i
Stress-Based Necking and Failure for
Incremental Sheet Forming by
Md Ziaul Haque
Advisor : Jeong Whan Yoon
Submitted in fulfillment of the requirements for the degree of Doctor of Philosophy
Faculty of Science, Engineering and Technology, Swinburne University of Technology
Australia
2014
ii
Authors Declaration
I hereby declare that this thesis contains no material which has been accepted for the
award to the candidate of any other degree or diploma, except where due reference is
made in the text of the examinable outcome. To the best of my knowledge it contains no
material previously published or written by another person except where due reference
is made in the text of the examinable outcome;
.
_______________________
Md Ziaul Haque
iii
Abstract
Incremental sheet forming (ISF) is a flexible process in which sheet metal is formed by
a progression of localized deformation. However, the overall mechanics is complicated
and special conditions, such as bending under tension, cyclic bending & unbending, and
shear deformation occur during the process which contribute to the overall enhancement
of formability. The research investigates the deformation mechanisms in ISF with
relation to necking and failure. A strain-based forming limit criterion is widely used in
sheet-metal forming industry to predict necking. However, this criterion is strictly valid
only when the strain path is linear throughout the deformation process. Strain path in
ISF is often found to be severely nonlinear throughout the deformation history.
Therefore, the practice of using a strain-based forming limit criterion often leads to
erroneous assessments of formability and failure prediction. On the other hands, stress-
based forming limit is insensitive against any changes in the strain path and hence it is
used to model the necking limit which is combined with the fracture limit based on
maximum shear stress (MSS) criterion (Stoughton and Yoon, 2011). Simulation model
is evaluated for a single point incremental forming using AA 6022-T43, and checked
the accuracy against experiments carried out with an ABB robot. The proposed model
has given a good scientific basis for the development of ISF and its usability over
conventional sheet forming process.
iv
Acknowledgements
First, I would like to express my sincere gratitude to my advisor, Professor Jeong Whan
Yoon for his invaluable advice, intellectual guidance, and encouragement throughout
my PhD. I always appreciate the trust he bestowed on me at the time of my struggle
with research challenges and the opportunities and experience he afforded me to direct
and face the issues independently thus fostering my growth as a researcher.
I am very grateful to Prof. John Beynon for having facilitated my research facilities
from AusAMRC(Australian Advanced Manufacturing Centre). Also, I am grateful to
Boeing for providing the financial support for the required experimental tests.
I would also like to thank Dr.Daeyong Seong for his support on my work through
various discussion, suggestions and friendly advices. I am also indebted to Prof. Jong-
Bong Kim, Drs. Thomas Stoughton and Dr Yanshan Lou for their fruitful discussion on
the topic of my thesis.
Many thanks to David Vass, Jawson Meredith, Alec Papanicolaou, Walter Chetcutiand
Krys Stachowicz, who continuously supported me to build experimental setup in the
workshop and always accepted my request with a smile. Special thanks due to Girish
Thipperudrappawho always make time for me to run the CNC machine.
I would like to thank all my colleagues at AusAMRC for creating a collaborative
working environment. Special thanks to Mariana Paulino for her kind assistance.
Finally, I am truly grateful to my parents, parents-in-laws, brothers and sisters and
friends for their priceless support and prayer. Special thanks to my little son Zayan for
cheering me up through his world best smiles. I would like to give my most valuable
thanks to my wife Nazia, who supported my baby and the home duties and continuously
encouraged me with her care and love.
v
Dedication
To my precious son “Zayan” on his first birthday.
vi
Table of Contents
STRESS-BASED NECKING AND FAILURE FOR INCREMENTAL SHEET FORMING .............. I
AUTHORS DECLARATION ..................................................................................................................... II
ABSTRACT .................................................................................................................................................III
ACKNOWLEDGEMENTS ........................................................................................................................ IV
DEDICATION .............................................................................................................................................. V
TABLE OF CONTENTS ............................................................................................................................ VI
LIST OF FIGURES ...................................................................................................................................... 9
LIST OF TABLES ...................................................................................................................................... 13
CHAPTER 1 INTRODUCTION ............................................................................................................... 14
1.1 Background ......................................................................................................................................... 14
1.2 Research Progress and Recent Trends in ISF .................................................................................... 16
1.3 Scope of Implementation: .................................................................................................................. 18
1.3.1 Increased Formability: ............................................................................................................... 21
1.3.2 Major drawbacks of ISF: ........................................................................................................... 21
1.4 Motivation and Objective: .................................................................................................................. 22
1.5 Outline of the Thesis .......................................................................................................................... 24
CHAPTER 2 LITERATURE REVIEW ................................................................................................... 25
2.1 Introduction ........................................................................................................................................ 25
2.2 Basic Concept of Forming Limit: ...................................................................................................... 25
2.2.1 Development of Experimental Forming Limit Diagram ........................................................... 25
2.2.2 Theoretical Models for FLD: .................................................................................................... 27
2.3 Marciniak-Kuczynski (M-K) model : ................................................................................................ 29
2.3.1 Sensitivity of MK Model ............................................................................................................. 32
2.4 Review of Formability Study in ISF: ................................................................................................. 36
2.4.1 2.4.3 Non Conventional approaches for ISF: ............................................................................ 39
2.5 Stress Based FLD: .............................................................................................................................. 42
2.6 Forming Limit Curve at Fracture (FLC-F): ....................................................................................... 45
2.6.1 Ductile Fracture Criteria (DFC) and Shear Fracture Criteria ............................................... 47
2.7 Formability Analysis based on through thickness Stress/Strain Gradient: ....................................... 48
CHAPTER 3 MAPPING OF FLC-N AND FLC-F BETWEEN STRESS AND STRAIN SPACE .. 53
3.1 Introduction ........................................................................................................................................ 53
vii
