experimental and numerical investigations of the bending characteristics of laminated steel

5/21/2018 Experimental and Numerical Investigations of the Bending Characteristics of Laminated Steel

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EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF THE

BENDING CHARACTERISTICS OF LAMINATED STEEL

A Dissertation

Submitted to the Graduate School

of the University of Notre Dame

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

by

Travis A. Eisenhour, B.S., M.S.

Edmundo Corona, Director

Graduate Program in Aerospace and Mechanical Engineering

Notre Dame, Indiana

July 2007


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This document is in the public domain.


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EXPERIMENTAL AND NUMERICAL INVESTIGATIONS OF THEBENDING CHARACTERISTICS OF LAMINATED STEEL

Abstract

by

Travis A. Eisenhour

Laminated steel, as studied in this work, is a sheet metal product that consists

of two steel sheets with a relatively thin polymer layer between them. It has

been developed for applications requiring sound and vibration damping. The

objective of the study concerns bending operations, in particular how tooling and

sheet geometry, material properties, and process parameters affect the shape and

integrity of formed parts. Two cases are being considered: wipe bending and draw

bending. Investigation of each case has experimental and numerical components,

the latter using the finite element method.

Because of the particular construction of laminated steel, bending can involve

some issues that are not of concern when bending solid steel sheet. In wipe bending

an undesirable, permanent curl is present upon completion of the operation, while

in draw bending the shape of the part and the state of the polymer layer are of

concern. Therefore, it was important to establish how the geometry and material

state after bending depend on the process parameters. Experimentally, it was

necessary to design and build a set-up for each type of bending that had the

required flexibility to change tooling parameters. Numerically, the problem is

highly nonlinear because it involves large deformations in the steel sheets and


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Travis A. Eisenhour

the polymer layer as well as nonlinear material models for both. In addition, for

draw bending the loading is complicated by factors such as localized deformations

under the draw beads and the sliding motion of the sheet with respect to the

tooling, which introduce loading/unloading cycles. Modeling of the mechanical

response of the polymer layer was particularly challenging, but appropriate models

have been implemented that give reasonable comparisons between experimental

measurements and numerical results. In addition, parametric studies have shown

how the curl and state of the polymer layer depend on various parameters that

are not easily varied in the experiments.

This work presents and explains the mechanics of bending laminated steel. It

is anticipated that the results can be used to improve processing of this material

during bending operations.


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To Rex and Jo Ellen Eisenhour

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CONTENTS

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 11.1 Sound Absorbing Laminated Steel . . . . . . . . . . . . . . . . . . 21.2 Bending of Laminated Steel . . . . . . . . . . . . . . . . . . . . . 51.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

CHAPTER 2: WIPING DIE BENDING . . . . . . . . . . . . . . . . . . . 132.1 Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Finite Element Model . . . . . . . . . . . . . . . . . . . . 262.3.2 Simulation of Experiments . . . . . . . . . . . . . . . . . . 302.3.3 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . 332.3.4 Variation of Thickness and Length . . . . . . . . . . . . . 39

CHAPTER 3: DRAW BENDING . . . . . . . . . . . . . . . . . . . . . . . 463.1 Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.1 Experimental Set-up and Procedure . . . . . . . . . . . . . 473.1.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . 54

3.2 Finite Element Analysis . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.1 Finite Element Model . . . . . . . . . . . . . . . . . . . . 673.2.2 Material Properties . . . . . . . . . . . . . . . . . . . . . . 683.2.3 Simulation of Experiments . . . . . . . . . . . . . . . . . . 813.2.4 Parametric Study . . . . . . . . . . . . . . . . . . . . . . . 96

CHAPTER 4: SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . 101

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APPENDIX A: DETERMINATION OF THE STATIC COEFFICIENT OFFRICTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

APPENDIX B: IMPLEMENTATION OF INTERFACE MODELS USINGABAQUS UINTER USER SUBROUTINE . . . . . . . . . . . . . . . 112

APPENDIX C: IMPLEMENTATION SUMMARIES OF MONOTONICAND CYCLIC INTERFACE MODELS IN ABAQUS . . . . . . . . . . 118C.1 Monotonic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 118C.2 Cyclic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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FIGURES

1.1 Frequency responses for laminated steel and solid steel showing thedamping effect of the polymer layer. . . . . . . . . . . . . . . . . . 3

1.2 Sequences of photos during bending in a wiping configuration. (a)Solid steel test, (b) laminated steel sheet test, and (c) finite elementmodel predictions for laminated steel sheet. . . . . . . . . . . . . 6

1.3 Sketches that illustrate what causes the curl. (a) Each sheet bendsindependently, causing a relative displacement between them. (b)The shear stress in the polymer layer acts on the surfaces of thesheets. (c) The shear stress induces a distributed moment thatresults in each sheet bending. (d) The final shape with curling. . . 8

1.4 Sketches of different types of bending: (a) 3-point bending, (b) 4-point bending, (c) wipe bending, (d) V-bending, and draw bendingwithout (e) and with beads (f). . . . . . . . . . . . . . . . . . . . 9

2.1 (a) Experimental wiping set-up mounted on a testing machine and

(b) schematic of the wiping apparatus. . . . . . . . . . . . . . . . 14

2.2 Definitions of wipe bending geometric parameters. . . . . . . . . . 16

2.3 Photograph of a typical specimen after wipe bending. . . . . . . . 17

2.4 Definition of bending angles. The deformation has been exagger-ated for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Variation of punch and clamping forces during a typical experiment. 19

2.6 Overall bending angles measured experimentally for (a) Rd= 0.01 in.and (b)Rd = 1/8 in. . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Measured curl in specimens bent with die radiiRd = 0.01 and 1/8 in. 222.8 (a) Variation of overall bending angle with time for specimen bent

under identical conditions and (b) corresponding curl variation. . 23

2.9 (a) Variation of overall bending angle with time for specimens bentwith a die radius 1/8 in. and various strokes and (b) correspondingcurl variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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2.10 Measured steel engineering stress-stain curve and points in piece-wise linear fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 Measured force-displacement curves from the lap-shear tests on thepolymer and points in piecewise linear fit. . . . . . . . . . . . . . 27

2.12 (a) Numerical model showing the specimen and contact surfaces,(b) mesh near the bend region, and (c) detail of polymer modelusing springs (the two steel layers are shown artificially separatedfor clarity). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.13 Comparison of experimental results and numerical predictions forRd = 0.01 in. (a) overall bend angle and (b) curl. . . . . . . . . . 32

2.14 Residual shear stress distribution in polymer layer for three bound-ary conditions (a) Both, (b) Top, and (c) Bottom. . . . . . . . 34

2.15 Comparison of experimental results and numerical predictions for

Rd = 1/8 in. (a) overall bend angle and (b) curl. . . . . . . . . . . 352.16 Parametric study results of varying steel sheet yield stress on the

(a) bend angle and (b) curl. . . . . . . . . . . . . . . . . . . . . . 36

2.17 Parametric study results of varying the thickness of the laminateon the (a) bend angle and (b) curl. . . . . . . . . . . . . . . . . . 37

2.18 Parametric study results of varying the polymer layer stiffness onthe (a) bend angle and (b) curl. . . . . . . . . . . . . . . . . . . . 38

2.19 Photograph of two thicker (t = 0.084 in.) specimen after wipebending. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.20 Interface model for shear behavior of the polymer layer. . . . . . . 412.21 Comparison of UINTER models with and without damage for (a)

the bend angle and (b) the curl. (c) Plot of length of broken regionof specimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.22 Plot of the curl for specimen of different lengths. . . . . . . . . . . 45

3.1 Photographs of draw bending fixture, (a) front view and (b) obliqueview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Sectional view of fixture to illustrate internal features. . . . . . . . 49

3.3 Orthographic detail of binder block/die interface. . . . . . . . . . 50

3.4 Definitions of geometric parameters for wipe bending. . . . . . . . 52

3.5 Photograph of complete fixture, including punch guides. . . . . . 53

3.6 Photograph with punch guides in place. . . . . . . . . . . . . . . . 53

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3.7 Measured steel engineering stress-stain curves for two material setsused in the draw bending study. . . . . . . . . . . . . . . . . . . . 56

