ONE STEP SPRINGBACK STRATEGIES IN SHEET METAL · PDF fileM.C. Oliveira, J.L. Alves and L.F. Menezes. 1 INTRODUCTION Sheet metal forming is a wide spread technology in almost all kind

VII International Conference on Computational Plasticity COMPLAS 2003

E. Oñate and D. R. J. Owen (Eds) CIMNE, Barcelona, 2003

ONE STEP SPRINGBACK STRATEGIES IN SHEET METAL FORMING

M.C Oliveira*, J.L. Alves† and L.F. Menezes* * Department of Mechanical Engineering

University of Coimbra Polo II, 3030 Coimbra, Portugal

e-mail: [email protected], web page: http://www.dem.uc.pt

† Department of Mechanical Engineering University of Minho

Campus de Azurém, 4080 -058 Guimarães, Portugal email: [email protected], web page: http://www.dem.uminho.pt

Key words: Implicit algorithms, Springback, DD3IMP.

Abstract. The near net shape characteristics of the formed component dictates the success of a sheet metal forming operation, for which one of the main sources of the lack of geometric and dimensional accuracy are the springback deviations. The application of numerical simulations to predict springback of sheet metal forming parts becomes more and more important, since these deviations often lead to tool modifications at the late stage of the development process. In this work the prediction of springback defects will be performed with CEMUC’s home code DD3IMP (contraction of ‘Deep Drawing 3-D IMPlicit code’) and the springback devoted code DD3OSS (contraction of ‘Deep Drawing 3-D One Step Springback’). This is a 3-D elastoplastic finite element code following a full implicit time integration scheme. This formulation allows to take into account large elastoplastic strains and rotations, and it has several constitutive laws and yielding criteria implemented. The Coulomb’ law models the frictional contact problem, which is treated with an augmented Lagrangian approach. The code considers the sheet as a three-dimensional domain described with solid finite elements. The use of an implicit method for the simulation of the deep-drawing process guarantees the structural balance at any given instant of the calculation, which includes the end of the forming process, before removing the tools. Three different numerical strategies for the springback simulation will be presented and discussed, including a numerical algorithm for the removing of all tools in a single step (DD3OSS). These numerical strategies will be used to perform the springback prediction of a complex geometry part.

1

M.C. Oliveira, J.L. Alves and L.F. Menezes.

1 INTRODUCTION Sheet metal forming is a wide spread technology in almost all kind of industrial domains

for the production of an enormous variety of parts and shapes. This forming technology is controlled by an enormous amount of parameters such as the material properties or the tribological behaviour and lubrication conditions. The growing number of materials and design variants, associated with the stringent requirements on precision also contribute for the increasing complexity of this forming technology. In this context the numerical simulation became a tool of increasing importance for the process analysis.

Several properties and phenomena like the tensile strength and elastic behaviour, anisotropy resulting from sheet rolling conditions and lubrication are accountable for geometrical and dimensional precision. Besides the material properties, the springback is also influenced by the tools design, so it is necessary to develop finite element codes that consistently predict this type of geometrical defect. Several computer codes have been developed in order to reduce the number of try-outs in the optimisation of the forming tools. However, a very precise prediction of springback deviation by means of a finite element analysis is still not available. The introduction of new materials such as aluminium alloys and high strength steels in order to improve fuel economy, emissions and safety requirements, increases this challenge since they present in general larger springback deviations.

The finite element simulations of springback are more sensitive to numerical tolerances than the forming simulations, including effects of element type as well as shape and size of finite elements mesh, integration scheme and unloading scheme 1,2. In terms of integration schemes almost every approaches have been tried. Typically explicit methods are used for the forming operations for which they are less expensive in terms of CPU time. For the springback simulations this relation inverts, and that is why many commercial codes couple explicit forming with implicit springback simulations (such as PAMSTAMP from ESI Group)3. However, coupling explicit to implicit methods may be efficient in terms of CPU time, but a bad option in terms of reliability and accuracy of results. In fact, explicit methods do not guarantee the final equilibrium of the deformable body, implying that the final stresses and strains may be completely wrong. Since the shape of the final part in a sheet metal forming operation depends mainly on the amount of elastic energy stored during the forming stage it seems consensual that better results will be attained with implicit methods coupled for both forming and springback. As mentioned before, the main drawback of this integration scheme is the CPU time.

