Predicting and Reducing Springback in Bending of an Aluminum Alloy and Selected

Advanced High Strength Steels (AHSS)

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in

the Graduate School of The Ohio State University

Tanmay Gupta

Graduate Program in Mechanical Engineering

The Ohio State University

2019

Master's Examination Committee:

Taylan Altan, Advisor

Jerald Brevick

Copyrighted by

Tanmay Gupta

2019

Abstract

Sheet metal forming or stamping is the process of shaping thin sheets of metal into a complex-shaped part. Part quality, which includes forming the part shape without any defects and within the required tolerances of the desired part geometry, is of major concern in the stamping industry. Major defects in sheet metal forming are splitting, wrinkling, and springback. The demand for improved safety and increased fuel efficiency in the automotive industry has led to an increased use of the Advanced High Strength

Steels (AHSS) and high strength aluminum alloys. However, these high strength materials tend to exhibit low formability and higher springback compared to the conventional mild steels due to thinner gauges and higher strength. Springback significantly affects the dimensional accuracy of stamped parts and accurate prediction of springback in high strength materials becomes a challenge and necessity in the die design stage to reduce the die-recut costs. Springback occurs after stamping operation is completed and part is unloaded from the tools as a result of the elastic recovery of the deformed material. Material model defined in the finite element simulation affects the springback predicted after unloading the tools. Current mathematical models for springback prediction are complex and require material parameters which are costly and tedious to determine. Young’s modulus is one of the most significant parameter affecting the springback prediction. However, the accurate determination of E-modulus is a challenge due to its variation with plastic strain and non-linear elastic unloading behavior

ii in AHSS. This study aims to develop a pragmatic approach for accurate determination of

E-modulus variation with strain using in-house 4-point bending tests combined with the inverse analysis approach using commercial finite element codes (AutoForm and

DEFORM). A similar approach done using wipe bending tests is also reviewed in this study and results from both the methods are compared and applied to real production part. Use of post-stretching to reduce springback is also investigated in the latter half of the study. Post-stretching using a variable blank holder force achieved via a servo hydraulic cushion to reduce springback and residual stresses in aluminum alloy is studied both experimentally and numerically. Finally, post-stretching using concept of “stake beads” is investigated numerically using AutoForm to see the effect of geometrical design of the beads on the forming forces and springback of the part.

iii

Acknowledgments

I would like to thank my advisor, Dr. Taylan Altan, for providing me with continuous encouragement, support and inspiration.

I am also grateful to Dr. Brevick for serving as member of my thesis committee.

I wish to thank the sponsors of the following research: GM, Bowman, Hyson and the other CPF member companies, as well as all of the personnel of those companies which assisted me.

I also thankful to my fellow colleagues at the Center for Precision Forming (CPF) – Ali

Fallahiarezoodar, David Diaz-Infante Hernandez, Advaith Narayanan, Cliff Goertemiller

Berk Aykas, Hitansh Singhal, Esmeray Ustunyagiz, and, Saul Hernandez for their assistance in my work.

I also greatly appreciate Ms. Linda Anastasi for her administrative support.

Lastly, I would like to thank my parents, for their enormous support and love throughout my journey.

Vita

2012...... High School, Delhi Public School,

Faridabad

2017...... B.E. (Hons.) Mechanical Engineering, BITS

Pilani, Goa

2017...... M.Sc. (Hons.) Chemistry, BITS Pilani, Goa

2017 to present ...... Graduate Research Associate, Center for

Precision Forming, The Ohio State University

Publications

1. Tanmay Gupta, Ali Fallahiarezoodar, Ethan Mclaughlin, Dr. Taylan Altan (2018),

R&D Update: Reducing springback in hat-shape bending with variable BHF using

a servo-hydraulic cushion, Stamping Journal March/April 2018, Pg. 16-17

2. Tanmay Gupta, Ali Fallahiarezoodar, Dr. Taylan Altan (2018), R&D Update:

Reducing springback using post-stretching with stake beads, Stamping Journal

May/June 2018, Pg. 20-22

3. Fallahiarezoodar, A., Gupta, T., Goertemiller, C., & Altan, T. (2019). Residual

stresses and springback reduction in U-channel drawing of Al5182-O by using a

servo press and a servo hydraulic cushion. Production Engineering, 1-8.

Fields of Study

Major Field: Mechanical Engineering

Table of Contents

Abstract ...... ii

Acknowledgments...... iv

Vita ...... v

Publications ...... v

Fields of Study ...... vi

Table of Contents ...... vii

List of Tables ...... xii

List of Figures ...... xiii

Chapter 1: Introduction ...... 1

Background ...... 1

Material Properties ...... 2

Bending ...... 4

Springback ...... 6

Chapter 2: Objective and Outline ...... 8

Chapter 3: Review on determination of material properties and formability ...... 9

Introduction ...... 9

Tensile test...... 10

vii

Biaxial Bulge test ...... 14

Combined – tensile and bulge test - method for obtaining accurate flow stress data ... 16

Frictionless dome test ...... 17

Chapter 4: Mechanics and Springback in Bending – A literature review ...... 21

Introduction ...... 21

Fundamentals of bending and springback ...... 22

Mechanical properties affecting springback ...... 25

Effect of Stress-strain data and Constitutive model on springback prediction ...... 26

Isotropic Hardening (IH) ...... 28

Kinematic Hardening (KH) ...... 30

Combined Isotropic and Kinematic Hardening (IH+KH) ...... 31

Effect of E-modulus ...... 33

Chapter 5: Effect of E-modulus Variation on Springback, Inverse Analysis Method and

The Wipe Bending Test – A Review ...... 34

Introduction ...... 34

Effect of E-modulus on springback prediction ...... 35

Nonlinear elastic unloading behavior ...... 37

Variation of elastic unloading modulus with plastic strain ...... 38

Strain path dependency of elastic unloading modulus ...... 39

viii

Inverse analysis method for determination of an apparent E-modulus ...... 39

Review on determination of variation of E-modulus through wipe bending test ...... 40

Experimental Setup...... 41

Calculation of the E-modulus variation through the Inverse analysis method ...... 42

Elastic modulus degradation with plastic strain ...... 43

Chapter 6: Determination of variation of E-modulus through a four-point bending test and application to springback prediction ...... 45

Introduction ...... 45

4-point bending ...... 47

Plane strain semi-pure bending...... 47

Tooling and Experiments...... 49

FE simulations ...... 53

Calculation of variation of E-modulus from inverse analysis ...... 55

E-modulus degradation with effective plastic strain ...... 56

Improvement in springback prediction with variable E-curve obtained using 4-point

bending ...... 59

Springback Prediction in Crash forming of an A-pillar – A case study ...... 60

Conclusions and Future Work ...... 62

Acknowledgements ...... 63

Chapter 7: Reduction of Springback and Residual Stresses in U-Channel Drawing of

Al5182-O by Using a Servo Press and a Servo Hydraulic Cushion ...... 64

Introduction ...... 64

Methods ...... 66

Servo hydraulic cushion ...... 66

Experiments ...... 67

FE simulation ...... 69

Results and discussion ...... 71

Effect of friction and lubrication ...... 71

Effect of material hardening model ...... 73

Inverse analysis and determination of average E-modulus ...... 75

Effect of post stretching method on reduction of springback ...... 76

Conclusions ...... 81

Acknowledgement ...... 83

Chapter 8: Springback Reduction Using Post-Stretching Concept of Stake Beads...... 84

Introduction ...... 84

Use of Stake Bead in hat-bending of Al5182-O/1.2 mm ...... 90

FE simulation ...... 91

Results and Discussion ...... 92

Case- Study: Optimizing Stake Bead Design in Draw-Bending of DP800/2mm ...... 96

FE simulation ...... 97

Results and Discussion ...... 97

Conclusions and Future Work ...... 100

References ...... 101

List of Tables

Table 1 Values of the tool dimensions in four-point bending ...... 50

Table 2 Yoshida-Uemori model parameters obtained using different methods ...... 59

Table 3 Selected Simulation Results (Al5182-O, 1.2 mm) ...... 94

Table 4 Predicted springback and die reaction force for the 2-D hat-shaped forming operation after optimization of stake bead design, *the die reaction force is multiplied by a factor of 7 to account for actual 3-D part length ...... 99

xii

List of Figures

Figure 1 Categorization of different classes of steels by strength and elongation [Billur,

2013] ...... 4

Figure 2 Wipe bending or straight flanging setup [Livatyali et al., 2003]...... 5

Figure 3 Air bending setup [Sever et al., 2012] ...... 5

Figure 4 U-die bending setup [Sever et al., 2012] ...... 6

Figure 5 Schematic of the dog-bone shape specimen for tensile test and a typical engineering stress-strain curve [Fallahiarezoodar, 2018] ...... 13

Figure 6 Schematic of the tooling and process of the viscous pressure bulge test

[Gutscher et al., 2004] ...... 15

Figure 7 Flow stress data obtained from the tensile test and the bulge test

[Fallahiarezoodar, 2018] ...... 16

Figure 8 The flow stress data obtained from the combined tensile test and the bulge test

[Fallahiarezoodar, 2018] ...... 17

Figure 9 Schematic of the frictionless dome test with major dimensions [Grote and

Antonsson, 2009] ...... 18

Figure 10 Sequence of operations in the Frictionless Dome Test ...... 19

Figure 11 Comparison of flow stress curves obtained from Frictionless Dome Test, Bulge

Test and Tensile Test for selected materials [Groseclose et al., 2014] ...... 20

xiii

Figure 12 Schematic of typical engineering stress-strain graph for three materials describing the effect of E-modulus and strength of the material on elastic recovery

(springback) [Fallahiarezoodar, 2018] ...... 22

Figure 13 Stress distribution in bending theory ...... 24

Figure 14 Schematic of stress-strain response of a sheet metal under tension-compression loading [Yoshida, 2002]...... 27

Figure 15 Schematic of yield surface expansion in isotropic hardening model ...... 29

Figure 16 Schematic of yield surface translation in kinematic hardening model ...... 31

Figure 17 Yoshida-Uemori two surface model. 훼 is the center of the yield surface, 훽 is the center of the bounding surface, and 훼 ∗ is the relative motion of the yield surface with respect to the bounding surface...... 32

퐸25 % and 퐸50 % are the calculated E-modulus using the line which connects the point of maximum stress before unloading and the point of 25% and 50 % of that maximum stress value. [c] E-modulus variation with plastic strain; Test results from [Xue et al.,

2016]...... 36

Figure 19 Flowchart describing the inverse analysis method for determining an apparent

E-modulus which can be used to obtain springback values in FE simulations, comparable to experimental measurement ...... 40

Figure 20 A schematic view of tools and dimensions used in the wipe bending tests ..... 42

xiv

Figure 21 Comparison of the E-modulus versus plastic strain calculated from the inverse analysis method and the LUL method ...... 44

Figure 22 Strain distribution in the bent part in wipe bending test ...... 46

Figure 23 Pure bending in a beam with constant applied moment throughout the length 47

Figure 24 Plane strain 4-point bending schematic of a sheet...... 48

Figure 25 Isometric view of the tooling with the sectional view ...... 49

Figure 26 2-D schematic of the four-point bending with tool dimensions ...... 50

Figure 27 Blank geometry used for four-point bending tests – MP980/1.2 mm, dimensions in mm ...... 52

Figure 28 Comparison of bending angle under load obtained from experimental tests and

FE simulations ...... 54

Figure 29 Comparison of the springback angles in experimental four-point bending tests and FE simulations ...... 55

Figure 30 Variation of average effective plastic strain with input punch stroke ...... 56