3.8 Measured shear stress-displacement curves of the polymer layer inlap-shear tests for the two material sets used in the draw bending

study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.9 Photograph of a typical specimen after draw bending and definitionof the bend angles. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.10 Detail of region around the draw bead that illustrates the separationthat occurs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.11 Plot of the punch and bead forces for an experiment withRd = 6mm andhb = 6 mm for Set 2. . . . . . . . . . . . . . . . . . . . . 59

3.12 Plots of the three bend angles for different die radii and bead heightson each side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.13 Plot of the thickness at different points on the bent specimen withthe largest thickness and spread in values. . . . . . . . . . . . . . 62

3.14 Location of the points where thickness measurements were made. 63

3.15 Plot of thickness extrema for all the draw tests. . . . . . . . . . . 64

3.16 Schematic illustrating where a mini-lap specimen was extractedfrom a formed bending specimen. . . . . . . . . . . . . . . . . . . 65

3.17 Plot showing the polymer shear response after forming at locationa compared to straight and curved mini-lap specimens for Set 2. . 66

3.18 Plot showing the polymer shear response after forming at three

locations along a specimen for Rd= 6 mm, hb= 6 mm, and Set 2. 673.19 Illustration of the finite element model with the location of sym-

metry noted and the divisions between element size zones marked. 69

3.20 Measured steel engineering stress-strain curve and bilinear fit forSet 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.21 Measured shear stress-displacement curves of the polymer layer inlap-shear tests and its multi-linear fit. . . . . . . . . . . . . . . . . 72

3.22 Shear stress-displacement curves of the polymer layer in lap-sheartests of 0.006 in. amplitude. Comparison of (a) cyclic response and

(b) load-reverse to failure response to the monotonic response. . . 733.23 Shear stress-displacement curves for lap-shear tests of 0.020 in. am-

plitude showing (a) comparison of cyclic and monotonic responsesplus a cyclic multi-linear fit and (b) the response for two loadingcycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

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3.24 Shear stress-displacement curves of the polymer layer in lap-sheartests of 0.015 in. amplitude, including the load-reverse to failureand monotonic responses plus a cyclic, multi-linear fit. . . . . . . 75

3.25 Interface model for shear behavior of the polymer layer. . . . . . . 76

3.26 Sketch showing key points in the cyclic polymer behavior used inmodeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.27 The failure envelope in shear/normal stress space. . . . . . . . . . 80

3.28 Comparison of the analytical and experimental results of 1, 2,and 3 forhb= 6 mm for Set 2. . . . . . . . . . . . . . . . . . . . 83

3.29 Comparison of the analytical and experimental results of 1, 2,and 3 forhb= 8 mm for Set 2. . . . . . . . . . . . . . . . . . . . 85

3.30 Experimental results that demonstrate the affect of lubrication onthe punch force for Rd = 6 mm andhb= 6 mm for Set 1. . . . . . 86

3.31 Comparison of thickness at different points along specimen for thestandard model and experimental results for Rd = 3 mm, hb = 8mm, and Set 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.32 History of (a)and (b) in the polymer for a point that experi-enced the complete loading history possible. . . . . . . . . . . . . 90

3.33 History of (a) vs. and (b) s and d vs. in the polymer for apoint that experienced the complete loading history possible. . . . 91

3.34 History of (a) n and (b) n in the polymer for a point that expe-rienced the complete loading history possible. . . . . . . . . . . . 92

3.35 History of11 vsp11 in steel for a point that experienced the com-plete loading history possible. . . . . . . . . . . . . . . . . . . . . 93

3.36 History of (a) p11

and (b) 11 in steel for a point that experiencedthe complete loading history possible. . . . . . . . . . . . . . . . . 94

3.37 Final state of polymer forRd = 6 mm and hb = 6 mm. . . . . . . 95

3.38 Parametric study results of varying steel sheet yield stress on (a)1, (b)2, and (c)3. . . . . . . . . . . . . . . . . . . . . . . . . 98

3.39 Final shape for (a) o/o = 0.90 and (b) o/o = 2.70. . . . . . . 99

3.40 Parametric study results of varying steel sheet yield stress on (a)punch and (b) bead forces. . . . . . . . . . . . . . . . . . . . . . . 99

3.41 Parametric study on the effect of varying hb on the angles for asingle die radius Rd = 6 mm. . . . . . . . . . . . . . . . . . . . . 100

A.1 Picture of experiment fixture for friction tests. . . . . . . . . . . . 109

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A.2 Picture of experiment fixture for friction tests. . . . . . . . . . . . 110

A.3 Coefficient of friction for test with clamping force of 300 lb. . . . . 111

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TABLES

3.1 FAILURE LENGTH FOR THEhb = 6 AND 8 mm CASES . . . 96

C.1 INFORMATION ON STATEV(i) . . . . . . . . . . . . . . . . . . 121

C.2 LIST OF PROPS(i) . . . . . . . . . . . . . . . . . . . . . . . . . 124

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ACKNOWLEDGMENTS

I would first like to thank my advisor and mentor, Dr. Edmundo Corona. I

am very grateful for and appreciative of his guidance, enthusiasm, supportiveness,

and friendship. This work would be in shambles without his desire for detail and

constant support. It was nice to see firsthand how he balanced his family, work,and other activities.

I am also extremely fortunate to have great parents, Rex and Jo Ellen Eisen-

hour. Without their love, guidance (slaps to the side of the head), example, and

teaching I would not be the person that I am today. They were the ones who

originally suggested engineering to me, and then they even paid for a lot of my

education. Which is especially nice because typically you have to pay people for

that kind of advice. I am grateful to my grandparents, Bud and Francie Haines,

for the love, humor, and visits.

I appreciate the time and help of my committee members Dr. David Kirkner,

Dr. James Mason, and Dr. Steven Schmid. For the wipe bending work, I would like

to thank Dr. Daniel Boss and Mr. Rich Williams both formerly of MSC Laminates

and Composites. For the draw bending work, I appreciate the contributions made

by Dr. Tom Stoughton, Dr. Siguang Xu, Dr. C.T. Wang, and Mr. Gary Telleck of

General Motors. I would also like to thank Shengjun Yin for pointing me in the

right direction when I was first learning how to use Abaqus.

The contributions and help that were provided by Leon Hluchota, Richard

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Strebinger, and Kevin Peters were instrumental in completing my experimental

work. I would like to thank Nancy Davis, Evelyn Addington, Judy Kenna, and

Nancy OConnor for answering my (sometime random) questions and getting me

the necessary paperwork during my time here.

I also appreciate my fellow graduates and other friends for making my experi-

ence more enjoyable. (If I tried to list them I would probably forget someone and

never hear the end of it.)

The work was carried out with support from the Arthur J. Schmitt Foundation

and the Aerospace and Mechanical Engineering Department at Notre Dame. This

support is gratefully acknowledged.

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CHAPTER 1

INTRODUCTION

Laminated sheet metals come in many varieties and have a myriad of appli-

cations. They can range from simple sheet metals with decorative films used

in office partitions, ceiling panels, marine applications, and home appliances, to

aerospace-grade fiber-reinforced materials being used as fuselage skin in the new

Airbus A380 [1].

While the simplest laminates consist of two material layers, multi-layer lam-

inates are widely used as well. ARALL (Aramid fiber Reinforced ALuminum

Laminate) and GLARE (GLAss fiber REinforced laminate) are two examples of

laminates that have multiple layers and are used in the aerospace industry. Theyboth have fiber reinforced polymer layers between aluminum sheets and combine

good qualities of both metals and fiber composites, yielding favorable properties

such as high strength, lengthened fatigue life, good machinability, acoustic damp-

ing, and impact resistance [2]. Other laminates do not contain fiber reinforce-

ment. Rather, the metal sheets are separated by layers of homogeneous polymeric

materials. These laminates have qualities such as sound damping, reduction of

vibration, thermal insulation, and increased stiffness that make them desirable

for many applications. Laminates that consist of two metal sheets sandwiching

a thick polymer layer (normally 40-60% of total thickness) with low density are

designed mainly to achieve weight savings, when substituted for monolithic sheet

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with comparable stiffness. Similar laminates, but with thin polymer layers (less

than 20% of the total thickness), are designed for sound-damping applications [3].