In terms of unloading scheme two different strategies are known to conduct to similar results: the first one corresponds to reverse the tools movement until the loss of contact and can be understood as a simple continuation of the forming process; the second corresponds to replace the tools by the corresponding forces and these are consecutively decreased until vanished 4. The first method has a better agreement with the real processes allowing to take into account changes in contact areas between the blank sheet and tools during the removing. However, this procedure leads to a duplication of the CPU computation time due to the phases necessary to reverse the tools movement. With this in mind, two other strategies are here analysed. The first consists in the removal tool by tool, in one step per tool, forcing the

2


equilibrium in each step by an implicit equilibrium iterative loop. The second tries to perform springback in only one step, removing all the tools simultaneously and forcing the blank sheet to attain equilibrium.

These springback strategies were tested in the deep drawing implicit finite element code DD3IMP. The simulated part consists in a rail specifically developed to emphasize the springback defect, according to the IMS project “Digital Die Design System – 3DS” 5.

2 THE FINITE ELEMENT CODE – DD3IMP The code DD3IMP uses a mechanical model that takes into account the large elastoplastic

strains and rotations. The plastic behaviour of the material is described by the Hill’s yield criterion with isotropic and kinematic work hardening, and by an associated flow rule. Several work hardening constitutive models have been implemented in order to allow a better description of the different material mechanical behaviour 6. The Coulomb classical law models the frictional contact problem, which is treated with an augmented Lagragian approach. A fully implicit algorithm of Newton-Raphson type is used to solve the non-linearities related with the frictional contact problems and the elastoplastic behaviour of the deformable body. The code uses solid finite elements. This represents an enormous cost in terms of CPU time, but they allow an accurate calculation of the stress gradients through thickness of the sheet as well as the thickness evolution 7,8. The global algorithm of this code is summarized in table 1.

Input and check data Repeat Impose a trial increment Calculate the tangent stiffness matrix - Using an explicit algorithm Solve the linear system Choose the trial increments by a rmin strategy Update sheet configuration and tools position Change contact/uncontact boundary conditions Repeat - Using an implicit algorithm Calculate incremental strains and rotations Integrate the constitutive law Calculate residual forces Calculate the consistent stiffness matrix Solve the linear system Update sheet configuration, state variables and adjust boundary conditions Until residual non-equilibrated forces are close to zero Until the end of the process Output results

Table 1: Global algorithm of DD3IMP code.

The evolution of the deformation process is described by an updated Lagrangian formulation. In each load step, first solution for the incremental displacements, stresses and frictional contact forces is calculated with an explicit approach. An rmin strategy is employed

3


to impose limitations on the size of the time increments. Nevertheless, this first solution satisfies neither the variational principle nor the coherence condition. In order to guarantee that the equilibrium of the deformable body is satisfied this trial configuration is successively corrected using an implicit method. The configuration of the blank sheet and tools as well as all state variables are only updated when the structural balance is satisfied. The main characteristic of the global algorithm of DD3IMP code is the use of a single iterative loop to solve both non-linearities related with the mechanical behaviour and the contact with friction problem 9.

In DD3IMP the geometry of the forming tools is modelled by parametric Bézier surfaces. The numerical schemes followed rely on a frictional contact algorithm that operates directly on the parametric Bézier surfaces.

3 UNLOADING STRATEGIES In sheet metal forming, at the end of the forming phase the blank sheet is subjected to

internal stresses due to the restrictions imposed by the tools. Once the tools are removed, the stress state tries to relax until a residual stress state is achieved. Springback, induced by the elastic or elastoplastic recovery make the part shape distort from the expected product shape. This is a key factor for the validation of a forming process.

In this section three different numerical strategies are discussed for the removing of the tools. The common ground for all three cases is that they start from a forming shape for each the equilibrium of stress and strain is guarantee, due to the use of an implicit algorithm, presented in the former section.

