. The workpiece eforme by bening is prevente from tearing by excessive stretching. Since the punch set oes not involve the blank holer when the workpiece is presse by the centre punch, the loss of material is minimize. The process is an unconstraine forming process since there is no blank holing of a sheet. The new process is base on incremental sheet forming. In this process, bening eformation occurs in a local contact region an the sheet is eforme incrementally as eformation spreas throughout the whole region by moving the workpiece. Accoring to the forming scheule, the sheet can be forme repeately to attain further epth in shape. 2.2 Apparatus for experiment The apparatus for experiment consists of a punch set consisting of one centre punch an two pairs of support punches, the x± y± z table ajusting the relative location between the centre punch an the support punches, an the riving program controlling the motors of the table (Fig. 2). The forming process in the sequence is illustrate from the initial conition to the secon step in Fig. 3. In the experiment, the workpiece is move manually along the forming paths in accorance with the regular given scheule. The apparatus is further Sheet Center Punch Support Punch Punch Set Fig. 2 Schematic iagram of the apparatus for the experiment B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
994 S J YOON AND D Y YANG Fig. 3 Forming process from the initial state to the secon step evelope to enable automatic movement of the workpiece for continue investigation. 3 SIMULATION OF THE PROCESS 3.1 Purpose of simulation Before carrying out an experiment, it is necessary to verify the forming mechanism of this process, i.e. to check if a oubly curve sheet metal can be mae by the process. Then, if possible, it is important to e ne the key parameters avecting the nal shape of the prouct, i.e. controlling its curvature. For these purposes, as a preliminary step before the experiment, nite element metho (FEM) analysis is implemente by an implicit coe using the shell element [11, 12]. 3.2 Conition of simulation The parameters use in the experiment are shown in Fig. 4. In orer to reuce the number of cases of simulation, the values of R s, t w an L are xe. The conitions of simulation are liste in Table 1 using the same parameters shown in Fig. 4. The conitions of simulation are classi e to n the relationship between the raius of curvature of the eforme workpiece an one of the other parameters, which inclue the istance between the support punches, the forming epth, the raius of curvature of the centre punch, etc. As the new process is an incremental sheet metal forming process, the local eformation is important. Therefore, the raii of curvature of the locally eforme region uring one stroke of the centre punch are calculate through FEM analysis. 3.3 Results of simulation Raii R x an R y are the raii of curvature at a point erive from the ata on the x axis an the y axis of the workpiece respectively (Fig. 4b). It is shown that they increase as one of the parameters 2a, 2b or R c increases an ecrease as the forming epth, z, increases in Fig. 5. It is notice that a oubly curve sheet metal can be forme by setting up two values of the istance between support punches in pairs (2a, 2b). From the results of simulation, the parameters 2a, 2b, z an R c are the factors having an evect on the curvature of the eforme shape. There are other geometric parameters avecting the curvature besies these. Because the experimental apparatus is set up an the imensions of the workpiece are ecie for simple veri cation, the parameters 2a, 2b an z are, however, regare as the only geometric factors in the experiment. Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 995 L 50 L R c 2a 2b R s t w 2a (a) Dz R x R y Fig. 4 (b) (c) De nition of the parameters: (a) geometric parameters; (b) e nition of the raii of curvature along the x axis an y axis; (c) e nition of the forming epth Table 1 4 EXPERIMENTS Conition of simulation Variable values (mm) Fixe values (mm) 2a 2b z R c R s t w L Simulation 1 30± 60 30 1.0± 1.2 50 Simulation 2 60 30 1.0± 1.4 50 7.5 2 100 Simulation 3 60 30 1.0± 1.4 40± 50 4.1 Geometry of the punch set an the workpiece The imensions of the punch set which is the core of the apparatus an the imensions of the initial blank use for the experiment are illustrate in Fig. 4a an Table 2. All support punches of the punch set have the same hemispherical surface an the centre punch has part of a spherical surface whose raius of curvature is larger than that of a support punch. The geometry of the centre punch is so esigne as to minimize the evect of concentrate loa on the blank when it is presse by the centre punch. Since the concentrate loa causes severe stretching of the workpiece, it is necessary to make the raius of curvature of the centre punch large enough to isregar stretching eformation. The value of the raius of curvature of the centre punch is ecie by the preliminary experiment an simulation an is shown in Table 2. The anneale aluminium (Al 1050) is use as the material of the blank for ning B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
996 S J YOON AND D Y YANG Fig. 5 Results of simulation: (a) simulation 1; (b) simulation 2; (c) simulation 3 out whether this new process is applicable to the col working process. 4.2 Forming stages, forming istance an forming paths for each forming stage Because the propose process is base on the principle of incremental forming, it is carrie out as a multistage process. The stages are etermine by the forming epth of the centre punch. One stage is nishe when the whole workpiece is locally eforme just once, region by region, within the speci c forming epth. The particular values of forming epth etermining the state of the stage are given in Table 3. They represent the values at which the forming loa increases suenly or the shape of the workpiece changes excessively. These phenomena are examine by the preliminary experiment. Table 2 Punch set Geometry of the punch set an the initial blank Initial blank R c (mm) 50 Shape Square R s (mm) 7.5 L (mm) 100 2a (mm) 30 or 40 t w (mm) 2 2b (mm) 30 Material Anneale Al (1050) Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 997 Table 3 Values of z etermining the forming stages Forming stages Stage I Stage II Stage III Stage IV z (mm) 0.4 0.7 1.0 1.2 The centre punch moves in a iscrete manner along the forming path which consists of a series of centre punch points; the constant istance between two ajacent centre punch points is e ne as the forming istance,. The local an successive eformation of the workpiece forme by the centre punch at each centre punch point along the forming path results in the global curvatures of the prouct. Thus, it shoul be note that the forming istance has a irect in uence on the curvatures of the prouct so that careful selection of the forming istance is vital for the prouction of smooth proucts. In etermining the forming istance, it is necessary to examine the increase in the contact region of the centre punch with the workpiece as it presses own the workpiece to the forming epth, z. The nite element results show that the spherical contact region ha a maximum raius of 10 mm. The maximum raius is obtaine uner conitions of the greatest istance between the support punches in pairs an the smallest forming epth, as in experiment 2A. With this result an Fig. 6a, it is possible to etermine an upper limit of the forming istance of 10 mm, an it is reasonable to choose a forming istance within the limit. In this work, the forming istance has been set at 5 mm in orer to manufacture smoother proucts. The forming paths for each forming stage are e ne in Fig. 6. Firstly, the initial blank or the workpiece forme in the previous stage is forme crosswise from Contact region Center punch point (a) (b) (c) ( forming istance : = 5 mm) h () (e) Fig. 6 Forming paths for multistage forming: (a) forming istance; (b) rst path or nal path; (c) main path; () path set for rst-stage forming; (e) path set for secon stage B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
