FINITE ELEMENT AND EXPERIMENTAL STUDIES OF

FINITE ELEMENT AND EXPERIMENTAL STUDIES OF SPRINGBACK IN SHEET METAL FORMING Fahd Fathi Ahmed Abd El AU A thesis subrnitted in conformity with the r...
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FINITE ELEMENT AND EXPERIMENTAL STUDIES OF SPRINGBACK IN SHEET METAL FORMING

Fahd Fathi Ahmed Abd El AU

A thesis subrnitted in conformity with the requirements for the degree of Master of Applied Science Department of Mechanical and Industrial Engineering University of Toronto

Q Copyright by Fahd Fathi Ahmed Abd El Al1 June, 2000

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Thesis Title:

Degree and Year: Narne: Department: University:

Finite Element and Experimental Studies of Springback in Sheet Metal Fonning Master of Appüed Science, 2000 Fahd Fathi Ahmed Abd El Al1 Graduate Department of Mechanical and Industrial Engineering University of Toronto

Abstract The ultimate goal of the metal forming industry is to form components made of a specific material into a required shape without experiencing wrinkling, bifurcation and springback. Of the aforementioned dBiculties,

attention in this thesis is devoted to

springback resulting from unloading foilowing stretch forming. In spite of the exhaustive application of the process to form many engineering components, the design and selection of sheet materials and twling geometry to minimize springback stili rely on the elusive trial and error approach. It is thus the objective of this thesis to avoid the ad hoc approach by analyzing the process using the finite element method. Three aspects of the current problem are accordingly investigated. The first is concemed with the development of a finite element mode1 of the stretch forming process accounting for material and geometric nonlinearities, and interface conditions using ANSYS software. The elastoplastic behavior of the blank materials is investigated using isotropie

and anisotropic constitutive models. Imposition of fnctionai contact constraints at the tooling-sheet interface is also introduced using contact elements. The second aspect of the work is concerned with the springback experienced by the deformed sheet following the punch release stroke. Several numericd techniques are introduced to define and quantiSr springback. For this purpose, use was made of

MATLM software to allow the determination of springback using four different accepted definitions.

In the third, attention was focuseci to the validation of the finite element models. Two alternative test rig designs were developed to test the axisymmeûic and the plane strain models. However, in view of cost and time limitations, only the plane strain stretch fonning ng was built to test the comsponding FE model. The Limiting Dome Height (LDH) benchmark test was also utibed as another verifkation

tool. The results obtained

from these tests reved close agreement with the finite element predictions of the deformed sheet springback and load d e k t i o n curves.

The technique was then applied to a case snidy conceming the stretch fonning of a rear bulkhead for an aircraft. The selection of this example was motivated by our desire to address some of the critical issues resulting from the crash of Japan airline m g h t 123), as a result of failure to account for the effect of springback in the assembly of the bulkhead to the fuselage. In Our study, we examine the effect of the mechanical properties, the geometry and interface parameten upon the resulting springback.

iii

Acknowledgements 1 offer my sincere gratitude to Dr. S. A. Meguid for his technical guidance, criticai discussions and support throughout the course of my research. Further recognition is due to the excellent staff members of the Engineering Mechanics and Design Laboratory. The

financial support of MM0 Ontario-Singapore Joint Research Programme and University

of Toronto is gratefidly acknowledged. Heartfelt thanks are due to my parents who sparked a young engineer interest in graduate studies and provided unflagging patience

and support,

Contents Abstract Acknowledgement Contents List of Figures List of Tables Notation 1 Justification for the Study 1.1

Stretch Forming in Manufacturing

1 -2

Justification for the Study

1.3

Objectives of the Study

1.4

Methcxi of Approach .

1.5

Layout of Thesis

1.6

References

.

.

.

2 Background and Literature Review

.

2.1

Sheet Metal Forming

2.2

Combined Fomiing and Wrinkiing Diagrams

2.3

Contact Conditions

2.4

Japan Airlines (Flight 123) Crash

2.5

References

.

.

-

3 Theoreticai Investigations: Finite Element Modeüing

of Stretch Forming 3.1

Stretch Forming Tooling Geometry .

3.2

Discretization of the Problem .

3.3

Loading and Constraints in Stretch Fonning

3.4

Yield Criteria

3.5

.

3.4.1

von Mises Yield Criterion

3.4.2

Barlat-Lian Yield Criterion

3.4.3

Tool Material

3.4.4

Hardening Rule

References

.

. .

.

.

4 Determination of S p ~ g b a c k in Stretch Forming 4.1

Detennining Springback in Stretch Forming

.

4.1.1

The Displacement Deviation Curve .

4.1.2

The Average Function Method

4.1.3

The Virtual Deviation Area Method

.

4.1.4

The Mean Normal Spacing Method

.

4.2

Cornputer Implementation for Determination of Springback.

4.3

References

.

5 Experimental Investigations

5.1

ExperimentalSetup

.

5.1.1

Test RigDesign

5.1.2

Blank Design

.

5.2

Measuring Devices and Machine Calibration .

5.3

Test Rig Assembly

5.4

Punch Loading and Release Strokes

5.5

Springback and Load-Deflection Characteristics

5.6

References

-

.

.

6 Results and Discussions 6.1

6.2

Validation of the Finite EIement Model 6.1.1

Limiting Dome Height Test

.

6.1.2

Experimental Verifcation

. . .

Parameters Influencing Springback 6.2.1 EffectofBlankThickness 6.2.2

Effect of Punch Travel

6.2.3

Eff'ect of Punch Radius

6.2.4

Effect of Tooling-Blank Interface Friction

6.3

A Case Study: Fomiing of Aircraft Rear Bulkheads

6.4

References

. .

.

Conclusions and Future Work

.

Statement of the Problem Objectives

.

General Conclusions

.

7.3.1

Finite Element Model of Stretch Forming

7.3.2

Determination of Springback .

.

7.3 -3 Experimentai Simulation of Stretch Forming .

Thesis Contribution

.

Future Work . Appendices A

Chernical Composition of Aluminium Alloys

B

Listing of a Sample MATLAB Code

C

Experimental Test Rigs C. 1

Axisyrnrnetric Test Rig

C.2

Plane Strain Test Rig

.

vii

.

List of Figures Stretch fonning operation.

.

Stretch fonned parts: (a) Hernispherical dome for the cryogenic tank of expendable launch vehicles, and (b) Automobile fuel tank. Stretch formed flange exhibits wrinWing.

.

-

Rear bulkhead assembly with the aircraft fairing.

.

Flow Chartof methodof approach. . Blank fonning regions. Blank deformation caused by punch loading. Critical Aluminium ailoy forming and wrinkling limit diagrarn. Jetliner ruptured left wing on the crash site.

.

Blank configuration: (a) Axisymmetric geometry, and (b) PIane strain geometry. Basic geometry parameters of the stretch fonning tooling. Element size refinement convergence test.

.

.

Meshing of the blank with six elements across thickness.

.

Punch stretch forming strokes: (a) Loading stroke with a specified incremental vertical displacement w i , i = [O,n], and (b) Release strokewithw=w,.

.

A typicai node-to-node contact element.

.

Force-deflection relationship (Optimal contact condition at k, = k, = 0 ) : (a) Variation of normal stiffhess k,

stiffness k,.

,and (b) Variation of tangentid

. viii

Punch-die and blank constraïnts imposed during the specified loading

-

and release strokes.

Springback depicted in an Aluminium alloy blank after punch release. Springback defined by the displacement deviation curve.

.

Blank loading and release configurations segmenteci at their nodes.

.

Sprîngback defined by the average function method. Numerical integration. Virtual deviation area between the blank loading and release [email protected] Parameters defining springback by the mean normal spacing method. Main modules and sub-modules of the developed MATLAB code.

.

Exploded view of the test rîg designed to form axisymmetric parts: (a) Lower mount, and (b) Upper mount.

.

Exploded view of the test rig designed to form plane strain parts. Calibration of the testing machine.

-

.

Load-deflection curve of the testing machine. Plane strain test rig assembly. Geometry and dimensions of the Limiting Dome Height test tooling.

Blank deformed geometry at the end of the loading stroke (w,

= 0.03 m).

Major strains obtained from the Limiting Dome Height test. Deformed blank geometry.

.

Blank loading/release displacement history. . Successive blank stretch forming loadinglrelease curve (Wnlubricated contact surfaces). Loading and release strokes of tested blanks. . Variation of the stretch fonning force with punch travel for a blank thickness t = 1 mm.

.

Variation of the stretch forming force with punch travel for a blank thickness t = 1.5 mm.

.

Variation of the stretch fomiing force with punch travel for a blank thickness t = 2 mm.

.

Variation of springback with blank thickness (punch radius rp= 17.5 mm, punch travel w = 20 mm,and p = 0.5). Wrinkiing a a punch travel w = 30 mm. Variation of the stretch fomiing force with punch travel for a maximum punch travel w , = 15 mm.

-

Effect of varying punch travel w on blank springback (punch radius r, = 17.5 mm, blank thickness t = 2 mm, and p = 0.5).

Effect of varying punch travel w on blank springback (punch radius r, = 12.5 mm, blank thickness t = 2 mm, and p = 0.5). Predicted dependence of springback on punch radius. Variation of the stretch forming force with punch travel under a wet interface condition using Grease NLGI Grade 2.

.

Predicted dependence of springback on friction at the twling-blank contact surfaces. Aircraft rear bulkhead: (a) BuMead geometry, and (b) Basic

bulkhead shape parameters.

.

Variation of normal contact stresses dong the normalized contact length of a stretch forrned Al-Zn 7075-T6bulkhead: (a) Punch-blank mating

surfaces, and (b) Die-blank mating surfaces.

Normalized bulkhead springback of two Aluminium alloys.

List of Tables Definition of the tooling geometry key parameters. Contact parameters.

.

.

. Mechanical properties of Al-Li 2090-T8 Aluminium alloy. . Mechanical properties of Al-Zn 7075-T6Aluminium d o y .

Description of the parts used in the axisymmetnç test rig.

-

Details of the components involved in the plane strain test rig. Geometrical and mechanical properties of tested blank specimens.

.

Punch loading specifications. . Mechanical and geometrical parameters of the Limiting Dome Height test.

.

Defonned blank geometric parameters. Tooling-bIank interfacial friction conditions. .

Geometric parameters of reference bulkhead.

Notation a

Bulkhead circumferential radius

AL

Aluminium

E

Elastic modulus

ET

Strain hardening tangential elastic modulus

h

Segment width

k

Contact stiffiiess parameter

KK

Critical fracture toughness

L L

Arc Length Lankford coefficient

M P

Fully plastic bending moment

P

Punch load

=,

Punch radius

SPB

Springback

t

Blank thickness

W

Vertical z-direction displacement

Y

Yield locus function

E1

Major principal strain

E

Minor principal strain

2

Q>

Blank deformation slope angle

CL

Coulomb coefficient of fiction

v

Poisson's ratio

P

Punch radius

6 1

Major principal stress

6 2

Minor principal stress

xii

oc

Normal contact stress

GOn

Offset strength

a,

Ultimate strength

o,

Yield strength

T

Time

AV

Average

DEV Deviation

L

Limit

L

Loading

N

Normal component

R

Release

T

Tangent component

xiii

Chapter 1

Justification for the Study 1.1

Stretch Forming in Manufachiring

Stretch forming is a metal fonning process in which the blank is formed by the application of primarily tensile forces in such a way as to stretch the material over a form. In view of the f a t that tensile stresses dominate the loading of the blank, large plastic

strains can be obtained in ductile maîerials. The process is used extensively in the aircraft and automotive industries to produce components with large radius of curvature. Stretch forniing equipment consists basically of a hydrauliçally dnven ram, which carries a form block and two blank holders or grippers for holding the ends of the blank. Figure 1.1 shows the tooling used in stretch fonning. The advantages of stretch forming are: (i) the ability to form double curvatures with ease, and (ii) large plastic strains can be

attained.