3.2 Review of the M-K model .................................................................................................................. 53
3.3 Constitutive Modeling of Stress and Strain forming limits: ............................................................. 56
3.3.1 Hardening law description: ....................................................................................................... 59
3.3.2 Yield Criteria Description: ........................................................................................................ 60
3.4 Forming Limit Modeling for AA 6022-T4E32 with Different Yield Criteria : ................................ 64
3.5 Modeling of Ductile and Shear Fracture Criteria: ............................................................................. 70
3.5.1 Shear Fracture Modeling using Advanced Constitutive Equations : ........................................ 73
3.6 Summary:............................................................................................................................................ 74
CHAPTER 4 EXPERIMENTAL OBSERVATIONS AND DATA ANALYSIS ................................. 75
4.1 Introduction: ....................................................................................................................................... 75
4.2 Design of Experiment: ....................................................................................................................... 75
4.2.1 CAD System and Tool Path Design: .......................................................................................... 75
4.2.2 Forming Tool Design: ................................................................................................................ 76
4.2.3 Fixture Plate, Die Design: ......................................................................................................... 78
4.2.4 Forming Machines: .................................................................................................................... 78
4.3 Measurement of Strain: ...................................................................................................................... 81
4.3.1 Strain Measurement by CMM, GPA, and ASAME Target Model :........................................... 81
4.4 Conclusions: ....................................................................................................................................... 86
CHAPTER 5 DEVELOPMENT OF STRESS FLD THROUGH FE APPROACHES ...................... 87
5.1 Introduction ........................................................................................................................................ 87
5.2 Implicit FE analysis of Incremental Sheet Forming : ........................................................................ 87
5.2.1 Tool Path Generation: ............................................................................................................... 88
5.2.2 Element Selection : ..................................................................................................................... 88
5.2.3 Mesh Sensitivity Analysis: .......................................................................................................... 91
5.2.4 :Effect of yield criterion: ............................................................................................................ 93
5.2.5 Prediction of Punch (Tool) Force: ............................................................................................. 94
5.3 Forming Limit Analysis from FE Results ......................................................................................... 95
5.3.1 Pyramid Shape Results(Yld2000): ............................................................................................. 99
5.3.2 Cone shape results from Yld 89 and Hill 48(mid plane only) : ............................................... 102
5.4 Non-planer stress analysis based on stress-based FLC: .................................................................. 103
5.4.1 :Nominal stress analysis using Hill’s (1948) quadratic function ........................................... 104
5.5 Conclusions ...................................................................................................................................... 106
CHAPTER 6 MECHANISM TO SUPPRESS NECKING IN INCREMENTAL SHEET FORMING
..................................................................................................................................................................... 107
6.1 Introduction ...................................................................................................................................... 107
6.2 Investigation of Process Mechanism in ISF .................................................................................... 108
viii
6.2.1 Effect of Tool Force on Deformation: ..................................................................................... 109
6.3 Mechanics of Necking Suppression ................................................................................................. 110
6.3.1 Strain-Based Analysis: ............................................................................................................. 110
6.3.2 Stress-Based Analysis: ............................................................................................................. 116
6.4 Summary:.......................................................................................................................................... 120
CHAPTER 7 OPTIMIZATION OF PROCESS PARAMETERS FOR INCREMENTAL SHEET
FORMING ................................................................................................................................................. 121
7.1 Introduction: ..................................................................................................................................... 121
7.2 Advanced ISF Process Development ............................................................................................... 123
7.2.1 Numerical Simulation of a Complex Shape in ISF .................................................................. 123
7.2.2 Multistage Forming .................................................................................................................. 125
7.2.3 Developing process plan for a cup forming: ........................................................................... 125
7.3 Summary:.......................................................................................................................................... 128
CHAPTER 8 CONCLUSIONS AND RECOMMENDATIONS ......................................................... 129
8.1 Overview and Conclusions .............................................................................................................. 129
8.2 Recommendations for Future Study ................................................................................................ 130
BIBLIOGRAPHY ..................................................................................................................................... 131
9
List of Figures
Figure 1.1:Principle of SPIF for a non-axi-symmetric shell, originally realized by Iseki . (Emmens et
al,2010). ..................................................................................................................................... 15
Figure 1.2 :TPIF as originally proposed by Matsubara(Emmens et al,2010). ..................................... 15
Figure 1.3: Process types of asymmetric incremental sheet forming (ASIF). ...................................... 15
Figure 1.4: A CAD ,CAM, Robot based System producing customized part in ISF (Meier et
al,2009) ...................................................................................................................................... 18
Figure 1.5Customized medical prototypes by ISF: (left) Canio-facial implants, (right) Steps to
generate the CAD model of a frontal orbit implant. (i) CT scan of the skull with
defects,(ii) Clay model of the skull,(iii) STL file of the implant generated using reverse
engineering,(iv) Final CAD model with the implant integrated into the work piece
definition,(v) Uncompensated titanium cranial implant made at UFRGS, Porto ,Alegre.(
Duflou et al. 2013) .................................................................................................................... 19
Figure 1.6: Change of pressure requirement with respect to the curvature in stamping process.
