square cup deep drawing using forming limit diagram

Chapter 55: Square Cup Deep Drawing using Forming Limit Diagram

55 Square Cup Deep Drawing using Forming Limit Diagram

PART 1. Explicit Forming Summary 1098

Introduction 1099

Modeling Details 1101

Results 1104

PART 2. Implicit Spring Back Introduction 1108

Modeling Details 1108

Results 1110

Input File(s) 1112

Reference 1112

MD Demonstration Problems

CHAPTER 551098

SummaryTitle Chapter 55: Square Cup Deep Drawing using Forming Limit Diagram

Features • Failure criterion based on the Forming Limit Diagram• Springback: Explicit -> Implicit switching

Geometry

Material properties • Sheet Metal (aluminum sheet): Anisotropic Materials under Plane Stress Conditions Exx = 71.0 GPa, = 0.33Stress constant = 0.0 MPa, Hardening modulus = 576.79 MPaStrain offset = 0.01658, Exponent for power-law hardening = 0.3593Lankford parameters: R0 = 0.71, R45 = 0.58, R90 = 0.70

• Punch, Die, and Clamp: Rigid

Analysis characteristics Transient explicit dynamic analysis (SOL 700 explicit single precision)Nonlinear implicit static analysis (SOL 700 implicit double precision)

Boundary conditions • Explicit: Fixed boundary condition of Die and Clamp• Implicit Springback: Fixed at the center point of the plate

Element types 4-node shell elements

FE results Stress Contour Plot, Forming Limit Diagram and more

Punch

Sheet

Clamp

Die

Implicit Spring BackExplicit Forming

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00%

Maj

or T

rue

Stra

in (%

)

Minor True Strain (%)

FLD at Mid. Surface

FLD with Safety margin

Element will fail at next step

1099CHAPTER 55

Square Cup Deep Drawing using Forming Limit Diagram

PART 1. Explicit Forming

IntroductionThis is a sheet metal forming example of a plate with anisotropic behavior that is drawn through a square hole by means of a punch. This particular example has experimental results from a verification problem of the 1993 NUMISHEET Conference held in Japan. The results are obtained at single punch depth (20 mm punch travel) for an aluminum alloy plate. The material is seen to be anisotropic in its planar directions; i.e., the material behavior is different for all directions in the plane of the sheet metal as well as in the out of plane direction. The data obtained from the NUMISHEET Conference is as follows:

Aluminum Alloy

Thickness = 0.81 mmYoung’s modulus = 71 GPaPoisson’s ratio = 0.33Density = 2700 kg/m3

Yield stress = 135.3 MPaStress = 576.79 * (0.01658 + p)0.3593 MPaLankford parameters: R0 = 0.71, R45 = 0.58, R90 = 0.70Friction coefficient = 0.162

The size of the plate modeled was 0.15 x 0.15 (in meters). No strain-rate dependency effects were included in the material data, so the metal sheet was analyzed without these effects. The dimensions of the plate, die, punch, and clamp are all given in Figure 55-1.

SOL 700 Entries IncludedSOL 700TSTEPNLDYPARAM,LSDYNA,BINARY,D3PLOTCSPHPSPHEOSGRUNSPHDEFTICMATD010PSOLIDDMATD003


CHAPTER 551100

Figure 55-1 Dimensions of Plate, Die, Punch, and Clamp (in Millimeters)

1101CHAPTER 55


Modeling Details

Figure 55-2 SOL 700 Model (Exploded View)

The SOL 700 model is shown in Figure 55-2. The main parts in the finite element model are:

• sheet metal• punch• die• clamp

Sheet Metal

The SOL 700 material model for sheet metals is a highly sophisticated model and includes full anisotropic behavior, strain-rate effects, and customized output options that are dependent on material choice. Since not all of the materials can be derived from the simplified set given by the NUMISHEET organization, most participants in the conference used an isotropic material model. In reality, the process is definitely anisotropic and effects due to these differences can be seen in the transverse direction. For materials displaying in-plane anisotropic behavior, the effect would be even more noticeable. The parameters on the MAT190 (refer to the MD Nastran Quick Reference Guide) specify planar anisotropic behavior and are as follows (for the aluminum sheet):