998 S J YOON AND D Y YANG Fig. 7 Forme regions: (a) without ege forming; (b) with ege forming its centre point (Fig. 6b), an then it is forme from the centre to the region near the ege along straight lines symmetric to its centre-line (Fig. 6c). Finally, it is forme once again along the same path as the rst step. In the next stage, the irection of the main path is change at a right angle to that of the main path in the present stage (Fig. 6e). The workpiece is forme by the centre punch at a regular interval, 5 mm, so the pressing points of the centre punch are marke on the surface of the initial blank before forming. 4.3 Metho for forming the ege region of the workpiece It is foun that the region near the eges of the workpiece is iycult to form (Fig. 7). It avects the shape of the prouct an also causes loss of the material just as the ange of the blank in the eep-rawing process. In orer to solve this problem, an ege forming metho is propose. The propose ege forming metho is iverent from the usual forming metho of this process an is applie only in the region near the eges of the workpiece. The iverences between the two methos are the number of support punches use, the location of the centre punch relative to the support punches an the value of the forming epth uring a stage. To exten the forme region of the workpiece, only three support punches are use (Fig. 8), so that the location of the centre punch is change from the centre of the rhombus mae by all four support punches to that of the isosceles triangle by only three support punches (from D to E). Because of the change, the value of the forming epth shoul also be change. The usual value uring a stage, z, is change into the moi e value, z 0, in Fig. 8 Schematic iagram of the ege forming metho Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 999 e Fig. 9 Moi e path set for one-stage forming the ege region of the workpiece. Accoring to the assumption that the ratio of BD to z an that of BE to z 0 are ientical, z 0 is e ne as 2 3 z. Ege forming is carrie out after forming along the main path (Fig. 6b). The moi e forming paths for each stage are illustrate in Fig. 9. 4.4 Results of the experiment The conitions of the experiment are shown in Table 4. The rst conition with the same values of 2a an 2b is Table 4 Conition of experiments 2a (mm) 2b (mm) z (mm) Experiment 1A 30 30 1.0 Experiment 1B 30 30 1.2 Experiment 2A 40 30 1.0 Experiment 2B 40 30 1.2 set up for experiment 1 where the values of R x an R y of the eforme workpiece are expecte to be the same, an the secon conition is for experiment 2 where the initial blank is expecte to be a prouct with iverent values of R x an R y such as a oubly curve sheet. The completely forme workpieces are shown in Fig. 10. 4.4.1 Comparison of prowle an curve Wtting In Fig. 11, the pro les of the forme sheet metal along the x axis an the y axis, mae from point ata of its outer surface measure by a three-imensional coorinate measuring machine, are compare with the curve tting constructe by the least-squares metho using the same point ata. The iverences between their z values are also shown in Fig. 11. The calculate raii of curvature of the curve tting are tabulate (Table 5). It is possible that the values of R x an R y in experiment 1 are almost the same because the ratios of R x to R y in experiment 1 are close to unity. The error is mainly the manual movement of the workpiece. It is B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
1000 S J YOON AND D Y YANG y ue to the fact that, the wier the istance between support punches, the more iycult it becomes to generate the curvature. (a) t x y x y 4.4.2 Distribution of the raius of curvature The istribution of the raius of curvature is isplaye in Fig. 12, which shows all the raius of curvature values calculate by the least-squares metho using a group of only ve successive point ata measure previously an marke over their central point. In Fig. 12, the raii of curvature obtaine locally near the centre of the workpiece are larger than the value calculate by total measure point ata. This means that the amount of curvature generate varies accoring to the location of the workpiece presse by the centre punch. It is foun that, because geometric constraint of the workpiece exists when the process is carrie out an the extent of that constraint varies accoring to the location of the workpiece, the generate curvature also varies. In particular, at the centre of the workpiece it avects curvature generation signi cantly. This is an important characteristic for forming a oubly curve sheet metal. x 5 PREDICTION OF THE RADIUS OF CURVATURE 5.1 Formulation (b) Fig. 10 x Forme workpieces: (a) experiment 1B; (b) experiment 2B y Since the propose new forming process aims at manufacturing a oubly curve sheet metal, it is important