The prediction of necking and wrinkles in deep drawing processes using the FEM

Materials Processing Dcfects S.K. Ghosh and M. Predeleanu (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

91

The Prediction of Necking and Wrinkles in Deep Drawing Processes Using the FEM DOEGE, E.; EL-DSOKI, T. and

SEIBERT,

D.

Institute for Metal Forming and Metal Forming Machine Tools, University of Hannover, Welfengarten 1A, D-30167 Hannover, Germany Abstract

Starting out from elementary analytical approaches, the authors discuss the main factors affecting failure by necking and wrinkles. To discuss necking, a large number of macroscopic criteria is evaluated in the light of recent results obtained with the Finite Element Method (FEM). The section on the prediction of necking closes with an evaluation of damage mechanics as a means to analyze failure. Parameters that influence wrinkling such as the blank holder force are discussed. Wrinkling in sheetmetal forming operation are considered either by an implicit FE-Code or an explicit FE-Code.

Introduction One of the main reasons for the FEM increasingly to attract the interest of the sheet metal working industry is that this numerical tool can indeed help to reduce the number of try-outs needed for die design. However, this requires criteria when analyzing FE-plots which allow to predict whether a deep drawing operation is feasible or not, necking and wrinkling representing the most important failure types.

2 2.1

Failure by Necking Analytical Approach

Generally spoken, failure by necking is said to take place when 9 the deep drawing ratio, i.e. ratio of blank diameter to punch diameter, is too large 9 the radii of the die are too small *The authors wish to express their appreciation to the "Deutsche Forschungsgemeinschaft (DFG)" for their financial support of the projects Do/75-2 and SFB 300/B5. Greatfully acknowledged are further the provision of the FE program ABAQUS from Hibbitt, Karlsson and Sorensen, Inc. and the successfull cooperation with the German agency ABACOM as well as the "Regionale Rechenzentrum fiir Niedersachsen (RRZN)"

92 9 the blankholder force is too high 9 lubrication is insufficient 9 the deep drawing gap, i.e. the gap between die and punch, is too small A simple equation first proposed by SIEBEL and PANKNIN [1] may help to understand this: Consider an axisymmetric cup, with the bottom already formed. The punch force Ft, which is in equilibrium with all forces acting on the cup, must be transmitted through its wall. If the punch force is larger than the transmittable force, then rupture will take place. Refection will show that for equilibrium conditions, the punch force is given by Ft -- Fid "~- Fbend"~- tPfric,die "~" Ffric,bh

,

(1)

Fid representing the ideal forming force, Fb~nd the bending force, Ffric,die the accumulated friction force between die radius and blank and Ffric,bh the friction force between blankholder and sheet. From the geometry and the yield behaviour of the cup one can deduce the load carrying capacity of its wall Fbt, the force at which bottom tearing will occur, and one can readily see that F, < Fbt

(2)

must hold as to avoid rupture. The research activities on the prediction of rupture following the analytical approach aim at improving the description of the terms in equation 1, extending them to general geometries and implementing them in fast PC runnable programs [2]. The main advantage of the analytical technique is its quickness in delivering results, while its main drawback is the lack of accuracy and the poor local resolution- for general part geometries, the method is not able to give stress and strain distributions in a sheet.

2.2

General R e m a r k s on the P r e d i c t i o n of N e c k i n g U s i n g the FEM

At the present stage, advantages and disadvantages of the FEM can be judged as opposite to the analytical approach: It is a slow method, yet though both hardware and software ~re becoming increasingly efficient, but it offers a good local resolution, giving realistic strain and stress distributions in the sheet. The accuracy of the FEM relies heavily on the knowledge of the boundary conditions one ha~-to prescribe. In particular, this involves the description of friction and yield behaviour which are both difficult to measure. Friction is a highly local phenomenon, depending on the lubrication conditions [3], the evolution of surface asperities during the forming operation and correct contact search, which in turn requires a shell element formulation which is able to incorporate the thickness in the contact algorithm. The yield behaviour is both history and stress state dependant - measuring the flowcurve for example by hydraulic bulging corresponding to a biaxial stress state will result in values 1 5 - 20% higher than those obtained by a uniaxial test [4]. Strictly spoken, one would