Figure 31 Variable E vs strain curve obtained by inverse analysis in AutoForm and

DEFORM 2-D ...... 58

Figure 32 Comparison of Variable E vs strain curves fitted to Y-U model, obtained using different tests (MP980/1.2 mm) ...... 58

Figure 33 Improvement of springback prediction in four-point bending using variable E curve ...... 60

Figure 34 Normal distance deviation between the simulation results and 3-D scans of the production formed part ...... 61

Figure 35 A schematic of tools and dimensions for the U-draw bending ...... 68

Figure 36 Three different BHFs used in the tests (Top) constant 100 kN, (middle) constant 400 kN, (bottom) variable 100 to 700 kN ...... 69

Figure 37 Method used in this study for measurement of springback ...... 71

Figure 38 Effect of Coefficient of friction (COF) on draw-in and flange length [400 kN blank holder force, 66 mm drawing depth] ...... 72

Figure 39 Effect of COF on springback prediction. Results are for simulations with 400 kN blank holder force and 70 GPa E-modulus ...... 73

Figure 40 Complex material hardening behavior in tension-compression condition\ ...... 74

Figure 41 Effect of kinematic and isotropic material hardening model on springback prediction ...... 75

Figure 42 Experimental results and simulation predictions of springback for three BHF values used in this study ...... 78

Figure 43 Stress distribution along the sheet thickness at wall area for two different BHF values ...... 79

Figure 44 Stress state of elements which undergo reverse loading condition when they pass the die corner radius and enter the die cavity ...... 80

Figure 45 Residual stress distribution along the sheet thickness ...... 81

Figure 46 Two-stage forming process with stage II showing multiple locking beads for post-stretching [Ayres 1984]...... 85

Figure 47 Schematic showing different stages in hat-bending when stake bead is used .. 86

Figure 48 Different bead geometries with important dimensions ...... 87

xvi

Figure 49 Variation of die force versus forming stroke (simulation), example material –

DP800/ 2mm, 70 mm stroke ...... 89

Figure 50 Tool dimensions used in the draw-bending of Al5182-O/1.2 mm ...... 91

Figure 51 Strategy used for optimization of bead dimensions to minimize springback and tonnage ...... 93

Figure 52 Predicted springback in simulation ...... 93

Figure 53 Comparison of the profiles after springback for post-stretching with Stake bead and variable BHF using Servo Hydraulic Cushion ...... 95

Figure 54 Tool dimensions for the draw-bending process of DP800/2mm ...... 96

Figure 55 Predicted springback in simulation ...... 98

Figure 56 Comparison of the springback profiles with the initial bead geometry and the optimized bead geometry ...... 98

xvii

Chapter 1: Introduction

Background

Sheet metal forming or stamping is an operation of plastically deforming a relatively simple metal blank using tools/dies in a press, into a useful part with a relatively complex geometry. The overall objective is to form the part without any defects and within the required tolerances and functional properties [Altan and Tekkaya, 2012]. Bending, flanging, deep drawing, and stretch forming are examples of sheet metal forming processes. Sheet metal forming process is widely used in the automotive industry to produce structural components in a short time with one stroke of press. There are many variables that affect stamping:

1. Material Properties and formability of the sheet material

2. Die geometry, material and coating

3. Friction and lubrication between the sheet/tool interfaces

4. Mechanics of deformation and forces

5. Characteristics of forming press – speed and motion

Design, analysis, and optimization of the forming processes require (a) fundamental knowledge regarding material behavior in the elastic and plastic region, metal flow, contact conditions at tool/material interface, and heat transfer as well as (b) technological information related to lubrication, heating and cooling techniques, material handling, die design and manufacture, and forming equipment. 1

Material Properties

Over the last decade, the Advanced High Strength Steels (AHSS) and high strength aluminum alloys are increasingly used in automotive industry to satisfy the demands for improved safety, stricter fuel efficiency standards and lower emission standards.

Replacing the conventional steel components with AHSS can help to achieve the earlier mentioned goals by reducing the structural weight of the automotive body. Higher strength of the material makes use of thinner sheets possible while increasing the crashworthiness [Smith, 2011]. In general, the formability of high strength steels and aluminum alloys is lower than milder grade steel. In addition, springback and die wear is more severe in forming high strength materials since the forming stresses and contact pressure is higher [Billur, 2013]. GEN3 materials are the latest class of steels to be developed by the steel industry.

Figure 1 illustrates general information about the strength and formability of sheet materials used in automotive industry. Accurate determination of material properties (i.e., the flow stress data, E-modulus, and uniform elongation etc.) is crucial for designing the sheet forming process. These properties are conventionally determined using a standard tensile test. Despite the tensile test being simple and inexpensive to conduct, it has limitations in the sense that it only provides material properties in uniaxial strain state.

However, in actual stamping operation, the strain state at the part can be nonlinear and multi-axial. In this condition, the true strain at the part may reach to the magnitude that cannot be determined through the tensile test. To obtain the flow stress data for larger

2 strains, biaxial tests such as bulge test and the frictionless dome test are used where the sheet material is under balance biaxial strain state during the deformation.

In addition to mechanical properties, forming limit of the sheet material is another important parameter influencing the design of the forming process. Forming limit is the maximum strain that can be attained in sheet material before onset of necking [Kumar et al., 1994]. Uniform elongation obtained from the tensile test is the simplest indicator of material formability. However, the uniform elongation provides the formability data in uniaxial loading state.

Forming Limit Diagram (FLD), a plot of major and minor limit strains in the principal strain space, is the other indicator of material formability [Werber et al., 2013]. It measures the material resistance to localized necking at different strain states ranging from uniaxial tension to balance biaxial tension. In practice, FLD is a standard tool to determine the feasibility of the forming process.

Figure 1 Categorization of different classes of steels by strength and elongation [Billur, 2013]

Bending

Bending is one of the common operations in the stamping industry, used to produce angled or complex profiled parts. Managing and design for bending operation can be complicated as elastic springback and residual stresses generally follow the process.

Different types of bending operations generally used in the industry are air bending

(Figure 3), wipe bending (Figure 2) , U-die bending (Figure 4), V-die bending, rotary bending etc. From the deformation point of view, bending of the material could be accompanied by drawing of the material and stretching of the material.

Figure 2 Wipe bending or straight flanging setup [Livatyali et al., 2003]

Figure 3 Air bending setup [Sever et al., 2012]

Figure 4 U-die bending setup [Sever et al., 2012]

Springback

Springback which is generally defined as the elastic recovery of material after unloading and tools removal is stress driven and therefore becomes more critical in forming high strength materials. High strength materials experience higher springback than traditional steels under the same forming conditions [AS/P, 2009]. Thus, it is very important to improve the accuracy of springback prediction to reduce the die development time and cost for springback compensation.

Nowadays, numerical simulation of springback prediction through finite element methods is helping the die makers to reduce the extensive trial-and-error method for final die modification. Several parameters affect the prediction of springback. In general, accuracy of springback calculation depends on the accuracy of stress distribution prediction in the part [Gau and Kinzel, 2001; Geng and Wagoner, 2002; Yoshida and Uemori, 2003].

Thus, all the parameters that influence the calculation of stress in finite element method affect the springback prediction.

Among the parameters affecting the springback, flow stress data and E-modulus are two most important material properties influencing the prediction results. Flow stress data provides the effective stress distribution at the part during the deformation and E- modulus controls the material unloading behavior after removal of tools. Also, the selected yield function determines the yield limits at different stress state affect the simulation of material behavior and springback. Hardening rules such as isotropic, kinematic or combined hardening rules also affect the springback prediction.

Additionally, it is shown by several researchers that the assumption of a constant and linear E-modulus was not accurate [Morestin and Boivin, 1996; Cleveland and Ghosh,

2002].

Chapter 2: Objective and Outline

The overall objective of this study is to develop practical methods to predict and reduce springback through Finite Element (FE) simulations and experimental tests of four-point bending, wipe bending and hat-shaped bending.

Chapter 1 gives a brief overview about sheet metal forming, material properties, bending and springback in forming. Chapter 2 provides a brief literature review on determination of material properties of sheets for stamping applications. Chapter 3 discusses the mechanics of bending, with a focus in the plastic region of deformation along with detailed material models affecting springback. Chapter 4 focuses on E-modulus and its effect on springback prediction and brief introduction on the inverse analysis methodology. Chapter 5 introduces a 4-point bending test to evaluate variation of E- modulus with strain. Chapter 6 and 7 are mainly concerned about post-stretching methods to reduce springback in a draw-bending in U-shaped dies using servo hydraulic cushion to provide variable Blank Holder Force and, stake beads to lock the material towards the end of the forming stroke.

Chapter 3: Review on determination of material properties and formability

Introduction

Advanced High Strength Steels (AHSS) are complex materials with carefully selected chemical compositions and multiphase microstructures which are a result of precise control of heating and cooling processes. The 1st and 2nd generations of AHSS grades include Dual Phase (DP), Complex-Phase (CP), Ferritic-Bainitic (FB), Martensitic (MS or MART), Transformation-Induced Plasticity (TRIP), Hot-Formed (HF), and Twinning-

Induced Plasticity (TWIP). The 3rd generation grades of AHSS with improved strength- ductility combinations are still under research and development and gradually are becoming available for wide use in industrial applications.

The unique strength and environmental advantages of aluminum alloys cause the steadily growing interest of this material in automotive industry. Automakers usually use two types of aluminum alloys for body parts as non-heat treatable and heat treatable alloys.

Al-Mn of series 3000 and 5000, are the non-heat treatable alloys and Al-Cu, Al-Cu-Mg,

A-Mg-Si, and Al-Mg-Si- Cu are some of the heat treatable alloys of 2000 and 6000 series used in automotive industry. One of the most popular alloy is 5182-O that have a fine- grained structure and good formability [Fridlyander et al., 2002].

Metal manufacturers evaluate properties, using certain standards. In AHSS, the batch to batch variation of material properties can be significant and it can cause challenges in process development. To overcome this problem, the die makers have to consider a large

9 safety factor and reduce the complexity of the tooling to avoid the failure of the stamped part.

Large variation in material properties and complex material behavior unique to AHSS show an essential need for more accurate tooling and process design. Therefore, it becomes imperative to improve the material characterization methods to have a more advanced material model.

Tensile test

The uniaxial tensile test, an industry standard [ASTM E8, 2016] is the most common and cost effective test method for determination of mechanical properties of a sheet material.

A dog bone shaped specimen, Figure 5, is used where the ends are gripped and the specimen is pulled at a constant rate until fracture. During the test, the load and elongation is measured and several mechanical properties as listed below can be obtained:

1) Engineering stress-strain curve,

2) Young’s modulus

3) Total and uniform elongation

4) Yield and ultimate tensile strength

5) Plastic strain ratio (R-values)

The true stress-true strain data that is used to describe the flow stress data of the material can be calculated as:

퐿 푑푙 퐿 휀푡푟푢푒 = ∫ = ln ( ) = ln (휀푒푛푔 + 1) (2-1) 퐿0 푙 퐿0 10

휎푡푟푢푒 = 휎푒푛푔(휀푒푛푔 + 1) (2-2)

In a tensile test, most of the materials experience a localization of strains which occurs towards the end of the test, known as necking. At this stage, strains and stresses are no longer uniform over the length of measurement or gage length. The elongation or uniaxial directional strain at which necking begins is called as uniform elongation, which is often considered as the simplest formability limit in plasticity. The Considere criterion gives an estimate of the end of uniform elongation and the beginning of necking. As per the criterion, onset of necking and the end of uniform elongation occur when the true work hardening rate exactly equals the true strain. The general form of Considere Criterion is mathematically stated as [Wagoner and Chenot, 1997]:

푑휎 = σ (2-3) 푑휀

The value of stress at the point on the engineering stress-strain curve at which the plastic deformation begins is known as the yield strength. For most of the materials the yield point is not clearly defined. A parallel line to the linear elastic region is drawn at an offset of 0.2% strain and the stress at the point of intersection of this line with the stress-strain curve is the yield strength.

