Experimental and numerical optimisation of the sheet products geometry using response surface methodology

Journal of Materials Processing Technology 189 (2007) 441–449

Experimental and numerical optimisation of the sheet products geometry using response surface methodology Ali Mkaddem a,∗ , Riadh Bahloul b a

LMPF-EA4106, Ecole Nationale Sup´erieure d’Arts et M´etiers, rue saint Dominique, 51006 Chˆalons-en-Champagne, France b LGM, Ecole Nationale d’Ing´ enieurs de Monastir, Avenue Ibn Eljazar, 5019 Monastir, Tunisia Received 5 November 2004; received in revised form 7 February 2007; accepted 8 February 2007

Abstract In this paper, two types of shapes were retained in order to investigate the behaviour of automotive safety parts that are obtained by successive sequences of blanking and bending. Firstly, experiments have been conducted in press tools for a sufficient number of process parameters combinations, particularly, die radius and clearance. Design of experiments and response surface methodology (RSM) were adopted to plot results obtained using the two-specimen geometries. Secondly, numerical model based on elastic plastic theory and ductile damage has been developed for the prediction of material behaviour during forming. The numerical approach was applied to study the mechanical responses of bent parts obtained by using each specimen shapes. The same parameters used for conducting experiments were retained for numerical simulation. However, the maximum bending load obtained for the two investigated cases were treated by application of response surface method. The damage values show a clear difference between the two considered specimen shapes. Numerical bending results compared to experimental values show the reliability of the proposed model for each case. The effects of geometry parameters on the bent parts and convenience of the obtained graphics were discussed in details. © 2007 Elsevier B.V. All rights reserved. Keywords: Steels; Design of experiments; Simulation; Damage; RSM

1. Introduction Most sheet steels commonly used in automotive and other manufacturing applications have high properties, such as ductility, resistance and Young’s modulus. These materials can be bent under very severe parameters without fracture or cracking. In a pioneer works [1], it has been laid down that the final mechanical characteristics of parts do not depend only in material properties but also in the geometry retained for producing workpieces. The optimisation of forming processes aimed to the production of net-shape components and high resistant products is nowadays one of the fundamental topics on which the interest of automotive research groups is focused. Several contributions [2–5] have been conducted in order to reduce stretching and springback phenomena by combining process parameters and sometimes by the choice of materials type having good properties, whereas, works treating the effect of

∗

Corresponding author. Tel.: +33 326699135; fax: +33 326699176. E-mail address: [email protected] (A. Mkaddem).

0924-0136/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2007.02.026

geometry in the final mechanical state of product remain nowadays insufficient. As a preferment procedure, response surface technique is retained in several cases for treating and then, for predicting sheet metal forming problems. It is a useful engineering method to analyse and to supervise the mechanical and geometrical behaviour of workpieces, such as bent parts. Todoroki and Ishikawa [6] have described in their works a new experimental method to optimise stacking sequence by applying a response surface method to composite cylinder products. Unfortunately, the two-dimensional plot still limited to overcome all understanding difficulties and remains far from the required fine model that can describes problems accurately. In addition, it is a consuming time to use because it is unable to present results versus more than one variable. Similarly, Chou and Hung [7] have used the response surface technique for analysing springback that they have taken as a function of both material properties and tools parameters. However, extensive works [8] has adopted the same procedure for optimisation of vehicle crashworthiness design. Several