• MATD190 elastic material properties.• Isotropic behavior was assumed in the elastic range:

Exx = 71.0 GPa = 0.33

Clamp

Die

Punch

Sheet

Z

YX


CHAPTER 551102

• Planar anisotropic yielding and isotropic hardening were assumed in the plastic range:

A = Stress constant = 0.0 MPa

B = Hardening modulus = 576.79 MPa

C = Strain offset = 0.01658

n = Exponent for power-law hardening = 0.3593

• Lankford parameters:

R0 = 0.71

R45 = 0.58

R90 = 0.70

Punch, Die, and Clamp

These three components provide the constraints and driving displacement for the analysis and are modeled as rigid bodies. Contact is then specified with the metal sheet using the friction coefficient values provided. The three contact types are specified as following:

• Contact between the punch and the sheet• Contact between the die and sheet• Contact between the clamp and sheet

Finally, the punch is given a scaled downward velocity providing the driving displacement for the analysis.

Input File

SOL 700 is an executive control entry and activates an explicit nonlinear transient analysis.

Case control section is below:

The bulk entry section starts:

SOL 700,NLTRAN stop=1

DLOAD = 1IC = 1SPC = 1BCONTACT = 1TSTEPNL = 1

BEGIN BULK$TSTEPNL 1 20 2.0E-3$DYPARAM LSDYNA BINARY D3PLOT 0.002

1103CHAPTER 55


TSTEPNL is a SOL 700 bulk data entry which describes the number of Time Steps (20) and Time Increment (2.00 ms) of the simulation. The end time is the product of the two entries. Notice here the Time Increment is only used for the first step. The actual number of Time Increments and the exact value of the Time Steps are determined by SOL 700 during the analysis. The time step is a function of the smallest element dimension during the simulation.

LSDYNA,BINARY,D3PLOT option of DYPARAM entry controls the output time steps of d3plot binary file. The result plots at every 0.002 seconds are stored in d3plot binary file.

Bulk data entries that define properties for shell elements

The MATD020 entry defines the rigid material property. In the example, the clamp, die, and punch are modeled by the rigid materials.

The MATD190 entry defines an anisotropic material developed by Barlat and Lian (1989) for modeling sheets under plane stress conditions and with Forming Limit Diagram failure criteria. This material allows the use of the Lankford parameters for the definition of the anisotropy.

In the model, Gosh’s hardening rule is used:

The forming limit diagram is defined in by TABLED1 as shown above.

All fields are set for the coefficients of equations. See MD Nastran Quick Reference Guide for details.

PSHELL1 1 1 BLT Gauss ++ .81

MATD020 2 1.0 210.E9 0.3 1 4 7

MATD190 1 2.7E-4 7.1E7 0.33 2.0 576.79E3.3593 0 ++ 6.0 .71 .58 .70 .01658 ++ 2.0 77 ++ 1.0 0.0 0.0 ++ 0.0 1.0 0.0TABLED1,77,,,,,,,,++,-100.0,196.67,0.0,30.,30.,45.,40.,47.,++,50.,45.,ENDT

SPCD2,1,RIGID,MR2,3,0,100,1.0,,++TABLED1,100,,,,,,,,++,0.0,-1000.,0.02,-1000.,ENDT

Y p k 0 p+ np–=


CHAPTER 551104

The SPCD2 entry defines imposed nodal motion on a node, a set of nodes or nodes of a rigid body. The rigid punch is moving downward at 1000 m/s from 0 to 0.02 seconds.

The FORCE entry defines a force on the grid point as well as rigids. Since the forces on the rigid body are not yet supported by the Nastran input processor, TODYNA and ENDDYNA entries are used in conjunction with the FORCE entry to by-pass the IFP (Input File Processor) and directly access SOL 700.

The BCBODY entry defines a flexible or rigid contact body in 2-D or 3-D. Although SOL 700 only supports flexible contact in BCTABLE, the rigid contact can be applied using the rigid material of contact bodies. In this example, all contact body pairs are given 0.162 static and kinetic friction coefficients. The surface-to-surface, one way contact method is used for all contact definitions.

The BCBODY entry defines a flexible or rigid contact body in 2-D and 3-D.