Forming of parts with small radii, a high pressure is necessary ....................................... 19
Figure 1.7: Comparison of ISF with a conventional forming based on Exergy analysis. (Dittrich et
al.,2012) ..................................................................................................................................... 20
Figure 1.8: Major and minor strains distribution in several regions of an automobile body
,presented based on diagram available at (Kalpakjian,2008) ............................................ 21
Figure 1.9: Change of strain path concept observation by Toyota and applied to tryout of a quarter
panel stamped from a deep draw quality steel(Stoughton and Yoon,2012) .................... 23
Figure 2.1Materials testing procedures to develop forming limit curves(Allwood and Shouler,2009)
................................................................................................................................................... 26
Figure 2.2:Typical strain path for forming limit diagram ..................................................................... 26
Figure 2.3: Changes to the forming limit curves after pre-strain to several levels of strain in uni-
axial, plane-strain, and equi-biaxial conditions. .................................................................. 33
Figure 2.4: Four forming limit curves in figure (a) are taken from Figure 2.3 for uniaxial tension
along the transverse direction (1) black FLC is FLDo at a longitudinal strain of about
0.19, (2) the tan-colored FLC with a cusp close to the horizontal axis at a transverse
strain of 0.07, (3) the blue FLC with a cusp close to a transverse strain of 0.13, and (4)
the red FLC with a cusp at a transverse strain of about 0.17. The set of all four of these
curves defines the evolution of the ‘‘single’’ FLC for a linear strain path corresponding
to uniaxial strain along the transverse direction. Figure (b) shows the evolution of the
stain FLC for a linear strain path corresponding to uniaxial strain along the rolling
direction for the four curves taken from Figure 2.3 (Stoughton and Yoon,2012) .......... 33
Figure 2.5 :FLC obtained by Ball tool test(Shim and Park,2001) ......................................................... 39
Figure 2.6: Maximum observed uniform strain in both tensile test and CBT test for all materials.
The dashed line shows the 1:1 relation. (Emmens and Boogaard,2011) ........................... 41
10
Figure 2.7 :Representation of strain based forming limit curve, the stress-based forming limit
curve, and the extended stress-based forming limit curve (XSFLC) (Simha et al.,2007)
................................................................................................................................................... 42
Figure 2.8:Path Independency of experimental strain-FLC if plotted as stress-FLC ............................ 43
Figure 2.9: Typical evolution of the FLD at necking and at fracture: (left) high-ductility materials;
(right) low-ductility materials. ............................................................................................... 46
Figure 2.10: Wierzbicki’s experiments on calibration of seven fracture model for Al 2024-T351.
Test carried on un-notched round bar(1) ; two notched bars (2 and 3), and flat grooved
specimens used for calibration of seven fracture models(4) . Upsetting (5–9), shear (10;
11), and tensile (12–15 ) (Wierzbicki et al., 2005) ................................................................ 48
Figure 2.11:For different starting geometries, Nakazima strips are deformed in a hemispherical
punch test to generate the different strain paths in the canter of the test specimens. .... 49
Figure 2.12 :Non-linear behaviour of strain paths in stretch-bending with a cylindrical punch ...... 49
Figure 2.13 :Sum of the principal strains for a 50 wide strip of 1008 AK steel stretch-bent over a
punch wedge with a ¼ inch radius to the depth at which onset of necking occurs, as
reported(Tharrett and Stoughton,2003). The forming limit is characterized as a simple
limit on the sum of the principals because the minor strain was less than or equal to
zero at all points along the strip in a region of the FLD characterized by a limit on
thinning strain for this metal. The FLC and FLDo was obtained from standard FLD
tests independent of the stretch-bend test (Stoughton and Yoon, 2011) ........................... 50
Figure 3.1:Schematic View of MK model ................................................................................................. 53
Figure 3.2:Structure of FLD code for strain and stress FLC : The Subroutine Structure for
Forming Limit Curve Prediction in Strain Space : a.Hardening law b. Yield Function c.
Flow Rule . ............................................................................................................................... 62
Figure 3.3: Stress –Strain relation for Isotropic Yield Criteria (Von Mises) ....................................... 63
Figure 3.4: Stress-Strain relation for Quadratic Model(Hill Normal Anisotropy): ............................ 63
Figure 3.5: Stress-Strain Relation for Non Quadratic Model(Yld2000-2d) ......................................... 63
Figure 3.6:Unique plastic work for different hardening curves (W1=W2) ........................................... 65
Figure 3.7:( a) Hardening curves are plotted for uniaxial tension and biaxial data using Swift law
and compared with experiment uniaxial tension test for 450curve. (b) Fitting Swift and
Voce hardenings curve with experimental curve upto fracture stress. ............................ 66
Figure 3.8: Stress directionality predicted from various yield functions for AA 6022-T4E32 .......... 68
Figure 3.9:r-value directionality predicted from various yield functions for AA 6022-T4E32 ......... 68
Figure 3.10:Yield locus predicted from various yield functions for AA 6022-T4E32 ......................... 69
Figure 3.11:Predicted strain-based forming limit curves for AA 6022-T4E32 .................................... 69
Figure 3.12:Predicted stress-based forming limit curves for AA 6022-T4E32 .................................... 70
Figure 3.13: Mapping procedure for fracture criteria between strain and stress spaces ................... 71
Figure 3.14: Comparison of fracture limit curve presented in strain space for AA 6022
T4E32(reference maximum fracture strain . . ................................................... 72
11
Figure 3.15: Comparison of fracture limit curve presented in stress space for AA 6022
T4E32.(reference maximum fracture strain . . .................................................. 72
Figure 3.16:Mapping procedure of fracture surface from stress space to strain space using
Yld2000-2d : ............................................................................................................................. 73
Figure 3.17: Strain space presentation of forming limit (FLC-N) and fracture limit (FLC-F) with
Yld 2000-2d .............................................................................................................................. 74
Figure 4.1: (a) Tool Path generated for different shapes (b) Straight and Skim modes downward
step. (c)Reference dimensions for pyramid and cone shape .............................................. 76
Figure 4.2 : (a)Design of tool with lubrication channel, (b) Complete assembly of tool with force
sensor mountings. .................................................................................................................... 77
Figure 4.3: Fixture for experiment ............................................................................................................ 78
Figure 4.4: Robot(left) and CNC machine in ISF .................................................................................... 79
Figure 4.5: Wrinkling occurred while forming using Robot (Pyramid (left) and cone (right)). ........ 79
Figure 4.6: The complete CAD/CAM-Robot setup for the experiment and FE analysis. .................. 80
Figure 4.7 in-house 3D membrane/ shell strain measurement of pyramid part by CMM(Yoon et al.,
2002) .......................................................................................................................................... 81