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functional evaluations may be needed for solving this kind of problems. Therefore, the diversification of consumer needs lead to the changes from mass production to small quantity in which geometry is mostly different. Hence, the reduction of time for product design is crucial. The response surface technique is showed to be so reliable to solve the problem not only from a practical point of view but also by adopting a numerical simulation strategy. In the way of the response surface method application, Ohata et al. [9] have developed an optimum design system to assist the decision of material process condition for making best sheet formability for stamping. They combined finite element analysis and discretised optimisation algorithms for this purpose. The response surface method has been applied in order to attempt search the optimum conditions quickly and to help for deciding a sheet forming process. Optimisation of the process by classical methods based on empirical rule adjustments is not usually applicable to complex geometries or materials without a large database of experience. It involves changing one independent variable of the problem while all others are a fixed level [10,11]. This is extremely timeconsuming and expensive for a large number of variables. To overcome this difficulty, factorial design of experiments and response surface methodology can be employed to optimise medium components [12–15]. In this work, comparison between experiments and simulation has been carried out for deciding suitable process parameters in bending operation. The initial shape of sheet specimens is chosen in such a manner that the bent section would be the same for the two cases and representative as soon as possible of the real geometry of safety parts. The first specimen consists in an oblong-hole at the bent section whereas the second is a full-section specimen. Experimental and numerical results are conducted for an extensive combination of die radius values and punch–blank clearance values. The maximum bending load and maximum damage were plotted by applying the response surface technique allowing a reliable prediction of parameters that avoid the appearing of fracture or cracking in bent product. 2. Design procedure 2.1. Experimental approach Sheet metal bending processes are widely used for mass production. In spite of their cost and their consuming time, experimental procedures still yet crucial to prove reliability of such model or formulation. Especially, experiment data are required to establish the objective function for an optimisation concept. The response surface methodology depends strongly on the measurement accuracy. In order to obtain suitable process parameters for specific bending condition, a method has been laid down for initial sheet design by an extended experimental study based on varying the clearance and the die radius. Lubrication is retained for all tests and, bending force is recorded for each design variables combination thanks to cell

Fig. 1. Initial (a1 and b1 ) and final (a2 and b2 ) steps of bending tests.

load and acquisition equipment. The considered steel is 4-mm thickness H.S.L.A. sheet material. The cut specimens used to examine the geometry effect on the strength of bent parts are given in Fig. 1. The experimental procedure has need 21 cases of tests in press tools. All parameters combinations are reported in the following Table 1. Although, much progress has been achieved for improved experimental strategy of sheet bending, there are yet still needs for further advancing the existing methods, in particular, for verifying the reliability of numerical models. This analysis consists, firstly, in an examination of the final state of the wiping-die bent sheet by adopting an experimental Table 1 Experimental combinations considered for bending tests HG = DR /εR − εD

Die radius/thickness: ¯ d = Rd /t R

Clearance/thickness: J¯ = J/t

0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596 0.596

0.5 0.5 0.5 0.5 0.5 0.5 0.5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5

−0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15 −0.15 −0.10 −0.05 0.00 0.05 0.10 0.15

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procedure. The combination of process parameters, such as clearance and die radius performed a large number of tests. Secondly, it is devoted to establish theoretical development for response surface methodology to predict the mechanical solution without carrying out additional tests. 2.2. Numerical approach The numerical simulation of the damage evolution has been modelled by means of continuum damage approach. The Lemaˆıtre damage model [16] taking into account the influence of triaxiality and the stress state into the fibres has been implemented into ABAQUS/Standard code in order to predict the risk within the bent section during the process. The application of damage theories allows to the identification of the cause of defects, the prediction of ductile fracture during forming processes and the evaluation of the material properties variation generated by the damage. The development of constitutive equations is based on the von Mises criteria, an isotropic behaviour hypothesis and an isotropic damage field [17,18]. The material damage rise is described by the scalar D, which leads to the following stress–strain relationship: ␴ = (1 − D)C0 ␧

(1)

␴ is the Cauchy stress tensor and ␧ is the true strain tensor. C0 is the Hooke operator matrix of the undamaged material. Moreover, the yield function f that depends on plastic strain and hardening law is coupled with isotropic damage in the following form: f (σeq , e¯ Pl , D) =

σeq ˜ ePl , D) − σy − R(¯ 1−D

(2)

˜ = (1 − D)k(¯ePl )n is the hardening power law, σ y the yield R stress, σ eq the equivalent stress and D is the damage variable computed as follows [17]:     2 DR σm 2 ␦¯ePl ␦D = (1 + ν) + 3(1 − 2ν) (3) εR − ε D 3 σeq DR is the critical damage value, εD and εR are, respectively, the initial strain value at which damage starts and the strain value at failure and ν is the Poisson’s ratio. σ m is the hydrostatic stress and ␦¯ePl the equivalent plastic strain. The normal to the frontier of the yield function is deduced from the derivative form of f by σ. It is generally noticed by a and written as given in Eq. (4): a=

∂f ∂f 3 = = Ls ∂σ ∂s 2σeq

(4)

where s is the deviator stress and L is a diagonal matrix. The normality rule implies that the plastic strain increment is calculated normally to the yield function frontier. Moreover, it can be written: ␦ePl = ␦¯ePl · a

(5)

443

δ¯ePl is the Lagrange multiplier defined by Eq. (6):  2 Pl T −1 Pl Pl e˙ L e˙ ␦¯e = 3