The BSURF entry defines a contact surface or body by element IDs. All elements with the specified IDs define a contact body.

ResultsTo verify the result of MD Nastran, the major and minor principal strains at 0.015seconds are compared with those of Numisheet and Dytran results in Figure 55-3 and Figure 55-4. Left plots of each figure were represented by

FORCE 9999 MR3 -19.6E6 1.

BCTABLE 1 3 SLAVE 1 0. 0. 0.162 0. 0 0 0 0 0.162 SS1WAY++

BCBODY 1 DEFORM 1..$BSURF 1 1 THRU 1600..

$GRID 1 -75. 75. 0.0..GRID 4528 -8.33333-37.0067-75.405$CQUAD4 1 1 1 2 43 42..CQUAD4 4468 63 4527 4273 4274 4528

1105CHAPTER 55


Makinouchi et al. (1993). The data in the plots were obtained from several companies which did the same test. MD Nastran gave a solution well within the spread of experimental values.

Figure 55-3 Comparison of Major Principal Strain Along Line OB(Numisheet and Dytran Results vs. MD Nastran SOL 700)

Figure 55-4 Comparison of Minor Principal Strain Along Line OB(Numisheet and Dytran Results vs. MD Nastran SOL 700)

Major Principal Strain

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

0 20 40 60 80 100 120

Distance from Center Along Line OB

Stra

in

Minor Principal Strain

-2.50E-01

-2.00E-01

-1.50E-01

-1.00E-01

-5.00E-02

0.00E+000 20 40 60 80 100 120

Distance from Center Along Line OB

Stra

in


CHAPTER 551106

Figure 55-5 Forming Limit Diagram Along Line OB at 0.019 Seconds

-20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

-30.00% -20.00% -10.00% 0.00% 10.00% 20.00%

Minor True Strain (%)

Maj

or T

rue

Stra

in (%

)

FLD at Mid. SurfaceFLD with Safety margin

Element will fail at next step

1107CHAPTER 55


Figure 55-6 Maximum Principal Strain Contour Plots at Mid Surface at Various Times

Note that the FLD diagram correctly predicts the failure of elements at t = 0.019 as shown in the stress fringe plots.

t = 0.000 seconds t = 0.004 seconds




CHAPTER 551108

PART 2. Implicit Spring Back

IntroductionSpringback refers to an event in which there is elastic strain recovery after the punch is removed. This deformation can alter the final desired shape significantly. In an explicit dynamic analysis, it can take some time before the workpiece comes to a rest, so the springback simulation is performed using the implicit solver to speed up this part of the analysis. Using explicit-implicit switching available in SOL 700, the residual deformations after sheet metal forming are computed and used as a pre-condition for springback analysis. Because, in this example, there was a failure at around 0.019 seconds in the sheet metal as shown in Part 1, the explicit simulation was terminated at 0.018 seconds. The initial condition, including the final stresses and deformation and the element connectivity of the explicit run are transferred to the implicit run. The analysis scheme is described below.

Figure 55-7 Analysis Scheme

SOL 700 Entries IncludedSOL 700MATD036SEQROUTSPRBCK

Modeling DetailsThe model of explicit run is the same as Part 1. In the implicit run, only the sheet metal is used.

Input File

Explicit Input File

BEGIN BULK$TSTEPNL 1 10 1.8E-3

SOL 700 Explicit

(Use SEQROUT Entry)

Generate jid.dytr.nastin

SOL 700 Implicit

(Include jid.dytr.nastin)

(Use SPRBCK Entry)

1109CHAPTER 55


As mentioned above, the end time of simulation is assigned to 0.018 seconds.

The SEQROUT entry generates the jid.dytr.nastin file at the end of simulation. The nastin file includes the final deformations and stresses of the assigned part. The nastin file can be used for a subsequent explicit or implicit SOL 700 run. In the example, only the result for Part 10 which includes the sheet metal is written out to the nastin file.

Implicit Input File

As mentioned above, the end time of simulation is assigned to 0.018 seconds.

Because all information of nodes and element connectivity is in jid.dytr.nastin file, Grid and CQUAD entries are removed in the implicit input. Only one point boundary condition at the center and SPRBCK entry are added in the input file.