3.1 Unloading based on tool displacement

This scheme can be understood as a simple continuation of the forming process. The removing of the tools is divided in the same number of steps, as there are tools to remove. In each of these steps, the movement of the tools is reversed. Lets consider, for instance, a deformation process involving a blank holder, a punch and a die. In the deformation process the blank holder was moved in the direction of the blank sheet until it reaches an imposed force. Then the punch starts its movement until a certain depth. The first step, in order to predict springback, is to reverse the movement of the punch until complete lost of contact. Then invert the previous movement of the blank holder to lose contact. At the end of these steps, the blank sheet can still be trapped in the die walls. An extra step is needed to complete springback evaluation; the die must move away from the blank sheet. In this last step an extra care is also needed, supplementary boundary conditions should be introduced in order to prevent rigid body motion and avoid convergence problems. For this unloading scheme it is also important to guarantee that the tool, after its removal, do not interfere in further simulation phases. For instance, during the die displacement the blank sheet must not enter in contact with the blank holder again.

This strategy can correspond to an enormous increase of CPU time, since now one needs to reverse the complete movement of the punch and blank holder, but also to perform the movement of the die as an extra step at the end of the process.

4


The algorithm followed for each step of the removing of the tools is the same as for the forming phase, already presented in table 1.

3.2 Unloading based on tool-by-tool removal The main disadvantage of the unloading scheme based on tool displacement is the increase

in CPU time due to the need of reversing tool displacements. In order to overtake this problem, another unloading scheme is tested. In this case the tools are also removed one by one, but now they simply vanish at the beginning of each unloading phase, i.e., all patches associated with the tool to be removed are no longer candidates to establish contact with the blank sheet.

Once again the algorithm followed is the one presented in table 1. The trial solution is now obtained admitting that all the nodes that were in contact with the tool to be removed lose that contact, and no new nodes establish contact with it. There is no replacement of the tool by forces, neither a slight decrease of the contact forces. The new equilibrium solution must be iteratively determined in the absence of that tool. Recovering the example from last section, once again the sequence for removing the tools is the punch, the blank holder and finally the die. In the last step supplementary boundary conditions should be introduced in order to prevent rigid body motion and avoid convergence problems.

3.3 One step springback Following the unloading scheme based on the removing of the tools, one after the other, an

idea that arises is the simultaneous removal of all tools. That corresponds to admit that all constraints imposed by the tools disappear at the beginning of the unloading. This corresponds to eliminate all the patches that define the tools as candidates to contact with the blank sheet, and consequently set to zero all the contact forces. There is no need to perform the trial solution since the initial solution for the correction phase corresponds to the configuration of the end of the forming phase, with all contact forces and displacements set to zero. So a new algorithm was developed in order to perform the, from now on designated, one step springback. This new code was designated DD3OSS and is dedicated to simulate springback in just an increment. The global algorithm of this new code is presented in table 2.

Input and check data All contact forces are set to zero Repeat - Using an implicit algorithm

Calculate incremental strains and rotations Integrate the constitutive law Calculate residual forces Calculate the consistent stiffness matrix Solve the system for the incremental displacements

Update sheet configuration, state variables Until residual non-equilibrated forces are close to zero Output results

Table 2: Global algorithm of DD3OSS code.

5


As for the other unloading schemes it is necessary to introduce supplementary boundary conditions in order to prevent rigid body motion and avoid convergence problems. However, it is important to notice that the new equilibrium solution must be iteratively determined in the absence of all tools. So, even with control on boundary conditions some convergence problems may arise. To stabilize the equilibrium loop, and avoid convergence difficulties a modified Newton-Raphson algorithm is used. Basically, two numerical strategies are used to minimize these convergence problems. One deals directly with the correction to the incremental displacements, inducing a control on this parameter in order to prevent strong vibrations in the correction procedure. This control acts reducing the estimated corrections to the incremental displacements:

CalculatedCorrected ζ∆∆ uu = , [ ]0,1ζ ∈ (1)

where the parameter ζ can be interpreted as a smoothing coefficient, and it is calculated in function of the convergence criteria C : Crit