Figure 4.8 Thickness measurement of laser marked pyramid with GPA system ................................ 82
Figure 4.9: Major Stain distribution for pyramid as measured with GPA .......................................... 82
Figure 4.10: Minor Strain Distribution for Pyramid as measured by GPA ........................................ 83
Figure 4.11: Experimental Strain FLD plot for Pyramid shape part. .................................................. 83
Figure 4.12 Strain measurment of a cone shape using the ASAME. .................................................... 84
Figure 4.13: Major strain distribution for a cone shape part. ............................................................... 85
Figure 4.14: Minor strain distribution for a cone shape measured in ASAME. ................................. 85
Figure 4.15: Strain-based forming limit plot for a cone shape measured in ASAME ........................ 86
Figure5.1:(left) Increase in displacement of BTL element due absence of warping stiffness (right):
Warping occurred in a pyramid corner with BWC element ............................................. 89
Figure 5.2: Comparison of effective plastic strain in a critical segment along an inner circle. ......... 90
Figure 5.3Comparison of thickness strain distribution along a side wall of x-axis . ........................... 90
Figure 5.4: Comparisons of the two principle strains and thickness strain predicted from different
mesh sizes with experimental result for a cone shape ......................................................... 92
Figure 5.5: Comparisons of the two principle strains and thickness strain predicted from different
mesh sizes with experimental result for a pyramid shape. ................................................. 93
Figure 5.6: Tool force sensitivity from yield criteria .............................................................................. 94
Figure 5.7: Tool force sensitivity from element types. ............................................................................ 95
Figure 5.8: Thickness strain distribution for a cone shape simulation with Yld 2000-2d .................. 96
Figure 5.9: Predicted strains from Yld2000-2d for a cone shape in strain-based forming limit: mid
layer(left);Bottom layer (middle) and top layer (right) ...................................................... 97
Figure 5.10: Stress plots from Yld2000-2d for a cone shape(top, mid and bottom layers) ................ 98
Figure 5.11: Thickness strain distribution for a pyramid shape simulation with Yld 2000-2d ......... 99
12
Figure 5.12: Predicted strains from Yld2000-2d for a pyramid shape in strain-based forming limit:
mid layer(left);Bottom layer (middle) and top layer (right) ............................................ 100
Figure 5.13: Stress plots from Yld2000-2d for a pyramid shape(top, mid and bottom layers) ....... 101
Figure 5.14:Stress plots for a cone shape (mid layer): (a) Yld89 (b) Hill48 ..................................... 102
Figure 5.15: Effect of nominal and transverse shear stresses presented in von Mises yield surface.
................................................................................................................................................. 103
Figure 6.1: Stress states occurring in incremental sheet forming. ...................................................... 107
Figure 6.2: (a) Influence of bending : Bending increases formability .Influence of in-plane shear
decreases formability. (b) Combined effects of stretch and transverse shear shown
based on 3D FLD (Allwood et al.,2007) .............................................................................. 108
Figure 6.3: Evolution of thickness strain and effective plastic strain for a selected element. .......... 111
Figure 6.4 : Strain state change of contacting element for tool in-plane motion. .............................. 112
Figure 6.5: Change of effective plastic strain on an element before and after contact for in-plane
tool motion. ............................................................................................................................ 112
Figure 6.6: Change of effective plastic strain for three consecutive downward steps. ..................... 113
Figure 6.7: Strain path change in three consecutive steps.................................................................... 114
Figure 6.8: Overall strain path change for selected element. ............................................................... 114
Figure 6.9: Two and three-dimensional representation of the deformation mechanism:(i) and (ii)
Deformation mechanism at in-plane motion (iii) and (iv) deformation mechanism
during downward step . ........................................................................................................ 115
Figure 6.10: Stress path change before and after contact (1: before contact , 2: during contact, 3:
after contact) for three consecutive downward steps. ....................................................... 117
Figure 6.11:Stresspath for top (left) and bottom (right) planes for a selected element C for the
overall process of forming to show the complete stress change. ...................................... 119
Figure 6.12: Step wise yield surface evolution of a selected element C. .............................................. 120
Figure 7.1:Factors that need to be considered for design incremental sheet forming process. ....... 122
Figure 7.2: a process table on the contributions from manufacturing and material parameters
according to the typical deformation modes. ..................................................................... 122
Figure 7.3: Three-dimensional representation of the different patches analysed. ............................ 124
Figure 7.4: Strain and stress-based analyses for a complex shape forming composed of three
patches. ................................................................................................................................... 124
Figure 7.5 Different process strategies for cup forming :Strategy1: Two step incremental forming;
Strategy2: Four step incremental forming ; Strategy 3. Progressive step incremental
forming ................................................................................................................................... 126
13
List of Tables
Table 1.1: An Overview of ISF development ........................................................................................... 17
Table 1.2: Material Saving in ISF process (Giuseppe Ingarao 2012). ................................................... 20
Table 2.1: Chronological list of work on theoretical approach for necking prediction ...................... 30
Table 3.1: Material Properties for AA6022-T4E32 ................................................................................. 65
Table 3.2: Material Constants for Different Yield Criteria ................................................................... 65
Table 3.3 : Selecetd fracture criteria ......................................................................................................... 71
Table 4.1: Experimental Conditions for Pyramid and Cone Shape…………………………………..82
Table 5.1: Effect of shell element on thickness prediction with element size of 2.5 x 2.5 for a cone
shape (Yld 2000-2d) ................................................................................................................ 91
Table 5.2: Effect of mesh size with a cone shape ..................................................................................... 92
Table 5.3: Effect of different Yield criteria on performance of FE of ISF ........................................... 93
Table:5.4 Properties of FE model for Cone Shape…………………………………………………....98
Table: 5.5 Properties of FE model for Pyramid Shape……………………………………………102
14
Chapter 1
Introduction
1.1 Background
Incremental sheet metal forming(ISF) is a flexible process in which sheet metal is
formed by a progression of localized deformation. It is flexible as specialized tooling is
not required; a simple tool moves over the sheet surface such that a highly localized
plastic deformation is caused. Hence a wide range of three dimensional shapes can be
formed by moving the tool along a correctly designed path (Jackson and
Allwood,2009).