Die

Form B lock

Gripper or Blank Holder

Figure 1.1 Stretch forming operation.

1.2

Justification for the Study

Stretch forming is considered to be essential for several industries, including the manufacturïng of fairings and panels in the aircraft and automotive fields. Most of the parts contained in an automobile panel such as doors, finders, and fuel ianks require accurate fonning. This is due to the fact that their assembly requires certain Iimits on their geometrical tolerances. Figure 1.2 shows illustrative examples of a hemispherical dome for the cryogenic tank of

expendable launch vehicle, and an automobile fuel tank that had been stretch formed

[l.11. These components experience springback after k i n g formed, thus affecting the specified tolerances needed for their assembly. If the change in geometry resulting fiom springback is not accounted for, difficulties may be experienced during the assembly of the part. in addition, wrinkling and bifurcation are some of the drawbacks that c m result

from stretch fonning.Figure 1.3 [l. 11 shows surface wrinkling in a stretch formed flange. At the present time, the design of many metal-fomed parts is canied out on trial and error basis. This emphasizes the need for accurate modeling accounting for geometrical, material, and interface conditions present in stretch fonning. In view of its versatility and ability to treat nonlinearities with ease, the finite element analysis technique is used in

analyzing many metal forming processes [1.2,1.3].

Figure 1.2 Stretch formed parts: (a) Hemisphencal dome for the cryogenic tank of expendable launch vehicles, and @) Automobile fuel tank (After [1.1]).

2

Figure 1.3 Stretch formed flange exhibits wrinkling (After Cl.11).

Springback represents a challenge for manufacturers who desire to meet specific dimensions. The accurate and d i a b l e assembly of components in the automotive or aircraft industry necessitates that the parts meet certain tolerances. Controlling and/or minirnizing springback would enable designers to achieve better process control and reduce rejects.

In this thesis, the finite element technique is utilized in modeiing smtch forming. The analysis accounts for geometrical and material nonlinearities, and interface conditions of the forming process. The results will provide load-deflection history, the stress and strain States caused by stretch forming, and quantify springback. Geometrical deviations caused by spnngback after forming aircraft rear bulkheads will be

analyzed as a representative case study. The resulting springback inhibits the accurate assembly of the bulkhead with the aircraft rear fairing. Thus, reduction in the margins around rivet holes at the splice of the upper and lower webs causes inaccurate fitting of the bulkhead. The explodeci view of the assembled fairing parts is shown in Figure 1.4.

Upper FaÜing Section

Bulkhead

Lower Fairing Section

mbled Fairing

Figure 1.4 Rear bulkhead assembly with the aircraft fairing.

1.3

ObjectivesoftheStudy

The aim of the present thesis can be stated as follows: To use ANSYS FE code to mode1 the stretch forming process accounting for matenal nonlinearity geometrical

using isotropie and anisotropic constitutive laws;

nonlinearity

for

large

deformations

and

rotations;

and

interface/contact conditions. To establish the final configuration of the sheet after releasing the punch load in order to [email protected] and define the resulting spnngback.

To study the effect of the tooling geometry and interface characteristics on springback. To validate some aspects of the results experimentally.

1.4

Method of Approach

The method of approach adopted to achieve the above stated objectives is presented in

the flowchart depicted in Figure 1S.

C

Geomeîrïcai

Nonüntlr Fite Elemcat -sis Contour Strctch Forming Using ANSYS

of

Displacemcmt, Strrila and Stirss Distiiboîiom

Forming Limit Diagram

.D

Modirj. Geomctry, Material, Interface or Boundary C0nditi0~

YE!3 Determinirig and Qliantifying Sprbgback

Figure 1.S Flow Chart of method of approach.

ANSYS FE code was used to model the stretch forming process accounting for material,

geometric and interfacdcontact nonlinearities. Strains obtained from the FE analysis are checked for thinning and wrinkling using the fomiing limit diagram (FLD) of the material under consideration [1.4.1.5]. The punch was then released to determine the resulting spnngback caused by a selfcquilibrating state of residual stresses. Finally, experiments using a specially designed rig were used to [email protected] the predictions attained from the finite element models,

1.5

Layout of Thesis

This thesis comprises a total of seven chapters. Following this introductory chapter, Chapter 2 presents a critical literahire review of sheet metal fonning and the finite element modeling of forming processes. It contains the pertinent work on the subject, the techniques adopted in treating the diffferent classes of problems and the limitations and difficulties encountered- Chapter 3 presents the theoretical investigations conducted using a finite element mode1 suitable for stretch fonning. Chapter 4 discusses the various methods used in defining and quantiQing springback. Chapter 5 outlines the experimental investigations conducted to validate the finite element predictions. Analysis of results obtained fiom the experimental work and the finite element analysis of the stretch fonning prccess together with the conclusions obtained from the springback determination methods are presented and discussed in chapter 6. We also discuss the forming of a rear bulkhead of an aircraft as a case study in the same chapter. Finaily, in chapter 7 we conclude the thesis and provide recornmendations for future work.

1.1

Peters, M. and Winkler, P. -J. Aluminium-Lithium. Germany: DGM, 1992.

1.2

Atlan, T., Kobayashi, S. and Oh, S. Metal Fonning and The Finite Element

Method. New York: Oxford University Press, 1989. 1.3

Tang S. C. and Wang, N. M. Computer Modeling of Sheet Metai Forming Process: Theory, Verifkation and Application. Pemsylvania: The Metaliurgical

Socieîy, 1985. 1.4

Johnson, W. and Mellor, P. B. Engineering Plasticity. England: Ellis Honuood Limited, 1983.

1.5

Mguil, S., Brunet, M. and Morestin, F. "Cornparison between Experimental and Theoretical Fonning Limit Diagrams for Aluminum Sheets." Simulation of Materials Processing: Theory, Methods and Applications, Numiform 98 (1998): pp. 739-744.

Chapter 2

Background and Literature Review This thesis is concemed with the analysis of spnngback in sheet metal stretch forming processes. The foliowing relevant areas are reviewed: (i) mechanics of sheet metal forming and spnngback, (ii) fonning limit diagram for Aluminium sheets, (iii) finite element modelling of stretch forming, and (iv) aircraft crash disasters caused by inaccurate assembly of stretch fomed components experiencing excessive springback.

2.1

Sheet Metal Forming

In sheet metal stretch forming, the blank under consideration undergoes complex conditions of elastoplastic straining history [2.1,2.2]. In order to simplify the analysis of the semi-spherical deformation, three regions in the blank are specified, as shown in

Figure 2.1. Annular region A initially undergoes radial drawing towards the throat of the die. As the metal passes over the die radius, it is first bent and then straightened, while at the same time k i n g subjected to a tensile stress. Simultaneously, zone B stretches in the die gap between the die shoulder and the punch nm leading to a frustum of a cone profile. The metal in this region is subjected only to simple tensile loading throughout the stretching operation. The punch profile from nose to nrn wül be in contact with region C, causing it to bend, slide and stretch. In so doing, the blank in this region experiences a reduction in thickness. FinalIy, region C mimors the punch sphericai shape leading to the formation of

the specified component geometry.

Figure 2.1 Blank forming regions.

Reviewing the literaîure. it is found that researchers have been studying sheet metal forrning and springback for more than four decades. As early as 1958, Gardiner [2.3] worked on a mathematical investigation on spnngback corrections under pure bending conditions. His research was concemed with springback in aluminium, titanium and ferrous alloys. Johnson and Yu [2.4,2.5] then presented an analogy between the deformation of the rigidninear work-hardening sheet beyond yield and the linear elastic beam. According to their work, the blank under consideration can be treated by assuming that the thickness of the blank is much smaller than the punch radius, i.e. t cc p. Figure 2.2 shows the €ree body diagram of the deformed blank, as given by Johnson and Yu 12.41. According to Figure 2.2 and by applying equilibnum in segment ni;,the following relationship for the blank deflection slope is obtained:

o,tL

M, =-

w here

and

O,

4

plastic bending moment of the blank per unit width,

is the yield stress.

Die

- Figure 2.2 Blank defonnation caused by punch loading (Afier [2.4]).

In order to obtain an expression for the deflection in segment

K,Johnson and Yu 12.41

used a cubic curve to fit the deflection in this segment of the deformed blank.

Kobayashi and Oh [2.6] have also examineci different sheet bending problems using numerical techniques. They have pointed out that in the case where the punch release conditions can be well defined, it is possible to analyze residual stress distributions and calculate springback by employing a rigid-plastic solution at the end of the loading stroke. Springback in their work was formulated as follows:

1 -

where

blank final curvature,

Pr and

1 -

blank plastic curvature.

PP Equation (2.2) suggests that springback is proportional to %,/Et.

Accordingly, the

parameters which govem springback can be divided into material parameters such as the yield stress and elastic modulus, and geometrical variables such as blank thickness and diameter. Pourboghrat and Chandorkar [2.7] evaluated springback for plane strain sheet metal forming problems. In their analysis, they performed FE modeling of deep drawing and stretch forming operations for inner hood sections of an automobile. The sarne punch travel was used to analyze and compare both operations. It was found that sections

manufactured by stretch forming had a smaiier springback than deep drawn sections. Over-stretching, which cm cause tears in formed blanks, was avoided by checking the principal strains with the Focming Limit Diagrams @TD)of tested blanks. Boyce etal. [2.8,2.9] explored the cup forming of AI 2008-T4Aluminium alloy under both constant and variable blank holder force conditions by carrying out experimental and finite element investigations. The results obtained showed improvement of cup

formability by using both trial and wellaefined variable blank holder force control. Repeatability of end part quality in ternis of final geometry and failure modes was also

established. In a recent work by Sunserï etal. 12.101, springback prediction in channel forming was also carried out using active blank holder force control. Kinsey etal. [2.11] proposeci a neural network system, dong with a stepped blank holder force trajectory to be used in controlling springback. In order to minimize springback, artificiai neural networks provided values for the blank holder force when faced with large variations in sheet thickness and interface friction. The results obtained established that neural networks were successful in determining the high blank holder force

necessary to reduce springback in the formed part. Love11 and Narasimhan [2.12] utilized FE analysis techniques to simulate the sheet metal stamping process of automotive components. Ra-

than using a blank holder to support

the formed blanks, they used two pins, which fit into holes in the tested blank during

forming in order to maintain its stability. Springback prediction results obtained fiom the

FE analysis reduced the required die set prototypes hom eight to two. Kutt etal. [2.13] analyzed springback in forming of doubly curved sheet metal parts using

a smooth saddle shaped die. They conducted laboratory experiments on two Titanium alloys: (i) Ti-B2 1S (Ti- 1SMo-3AI-3Nb), and (ü) Ti- 15-3 (Ti- 1SV-3AL3Cr-3Sn). Finite element modeling was also carried out in their work to analyze sheet metal forming with the saddle shaped die using ABAQUS software [2.14]. Evaiuations of the springback

displacement and the shortening of the blank chord resulting after punch release were attained form their FE analysis.

