3-d Sheet Metal Forming

Chapter 3: 3-D Sheet Metal Forming

3

3-D Sheet Metal Forming



Summary



Introduction



Solution Requirements



FEM Solutions



Modeling Tips



Input File(s)



Video

93

80 81

82 92 93

81

80 MD Demonstration Problems CHAPTER 3

Summary Title

Chapter 3: 3-D Sheet Metal Forming

Contact features

• • • •

Geometry

2-D Plane strain elements or shell elements (units: mm) • Punch radius = 23.5 • Die radius R2 = 25.0 • Die shoulder R3 = 4.0 • Width of tools = 50.0 • Length of sheet (initially) =120.0 • Thickness of sheet = 1.0 • Width of sheet = 30.0 • Punch stroke = 28.5

Material properties

Rigid and deformable bodies Mesh dependency Elasticity, plasticity and spring back Sliding contact around circular surface

• Young’s modulus: E = 70.5kN  mm 2 • Poisson’s ratio:  = 0.342 • Initial yield stress:  0 = 194N  mm 2

Original Position

Punch

Sheet Final Position W R2 R3

Die

• Hollomon hardening:  = K n K = 550.4N  mm 2 n = 0.223

Analysis type

• • • •

Quasi-static analysis Elastic plastic material (isotropic hardening) Geometric nonlinearity Nonlinear boundary conditions

Displacement boundary conditions

• Symmetric displacement restraints (half symmetry). • Bottom surface fixed. • Prescribed vertical displacement for the punch.

Element type

2-D Plane strain - 4-node linear elements; 3-D Shell - 4-node shell elements

Contact properties

Coefficient of friction  = 0.1342

FE results

1. Forming angle and angle after release 2. Plot of punch force versus punch displacement compared to experimental values 2D Plane Strain With Friction Punch Force (N)

300

SOL 400 Marc

250 200 150

Experimental

100 50 0

0

5

10

15

20

25

30

Punch Displacement (mm)

CHAPTER 3 81 3-D Sheet Metal Forming

Introduction This benchmark problem is an approximation of the Numisheet 2002 – Benchmark B problem. Simulations are carried out using MD Nastran solution sequence 400 to find the angles before and after spring back. Experimental results are available for this benchmark, but it is noted that the sheet is slightly anisotropic. The text setup and reference details of these experimental results are given in Figure 3-1. The current problem uses an isotropic elastic-plastic hardening behavior.

SOURCE FREE BENDING BENCHMARK TESTING OF 6111-T4 ALUMINUM ALLOY SAMPLE John C. Brem*, Frederic Barlat**, Joseph M. Fridy** Alcoa Technical Center, Pennsylvania, Numisheet 2002 Conference, Korea

Figure 3-1

Test Setup for Numisheet 2002 - Benchmark B Problem

Solution Requirements Two solutions: one using friction coefficient 0.1342 (bilinear Coulomb friction model) between the sheet and both tools, and one frictionless solution are requested for: • Forming angle (the angle  at the end of the punch stroke) • Angle after release (the angle after tool removal) • Punch force - punch displacement diagram Figure 3-2 shows the definition of angle  . The solutions, obtained with shell elements and plane strain elements, include the following: • • • •

Element size (in particular near the curved zones) Method used in discretization of the tools Method for normal contact detection (hard/direct contact) Method for stick slip approximation (bilinear Coulomb friction model)

82 MD Demonstration Problems CHAPTER 3

Unit: mm A

C

20 B

D

20

C 20

y x

Figure 3-2

D

θ

Requested Angles for Benchmark 3

FEM Solutions FEM solutions have been obtained with MD Nastran’s solution sequence 400 for the 2-D plane strain and 3-D shell representations of the present sheet metal forming problem. The details of finite element models, contact simulations, material, load, boundary conditions, and solution procedure of both the 2-D plane strain and 3-D shell approaches are discussed.