Early concepts of ISF were patented in USA by Roux in 1960 although the process was
first envisaged by Lezak who patented it in 1967(Leszak,1967). However, the process
was not viable at the time because computer numerical control (CNC) systems and
associated software were still in their infancy(Jeswiet et al.,2008). Pioneering work
began early 90’s in Japan by Iseki (Iseki,1989) and his co-workers using a simple tool
and a path of the contour line (Fig 1.1). His paper(Iseki,1989), referring to Mason’s
work and to his intuitive thinking from the tool-path of a three-dimensional CNC
milling machine, showed the first manufacturing of non-symmetrical parts. Later Iseki
and Naganawa in 2002 proposed a three-tool incremental forming method, and obtained
a Japanese patent in 2003 [JP 10-180365, P3445988] on the three tool incremental
bulging machine(Emmens et al,2010).
Figure 1.
axi-symm
by Iseki .(
Figure 1.3
Four varia
(a) Single
Point Incr
sheet shap
and upwar
for clarity
1:Principle
metric shell
(Emmens e
3: Process t
ations in inc
e Point Inc
remental Fo
pe in (a-d) i
rds in (c-d)
(Jeswiet et
15
e of SPIF f
l, originally
et al,2010).
types of asy
cremental sh
cremental F
rming (TPI
is a truncat
. One quart
al.,2005)
5
for a non-
y realized
ymmetric i
heet forming
Forming (SP
IF) with (c)
ted pyramid
ter of the sh
Figure 1.
by Matsu
incrementa
g are shown
PIF), (b) IS
a partial an
d, with the p
heet and fac
2 :TPIF as
ubara(Emm
al sheet form
n as
SF with co
nd (d) a full
pyramid top
ceplate is de
s originally
mens et al,20
ming (ASIF
ounter tool,
l counter die
p downwar
esigned to b
y proposed
010).
F).
(c-d) Two
e. The final
rds in (a-b),
be invisible
d
o
l
,
e
16
Most simple and common Incremental forming usually means SPIF (single point
incremental forming). In another type of the process, known as TPIF(Two-Point
Incremental Forming), the perimeter moves vertically and synchronously with the
punch, while the product is supported at its centre. The punch is drawing contours from
the inside to outwards, moving the perimeter of the blank gradually downwards (Figure
1.3). TPIF (Two-Point Incremental Forming), was first presented by
Matsubara(Matsubara,1994)in a set-up shown in Figure 1.1 . Research activities were
propagated from the Asia to the Western world, mainly Europe only 15 years ago. The
interest of the Western world was only aroused when the process was presented at a
CIRP meeting in 1997 and the first major publication appeared in 2001 (Emmens et
al,2010). A comprehensive review of the development of the process is given by
Jeswiet et al.(2005).The paper handles asymmetric single point incremental forming.
Another more updated review presented by Emmens et al.(2010)presented almost all
technological development of Incremental Sheet forming.
Figure 1.3shows the different configuration of asymmetric incremental sheet forming
(ASIF) techniques. ISF process is not always “dieless” as sometime dies made of wood
or softer material is used especially for rapid prototype shapes. TPIF requires especially
build dedicated machine with advanced control method while SPIF can be conveniently
carried out with a simple 3-axis CNC machine, then hence it attracted more attention
from manufacturers and researchers.
1.2 Research Progress and Recent Trends in ISF
Since the last decade dieless process got immense interest to form a specialized part
required usually in small scale and economically unprofitable with conventional
forming process. Simultaneously researchers paid their attention to its easy
experimental setup compared to other contemporary processes. Simple statistical
representation on the distributions, application areas, and publications are presented in
Table 1.1, this table highlights the research trends in the field of ISF. The chart is
mainly based on the information taken from conference and journal articles with the
relevant years.
17
Table 1.1: An Overview of ISF development
“x” is mentioned to show relative increase of work in the field based on literature
available.
Screening trends Pre
2004
2004 2005 2006 2007 2008 2009 2010 2011 2012
Forming method SPIF x x xxx xx xxxx xxxx x x x
TPIF x X xxx x x x x
Hybrid x
Formed sheet Aluminium xxx xx xxx Xxxx xxxx xxxxxx xxxxx xxx xx xxx
Titanium x x x x xxx
Magnesium x xx xx x x
Polymer x xx x x
Copper x
Steel xx xx
Tool path strategies xx x X x xxxxx x x xxx xxxxx
Machine CNC xx xx xx
Robot x xx xx
FE analysis x xx X xxx xx xx
Forming tools Rigid x X x xxx x
WJ X x x x x
Laser
assisted
x x xx
Forming limits x x xxx x x xx x
Deformation
Mechanism
x x xx xx xxx
The statistics shows clearly the rapid increase of the research attention for incremental
sheet forming process. Early researches were mainly concentrated on performing
successful experiments with different process parameters, tool path, and material.
Recently strong attention is made for the development of FE approaches. Recent trends
of researches are focused on the two major directions:(i)Overcoming the shortcoming of
the ISF process through the development of optimum forming path and tooling process
with FE approach,(ii)Development of forming limit tool to successfully characterize the
formability in incremental sheet forming.