( ) ( )RRuu ..∆∆ +=CritC (2)

This convergence criterion is a mixed combination of the relative norms of the primal displacement increment and the residual internal forces vector R . The admissible value for the convergence criterion is usually set to 1 .

u∆ ( )u210−×

The smoothing parameter is calculated according with the following expression:

( )

=⇒≤

−−=⇒<<

=⇒>

0.1ζ0.10000.1000

0.10000exp25.00.1ζ0.1500000.1000

75.0ζ0.150000

Crit

CritCrit

Crit

CC

C

C

(3)

The other strategy is associated with the global stiffness matrix. In fact, after the removal of all tools the global stiffness matrix should be symmetric. However, due to residual values and small numerical errors, which increase with the problem dimension, this does not occurs. In order to prevent convergence problems within the solver, the global stiffness matrix is imposed to be symmetric.

4 NUMERICAL EXAMPLE

The deep drawing problem selected for the comparison of the different springback algorithms is the rail presented in figure 1 a). This is a rail specially conceived to emphasize springback defects 5. As described in this figure, its deep drawing involves three different tools: the punch, the blank holder and the die. The blank sheet is a square with 300.0 mm length and 1.0 mm thickness. Due to the symmetry only a half of the problem was considered.

The blank sheet is modelled with a non-uniform mesh of solid finite elements. Following previous works the elements used are the 8-node hexahedron solid finite elements with a selective reduced integration method 10. The average element size in the flat contact zone between the blank holder and the die is 6 mm, and it is reduced to 3 mm in the other areas.

6


The mesh is also presented in figure 1 b). In thickness only one layer was used, although previous works indicate that this may affect the accuracy of the results 1.

α

R

Die

Punch

Blank Holder

Blank HolderPunch

Blank Sheet

y

x

a) b)

Figure 1: a) Forming tools geometry; b) Blank sheet discretization.

The process conditions are summarized in table 3. An aluminium alloy (AA5182) was selected for the blank sheet 11. In this work this material is assumed to have an isotropic and kinematic work hardening. The isotropic work hardening is described by the Voce equation:

( )[ ]pε+= Rsat0 C-exp-1RYY (4)

where Y0, Rsat and CR are the material parameters determined from standard mechanical tests; Y is the flow stress while pε is the equivalent plastic strain given by

dtt pp∫ ε=ε 0& (5)

The Lemaître and Chaboche saturation law describes the kinematic hardening:

( ) ( ) 0XXXX =ε

−−′= 0,σ psatX

XC &o

σ

(6)

where is an objective rate of the back stress tensor , characterizes the saturation value of the kinematic hardening, while the material parameter C characterizes the rate of approaching the saturation. The equivalent stress definition assumes the form:

oX X satX

X

7


( ) ( )XσMXσ −′−′=σ :: (7)

where is a fourth-order symmetric anisotropy tensor associated to the Hill’s 48 yield criterion, used to describe the orthotropic anisotropy of the rolled sheet.

M

Initial sheet geometry Width/2 150 mm Length 300 mm Thickness 1 mm Material Data Young’s modulus, Poisson ratio 70000 MPa, 0.29

Y0 148.5 MPa CR 9.7

Isotropic hardening Parameters

Rsat 192.4 MPa CX 152.7 Kinematic hardening ParametersXsat 26.0 MPa F 0.63049 G 0.55866 H 0.44134

Hill 48 Anisotropy Coefficients

L=M=N 1.60654 Process Parameters Friction Coefficient 0.15 Initial Blank holder Force 90000 N

Table 3: Summary of the input data for the numerical simulation of a rail.

The deep drawing process is divided in three phases. First the blank holder is moved until it reaches the initial prescribed force. Then the punch starts its movement until it reaches 60 mm displacement. During this phase the blank holder force increases proportionally to the punch displacement to a maximum value of 118 kN. The final phase is the removing of the tools. For this phase the three different strategies presented were compared. It is important to notice that all springback simulations start from the same configuration, at the end of the forming phase, and that the same supplementary boundary conditions were introduced in order to control the rigid body motion (figure 2 a)). These boundary conditions are imposed only in one node since only one half of the structure was simulated.