Finite Element Models The finite element model used for the 2-D plane strain approach is shown in Figure 3-3. The punch and die are modeled in analytical form. The finite element mesh for the sheet contains 850 elements with 5 elements over the thickness. Only half of the sheet is modeled. The applied element lengths can be determined from Table 3-1. MD Nastran’s 2-D plane strain solid elements with material ID 1 are selected using the following PLPLANE and PSHLN2 entries. The 30 mm for the width of the sheet is specified in PSHLN2 option. PLPLANE 1 PSHLN2 1 + C4

1 1 PLSTRN

1 L

30.0

+

CHAPTER 3 83 3-D Sheet Metal Forming

Figure 3-3

FE Model for 2-D Plane Strain Approach

Table 3-1

Number of Elements in Length Direction (2-D Plane Strain Model)

Position

Number of Elements

0  x  27mm

50

27  x  40.2mm

100

40.2  x  60mm

20

The finite element model used for the 3-D shell approach is presented in Figure 3-3. Also, in this case, only half of the plate has been modeled with appropriate symmetry conditions at the middle of the plate. The sheet is modeled using 1020 thick shell elements with 6 elements across the width and 170 elements along the length (as in Table 3-2). MD Nastran’s thick shell elements with material ID 1 are selected using the following PSHELL and PSHLN1 entries. The thickness 1 mm for the sheet is specified in PSHELL option. PSHELL PSHLN1 +

1 1 C4

Figure 3-4

1 1 DCT

1. 1 L

1 NO

FE Model for 3-D Shell Approach

1 +

84 MD Demonstration Problems CHAPTER 3

Table 3-2

Number of Elements in Length Direction (Benchmark 3)

Position 0  x  40mm

Number of Elements 160

40  x  60mm

10

Contact Models In defining the contact model for the 2-D plane strain case, the sheet is modeled as a deformable body and the punch and die are modeled as rigid bodies. Elements comprising the sheet are used to generate a deformable contact body with ID 4 using the following BCBODY and BSURF entries. Contact body ID 5 is used to define the load controlled rigid body with a control node ID 1 for the punch and contact body ID 6 is used to define the position controlled rigid body for the die. The geometry profiles of these rigid bodies are defined using 2-D NURB curves that describe the true surface geometry and most accurately represent the punch and die geometry. The friction factor of 0.1342 is defined for all these contact bodies. BCBODY BSURF ... BCBODY

... BCBODY

4 4

2D 1

5 2D 0 0. RIGID 1 NURBS2D -3 6 2D 0 0. RIGID 0 NURBS2D -2

DEFORM 2 RIGID 0. 4 3 RIGID 0. 6 2

4 3 0. CBODY2 50 0. CBODY3 50

0 4

.1342 5

6

7

0 0.

.1342 0.

1 0.

1 0.

0 0.

.1342 0.

1 0.

-1 0.

...

The contact bodies for the 3-D shell models are also defined in similar way with the punch and die surfaces defined using 3-D NURB surfaces. The following BCBODY entries are used to define contact bodies for 3-D shell model. The control node ID 1198 is used in this case to define the load controlled rigid body for the punch. BCBODY BSURF ... BCBODY

... BCBODY

1 1

3D 1

2 0 RIGID NURBS

3D 0. 1198 -19

3 0 RIGID NURBS

3D 0. 0

DEFORM 2 RIGID 0. 1 4 RIGID 0. 5

-7

13

1 3 0. CBODY2 4 0. CBODY3 4

0 4

.1342 5

6

7

0 1.

.1342 0.

1 0.

1198 0.

50

14

1 0.

-1 0.

50

0

4 0 1. 4

50 .1342 0. 50

...

The following BCPARA bulk data entry defines the general contact parameters to be used in the analysis. The ID 0 on the BCPARA option indicates that the parameters specified herein are applied right at the start of the analysis and are maintained through the analysis unless some of these parameters are redefined through the BCTABLE option. Important entries under BCPARA option include: FTYPE – the friction type and the BIAS - the distance tolerance bias. For all the models, the bias factor, BIAS, is set to 0.99. The bilinear Coulomb friction model is activated by setting FTYPE to 6. For the models without friction, FTYPE is set as 0. BCPARA