18
1.3 Scope of Implementation:
Several advantages in incremental sheet forming make it an attractive choice for
research and industrial applications. These include(i) Flexible production with the direct
CAD data,(ii) Product can be produced without dies or with minimum supporting
dies,(iii)Rapid prototyping and reverse manufacturing capacity,(iv) Conventional CNC,
or robot can be used as a forming tool,(v) The localized plastic zone and incremental
nature of the process increase formability,(vi) The limitation of part size usually does
not depend on machine force. These advantages of incremental sheet forming are
further explored in the next sections.
i. CAD/CAM/Robot based flexible production capability:
This integrated CAD/CAM system allows a manufacturer to produce highly
customized parts with prompt variation. The recent trend of incremental sheet forming
is to utilize robots and CAD/CAM data directly. In other conventional processes, die or
dedicated machine depending on size and shapes prevent them to be integrated in the
flexible manufacturing system.
Figure 1.4: A CAD ,CAM, Robot based System producing customized part in ISF
(Meier et al,2009)
ii. Manufacture of Customized Rapid Prototypes:
19
For its customized and complex shape producing capability through CAD data, the
process got huge interest to produce prototypes for human body implants. CAD model
are generated from scanned surface of the implant which gives perfect match of the
required implants. Fig 1.5 shows a recent example of producing titanium Canio-facial
implants through incremental forming at UFRGS. Porto Alegre(Duflou et al. 2013)
Figure 1.5Customized medical prototypes by ISF: (left) Canio-facial implants,
(right) Steps to generate the CAD model of a frontal orbit implant. (i) CT scan of
the skull with defects,(ii) Clay model of the skull,(iii) STL file of the implant
generated using reverse engineering,(iv) Final CAD model with the implant
integrated into the work piece definition,(v) Uncompensated titanium cranial
implant made at UFRGS, Porto ,Alegre. (Duflou et al. 2013)
iii. Tool cost reduction for parts with small radius:
As mentioned earlier, incremental sheet forming allows the process to be implemented
in various targets of lightweight forming components, which are not feasible to be
produced in lots or batch due to technical limitations and cost. For example, forming of
a part with small curvature with high pressure is required to increases the tool cost
significantly as the production volume is reduced (Figure 1.6)
Figure 1.6: Change of pressure requirement with respect to the curvature in
stamping process. Forming of parts with small radii, a high pressure is necessary
iv. Tr
pro
Limited sp
significant
large volu
production
saving is
compared
Table 1.2
Figure 1.
analysis. (
rade off
oduction:
peed of ISF
tly compar
ume of prod
n, the energ
considered
to stamping
: Material S
.7: Compa
(Dittrich et
20
between e
process req
ed to other
duction, this
gy differen
. Table 1.
g(Giuseppe
Saving in IS
arison of I
t al.,2012)
0
energy cos
quires high
r forming t
s is a major
ce can be r
.2, shows t
e Ingarao 20
SF process(G
SF with a
st and ma
process tim
technologie
r drawback
reduced to
the compari
012).
Giuseppe In
a conventio
aterial savi
me, increasin
s especially
k in ISF. Ho
a reasonab
ison of ma
ngarao 2012
onal formin
ings for s
ng energy c
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owever, in
ble amount
aterial savin
2).
ng based o
mall scale
cost per part
ping. So in
small scale
if material
ng in SISF
on Exergy
e
t
n
e
l
F
y
21
Based on energy analysis ISF always require a higher processing energy. But if exergy
analysis is taken into account, ISF takes its superiority(Dittrich et al.,2012).The concept
of exergy analysis can be used to characterize and accumulate work, heat, and material
streams entering and leaving manufacturing systems.
1.3.1 Increased Formability:
Aluminium alloy has a limited forming limit. It constrains the design of stamped part.
Although advanced high strength steel has more degree of freedom to select high
ductility material, forming limit is still a major concern. Figure 1.8 shows a typical
range of forming limit in major and minor strains in several zone of auto body. The
limitation of forming limit can be dramatically improved if ISF is adopted. ISF process
gives a new lifted limit and allows to design a complex part far beyond a conventional
forming limit.
Figure 1.8:Major and minor strains distribution in several regions of an
automobile body, figure produced based on diagram at (Kalpakjian,2008)
1.3.2 Major drawbacks of ISF:
A major drawback of incremental sheet forming process is mainly forming time which
is much longer than competitive processes such as deep drawing or stamping. The
drawback of ISF limits its application to small size batch production. The shape is also
22
limited to a certain capacity. The forming of deep angle cannot be done in one step, but
requires multi-step processes. Spring back for certain shapes is high and causes much
distortion. Surface roughness also depends on the number of steps. In order to produce a
smooth surface, smaller tool radius and refined incremental steps are required. Although
the drawbacks prevent a wider application, the process is still attractive for small
volume production which needs a high forming and fracture limits. With the advance in
modelling methodology, good understanding on the process condition and forming
mechanics is continuing to overcome some of drawbacks.
1.4 Motivation and Objective:
Remarkable formability in ISF is a well proven phenomenon and this is one of the
driving forces for the ISF research. The question arises if it is possible to use the
conventional forming limit diagram for incremental sheet forming. The conventional
forming limit measured in the strain space is assumed to be static i.e., insensitive to
strain path change. However, in incremental sheet forming, sheet metal undergoes
significant strain path changes during the entire process .For this reason accurate
prediction of necking in ISF is very challenging using the conventional forming limit
diagram.
Early investigation by Isigaki(1977) at Toyota Motors Company first identified this
interesting phenomenon to occur in multi step stamping processes. Then they used the
dynamic behaviour and thus Toyota achieved remarkable improvement of the
formability, reaching thinning strains up to 60% with a net minor strain near to zero in a
multi-stage forming process of a quarter panel, although the FLDo for the zero pre-
strained condition of the metal is only 37 %. The process is explained in Figure 1.9. The
grey line in the figure denotes the initial forming limit curve and Toyota engineers
found that this line is not true, since it predicted tearing at the end of stage 6. Based on
this observation Toyota engineers remeasured the forming limit with pre-strained large
sheet up to the stage 4 point. The modified forming limit curve (as shown in red line)
was used as an estimate of the residual formability of the metal at the stage of 4 in the
critical location and accordingly the deformation process was modified to drive the
strain to follow a new biaxial path instead of the original path leading to failure. The
detailed story is summarized in Stoughton and Yoon (2012).