2.2

Combined Forming and Wrinkling Diagram

Displacements and strains obtained from the finite element solution for the stretch forming process need to be checked to avoid sheet wrinkling and failure. Stretch formed blanks undergo severe combination of strains in any of the regions rnarked A, B, or C discussed previously in Figure 2.1. In a recent work by Mguil etal. [2.15], cornparison between exprimental and theoretical forming limit diagrams for Aluminium blanks was

carried out. Forming limit diagrams were utilized throughout the thesis in order to assess the fonning behaviour of the Aluminium blanks under consideration. In a forming lirnit curve (FLC), the strain states are presented using data pairs in ternis of the major and rninor principal suains. The forming Limit curve provides a distinct separation between the safe and unsafe strain regions that blanks undergo dunng forming. In the safe domain,

blanks do not show any instability, while the unsafe region defines strain states leading to

necking and cracking. Figure 2.3 shows the critical combined forming and wrinkiing strain percentages of an Aluminium ailoy.

50 -

Unsafe Domain

-------- Wiinkling curve Forming limit curve

Unsafe Domain

-

Safe Domain

-30

-20

=

-10

10

-

O

Safe Domain

10

20

30

Minor strain, a,(96) Figure 2.3 Critical Aluminium ailoy forming and wrinkling limit diagram (After [2.15]).

2.3

Contact Conditions

In stretch forming, some of the restrictions established by Hertz [2.16.2.17], such as elasticity, small deformation, and ftictionless contact, cannot be accornrnodated in the actual process. In finite element analysis of stretch forming, contact is mostly modeled

using contact eiements at the tooling-blank interface [2.18]. These contact elements are used in transmitting tool loading to the blank. Wagoner and Kaiping [2.19] conducted a series of FE simulations for the draw bend test of Al 6022-T4Aluminium alloy sheets. The FE analysis of springback was determined to be very sensitive to numerical parameters, such as the number of through thickness elements, angle of contact and tolerances for equilibrium and contact. A sufficient

number of contact elements at the tooling-sheet interface was found to be cntical for the accurate prediction of springback.

2.4

Japan Airlines (Flight 123)Crash

In 1985, Japan Airlines suffered an unexpected disaster due to the crash of its Boeing 747 flight 123 at Gumma, Japan [2.20]. A dimensional flaw due to springback in the manufactured and assembled rem pressure bulkhead was established as the primary factor of the accident. The disaster resulted in severe repercussions for the airline.

The jetliner rear pressure bulkhead disintegrated and ruptured at an elevation of 32000 ft, allowing air to flow through the aircraft deck and pressure to be lost. Just before the

crash, the aircraft conducted a 360-degree right tum and brushed against a tree-covered area bursting into flames. Only four passengers among the 524 people aboard including the flight crew survived, leading to one of the highest deaths ever in a flight crash.

The Boeing jetliner maintenance history reports revealed that the rear bulkhead had previously been replaced. However, the newly manufactured bulkhead did not conform to the required dimensional specifications mainly due to excessive springback after the

stretch forming process. The margins around the bulkhead rivet holes at the splice of the upper and lower webs were less than the design specifications. Figure 2.4 shows the

jetliner ruptured left wing.

Figure 2.4 Jetliner ruptured left wing on the crash site (After [2.20]).

2.5

References Johnson, W. and Mellor, P. B. Engineering Plasticity. England: Ellis Horwood Limited, 1983. Zharkov, V. A. Theory and Practice of Deep Drawing. London: Mechanical Engineering Publications Limited, 1995. Gardiner, A. G. "The Spnngback of Metals." ASME Transactions. (1957): pp. 1-9. Johnson, W. and Yu, T. X. "The Press-Brake Bending of RigidLinear WorkHardening Plates." International Journal of Mechanical Sciences. vol. 23 (1981):

pp.307-318. Johnson, W., Stronge, W. J. and Yu, T. X. "Stamping and Springback for Circular Plates Deformed in Hemisphericai Dies." International Journal of Mechanical Sciences. vol. 26, no. 2 (1984):pp.13 1-148. Kobayashi, S. and Oh, S. 1. "Finite Element Andysis of Plane-Strain Sheet Bending." International Journal of Mechanical Sciences vol. 22 (1980): pp. 583-

594. Pourboghrat, F. and Chandorkar, K. "Springback Calculation for Plane Strain Sheet Forrning Using Finite Element Membrane Solution." ASME, Numerical Methods for Simulation of Industriai Metal Fonning Processes. (1992): pp. 85-93. Boyce, M. C. and Sim, H. B. "Finite Element Analysis of real time stability control in sheet metal forming processes." ASME, Journal of Engineering Material Technology. (1992): pp. 180-188.

2.9

Boyce, M. C., Hardt, D., Cao, J. and Jalkh, P. "Optimal Forming of Aluminium

2008-T4Conicai Cups Using Force Trajectory Control." SAE, Sheet Metai and Stamping Symposium. (1993): pp. 101-1 12. 2.10

Sunseri, M., Cao, J., brafillis,

A. P. and Boyce, M. C. "Accomodation of

Springback error in Channel Forming Using Active Binder Force Control." ASME, Journal of Engineering Material Technology. (1996): pp. 426434. 2.1 1

Kinsey, B.,SoUa, S. A. and Cao, J. "Consistent and Minimal Springback Using A Stepped Binder Force Trajectory and Neural Network Control." ASME, Journal of Engineering Materials and Technology. vol. 122 (2000): pp. 113- 118.

2.12

Lovell, M. and Narasirnhan, N. **PredictingSpringback in Sheet Metal Forming: An Explicit to Irnplicit Sequential Solution Procedure." Elsevier, Finite Elements

in Analysis and Design. vol. 33 (1999): pp. 29-42. 2.13

Kutt, L. M., etal "Non-Linear Finite Element Analysis of Springback." Communications in Numerical Methods in Engineering. vol. 15 (1999): pp. 3342.

2.14

ABAQUS/Standard User's Manual, Hibbitt, Karlsson, and Sorensen, Inc., 1997.

2.15

Mguil, S., Brunet, M. and Morestin, F. "Cornparison Between Experimental and Theoretical Fonning Limit Diagrams for Aluminium Sheets." Simulation of Materials Processing: Theory, Methods and Applications, Numifonn 98. (1998): pp. 739-744.

2.16

Hertz, H." Über die ~ e r hmng u Fester Elastischer K; rper (On The Contact of

Elastic Solids)." J. Reine and Angewandte Mathematik. vol. 92 (1882): pp. 156171.

2.17

Johnson, K. L. Contact Mechanics. Cambridge: Cambridge University Press, 1985.

2.18

Swanson Anaiysis Systems. ANSYS, Engineering Analysis Systems. Release 5.5, 1999.

2.19

Wagoner, R. and Kaiping, L. "Simulationof Springback." Simulation o f Materiais Processing: Theory, Methods and Applications, Nurniform 98. (1998): pp. 2 1-3 1.

2.20

Gero, D. Aviation Disasters: The World's Major Civil Airliner Crashes Since 1950. London: Patrick Stephens Limited, 1998.

Chapter 3 Theoretical Investigations: Finite Element Modelling of Stretch Forming In this chapter, we discuss the finite element model used in simuiating the stretch forming process. The àifferent geometries and material models used in the analysis are discussed. Furthemore, the mesh design and refinement, which are used in descretizing the blank under consideration, are examined.

3.1

Stretch Fonning Tooliag Geometry

In the present finite element analysis, stretch forming is carried out by utilizing plane strain and axisymmetric conditions. Plane strain elements are used to represent a slice or unit thickness of the defonned blank whose geometry and loading conditions do not vary dramatically in the out of plane direction. Axisymrnetric elements are used to model blanks that are rotationally symmetric about an axis. In this case, the deformed blanks are subjected to axisymmetric loads from the forming punch and supporting die. A two dimensional analysis of a sector of the deformed blank is carried out in order to yield the complete stress and strain distributions.

Stretch formed blanks exhibiting axisymrnetric and plane strain conditions are presented in Figures 3.1 (a) and 3.1 (b), respectively.

Figure 3.1 Blank configuration: (a) Axisymmetric geometry, and (b) Plane strain geometry. Symmetry is further exploited by analyzing only one-half of the tooiing model. The basic parameters defining the stretch fonning tooling geometry are presented in Figure 3.2 and

detailed in Table 3.1.

1

1

Punch

t

Blank

I

I I I

I I

l

Die

Figure 3.2 Basic geometry parameters of the stretch fonning twling.

parameter

1

Description

1

a

Parr circumferential radius.

r

Part spherical curvature radius.

w

Part depth.

t

Part Blank thickness.

I

Table 3.1 Definition of the twling geometry key parameters.

3.2

Discretization of the Problem

A fundamental phase of the finite element modeLing is to mesh the blank by a finite

number of elements interconnected at their common nodes. The Aluminium blank under consideration is meshed by using isoparameteric quadrïlateral four noded elements. In order to improve the mode1 accuracy, element convergence checks were made. It is also important to emphasize that in nonlinear problems, which are solved iteratively, numerical errors resuking from the use of excessive elements may lead to numerical

diffkulties. The size and number of elements used in meshing the blank rely on the required accuracy, which in itself affects the computing time of the analysis. In order to obtain a converged finite element solution, it is required to Vary the size and number of elements within the blank mesh until a compromise between solution accuracy and computing time is achieved [3.1]. Since the blank undergoes stretching, bending and double sided contact from the punch and die shoulder, steep gradients in the stress and strain States are expected to exist in the blank. Thus, mesh cefinement is employed throughout the blank.

Mesh convergence tests were conducted by using the analogy ktween the deformation of sheets and beams, which is suggested by Johnson and Yu 13.21.For this purpose, an Al 3003-Hl4 Aluminium aüoy beam, which is fmed at both ends with the punch load

assumed to be concentrated at the beam centre, was analyzed. Beam deflection results obtained from the FE analysis were compared with the classical solution. Based on Figure 3.3 obtained from the conducted convergence tests, the slope of the deflection curve decays asyrnptotically, if six elements are used across the beam thickness.

Therefore, meshing of blanks was carried out using six elements across the blank thickness, as depicted in Figure 3.4.

-

-

1

1

1

1

1

1

1

1

1

2

3

4

5

6

7

8

9

Number of elements accross beam thickness Figure 3.3 Element size refinement convergence test-

4-noded quacirilateral element

O II I

Figure 3.4 Meshing of the blank with six elements across thickness.

3.3

Loading and Constraints in Stretch Forming

In the developed finite element mode1 of stretch forming, the blank is subjected to double-sided contact from: (i) the punch loading, and (ii) the die shoulder support. A prescribed vertical displacement w is assumed for the punch upper surface. In this case, the die provides the support necessary for the blank.

After achieving the required maximum punch travel corresponding to the specified part depth, the punch displacement pmcnbed in the loading stroke is released in substeps

allowing the blank to springback. In view of the plastic deformation associateci with the

process, the punch loading steps were exerted incrementally. Figure 3.5 shows the punch imposed displacement and its release.

Deformed Blank

(a) (b) Figure 3.5 Punch stretch forming strokes: (a) Loading stroke with a specified incremental vertical displacement w ,i = [O,n], and (b) Release stroke with w = w,

.