(6)

T

e˙ Pl is the vectorial form of the plastic strain rate. When the material flow occurs, consistency condition is expressed by: ˜ − D) = 0 f = G(2˜e − 3␦¯ePl ) − (σy + R)(1. where  e˜ =

(7)

3 T −1 eˆ L eˆ 2

(8)

Total and G are the shear modulus. eˆ = eEl n + ␦e The non-linear Eq. (6) can be resolved using the Newton–Raphson iterative method. The theory of isotropic plasticity leads to the relationship between the deviatoric stress and the deviatoric strain, such as:

∂s = AD ∂ˆe

(9)

s is the deviatoric form of stress and AD is the matrix depending in σ y , σ eq , D and the deviator stress s. Also, the relationship leading to the calculation of the stress increment from the strain increment can be founded by means of several substitutions. Already, knowing that: ε = e + jεm

(10)

σ = s + jσm

(11)

After some modifications, the use of these terms has to lead to the relationship between ∂␴ and ∂␧. The required formulations give the expression of the consistent operator in such a manner that: ∂σ = KtgD ∂ε

(12)

When damage is not considered, the consistent tangent operator reduces to the classical ones of [19]. Consequently, the numerical modelling of bending operations including material damage can be handled by an implicit scheme using the finite element method. Elastic parameters, hardening coefficients and damage characteristics retained for simulation are identified experimentally [20] and reported in Table 2. The real geometries of parts are accurately reproduced for mesh as a deformable workpieces. Tools are considered rigid bodies and lubrication was modelled by a low friction coefficient f at the interface between punch and blank. The both states during bending, initial and final, are shown in Fig. 2 for the two considered geometries. Table 2 Parameters used for simulation of bending E0 (GPa)

σ y (MPa)

k (MPa)

n

ν

εD

εR

DR

f

200

560

800

0.745

0.28

0.

0.35

0.21

0.09

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are insufficient to representing correctly the evolution of the maximum bending load and maximum damage when the clearance and the die radius vary. Contrary, a cubic order function is showed to be more adequate to fit the experimental data by using the response surface methodology with less error. For a general cubic polynomial approximation, the function will be: f = α0 +

n 

α i xi +

i=1

+

n  i=1

Fig. 2. Initial (a1 and b1 ) and final (a2 and b2 ) steps of bending simulation for the considered specimens.

The proposed numerical investigation of the sheet metal bending steps makes possibility to study the interaction effects between the punch–blank clearance and the die radius on the variation of the maximum bending force and the maximum damage for the two-specimen geometries. 3. Response surface methodology 3.1. Overview to the method The aim was to study the effects of the interaction between the clearance and the die radius on the evolution of the maximum bending load and damage risk. As it was previously mentioned, the final mechanical properties of material have a marked dependence upon the initial blank geometry to form. One method, which makes this possible, is called response surface methodology. It allows to constructing global approximations of the objective functions based on functional evaluations at various points in the design space. The use of structural optimisation based on this technique has increased remarkably during recent years, mainly due to faster script, better programming routines and more frequent use of numerical simulations. This technique consists in a useful tool to improve the design in a well-structured manner. In the response surface methodology, polynomial surfaces are fitted to evaluate the objective values in the considered design space. This avoids performing unlimited tests in order to find the solution. Thanks to the construction of the surfaces, all design points of the factorial intervals will be smoothened out. Then, the optimal solution is searched within all meshed surface and may be out the real design points that allowed to plotting the surface. The design domain is the space framed by the considered ¯ d and J. ¯ The selection of approximation functions variables R to represent accurately the material behaviour is essential. Here, it has been found that first and quadratic order interpolations

n  i
αij xi xj +

n  i=1

αii xi2 +

n  i=1,i=j

βii xi3 + ε

βij xi2 xj

(13)

where f is the interpolation function and xi one of the two retained ¯ d . α0 , αi , αii , αij , βii and βij are the regression variables J¯ or R coefficients, n the number of variables which is equal to 2 in our case and ε the approximation error. The proposed function offers the possibility to predict the optimal solution for any values of die radius and clearance included in the considered ranges without necessity to conduct additional coast tests. The theoretical development allows to minimising the square error χQuad between the real design points fr and the estimated values computed by using the cubic function f. One defines χQuad as follows:  χQuad = (f r − f )2 (14) ¯ R ¯m J,