The 2nd im
through th
inaccurate
Figure 1.9
tryout of
Yoon,201
Defining n
several re
Stoughton
dealing wi
The objec
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2)
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searchers (
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23
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Ignorance o
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Arrieux et
n,2005; Sto
ath change.
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al.,1982; S
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Sing et al.,1
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the limitatio
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24
post neck behaviour up to failure. This research provides a scientific basis to handle
nonlinear strain path with stress-based approach and to explain the necking suppression
mechanics to improve the forming limit in incremental sheet forming towards
successful industrial application.
1.5 Outline of the Thesis
Chapter 2: The literature review covers the range of topics relevant to formability and
failure. Necking criteria are reviewed and brief background of stress based formability
is discussed.
Chapter3: Modelling of stress based forming and failure limits is provided.
The original research is provided in the chapters 4,5,6, 7.
Chapter 4: presented the experiment carried out in CNC and Robot to validate the FE
results and to describe the details of the incremental sheet forming process.
Chapter 5:Implicit FE approach for symmetric shapes(pyramid and cone) and complex
shape geometry is described. The FE results are analysed through both strain-based and
stress-based limits. Stress-based limit is linked with a proper constitutive relation.
Chapter 6: Stress-based modelling is used to explain the mechanics of ISF compared to
conventional strain-based analysis. Deformation mechanism is clearly presented with
the aid of stress path changes which reveal the major factor to suppress necking in
incremental sheet forming.
Chapter 7: The developed stress-based approach is further used to analyse of the effect
of the process parameters on the partial contribution to control necking. Guideline is
provided for the process design avoiding early neck and failure.
25
Chapter 2
Literature Review
2.1 Introduction
In sheet forming operation plastic deformation becomes unstable at some point which
leads to localized necking followed by failure. Reliable prediction of this necking and
failure limits always is required to successfully design forming operation. Prediction of
forming limit diagram (FLD) experimentally requires series of test to cover various
triaxiality. FLD can be also predicted theoretically. The following chapter reviews
strain-based and stress-based forming limits followed by its application to nonlinear
strain path.
2.2 Basic Concept of Forming Limit:
2.2.1 Development of Experimental Forming Limit Diagram
Keeler and Backofen(1964) tested on several materials including steel, copper, brass
and aluminium by stretching with solid punch. They (Keeler and
Backhofen,1964)introduced the forming limit diagram (FLD) to show the strain limit of
largest principle strains to be stretched. Keeler FLD’s show the strain space (ε1, ε2) to
safe strain states achievable from a material. The experimental technique of Keeler were
further developed by Goodwin(1968) to produce a successful FLD for a mild steel used
for stamping process with the earliest FLD named as Keeler-Goodwin diagram.
FLD is determined by series of experiments. Javignot(1930)used hydraulic pressure to
form a sheet metal. This doesn’t have a friction effect as punch is not in contact and
equi-biaxial stress can be obtained as a function of pressure and die geometry.
Nakazima test (Nakazima et al.,1971) uses hemispherical punch with a circular die to
form rectangular blanks. Marciniak test(Marciniak et al.,1973) or in-plane stretching
used flat bottom punch and deforms blank until rupture at the flat bottom punch
although complex in-plane stretch test was demonstrated by Tadros and Mellor (1978)
Figure 2.
and Shou
Figure 2.2
Later fric
simple ten
by electro
radius wh
1Materials
uler,2009)
2:Typical s
ctionless in
nsile test by
chemical m
hich is used
26
s testing pr
strain path
n-plane form
y Sing and R
method. A gr
to measure
6
rocedures
for formin
ming limit t
Rao(1993).
rid is forme
e strain. Af
to develop
ng limit dia
test was car
Initially cir
ed to an ellip
fter 1985, s
p forming l
gram
rried out by
rcular grid m
ptical shape
quare grids
limit curve
y Raghavan
method is w
e with major
s is more po
es(Allwood
n(1995) and
widely used
r and minor
opular with
d
d
d
r
h
27
the introduction of high capacity CCD camera incorporating computer packages that
enables the measurement of many squares for wider area of the sheet at one time.
Recently developed experimental FLD tests include a bulge tester to measure biaxial
strain path with an optical online strain measurement by Keller et al.(2009), biaxial
tension test with different cruciform specimen by Hannon(2008) , biaxial tension test
using comb-shaped specimen installed into a biaxial testing machine in order to apply
for combined tension-compression by Kuwabara(2008).
2.2.2 Theoretical Models for FLD:
Hill (1952) first proposed a general criterion for localized necking under plane stress
condition. Marciniak and Kuckzinsky(1967)proposed a mathematical model to
determine forming limit with local geometric imperfection, where heterogeneous plastic
flow develops and eventually localizes. The overall approaches developed can be
identified with three groups as mentioned by Yao and Cao. (2002) : (a) Bifurcation
analysis, (b) Local instability at a defect (M-K),(c) Accumulation of damage.
i. Bifurcation:
In this criteria localization is viewed as bifurcation from quasi homogenous strain which
leads to localized strain. The general bifurcation criteria was introduced by
Drucker(1950)as the necessary condition for the loss of uniqueness of the solution to
the boundary value problem for rate independent materials, except for any elastic
unloading. Hence the criterion of non-bifurcation corresponds to the positivity of the
second order work:
Δσ: ΔεdV 0 (2-1)
For small deformation, this criterion may be achieved with null or negative hardening in
case of associative plastic law. The criterion is a lower bound of localization that
predicts the possible occurrence of the first diffused neck. Therefore, the criterion is too
much conservative.