Laminates designed for weight saving have been found to save 3% of total vehicle

weight while maintaining similar torsional rigidity, bending stiffness, and stress

induced by seat loads management [4]. Laminates with dissimilar metal layers

can also be constructed to achieve properties that cannot be achieved by a single

kind of metal.

1.1 Sound Absorbing Laminated Steel

The laminates of interest in this work have steel sheets and belong to the

sound absorbing category above. Several patents have been filed in relation to

this type of product, for example [57]. Since sound and vibration are undesirable

in many applications, laminated steel has found uses in sheet metal products

that are in environments where sound or vibration must be minimized. The

sound and vibration absorbing characteristics of a laminated steel sample are

demonstrated in Fig. 1.1 [8], which compares the frequency responses of laminated

and solid steel sheets. The smoother response of the laminated sheet, without the

sharp resonance peaks apparent for the solid sheet, illustrates the relative damping

properties of the laminate.

The formulation of the polymer layer in laminated steel is of paramount im-

portance. Different polymer cores are available, such as acrylics and silicones,

that can be tuned to maximize damping under certain operating temperatures

and frequencies. As a part vibrates, resonant modes in the structure create local

regions of bending. These regions of bending cause shearing in the viscoelastic

polymer layer, which dampens the vibration. The shearing occurs due to the low

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Figure 1.1. Frequency responses for laminated steel and solid steelshowing the damping effect of the polymer layer.

shear modulus of the polymer compared to that of the steel, which allows relative

displacement between the two sheets.

Laminated steel is produced by specialized manufacturers such as MSC Lam-

inates and Composites (tradename Quiet Steel), Roush (tradename Dynalam),

and others. In the case of MSC, the manufacturer receives steel coils of the

desired thickness. During the laminating process, steel sheet is simultaneously

drawn from two steel coils. The polymer is continuously applied to one side of the

sheets, which are then pressed together to create the laminate. Subsequently, the

laminated sheet is rolled into a coil and prepared for delivery to their customers.

The availability of laminated steel makes it possible to achieve weight and cost

savings in several applications. For example, since laminated steel sheet absorbs

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sound and vibration it does not have to be further treated with damping materials

(solid steel sheet is usually treated with baked-on damping treatments and fibrous

sound absorption materials). This reduces the number of manufacturing steps

required to make a given part. Another advantage of laminated steel is that

it is easily recyclable because it can be put into a re-melt furnace without any

preparation [8].

Examples of current automotive applications of laminated steel include valve

covers, oil pans, brake shims, firewalls, and body panels. The 1999 Ford Explorer

used the laminate as a structural component, and by 2002 Ford expanded its use

to the Ford Ranger, Mercury Mountaineer, and Lincoln Navigator [9]. Since then,

the material has been used by other car manufacturers such as General Motors

and DaimlerChrysler.

Hard disk drive covers is an application for the laminate that has been de-

veloped in the computer industry. The benefits of using laminated over solid

steel sheet include eliminating 12 steps in manufacturing, reducing overall prod-

uct costs, improving sound level by 2 dB, and saving space [10].Since laminated steel is a relatively new material, its manufacturing and form-

ing characteristics have not been fully understood yet. Issues that have been

previously addressed for solid steel sheet have had to be revisited for laminated

steel. These include, for example, studies of wrinkling in stamping processes [11]

and studies of weldability [12]. Issues related to bending of laminated steel have

also been of interest and are of particular concern in the present work.

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1.2 Bending of Laminated Steel

Many components manufactured using laminated steel to date have been rel-

atively flat, stamped shapes. For this case no significant problems are present.

On the other hand, bending has proven to be a more challenging operation. Key

issues include springback, curl, and delamination. Springback is the recovery of

elastic deformation and occurs with solid sheets as well. On the other hand, curl

occurs when the walls of a part away from the locally bent region, which ideally

should remain straight, become curved. This makes subsequent assembly of sheet

metal products more difficult and reduces the aesthetic appeal of the product.

To illustrate springback and the development of sidewall curl during wipe

bending operations, Figure 1.2 shows sequences of photographs taken during the

bending of (a) a solid steel strip and (b) a laminated steel strip subjected to 90

bends. In these photographs, sheet samples of 4 in. in length, 1 in. in width, and

0.04 in. in thickness are clamped between the bending die and the blank holder,

located on the right. A section 1.5 in. long protrudes from the die and is bent

90 as the punch moves up to a maximum stoke of 0.75 in. (third photograph in

both sequences). The punch subsequently retracts to its original position. Note

that while the solid sheet bends with a straight wall, the laminated sheet curls

substantially during the bending process, and that this curl remains after bending.

Springback is clearly visible for both materials after retracting the punch, but

the curl present in the laminated steel sample gives the appearance of a larger

springback. Sidewall curl has also been experimentally observed in three- andfour-point bend tests [13, 14] and in V-press bending [15, 16].

Qualitatively, the source of wall curl is well understood. It comes from the

relatively weak resistance to shear of the polymer. In the bend region, each sheet

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(a)

(b)

(c)

Figure 1.2. Sequences of photos during bending in a wipingconfiguration. (a) Solid steel test, (b) laminated steel sheet test, and (c)

finite element model predictions for laminated steel sheet.

basically bends independently of the other. Therefore, the sheets have different

bending radii and develop a relative displacement between them as shown in

Fig. 1.3 (a). For simplicity, the relative displacement is taken as zero at the left

end of the sheets in the figure. If the polymer stiffness is negligible, the offset

would also appear at the top end as shown. However, for a finite stiffness of

the polymer this displacement causes a shear stress to develop and act on the

steel sheets as illustrated in Fig. 1.3 (b). The shear stress induces a distributed

momentm (of magnitude t2

for unit width) in the same direction on both sheets,

causing them to bend as illustrated in Fig. 1.3 (c). Thus the laminate exhibits

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curl, and the offset between the two sheets reduces to at the top edge as shown

in Fig. 1.3 (d). At one extreme, a very compliant polymer allows the sheets to

slide past each other with little resistance, the induced moment is small, little curl

occurs, and . On the other hand, a very stiff polymer will make the two

steel sheets tend to bend as one and therefore reduce the curl. Maximum curl will

occur for intermediate polymer layer compliance, as will be demonstrated in Sect.

2.3.3.

Industrial bending operations are usually conducted by wipe bending, V-

bending, or draw bending. In addition, three- and four-point bending are simple

arrangements used in laboratory settings. Figures 1.4 (a) and (b) show three- and

four-point bending, respectively. In wipe bending the blank is clamped between

the die and the blank holder with clamping force Fc. Then the punch moves and

wipes the blank around the die; see Fig. 1.4 (c) (also see Ch. 2 for more details

on wipe bending). In V-bending the punch pushes the blank into the die (both

are V-shaped) as shown in Fig. 1.4 (d). Draw bending, shown in Fig. 1.4 (e),

is very different from the other types of bending because the blank is in tensionduring bending. The tension is generated by friction between the blank, the die,

and the blank holders due to the clamping force Fc. The friction can be regulated

by the clamping force applied to the holders. In some cases, draw beads are added

to the holders to increase the tension, as shown in Fig. 1.4 (f). (See Ch. 3 for

more details on draw bending.) Manufacturing textbooks, such as [17], discuss

the bending operations mentioned here in more detail.

Work on bending of laminated steel has been conducted by several researchers

in the past few years. Combined experimental and finite element work was done in

[14] on the bending characteristics of laminated steel under three-point bending.

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Rd

A A A

B B B

(a)

A A A

B B B

Rd

(b)

(d)(c)

<

inside

sheet

outside

sheet

inside

sheet

outside

sheet

m

Figure 1.3. Sketches that illustrate what causes the curl. (a) Each sheetbends independently, causing a relative displacement between them. (b)

The shear stress in the polymer layer acts on the surfaces of the sheets.(c) The shear stress induces a distributed moment that results in each

sheet bending. (d) The final shape with curling.