The incremental load is transmitted from the punch to the blank by using contact

elements. The normal k, and tangential k, stifbess parameters are used to form the stiffhess matrix for each contact element [3.3]. The tangential sticking contact stifhess k, is used as a penalty parameter in cases where

friction is described by Coulomb's law. It enforces the sticking component of the toolingblank contact friction, and corresponds to a contact stiffness in a direction tangential to the interface region. The normal stifiess k, is also used as a penalty parameter to

prevent interpenetration at the contact surfaces between the twling and the stretch

fonned blank. Figure 3.6 shows a typicai contact element.

Figure 3.6 A typical node-to-node contact element. The values of k, and k, are required to be very iarge. However, the use of excessively high values of k,

and k, leads to ill-conditioned global stiffness matrices, causing

numerical errors and divergence. On the other hand, utilizing smaller values of k, and

k, results in convergence to the wrong solution due to interpenetration and wrong estimates of the stick and slip regions. The variation in normal and tangential contact stiffness parameters is presented in Figure 3.7.

(a)

(b)

Figure 3.7 Force-deflection reiationship (Optimal contact condition at k N = k, = 0 ) : (a) Variation of nomal stiffhess k, ,and (b) Variation of tangential stiffbess k, .

Specification of the normal k, and tangentiai k, stiffhess parameters is very critical for the convergence of the analysis. The convergence criterion relies on the ciifference in the

values of the normal force F, on two successive iterations. If the difference is less than 5%, the solution substep is assumed to converge. Contact parameters used in the FE

analysis are sumrnarized in Table 3.2.

Table 3.2 Contact parameters.

As discussed previously in section 3.1. tooling and blank symmetry is exploited to

analyze only one-haif of the modeI. Thus, symmetry is carried out by restraining the radial displacement of the punch and blank at their centerlines. Simultaneously, the rigid die is fixed with respect to the frame of reference by imposing zero displacement values

on its radial and axial boundaries.

In order to support the defonned blank, the draw bead and clamp effects are simulated in the mode1 by using a cantilever constraint on the circumferentid boundary of the blank.

Hence, restraining the circumferential blank displacement in the radial and axial directions. The detailed tooling and blank displacement constraints are presented in Figure 3.8.

I

W

-----

Contact Ekment

Figure 3.8 Punch-die and blank constraints imposed during the specified loading and release strokes.

3.4

Yield Criteria

The yield criteria used in the finite element anaiysis of the stretch forming process, are described in this section. The selection of these critena is govemed by the behavior of the different Numinium alIoys examinai in the FE analysis.

3.4.1

von Mises Yield Criterion

The von Mises yield critenon is employed in characterizing the isotropic materid behavior. This model can be employed in the stretch forming of some Aluminium alloys such as isotropic Al-Zn 7075-T6 Aluminium alloy. The von Mises yield critenon is represented by the following expression: Y = a; =a: -a,-+a,

where

2

Y

yield locus function,

=Y

matenal yield stress, and

a,and a,

major and minor principle stresses.

(3.1)

In view of its high fatigue resistance, resistance to corrosion and high specific strength,

Aï-Zn 7075-T6 is currentiy being used in fonning of aircraft structures. Thus, N-Zn

7075-T6was analyzed by using an isotropic material model. The chemicai composition of Ai-Zn 7075-T6Aluminium alloy is given in Table A. 1 of Appendix A. The measured

mechanical properties of this type of Aluminium are provided in Table 3.3.

Table 3.3 Mechanical properties of Al-Zn 7075-T6Aluminium alloy (After [3.4,3.5]).

If the material under consideration exhibits an anisotropic behavior, other constitutive Iaws should be used. These anisotropic models are presented below.

3.4.2

Barlat-Lian Yield Criterion

Hill's (1948) yield model is employed in analyzing anisotropic material behavior in sheet metal fonning processes [3.6,3.7].This model relies on the Lankford coefficients b,

and b 0 -

where

Le

Lankford coefficient in 0 direction to the main rolling direction to the bIank,

E,

the blank width strain, and

Et

the blank thickness strain.

Barlat and Lian (1989) extended Hill's yield criterion and presented the following yield

surface taking into account anisotropy L3.81.

where

According to the experirnents conducted by Barlat and Lian, their model is more accurate in estimating the flow stress in Aluminium alloys than the model previously devehped by Hill f3.91.

The exponent n in the yield surface function is subject to a range of values starting from 2. For fcc crystal based materials such as Aluminium alloys, Bariat and Lian used an exponent value n = 8. By using the Lankford coefficients, the values for ai, follows:

a2,

b and c can be defined as

Ai-Li 2090-T8 Aluminium alloy, which has been recently used in various airfiame structures, has a characteristic anisotmpic material behavior [3.10.3.11]. This type of Aluminium was developed with 10% higher elastic modulus and 896 lower density than the previously discussed Al-Zn 7075-T6.Microstructural analysis of &Li

2090-T8

establishes a highly anisotmpic domain in the alloy. The average grain siw is 50 pm. 500

Pm, 3 mm in the short transverse (ST), long transverse (LT), and longitudinal (L) directions, respectively. Thus, an anisotropic material mode1 is essential when employing this alloy in the analysis. Weight percentages of Al-Li 2090-T8 indicating its chemical composition are presented in Table A.2 of Appendix A. The measured mechanical properties of Ai-Li 2oSeT8 Aluminium alloy are provided in Table 3.4.

Angle, 8

E (GPa)

a,n (MPa)

ouWa)

O

78.8

552

586

38

45

78.8

468

534

27

90

80

548

586

29

K,(MP~&)

Table 3.4 Mechanical properties of AI-Li 2090-T8Aluminium alloy (After [3.4,3.5]).

Tooling used in the finite element analysis consists of a punch and die system. Since the conducted analysis involves stretch forming of Aluminium alloy sheets. the material used in tooling should be tool steel. For this purpose, the punch and die used are specified to be rigid throughout the finite element modelling and analysis.

Material strain hardening is modelled by employing a bilinear elastoplastic hardening rule. Thus, in addition to specifying the material elastic rnodulus E, a tangentid modulus

Er representing the deelement software 13.121.

of work hardening must be provided to the ANSYS finite

3.5

References

3.1

Atlan, T., Kobayashi, S. and Oh, S. Meta1 Fonning and The Finite Element Method. New York: Oxford University Press, 1989.

3.2

Johnson, W. and Yu, T. X. "The Press-Brake Bending of RigidLinear WorkHardening Plates." International Journal of Mechanical Sciences. vol. 23 (198 1): pp.307-3 18.

3.3

Bathe, K. Finite Element Procedures. New Jersey: Prentice-Hall, 1996.

3.4

Ferahi, M. and Meguid, S. A. "Fatigue Fracture Behavior of Al-Li 2090-T8in the Presence of the Residual Stresses." International Conference on Fatigue. vol. 1 (1 993): pp. 427-432.

3.5

Ferahi, M. and Meguid, S. A. "Effect of Residual Stresses upon the Structurai Integrity of Advanced Ailoys." In Recent Advances in Experimental Mechanics. (1994): pp. 799-803.

3.6

Hill, R. "A Theory of The Yielding and Plastic Flow of Anisotropic Metals." Proc. Roy. Society. (1948): pp. 28 1-297.

3.7

Hill, R. "Constitutive Modelling of Orthotropic Plasticity in Sheet Metals." Journal of Mechanics and Physics of Solids. vol. 38 no. 3 (1990): pp. 405417.

3.8

Barlat, F. and Lian, J. "Plastic Behavior and Strechability of Sheet Metals: Part 1: A Yield Function for Orthotropic Sheets under Plane Stress Conditions."

International Joumal of Plasticity, Pergarnon Press. vol. 5 (1989): pp. 51-66.

3.9

Berg, H., Hora, P. and Reissner, J. "Simulationof Sheet Metal Fonning Processes Using Different Anisotropic Constitutive Modeb." Simulation of Materials Processing: Theory, Methods and Applications, Numiform 98. (1998): pp. 775780.

3.10

Noor, Ahmed K. and Venneri, Samuel L. Flight-Vehicle Materials, Structures,

and Dynamics. New York: The American Society of Mechanicd Engineers, 1994. 3.1 1

Smith, W. Structure and Properties of Engineering Alloys. New York: McGrawHill, 1981.

3.12

Swanson Andysis Systems. ANSYS, Engineering Analysis Systems. Release 5.5, 1999.

Chapter 4 Determination of Springback in Stretch Forming This chapter presents several methods used in quantifying springback. The chapter also discusses the MATLAB code used in determinhg springback in stretch formed components.

4.1

Determinhg S p ~ g b a c k in Stretch Forming

In this section, an attempt is made to define and numerically [email protected] spnngback. The surface of a general sheet metal workpiece can be defined by several radü of curvature, which Vary in sign and magnitude from point to point [4.14.3]. Figure 4.1 depicts a

blank exhibiting springback afler punch release. In the following, we sumrnarize several

numerical methods used to [email protected] springback.

Loading ~ontiguraiion

Figure 4.1 Springback depicted in an Aluminium dloy blank after punch release.

4.1.1

The Displacement Deviation Curve

After achieving the maximum punch travel, the blank oeutral plane nodal displacements define a deformation pattern or a loading curve. The punch is then released ailowing the neutral plane nodes to establish a different deformation trend or release curve due to springback. Displacement loading and release curves are attained by utilizing a cubic polynomial function fitting the blank neutral axis nodal displacements. The cubic polynomial can readily fit the nodal displacements as foff ows: 3

General Cubic Polynomial

z =ç a i r i i=O

Loading Curve Release Curve

where

coefficients of a cubic polynomial,

ai a

L

coefficients of the loading curve, and

R

coefficients of the release curve.

ai

A displacement deviation curve is then obtained by subtracting the reIease curve z from the loading curve z

,.The deviation curve defines the blank springback pattern dong the

neutral axis. The maximum value attained from the deviation curve is then taken as a discrete value for spnngback. Springback value as defined by this method is depicted in Figure 4.2, such that:

where

z

.,

displacement deviation curve function, and

SPB1 spnngback obtained h m the displacement deviation curve.

Figure 4.2 Springback defined by the displacement deviation curve. 4.1.2

The Average Function Method

The average values of the cubic loading z, and release z, curves discussed in the previous subsection can be obtained by using the foiiowing definite integrai:

where

z , average function value, and b

integration limit equal to the blank radius.

A multiple-application of the Simpson's mle is used to numencally integrate equation 4.6 [4.4]. Figure 4.3 shows the loading and release curves segmentation at their nodes.

number of equal width segments in the [O,b] range, and

w here

segment width.

z n segments

Figure 4.3 Blank loading and release configurations segmented at their nodes.

Equation 4.6 is applied to the cubic loading z, and release z, curves to attain their average function values z

and z

tV, respectively. Blank springback is then defined by

subtracting the average function values of the loading and release curves, as presented in Figure 4.4.

SPBZ = Izy -=FI w here

SPBZ springback obtained by using the average function method.

Figure 4.4 Spnngback defined by the average function method.

4.1.3

The VoirtualDeviationArea Method

Spnngback can also be defined as the virtual area extending between the cubic loading z

and release z

curves. Figures 4.5 and 4.6 depict the numencal integration utilized to

attain the springback value and the springback virtual area, respectively. The Simpson's rule introduced in the previous subsection is also utilized in detennining the value of the following springback definite integrai:

SPB3 = Springback Area =

(Z

-z

)dr

O

w here

SPB3 springback obtained by using the virtual deviation area method.