Often, material suppliers do not provide the true or the engineering stress-strain data.

Only the yield stress and ultimate tensile strength of the material is available. Holloman’s power law is a commonly used empirical stress-strain relationship. It is obtained by fitting an exponential curve to the experimental data points of the flow stress curve and is defined by the power law:

휎 = 퐾휀푛 (2-4)

Where K = strength coefficient, n = strain hardening coefficient.

FE simulation input requires the data in form of stress strain curve. Thus, the values of

UTS and Yield Strength (Y) are used to obtain the value of K and n. In this method

[Sever et al., 2011], there are two assumptions:

1. Effect of strain is neglected

2. Strain hardening exponent, n remains constant for all strains

Since both Y and UTS lie in the plastic region of the flow stress curve, the Hollomon power law is applicable to both points as shown in equations below:

푛 휎푈푇푆 = 퐾휀푢 (2-5)

푛 휎푌 = 퐾휀0 (2-6)

Where, ɛu = uniform elongation and ɛ0 = elastic strain at yield point

Using Considere criterion, it can be easily substituted and simplified to show that the uniform elongation (휀푢) equals the strain hardening exponent n.

After simplication and substitution we get,

푛 휎푈푇푆 = 퐾푛 (2-7)

푛 휎푌 = 퐾휀0 (2-8)

The values of K and n can be found using above equations using trial and error.

Figure 5 shows typical stress-strain curve obtained from the tensile test and the mechanical properties for the selected sheet materials. The tensile test is simple and relatively inexpensive. However, in this test, the specimen is under in-plane uniaxial strain state. Therefore, the formability of the material is limited and the material data can only be obtained up to a small strain value compared to the strains observed in industrial

12 stamping operations. In addition, in industrial stamping, the strain state may not be linear uniaxial and can cover different linear or non-linear strain path from pure shear to balanced biaxial. For numerical simulation of a sheet metal forming process extrapolation of the flow stress curve obtained from the tensile test is required to simulate the material behavior in strain values higher than what is obtained from the tensile test. Since the extrapolation is based on mathematical formula, it may not describe the actual hardening behavior of material in high strain values.

Figure 5 Schematic of the dog-bone shape specimen for tensile test and a typical engineering stress-strain curve [Fallahiarezoodar, 2018]

Biaxial Bulge test

Hydraulic Bulge (HB) test and Viscous Pressure Bulge (VPB) test are other test methods for determination of flow stress data. In these tests, a sheet is clamped around its edge and stretched against a circular die using hydraulic fluid or a viscous material as a pressure medium, Figure 6. The sheet material is deformed under balanced biaxial tension until it ruptures. Compared to uniaxial tensile test, higher range of strain can be obtained under biaxial tensile condition [Gutscher et al., 2004; Sigvant et al., 2009].

Gutscher et al., obtained the flow stress data of an aluminum killed drawing quality

(AKDQ) steel up to 0.8 accumulated strain with viscous pressure bulge (VPB) test while this material showed only 0.35 strain in uniaxial tensile test [Gutscher et al., 2004]. The higher strain range obtained from the biaxial test eliminates the need for extrapolating the flow stress data, thereby reducing the approximation in the material behavior for accurate simulation of industrial stamping operation.

Most of the yield functions (Hill 48, Yld2000-2d, or BBC2003) which account for plastic anisotropic behavior of the material require the data from biaxial tests in addition to the uniaxial tensile test [Banabic, 2010].

The VPB test is developed to make the tooling design and equipment setup simple and easy to use compared to the HB test. In the VPB, a semi solid punch made from viscous material such as polyurethane or silicone is used as pressure medium. Friction between the viscous material and blank is negligible. The dome height and the pressure of the viscous material are measured during the test. The pressure versus stroke curve obtained from the experiment is compared with a data base developed from Finite Element (FE) 14 simulation, and the flow stress curve is calculated accordingly. A full description of the viscous pressure bulge test is published by [Gutscher et al., 2004].

Figure 6 Schematic of the tooling and process of the viscous pressure bulge test [Gutscher et al., 2004]

Determination of the yield stress and also the flow stress data at strains close to the yield point are difficult to obtain, Figure 7. A more reliable flow stress curve is obtained by combining the tensile test and bulge test data.

DP980/1.2mm 1400 cannot determine data for strain values less than this point 1300

1200 Bulge test

1100 cannot determine data for 1000 strain values higher than this

900

True stress (MPa) stress True 800 Tensile 700 test

600 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 True strain

Figure 7 Flow stress data obtained from the tensile test and the bulge test [Fallahiarezoodar, 2018]

Combined – tensile and bulge test - method for obtaining accurate flow stress data

In this method both the tensile test and the bulge test results are used to define the hardening behavior of the material. The yield strength of a selected material is determined through the tensile test. Then, a power law equation (σ = Kεn) is fitted to the strain-stress data obtained from the bulge test and the curve is extrapolated from left side until the yield stress is achieved, Figure 8. With this methodology, a reasonably reliable flow stress data can be determined from the yield point up to the maximum strain values obtained from the VPB test.

Al5182-O, 1.2 mm 400

350

300

250 data obtained from the Bulge test 200

Extrapolation of bulge True stress (MPa)stress True 150 test result using power law 100 Yield stress from 50 the Tensile test 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 True strain

Figure 8 The flow stress data obtained from the combined tensile test and the bulge test [Fallahiarezoodar, 2018]

Frictionless dome test

Use of Viscous Pressure Bulge (VPB) Test is not very practical to evaluate flow stress data or conduct formability studies of the sheet material, especially in a small production environment. Frictionless Dome Test is another method to obtain biaxial flow stress data and based on Limiting Dome Height Test. Figure 9 shows a schematic of the test. The friction needs to be eliminated between punch and blank so that the specimen fractures at the apex of the dome. The dome height and deformation force during the process is used 17 to obtain the flow stress data and K, n values (for power law) for the simulation input using a MATLAB code PRODOME, which is developed by CPF [Groseclose et al.,

2014].

Figure 9 Schematic of the frictionless dome test with major dimensions [Grote and Antonsson, 2009]

The design of the lockbead should be such that a locking condition to prevent any flow of the material into the die cavity is achieved and we get a biaxial tension strain path. The first step involves bending of the material to form the bead by crash forming. The second stage involves movement of the punch (dome) to bi-axially stretch the material (with

18 stretch-bending at the die radius) with “perfect lubrication” condition so that we obtain a crack in the material at the punch apex (Figure 10).

Figure 10 Sequence of operations in the Frictionless Dome Test

Care must be taken while designing the lockbead and the die radii such that no fracture in the material is observed in these locations by using the bending and stretch-bending design guidelines (critical R/T ratios) for AHSS [Shih et al., 2018].

Figure 11 shows comparison of the flow stress data obtained using tensile test, VPB test and Frictionless Dome Test for Al5182-O /1.5 mm and TRIP980/1.2 mm.

Figure 11 Comparison of flow stress curves obtained from Frictionless Dome Test, Bulge Test and Tensile Test for selected materials [Groseclose et al., 2014]

Chapter 4: Mechanics and Springback in Bending – A literature review

Introduction

Light-weighting is one of the most popular trend word in the automotive companies, which are pushing to develop lighter vehicles by replacing carbon steels with Advanced

High Strength Steels (AHSS) and aluminum alloys. One of the major challenges of using

AHSS and aluminum alloys is their high tendency to springback, as seen in Figure 12.

Springback is defined as the elastic recovery that a part exhibits upon unloading, and is the result of heterogeneous redistribution of strains and stresses in sheet thickness. The large ultimate strength of AHSS results in a larger bending moment and stress distributions during the cold forming process. Higher stresses result in a larger amount of elastic recovery and consequently a higher springback. In aluminum alloys, the lower elastic modulus compared to low carbon steels is responsible for the large springback.

Most of the joining and assembly processes between components are handled by automatic robotic arms which require the geometrical tolerances to be strict in order to avoid further re-working. The demand to minimize the lead time is high by the automotive companies, which makes springback as one of the major challenges for tool and die designer for press forming of AHSS.

Figure 12 Schematic of typical engineering stress-strain graph for three materials describing the effect of E-modulus and strength of the material on elastic recovery (springback) [Fallahiarezoodar, 2018]

Fundamentals of bending and springback

Bending operation defined as a sheet forming operation to produce angled parts, is widely used in sheet metal forming where parts require simple or complex bent profile. A sheet material is bent by an imposed moment, by stretching over a cylindrical form, or by combination of both moment and tension. In bending of a sheet material, area around the bending radius experiences simultaneous tension-compression loading state. In bending, a nonlinear strain distribution across the sheet thickness is introduced as:

푡푟푢푒 r y ε푋 = ln = ln (1 ± ) (3-1) Rn Rn

22 where Rn is the radius of the neutral axis and y is the distance from the neutral axis.

The amount of the stresses developed on the sheet during the forming process determines whether the area is in pure elastic or elastoplastic strain state Figure 13. Under plane strain condition, using Hooke’s law, the elastic stress component can be calculated as:

E E y σx = 2 εx = 2 (3-2) (1−ν ) (1−ν ) Rn where E is Young’s modulus and ν is Poisson’s ratio. The stress component for plastic deformation depends on the complexity of the material hardening model. For the Swift’s

n law (σ̅ = K(ε0 + ε̅) ), using Hill’s yield index F, the bending stress can be described as:

ε −εe σ = KFn+1[ 0 0 + ε ]n (3-3) x F x

Where:

1+R̅ F = (3-4a) √1+2R̅

ε̅ = Fεx (3-4b)

e R̅ is the normal anisotropy, ε0 is yield strain, K is the hardening coefficient, and n is the hardening exponent.

Figure 13 Stress distribution in bending theory

The integration of the bending stresses through the sheet thickness provides the internal bending moment per width. The bending moment can also be divided into the elastic

(Me) and plastic (Mp) bending moment. This bending moment can be calculated as:

t M = M + M = σ ydy (3-5) e p ∫0 x where

εe Me = ∫ σxydy (3-6) −εe and

−εe εmax Mp = ∫ σxydy + ∫ σxydy (3-7) εmin εe 24

Based on the classic elastic bending theory, the unloading moment can be expressed as:

1 1 M = ( − )E′I (3-8) r r′ where E′ = E⁄(1 − ν2) is the plane strain modulus, I is the second moment of area about

wt3 the middle axis (I = ), r and r′ are the radius of curvature of the sheet metal before 12 and after springback. E′I describes the stiffness of the bent sheet. If it is assumed that the unloading moment has the same magnitude but opposite sign to the applied bending moment, the change in curvature due to springback is:

1 1 M 12(1−ν2) ( − ) = unloading = (M + M ) (3-9) r r′ E′I wt3E e p

Substituting equation 3-2 and 3-3 into the above relations, it can be seen that the springback is a function of material properties, sheet thickness, bend radius, and the stress-strain state in the part. It should be noted that all of the above equations, correspond to specific section of the sheet. Therefore, the total springback is calculated as the summation of all the incremental springback angles of each individual section.