In particular, the photographs in Figs. 4 and 7 clearly show the relative motion

between the sheets, and the curl that develops during bending for laminates with

polymer layers of different stiffnesses. Further finite element simulations of the

same set of experiments were continued in [18]. Cantilever beam, three-point

bending, tensile, and shear tests on laminated steel were examined in [19], using

experimental, theoretical, and finite element work. Draw bending without beads

8


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(a)

4

2

3

1

(b)

(c)

1

2

3

4

Die

Blank Holder

Punch

Blank

1

2

3

(d)

2

3 1

5 5

(e)

load load load

2

3 1

5 5

(f)

Fc

Fc Fc

Fc Fc

5 Binder Block

Figure 1.4. Sketches of different types of bending: (a) 3-point bending,(b) 4-point bending, (c) wipe bending, (d) V-bending, and draw bending

without (e) and with beads (f).

was studied in [20] to see the effect of the polymer layers thickness and properties

on springback. These works used solid elements for both the steel sheets and

polymer layer or shell elements with dissimilar material properties through the

thickness to accommodate the polymer layer. Alternatively, if the polymer layer is

thin, it can be modeled using discrete springs or interface models, thus simplifying

9


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the modeling. In both cases, the thickness of the layer is neglected, but the

polymer influence on the mechanical coupling between the two steel sheets is

retained.

Springs are available in many commercially available finite element packages

and can be used to elastically connect the two steel sheets. Each spring connects

a node pair and applies a force on each node, which acts along the line between

the two nodes. The magnitude of the force is found from the relative displacement

between the nodes and the springs load-deflection characteristics. Springs were

first used in vibration analysis of laminates in [21, 22], and later for bending during

forming in [23].

Interface models are generally used to represent the interaction between two

bodies that are in contact. Models that treat the polymer layer as interfaces are

more flexible but require development and programming by the user. They can

be used for a variety of purposes. For example, in [24] an interface model was

used to implement a new tool/blank contact formulation for sheet metal bending

operations. In another example [25], which is somewhat closer to the presentstudy, an interface model was used to predict the initiation and propagation of

damage in a laminated fiber reinforced composite. In the current problem, an

interface model can be used to specify the mechanical interaction between the

two steel sheets including inelastic effects and damage. The last two items are

not usually included in spring elements found in the libraries of commercial finite

element codes.

Both the spring and interface methods are appropriate for the laminated steel

studied here because the polymer layer is very thin (0.001 in. for a 0.042 in. thick

laminate). Depending on the aspects to be modeled and the objectives of the

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study, one may be more appropriate than the other.

1.3 Objectives

The objectives of the work conducted in this project relate to the behavior of

sound and vibration absorbing laminated steel in industrial bending operations.

The operations considered are wipe and draw bending. In the case of wiping,

careful experimental and numerical studies have been conducted to study the

development of curl with the objective of assessing its dependence on bending

parameters such as the die radius, the die-punch clearance, and the length of the

punch stroke. The objective of the numerical effort is to develop and use a finite

element model (using the commercial program ABAQUS) of the laminate that

can replicate the observations made in the experiments, which will be detailed in

Ch. 2. The determination of the material properties of the steel and the polymer

are ancillary experimental projects conducted in parallel to make the model as

realistic as possible. Comparison between the experimental and numerical results

reveals the fitness of the model to replicate the experimental results and hence

model the bending process. Once the predictive capabilities of the model are

established, it can also be used to conduct numerical studies of the response of

the laminate to changes in parameters that are difficult to vary experimentally,

such as the material properties of the steel sheets and the polymer layer. These

numerical experiments can help further understand the behavior of the laminates.

The objective and methods of work are similar for the part of the study con-

cerned with draw bending. As in the previous case one objective is to establish the

final geometry of the specimen and its dependence on parameters similar to the

ones mentioned above, plus the level of tension in the specimen. The tension in

11


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the specimen is controlled with the use of draw beads, as will be discussed in Ch. 3.

The draw beads and the radius of the die introduce a complex bending-unbending

loading history as the material passes through these regions, and that can lead

to delamination. As a result, the state of the polymer layer after completing

the bending operation is also of interest in this case, and will be studied exper-

imentally and numerically. A finite element model of the draw bending process

is also developed with the objective of replicating the experimental observations

and conducting parametric studies of the bending characteristics of the laminate

for different material and geometric parameters.

The combination of experimental and numerical methods used in this study

give a reasonably clear picture of the behavior of laminated steel under two very

different bending operations. Although some of the characteristics of laminated

steel under bending without tension (wiping, V-bending, etc) had been discussed

in the literature prior to the current work, both the experiments and analysis

presented here have been carried out with a greater attention to detail, as was

briefly discussed above. For draw bending, the results presented here appear tobe the first where draw beads and their effect on the steel and polymer response

have been addressed. The results of the investigation are presented in the next

two chapters, which are followed by a summary of the main findings.

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CHAPTER 2

WIPING DIE BENDING

The first problem addressed in this work is wiping die bending, which was

introduced in Figs. 1.2 and 1.4. This bending operation can be used in the man-

ufacture of box-like products. In these cases, starting from a flat sheet, four

segments are bent 90 with respect to the base to form the sides. Sidewall curl,

which arises when bending laminated steel, can significantly complicate assembly

and degrades the appearance of the box, as the sides are not straight. This chapter

addresses the behavior of laminated steel, including sidewall curl, when subjected

to wipe bending.

2.1 Experimental Work

In order to conduct the experiments, a wiping die set-up was designed and

constructed [26, 27]. The set-up is mounted in a 20-kip servo-hydraulic testing

machine as shown in Fig. 2.1 (a). A schematic showing the components of the

set-up is shown in Fig. 2.1 (b). The 90 bending die 2 and the punch guide block

3 are attached to the fixed cross-head of the testing machine through a base plate

1. The blank (specimen being tested) 10 is a rectangular strip clamped between

the die and the blank holder 5 using a screw clamping mechanism consisting

of parts 4 and 6 through 8 . The clamping mechanism contains a load cell

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6 to measure the clamping force. The punch 9 is attached to the actuator of

the testing machine. The specimen is bent by moving the punch up. The control

system of the testing machine allows the punch stroke and speed (up to 3.5 in/sec)

to be prescribed. The set-up was designed to test specimens with widths of up to

2 in. and free lengths (the length protruding from the die) of up to 4 in.

7

6

5

4

3

2

1

8

9

10

1

2

3

4

5

6

7

8

9

Base Plate

Die

Punch Guide Block

Pivot

Blank Holder

Load Cell

Clamping Reaction Plate

Clamping Screw

Punch

Specimen

10

(a) (b)

Figure 2.1. (a) Experimental wiping set-up mounted on a testing

machine and (b) schematic of the wiping apparatus.

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The main geometric parameters of the wipe bending set-up are defined in

Fig. 2.2. The radius of the die and the punch areRd and Rp, respectively. The

specimen has thicknesst and free lengthL. The clearance between the punch and

the die is given by t+g. Other parameters in the experiment not shown in the

figure include the punch stroke S, the punch speed V, and the clamping force Fc.

Three dies with radii of 1/8, 1/16, and 0.01 in. were used in this study. In the

interest of brevity, results for 1/8 and 0.01 in. only will be presented. The punch

radius was kept fixed at 1/8 in. All experiments were conducted using specimens

with thicknesst = 0.042 in. in a quasi-static manner with V= 4 in/min andFc =

300 lb unless otherwise noted. The specimens used in the study had width of 1 in.

In most cases, the free length L was 1.5 in. except when the effect of specimen

length was considered (see Sect. 2.3.4). In all cases, a segment approximately

2.5 in. long resided between the die and the blank holder.

After cutting the specimens to the desired size, measuring them, choosing the

die radius, and fixing the tooling to a prescribed value ofg, the following testing

procedure was conducted:

Step I: Mark the free length on the specimen.