In the first algorithm, based on the tools removal by reversing the displacement until the complete loss of contact, the sequence of removing is: first the punch, then the blank holder and finally the die. For the second algorithm, based on the disappearing of each tool, two sequences were tested, in order to evaluate the influence of the order. The first sequence is the same as for the first algorithm. In the second sequence the order between the punch and the blank holder is inverted. In the third algorithm the lost of contact is imposed for all tools in the same step, as described earlier. Table 4 summarizes the different steps for each simulation.

8


x

y

Node for supplementary boundary conditions

Section 3

Section 1 Section 2

Section 3

Section 1 Section 2

a) b) Figure 2: a) Boundary conditions for the simulation process, and supplementary boundary conditions.

b) Geometry after deformation: definition of the sections for springback analysis.

Unloading Scheme Tools

Displacement Tools removal

(1) Tools removal

(2) One step

Springback Total number of steps 3 3 3 1

Step 1 Punch Punch Blank holder All tools Step 2 Blank holder Blank holder Punch

Tool controlling: Step 3 Die Die Die

Table 4: Summary of the unloading schemes tested in the springback simulation.

5 RESULTS AND DISCUSSION In order to evaluate the springback results three sections were established in the rail as

presented in figure 2b). To correctly evaluate shape errors the “NXT Post Processor II” 12 was used in order to guarantee a correct comparison between the predicted profiles. The evaluation methods used are based on the differences between the original CAD data and actual surfaces13.

The first results to be compared correspond to the removal of the punch and blank holder, obtained with displacement or by elimination. The different removal order of these tools is also analysed. The section profiles as well as the predicted angles are presented in figure 3. If the first step corresponds to the removal of the punch the section profiles obtained are identical (tool displacement and tool removal (1)), presenting some twist resulting from the different constraints imposed on both sides of the blank sheet, resulting from the curved shape of the rail. The flange moves down, but as it movement is restrained by the blank holder they

9


remain flat (see section 3). After the punch removal, in the second step the sections remain almost identical in particular in the outward flange. There is a slight change in the springback angle for the inward flange mainly in section 1. This difference is enhanced in the global shape on section 3, although the springback angle predicted is the same. On the other hand, if the first tool to be removed is the blank holder the constraint imposed by this tool disappears increasing the angle of springback of the flanges after the first step, and the flange presents already same distortion (see section 3). After the punch removal it is clear, from the geometrical section comparison, that there are some differences between the unloading schemes. Once again the maximum difference occurs in the inward flange for section 1. In this case, admitting again as reference the profile obtained by tools displacement, the difference in springback angle increase to 1.3º. For section 2 the difference in the springback angle of the flanges is always inferior to 0.5º. Regarding section 3 this presents a different global shape, with a lower springback depth. In the end of step 2, in the three cases the sheet remains inside the die. That is why the vertical walls remain flat with no curvature.

( º7.0< )

)

)

The final springback predicted for each section with the different unloading schemes is presented in figure 4. Globally the profiles present the same behaviour, with some twist visible from the comparison between section 1 and 2. For section 1 the major differences are located in the springback angle in the inward flange. In fact, the unloading scheme based on tools displacement predicts an angle of 5.7º. The tool by tool method using the same tools removal sequence shows an angle of 6.5º (tool removal (1)). Reversing the tools removal order the springback angle for this flange increases slightly to 7.5º (tool removal (2)). The one step springback leads to an angle of 7.8º. This corresponds to a maximum difference of 2.1º in the inward flange, for section 1. For section 2 this differences are lower . The z coordinate of section 3 is influenced by the springback angle in the inward flange. However, the obtained springback angle along section 3 is similar in all cases (differences lower then

.

( º0.1<

( )º6.0<Figure 5 presents the equivalent stress distribution as obtained with the different strategies.

At the end of the deformation process the range for the equivalent stress is between 0 and 300 MPa. After springback a global reduction of the equivalent stress occurs. No major differences induced by the unloading strategies can be detected. Comparing the equivalent stress evolution according with the steps it is possible to verify the resemblance between the tools displacement and the tools removal, with the same sequence. Also, the tool removal (2) sequence, presents a similar distribution, before the die removal.