Figure 4.5 Numerical integration.

38

Figure 4.6 Virtual deviation area between the blank loading and release confr~gurations. 4.1.4

The Mean Normai Spacing Methoà

The Mean Normal Spacing is another method utilized in this thesis to define spnngback as the average distance ktween the loading z, and release z, curves. This average

distance is obtained by dividing the area between the loading and release curves by the average arc length of the two curves as presented in the following equation: SPB4 = mean normal spacing = w here

2A

(4.1 1)

LL +LR

SPB4 spcingback defined by the mean normal spacing method, A

area between the loading and release curves defined in the previous subsection,

,

..

L,

arc length of the displacement loading curve z ,and

L,

arc length of the displacement release curve z

The arc length values for the loading z and release z, curves are obtained by using the

following definite integral:

The integral in equation 4.12 is carrieci out by employing the Simpson's rule. Figure 4.7 depicts the blank springback, as defined by this method.

Figure 4.7 Parameters defining springback by the mean normal spacing method.

In three-dimensional geometries, springback will vary depending on the position. The use of a specific criterion wiii depend on the complexity of geometry and type of loading.

Hence, the use of the four previously discussed methods.

4.2

Cornputer Implementation for Determination of Springback

The springback detennination te~hniques discussed in the previous section are

incorporateci in a specially developed cornputer code. The developed code uses MATLAB in a rnodular structureci fashion [4.5]. Within the program, the modules are reaiized as procedures or functions with the main program invoking subprograms.

A hierarchy chart of the MATLAB program modules is presented in Figure 4.8. At the highest level a rectangle represents the main program. Below this rectangle are the

prirnary modules invoked by the main program, listed in order fiom left to right. Like-

wise each of these modules has its sub-modules displayed in a hierarchal fashion.

f

L

\

f

\

3

f

Results and Plots

Procedures and Functions

Input Parameters

/

\

&

Figure 4.8 Main modules and sub-modules of the developed MATLAB code. A detailed listing of the developed springback MATLAB code including the functions

and procedures utilized is provided in Appendix B.

4.3

References

4.1

Pearce, R. Sheet Metal Forming. New York: The Adam Hilger Series, 1991.

4.2

Loveil, M. and Narasimhan, N. "Predicting Springback in Sheet Metal Fomiing: An Explicit to Implicit Sequential Solution Procedure." Finite Elements in

Analysis and Design. vol. 33 (1999): pp. 29-42. 4.3

Kutt, L. M., etal "Non-Linear Finite Element Anaiysis of Springback."

Communications in Numerical Methods in Engineering. vol. 15 (1999): pp. 3342.

4.4

Canale, R. P. and Chapra, S. C. Numericd Methods for Engineers. New York:

McGraw-Hill, 1990. 4.5

Math Works, Inc. MATLAB. Natick, MA. 1999.

Chapter 5

Experimental Investigations This chapter is devoted to the experimental investigations employed in verifjhg the finite element mode1 of the stretch forming process. Whilst two experimental rigs were designed, covering plane strain and axisymmetric loading conditions, only the plane strain experiments were carried out. This is due partly to its significance to many fonning applications as weil as cost and time limitations. In this chapter, we provide a summary

of both designs.

5.1

Experimental Setup

This section discusses the alternative test rigs designed for the purpose of carrying out the experimental investigations. Details of the selected candidate test rigs are presented with their specific geometry. Furthemore, the geometrical and mechanical properties of the

blanks utilized in the experiments are provided.

Two candidate test rigs representative of the stretch forming process are designed to validate the results obtained numerically. One of them is designed to carry out the süetch forming process of axisymmetric parts. The designed test ng consists of several components which are detailed in Figure 5.1 and Table 5.1.

Figure 5.1 Exploded view of the test rig designed to form axisymmeaic parts: (a) Lower mount, and (b) Upper mount.

"

I

umber Component

1

I

I

Description used to c o ~ e cthe t punch to the head of the Instron Machine.

\Top Base

can be changed dlowing for flexibility in part radius. 1

1Blank Holder

3

employed in clamping siretch formed aluminium blanks, provides a smooth Nlet radius for the stretch formed blanks, used to carry the Die - Blank Holder assembly,

1

1 Die Carrier Support Angles

6

1 provide fiont and rear support to the Die Carrier. -

1

provide side-to-side support to the Die Carrier.

7

T OSide-to-Side ~ Angles

8

Plunger-Cylinder Subassembly measures springback using a helical spring and a clip gauge.

9

Channel

used as the primary support to the Die Carrier.

10

Channel Support Angles

provide side-to-side support to the Channel.

IBottom Base

11

1 used to a w h the Channel to the Instron Bed.

Table 5.1 Description of the parts used in the axisymmetric test ng.

In view of cost and time limitations, an alternative plane strain test rig was designed, built and used in the study. This plane strain test rig is presented in Figure 5.2 and its components are detailed in Table 5.2.

Pm Number Component

Description designed to have a cylindrical shape.

1

Punch

2

Blank Holder used to support and align the stretch fonned blanks.

3

Die

designed to have a generous fiilet to provide smooth stretch forming.

4

Base

carries and supports the test rig assembly.

Table 5.2 Details of the components involved in the plane strain test ng.

(2) Blank

(3) Die

Figure 5.2 Exploded view of the test ng designed to form plane strain parts. Details of the plane strain and axisymmeîric test rigs are presented in Appendix C.

5.1.2

Bïank Design

The details of the geometrical and mechanical properties of the tested blank spccimens are sumrnarized in Table 5.3.

Properties of Tested Blanks

Description

Length (mm)

125

F

Thickness (mm)

1, 15, and 2

Material

Al 3003-Hl4 Aluminium AIloy

1Yield Streagth, a, .

(MPa)

1

125

Ultimate Strength, a, ,(MW

155

Young's Modulus, E,(GPa)

69

Poisson's Ratio, v

0.35

Table 5.3 Geometrical and mechanical properties of tested blank specimens.

5.2

Messuring Devices and Machine Caiibration

In order to obtain the load-deflection cwve of the tested blanks, Instron uniaxial testing system, with its load cell, was utiiized to measure the fomiing forces required, while the

LVDT device traces the blank vertical deflection t5.11. The stiffbess of the testing machine affects the load-deflection curves attained from the LVDT and the machine load cell. Hence, calibration of the testing machine was carried out by applying compressive loads on a very rigid block of dimensions (5 x 5 x 2.5 cm), which acted as a very stiff spnng. Figure 5.3 shows the machine calibration procedure.

Results obtained from the compression test are presented in Figure 5.4. At the end of the loading stroke, an accurate dia1 gauge with a resolution of 10 pm is positioned beneath the defomed blanks and preloaded to a displacement of 5 mm. The punch is then released allowing the blank to spnngback and the dial gauge preloaded displacement to be relaxed to a lower value. By subtracting the final relaxed displacement

from the preloaded one, the maximum unloading displacement of the blank is obtained 15.21.

i Loading Ram

Figure 5.3 Calibration of the testing machine.

-

load used in stretch fonning experiments = 3.53 kN

0 Maximum

-

-

O

0.006 0.012 0.018 0.024 0.03 0.036 0.042 0.048 0.054 0.06

Testing niachine deflection, x, (m)

Figure 5.4 Load-deflection curve of the testing machine.

5.3

Test Rig Assembly

The test rig cylindrical punch is fmt ciamped into the Instron crosshead using special

grippers. The die is then attached to the Instron bed. Carefully machined Aluminium blanks are then positioned and held between the blank holder and die assembly with the

aid of assembly bolts. The test rig assembly is illustrateci in Figure 5.5.

Load

Punch Blank Die

auge

Instron

Figure 5.5 Plane strain test rig assembly.

5.4

Punch Loading and Release Strokes

Specified punch displacements are fed into the electronic panel of the test system causing the punch to load the Aluminium blanls. During the punch loading step, the loaddeflection curve of the deformed blank is plotted instantaneously until the maximum

specified displacement is reached. The displacement is maintained, while positionhg a high-resolution dial gauge beneath the deformed blank. Before executing the punch release stroke, the dial gauge is preloaded to a specific displacement, as discussed previously in section 5.2. It is significant to note that punch loading and release strokes were conducted at a specified loading rate of 0.65 mm/ssec and 0.75 mm/sec, respectively. Table 5.4 presents the punch loading and release parameters used in the tests.

Loading Rate,

Punch Travel, w, (mm) 10

Release Rate,

Dia1 Gauge Reload Displacement,

(d-) ( d = c )

1

0.65

(mm)

0-75

5

Table 5.4 Punch loading specifications.

5.5

Springback and Load-Deflection Characteristics

The springback results acquired fiom the experiments were compared with the finite

element predictions. Furthemore, the load-deflection curves obtained from the numerical

and experimental models were also compared. Conclusions and discussions on the FE predictions and experimental findings are provided in chapter 6.

5.6

References

5.1

Allison, 1 M. Experimental Mechanics: Advances in Design, Testing and Analysis. Netherlands: A. A. Bakema, 1998.

5.2

Metals Handbook: Mechanical Testing. American Society of Metals vol. 8 (1985): pp. 5 6 5 6 5 .

Chapter 6 Resuits and Discussions This chapter is devoted to the analysis and discussion of the results obtained from the conducted finite element analysis and experimental investigations. The developed finite element model is validated prior to implementation by utiking two different approaches: (i) carrying out the Limiting Dome Height benchmark testing, and (ii) conducting an

extensive experimental work to provide confidence in the developed finite element models. In the remainder of this chapter, the effect of shape and interface parameters on the springback is also examined. We will also apply the developed finite elements model to analyze the stretch fonning process of an aircraft bulkheaâ, which is discussed as a

relevant case study.

6.1

Validation of the Finite Element Mode1

Two methods were used to validate the finite element results. In the first, the FE predictions were compared against previously established standard tests as will be presented in the Limiting Dome Height benchmark testkg. In the second method, we compare the experimental findings with the finite element predicnons.

6.1.1

Limiting Dome Height Test

The Limiting Dome Height (LDH) test was fmt introduced as a benchmark at the Numisheet 96 conference [6.1-6.4]. 1t is effective in comparing finite element models and experiments for the validation of the attained results. The test tooling is detailed in Figure 6.1.

Dimensions in mm 132.6

Figure 6.1 Geometry and dimensions of the Limiting Dome Height test tooling.

The model can be reduced in size by exploiting the tooling symmetry evident in Figure 6.1. Therefore, only one-half of the model is analyzed using four noded axisymmetric

quadnlateral elements. Meshing is conducted using six elements across the blank

thickness in the range from the axis of symmetry to the lockbead position. The lockbead constraint is modeled with an equivalent cantilever constraint imposed on the blank periphery. Table 6.1 summarizes the deformeci blank mechanical properties.

'Parameter Elastic Modulus, E, (GPa) poisson Ratio, v

Magnitude

1

206 0.3

1

Table 6.1 Mechanical and geometncal parameters of the Limiting Dome Height test (After [6.2]). Employing ANSYS finite element software, the Limiting Dome Height test was carried out by incrementally applying a net upward displacement of 30 mm on the punch bottom surface. The analysis was conducted at 40 equal displacement substeps (each substep =

0.75 mm). The deformed blank geometry at the completion of the loading stroke is presented in Figure 6.2. This Figure illustrates that at a maximum punch travel of 30 mm, the deformed blank exactly rnirrors the twling configuration. However, if the punch is incrementally released the defonned blank wiil undergo geometric deviations caused by elastic recovery.