93 have to measure the full yield surface taking path dependancy into account - a very timeconsuming task. For the experimental determination of the boundary conditions, the approach chosen in this work is to measure the yield stress in a hydraulic bulging test and to perform simulations of the bulging test, where no friction develops. The friction is determined as the unknown quantity when simulating for example an Erichsen test and is adapted such as to give optimum agreement of punch force and strain distribution in both experiment and simulation. The friction parameter found thus is then also used for other geometries, when experimental data is not available. 2.3

Macroscopic

Fracture

Criteria

The term "macroscopic fracture criteria" was proposed by GROCHE [4, 5, 6] and implies criteria consisting of products, integrals and sums of macroscopic stresses and strains. To determine the value of this criteria at the onset of failure, both experiments and FE-simulations of hydraulic deep drawing processes, simple stretch and deep drawing operations were conducted. In the simulations, standard LEvY-MISES-plasticity was used, anisotropy effects taken into account through a quasi-isotropic flowcurve after SEYDEL

[7]. After determining characteristic values of the different criteria, their accuracy in predicting the critical punch stroke at which rupture would take place was investigated. It was found that the main factor affecting the accuracy is the mode in which failure takes place, whether under deep drawing or under stretching conditions. The deep drawing condition is characterized by a halt of the flange draw-in in spite of an increasing punch stroke, while deep drawing condition can be recognized by the monotonic flange draw-in punch stroke curve. The results are summarized in figures 1 and 2, indicating the deviation of the predicted punchstroke from the value determined experimentally. These results are confined to deep drawing cracks, which reveals a severe drawback of these criteria: One must know beforehand what type of crack will take place, i.e. whether failure will occur under deep drawing or stretch drawing conditions, [4, 6]. The equivalent MISES stress was judged best for the prediction of both deep drawing and stretch drawing cracks. It turned out, however, that the locus of maximum equivalent MISES stress does not necessarily coincide with the locus of failure in the sheet [4, 6]. At this stage, some remarks on implicit and explicit FE integration schemes seem appropriate. The results above mentioned were obtained using the implicit Finite Element Method. In industrial applications involving large models however, the explicit integration scheme is becoming increasingly important [8], as long as elastic springback prediction is not involved. In the explicit integration scheme, dynamic effects may superpose the solution and will be very noticeable especially in the stress distribution plots. Thus, the thickness strain and the sheet thickness distribution are currently the most widely spread variables used when evaluating a FE-simulation of a sheet metal forming process. In spite of its popularity, however, this kinematic criterion also has several shortcomings: There is no material-dependant critical sheet thickness reduction, since this parameter is operation-dependant. As an example, the reader may refer to the results of -

94

Figure 1" Errors in the prediction of the critical punch stroke using diverse instantaneous macromechanical fracture criteria, after GROCItE [5]

Figure 2: Errors in the prediction of the critical punch stroke using diverse integral macromechanical fracture criteria, after GROCIIE [5]

95 the INPRO group [9], where major strains of over 180% were obtained in the actual multi stage forming and simulation of an oil pan out of mild steel. Moreover, the thickness distribution may also indicate the wrong locus of failure, [6]. For two processes A and B, figures 3 and 4 show the equivalent plastic strain and thickness distribution, respectively. Both processes lead to fracture, process A under deep drawing conditions, process B under stretching conditions. From the diagrams 3 and 4, however, one would presume that only operation B is not feasible, whereas operation A is, which is not confirmed by the experimental findings. Moreover, knowing that process A leads to failure, one would erroneously deduce failure to take place at about 40ram from the center, which is near the die radius instead of the punch edge radius. Therefore, sheet thickness distribution and equivalent plastic strain must also be interpreted with great care and experience when attempting to predict failure.