Step II: Install the specimen between the die and the blank holder

and align the mark made above with the edge of the die.

Step III: Increase the clamping force Fc to the desired value using

the clamping screw.

Step IV: Move the punch up so it just touches the specimen.

Step V: Program the stroke length and punch speed information into

the testing machine controller.

Step VI: Start the data acquisition system (which records clamping

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Rp

Rdt

t+g

Die

Punch

Punch

Guide

Blank Holder

L

Figure 2.2. Definitions of wipe bending geometric parameters.

force, punch force, and punch displacement).

Step VII: Start the program to move the punch to its full stroke at

constant velocity and then retract to its initial position.

Step VIII: Stop the data acquisition.

Step IX: Release the clamping force and remove the specimen.

After each experiment, the geometry of the specimen was similar to the one

shown in Fig. 2.3 and was measured using an optical comparator. Two parameters

were defined to characterize the geometry of the specimens after bending. The

first is the overall bending angle, . This angle is defined in Fig. 2.4 as the angle

between the lineBB and the tangent to the specimen at pointA, right at the start

16


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of the bend. When the specimen was released from the blank holder the section

to the left of A, which had been clamped, also developed a small amount of curl.

That is why the tangent at Ais not shown as a horizontal line in the figure. The

second is the curl parameterC. It is defined as

C=b t

b. (2.1)

Herebrepresents the local bending angle measured between the tangent at point

C (just above the bend) and the tangent at A. The bending angle at the tip, t,

is measured between the tangent at point D and the line EE, which is parallel tothe tangent at A. A value C= 0 indicates no curl, and increasing values correlate

with increasing curl.

Figure 2.3. Photograph of a typical specimen after wipe bending.

Figure 2.5 shows the load exerted by the punch Fp and the variation of the

17


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b

t

B

B

A

C

D

EE

Figure 2.4. Definition of bending angles. The deformation has beenexaggerated for clarity.

clamping force Fcas functions of the punch travel pduring a typical experimentwith Rd = 0.01 in. and g = 0 (experimental parameters similar to those in Fig.

1.2 (b)). Both Fp and Fc rise rapidly as the specimen is bent over the die.

Once the specimen is bent,Fpdrops rapidly and levels off at approximately 40 lb,

mostly due to friction at the punch/specimen and the punch/guide contacting

surfaces while Fc remains essentially constant at a slightly lower load. As the

punch is retracted, Fp changes sign but Fc remains at approximately the same

level because it is essentially a reaction holding the springback of the specimen.

Further retraction of the punch allows the specimen to springback, hence both

loads decrease in absolute value. Once the punch retracts to p/L= 0.11 contact

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is lost with the specimen, and Fp goes to zero. A small, residual Fc remains.

This residual value is due to the fact that the part of the specimen under the blank

holder also develops a small amount of curl. Therefore, a small load is required

to keep that part flat. Experiments conducted with different values of die radius

or gap parameter have similar curves in character but different load values. For

example, ifRd = 1/8 in. the peak loads decrease approximately by 50%.

50

0

50

100

150

200

0 0.1 0.2 0.3 0.4 0.5 /Lp

Fp Fc,

(lb)

= 0

L= 1.5 in

t= 0.042 inR = 0.01 ind

S/L= 0.5

g

Fp

Fc

F = 300 lbc

Figure 2.5. Variation of punch and clamping forces during a typicalexperiment.

The first aspect of the problem to be examined is the dependence of the overall

bending angle on the various parameters of the problem. Figure 2.6 shows

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measured values of for die radii Rd = 0.01 in. and 1/8 in. Results for Rd =

0.01 in., a rather sharp bend, are shown in Fig. 2.6 (a) as a function of the punch

stroke S(normalized by L) for three gap parameters g . Ideally, the closer is to

90 the better, but all experiments show lower values because of springback, wall

curl, and in some cases the fact that g was not zero. It is clear that increases

with increasing S, although most of the increase occurs in the range S/L < 0.5.

The maximum bending angles achieved are on the order of 85. These angles are

smaller than those that would be obtained when solid steel specimens of similar

thickness are bent in the same set-up.

Similar patterns can be seen from the results in Fig. 2.6 (b) for the blunter

die withRd = 1/8 in. In this case, however, the values of are uniformly smaller

than in the previous case, with the maximum values being on the order of 80. In

the main, longer strokes and smaller die-punch clearances result in larger bending

angles.

The sidewall curl parameterC, defined in Eq. 2.1, was calculated for the cases

with g = 0 and the two die radii above (in a separate series of tests for Rd =0.01 in.). The results are shown in Fig. 2.7. The measurements indicate that the

curl is larger for the sharper die and that it decreases with increasing stroke length

(with the exception of one point in the series with Rd = 1/8 in.). Yet, the curl

cannot be eliminated by simply extending the punch travel.

In order to study the repeatability of the two bending parameters studied

above, a set of five specimens were consecutively bent under identical conditions:

Rd = 0.01 in. g = 0, and S/L= 0.5. The maximum variations in andCwere

0.25 and 0.01 from the mean, respectively.

Because the polymer layer is visco-elastic, it can be expected that both and

20


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0 0.2 0.4 0.6 0.8 1.065

70

75

80

85

90

00.120.24

L= 1.5 int= 0.042 inR = 0.01 ind

g/t

S/L

(Deg)

0 0.2 0.4 0.6 0.8 1.050

60

70

80

90

00.120.22

L= 1.5 int= 0.042 inR = 1/8 ind

g/t

S/L

(Deg)

(a)

(b)

Figure 2.6. Overall bending angles measured experimentally for (a)Rd= 0.01 in. and (b)Rd = 1/8 in.

C will change with time after the bending operation [28]. The deformations of

several specimens were tracked over time in order to assess the changes in both

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0 0.2 0.4 0.6 0.8 1.00

0.02

0.04

0.06

0.08

0.10

L= 1.5 int= 0.042 ing= 0

S/L

C

0.01

1/8

R (in)d

Figure 2.7. Measured curl in specimens bent with die radii Rd = 0.01and 1/8 in.

and C. Figure 2.8 (a) shows the variation of the overall bend angle over a period

of approximately 200 days for three of the specimens used in the repeatability

study mentioned above. The results show that increased approximately 2.5

over this time period. Changes beyond this value are expected to be negligible

because the curves are relatively flat in the vicinity of 200 days. Similarly, the

curl of the specimens decreased by approximately 0.02 over the same period in

time, as shown in Fig. 2.8 (b). Again, the change in curl had become negligible

by the end of the measurement period.

Fig. 2.9 shows similar measurements for specimens bent using a die with Rd

= 1/8 in. and three strokes. The results show that the absolute changes in bend

angle are approximately the same for the three specimens. The curl change over

time is more pronounced for the shorter punch stroke, but it is not enough to

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= 0

103

102

101

100

101

102

10380

81

82

83

84

85

L= 1.5 int= 0.042 inR = 0.01 ind

t (days)

(Deg)

(a)

103

102

101

100

101

102

103

0

0.02

0.04

0.06

0.08

l

t (days)

C

(b)

S/L= 0.5

g

= 0

L= 1.5 int= 0.042 in

R = 0.01 ind

S/L= 0.5

g

Figure 2.8. (a) Variation of overall bending angle with time for specimenbent under identical conditions and (b) corresponding curl variation.

decrease the curl to the level of the other two experiments.

Since the parameters and Cchange with time, it is important to compare

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103

102

101

100

101

102

103

76

78

80

82

84

L= 1.5 int= 0.042 inR = 1/8 ind

t (days)

(Deg)0.330.51.0

S/L

103

102

101

100

101

102

103

0

0.02

0.04

0.06

0.08

L= 1.5 int= 0.042 inR = 1/8 ind

t (days)

C0.330.51.0

S/L

(a)

(b)

Figure 2.9. (a) Variation of overall bending angle with time forspecimens bent with a die radius 1/8 in. and various strokes and (b)

corresponding curl variation.

their measured values between different cases at a common time. The results

previously shown in Figs. 2.6 and 2.7 were measured immediately after the bending

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operation was completed.