The minimisation of the quadratic error requires the derivation of χQuad with respect to the constants that appears in Eq. (13). A linear system of ten equations leads to the identification of all factors αi , αii , αij , βii and βij . 3.2. Design model Response surface methodology is applied to obtain an approximation of a response function, which is here the maximum relative bending load, in terms of predictor variables. Referring to Eq. (13), the response model is written as shown in Eq. (15): ¯ m + α3 J¯ · R ¯ m + α4 J¯ 2 + α5 R ¯ 2m f = α0 + α1 J¯ + α2 R ¯ m + α8 J¯ 3 + α9 R ¯ 2m + α7 J¯ 2 · R ¯ 3m + ε + α6 J¯ · R

(15)

As response surface technique is a statistical and mathematical method, objective function should be determined with so low error that can lead to the best fitting of the investigated behaviour. Thus, the selection set of the objective function, which is strongly sensitive to the data points, must be made with high attention for deciding the behaviour map. Thus, it is desirable to choose an objective function with the less constants number that can represent the evolution details of studied response as non-linearity and local solutions at which focus is generally put.

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Table 3 Regression coefficients of bending approximation functions Constants

a1 b1 a2 b2

α0

α1

α2

α3

α4

α5

α6

α7

α8

α9

0.3750 0.4008 0.2714 0.3258

−0.0560 −0.4355 −0.1735 −0.7457

−0.0607 −0.0626 −0.0106 −0.0191

−0.6968 0.1254 −0.2445 0.9617

0.2832 0.5333 −0.0360 −0.7068

−0.1267 −0.1339 −0.0672 −0.0843

0.2659 0.0646 0.0713 −0.3514

0.5840 −0.2146 −0.2375 0.4723

−0.4799 −2.0499 3.5273 −0.6807

0.0682 0.0710 0.0311 0.0463

3.3. Bending load response The theoretical development based on experimental and numerical data points corresponding to the factorial plan of Table 1, has lead to determine the cubic objective functions for the two considered geometries. Results that are deduced

from experimental tests and numerical simulations were treated carefully for comparison purposes. Results are computed for all data points relatively to the elastic force Fy deduced from standard tensile test. Response surfaces are obtained from the polynomial approximations described for each case by the αi -constants shown in the following Table 3.

Fig. 3. Numerical (a1 and b1 ) and experimental (a2 and b2 ) bending load for the considered specimens.

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a1 and b1 are the experimental results of maximum bending loads, respectively, for the a1 -geometry and b1 -geometry shown in Fig. 2. The graphics of both cases obtained by using the suitable approximation functions are shown in Fig. 3. From a global point of view, it is worth noting in a1 and a2 that maximum bending loads evolves similarly. Thus, the obtained maps give a same tendency at the domain of die radius and clearance. It can be evoked, firstly, that the low values are located at the high ranges of die radius and clearance whereas, high values of maximum bending load are reached for the most severe conditions of forming, such as less die radius and negative clearance which induces a marked stress state especially at the interface between punch and blank. In addition, it has to note that bending load is strongly dependent in clearance as in die radius evolutions. However, for negative values of clearance, the bending load seems to be much less sensitive to the evolution of the die radius than for positive values of clearance as can be shown for numerical as well as for experimental distribution. Then, it is interesting to evoke the non-linearity dependency of the maximum load to the two retained variables. Even if results of numerical prediction model are higher than those deduced from experimental tests for all variables ranges, it can be observed at least that the response surfaces a1 and a2 have almost the same difference gap for all data points. Furthermore, less die radius and less clearance show an opposite configuration of the studied domain corresponding to highest die radius and highest clearance. It means simply that severe conditions of bending have to be confused whereas careful must be made for the less severe conditions where results are completely opposites. Secondly, numerical and experimental response surfaces were constructed for the oblong-hole specimens as shown in b1 and b2 of Fig. 3. Certainly, results have to be affected by the stress concentration due to the oblong-hole where bending is performed. Moreover, it can be seen that the maps have globally similar evolutions. The less values of maximum bending load are located particularly at the ranges of high die radius and clearance. As noted for the first geometry (results given by a1 and a2 ), bending load is included in a little domain if we compare results obtained for less die radius – highest clearance and high die radius – less clearance. Similarity of bending conditions is associated, especially, to these combinations. In the same way, it is worth noting for this case that numerical prediction conducts globally to an over-estimated values comparing to the measured values. This is essentially attributed to the errors probably caused during identification of material and

