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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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top related