Mechanical properties affecting springback

Regarding the material properties, equation 3-9 shows that the springback is highly dependent on E-modulus and the flow stress data of the material. Generally: a) Springback is more in material with higher yield stress (YS), work hardening, and

anisotropy due to greater resistance to plastic yielding

25 b) Springback is more in materials with lower E-modulus as the resistance to elastic

bending increases with E-modulus.

Effect of Stress-strain data and Constitutive model on springback prediction

During forming process the sheet metal experiences stretch bending, stretch unbending, and reverse bending – also known as reverse loading. The stress-strain response of sheet metal under cyclic tension and compression loading is complex and may not follow a linear isotropic hardening rule [Yoshida, 2002]. Figure 14 represents a schematic of stress-strain behavior of a sheet metal under tension-compression loading. The

Bauschinger effect, transient behavior, work hardening stagnation, and permanent softening, are the four main characteristics indicated in Figure 14. Therefore, the classical hardening laws cannot accurately predict stress and strain magnitudes and distributions, during reverse loading.

Figure 14 Schematic of stress-strain response of a sheet metal under tension-compression

loading [Yoshida, 2002]

There are extensive studies on developing material models that are able to capture the characteristics illustrated in Figure 14. The main focus of these models is the yield function and hardening behavior of the material. In materials with work hardening characteristic, once yielding occurs, the stress needs to be continually increased in order to drive the plastic deformation. In multiaxial loading case, the initial yield surface is usually defined as:

푓(휎푖푗) = 0 (3-10)

However, in materials with strain hardening behavior, the size, the shape, and the position of the yield surface can be changed during the plastic deformation. Therefore, the yield surface can be described by:

푓(휎푖푗, 퐾푖) = 0 (3-11) where, 퐾푖 represents one or more hardening parameters which determine the evolution of the yield surface. Different hardening models are shortly discussed in the next sub- sections.

Isotropic Hardening (IH)

In isotropic hardening model, the shape and position of the yield surface remains unchanged but expands with increasing stress, Figure 15. In this model, the yield function is described as:

푓(휎푖푗, 퐾푖) = 푓0(휎푖푗) − 퐾 = 0 (3-12) where, 푓0(휎푖푗) determines the shape and the size of the initial yield, and the hardening parameter 퐾 controls the expansion of the yield surface.

Figure 15 Schematic of yield surface expansion in isotropic hardening model

Most sheet metals are anisotropic as the result of rolling process used in making the sheets. It means that the properties of the sheet material are dependent to the direction that the property is measured with respect to the rolling direction. Therefore, anisotropic yield surfaces are introduced to more accurately represent the plastic behavior of the material. Hill’s quadratic yield function [Hill, 1948] and Barlat non-quadratic Yld2000-

2d [Barlat et al., 2003] are two conventional anisotropic yield functions used in simulation of sheet metal forming.

Kinematic Hardening (KH)

In the isotropic hardening model the yield surface is symmetric about the stress axes, and remains equal as it develops with plastic strain. Therefore, it implies that the yield strength in tension and compression are the same. However, in reverse loading the behavior of material may not be identical in tension and compression stress state, i.e.

Bauschinger effect and similar responses. Kinematic hardening is mainly of interest in cases where a hardening in tension will lead to a softening in a subsequent compression.

In Kinematic hardening model, the yield surface remains the same shape and size but translates in stress space, Figure 16.

The general form of the yield function in kinematic hardening model can be described as:

푓(휎푖푗, 퐾푖) = 푓0(휎푖푗 − 훼푖푗) = 0 (3-13)

The hardening parameter훼푖푗, known as back-stress, determines the relative motion of the yield surface to the stress space axes. In kinematic hardening model, the material yields earlier in subsequent compression after tension loading, compared to the isotropic hardening model.

Figure 16 Schematic of yield surface translation in kinematic hardening model

Combined Isotropic and Kinematic Hardening (IH+KH)

Neither the isotropic nor the kinematic hardening model can describe accurately the behavior of sheet material in cyclic tension compression loading state. Therefore, more complex hardening rules such as combined isotropic-kinematic hardening (IH+KH) model are introduced [Krieg, 1975; Jiang and Sehitoglu, 1996; Yoshida et al., 2002;

Fredrick and Armstrong, 2007]. The IH+KH model combines features of both the isotropic and kinematic hardening models and the yield function takes the general form:

푓(휎푖푗, 퐾푖) = 푓0(휎푖푗 − 훼푖푗) − 퐾 = 0 (3-14)

Multi surface hardening models are the improved versions of the IH+KH model. In a multi surface hardening model, there are two surfaces: the yield surface and the bounding surface. The bounding surface expands and also slightly moves with increasing plastic strain. The yield surface moves within the bounding surface. The Yoshida-Uemori (Y-U) model is one of the conventional two surface (yield surface and boundary surface) plasticity hardening models [Yoshida et al., 2002]. In this model, the relative kinematic motion of the two surfaces is a function of the difference between their sizes and the yield surface never crosses the bounding surface, Figure 17.

Figure 17 Yoshida-Uemori two surface model. 훼 is the center of the yield surface, 훽 is the center of the bounding surface, and 훼∗ is the relative motion of the yield surface with respect to the bounding surface

[Komgrit et al. (2016)] uses the Y-U material model to predict the springback in U- bending of 980Y high strength steel with additional bending with counter punch to 32 reduce the springback. They investigated the effect of the counter punch force on springback reduction and showed that the springback predictions are more accurate in all testing conditions when using Y-U model compared to pure isotropic hardening model.

[Ogawa and Yoshida (2011)] investigated the effect of the bottoming in reduction of springback in U-shaped bending process of 590MPa high yield stress steel material. They concluded that the Y-U model can appropriately predict the bending moment and springback in FE simulation of bottoming.

Effect of E-modulus

In addition to the flow stress of the material, the E-modulus (Young’s modulus) is the other material property that significantly affects the springback. In fact, unloading elastic modulus that determines the elastic recovery of the material after load removal is responsible for springback. In practice, the unloading elastic modulus is always assumed to be the same as the E-modulus of the material. Researchers have shown that the unloading elastic modulus of material is not constant and is a function of plastic strain.

[Morestin and Boivin (1996)] discussed the effect of work hardening on E-modulus and implementation of accurate E-modulus in elastic-plastic FE simulation of metal forming.

A detailed discussion of the effect of the unloading elastic modulus on springback prediction is presented in the next chapter. Also, the challenges in determination of the unloading elastic modulus are described.

Chapter 5: Effect of E-modulus Variation on Springback, Inverse Analysis Method and

The Wipe Bending Test – A Review

Introduction

The conventional approach is to simulate the forming operation using Finite Element

(FE) discretization, and utilizing the simulation result to modify the initial die design for springback compensation. The die is then re-cut and the final modification of the tool geometry is performed through successive iterations. These additional trials lead to an increase in development cost and time. Thus, accurate springback prediction through FE simulation is vital to the stamping industry.

A thorough review of FE technique and constitutive modeling for springback prediction can be found in [Wagoner et al., 2013]. Material models play a major role in simulating sheet metal stamping and springback. Novel high strength materials such as AHSS exhibit behaviors such as Bauschinger effect, transient behavior, work hardening stagnation, and permanent softening when they are formed under reverse loading condition which follows a strain path into both tension and compression [Yoshida et al.,

2001]. Therefore, the conventional isotropic hardening model may not be an accurate representation of the real phenomena. Most significant parameters affecting the springback prediction are:

1) Flow stress data, relating the stress state of each element in the formed part to its

strain state in plastic region

2) Young’s modulus or E-modulus, which affects the proportion of elastic and plastic

deformation and amount of elastic recovery

3) Hardening rule, which represents complex evolution of the yield function and capture

the accurate strain hardening behavior of material such as the Bauschinger effect,

transient behavior, work hardening stagnation, and permanent softening

Effect of E-modulus on springback prediction

During the unloading process, the elastic recovery of material after plastic deformation is usually assumed to be linear with stiffness equal to a constant value of E-modulus.

Therefore, an accurate E-modulus or unloading elastic modulus is necessary for accurate springback prediction [Morestin and Boivin, 1996]. Figure 18 shows the results of a load- unload-load (LUL) tensile test, one of the conventional methods for determining the unloading elastic modulus. Conventional methods have the following challenges determining the elastic unloading modulus:

1) The elastic unloading behavior of material is nonlinear

2) The elastic unloading modulus is a function of plastic strain (not constant)

3) The elastic unloading modulus is strain path dependent

Figure 18 [a] Example of a loading-unloading tensile test result for determining E- modulus variation with plastic strain for DP780/0.8 mm. [b] Expanded view of the last unloaded cycle indicating the nonlinear elastic unloading behavior of the material. 퐸25 % and 퐸50 % are the calculated E-modulus using the line which connects the point of maximum stress before unloading and the point of 25% and 50 % of that maximum stress value. [c] E-modulus variation with plastic strain; Test results from [Xue et al., 2016].

Nonlinear elastic unloading behavior

Experimental observations show that dislocations pile-up and relaxation cause nonlinear unloading behavior in metals [Sun and Wagoner, 2011; Perez et al., 2005; Xue et al.,

2016]. Nonlinearity of the unloading behavior makes determination of the unloading elastic modulus to be inaccurate. Figure 18 shows an example of nonlinear loading and unloading elastic behavior of DP780 steel. It can be observed that determining the E– modulus is with some approximation and depends on the stress points selected along the unloading curve. The E25 % and E50 % are the calculated E-moduli using the lines which connect the point of maximum stress before start of unloading and the points of 25% and

50% of that maximum stress. Several approaches have been suggested to describe the elastic unloading behavior of the material using the tensile test. Most of the practical approaches propose an average E-modulus obtained using a chord line that connects two points obtained from the start of unloading and the end of unloading [Yoshida et al.,

2002].

[Sun and Wagoner, 2011] have introduced a new strain component named Quasi-Plastic-

Elastic (QPE) strain in addition to the elastic and plastic strains to describe the nonlinearity of elastic unloading. They implemented the QPE to a multi surface yield function approach (combined kinematic and isotropic hardening) for DP980 material and successfully observed improvement in springback prediction using the new model compared to the simulations with standard model or with variable E-modulus.

Variation of elastic unloading modulus with plastic strain

In addition to the nonlinearity of the unloading elastic modulus, the average unloading modulus decreases with increasing the plastic strain [Xue et al., 2016; Kim et al., 2013].

As shown in Figure 18, the reduction of the unloading elastic modulus with plastic strain can be determined through the loading-unloading tensile test using the chord line. For many types of steel, the unloading elastic modulus decreases rapidly with increasing the plastic strain, though this effect tends to saturate at a tensile strain of about 0.2 [Lee at al.,

2012]. As illustrated in the example shown in Figure 18, when unloading at about 0.09 plastic strain, the unloading elastic modulus of the DP780 material is about 157 GPa.

This is about 25% less than the original value (207 GPa) and could potentially produce more springback compared to using the standard E-modulus of 207 GPa.

Although the constitutive model with the QPE strain component shows some potential to describe more accurately the elastic behavior of the material, for practical modeling, it is easier to define the elastic portion of the deformation using a constant E-modulus.

Yoshida et al. (2002) suggested to use an average E-modulus, 퐸푎푣, using a chord line during the unloading process. They introduced the following equation to express the variation of E-modulus with plastic strain:

푃 퐸푎푣 = 퐸0 − (퐸0 − 퐸푎)[1 − exp(−휉휀0 )] (4-1)

Where 퐸0 and 퐸푎 are the E-modulus for virgin and approximately large pre-strained materials, respectively, and 휉 is a material constant.