to make a nal prouct with the esire raii of curvature. In this paper, to control the raii of curvature along the x axis an y axis of the workpiece, the relationship between the parameter R w, which is either R x or R y (Fig. 4b), an other experimental parameters (2a, 2b, z) is erive. The equations are base on the geometric states of both the uneforme sheet an the eforme sheet (Fig. 13). The formulation is as follows: R w a; ˆ a R w 1 expecte that such an error will be reuce if an automatic workpiece movement system is evelope in future work. In Fig. 10b, the values of R x an R y iver far more than in Fig. 10a. This suggests that a oubly curve sheet metal can be manufacture by setting up the values of 2a an 2b iverently. Comparing the result of experiment 2A with that of experiment 2B, it is foun that, as the forming epth increases, the raius of curvature of the forme workpiece ecreases but the ratios of R x to R y increase. As a result, R x = z is larger than R y = z. This is ˆ p 2 ˆ p 2 a 2R w ˆ tan 1 z p a 2 a ˆ p 2R w 2 ; tan 1 z a ˆ a 2R w z a ˆ tan a 2R w a 2R w, a 1 2R w 2 3 Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 1001 Fig. 11 Comparison of pro les an curve tting: (a) experiment 1B; (b) experiment 2B Table 5 Raii of curvature of the curve tting by the least-squares metho Raius of curvature of the forme sheet, R w (mm) Along the x axis, R x Along the y axis, R y R x =R y Experiment 1A 257.5 247.4 1.04 Experiment 1B 162.0 181.0 0.90 Experiment 2A 339.3 235.9 1.44 Experiment 2B 284.3 140.0 2.03 B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
1002 S J YOON AND D Y YANG Fig. 12 Distribution of the raius of curvature: (a) experiment 1B; (b) experiment 2B Fig. 13 Moelling for computing the raius of curvature using the geometry: (a) real moel; (b) simpli e moel Finally, the raius of curvature is expresse as R w a 2 2 z 4 where R w enotes R x. Since parameter a in equation (4) represents the half-istance between a pair of support punches, in orer to preict R y, parameter a in equation (4) shoul be replace with parameter b. The preicte results of the raii of curvature using equation (4) are liste in Table 6. 5.2 Moi cation of the equation Comparing Table 5 with Table 6, it is foun that the Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001
INVESTIGATION INTO A NEW INCREMENTAL FORMING PROCESS 1003 Table 6 Raii of curvature using equation (4) Table 8 Simulation results of springback Raius of curvature of the forme sheet, R w (mm) Along the x axis, R x Experiment 1A 113.5 113.5 Experiment 1B 94.8 94.8 Experiment 2A 201.0 113.5 Experiment 2B 167.7 94.8 Along the y axis, R y 2a (mm) 2b (mm) z (mm) z after springback z before springback Ð 30 30 0.8 3:19 10 2 Experiment 1A 30 30 1.0 8:56 10 2 Experiment 1B 30 30 1.2 1:69 10 2 Experiment 2A 40 30 1.0 1:41 10 2 Experiment 2B 40 30 1.2 4:69 10 3 preicte values of Table 6 are smaller than the measure values of Table 5. In orer to moify equation (4), the ratios, k, of the measure values to those preicte are calculate in Table 7. The values of R w in Table 7 are the mean values of all the same cases. In Table 8, the ratios are almost the same except the value of case 1. Ratio k represents other factors unconsiere uring the previous formulation process. That is, the properties of the material, the characteristics of the process using two pairs of support punches simultaneously, the evect of springback, an so forth, are incorporate in ratio k. In orer to n which factor causes the iverence between the ratio of case 1 an the ratios of other cases, springback is analyse by the nite element metho, which is implemente by the implicit coe using the shell element [11, 12]. The results of simulation in the previous section, incluing the evect of springback, are compare with those without consiering the springback evect. The ata in Table 8 are checke uring the FEM analysis immeiately after the initial blank is rst presse by the centre punch at stage I. From the results of Table 8 it is foun that springback occurs in case 1 more than 5 times as much as in other cases. This means that there exist some conitions that re ect the istances between support punches in pairs an the forming epth of the centre punch that bring about springback more severely. It is expecte that, as the istance between support punches in pairs an the forming epth of the centre punch become smaller, springback takes place more excessively. Consiering the previous analysis, equation (4) is moi e to equation (5). Equation (5) is employe when an anneale aluminium (1050) sheet with a thickness of 2 mm is use an the experiments are carrie out. In the experiments, springback oes not take place signi cantly: R w k a2 2 z 5 where k ˆ 1:7 for anneale Al(1050) with 2t. It has been foun that equation (5) can be conveniently employe for the esign of the process. 