2.2 Material Properties

To better understand the experimental work and to get parameters needed for

the finite element analysis to be presented in Sect. 2.3, the material properties of

the steel sheets and polymer layer had to be determined experimentally. The ma-

terial properties of the steel were obtained from uniaxial tension tests on coupons

cut in the rolling direction of the sheet. The stress was calculated based on the

tensile load, measured with a calibrated load cell, and the strain was measured

directly on the specimen with an axial extensometer. The uniaxial engineering

stress-strain (-) curve is shown in Fig. 2.10. Youngs modulus and the 0.2 %

offset yield stress are given in the figure.

Several lap-shear tests were conducted on appropriately machined laminated

steel specimens to better understand the mechanical behavior of the polymer layer.

The insert in Fig. 2.11 shows a sketch of the specimen. In the actual tests the

dimensions were W = 0.75 in. and b = 1 in. The relative motion between the

sheets was measured directly on the specimen via an axial extensometer. The

displacement rate in the test was 0.03 in/min, which is in the vicinity of shear

rates estimated to occur in the bending experiments. The three force-displacement

curves shown in Fig. 2.11 exhibited some scatter, even for specimens identically

machined from the same mother sheet and tested under identical conditions. The

range of scatter is shown by the dashed lines in the figure. The curves show a

hardening shear response at low strains, followed by a nearly linear region that

terminates in a relatively sharp peak as the polymer fails and the force drops

precipitously.

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40

0 0.05 0.1 0.15 0.2 0.250

(ksi)

Deep draw quality steel

E = 30.0 x 10 ksi

Test

Fit

= 25.8 ksio

3

10

30

20

Figure 2.10. Measured steel engineering stress-stain curve and points inpiecewise linear fit.

2.3 Finite Element Analysis

The second part of the investigation consisted of the development and use of

a finite element model to simulate the experiments and to conduct parametricstudies of the problem.

2.3.1 Finite Element Model

In order to reduce computation time, the problem was reduced to two dimen-

sions by assuming a state of plane strain. This assumption is justified by the large

width-to-thickness ratio of the specimens tested. A few cases were simulated using

plane stress to test the sensitivity of the results to the choice of two-dimensional

model. The differences between the results produced by the two models were very

small.

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0 0.01 0.02 0.03 0.04 0.050

50

100

150

200

250

300

350

P

(lb)

(in)

TestFit

W

b

P

P

Figure 2.11. Measured force-displacement curves from the lap-sheartests on the polymer and points in piecewise linear fit.

A plane strain model with four-noded, bilinear, reduced-integration, quadri-

lateral elements (Abaqus CPE4R [29]) for the steel sheets was considered. Figure

2.12 (a) shows the overall view of the model, with the specimen in the initial,

straight configuration. The die, punch, and blank holder were modeled as rigid

surfaces that made contact with the specimen. The contact conditions between

the specimen, the die, and the blank holder were modeled with Coulomb friction

and a coefficient of friction of 0.17, as measured in an ancillary study presented

in Appendix A. Figure 2.12 (b) shows a close up view of the bend region that re-

veals the mesh used. The sheet in contact with the blank holder had two elements

through the thickness while the one in contact with the die had four. The element

density in the longitudinal direction varied along the specimen. Away from the

bend, the element length was 0.075 in. to the far left and 1/16 in. to the far right.

A region 0.5 in. near the bend had a refined mesh in both sheets, where the length

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of the elements was 0.01 in. Further refinement in the upper sheet at the bend

was required for cases with Rd = 0.01 in. as shown. The optimum element sizes

and the extent of the refined regions were determined by parametric studies of

the effect of the mesh on the predicted value of the overall bend angle . The

studies demonstrated that the value of calculated using the adopted mesh has

converged within 0.5.

The shear response of the polymer layer has been modeled using two methods:

nonlinearly elastic springs (discussed here) and an interface model (discussed in

Sect. 2.3.4). Chronologically, the elastic spring model was developed much earlier

than the interface model. At the time, it was used in the simulation of the exper-

imental results in Sect. 2.1. Interface models have been used in the simulation of

later experiments that will be discussed in Sect. 2.3.4 as well as in the simulation

of draw bending to be presented in Ch. 3.

The springs (Abaqus axial connector [29]) were attached to nodes in the steel

sheets as shown in Fig. 2.12 (c). It is important to attach the springs exactly as

shown to ensure that the most severely deformed springs are stretched. Otherwisecompression larger than the length of the springs may occur, which yields erro-

neous results. The length of the springs was equal to the spacing of the nodes in

the lower sheet in Fig. 2.12 (b). For clarity, the sheets are shown separated in Fig.

2.12 (c). In the model, however, the two steel sheets were constrained to remain

in contact at every point along the specimen (no separation was observed in the

wipe bending experiments presented until now). As it stands, the model cannot

account for the geometric changes as function of time shown in Fig. 2.8. The

numerical results will be compared only to experimental measurements conducted

within minutes after each experiment.

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(a)

(b)

(c)

Punch

Die Specimen

A

a

Blank Holder

Figure 2.12. (a) Numerical model showing the specimen and contactsurfaces, (b) mesh near the bend region, and (c) detail of polymer model

using springs (the two steel layers are shown artificially separated forclarity).

The constitutive model for the steel sheets was J2 flow theory plasticity with

isotropic hardening. The material properties were obtained from the uniaxial

tension tests discussed in Sect. 2.2, using the piecewise linear fit shown in Fig.

2.10. A Poisson ratio of 0.3 was assumed. The load-displacement relation for

the springs modeling the polymer core was obtained from the piecewise fit of the

29


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lap-shear test results shown in Fig. 2.11, as will be discussed next.

Because the polymer layer was modeled by discrete spring elements of different

initial lengths, their force-displacement response had to be related to the data in

Fig. 2.11. For each value ofthe force of the spring,f, is given by

f=Pa

b

WbW

, (2.2)

whereais the initial length of the spring as shown in Fig. 2.12 (c),Wbis the width

of the bending specimens (unit width), W is the width of the lap specimen, and

b is the length of the overlap (see insert in Fig. 2.11). Analysis of the numericalresults indicated that the shear displacements in the polymer layer remained below

the value at the load peak throughout the bending process for the materials and

geometries considered in Sect. 2.1.

2.3.2 Simulation of Experiments

Figure 1.2 (c) shows the predicted specimen shapes that correspond to the

photographs in Fig. 1.2 (b). Qualitatively, the observed and predicted shapes

are in very close agreement. In the following, quantitative comparisons between

experiments and analysis will be carried out.

Figure 2.13 shows results of the analysis, compared with experimental mea-

surements for both the bending angle and curl parameter C as a function of

punch stroke for bending with die radius Rd = 0.01 in. and g = 0. Two experi-

mental series are shown in Fig. 2.13 (a). In general, some spread of experimental

results as shown would result if the set-up was removed and the reinstalled. The

differences between the two series decrease for larger punch stokes. This indi-

cates that in actual production scenarios larger punch strokes may lead to more

30


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consistent geometries after bending. The curl, shown in Fig. 2.13 (b), was only

measured for series 1. Three series of predictions are also shown in the figure.

Each series corresponds to one boundary condition implementation at the end A

of the specimen (shown in Fig. 2.12 (a)). In the series labeled Both, the horizon-

tal displacement of both sheets was restricted at that point. In the series labeled

Bottom only the lower sheet was restricted, while in the series Top only the

upper sheet was restricted. In all three cases, the comparisons between analyt-

ical and experimental results are reasonable once S/L > 0.3 for both and C.

The implementation of the boundary conditions makes the most difference for the

shortest stroke. Here, the curl predicted by the Top series is much closer to the

experimental results, although the prediction for is highest.

Because the curl is a direct result of the shear stress that develops in the

polymer, which is calculated from

= f

Wba, (2.3)

it is interesting to look at its distribution along the length of the specimen at the

conclusion of the bending simulation. Figure 2.14 shows these results (has been

normalized by p, the shear stress corresponding to the peak force shown in Fig.