Strain path dependency of elastic unloading modulus

Xue et al. (2016) investigated the effect of the loading path on elastic unloading behavior of dual-phase steels. They used three test methods: uniaxial loading-unloading test, biaxial loading-unloading test, and three-point bending test. They concluded that the initial and the degradation of the elastic modulus depend on the loading strategy.

Therefore, an E-modulus determined with the tensile test may not be sufficient for accurate springback prediction when the strain state at the part is not uniaxial.

Inverse analysis method for determination of an apparent E-modulus

Considering the important role of the E-modulus on springback prediction and the challenges reviewed in previous section regarding the determination of the E-modulus through the tensile test, it is desirable to develop a simple practical method for obtaining a value of the E-modulus which can provide more accurate springback prediction. In the current study, inverse analysis method, Figure 19, is introduced for determining an average apparent E-modulus for a given material, thickness, and a bending operation.

This E-modulus can provide reliable springback prediction. The assumption is that a constant apparent value of the E-modulus over the entire part can provide accurate springback prediction. This apparent E-modulus represents the average of the actual values of the E-moduli at different locations in the part.

Figure 19 Flowchart describing the inverse analysis method for determining an apparent E-modulus which can be used to obtain springback values in FE simulations, comparable to experimental measurement

Review on determination of variation of E-modulus through wipe bending test

In order to predict springback correctly, accurate behavior of the material needs to be captured. Cyclic tension and compression tests along with Loading-Unloading-Loading

(LUL) tests provide helps to capture the material behavior in tension and compression, reverse loading scenario and variation of E-modulus with strain. However, the tooling for cyclic tension compression test is expensive and complicated due to possible buckling of the sheet in the compressive state. Further, it is difficult to obtain the correct E-modulus at different strains in the LUL test due to non-linear unloading behavior of high strength steels. Thus, as a consequence, there is a need to develop a bending test with strains in tensile and compressive direction, which better emulates the strain evolution during industrial bending operations, in order to establish the variation of E-modulus versus

40 strain. A review on obtaining variation of E-modulus with plastic strain from the wipe bending test is mentioned below.

Experimental Setup

Wipe bending tests were performed using a 5500 series Instron machine. Schematic of the tool geometry and dimensions are shown in Figure 20. A MP980 steel sheet with 1.2 mm thickness was considered and its basic mechanical properties were provided by

General Motors. The flow stress data obtained from the tensile test was fitted by Swift

n law, σ̅ = K(ε0 + ε̅) . The degradation of unloading elastic modulus with plastic strain was initially determined by the uniaxial loading-unloading-loading test. The experimental data was fitted by Y-U model (퐸0= 207 GPa, 퐸푎= 156 GPa, 휉 = 15).

The bending specimens were waterjet cut to 70×100 mm rectangular geometry with two extra flange areas which allows measuring the bending angle under load using a digital protractor. Blank holder force was applied by four M10 screws and it was carefully controlled during the test that the blank holder does not move upward due to reaction force. No lubricant was applied on the specimen surfaces. Specimens were subjected to wipe bending with a punch speed of 10 mm/min. A clearance of 1.85 mm (54% additional to the sheet thickness) was present between the punch and the die.

Due to the nature of the wipe bending operation, during the deformation, a horizontal reaction force is applied to the punch. Therefore, a punch guide was designed to eliminate the elastic deflection of the punch and keep the clearance between the die and the punch constant throughout the deformation. The elastic deflections of the tools were measured using dial indicators and it was confirmed that the elastic deflection of the tools is small 41 enough to be neglected in the computer simulation. Seven different punch displacement strokes i.e. 3 mm, 5 mm, 10 mm, 15 mm, 20 mm, 25 mm, and 30 mm were considered to provide seven different bending angles. The maximum punch stroke considered in the test was 30 mm which provides 90 degree bending angle. Any punch stroke more than 30 mm does not increase the bending angle. After each test, the springback was calculated as the difference between the bending angle under load and angle after unloading. Three tests were repeated for each punch stroke and it was confirmed that the results were reproducible.

Figure 20 A schematic view of tools and dimensions used in the wipe bending tests

Calculation of the E-modulus variation through the Inverse analysis method

The inverse analysis method was used to determine an apparent E-modulus for each punch stroke. To correlate the punch stroke to strain, and determine the variation of E-

42 modulus as a function of strain, average strain in the part at each punch stroke was calculated as:

∑ 휀̅ 휀̅ = 푖 (푖 = 1 푡표 푛) (5-1) 푎푣 푛

th Where 휀푖̅ is the effective strain of the 푖 element and 푛 is the total number of the elements which have strain value more than zero. The average strain in the part rose up until the punch strokes reaches to 20 mm and then by continuing the punch movement, the average strain is reduced.

Elastic modulus degradation with plastic strain

In general, by increasing the punch stroke the apparent E-modulus decreases and reaches a saturation value. This is consistent with the results of the LUL tensile test. The reduction of E-modulus by increasing the punch stroke is due to increase of plastic strain.

The minimum calculated apparent E-modulus, 155 GPa, was considered as the saturation value 퐸푎 in the Y-U model.

The apparent E-modulus calculated for each punch stroke was used to create the variable

E-modulus versus strain curve. Since the average strain at the part starts to decrease after about 20 mm punch stroke, the apparent E-modulus for the strokes more than 20 mm is eliminated from the data. Figure 21 shows the comparison between the Y-U curve obtained from the wipe bending test and the inverse analysis method and the curve obtained from the LUL tensile test. The saturated value of the E-modulus (퐸푎) was about

155 GPa in both methods. However, the reduction rate of the E-modulus with strain (휉) is more abrupt in the model obtained from the inverse analysis method than the model obtained from the LUL tensile test. In order to investigate the improvement in springback 43 prediction using the variable E-modulus, the calculated curves from the inverse analysis method and the LUL method were applied in the simulations and results were compared with experimental measurements.

220 210 Inverse analysis method 200 LUL method 190 180 170

160

modulus (GPa) modulus -

E 150 140 130 120 0 0.05 0.1 0.15 0.2 0.25 Plastic strain

Test method 퐸0 [GPa] 퐸푎 [GPa] 휉

LUL Method 207 156 15

Inverse analysis method 207 155 120

Figure 21 Comparison of the E-modulus versus plastic strain calculated from the inverse analysis method and the LUL method

Results of this study showed a significant improvement in springback prediction in both simple bending operation and bending of a practical part [Fallahiarezoodar, 2018].

Chapter 6: Determination of variation of E-modulus through a four-point bending test and

application to springback prediction

Introduction

Despite the variable E curve obtained using wipe bending test method offered an improvement in the accuracy of springback prediction, there were some limitations related to the wipe bending operation that was used for determining the E-modulus variation data. In the wipe bending, the bending operation is not pure bending, which causes non-uniform strain distribution through the bending area, Figure 22.

Figure 22 Strain distribution in the bent part in wipe bending test

Therefore, to correlate the bending angle and the apparent E-modulus at each bending angle to a specific value of strain, an average of the strain values in the elements present through the bending area were used. This averaging method is found reduce the accuracy of the method.

The objective of this chapter is to investigate a method for calculating the unloading elastic modulus degradation with plastic strain using a pure bending test.

4-point bending

Plane strain semi-pure bending

Pure bending occurs when a constant bending moment is experienced by a workpiece throughout the desired length and shear force is zero (Figure 23). A pure bending condition is only theoretical and practically does not exist because it requires a weightless member. Thus, pure bending in practicality is only an approximation. Pure bending devices require customized tooling, which is not standardized or commonly available.

Figure 23 Pure bending in a beam with constant applied moment throughout the length

Figure 24 Plane strain 4-point bending schematic of a sheet

For small strains, 4-point bending gives a ‘semi-pure bending’ situation, which means that pure bending is achieved only in a part of the complete bent part (Figure 24). 4-point bending provides close to pure bending condition for large strains as the shear forces start to dominate. An easier determination of the E-modulus and effective strain relationship is expected due to constant strains produced throughout the length of the bent workpiece.

As a result, springback calculations are expected to be more accurate.

Tooling and Experiments

The 4-point bending tooling design needed to meet the following constraints:

1. Tools should to be compact enough in size so as to be used in the Instron 5500

series testing machine available (capacity 50kN).

2. The sample should be able to achieve an effective strain of around 0.1 since most

of the E-modulus degradation in AHSS is observed till 0.1 strains.

3. The sample should achieve a bending condition which is close to pure bending i.e.

strains along sections along the length of the bent area should be the same.

Figure 25 Isometric view of the tooling with the sectional view

Figure 26 2-D schematic of the four-point bending with tool dimensions

Specimen Material MP 980 (1.2 mm)

c (Punch-die clearance) 2.5 mm

rd (Die corner radius) 1 mm

rp (Punch Corner radius) 1 mm

t (Blank Thickness) 1.2 mm

Wp (Punch Width) 5 mm

Wd (Distance between dies) 10 mm

Table 1 Values of the tool dimensions in four-point bending

4-point bending tests were performed using a 5500 series Instron machine. Schematic of the tool geometry and dimensions are shown in Figure 25, 26 with dimensions in Table 1.

A MP980 steel sheet with 1.2 mm thickness was considered and its basic mechanical properties were provided by General Motors. The flow stress data obtained from the

n tensile test was fitted by Swift law, σ̅ = K(ε0 + ε̅) . The degradation of unloading elastic modulus with plastic strain was initially determined by the uniaxial loading-unloading- loading test. The experimental data was fitted by Y-U model (퐸0= 207 GPa, 퐸푎= 156

GPa, 휉 = 15).

The bending specimens were waterjet cut to 30×100 mm rectangular geometry with two extra flange areas, which allows measuring the bending angle under load using a digital protractor (Figure 27). Size was chosen so as to fit in the space available as well as satisfy the limitations of the maximum load capacity requirement. No lubricant was applied on the specimen surfaces. Specimens were subjected to bending with a punch speed of 30 mm/min. A clearance of 2.5 mm was present between the punch and the die.

Figure 27 Blank geometry used for four-point bending tests – MP980/1.2 mm, dimensions in mm

Due to high vertical reaction forces on punch, measurement of elastic deflections was a significant issue. The elastic deflections of the tools were measured using dial indicators and these deflections were compensated in the input punch strokes to calculate an effective stroke of the punch. Five different punch displacement strokes i.e. 0.5 mm, 1.5 mm, 2.5 mm, 3.5 mm, and 4.5 mm were considered to provide five different bending angles. Bending angle versus punch stroke data obtained from the experiment is shown in

Figure 28. After each test, the springback was calculated as the difference between the bending angle under load and angle after unloading. Three tests were repeated for each punch stroke for reproducibility of the results.

FE simulations

AutoForm and DEFORM 2-D commercial FEM codes were used in this study for inverse analysis.

In DEFORM, the blank was modeled using plane strain quadrilateral elements with 9 elements through the thickness direction. In AutoForm, the blank was modeled with adaptive triangular shell elements with 11 integration points along the thickness. The tools were modeled using rigid analytical surfaces. The effect of material anisotropy was neglected and the von mises yield criterion was used with isotropic hardening since there is no reverse loading. A constant E-modulus measured from the tensile test was used as the initial value in the simulation. The Coulomb friction law was used with a coefficient of 0.1 for all contacts between tool and blank. The flow stress data of material were defined using the Swift law. A constant speed was input for the punch to move downward and form the part. Simulations were stopped at punch strokes similar to the experiment and springback was predicted at each punch stroke. The predicted bending angle under load at each punch stroke was compared with the experimental measurement to validate the simulation model, Figure 28. Results showed that the simulation model predicted the bending angle underload with less than ±2 degree variation.