Figure 6.2 Blank defonned geometry at the end of the loading stroke (w- = 0.03 m).

Major strains dong the blank radial r-axis are depicted in Figure 6.3 at a net punch travel of 30 mm. The experimental results of Numisheet 96 [6.1] and the FE predictions of

Carleer etal. [6.2] are also included for the sake of mode1 validation. The major saains

are traced at the outer blank path opposite to the punch-blank contact region. It can be established from Figure 6.3 that the finite elemnt solution agrees with the experimental results dong the blank radius. A maximum discrepancy of approxirnately 14.6% at R/t =

28 exists between the fMte element model predictions and the experimental results of

Numisheet 96 [6.1].

0.25 -

-F

i i Ekment Soiution (Cwrent)

0.05 - -------- Experimental Resuhs (Numisheet % [6.1])

O

10

20

30

40

50

Nornialized blank radius w ith respect to thickness, Rit Figure 6.3 Major strains obtained from the Limiting Dome Height test.

The finite element model was further verified by conducting stretch forming experïments carried out using the plane strain test rig outlined in Chapter 5, and detailed in Appendix C. The selection of the plane strain model was governed by its signifcance to different

sheet metal forming applications. Mechanical properties obtained for the tested Al 3003H l 4 Aluminium blank specimens were introduced previously in subsection 5.1.2. The

defonned Aluminium blank geometry at the end of the loading stroke is presented in Figure 6.4 and the blank geometric parameters are detailed in Table 6.2.

Figure 6.4 Deformed blank geometry.

Description

Parameter

Blank initial length = 125 mm Punch travel (variable, { 10, 15,20 mm))

L, w w

,

.

Blank width = 44 m m

,

Wd

Die width = 9.5 mm

rd

Punch radius (variable, ( 12.5, 17.5 mm}) Die radius = 6.35 mm

Table 6.2 Defomed blank geometrk parameters. Expenmentai testing was carried out using different interfacial friction conditions at the Steel tooling - Aluminium blank mating surfaces. The interface friction cases are given in Table 6.3.

1 bnlubricated dry interface 1

luiterface lubrication

Coefficient of friction, p

1

0.5

1 1

Table 6.3 Tooling-blank interfacial friction conditions (After [6.5.6.6]).

Aluminium blanks with a thickness of 1 mm undenvent successive loading and release strokes, designed to provide an insight into the material behavior. The testing was conducted under an unlubricami dry interfaa condition (p = 0.5) at the toolinghlank contact surfaces [6.5,6.6]. In order to carry out this loading/release test, a spacified displacement history is fed into the control panel of the Instron testing sysiem, as presented in Figure 6.5. The successive loadinglrelease stretch forming curve is shown in

Figure 6.6.

Tim, T, (sec) Figure 6.5 Blank Ioading/release displacement history.

-

Uniubricated interface ( CL= 0.5) Blank thickness t = 1 mm

-

Punch travel, w,(nnn) Figure 6.6 Successive blank stretch fonning loading/release curve (Unlubricated contact surfaces).

Results obtained fiom Figure 6.6 establishes that due to minor hysteresis effects, the reloading curves 3-2 and 5 4 closely followed the eariier release curve. Tt can aïs0 be noted that for the reloading curves 3-2 and 5-4, the elastic Limit or the cumnt yield has increased as a result of strain hardening, which transpired during the earlier loading of the tested blank specimen. The successive loadinghlease simulation is illustrated in Figure

6.7.

Scaie 1:2.82

Scale 1:2.49

Scale 1:2.65

(a) Initial state.

(b) Loading stroke (Punch travel = 20 mm).

(c) Release stroke.

Figure 6.7 Loading and release strokes of tested blanks.

In order to v e r w the finite element findings, experimental and FE results were compared for the case of a prescribed punch travel of 20 mm. The experiments were conducted using a cylindrical punch having a radius r, = 17.5 mm, and an unlubncated tooling-blank interface condition. For this case, a coefficient of friction p = 0.5 for Aluminium-Steel

mating surfaces was assumed [6.5,6.6]. The Results obtained from the FE analysis and the conducted experiments are presented in Figure 6.8. Figure 6.8 shows that a maximum discrepancy of 11% exists between the FE predictions

and the experimental results. Discrepancies between the predictions of the finite elements analysis and the experimental measurements can be attributed to the effect of the exact constitutive law of the material, the friction conditions, and the coaxiality of the applied load. With regard to the conducted expenments, due to the flexibility of the thin sheet metal parts, blanks may have been subjected to pre- or pst-forming deformations durhg clarnping and removing from the stretch forming test rig. Considenng these factors, the

agreement between the FE prcdictions and experimentai results can be considered satisfactory.

2.5

3 w

a

2 -

1.5 -

Experiment (unhibrkated interface = 0.5)

-----Fmite Element Solution (bhnk thicicness t = 1 mm, and punch

radius rp = 17.5 mm)

2 O

e

= O

1 -

2 O

2.5

5

7.5

10

12.5

15

17.5

20

22.5

Punch travel, w ,(mm) Figure 6.8 Variation of the stretch forming force with punch travel for a blank thickness t = 1 mm.

6.2

Parameters Inf'luencingSpringback

The effect of the geometric features and the tooling-blank interface fiction conditions upon the resulting blank springback are discussed in this section.

in order to evaluate the effect of Al 3003-Hl4Aluminium alloy blank thickness upon the resulting springback, three specimen thickness values of 1 mm, 1.5 mm, and 2 mm were

chosen. The finite element solutions and experimental investigations were conducted

using a punch radius r, = 17.5 mm. An unlubricated tooling-blank interface condition with a coefficient of friction p = 0.5 was assumed [6.5,6.6]. The results obtained are

given in Figures 6.9 and 6.10.

-

Experïment (unhrbricated interface p.= 0.5)

-----Fmite Eiement Soiution (bhnk radius rp = 17.5 mm)

-

O

2.5

5

7.5

10

12.5

15

17.5

20

22.5

Punch travel w, (mm)

Figure 6.9 Variation of the stretch fonning force with punch travel for a bIank thickness t = 1.5 mm.

-

-

-

Experiment (uniubkated mterface p= 0.5)

-----F i Element Soiution (bhnk radius rp = 17.5 mm)

-

-

Punch travel, w, (mm)

Figure 6.10 Variation of the stretch forming force with punch travel for a blank thickness t = 2 mm.

From Figures 6.9 and 6.10, it is evident that the stretch fonning force increases with the increased blank thickness. The punch loading force has increased by 29.7% due to an increase in the blank thickness from 1.5 to 2 mm. Springback results obtained by varying the blank thickness are given in Figure 6.1 1.

-

Expriment

-

-----FE (Maximum Deviation, SPB 1)

-

FE (Averagie Deviation, SPBS) FE (Mean Nomai Spacmg, SPB4)

---.-a

-

-

- -.--.- - --

-.

-

s

-

-

-.- - -. ------_-_ ----_-----.___ -_ -*

-O----

---------_________

-

0.75

1

1.25

1.5

1.75

2

2.25

Blank thickness, t, (mm) Figure 6.1 1 Variation of springback with blank thickness (punch radius r, = 17.5 mm. punch travel w = 20 mm,and p = 0.5).

It can be extracted from Figure 6.1 1 that a maximum discrepancy of 10.3% exists between the springback experimental results and the FE maximum deviation prediction SPB 1. It also shows that the amount of springback, as given by the conducted

experiments or numerically detennined by the FE maximum deviation SPB 1, the average deviation SPB2 and the mean normal spacing SPB4, decreases by increasing the blank thickness. This finding can be attributed to the increased blank stifiess caused by increasing the blank thickness. Based on the mean normal spacing method, by increasing the blank thickness from 1.5 to 2 mm, a springback reduction of 18%is attained.

6.2.2

EEect of Punch Travel

In this subsection, we concentrate on the sensitivity of blank springback to the amount of plastic deformation caused by the prescribed punch travel. Blanlcs with 2 mm thickness were chosen for testing, since it was established in the previous subsection that they provided lower springback values than the thinner blanks. The punch travel w was varied

from 10 mm to 20 mm in steps of 2.5 mm. A limiting value of w = 20 mm was used, since values of w 2 25 mm resulted in inappropriate wrinkling and earing in the vicinity

of the blank holder. Wrinkling of tested blanks at a punch travel w = 30 mm is shown in Figure 6.12. Edge Wrinkli

Figure 6.12 Wrinkling at a punch travel w = 30 mm. The analysis was carried out using the same punch, which has a radius r, = 17.5 mm, and

under a tooling-blank unlubricated dry interface condition with a coefficient of friction p = 0.5 [6.5,6.6]. The results obtained from the experiments and FE analysis for a punch

travel w = 15 mm are presented in Figure 6.13. Springback findings, as defined by the

maximum deviation SPBf and the average deviation SPB2, are also given in Figure 6.14.

-Experiment (uniubricated -

mterface jL = 0.5)

- -----F i Ekment Solution (bhnk tbickness t = 2 mm, and punch radius rp = 17.5 mm) -

O

2.5

7.5

5

12.5

10

15

17.5

Punch travel, w, (m) Figure 6.13 Variation of the stretch forming force with punch travel for a maximum punch travel w , = 15 mm.

FE (Maximum Deviation, SPB 1)

a--

7.5

10

12.5

15

17.5

20

22.5

Punch travel, w ,(mm)

Figure 6.14 Effect of varying punch travel w on blank springback (punch radius r, = 17.5 mm, blank thickness t = 2 mm, and p = 0.5).

The findings extracted h m Figures 6.13 and 6.14 illustrate that the blank springback

The springback was reduced by approximately 27.7% as a result of decreasing the punch travel w €rom 20 to 10mm. This

decreases by reducing the prescribed punch travel

W.

trend of the blank springback is due to the reduction in the elastic recovery energy at lower plastic deformations.

6.2.3

Effect of Punch Radius

The analysis was further extended to examine the effect of varying the punch radius on springback. For this purpose. a smaller punch having a radius r, = 12.5 mm was utilized.

Experiments were also carried out on blanks having a thickness of 2 mm and by assuming a coefficient of friction p = 0.5 for an unlubricated dry tooling-blank interface condition [6.5,6.6]. The springback results obtained herein are given in Figure 6.15. Figure 6.16 also presents the variation of the springback, as defined by the mean normal spacing SPB4,with the punch travel for two dinerent punch radii.

Expriment FE (Maximum Deviation, SPB 1)

-0

7.5

10

12.5

15

17.5

20

22.5

Punch travel, w, (mm) Figure 6.15 Effect of varying punch travel w on blank springback (punch radius r, = 12.5 mm, blank thickness t = 2 mm,and p = 0.5).

.-

.-.. ---

-.O--

a-------

*-

-0

FE Mean Normal Spacing, SPB4 (punch radius rp = 12.5 mm) FE Mean Normal Spacing, SPB4 (punch radius rp = 17.5)

7.5

10

12.5

15

17.5

20

22.5

Punch travel, w, (mn) Figure 6.16 Predicted dependence of springback on punch radius.