2.11) for the three boundary conditions considered and S/L = 0.5. The bend is

located atx/L= 0. The Both and Bottom cases have very similar distributions,

with negligible stress in the part of the specimen under the blank holder, relatively

drastic changes near the bend and then a smoothly decreasing shear stress as the

tip of the specimen is approached. The Top case allows the sheets to slide under

the blank holder and therefore displays a slightly larger stress (in absolute value)

in that region. The peak stress near x/L = 0 is smaller than in the other two

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0 0.2 0.4 0.6 0.8 165

70

75

80

85

90

(Deg)

S/L

= 0

L= 1.5 int= 0.042 inR = 0.01 indg

Exp. series 1Exp. series 2BothTopBottom

Analysis

0

0.04

0.08

0.12

0.16

0.2

0 0.2 0.4 0.6 0.8 1

C

S/L

= 0

L= 1.5 int= 0.042 inR = 0.01 indg

ExperimentBothTopBottom

Analysis

(a)

(b)

Figure 2.13. Comparison of experimental results and numericalpredictions forRd = 0.01 in. (a) overall bend angle and (b) curl.

cases, but the stress levels for x/L >0 are about the same in all cases. One would

expect that the Top and Bottom cases would allow the sheets to slide relative

32


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to each other under the blank holder. In the Bottom case, however, the top sheet

tends to get locked at the bend due to the sharp die radius. Therefore, it gives

essentially the same results as the Both case. Only in the Top case, the lower

sheet is free to slide, and the results are different. Although the Top case seems

to be more physically correct, numerical convergence was at times problematic,

so it was not pursued further.

Clearly, high residual shear stress in the polymer can be expected. Given the

viscoelastic characteristics of the polymer, it is not surprising that some of this

stress will be relaxed with time by changes in geometry with the results shown in

Figs. 2.8 and 2.9.

Figure 2.15 shows results similar to those in Fig. 2.13 but for Rd = 1/8 in.

As in the Rd = 0.01 in. case, is reasonably predicted, although the predictions

are somewhat high. The predictions for C are also reasonable. Note that while

the Bottom case yields more favorable predictions for the curl, the bend angle is

slightly better predicted by the Both case. This contrasts with the fact that both

of these cases produced almost identical results for Rd = 0.01. A possible reasonfor this difference is that the blunter die radius does not restrain the motion of

the upper sheet as much, which can then slide more freely over the die.

2.3.3 Parametric Study

The finite element model developed above yielded results that were reason-

ably representative of the experiments. In this section, the same model is used

to perform parametric studies of the dependence of the bend angle and curl on

variations of the yield stress of the steel, the thickness of the steel sheets, and

the stiffness of the polymer layer. The boundary condition Both was used in all

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1.5 1 0.5 0 0.5 10.2

0

0.2

0.4

0.6

0.8

1

(c)

1.5 1 0.5 0 0.5 10.2

0

0.2

0.4

0.6

0.8

1

(b)

1.5 1 0.5 0 0.5 10.2

00.2

0.4

0.6

0.8

1

(a)

S/L

= 0

L= 1.5 int= 0.042 inR = 0.01 indg

= 0.5

Both

Top

Bottom

/ p

x/L

/ p

x/L

/ p

x/L

Figure 2.14. Residual shear stress distribution in polymer layer for threeboundary conditions (a) Both, (b) Top, and (c) Bottom.

cases.

The dependences of the bend angle and curl on variations of the yield stress

of the steel sheets are shown in Fig. 2.16. The values of the geometric parameters

used are shown in the insert. In this study the shape of the steel stress-plastic

strain curve was kept constant, but the stress was raised or lowered at each plastic

strain value by the same amount to match the desired yield stress. The horizontal

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i

0 0.2 0.4 0.6 0.8 150

55

60

65

70

75

80

85

90

S/L

(Deg)

= 0

L= 1.5 in= 0.042 int

R = 1/8 ing

Experiment

BottomAnalysis

Both

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

C

S/L

= 0

L= 1.5 int= 0.042 inR = 1/8 indg

ExperimentBothBottom

Analysis

(a)

(b)

Figure 2.15. Comparison of experimental results and numericalpredictions forRd = 1/8 in. (a) overall bend angle and (b) curl.

axes show the ratio of the yield stress (o) to the yield stress of the curve in Fig.

2.10 (o = 25.8 ksi). The response of the polymer layer was still dictated by the

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curve in Fig. 2.11. The results indicate that increasing the yield stress causes a

decrease in the bend angle and an increase in the curl. Physically, the bending

operation is essentially deformation controlled. Therefore, the strains generated

during the bending phase are independent of the yield stress, but the elastic strain

is larger for materials with higher yield stress. Upon unloading (punch retraction),

the larger amount of recoverable strain leads to lager springback, which results

in smaller values for the local bending angles b and t, and yield smaller and

largerC.

0.5 1 1.5 2 2.580

82

84

86

88

90

(Deg)

o

o/

= 0

L= 1.5 in

t= 0.042 in

R = 0.01 indg

o

S/L= 0.5= 25.8 ksi

0.5 1 1.5 2 2.5

o

o/

= 0

L= 1.5 in

t= 0.042 in

R = 0.01 indg

o

S/L= 0.5= 25.8 ksi

0

0.02

0.04

0.06

0.08

0.1

C

(a) (b)

Figure 2.16. Parametric study results of varying steel sheet yield stresson the (a) bend angle and (b) curl.

Figure 2.17 shows the effect of varying the thickness of the laminate ( t). Both

steel sheets had equal thickness. The results indicate that the bend angle increases

and the curl decreases as the thickness of the laminate increases. These trends are

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the result of two competing effects. On one hand, a thicker sheet results in larger

relative displacements between the sheets at the bend zone. This has the effect of

raising the stress in the polymer layer and therefore increasing the moment on the

sheets, which tends to increase C. On the other hand, thicker sheets are stiffer

and present more resistance against the development of curl. The trends in the

figure represent the combined effects. Geometrically, an increase in Cis generally

accompanied by a reduction in . Raising t/t above one results in shear strains

in the polymer exceeding the failure point. Because the elastic spring polymer

layer model does not account for failure, results are not included for t/t > 1 in

Fig. 2.17. Cases with t/t >1 will be discussed in Sect. 2.3.4, where the interface

model will be used in the predictions.

(a) (b)

80

82

84

86

88

90

(Deg)

t/t

= 0

L= 1.5 int= 0.042 inR = 0.01 indg

S/L= 0.5

0.5 0.625 0.75 0.875 1

t/t

= 0

L= 1.5 int= 0.042 in

R = 0.01 indg

S/L= 0.5

0

0.02

0.04

0.06

0.08

0.1

C

0.5 0.625 0.75 0.875 1

Figure 2.17. Parametric study results of varying the thickness of the

laminate on the (a) bend angle and (b) curl.

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The third parametric study addressed the effect of the stiffness of the polymer

layer. For simplicity, the response of the polymer was set to be strictly linear with

slope Kas shown in the inserts in Fig. 2.18. The horizontal axes present the ratio

of the stiffness to the average slope of the ascending curve in Fig. 2.11 between 50

and 325 lb. The results indicate a high bending angle and zero curl for K = 0.

As the stiffness increases, the bend angle decreases and the curl increases rapidly.

The maximum curl occurs at K/K= 1. Further increase in Kresults in relatively

slow changes in bend angle and decreasing curl as shown. The physical reasons

for the observed variation in curl were explained in Sect. 1.2. Briefly, no curl is

expected when K= 0 or , soCmust increase to a maximum and then decrease

with increasing K.

0 2 4 6 8 1080

82

84

86

88

90

(Deg)

K

= 0

L= 1.5 in

t= 0.042 in

R = 0.01 indg

S/L= 0.5= 23.2 ksi/inK

K/

0 2 4 6 8 10

K K/

0

0.02

0.04

0.06

0.08

0.1

C

= 0

L= 1.5 in

t= 0.042 in

R = 0.01 indg

S/L= 0.5= 23.2 ksi/inK

(a) (b)

K1

K1

Figure 2.18. Parametric study results of varying the polymer layerstiffness on the (a) bend angle and (b) curl.