damage parameters used in numerical model. It may be results from the simplification of the contact definition between blank and tools, which is modelled here by adopting coulomb friction law. The friction phenomenon is generally non-uniform and so complicated to formulate and to model that leads to some errors in prediction [21]. Observing Fig. 3b1 and b2 , it can be noted that non-linearity according to the clearance variation seems to be not so marked as according to the die radius variation. For low die radius values, bending load showed to increase more rapidly versus the clearance than for high die radius values. From Fig. 3, it can be said finally, that the initial specimen design has a marked influence on the bending load. Moreover, it has been establish experimentally as well as numerically that the maximum bending load increases when the oblong-hole geometry is retained instead of the full-section geometry. This is essentially, due to the plastic flow localisation and the stress concentration in the bent section caused by the hole. However, numerical maximum load obtained for oblong-hole sheet geometry is almost 1.06 times higher than one obtained for the full-section sheet geometry. In the same way, the ratio computed for measurement values is about 1.22. Consequently, it can be concluded that response surface method covers at least the observation data for measurement or prediction within the variables ranges. 3.4. Damage response Bending process is well known as a high straining process that during forming material flow would be occurs and irreversible strain would be induced within the sheet fibres. When the magnitude of plastic deformation reaches the strain value εR , damage takes its critical value DR at which failure occurs. The proposed model based on ductile damage approach is aimed to predict the aptitude limit of material to be bent without cracking risk. In this way, simulations have been conducted according to the factorial design given in Table 1. The regression coefficients of polynomials are reported in Table 4. Response surfaces are plotted for the two geometry cases, in order to examine the sheet strength state at the end of each bending operation. Results are provided in Fig. 4. It is worth noting from Fig. 4a and b that damage is as high as die radius decreases. For the two cases, the highest values are located at the tensile fibres of the fold zone. Especially, for the oblong-hole specimens, localisation zones of damage are observed at the round-offs of the oblong-hole. Damage localisation seems to be more sensitive to the die radius variation than to the clearance variation. Moreover, dam-

Table 4 Regression coefficients of the damage approximation functions Constants

a b

α0

α1

α2

α3

α4

α5

α6

α7

α8

α9

0.1626 1.8983

0.0567 −0.1034

−0.0289 −1.3715

−0.0635 0.2406

−0.0590 −0.0923

−0.0577 −1.3943

0.0228 −0.0923

0.0518 0.0803

−0.5668 0.3568

0.0284 1.0125

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Fig. 4. Damage response surfaces plotted for (a) full-section sheet and (b) oblong-hole sheet.

age varies very slightly according to the clearance values for the considered die radius range. The worst bending conditions were found for the less die radius values independently of used clearance. Thus, it is interesting to note that in the same conditions, failure risks are more marked when oblong-hole geometry is used to simulate the real safety parts than in the second case. Solely, it is strongly desadvised to produce bent parts by using low die radius ranges for any clearance value. When full-section geometry is used, numerical results are confirmed by experiment that the maximum damage still lower than the critical value which draw aside all processes problems. Contrary, damage seems to increase rapidly with the use of oblong-hole geometry that is essentially due to the localisation phenomenon of plastic strain at the edge of the oblong-hole.

Thus, maximum damage reaches the critical value; cracks start and then propagate within the sheet thickness as shown in Fig. 4b. Experimentally, for the full-section model, no cracks have been detected for the most severe conditions of bending and observations have proved that critical damage DR has not been reached. However, bendability expressed by the lowest usable die radius that should not be exceeded in order to prevent cracks to appear in the oblong-hole bent sheet, was defined by Rd /t = 0.943. Numerical model seems to be so reliable that damage localisation deduced from simulation agrees very well with experiment tests. As observed in Fig. 5b, highest damage iso-values are located at the failure regions detected in real specimens as previously mentioned in Fig. 4.

Fig. 5. Damage contours for (a) full-section sheet and (b) oblong-hole sheet.