180 160 140 120 Experimental angles under load 100 Simulation angles under 80 load (AutoForm) 60 Simulation angles under 40 load (DEFORM 2D)

Angles under Load [degrees] Load under Angles 20 0 0.5 1.5 2.5 3.5 4.5 Input Punch Stroke [mm]

Figure 28 Comparison of bending angle under load obtained from experimental tests and FE simulations

In addition to the simulation model with constant E-modulus, two other simulation models were also conducted with variable E-modulus as a function of plastic strain. One model is based on the data obtained from the loading-unloading-loading tensile test and the other model is based on the data obtained from the inverse analysis method described in next section. Results of springback predicted by these three simulation models are compared with experimental measurements for all seven different punch strokes.

Figure 29 Comparison of the springback angles in experimental four-point bending tests and FE simulations

Calculation of variation of E-modulus from inverse analysis

Figure 29 shows the simulation model under-predicts the springback angles as we increase the punch stroke. Thus, the E-modulus needs to be adjusted in the simulation for each case so as to match the springback angles obtained in the experiments. The adjusted

E-modulus is called as the “apparent E-modulus” in the inverse analysis method. In order to correlate the punch stroke to strain, and determine the variation of E-modulus as a function of strain, average strain in the part at each punch stroke was calculated as:

∑ 휀̅ 휀̅ = 푖 (푖 = 1 − 푛), (5-2) 푎푣 푛

th where 휀푖̅ is the effective strain of the 푖 element/integration point and 푛 is the total number of the elements/integration through the thickness of the bent sheet. Variation of the average effective strain with input punch stroke is shown in the figure 30. Also, for comparison purposes, the apparent E-modulus was also correlated with the max. tensile effective plastic strain.

0.1

0.09

0.08 train S 0.07 0.06 Plastic 0.05 0.04 0.03 0.02

0.01 Average Effective Average 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Input Punch Stroke [mm]

Figure 30 Variation of average effective plastic strain with input punch stroke

E-modulus degradation with effective plastic strain

As previously established, E-modulus decreases with plastic strain and reaches a saturation value, which we observe in this study as well. This is consistent with the results of the traditional Loading Unloading Loading tensile test. The reduction of E-

56 modulus by increasing the punch stroke is due to increase of plastic strain. The minimum calculated apparent E-modulus, 145 GPa, was considered as the saturation value 퐸푎 in the

Y-U model.

The apparent E-modulus calculated for each punch stroke was used to create the variable

E-modulus versus strain curve. Figure 31 shows the comparison between the Y-U curve

(variable E) obtained from the four-point bending test and the inverse analysis method using AutoForm and DEFORM 2-D. Figure 32 shows the comparison of the variable E curve using four-point bending, wipe bending and LUL tests. The variable E curve using four-point bending was calculated using average effective strains through the thickness and using the maximum effective tensile strain. We see a large degradation rate (휉) in case of variable E curve obtained from wipe bending compared to rest of the tests. In order to investigate the improvement in springback prediction using the variable E- modulus, the calculated curves from the inverse analysis method and the LUL method were applied in the simulations and results were compared with experimental measurements.

Figure 31 Variable E vs strain curve obtained by inverse analysis in AutoForm and DEFORM 2-D

Figure 32 Comparison of Variable E vs strain curves fitted to Y-U model, obtained using different tests (MP980/1.2 mm) 58

4-point Bending 4-point Bending Wipe Bending Yoshida equation LUL Test Test Test Test parameters (Avg. Strains) (Max. Strains) (Avg. Strains)

퐸0 207 207 207 207

퐸푎 156 145 145 155

휉 15 17.8 9.3 120

Table 2 Yoshida-Uemori model parameters obtained using different methods

Improvement in springback prediction with variable E-curve obtained using 4-point bending

Springback prediction using the variable E curve obtained from four-point bending test and inverse analysis showed a significant improvement for all the punch strokes when compared to the springback prediction made with a constant E-modulus (figure 33).

12.5 Experimental 11.5 Springback

10.5

9.5 Simulated Springback degrees] 8.5 (AutoForm) - Constant E = 207 7.5 GPa

6.5 Simulated Springback

Springback [ Springback (AutoForm) - 5.5 Variable E - 4pt bending 4.5 0.5 1.5 2.5 3.5 4.5 Input Punch Stroke [mm]

Figure 33 Improvement of springback prediction in four-point bending using variable E curve

Springback Prediction in Crash forming of an A-pillar – A case study

In order to check the validation of the methodology developed, the variable E-curve was used to predict the springback in crash forming of a production part (A-pillar) made out of MP980/1.2 mm. The springback for the A-pillar was predicted using three variable E vs strain models obtained using four-point bending, wipe bending test, and LUL test.

Isotropic hardening (IH) was assumed in the FE simulations since the study by

Fallahirezoodar et al. (2018) showed that the kinematic hardening does not provide any reasonable improvement in the springback prediction. Anisotropy was considered with r-

60 values used in the Hill 48 yield function. The 3-D CAD scan of the formed part was provided by the sponsor and was digitally correlated with the extracted mesh of the simulated formed part after springback. The normal distance deviation of the simulated part after springback for each case was computed with respect to the 3D CAD scan of the actual production part and shown in Figure 34. The largest maximum deviation from experiment is observed for the springback prediction with constant E-modulus. Use of a variable E-modulus with plastic strain shows good improvement over use of constant modulus. Variable E using four-point bending improves the springback prediction in the crash forming of the A-pillar by about 9%. However, wipe bending results showed a much better improvement in the prediction (about 35%).

Figure 34 Normal distance deviation between the simulation results and 3-D scans of the production formed part

Conclusions and Future Work

Accuracy of springback prediction is a major concern in the stamping industry. Four- point bending provides a case of semi-pure bending with constant strain distribution in the bent part with a relatively inexpensive and easy-to-manufacture tooling for in-house testing. The strains are averaged along the thickness of the bent sheet which makes for a better averaging technique than averaging strains in the whole bent part, as is done in the wipe bending study. Wipe bending showed promising improvement (36%) in springback prediction in the A-pillar, but was presented with a limitation of correlating the correct strain with the apparent E-modulus obtained using inverse analysis at each stroke since strains are not uniformly distributed. Compared to using a constant E-modulus, using the calculated variable E-modulus versus effective strain improves the springback prediction for A-pillar when using the four-point bending test (9%). In order to use the variable E- modulus as a simulation input in commercial FE codes like AutoForm, a curve should be fitted to data obtained using the bending test by inverse analysis methodology, as presented in this study.

The four-point bending test however, did not offer improvement in the springback prediction over the wipe bending test. The present work and approach has certain limitations and future scope for improvement:

1. Only 5 punch strokes were tested to obtain the E-modulus vs strain curve which leads

to a curve which poorly fits to the data. Additional data points are required, especially

in the small strain region, to have a better fit of data to the Yoshida-Uemori equation. 62

2. The existing four-point bending tooling, due to its small, compact size leads to

difficulties in precise and accurate measurement of punch stroke with increased

proportion in elastic deflections in the tools and frame of the testing machine.

3. Only a single case study was conducted using the four-point bending and wipe

bending results. Plastic Strain upto 0.05 were obtained in the springback affected

areas of the part (side walls and corners). More materials, thicknesses and production

parts may need to be considered for validation of the inverse analysis approach.

Acknowledgements

This study was supported by General Motors. The authors gratefully appreciate General

Motors Company for providing the material used in this study and characterizing the material properties.

Chapter 7: Reduction of Springback and Residual Stresses in U-Channel Drawing of

Al5182-O by Using a Servo Press and a Servo Hydraulic Cushion

Note: This chapter is accepted for publication to Production Engineering (Springer

Journal).

Introduction

In sheet metal forming, elastic recovery after deformation and tool removal, known as springback, affects dimensional accuracy and geometric tolerances. To develop lighter vehicles, designers are turning toward new materials such as high strength steels and aluminum alloys. The issues caused by springback are more significant in parts made from aluminum alloys and high strength materials. Due to strict geometric tolerances and the need to reduce time to market, accurate prediction and reduction of springback is essential.

During the past decade, springback predictions have been conducted using numerical solutions and Finite Element (FE) methods. The accuracy of the prediction result is affected by several parameters such as material properties, material model, Coefficient of

Friction (COF), numerical procedures, and element type and size. In terms of material properties, flow stress data of sheet material and E-modulus are two most important properties affecting the prediction of springback. Regarding to material model, plastic constitutive equations [Komgrit et al., 2016, Lee et al., 2012] and degradation of

64 unloading elastic modulus [Yu, 2009; Xue et al., 2016] significantly affect the simulation results. The influence of COF on springback is also investigated by several researchers

[Gil et al. 2016; Wang et al., 2016].

In addition, there are some other parameters such as the elastic deflection of tools and presses during the forming process and the inertia and response time of the machine that can affect the springback of the part in an industrial stamping operation [Eggertsen and

Mattisson, 2012]. However, owing to their complexity, it is not practical to consider these parameters in simulation.

Regarding control and reduction of springback, in general, there are two practical ways

[Wagoner et al., 2013]:

a) Obtain the final part within geometric tolerances after compensating for springback

by altering the geometry of tooling b) Applying additional stretch/tension forces to the part during the forming operation

Tooling geometry compensation is based on simulated prediction of springback as well as trial and error. Effects of additional tension on reduction of springback has been shown both experimentally and numerically [Ayres, 1984; Liu et al., 2002]. Sidewall curl springback resulted from drawing process (process where the draw-in of the sheet material into die cavity is controlled by blank holder and material is under stretch force during the deformation) causes assembly difficulties and the post stretching method has

65 been suggested as a solution to this problem. In this method, the sheet material undergoes extra stretching toward the end of forming process. This extra tension reduces the heterogeneous distribution of stresses through the sheet thickness at the wall area and consequently reduces springback and residual stresses.

Post stretching with Servo Hydraulic Cushion (SHC) was discussed in our previous publication [Gupta et al., 2018]. In the present study, a more detail results of the use of this technology for reducing of springback and residual stresses in a U-channel part are presented experimentally and numerically. The effect of material work hardening and

COF on springback prediction is also studied.

Methods

Servo hydraulic cushion

Generally, in a deep drawing process, the Blank Holder Force (BHF) is generated by pneumatic cushions, nitrogen cylinders, or a hydraulic cushion. Hydraulic cushions allow for computer numerical control of the BHF by regulating the oil flow to the cushion cylinders. The main advantage of the SHC compared to other types of cushions is that the

SHC allows accurate control of the BHF during the deformation.

Experiments

U-channel drawing of 1.2 mm Al5182-O was conducted using a 300 ton servo press to investigate the possibility of reduction of springback using the post stretching method.

The variable blank holder force was applied with a 100 ton SHC. Information about the geometry and dimensions of the tooling is presented in Figure 35. The blank was rectangular with the dimensions of 720 mm × 120 mm and it was drawn to a target stroke position of 67 mm. However, due to the elastic deflection of the system, the actual drawing stroke was slightly different than the target value especially when the blank holder force was increased. During the test, the blank holder and the die displacement were recorded by using the servo press and cushion built-in sensors. This information was used to determine the exact final stroke and elastic deflection of the system for each

BHF.

Two constant blank holder forces (100 kN and 400 kN) and one variable with stroke (100 kN to 700 kN) were used, Figure 36. As shown in this figure, there is a slight difference between the target and actual blank holder force for each case. The forming speed, cushion response time, and inertia of the system influence the response of the system and lead the difference between the input and the output results.