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2.3.4 Variation of Thickness and Length

The results presented in Fig. 2.17 demonstrated that thickness plays a role

in the final geometry of the specimens. The reasons for this were pointed out inthe associated discussion. The trend presented indicated that as the thickness

increases, gradually increases and C decreases. While generating these results,

care was taken to ensure that the relative displacement between the sheets re-

mained below= 0.026 in., where the load peaks, to avoid failure of the polymer

layer. Although the fit presented in Fig. 2.11 models the sudden drop in strength

at failure, one must remember that the springs are elastic and follow the same

curve upon loading and unloading. This behavior was quite unrealistic since the

failure of the polymer layer has to be irreversible.

A small series of experiments was conducted on specimens with thickness of

0.084 in. The specimens had two identical steel sheets of thickness 0.0414 in.

and a thin polymer layer of essentially the same thickness as the specimens used

previously. The stress-strain curve of the steel had a yield stress 38 ksi, which is

47% higher than that shown in Fig. 2.10. The polymer layer had a very similar

response to that in Fig. 2.11. Figure 2.19 shows a photograph of two specimens

after bending. It is obvious that the curl is negligible in both cases. Note that

one specimen showns delamination, indicating that failure of the polymer layer

occurred.

In order to include irreversible failure in the polymer layer model, a simple

interface model was developed. The shear behavior of the model is modeled using

a spring-dashpot system as shown in Fig. 2.20, where is the stress developed by

the relative motion of the sheets . The stress in the springsobeys the nonlinear

relation shown in Fig. 2.11. It was necessary to add some damping to the model

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Figure 2.19. Photograph of two thicker (t= 0.084 in.) specimen afterwipe bending.

in order to obtain numerical convergence. As a result, a linear dashpot was added

to the model. The stress generated by the dashpot is given by

d= c. (2.4)

Here, is the rate of change ofwith respect to time. Note that here time

is just a parameter proportional to the prescribed loading and does not represent

real time. In all runs conducted, the time length to complete the analysis was

kept the same for consistency. The damping constant c was chosen to be the

smallest value that yielded converged solutions that closely agreed with those

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obtained with the spring model. A value of 200 psi/(in/unit of time). The total

stress in the model is given by the sum of the two contributions

=s+ d (2.5)

Figure 2.20. Interface model for shear behavior of the polymer layer.

The interface model must also address the interaction between the sheets in

the normal direction. To emulate the conditions used in the spring model that

prevent separation and penetration of the contacting surfaces, a simple linearly

elastic behavior given by

n= knn (2.6)

is adopted. Here,n andn are the stress and relative motion of the sheets in the

normal direction. The constantkn was chosen to have a large value 109 psi/in.

The polymer layer behavior in shear shown in Fig. 2.11 exhibits a sharp drop

in load if the relative displacement between the sheets exceeds a critical value.

This is the result of irreversible damage with the result that the load-carrying

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capacity of the layer is almost completely eliminated. In the interface model,

damage is implemented by setting s = 0 for all displacements after it exceeds

the critical value (0.026 in.) once. This model was implemented in Abaqus as a

user subroutine (UNITER [29]) as explained in Appendices B and C.

The results of this thickness study are shown in Fig. 2.21. Two sets of results

are shown for the range 0.5 t/t 2.25, which includes much thicker laminates

than Fig. 2.17. In one, the interface model for the polymer did not account for

damage, hences continues to increase indefinitely with increasing , yielding the

results shown by triangles. In this case the bend angle in Fig. 2.21 (a) showed little

dependence on thickness, and the curl in Fig. 2.21 (b) showed a small decrease,

but remained significant. As expected, these trends are continuations of those

presented in Fig. 2.17. The second interface model, which accounted for damage

of the polymer layer, yielded the results shown by circles. For specimens with

thickness up to thickness ratio of 1.75 the results are nearly coincidental with

those of the model without damage. For thicker specimens, however, the bend

angle suddenly jumps to the 88

level and the curl decreases to nearly zero.The reasons for these abrupt changes in the trends can be explained with the

help of Fig. 2.21 (c), which shows the extent of damage in the specimen at the

conclusion of the bending process. For t/t 1 no damage takes place and the

results for bend angle and curl coincide with those obtained without accounting

for damage. Starting with t/t= 1.25 damage starts appearing for a short length

of the polymer layer near the bend. The damaged length continues to increase

gradually for thicker laminates. Fort/t 1.875 the damage propagates along the

complete length of the polymer between the bend region and the tip. The total

failure of the polymer layer decouples the two sheets and allows them to bend

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0.5 0.75 1 1.25 1.5 1.75 2 2.2580

82

84

86

88

90RL

t S/L

L

L

d

b

0

0.02

0.04

0.06

0.08

0.1

Model with Damage

= 0.01 in

= 0.042 in = 0.5

= 1.5 in

(a)

(b)

(c)

Model without Damage

0

1

(Deg)

t/t

0.5 0.75 1 1.25 1.5 1.75 2 2.25

t/t

0.5 0.75 1 1.25 1.5 1.75 2 2.25

t/t

C

Figure 2.21. Comparison of UINTER models with and without damagefor (a) the bend angle and (b) the curl. (c) Plot of length of broken

region of specimen.

independently, thus eliminating the curl and increasing the bend angle. Since the

polymer has suffered damage, the damping and other properties of the laminate

may have been affected.

Another geometric parameter that has been held constant up until now is the

length of the specimen. Four-point bending experimental results presented in [13]

suggest that curl during wipe bending will increase with the free length of the

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specimen. Figure 2.22 shows how varying the free length (L) affects the curl. In

the figure, L has been normalized by L = 1.5 in. The values of all parameters

kept constant are given in the figure. The figure contains two experimental points

at L/L = 1 and 2, shown by circles, plus four numerical predictions with L/L

between 0.5 and 2 shown by squares. In all cases the section of the specimen

clamped between the die and the blank holder was constant and equal to 2.5 in.

The predictions show good agreement with the experimental points and indicate

a roughly linear increase in curl with specimen length, confirming the suggestions

inferred from four-point bending. The physical reason for the relationship between

curl and specimen length can be appreciated by first noting from Fig. 2.14 that

the shear stress in the polymer is relatively constant in the free length away from

the bend region. Since the shear stress induces a distributed moment and the

internal bending moment has to be zero at the tip of the specimen, the bending

moment in the sheets near the bend region has to increase with specimen length,

thus inducing more curl. The bend anglehas not been shown because it depends

on the length of the specimen even if the curl were to stay constant. Hence it isan appropriate parameter only when comparing cases with equal length.

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0.5 0.75 1 1.25 1.5 1.75 20

0.02

0.04

0.06

0.08

0.1

0.12

C

L L

= 0

L= 1.5 int= 0.042 inR = 0.01 indgExperiment

Analysis S/L = 0.5

Figure 2.22. Plot of the curl for specimen of different lengths.

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CHAPTER 3

DRAW BENDING

The second problem addressed in this work is that of draw bending. This

bending operation was introduced schematically in Figs. 1.4 (e) and (f). Here,

bending takes place in the presence of tension as explained in Sect. 1.2. Deep

drawing operations are commonly used in the automotive and other industries to

manufacture products such as car trunk wells, fuel tanks, kitchen sinks, etc. [17].

Many of these parts have complex, three-dimensional shapes, and their manufac-

turing operations can be affected by wall wrinkling and thinning instabilities that

are influenced by the level of tension in the blank. In addition, use of laminated

steel can influence the final shape of the part compared to solid sheet as discussed

in Ch. 2. Delamination is another issue that can arise when using this material.

The case to be studied here has a very simple geometry, and the objective of

the work is to make an initial assessment of the behavior of laminated steel in

deep drawing operations. As in the previous chapter, experimental results will be

presented first, followed by numerically generated results, comparisons between

experimental measurements and numerical predictions, and parametric studies.

46

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experimental and numerical investigations of the bending characteristics of laminated steel

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