0 BIAS

.99

FTYPE

6

CHAPTER 3 85 3-D Sheet Metal Forming

The following BCTABLE entries identify how the contact bodies can touch each other. The BCTABLE with ID 0 is used to define the touching conditions at the start of the analysis. This is a mandatory option required in SOL 400 for contact analysis and is flagged in the case control section through the optional BCONTACT = 0 option. Similar BCTABLE options with ID 1, 2 and 3 are used to define the touching conditions for later steps in the analysis, and it is flagged using the option BCONTACT = n (where n is the step number 1, 2 or 3) in the case control section. Two contact pairs are defined in the BCTABLE option: one between the sheet and punch and one between the sheet and die. Both the 2D plane strain and 3-D shell models have similar BCTABLE entries. BCTABLE

0 SLAVE

4 0 FBSH MASTERS 5 SLAVE 4 0 FBSH MASTERS 6

0. 0 1.+20

2 0. 0 .99

0. 0 1.+20

0. 0 .99

.1342

0.

0

0.

0

0. .1342 0.

Material The isotropic elastic and elastic- plastic material properties of the sheet are defined using the following MAT1, MATEP, and TABLES1 options. The Hollomon hardening behavior,  = K n with K = 550.4N  mm 2 ,and n = 0.223 is represented in the form of stress-strain data defined in TABLES1 option. MAT1 MATEP TABLES1

1 1 1 0. .08 .4 .8 1.2

70500. Table 2 194. 313.378 448.681 523.682 573.239

.02 .1 .5 .9 1.3

.342 1

1.

230.043 329.365 471.573 537.619 583.564

.04 .2 .6 1. 1.4

Isotrop Addmean 268.496 384.423 491.14 550.399 593.287

.06 .3 .7 1.1 ENDT

293.904 420.802 508.317 562.224

The following NLMOPTS entry enables large strain formulation using additive plasticity with mean normal return. NLMOPTS,LRGS,1

Loading and Boundary Conditions The following set of boundary conditions has been applied for both 2-D plane strain and 3-D shell models: • Symmetry conditions (i.e., no displacement in horizontal direction) have been applied to the left size of the strip • For the position controlled rigid body used for the die surface, all degrees of freedom have been suppressed. For the control node of the load controlled rigid body used for the punch surface, the displacement components in horizontal directions are suppressed, while the displacement in vertical direction is specified as a function of the time (refer to Table 3-3).

86 MD Demonstration Problems CHAPTER 3

Table 3-3 Vertical Displacement of Punch as a Function of Time Time 0.0 1.0 2.0 3.0

Vertical Displacement 0 -28.5 -28.5 0

The following data in the case control section of the input file defines the load and boundary conditions at the four different steps of the analysis. The bulk data entries SPCD, SPCR and SPC1 are used to define the loads in these steps. The SPCD data presented here shows the application of the imposed downward displacement of 28.5 in vertical direction in steps 1 and 2 at node 1 for the 2-D plane strain model. A similar imposed displacement is applied at node 1198 for the 3-D shell model. The SPCR data presented here shows the application of the imposed upward relative displacement of 10.0 in vertical direction in step 3 and its fixation in step 4 at node 927 for the 2-D plane strain model. A similar imposed relative displacement is applied at node 1167 for the 3-D shell model. SUBCASE 1 STEP 1 NLSTEP = BCONTACT SPC = 2 LOAD = 1 STEP 2 NLSTEP = BCONTACT SPC = 2 LOAD = 2 STEP 3 NLSTEP = BCONTACT SPC = 3 LOAD = 3

1 = 1

2 = 2

3 = 3

$ Loads for Load Case : step-1 SPCADD 2 7 9 SPCD 1 1 2 SPC1 7 1 2 SPC1 9 12 1 $ Loads for Load Case : step-2 SPCD 2 1 2 $ Loads for Load Case : step-3 SPCADD 3 7 8 SPCD 3 1 2 SPCR 3 927 2 SPC1 8 2 927

-28.5 3

4

5

6

7

-28.5 9 -18.5 10.