Figure 6.16 indicates that smailer values of the punch radius r, for the same blank

thickness and interface fiction condition result in smdler springback values in the deformed blanks. Springback reduction of 14.2% is evident at a punch travel w = 20 mm. as given by the mean normal spacing (SPB4)method. 6.2.4

Effect of Tooîing-Blaak Interface Friction

The main emphasis of this subsection is to focus on the effect of friction and lubrication

on the blank springback behavior. The stretch forming deformation process involves high loads and pressures at the tooling-blank contact surfaces. These forces and pressures were mitigated by using a fluid film to lubricate and cushion both the punch-blank and dieblank contact interfaces. For this purpose. grease NUiI Grade 2 was used in the conducted experiments. According to Bhushan and Szen [6.5,6.6], a coefficient of friction p = 0.05 can be assumed for this type of grease, as given previously in Table 6.3.

The loading was carried out using a prescribed punch travel w = 20 mm through a punch with a radius r, = 17.5 mm. The blank thickness was 2 mm. The results obtained are

depicted in Figure 6.17.

3 2.5 -

Experiment (wet interface usmg Grease NLGI Grade 2 p= 0.05)

-----Finite Ekment Solution (bhnk thickness t = 2 mm, and punch radius rp = 17.5 ~ i m )

a

O

2.5

5

7.5

12.5

10

15

17.5

20

22.5

Punch travel, w, (mm)

Figure 6.17 Variation of the stretch forming force with punch travel under a wet interface condition using Grease NLGI Grade 2.

It is evident from Figure 6.17 that the stretch forming force decreases with decreased

-

interfacial friction at the tooling-blank contact surfaces. Compared to Figure 6.10, which indicated a maximum punch loading force P

= 3.53 kN at p = 0.5, w = 20 mm, and for

the same blank thickness and punch radius, results obtained from applying grease as a

lubricant with p = 0.05 provided a maximum load P=,

2.4 kN;a decrease of 32%. In

order to investigate the effect of different interface fiction conditions at the tooling-blank contact regions upon the resulting springback, the coefficient of friction p was varied in the FE analysis from 0.05 to 0.5. Springback values obtained from the experiments and

FE predictions are given in Figure 6.18.

O

O. 1

0.2

0.3

0.4

0.5

0.6

Coulomb coefficient of friction, p Figure 6.18 Predicted dependence of springback on friction at the tooling-blank contact surfaces.

Figure 6.18 reveals that introducing interface lubrication by using grease NLGI Grade 2

(p = 0.05) reduces interfacial friction causing a decrease in the tangentid stretching force acting on the deformed blanks, and thus an increase in the amount of springback. Grease provided a springback increase of 22.3% relative to the unlubricated dry interface condition (p= 0.5) at t = 2 mm, r, = 17.5 mm, and w = 20 mm. The above results show clearly that increasing the blank holder force as a result of friction decreases springback. This is due to the fact that in stretch forming, two deformation mechanisms operate: bending deformation and tensile deformation. Increasing the tensile deionnation through friction decreases the bending component leading to an overall reduction in springback. This is possible, since the compressive component due to bending is greatly reduced.

63

A Case Study: Formhg of Aireraft Rear Bulkheads

In this section, we will apply the developed FE mode1 to the stretch fonning of an aircraft rear bulkhead, which is chosen as a case study. The Boeing 747 Al-Zn 7075-T6

Aluminium alloy bulkhead under consideration has the reference geometrical parameters presented in Figure 6.19 and Table 6.4[6.7,6.8].The anaiysis will focus on obtaining the

normal contact stresses as well as springback in the stretch formed bulkhead.

Figure 6.19 Aircraft rear bulkhead: (a) Bukhead geometry, and (b) Basic buikhead shape parameters.

Table 6.4 Geometric parameters of reference bulkhead (After [6.7,6.8]). The bulkhead blank was modeled using 4-noded quaddateral elements under the conditions of axisymmetric loading. In view of the large plastic deformation that the blank undergoes during forming, the punch loading stroke was exerted incrernentally.

Adjustment of the load step size was canied out using 55 substeps. The development of

the contact zones corresponding to the punch-blank and the die-blank mating interfixes

as welI as the resulting contact stress distributions are presented in Figure 6.20. The

length of the contact zone is normalized by the punch radius r.

&

575

.

-Punch-Blank Contact Interface

I

O

0.1

0.2

0.3

0-4

0.5

0.6

0.7

0.8

0.9

1

Normalized contact length, ah

-----Die-Bhnk Contact Interface

I/ @

'\ 7

Normalized contact iength, a(t

Figure 6.20 Variation of normal contact stresses dong the normalized contact length of a stretch formed Al-Zn 7075-T6bulkhead: (a) Punch-blank mating surfaces, and (b) Die-blank mating surfaces.

The results obtained fiom Figure 6.20 show that large contact stresses exist at the punchblank and die-blank interface regions. The maximum contact stress at the die-blank interface is comparable to that of the punch-blank contact stresses. Furthemore, the

maximum contact stresses do not appear within the contact area rather than the edges for both regions of contact. Springback results obtained for the formed bukhead using Al-Zn 7075-T6 is presented in Figure 6.21.The work was also extended to examine the springback behavior of a new

alloy Al-Li 209eT8, which is currently being considered as a replacement for Al-Zn

7075-T6.The results for the new Aluminium alloy are also s h o w in Figure 6.2 1.

-e m

vi

0.08 I Maximum Deviation, SPB 1

0.07 -

Average Deviation, SPB2

A l - a 7075-T6

Al-Li209eTS

Aluminium ailoys Figure 6.2 1 Normalized bulkhead springback of two Aluminium alloys.

Figure 6.21 establishes that Al-Li 2090-T8provides a maximum reduction in spnngback of 14.5% over the currently used Ai-Zn 7075-T6 Aluminium alloy. This reduction is due to the increase in the elastic modulus E of Al-Li 209eT8 over Al-Zn 7075-T6. This

increase in E results in a reduced unloading strain.

6.4

References Lee, J. K. etai. "Numerical Simulations of Sheet Metal Forming Processes, Verification of Simulations with Experiments." Numisheet 96 (1996). Carleer, B. D., ~ u e h n k J., , Pijlman, H. H. and Vegter, H. "Application of the Vegter Criterion and a Physically Based Hardening Rule on Simulation of Sheet Metal Fonning." Simulation of Materials Processing: Theory, Methods and Applications, Numiform 98 (1998): pp. 763-768. Batoz, I. L., Bouabdallah, S., Guo, Y. Q., Mercier, F. and Naceur, H. "On Some enhanced computational aspects of the inverse approach for sheet fonning

analysis." Simulation of Materials Processing: Theory, Methods and Applications, Numiform 98 (1998): pp. 807-812. Thompson, R. "The LDH Test to Evaluate Sheet Metal Formability - Final Report of the LDH Cornmittee of the North American Deep Drawing Research Group." Sheet Metal and Starnping Symposium, SAE SP-944 (1993): pp. 291301.

Bhushan, B. Principles and Applications of Tribology. New York: John Wiley &

Sons, 1999. Szeri, A. 2. Fiuid Film Lubrication, Theory and Design. Cambridge: Cambridge University Press, 1998. Raymer, D. P. Aircraft Design: A Conceptual Approach. California: Lockheed Aeronautical Systems, AIAA Education Series, 1989. Torenbeek, E. Synthesis of Subsonic Airplane Design. Delft University Press, 1976.

Chapter 7 Conclusions and Future Work 7.1

Statement of the Problem

Stretch forming of sheet metal is an important manufacturing process employed in several industries to fom panels of cornplex and varied curvatures. Despite the process ubiquitous application, the design of the tooling and the selaction of the sheet materials are still based on a trial and e m r approach. 1t is therefore the objective of this thesis to

carry out a systematic study of stretch forming using the finite element method. Of

particulai- interest is the springback resulting from unloading.

7.2

Objectives

To achieve this objective, the following tasks were necessary: develop a finite element mode1 accounting for material and geometrical nonlinearities and interface conditions, in order to obtain the stress and strain States resulting from the stretch forming process, obtain the blank final configuration after releasing the tool loading so as to determine springback, apply the developed finite element mode1 to treat the stretch fonning of the bukhead of an aircraft, and carry out an experimental work to veriw the developed FE models.

7.3

General Conclusions

The developed finite element model accounted for the materiai and geometric nonlinearities, and interface conditions present in the stretch forming process. The elastoplastic large deformation material models adopted used the von Mises yield criterion for isotropie materials, and the Barlat-Lien yield model for anisotropic materiais, such as Ai-Li 2090-T8 Aluminium ailoy. To account for work hardeniag, a bilinear strain-hardening model was utilized. Frictional contact at the twiing-blank interface was also introduced using node to surfixe contact elements. Different methods were used to verify the accuracy and reliability of the finite element model. Convergence tests were conducted to assess the validity of the eIement size used in the analysis. Results from the Limiting Dome Height (W)H) benchmark testing

showed close agreement with findings obtained from the FE model concerning the principal strains resulting during blank deformation. The conducted finite element analysis enabled the evaluation of the strain and stress states in the deformed blank d u h g forming as well as the residual stresses after punch release. Spnngback of the blank was also attained by releasing the punch loading.

7.3.2

Determination of Springback

Springback obtained from the deformed blanks at the end of the punch release stroke was defined using several techniques. These include: (i) the displacement deviation curve, (ii) the average function method, (iii) the virtual deviation area technique, and (iv) the mean

normal spacing method. The four techniques have been incorporated using MATLAB.

The effect of geometric parameters, such as blank thickness and punch radius, on springback was examined. The results also reveal the importance of interfacial friction conditions upon the resulting springback. Determination of springback in the stretch

forming of aircrafk rear bullcheads was also carried out as a case study, to demonstrate the utility of the developed FE d e l .

7.3.3

Experimental Simulation of Stretch Fomüng

Experimentai work was conducted to validate some of the finite element models used to simulate the stretch forming process. For this purpose, two test rigs were designed: (i) axisymmetric model, and (ii) plane strain model. In view of time and cost limitations, the plane strain test rig was selected for testing of springback. Findings obtained fiom experimental testing pertaining to blank springback and load-deflection curves proved to

be in agreement with the FE predictions.

7.4

Thesis Contribution

The primary contribution of the current thesis work can be summarized as follows: the development of a stretch forming finite element rnodel using ANSYS,

the development and implementation of a MATLcAB code to determine springback, employing several numerical techniques such as the displacement deviation curve and the mean normal spacing method, the determination of the effect of tooling/blank geometry and interface

conditions on springback, the application of the developed finite element model to treat a case study involving the stretch forming of a rear bulkhead of an aircraft, and the design and manufxture of a versatile test rig used to carry out the experimental investigations of the stretch forming process.

7.5

Future Work

The following areas are worthy of m e r research:

development of an adaptive mesh for elastoplastic sheet metai fonning finite ekment modeling, introduction of wrinkling anaiysis and bifurcation prediction into the developed springback code, and development of nonlinear constrained optimization algorithm in order to control springback, wrinkling and bifurcation in sheet metal forming.

Appendices Appendix A:

Chernical Composition of Aluminium Ailoys

77

Appendix B:

Listing of a Sample MATLAB Code

78

Appendix C:

Experimental Test Rigs C. 1

Axisymmetric Test Rig

C.2

Plane Strain Test Rig

Appendix A

Chernical Composition of Aluminium M o y s The chemical composition of Al-Zn 7075-T6 and AI-Li 209eT8 Aluminium alloys, which are used in the FE analysis, are provided in Tables A. 1 and A.2, respectively.