EP

1.0 0.8 0.6 0.4

\

0.2

process A

0

20

40

60 blank diameter [mm]

Figure 3: Distribution of the equivalent plastic strain in an axisymmetric cup, [6]

2.4

Microscopic

Fracture

Criteria

The drawbacks of the macroscopic fracture criteria gave rise to the idea of applying the concepts of damage mechanics to sheet metal forming. Describing the evolution of an initially flawless material to a microcrack, damage mechanics bridges the fields of continuum mechanics dedicated to the study of perfectly homogeneous deformable bodies, and fracture mechanics, the focus of which is crack propagation [10]. This is done by describing the microscopic processes that precede ductile failure, which is generally attributed to the growth and coalescence of voids nucleating at rigid second phase particles [11]. Some micrographs taken with a light optical and scanning electron microscope can be seen in the figures 5 and 6. They show void formation in the necking area close to the rupture surface. As one can see, outside the necking area hardly any voids can be found. For a more detailed discussion, the reader may refer to [13]. One plasticity model to account for interior damage is the GURSON model [12], which was derived in an attempt to model a plastic material containing randomly dispersed

96

0.9 sheet thickness [mm] 0.6

f

process A =..-~

"\

0.4 process B 0.2

20

40

60 r / [mml

Figure 4: Sheet thickness at initial failure, [6]

Figure 5: Micrograph of a ruptured X5 Cr Ni 18 10 sheet (light optical microscope)

97

Figure 6: Micrographs of a ruptured X5 Cr Ni 18 10 sheet (scanning electron microscope) voids. Studying a unit cell large enough to be statistically representative and applying admissible velocity fields, the yield surface was derived as

q)~ + 2qlf cosh(F = (-~I

) - (1 + q3f 2)

(3)

In equation 3, q is the root of the second stress deviator, p is the hydrostatic pressure, k/ is the yield stress and f is the void volume fraction. When interpreting f geometrically as a fraction of void volume to matrix volume, one can say that for sheet metal forming, the damage variable f is small [13]. When f is equal to zero, the GURSON model abridges to standard LEVY-MISES plasticity. A suggestion how to extend the Gurson model to anisotropic matrix behaviour so that it is suitable for simulating sheet metal forming is sketched in [14]. To implement this constitutive model in a commercial FE package, an integration algorithm due to ARAVAS [15] was utilized. Documentation of uniaxial and hydrostatic tests performed on an eight-node brick element is presented in [14]. When applying the algorithm to shell elements that use the plane stress assumption, modifications of the method are needed since the out-of-plane component is not defined kinematically. These modifications are briefly outlined in [15]. Further modifications are needed when applying the algorithm to explicit FE schemes. When the elastic predictor is very large, i.e. 3q2p/(2kt) > 30, difficulties may arise with calculating the cosh term. As a modification, the authors chose a subincrementation following OWEN and HINTON [16] in order to avoid premature abortion of the iteration process of the Backward Euler algorithm. Figures 7 and 8 shows contour plots of the MISES equivalent stress and the damage variable of a large rectangular cup. Though the calculations were performed at a very high punch speed, the damage variable distribution is still very reasonable, the maximum indicating well the locus of necking, while the MIsEs equivalent stress distribution leaves ample room for speculation. Ergo, the damage variable works successfully as a pointer to the endangered area. Whether the damage variable will also work as a failure criterion, has to be analyzed in future work.

98

Figure 7: MISESequivalent stress distribution in a rectangular cup. For symmetry reasons, only one quarter of the cup was modelled

Figure 8: Damage variable distribution in a rectangular cup

99

3

Failure

by

Wrinkling

Apart from cracks, wrinkling represents another important kind of failure in the area of sheet metal forming. Two different types of wrinkles are known: 9 wrinkles of first order in the flange (figure 9) 9 wrinkles of second order in the free forming zone between the punch radius and the die radius While wrinkles in the flange can be suppressed by the blank holder force, this is not possible for the secondary wrinkles.

Drawing Conditions: 'drawing ratio' Blankholder Force Punch Geometry Punch Stroke

= 1.77 = 81 kN = 220 mm * 110 mm = 70 mm

Figure 9: Undeformed and deformed mesh for a rectangular box

3.1

General R e m a r k s on the A p p e a r i n g of Wrinkles

When using thin sheets for drawing a cup, the flange may start to wrinkle. This tendency can be explained by considering an axisymmetrical cup. Concentric circles move inward and attain smaller radii. This movement results in a pressure stress in circumferencial direction and a tension stress in radial direction. The sheet starts to wrinkle for a critical ratio of both stresses. Pressure due to the blank-holder can help suppress the wrinkles somewhat, but if the force increases too much, wrinkles may be replaced by necking.