Since MATD190 is not available in the implicit analysis, MATD036 is used instead of MATD190. MATD036 and MATD190 are identical material models except that FLD is supported only in MATD190.

MATD036 is only different in the failure criteria using FLD. Others are the same as MATD190 in the explicit simulations of Part 1 and 2.

SPRBCK activates the implicit spring back analysis. Nonlinear with BFGS updates solver type is used in the example. See MD Nastran Quick Reference Guide for other fields.

Only one point at the center of the sheet metal is fixed to prevent singular condition in the implicit simulation.

SEQROUT 10BCPROP 10 1

BEGIN BULK$TSTEPNL 1 10 1.8E-3

MATD036 1 2.7E-4 7.1E7 0.33 2.0 576.79E3.3593 0 ++ 6.0 .71 .58 .70 .01658 ++ 2.0 ++ 1.0 0.0 0.0 ++ 0.0 1.0 0.0

SPRBCK 1 0.005 ++ 200 0.0 1.00E-3 ++ 2 1 100 1.0E-2 0.10 ++ 1 1

SPC1 1 123456 841


CHAPTER 551110

ResultsThe springback simulation from explicit to implicit runs works fine. The results of explicit and implicit analyses are shown in Figures 55-8 to 55-10. Figure 55-8 shows the displacement contours at the start of analysis and at the end of analysis. Note that the initial deformation of the plate grids in the implicit analysis is set to zero because the final deformation of explicit analysis is applied to the initial location of grid points in the springback implicit analysis. In Figure 55-9 the initial stress condition of springback implicit analysis is perfectly coincident with the final stage of explicit analysis. The initial stress of implicit analysis causes the additional deformation in the springback implicit analysis. :

Figure 55-8 Vertical (Z-direction) Displacement Contour Plot

t = 0.000 seconds t = 0.018 seconds (end of explicit run)

Initial condition of implicit run Final result of implicit run

Because the final results are applied asthe initial condition for implicitsimulation, the initial deformation ofimplicit simulation is set to 0.

Explicit Simulation

Implicit Simulation

1111CHAPTER 55


Figure 55-9 von Mises Stress Contour Plot

The location of each grid point along the diagonal line of the plate at the end of the explicit and the springback analysis is plotted in Figure 55-10; the maximum difference between these curves is around 0.756 mm. The centers of the implicit and explicit sheet are positioned to have the same position as a reference, hence the largest differences tend to appear at the ends of the sheet.

t = 0.000 seconds t = 0.018 seconds (end of explicit run)

Initial condition of implicit run Final residual stress of implicit run

Because the final results are applied as the initial condition of implicit simulation, the initial stress of implicit simulation is the sameas the final stress of the explicit simulation.

Explicit Simulation

Implicit Simulation


CHAPTER 551112

Figure 55-10 Comparison of Vertical Displacements (z-direction) After Explicit and Springback Simulations Along Diagonal Line of Plate

Input File(s)

ReferenceMakinouchi, A., Nakamachi, E., Onate, E., and Wagoner, R. H., “Numerical Simulation of 3-D Sheet Metal Forming Processes, Verification of Simulation with Experiment,” NUMISHEET 1993 2nd International Conference.

File Description

nug_55a.dat MD Nastran input file of explicit square cup deep drawing analysis using Forming Limit Diagram.

nug_55b.dat MD Nastran explicit input file for springback analysis.

nug_55c.dat MD Nastran implicit input file for springback analysis

nug_55d.dat MD Nastran stress and deformation information of explicit analysis for input to implicit analysis

-20

-15

-10

-5

0

5

-100 -80 -60 -40 -20 0 20 40 60 80 100

Distance from center

Def

orm

atio

n to

ver

tical

dire

ctio

n

at the end of explicit runat the end of implicit run

http://www.mscsoftware.com/doc/nastran/mdug/input_files/ch055/nug_55a.dat

http://www.mscsoftware.com/doc/nastran/mdug/input_files/ch055/nug_55b.dat

http://www.mscsoftware.com/doc/nastran/mdug/input_files/ch055/nug_55c.dat

http://www.mscsoftware.com/doc/nastran/mdug/input_files/ch055/nug_55d.dat

square cup deep drawing using forming limit diagram

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