AL-Zn 7075-T6

Zn

Fe Cu Mg 5.1-6.1 2.1-2-9 1.2-2.0 0.5

Si 0.4

Ti Cr Al 0.3 0.184.28 0.2 Remainder

Mn

Table A. 1 Ai-Zn 7075-T6 Aluminium alioy chemical composition (wt %) (After [A. 11).

Zr Fe Z n Si Cr M n Al 2.4-3.0 0.25 0.15 0.08-0.15 -0.12 0.1 0.1 0.05 0.05 Remainder

Cu

Mg

Ti

Table A.2 AI-Li 2090-T8 Aluminium alloy chemical composition (wt %) (After rA.21).

References A.l

Pao, P. S., Imam, M.A., Cooley, L. A. and Yoder, G. R. "Cornparison of Corrosion-Fatigue Cracking of Al-Li Ailoy AA 2090-T8 and Alloy AA 7075-T6 in Saltwater."Corrosion. vol. 45: pp.530-535.

A.2

Goodyear, M. D. "Alcoa Green Letter: Alcoa Alloy 2090." GL 226. 2* Edition. Alcoa Center. 1989.

Appendix B

Listing of a Sample MATLAB Code The following developed MATLAB code is utilized in detennining springback in a stretch fonning process.

The nodal displacements obtained from the blank loading z, and release z, curves are

required as an input to the code. The code carries out several procedures and functions. The primary functions used are as follows:

1 Function 1 numint

1 arien

Description

I

1 numericai intepation using Simpson's rule. as presented in equation 4.6. I 1 numerical intepation using a mdtiple-application of Simpson's nile, as given in equation 4.7. 1 numericai i n t e m o n usinp; Simpson's nile, as established in equation 4.12. 1 Table B. 1 Functions utilized as subroutines.

The code yields results and plots pertaining to different definitions of springback, as

discussed in section 4.1. These definitions are given below in Table B.2.

Sprin~backParameter SPB 1 SPB2 SPB3 SPB4

Definition Springback as defined by the Displacement Deviation Curve. Springback as quantified by the Average Function Meuiod. Springback as defined by the Virtual Deviation Area Method. Springback as quantifiai by the Mean Normal Spacing Method.

Table B.2 Parameters employed in quantifying springback.

% % % % % % %

******************************************************************** bdATLAB Coda -1-d i a Qumntifying Springback *** et* in tha Strotch Eoriirrg Procorr ***

*tt

**************************************t*****************************

*** **t

Programmer: Fahd Fathi Abd El All Supemisor: Prof, S - A , Meguid

+*t.

***

********************************************************************

%

for yy=l :10 fprintf(l\t***t****t***********t**tf**f*******~************************ \n '1

fprintf('\tx** Quantifying Springback in Stretch Forming

*** \n

')

fprintf(l\t*f***X***f******ttt*tt***ft*********************************

\n ' 1 fprintf('Enter number of points in the displacement loading/release curves \n') m = input(' m = ' ) ; n = m-1; % *** Initialization x = zeros(m,l) ; y1 = zeros (m,1); y2 = zeros (m,1); % % % & ! %

................................................................... *** *** Input Parameters

*** t**

***

***

...................................................................

fprintf('Enter r-coordinates for the loading/release curves (tmits:meteres) \n') for i = 1:l:m fprintf('Point %g)\n8,i) x(i,l)=input(' r = ' 1 end; fprintf('Enter z-coordinates for the loading curve (units:meters) for i = 1:l:m fprintf( ' Point %g)\nt,i) yl(i,l)=input(' z-loading = ' ) end; fprintf('Enter z-coordinates for the release curve (units:meters1 for i = 1:l:m fprintf ( ' Point %g)\nt,i) y2 (i,1)=input ( ' z-release = ' ) end; %

3 3 % $

% % %

.................................................................. **+ * k t

Procedures and Functions

*** *** ******************************************************************

fuiction f odsum] = oddsum (Y,m) * * * summation over odd indexes *** odsum = 0; for k = 1:2:m-2 odsum = odsum+y (k,1)

%

*** ***

end; function [evsurn] = evensum(y,m) % * * * Sununation over even indexes *** evsum = 0; for j = 2:2:m-1 evsum = evsum+y ( j ,1 ) end; function [nint] = numint (x,y,odsurn, evsumIm,n) % * * * Numerical integration using a multiple-application of the Simpson's 1/3 % technique * * * a = x(1,l); b = x(m,1); y0 = y(l,l); yn = y(m,l); c = 4*0dsum; d = 2*evsum; nint = (b-a)* ( (yO+c+d+yn)/ (3*n)) ; function [yd] = derlist (pd,x,m) % * * * Derivative List *** for 1 =l:m yd(l,l)=polyval(pd,x(l,l) end;

function [yddl = arlen (yd,m) 3 * * * Arc Length Function * * * for s =1:m ydd(s,l) = (((yd(s,l))"2)+1)^0.5 end; %function [devl = deviat (yl,y2,m) % * * * Displacernent Deviation Curve * * * for i = l:m ayl = abs(yl(i,l) ay2 = abs(yS(i,l) dev(i,1) = abs (ay2-ayl) end ;

function [fav] = avfun(nintt,x,m) % * * * Average Function Value *** a = x(1,l); b = x(m,l); c = l/ (b-a); fav = (c)*nintt; $

*************************************************************** *** *** Main Program tt.* t** *** *************************************************************** ***

g *** % g $

& ! %

**.t Area enclosed between loading/release curves odsumï = oddsum(y1,m) ; evsuml = evensum (y1,m) ; nintl = n d n t (x,yl,odsuml,evsuml ,m,n) ; odsum2 = oddsum (y2,ml ; evsum2 = evensum(y2,m) ; nint2 = numint (x,y2,odsum2, evsum2,m,n); ar = abs (nint2-nint1 ;

%

***

$ *a*

*te

% *kir Polynomial Manipulations pl = polyfit(x,yl,3); p2 = polyfit(x,y2,3); pdl = polyder (pl); pd2 = polyder (p2) ; den =[Il ; fprintf('\tLoading Curve Polynomial Fit \ n u ) printsys(p1,den) fprintftl\tRelease C m Polynomial Fit \ n l ) printsys (p2,den) fprintf('\tFirst Order Derivative of The Loading C u r v e Polynomial Function \ n o ) printsys ( p d l,den) fprintf('\tFirst Order Derivative of The Release Curve Polynomial Function \ n t ) printsys (pd2, den) ydl = derlist(pdï,x,m); yd2 = derlist (pd2,x,m); % % %

***

*** Path Length Function

+t+.

yddl = arlen (ydl,m) ; ydd2 = arlen ( y d 2 ,m) ;

***

% % %

***

$

***

k t * Mean Normal Spacing odsum3 = oddsum (yddl,m) ; odsum4 = oddsum(ydd2,m) ; e v s d = evensum (yddl,m) ; evsum4 = evensum (ydd2,m) ; nint3 = numint (x,yddl,odsum3,evsum3 ,rn,n) ; nint4 = numint (x,ydd2,odsum4,evsum4,m,n) ; 11 = nint3; Ir = nint4; r = (ll+lr)12; mns = ar/r;

g *** %

***

Extension of Displacement C u m e s

do = x ( m , l ) ; edl = abs(ll-do) ; edr = abs (lr-do); $

$

***

***.

Average Function Value favl = avfun (nintl,x, rn) ; fav2 = avfun(nint2,x,m); avr = abs(fav2-favl) ; % *t* Displacement Deviation Curve %

t+*

%

***

dev = deviat(yl,yZ,rn); mxdev = max(dev); %

.............................................................. Results

fprintf( ' \te** Cumes Extensions Results *** \nl) fprintf('\tFinal Loading Curve Length \no) fprintf('\tll = %gt,ll) fprintf('\t****\n') fprintf('\tExtension of Displacement-Loading C u r v e \nt) fprintf('\tedl= %gg,edl) fprintf( '\t****\nl) fprintf('\tFinal Release Curve Length \nt) fprintf( ' \tlr = %g ',Ir) fprintf('\t****\nt) fprintf('\tEktensionof Displacement-Release Curve \nt) fprintf('\tedr= %gt,eàr) ) fprintf( '\t****\no fprintf( '\te** *** \nt) fprintf(t\t******+****f**************************e*********************

*******

in')

fprintf( ' \t***

Press Enter to Get Springback Results

***

\nt) fprintf(l\t*********t*****************tf*******************+***********

*******

\no)

pause %

***

t**

fprintf('\n\n\n\tf** Springback Results *** \nt) fprintf( ' \tMaximum Displacement Deviation \no) fprintf( ' \tSPBl = %gl,mxdev) fprintf('\t****\no) fprintf('\tDiferenceof Average Curves Values \nt) fprintf('\tSPSS = %gl,avr) ) fprintf ( '\t****\no fprintf('\tVirtual Deviating Area between Curves \n') fprintf( '\tSPB3 = %g ' ar) fprintf( \t****\nt ) fprintf( ' \tMean Normal Spacing \no) fprintf('\tSPB4= %go,mns) fprintf('\t****\no) fprintf( '\te** *** \nt)

.

l

fprintf('\t********f***rt*************t*******~~************************

************

\no)

fprintf ( ' \t*** *** \nt)

Press Enter to Plot Figures

fprintf('\t*************f**********t*t********************************* **********te

)

qj

*******************+Xtf+******************************************

g

***

% % qj

*tt

+** PLotting

*+t

***

*** **************************************************************

pause subplot (211),plot(x,yl) title(' Sheet Displacernent under Load Application '),xlabel('Sheet Center Line Path,s,(m)'),ylabel('Sheet Z-Dir Displacement,w,(m)' ) ,grid subplot(212),plot(x,y2)

title(' Sheet Displacement under Load Release '),xlabel('Sheet Center Line Path,s,(m)'),ylabel('Sheet Z-Dir Displacement,w,(rn)'),grid pause plot (x,yl,Ir--' ,x,Y~, 'g-'1 title(' Sheet Diaplacement ~oading/Releasing '),xlabel('SheetCenter Line Path,s,(m)'),ylabel('Sheet 2-Dir Displacement,w,(m)' ) ,grid pause plot (x,dev) title(' Deviation Due to Springback '),xlabel('Sheet Center Line Path,~,(rn)'),ylabel(~ Displacement Deviation,dev, (m) '),grid pause y = menu('Do you want to utilize the code again ?','YESt,'NO') if y == 2 break else clc end ; %

end; % % $

$

% qj %

*** *** *** ************************************************************** *** *** *** END *** *** t** **************************************************************

Appendix C

Experimental Test Rigs Detailed design of axisynimetric and plane strain test rigs is presented with their detailed dimensions. The assembled test rigs as well as their individual components are included.

C.l

Axisymmetric Test Rig

The detailed geornetry and dimensions of the test ng designed to stretch f o m axisyrnmetric parts are provided in this section.

Figure C.7 Die Carrier (Top view).

111

.

111

1 1 111

Figure C.9 Channel Side (Righi side view).

C.2

Plane Strain Test Rig

The detailed geometry and dimensions of the test rig designed to stretch form plane strain parts are provided in this section.

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