3.2

T h e Blank-Holder-Force

As above mentioned the primary wrinkles can be suppressed by using a blank-holder during the deep-drawing process. SIEBEL [20, 21] was the first one who analyzed the connection between the occuring of wrinkles and the blank-holder-force on a theoretical

100 base.

Nearly the same investigation was made by GELEJI [22] in a more simple way. More complex mathematical relations were done by SENIOR [23], Yu and JOHNSON [24] as well as M E I E R a n d R E I S S N E R [25]. For the calculation of the blank-holder-force SIEBEL [20] suggested for rotational parts: 0.5Do] (~o - 1)2 + 100so

Pbh,Siebel "= (2...3) 10-3R~

P~ ~0 Do so

(4)

tensile strength forming limit ratio blank diameter initial blank thickness

While GELEJI [22] gave the relation Pbh,Geleji -" 0.02Rv0.2

/~.2 dp u Do

dp + 2u ] Do + dp + 2u

(5)

yield strength punch diameter gap between punch and die blank diameter

Both equations give nearly the same results. However practical investigations with a rigid blank-holder have shown, that wrinkles appear even if the upper limit of the force, calculated with one of the equations mentioned above, acts during the deep drawing process. The experience shows, that the force for suppressing wrinkles can be calculated by Pbh,exp -~ 1.5pbh,Geleji

(6)

For rectangular parts, SOMMER [26] suggests to calculate the needed force by Pbh,rec. -- k

k m

Ao/Ast

m

(ao) Ast-

1

Rm

(7)

parameter considering the thickness distribution in the flange parameter taking into account the workpiece geometry blank area/projected punch area

To which extent the blank-holder-force influences the success of the deep-drawing operation is illustrated in figure 10. The abscissa stands for the reduction ratio and the ordinate for the blank-holder-force. In the diagram there are three regions

101 9 region where wrinkling is expected 9 region where a successful draw is expected 9 region where necking is expected For a given reduction ratio there are two critical points. The first one is when wrinkling is eliminated and a successfull draw is expected. The second one is when necking is expected [27, 28]. The second region increases if either the friction between blank and die decreases or the friction between punch and blank increases. For a reduction ratio greater than the maximum ratio wrinkling and/or necking always occurs.

Figure 10: The domains of wrinkling and necking

3.3

Aspects

of

Stability

The wrinkling represents a so-called stability problem. The specimen under force deforms so that the new geometry is from the mathematical point of view a stable state of equilibrium [17]. This is characteristical for this kind of problems. By continous increase of the force the state of equilibrium is formally maintained, but at a certain time it becomes instable. At this critical point, even the smallest disturbance such as a non-centered point of application of force, inaccuracy due to manufacturing etc., will lead to instability. This holds for buckling of a bar as well as for wrinkling of sheet metals. The state of equilibrium is stable. The engineer's duty is to avoid a switching over to the stable equilibrium, since a drawing piece with such a geometry can not fulfill the requirements of the design nor its original function. 3.4

EULER's Formula

The wrinkling during sheet metal forming processes is similar to the mechanism of the buckling of a bar, as it was described by EULER when deriving his formula. This comparison is similar to the one of SIEBEL.

102 This process was simulated using the FE-package ABAQUS/Standard and ABAQUS/Explicit (figure 11). In order to reduce the needed CP-time, a plain strain condition was assumed. Another advantage of this assumption is that the discretisation of the model would not influence the results in a wrong way. The model in figure 11 was

Figure 11: Undeformed and deformed mesh for the buckling problem discretised using 8"100 linear elements. After a displacement of u = 21ram every code gives a different result: 9 for the implicit code the process will resemble an upsetting of the specimen, as it is well known from the forging process. 9 the explicit code shows the buckling of the model. For the engineer's point of view it suffices to know that wrinkling or buckling appears. The question of the quantity and the quality of the wrinkles is of a theoretical and academical nature. However it is possible to explain both results by the mathematical formulation of the used integration scheme [19, 29, 30]. For this reason it is also possible to gain the same results using an implicit code. Therefore imperfections have to be considered in the model: 9 geometrical imperfections, i.e. nonuniform sheet thickness 9 physical imperfections, i.e. nonuniform u