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From Fig. 5, damage distributions showed to be non-uniform according to the fold zone for the two considered geometries, which highlights the advantages of the 3D simulation model. Damage cards show that there is no damage localisation in the compressive fibres, which is supported by the hypothesis assuming that micro voids grow only under tensile stress conditions [22]. As a conclusion, one can said that the response surface methodology is a very efficient tool for optimisation context that can be applied successfully to predict material behaviour during bending. In the same time, it makes possible to prove reliability of numerical models referring to the measurements or observations. From practical point of view, oblong-hole geometry is more expensive and consuming time to obtain but it is more representative to characterise the real behaviour of safety parts. 4. Discussions and comparison The obtained results proved satisfactory between experimental strategy, numerical simulation and investigation methodology to predict the mechanical behaviour of the material during sheet bending. The choice of a full factorial design with high number of combination allows to an accurate prediction of the behaviour. The experimental procedure, which consists in a large number of tests, has conduct firstly, to compare measurements to the simulation and secondly, to make correlations between the damage localisation regions predicted by simulation and the detected cracks within the real produced parts. Essentially, focus is put on the influence of geometry on the material behaviour during bending. Agreement between experiment and simulation is well that for maximum bending load, response surfaces present a similar global evolution in the considered variables ranges. The mean error computed for each die radius between experiment and simulation is reported in the following Table 5. It is shown that with the use oblong-hole sheet, numerical model is more coherent to simulate the reality than with using a full-section sheet. Globally, maximum load deduced from the bending of the oblong-hole geometry is higher than one obtained by a fullsection geometry. Certainly, the specimen with oblong-hole is the most suitable shape to characterise the real safety parts but if great attention is paid to the deviation between the twospecimens design, full-section geometry can be also adopted with success for the prediction of the material behaviour and can lead to a very satisfactory results. Damage localisation is showed to agree very well with the observations, which confirm the reliability of the coupleddamage model. Table 5 Mean experiment-simulation errors (%) for the considered designs Designs/relative die radius

Rd /t = 0.5

Rd /t = 1

Rd /t = 1.5

Oblong-hole specimen Full-section specimen

18 30

6.6 18

5.7 16

5. Conclusion In this study, a comprehensive and accurate characterisation of material behaviour is conducted by means of two-specimen designs. Response surface methodology is principally, applied in order to analyse the behaviour of the automotive safety parts. Experimental approach and numerical simulations have been conducted through the same design of experiment, with two bending variables; the die radius and the clearance. The response surface method deals with an efficient and satisfactory choice of sheet design for a best characterisation of real bent parts. The following conclusion points can be retained: 1. Numerical results and experimental tests led to similar maps of maximum bending loads with the two considered specimen design whereas, coherence between experiment and numerical prediction is better when oblong-hole geometry is used. 2. The low die radius range and negative clearance range have to be avoided to produce parts by bending processes because these intervals lead to a highest bending load and highest risk of failure. 3. It has been found that the maximum bending load is sensitive to the considered parameters; whereas damage seems to depend much less on the clearance than on the die radius. 4. Finally, it is interesting to evoke that numerical simulation coupled to ductile damage is showed to be satisfactory to predict correctly the failure regions, particularly, when oblong-hole design is retained. Acknowledgements Authors are grateful to Deville Company for its technical supports and to the assistance of A. Potiron in this work. References [1] L. Kurt, Handbook of Metal Forming, first ed., McGraw-Hill Book Company, USA, 1985. [2] C. Wang, G. Kinzel, T. Altan, Mathematical modelling of plane-strain bending of sheet and plate, J. Mater. Process. Technol. 39 (1993) 9–304. [3] Z. Kampus, The influence of the shape of die inlet opening on bending, in: Proceedings of International Conference on Sheet Metal, vol. 8, 2000, pp. 497–504. [4] H. Livatyali, Gary L. Kinzel, Talan Altan, Computer aided design of straight flanging using approximate numerical analysis, J. Mater. Process. Technol. 142 (2003) 532–543. [5] Z. Tekiner, An experimental study on the examination of springback of sheet metals with several thickness and properties in bending dies, J. Mater. Process. Technol. 145 (2004) 109–117. [6] A. Todoroki, T. Ishikawa, Design of experiments for stacking sequence optimizations with genetic algorithm using surface approximation, Compos. Struct. 64 (2004) 349–357. [7] I. Chou, C. Hung, Finite element analysis and optimization on springback reduction, Int. J. Machine Tools Manuf. 39 (1999) 517–536. [8] R. Marcus, N. Larsgunnar, Using space mapping and surrogate models to optimize vehicle crashworthiness design, in: AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimisation, vol. 3, 2002, pp. 1–11. [9] T. Ohata, Y. Nakamura, T. Katayama, E. Nakamachi, Development of optimum process design system for sheet fabrication using response surface method, J. Mater. Process. Technol. 143–144 (2003) 667–672.

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