Solution Procedure The present analysis of metal forming and gradual spring back is carried out in four different steps on both the 2-D plane strain and 3-D shell models. In each of these models, the analysis has been carried out for the cases with and without friction using SOL 400 in MD Nastran. The first step analyses the metal forming process, the second step is used to achieve a more accurate solution before the spring back analysis starts in steps 3 and 4. In the first step, the metal forming operation is simulated by applying a vertical downward displacement of punch. The nonlinear procedure is defined through the following NLSTEP entry with ID 1. Here 100 indicates the total number

CHAPTER 3 87 3-D Sheet Metal Forming

of increments; PFNT represents Pure Full Newton-Raphson Technique wherein the stiffness is reformed at every iteration; 500 is the maximum number of allowed recycles for every increment. UP indicates that convergence will be checked on displacement (U) and residuals (P). The 0.01 defined in the fourth line of NLSTEP indicates the convergence tolerances of 0.01 for displacement and residual checking. The negative sign of displacement tolerance indicates that iteration on displacements will be checked against the incremental displacement quantity instead of total displacement. The second step is considered to be a dummy one in which the load applied in the first step is maintained with very fine convergence tolerances on displacement and residual. This step is used to ensure that the model reaches the good equilibrium condition at the end of step 2 and before starting step 3 involving the more complex spring back operation. It can be seen from the NLSTEP ID 3 that this spring back operation is done over 200 increments with a convergence check only on displacement. NLSTEP

1 1. GENERAL 500 FIXED 100 MECH UP 0 NLSTEP 2 1. GENERAL 500 FIXED 10 MECH UP 0 NLSTEP 3 1. GENERAL 500 FIXED 200 MECH U 0

1 1 -0.01 0

10 0.01

1 10 1 -0.0001 0.0001 0 1 1 -0.01 0

PFNT

-1

PFNT

-1

10 PFNT

-1

To restrict rigid body movement during the springback step-3, a spring with very small stiffness (1e-5) is added at the free end using the following CELAS1 and PELAS cards. CELAS1 PELAS

851 2

2 1.E-5

927

2

Results The characteristic deformed stages from the 2-D plane strain analysis without friction and with friction during the forming step are shown in Figure 3-5. The deformed shapes during the release in various stages are shown in Figure 3-6.

88 MD Demonstration Problems CHAPTER 3

Figure 3-5

Various Deformed Stages during Forming Step

CHAPTER 3 89 3-D Sheet Metal Forming

Figure 3-6

Various Deformed Stages during Spring Back Step

In the analysis without friction, contact is initially present between the sheet and the lower section of the punch. Near the end of the deformation, the sheet separates at the lower section of the punch and gets in contact with the lower section of the die. As soon as this contact is detected, the sheet is further bent into the final shape and the required force in the force displacement history curve increases (Figure 3-5). In the analysis with friction, the deformation behavior is different. The tangential forces due to friction result in a stretching of the sheet causing contact between the punch and the sheet to be present during the complete forming history. The characteristic load displacement curves for the analysis from SOL 400 without friction and with friction are shown in Figure 3-7. The differences in the shape of the curves are caused by the different contact conditions at the end of the forming stage.

90 MD Demonstration Problems CHAPTER 3

2D Plane Strain MD Sol 400 Punch Force (N)

350 300

No Friction

250 200 150

With Friction

100 50 0

0

5

10

-50

15

20

25

30

Punch Displacement (mm)

Figure 3-7

Load Displacement Diagram for 2-D Plane Strain Model

Observe that the unloading stage is analyzed in two steps. In the first unloading step the punch and the strip are moved simultaneously in upward direction. This releases the strip from the die, while it remains in contact with the punch. In the second unloading step the strip is fixed in vertical direction while the punch is moved further upward to its original position. This gradually releases the strip from the punch and allows it to spring back to its final configuration. Note that the fixation of the strip is such that there are no reaction forces after it has lost contact with both the die and the punch. This, of course, is a requirement in order to capture the proper spring back behavior. The fixation primarily serves to suppress rigid body motions of the model during the unloading stage. The characteristic values of the angles at the end of the forming stage and after removal of the tool are listed in Table 3-4. Table 3-4

Characteristic Angles during Forming and Release Process (2-D Plane Strain Model)

Friction Coefficient

Forming Angle

Angle After Release

0

20.42

46.24

0.1348

20.35

54.56

A comparison of the results obtained with Marc and SOL 400 of MD Nastran is shown in Figure 3-8 (no friction) and Figure 3-9 (friction). In the last figure, a comparison is also made with the experimental result. The results from SOL 400 are found to be on the higher side, particularly towards the end of forming. The results exhibit more oscillations in the load displacement curve and this is caused by the use of hard contact approach in Marc and SOL 400. It should be noted that no experimental data points are reported for the unloading.