4

modulus, nonuniform yield stress

Summary

Failure by necking and wrinkling are two important types of failure in deep drawing which can be predicted using the Finite Element Method. After a brief survey on analytical methods, a large number of macroscopic failure criteria are reviewed in the section devoted to the study of necking. In the framework of continuum mechanics, the highest accuracy in predicting the critical punch stroke is attained with the equivalent MISES stress, which

103 however falls short of indicating the locus of necking. The section on necking closes with an evaluation of damage mechanics. Focussing particularly on the GURSON model, the void volume fraction is prooved to work successfully as a pointer to the endangered area, regardless of geometry and type of operation. Wrinkles in the flange can be suppressed by an adequately chosen blank holder force. The friction behaviour at punch/sheet and die/sheet as well as the sheet thickness influence the succeeding of the deep-drawing operation. In order to produce very thin cups, a subsequent and separate ironing operation usually follows. Wrinkles can be simulated by either an implicit FF_,-Code or an explicit FE-Code.

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[2]

DOEGE, E. and SCHULTE,E.: Design of Deep Drawn Components with Elementary Calculation Methods. In: PIETRZYK, M. and KUSIAK, M. (Eds.): Proc. of the 4th Int. Conf. on Metal Forming, Krak6w, Poland, Sept. 20-24, 1992. Journal of Materials Processing Technology, Vol. 34, pp. 439-448 (1992)

[3]

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[5]

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[6]

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M.: Numerische Simulation der Blechumformung unter besonderer Berficksichtigung der Anisotropie. Fortschritt-Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 182, VDI Verlag Dfisseldorf (1989)

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and BOYCE, M." Numerical Investigations of Sheet Metal Forming Processes. In: MAKINOUCHI, E.; NAKAMACHI,E.;

104

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[19] NAGTEGAAL, J. C. and TAYLOR, L. M.: Comparision of implicit and explicit finite element methods for analysis of sheet metal forming problems. Proc. of the Int. Conf. FE-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991)

[20] SIEBEL, E.: Der Niederhalterdruck beim Tiefziehen. Stahl und Eisen 74, pp. 155-158 (1954) [21] SIEBEL, E. and BEISSWANGER, H.: Tiefziehen. Carl Hanser Verlag, Mfinchen (1955) [22] GELEJI, A.: Bildsame Formung der Metalle in Rechnung und Versuch. Berlin: Akademie (1960) [23] SENIOR, B. W.: Flange Wrinkling in Deep-Drawing-Operations. J. Mechanics and Physics of Solids 4, pp. 235-246, (1956)

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[24]

Yu, T. X. and JOHNSON, W.: The Buckling of Annular Plates in Relation to the Deep Drawing Process. Int. J. Mech. Sci. 3, pp. 175-188 (1982)

[251

MEIER, M. and REISSNER, J.: Instability of the Annular Ring as Deep-Drawn Flange under Real Conditions. Annals of the CIRP, Vol. 32/1, pp. 187-190 (1983)

[26] SOMMER,N.:

Niederhalterdruck und Gestaltung des Niederhalters beim Tiefziehen yon Feinblechen. Fortschritt- Berichte VDI, Reihe 2: Fertigungstechnik, Nr. 115, VDI Verlag, Dfisseldorf (1986)

[27] SCHEY,J.

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[28] AVITZUR, B.: Handbook of Metal-Forming Processes. A Wiley-Interscience Publication (1983) [29] TEODOSIU, C. et.al.: Implicit versus Explicit Methods in the Simulation of Sheet Metal Forming. Proc. of the Int. Conf. FF_,-Simulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991) [30] MATTIASSON, K. et. al.: On the Use of Explicit Time Integration in Finite Element Simulation of Industrial Sheet Forming Processes. Proc. of the Int. Conf. FESimulation of 3-D Sheet Metal Forming Processes in Automotive Industry, Zfirich, ibid (1991)