The values of the BHFs were selected through an iterative process using simulation predictions to avoid excessive thinning and failure on the part. For the case of variable

BHF, deformation starts with constant 100 kN BHF and then at stroke of 60 mm the BHF starts to raise up and reach to 700 kN at stroke of 65 mm. The variable BHF is selected to

67 provide the post stretching condition for reduction of springback. Blanks were coated by a dry lubricant. Three replications were considered to ensure repeatability of the results.

Figure 35 A schematic of tools and dimensions for the U-draw bending

Figure 36 Three different BHFs used in the tests (Top) constant 100 kN, (middle) constant 400 kN, (bottom) variable 100 to 700 kN

FE simulation

3-D FE simulation of the U-channel drawing process was developed using the AutoForm software package. The blank was created using shell elements with 11 integration points 69 and tools were modelled using rigid analytical surfaces. The flow stress of the material was determined from a tensile test and it was extrapolated using the power low (휎̅ =

퐾휀푛̅ ). Also, the yield surface was defined by Barlat 89 Yield function. An initial E- modulus of 70 GPa was used in simulations. However, more accurate average E-modulus was calculated for each blank holder force case, using inverse analysis method described in the next section . Effect of friction and lubrication on springback was investigated through numerical simulation and results are discussed in section 3.1. Friction coefficients for each blank holder force case were determined by comparing the predicted flange length at each case with experimental measurements.

Due to the inverse loading condition happens during the hat-shape drawing operation, effect of the material work hardening model on prediction of material behavior in this loading condition is investigated by conducting simulations with isotropic hardening material model and combined isotropic-kinematic hardening model. Results are discussed in the next section.

For each testing condition, the simulation was stopped at the exact forming stroke similar to what is measured through the experiment and springback was predicted.

Springback angle 휃 is defined as the angle between the position of the flange before and after unloading (Figure 37).

Figure 37 Method used in this study for measurement of springback

Results and discussion

Effect of friction and lubrication

Coulomb friction law is considered for contacts between blank and tool surfaces. In a draw forming process, blank holder force and COF determine the amount of the stretch force that controls the material flow into the die cavity. Therefore, COF can significantly affect the forming result and springback. In the post-stretching method, a low value of

COF can reduce the effect of the additional stretch force and consequently minimize springback reduction. On the other hand, high COF can cause excessive stretch at the wall and failure.

In this study, the COF for each testing condition was determined by comparing the predicted and measured flange length of the part. Simulations with different values of

COF were conducted and the value of the COF that predicted a similar flange length to 71 the measured value was considered as the COF for that test condition (Figure 38). Using this methodology, COF of 0.04 was predicted for experiments with 400 kN BHF and

COF of 0.05 was predicted for the test with 100 kN and variable BHF. This is consistent with the results reported by Wang et al. (2016) showing a reduction of COF by increasing the contact pressure.

Flange length (mm) length Flange 40

30 COF=0.02 COF=0.04 COF=0.05 COF=0.07 Experiment

Figure 38 Effect of Coefficient of friction (COF) on draw-in and flange length [400 kN blank holder force, 66 mm drawing depth]

Effect of COF on springback was investigated by comparing prediction results with different values of COF (Figure 39). Figure 39 illustrates that with a constant blank holder force, springback decreases by increasing the COF. This is because increasing the

COF leads to increase of tension force around the die corner radius and flange area where the material is in contact with the tool surfaces.

Figure 39 Effect of COF on springback prediction. Results are for simulations with 400 kN blank holder force and 70 GPa E-modulus

Effect of material hardening model

Accurate numerical simulation of springback is critical since it reduces the time and cost of tool design and re-modification for springback compensation. There are many diverse efforts made for improving the accuracy of simulation results. Several studies investigated the effect of the work hardening and material constitutive model on springback prediction. In a drawing process, the sheet material initially is bent over the die corner radius and then unbent after it passes the corners and enters the die cavity. This bending-unbending process causes reverse loading conditions. It is known that some materials show the Bauschinger effect, permanent softening, and transient behaviour in 73 reverse loading condition. Kinematic hardening models are developed to simulate this complex material hardening behavior when the sheet material undergoes reverse loading condition [Wagoner et al., 2013].

In the current study, kinematic hardening model was used to simulate the effect of transient softening and work hardening stagnation. Two additional parameters known as transient softening rate (K) and stagnation ratio (ξ) were added to the material model. The

AutoForm default values for 5000 series aluminum were used (k=0.005 and ξ=0.28).

Using these parameters, the material hardening behavior in reverse loading condition is described in Figure 40.

450

400

350

300

250 Work hardening in reverse loading (Isotropic+Kinematic 200 hardening model)

True stress (MPa)stress True 150 Work hardening in reverse 100 loading (Isotropic hardening model) 50

0 0 0.2 0.4 0.6 0.8 1 1.2 True strain

Figure 40 Complex material hardening behavior in tension-compression condition\

To investigate the effect of the material hardening model on springback prediction, simulations with isotropic and combined isotropic-kinematic material models were conducted. Figure 41 illustrates the effect of the kinematic hardening model on springback prediction. As shown in this figure, simulations with combined isotropic- kinematic material model predicts more side wall curl and springback compared to simulation results with only isotropic hardening model.

Figure 41 Effect of kinematic and isotropic material hardening model on springback prediction

Inverse analysis and determination of average E-modulus

It is known that determination of E-modulus is challenging as the elastic deformation of material is not perfectly linear [Li et al., 2013]. Degradation of elastic modulus by plastic strain also increases the complexity of determination of E-modulus. In this study, an

75 inverse analysis method is used to determine an average E-modulus called apparent E- modulus (퐸푎푣푔). In this method, for each test conditions, E-modulus in simulations was adjusted until the predicted springback matched the experimental measurements. The E- modulus value which provides most accurate springback prediction at each forming condition was considered the apparent E-modulus for that condition. Apparent E-moduli of 87 GPa and 80 GPa, were calculated for the case of 100 kN and 400 kN BHF, respectively. For the case of variable BHF, apparent E-modulus of 75 GPa was calculated. During the inverse analysis, simulations were conducted using the COF obtained from the flange length comparison method described in next sections. Also, combined kinematic-isotropic material model was considered in simulations.

Effect of post stretching method on reduction of springback

Figure 42 shows the profiles of the parts obtained from the experiments and comparison with the simulation results obtained with average E-modulus. As shown in this figure, there is a large side wall curl in the part for both constant 100 kN and 400 kN BHFs. The side wall curl is significantly reduced with the variable BHF (post stretching method).

The reduction of springback with post stretching is very well predicted. Also, results shown that by utilization of appropriate values of average E-modulus and COF along with the combined isotropic-kinematic material hardening model, accurate springback prediction can be achieved.

To describe how post stretching results reduction of springback and side wall curve, the predicted stress distribution along the sheet thickness at the wall area for the cases of constant 100 kN and variable BHFs is shown in Figure 43. The predicted stresses are at the end of forming process before tools removal and springback. Figure 43 shows that how with variable BHF the compression stresses on the wall, which are created as a result of unbending of the sheet material when it passes the die corner radius, are converted to tension stresses. Thus, springback that is the result of heterogeneous stress distribution through the sheet thickness is reduced with the post stretching method.

Figure 42 Experimental results and simulation predictions of springback for three BHF values used in this study

Figure 43 Stress distribution along the sheet thickness at wall area for two different BHF values

As shown in Figure 43, during the post stretching, there is a significant change of stresses for elements which are under compression stress state before the additional tension.

However, the increase of stress value in elements which are in tension stress state before applying the additional stretch force is not significant. The reason is described in Figure

44. As shown in Figure 44, when the addition stretch force is applied, first elastic deformation happens in those elements which were in compression stress state.

Therefore, due to the characteristic of elastic deformation in metals, there is large stress change for a small amount of strain increase. On the other hand, by the stretch force, those elements which were in tension stress state undergo plastic deformation and therefore not significant increase in the stresses is observed in those elements.

Figure 44 Stress state of elements which undergo reverse loading condition when they pass the die corner radius and enter the die cavity

In Figure 44, during the bending over the die corner radius, the inner elements with respect to the neutral axis are under compression stress (pink color) while the outer elements are in tension stress state (green color). After applying the additional stretch at the end of deformation all elements undergo tension stress state.

Figure 45 illustrates the distribution of the residual stresses at the wall of the part after part release and springback. Results showed that the residual stresses through the sheet 80 thickness at the wall are significantly reduced by applying post stretching. Lower residual stresses can provide higher load carrying capacity and increase the fatigue strength of the formed part.

Figure 45 Residual stress distribution along the sheet thickness

Conclusions

Results of this study indicate that the side wall curl in drawn aluminum parts can be significant. Servo hydraulic cushion provides the capability to control the BHF through the die stroke. Therefore, it allows accurately controlling and increasing the BHF toward the end of deformation and significantly reducing springback. By applying the post stretching method, the additional tension force converts the tension-compression stress

81 distribution to only tension which cause reduction of springback. In addition, post stretching method results reduction of residual stresses at the part after springback.

Although the result of this study shows that by using a SHC it is possible to apply the post stretching method and significantly reduce the springback, some limitations should be considered when this technique wants to be applied in real stamping operation of large

3D panels. First of all, the material used in this study was 1.2 mm 5000 aluminum alloy.

Recently, the use of advance high strength steels (AHSS) and Gen 3 steels is increasing in automotive industry. These materials are three to four times stronger than the aluminum alloy used in the current study. Therefore, the deformation force and stresses at the part are significantly larger that results significant amount of springback. As discussed in this paper, in post stretching by using SHC, the stretch force is resulted from the friction force between the contact surfaces of the sheet material and the die and blank holder at flange area. Considering higher stiffness of strong steels compared to aluminum, a large increase of blank holder force is required to be able to convert the compression stresses to tension and reduce the springback by post stretching method.

Therefore, a significantly large SHC is required. This can reduce the practicality of this method in real stamping operation.

The other conventional method for post stretching is to design the tools with lock bead which can be activated toward the end of the deformation. In this method, there is no need for SHC. However, it is not also easy to lock the draw-in of such strong materials and the lock bead needs to be designed with sharp edges and fillet radii. Therefore, design of the lock bead adds more complexity to tool design and sometime failure of the

82 material at the lock bead area is a difficult issue to solve. A combination of using SHC and lock bead can be a possible solution. With a SHC the lock bead designs can be more moderate and reduce the excessive deformation and possibility of failure of the sheet material at the lock bead area.

Results show that the prediction of springback is sensitive to E-modulus and COF. In the current study, an apparent E-modulus is determined through inverse analysis method for each forming condition (blank holder force). COF was also determined by measuring the flange length in experimental samples and compared with simulation.

Some limitations were observed when applying the BHF with SHC. The actual BHF is not always the same as the target values. Some parameters such as the forming speed, the range of BHF variation, the size of the cushion, and the speed of the closing and opening of the hydraulic valves can affect the accuracy of the actual force, compared to the input or target value.

Acknowledgement

The author gratefully appreciates Hyson Metal Forming Solutions for providing the servo press and servo cushion. Also, special thanks to Mr. Ethan McLaughlin for supporting the experiments.