It terms of CPU time, the springback performed with tools displacement lead to 35% of the time required for the forming process. However, due to the die geometry in this particular example it was possible to open the die, moving it in diagonal to the part. This allowed for a reduction of the CPU time compared to the one that would be necessary to move the die downward 60 mm. With the unloading scheme based on the tools removal the CPU time involved in springback reduces to around 15% of the time required for the forming phase. This reduction goes to less than 1% with the one step springback approach.

10


Section 1

-66

-56

-46

-36

-26

-16

-6-130 -80 -30 20 70

y(mm)

z(m

m)

outward flange inward flange

Angle Section 1

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200 250 300

Length(mm)

Deg

ree

Section 2

-66

-56

-46

-36

-26

-16

-6-130 -80 -30 20 70

y(mm)

z(m

m)


Angle Section 2

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200 250 300

Length(mm)

Deg

ree

Section 3

-63

-62

-61

-60

-59

-58

-57

-56

-55

-54

-53

y(mm)

z(m

m)

Angle Section 3

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300

Length(mm)

Deg

ree

CAD geometry Tool displacement - Step 1Tool removal (1) - Step 1 Tool removal (2) - Step 1Tool displacement - Step 2 Tool removal (1) - Step 2Tool removal (2) - Step 2

CAD geometry Tool displacement - Step 1Tool removal (1) - Step 1 Tool removal (2) - Step 1Tool displacement - Step 2 Tool removal (1) - Step 2Tool removal (2) - Step 2

Figure 3: Section profiles and angles after step 1 and 2 for springback prediction with the unloading schemes based on tool displacement and removal.

11


Section 1

-61

-51

-41

-31

-21

-11

-1-130 -80 -30 20 70

y(mm)

z(m

m)


Section 2

-61

-51

-41

-31

-21

-11

-1-130 -80 -30 20 70

y(mm)

z(m

m)


Section 3

-61

-60

-59

-58

-57

-56

-55

-54

-53

-52

-51

y(mm)

z(m

m)

Angle Section 1

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200 250 300

Length(mm)

Deg

ree

Angle Section 2

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 50 100 150 200 250 300

Length(mm)

Deg

ree

Angle Section 3

-3

-2

-1

0

1

2

3

0 50 100 150 200 250 300

Length(mm)

Deg

ree

CAD geometry Tool displacement - Step 3Tool removal (1) - Step 3 Tool removal (2) - Step 3One step

CAD geometry Tool displacement - Step 3Tool removal (1) - Step 3 Tool removal (2) - Step 3One step

Figure 4: Section profiles and angles for springback prediction with the unloading schemes based on tools displacement, tools removal and one step springback.

12


End of deformation

Tools Displacement

Punch Blank holder Die

Tools removal (1)

Punch Blank holder Die

Tools removal (2)

Blank holder Punch Die

One step Springback

Figure 5: Equivalent stress evolution predicted according with the unloading scheme.

13


6 CONCLUSIONS Springback is still one of the main issues in the numerical simulation of deep drawing

process. Although the enormous effort in last decade, no code has yet found wide acceptance for springback analysis. In this work, this problem is approached using the static implicit code DD3IMP and the springback devoted code DD3OSS. The use of implicit schemes is of paramount importance for the numerical prediction of the final stress and strain states.

Two algorithms for simulation of the springback were presented and compared. They were also compared with the usual unloading scheme based on reversing the tools displacement. This unloading scheme tends to increase enormously the CPU time required for the springback simulation, since it is necessary to reverse all tools displacement until the lost of contact is guaranteed. That is why a new unloading strategy based on tools removal is tested. The two algorithms presented are both based on this strategy: in the first the tools are removed step by step, following a sequence, in the second the tools are removed simultaneously in one single step. This second strategy leads to the new code designated DD3OSS. For the tool-by-tool removal two different sequences of tools removal were also tested.