6 CONCLUSION Using the propose tools set-up, a new incremental forming process for manufacturing a oubly curve sheet metal has been suggeste an evelope, which is base on the principle of incremental bening eformation. Some funamental experiments have been carrie out for eveloping the propose process an simulation using the nite element metho has been performe to verify the forming principle of that process. The parameters of experiments, 2a, 2b an z, have been foun to be the key factors avecting the raii of curvature of the forme workpiece. A general forming metho consisting of the forming stages an forming paths an the ege forming metho especially for the region near the eges of the workpiece have been propose. Through the experiments, the propose forming metho has been shown to be evective. The propose moi e equation for the preiction of the raii of curvature of the forme workpiece is useful for esign of the process. There exist particular conitions in which springback takes place rather signi cantly an the equation for the preiction of the raii of curvature of the forme workpiece is not reaily applicable. Table 7 Comparison of experimental results an preicte results 2a (mm) z (mm) R w exp (mm) R w pre (mm) k ˆ R w exp / R w pre Case 1 30 1.0 246.9 113.5 2.18 Case 2 30 1.2 161.0 94.8 1.70 Case 3 40 1.0 339.3 201.0 1.69 Case 4 40 1.2 284.3 167.7 1.70 B08900 # IMechE 2001 Proc Instn Mech Engrs Vol 215 Part B
1004 S J YOON AND D Y YANG It has been shown from the present stuy that the propose metho can be evectively employe in forming a general oubly curve sheet metal with some aitional improvement in process automation an preiction of process geometry. REFERENCES 1 Avitzur, B. an Yang, C. T. Analysis of power spinning of cones. Trans. ASME, J. Engng for Inustry, 1960, 82, 231± 245. 2 Kalpakcioglu, S. A stuy of shearð spinnability of metals. Trans. ASME, J. Engng for Inustry, 1961, 83, 485± 495. 3 Powell, N. N. an Anrew, C. Incremental forming of ange sheet metal components without eicate ies. Proc. Instn Mech. Engrs, Part B, Journal of Engineering Manufacture, 1992, 206(B1), 41± 47. 4 Matsubara, M., Tanaka, S. an Nakamura, T. Forming process of spherical surface by incremental press forming system. J. Jap. Soc. Technol. Plasticity, 1994, 35(406), 1330± 1335. 5 Matsubara, S. Incremental backwar bulge forming of a sheet metal with a hemispherical hea tool. J. Jap. Soc. Technol. Plasticity, 1994, 35(406), 1311± 1316. 6 Iseki, H. an Kumon, H. Forming limit of incremental sheet metal stretch forming using spherical rollers. J. Jap. Soc. Technol. Plasticity, 1994, 35(406), 1336± 1341. 7 Kitazawa, K., Wakabayashi, A., Murata, K. an Seino, J. A CNC incremental sheet metal forming metho for proucing the shell components having sharp corners. J. Jap. Soc. Technol. Plasticity, 1994, 35(406), 1348± 1353. 8 Shima, S., Kotera, H. an Murakami, H. Development of exible spin-forming metho. J. Jap. Soc. Technol. Plasticity, 1997, 38(440), 814± 818. 9 Cho, C. H. Development of a new NC-controlle trial manufacturing process for sheet metal forming. MS thesis, KAIST, 1997. 10 Kim, T. J. an Yang, D. Y. Improvement of formability for the incremental sheet metal forming process. Int. J. Mech. Sci., 2000, 42, 1271± 1286. 11 Yoon, J. W., Song, I. S., Yang, D. Y., Chung, K. an Barlat, F. Finite element metho for sheet forming base on an anisotropic strain-rate potential an the convecte coorinate system. Int. J. Mech. Sci., 1995, 37(7), 733± 752. 12 Yoon, J. W., Yang, D. Y., Chung, K. an Barlat, F. A general elasto-plastic nite element formulation base on incremental eformation theory for planar anisotropy an its application to sheet metal forming. Int. J. Plasticity, 1999, 15(1), 35± 67. Proc Instn Mech Engrs Vol 215 Part B B08900 # IMechE 2001