Chapter 8: Springback Reduction Using Post-Stretching Concept of Stake Beads

Introduction

Springback compensation is one of the most popular methods to reduce springback which may require multiple die recuts. However, it is difficult to fix side wall curl with compensation [Zhou et al. 2016]. Chapter 6 showed an effective way of inducing post- stretching for springback control using a variable blank holder force using a servo hydraulic cushion in a hat-bending operation. However, the costs associated with servo- hydraulic cushions makes them unaffordable for most the stamping industry in the United

States. Shape-set method by General Motors [Ayres, 1984] is the benchmark study involving a die process for post-stretching using locking-type beads. It involves a two- stage forming process as shown in the Figure 46. A preform is first stamped with a shim in the die cavity to limit the stroke of the punch. The second stage consists of stretching the preform to the final shape by clamping the flanges on both sides to lock the material flow. This method was observed to be providing a stretch of 2% which was effective in reducing the springback.

Figure 46 Two-stage forming process with stage II showing multiple locking beads for post-stretching [Ayres 1984]

So an alternative way to generate post-stretching in the part is by addition of beads

(known as stake beads) in the die design. Recent studies have been performed on using stake beads in stamping production [Zhou et al., 2016; Zhou et al., 2017] for springback control. Mechanism of stake beads works in such a way that the beads penetrate in part towards the end of the forming stroke which causes locking of the material, restricting its flow and thereby generating a stretch force in the side wall of the part. This additional stretch or tensile force reduces the heterogeneous stress distribution in the part and consequently the unbending moment is reduced leading to a reduced springback. Stake beads should not be confused with the drawbeads which are engaged throughout the

85 forming stroke and allow controlled flow of the material. A schematic depicting the different stages in the forming in hat-shaped bending is shown in Figure 47. We can see that the tool setup at the start of the deformation at position A. The upper die moves down, drawing and bending the material until a point when the bead starts to touch the material at position B. As the die continues to move downwards, the material forms around the bead which increasingly restricts the material flow and provides a stretch force in the side wall. Position D indicates the end of the forming operation.

Figure 47 Schematic showing different stages in hat-bending when stake bead is used

Previous studies by major OEMs have shown to have a positive effect on the springback with the concept already in production [Zhou et al., 2016].

This study investigates the effectiveness of the concept in reducing springback in Al alloys and AHSS numerically. Different bead designs and their effect on the springback are looked upon in this study to establish a methodology to optimize the bead geometry to reduce springback. Different bead geometries, (Rectangular, circular and edge bead) –

[Tufecki et al., 1994], considered for the stake bead design can be seen in the Figure 48.

Figure 48 Different bead geometries with important dimensions

The corner radius of the Stake Bead or the Stake Bead radius (Rs), Die Groove radius

(Rd) and Bead Penetration (Hs) are the significant dimensions that determine the ‘locking’ capacity of the bead design. For the stake bead concept to be successful, the die needs to be designed so that:

1. The stake bead will not break or wear out easily and should be within the

manufacturing capabilities

2. Stretching of the material in the bead itself does not cause local fracture in the

material

3. The stake bead design provides adequate stretch force in the side wall to reduce

springback

The strength analysis of the bead itself should be performed in the die design stage by calculating the maximum reaction forces on the bead during the forming process. Local failure due to local formability of the material i.e. bending and stretch-bending also requires extensive analysis [Hudgins et al., 2010; Shih and Gambon, 2017] which is beyond the scope of this study.

The shape of the bead not only influences the restraining force in the material but also affects the vertical reaction on the die during the deformation process. Thus, the forming force curve is affected by the bead design. In general, the forming force curve in a simple hat-bending operation involving use of stake bead is shown in Figure 49.

Figure 49 Variation of die force versus forming stroke (simulation), example material – DP800/ 2mm, 70 mm stroke

During the forming process, when the beads start touching the sheet material, higher BHF is required to prevent an opening between the die and the blank holder. As a result, the total load on the die and press ram increases dramatically. The load versus displacement curve shown in figure 4 shows the trend of reaction force applied on the blank holder during the deformation. The reaction force is calculated using simulation with a constant gap between the die and the blank holder. In reality, however, when nitrogen cylinders or air cushions are used, the BHF cannot be controlled or increased toward the end stroke like it can be done with a servo-hydraulic cushion. Therefore, a high BHF should be applied at the beginning of the deformation stroke. This is one of the limitations of using stake beads for post-stretching.

For the increase of force in forming with stake beads to be minimized, the die design requires a stake bead design that eliminates the possibility of fracture in the formed part wall and the bead region. In addition, the maximum force on the die should be as small as possible.

To investigate the concept of stake bead to reduce springback, initially hat-bending of

Al5182-O/1.2 mm will be considered similar to that in Chapter 6. The process parameters and tool dimensions will be kept same except the addition of the stake beads in the post.

The springback angles after unloading the tools will be compared with the springback angles obtained using variable blank holder force generated using SHC. Furthermore, a case study to reduce springback and minimize press force using stake beads in forming of a production part (A-pillar – DP800/2mm) will be discussed.

Use of Stake Bead in hat-bending of Al5182-O/1.2 mm

A study similar to the hat shaped draw-bending of Al5182-O in chapter 6 is presented here for convenient comparison to the experimental data available. Use of stake beads instead of the variable BHF using servo hydraulic cushion is proposed. A schematic of the setup is shown in figure 50.

Figure 50 Tool dimensions used in the draw-bending of Al5182-O/1.2 mm

FE simulation

A 3-D FE simulation of the U-channel drawing process was developed using the

AutoForm software package. The blank was created using adaptive (varying element size) triangular shell elements with 11 integration points normal to the shell and tools were modelled using rigid analytical surfaces. The flow stress of the material was determined from a tensile test and it was extrapolated using the combined Swift/Hockett-

Sherby power law defined as:

푚 푝 휎̅ = 훼 ∗ [퐶 ∗ (휀푝푙 + 휀0) ] + (1 − 훼) ∗ [휎푠푎푡 ∗ (휎푠푎푡 − 휎푖) ∗ exp(−훼휀푝푙 ) (7-1)

Also, the yield surface was defined by Barlat 89 yield function. Isotropic hardening was taken as the hardening rule for the yield function. Anisotropy was considered and r-

91 values were provided by the sponsor. Thinning limit of 15% was assumed for Al-5182O by measurements of the cracked samples in the previous study (see Chapter 6). A coefficient of friction value of 0.05 was (BHF) used in the simulation as was established in chapter 6. A minimum blank holder force was provided in each case considered such that there is no die opening observed.

Results and Discussion

Rectangular bead design would be explored for hat-bending of Al5182-O. Initial guess bead geometry dimensions were taken based on the initial studies made by Zhou et al.

Optimization of the bead dimensions were made based on a trial and error strategy described in Figure 51. Method to quantify springback for comparison is depicted in

Figure 52. Table 2 contains selected iteration results for different stake bead dimensions along with minimum required BHF, maximum thinning % and measured springback.

Figure 51 Strategy used for optimization of bead dimensions to minimize springback and tonnage

Figure 52 Predicted springback in simulation

Min. BHF Measured R R H Applied % Max. Case d s s required Springback (S) (mm) (mm) (mm) BHF (kN) Thinning (kN) (mm) 1 5 1 10 8.9 9 10 % 1.25

2 5 2 10 9 9.1 9.2 % 3.62

3 3 1 10 13.7 13.8 22.1 % 0.83

4 3 2 10 14.6 14.7 14.2 % 0.56

5 3 1 5 13.2 13.3 7.6 % 8.01

Table 3 Selected Simulation Results (Al5182-O, 1.2 mm)

Since the applied BHF is directly proportional to the forming tonnage required, BHF would be used in equivalence instead of die force to quantify and compare the press tonnage. The final part after springback for the best cases are compared with results chapter 6 which are shown in the Figure 53.

Figure 53 Comparison of the profiles after springback for post-stretching with Stake bead and variable BHF using Servo Hydraulic Cushion

The reduction in the side wall curl using stake bead is significant and comparable to the post stretching method of variable BHF. In addition to reduction in the side wall curl, the press tonnage required is also significantly reduced with the stretch force taking up much less energy from the tonnage increase.

Stake Bead methodology is not restricted by the press and cushion limitations in terms of their dynamic response and generates the stretch force by internal mechanism incorporated in the die design.

Case- Study: Optimizing Stake Bead Design in Draw-Bending of DP800/2mm

A case study on application of stake bead to reduce springback and station tonnage simultaneously for DP800/2mm is presented. A 2-D cross-section of a 3-D production part (A-pillar) was considered as provided by the sponsor. The tool and bead dimensions were given initially and objective was to optimize the bead geometry to minimize or maintain springback in part while reducing the force (press tonnage) required to form the part. A 2-D schematic with dimensions of the worst affected cross-section in the 3-D part is depicted in Figure 54. Initially a circular bead design was considered as it was provided in the initial design. Later, the edge bead design was also investigated.

Figure 54 Tool dimensions for the draw-bending process of DP800/2mm

FE simulation

A 3-D FE simulation of the hat-shaped drawing process was developed using the

AutoForm software package similar to the one mentioned the last section. The flow stress data of the material was provided by the sponsor and expressed as the combined

Swift/Hockett-Sherby power (equation 7-1).

Also, the yield surface was defined by Banabic steel yield function. Isotropic hardening was taken as the hardening rule for the yield function. Anisotropy was considered with r- values and directional yield stress values were provided by the sponsor. A Forming Limit

Curve was provided by the sponsor and was used as the fracture criterion. A coefficient of friction value of 0.1 was (BHF) used in the simulation with a blank size of 500 x 120 mm. A minimum blank holder force was provided in each case considered such that there is no die opening observed.

Results and Discussion

Optimization of the bead dimensions were made based on a trial and error strategy described in Figure 51. Method to quantify springback for comparison is depicted in

Figure 55, as the side wall is of major significance. Table 3 contains selected iteration results for different stake bead dimensions along with minimum required BHF, maximum thinning % and measured springback.

Figure 55 Predicted springback in simulation

Figure 56 Comparison of the springback profiles with the initial bead geometry and the optimized bead geometry

Stake bead geometry Minimum % Die parameters Predicted Required Change Reaction Case springback R R H BHF in die Force* d s s [mm] [mm] [mm] [mm] [tons] force [tons]

Initial Bead Referenc Geometry 3.5 5 7 58 2.23 692 e (Case 1) Optimized Circular Bead Geometry 5 5 7 46.7 -9.7% 2.51 625 (Case 2) Optimized Edge Bead Geometry 3.5 3 10 63 -33% 2.09 465 (Draw angle = 10° ) (Case 3) Table 4 Predicted springback and die reaction force for the 2-D hat-shaped forming operation after optimization of stake bead design, *the die reaction force is multiplied by a factor of 7 to account for actual 3-D part length

Reaction force on the die was taken as a measure of the tonnage comparison among the various cases. Edge Bead design was found to perform better (in terms of reduced springback and tonnage) according to simulations as compared to the circular bead design due to reduction in the number of bend radii for the material during the forming process.

Case 3 (Table 3) gives a reasonably large reduction in die force (33%) with simultaneous improvement in the springback value (6% decrease in the predicted springback value), having a draw angle of 10° for ease of manufacturability.

Conclusions and Future Work

Use of stake beads to induce post-stretching in a draw-bending forming operation to reduce side wall curl springback is discussed. Numerical studies on use of stake beads in hat-shaped draw-bending of an aluminum alloy (Al5182-O/1.2 mm) and an AHSS

(DP800/1.2) are presented. Both the case studies show promising improvement in the springback reduction using the stake bead methodology. Experimental studies to validate the findings of the numerical studies still need to be made. Additionally, different stake bead designs are discussed and utilized in the tool design in FE simulations to study the effects on final springback and press tonnage. According to the study, edge bead design provides the ideal combination of post-stretching and material deformation to reduce springback and ram force (tonnage) simultaneously.

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