The springback simulation of a rail was performed with all the presented strategies. Through the complex part shape used as example, it was possible to confirm the global accuracy between the geometries predicted with all unloading schemes, as well as the advantages in terms of CPU time of the one step springback strategy.

ACKNOWLEDGEMENTS The work presented here was funded by the European Community throughout the Growth

Programme (contract G1RD-CT-2000-00104) and by the Portuguese Foundation for the Science and the Technology throughout the Programme POCTI (contract EME/35945/99). The authors are grateful for this support.

REFERENCES [1] M.C. Oliveira, J.L. Alves and L.F. Menezes, Springback evaluation using 3-D finite

elements, Proceedings of the 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Verification of Simulation with Experiment, NUMISHEET’02, Jeju Island, Korea, 189-194 (2002).

[2] R.H. Wagoner, Fundamental Aspects of Springback in Sheet Metal Forming, Proceedings of the 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Verification of Simulation with Experiment, NUMISHEET’02, Jeju Island, Korea, 13-24 (2002).

[3] F. El Khaldi, M. Lambriks, New requirements and recent development in sheet metal

14


stamping simulations, Proceedings of the 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Verification of Simulation with Experiment, NUMISHEET’02, Jeju Island, Korea, 411-421 (2002).

[4] M. Kawka, T. Kahita and A. Makinouchi, Simulation of multi-step sheet metal forming processes by a static explicit FEM code, Journal of Materials Processing Technology, 80-81, 54-59 (1998).

[5] A. Col, First Results of the 3DS research project, Proceedings of the 22nd International Deep Drawing Research Group Congress and Working Group Meeting, Nagoya, Japan (2002).

[6] L.F. Menezes, S. Thuiller, P.Y. Manach, S. Bouvier, Influence of the work-hardening models on the numerical simulation of a reverse deep-drawing process. Plasticity, Damage and Fracture at Macro, Micro and Nanoscales, Proceedings of Plasticity’02: The ninth International Symposium on Plasticity and its Current Applications, Akhtar S. Khan and Oscar Lopez-Pamies (eds.), Neat Press, Maryland, USA, 331-333 (2001).

[7] L.F. Menezes, C. Teodosiu, A. Makinouchi, 3-D solid elasto-plastic elements for simulating sheet metal forming processes by the finite element method, FE simulation of 3D sheet metal forming process in automotive industry; Tagungsbericht der VDI-Gesellschaft Fahrzeugtechnik, VDI-Verlag, Zurich, Germany, 381-403 (1991).

[8] L.F. Menezes, C. Teodosiu, Three-dimensional numerical simulation of the deep-drawing process using solid finite elements. Journal of Materials Processing Technology, 97, 100-106 (2000).

[9] L.F. Menezes, C. Teodosiu, Improvement of the frictional contact treatment in a single loop iteration algorithm specific to deep-drawing simulations. Proceedings of the conference NUMISHEET’99; J.C. Gelin and P. Picart (eds.), Besançon, France, 197-202 (1999).

[10] J.L. Alves, L.F. Menezes, Application of tri-linear and tri-quadratic 3-D solid finite elements in sheet metal forming process simulation, in Ken-ichiro Mori (eds.), Proceedings of the 7th International Conference NUMIFORM 2001, Simulation of material processing: Theory, methods and applications, Japan, 639-644 (2001).

[11] C. Teodosiu, S. Bouvier, 2001, Selection and Identification of Elastoplastic Models for the Materials used in the Benchmarks, 18-Months Progress Report, "3DS, Digital Die Design System", contract G1RD-CT-2000-00104 (not yet published).

[12] M&M Research,Inc. http://ww.m.research.co.jp [13] K. Kase, A. Makinouchi, T. Nakagawa, H. Suzuki, F. Kimura, Shape error evaluation

method of free-form surfaces, Computer-Aided design, Vol.31, nº 8, 495-505 (1999).

15

Documents

ONE STEP SPRINGBACK STRATEGIES IN SHEET METAL · PDF fileM.C. Oliveira, J.L. Alves and L.F. Menezes. 1 INTRODUCTION Sheet metal forming is a wide spread technology in almost all kind