CHAPTER 3 91 3-D Sheet Metal Forming

2D Plane Strain No Friction Punch Force (N)

300 250

No Friction Marc 200 150 100

No Friction MD SOL 400

50 0

0

5

10

15

20

25

30

Punch Displacement (mm)

Figure 3-8

Load Displacement Curves from Marc and SOL 400 (without friction) 2D Plane Strain With Friction

Punch Force (N)

300

SOL 400 Marc

250 200 150

Experimental

100 50 0

0

5

10

15

20

25

30

Punch Displacement (mm)

Figure 3-9

Load Displacement Curves from Marc and SOL 400 (with friction)

The results of analyses from 3-D shell models have been compared with the plane strain analysis for both the cases with and without friction. The load displacement curves for these two models are shown in Figure 3-10 (no friction) and Figure 3-11 (friction=0.1348). 2D & 3D No Friction Punch Force (N)

300

3D

250 200 150

2D

100 50 0

0

5

10

15

20

25

30

Punch Displacement (mm)

Figure 3-10

Comparison of Plane Strain and Shell Analyses (no friction)

92 MD Demonstration Problems CHAPTER 3

2D & 3D With Friction Punch Force (N)

300 250 200 150

3D

100

2D

50 0

0

5

10

15

20

25

30

Punch Displacement (mm)

Figure 3-11

Comparison of Plane Strain and Shell Analyses (friction = 0.1348)

The resulting values of the characteristic angles are listed in Table 3-5 (no friction) and Table 3-6 (with friction). For the case with friction, the results are compared with experimental predictions from Numisheet 2002. The predictions of SOL 400 from both 2-D plane strain case and 3-D shell models are found to match well with the experiment. Table 3-5

Comparison of Angles for Plane Strain and Shell Approach (no friction) Forming Angle

Angle After Release

Plane strain

20.42

46.24

Shell

20.38

46.67

Table 3-6

Comparison of Angles for Plane Strain and Shell Approach (Friction 0.1348) Forming Angle

Angle After Release

Plane strain

20.35

54.56

Shell

20.45

54.07

19.6 to 21.0

53.4 to 55.8

Numisheet

Modeling Tips One of the complicating characteristics in this benchmark problem is a very local contact between the plate and the curved shoulders of the die. In fact, the contact is almost a point (2-D) or line (3-D) contact with a large amount of sliding. Contact is only verified between the nodes of the plate and the rigid dies. Hence, in the discrete steps of the displacement history, points can be identified where no contact is detected; especially, if large elements are used near the shoulder of the die. The following are some guidelines and tips for modeling this benchmark: • A fine mesh has to be used to describe the contact of the nodes of the sheet with the die properly • A smooth representation of the die has to be chosen, either in an analytical form or by a piecewise linear curve using a high number of segments

CHAPTER 3 93 3-D Sheet Metal Forming

• The unloading behavior is characterized by removal of the tools and at the same time adding boundary conditions preventing the possibility of rigid body movement. • The unloading behavior should preferably be done in a number of steps. Note that in these steps low values of the normal and, consequently, the friction forces are present which makes it difficult to obtain a converged solution • Numerical damping is often recommended to stabilize the solution, but it can be shown that this greatly influences the accuracy of the solution.

Input File(s) File

Description

nug_03a.dat

MD Nastran SOL 400 input for 2-D plane strain model (without friction)

nug_03b.dat

MD Nastran SOL 400 input for 2-D plane strain model (with friction)

nug_03c.dat

MD Nastran SOL 400 input for 3-D shell model (without friction)

nug_03d.dat

MD Nastran SOL 400 input for 3-D shell model (with friction)

Video Click on the image or caption below to view a streaming video of this problem; it lasts approximately 25 minutes and explains how the steps are performed. Original Position

Punch

Sheet Final Position W R2 R3

Figure 3-12

Die

Video of the Above Steps