28
ii. Load Bifurcation (Swift):
Swift (1952) predicted the onset of diffused necking by developing an instability
criterion based on the maximum load definition under proportional loading. He showed
that the major limit strain in diffused neck could be calculated as follows:
2 11 2 2
(2-2)
Where, is the strain ratio (ratio of the minor strain to the major strain). Swift’s
bifurcation theory can cover the entire tensile range of deformation modes encountered
during sheet metal forming between uniaxial tension 0. 5 and equibiaxial
tension 1 . Obviously, diffused neck cannot be observed in a deformed sheet,
therefore, the plastic limit strain predicted by Swift’s method are usually considered the
onset of localized necking rather than diffused necking. However, since diffused
necking appears at a lower strain than localized necking, the limit strain from Swift’s
bifurcation approach is conservative compared to the strains measured experimentally in
the localized neck especially for negative strain ratios. It can be concluded that Swift’s
method for FLC prediction only provides an approximate estimation of the limit strains
and, therefore, is not a reliable method for industrial applications.
iii. Bifurcation with Flow theory (Hill):
Flow Bifurcation analysis was initiated by Hill (Hill,1952) who assumed that once a
discontinuity appears in the Cauchy stress and the velocity, this indicates the onset of
failure. Hill (1952) then formulated the restrictions on the flow stress and the rate of
work hardening in the growth of the localized neck. He developed a method which
shows that a localized neck starts in the zero-extension direction during uniform
deformation and the magnitude of plastic work decreases below the minimum value at
the instability condition. Two conditions are then obtained: the maximum loading and
the orientation of the localization band: :
1
1/ (2-3)
where R is the plastic anisotropy parameter and is strain ratio.
29
However, equation 2-3 only has a real solution when the minor strain is negative;
covering the loading paths on the left hand side of the FLC. Therefore the drawback of
this theory is that it cannot predict the limiting strains on the right hand side of the FLC
where minor strains are positive. Using Power law of stress-strain relation, ,
(Where K is the material constant and is the plastic strain for plastic stress .) The
critical condition for localized necking for negative strain ratio becomes:
1
(2-4)
iv. Bifurcation with Vortex theory (Storen Rice):
The physical theory of plasticity based on the crystallographic slip by Lin (1971)
predicted the onset of a sharp vertex at the loading point on the yield locus of a
polycrystalline material. The creation of vertices or corners on a yield locus during
deformation has also been validated by the continuum theory of plasticity and has been
confirmed by experimental studies conducted by Hecker(1973).Stören and Rice
(1975)developed a new bifurcation model based on Vertex theory of plasticity to
predict the FLC for the entire range of strain paths between uniaxial tension and equi-
biaxial tension. They assumed that localized necking will occur for each strain path
when a corner appears in the yield locus at the forming limit. They also showed that on
the left hand side of the FLC (i.e. negative minor strains), the orientation of a local neck
is not parallel with the zero-strain direction, but on the right hand side of the FLC
(positive minor strains), the localized neck is parallel with the minor strain direction.
2.3 Marciniak-Kuczynski (M-K) model :
MK models was introduced first by Marciniak and Kuckzinsky(1967). This model is
based on the hypothesis of the existence of imperfections in sheet metal. Sheet metal
has geometrical imperfections (thickness variation) and/or structural imperfections
(inclusions, gaps). In the forming process these imperfections progressively are evolved
and the plastic forming of the sheet metal is localized, leading to the necking of the
sheet metal. This model took a great attention for its easy implementation and several
advantages which include: (a) intuitive physical background; (b) accuracy;(c) capability
to integrate various constitutive models and (d)easy implementation for Finite Element
30
simulation for sheet metal forming processes. Main drawbacks also include: prediction
insensitive to the constitutive equations and the non-homogeneity parameter. Marciniak
(1968)analysed the strain localization phenomenon from the right side of the FLD and
extended his original model to cover this area. MK model was implemented by many
authors with different yield criteria (Bassani et al.,1989;Gotoh,1985; Graf and
Hosford,1989; Hill,1979; Neale and Chater,1980; Parmar and Mellor,1978). Banabic’s
group also implemented various non-quadratic yield criteria using the MK
model(Banabic,1999; Banabic et al.,2004; Banabic and Dannenmann,2001).Butuc et al.
(2002, 2003) developed a theoretical code for advanced constitutive relations suitable
for non-quadratic yield functions and later implemented for Barlat(1997) criteria(Butuc
et al.,2002; Butuc et al.,2003).Barlat’s Yld2000 formulation was also included by
Aretz(2004) in the MK model for studying the influence of the biaxial coefficient of
plastic anisotropy on forming limit. A non-exhaustive and chronological list of works
on the theoretical approach to predict necking is presented in Table 2.1.
Table 2.1: Chronological list of work on theoretical approach for necking
prediction
Ref. Models Characteristics
(Swift,1952) Swift criterion i. First model for instability ii. Plastic instability at the
maximum load for proportional loading
(Hill,1952) Hill i.Necking based on plastic instability of homogenous sheet
metal from discontinuity of stress and velocity.
ii. Restriction on the flow stress, rate of work hardening on
localized neck.
(Marciniak and
Kuckzinsky,1967)
MK model i. Model based on heterogeneous continuum
ii. Necking criterion based on the comparison of the strain rate
in and out of the neck band
iii. Prediction for expansion domain only
(Storen and
Rice,1975)
Bifurcation
Theory
i. A model based on heterogeneous continuum
ii. Large strain formulation
iii. Associated or non-associated plasticity
iv. Smooth yield surface or yields surface with vertex
v. Mainly used in plane strain
(Brunet et
al.,1977)
Damaged
Based Neck
i. Analysis necking in 3D based on damage variable
31
(Hutchinson and
Neale,1978. ;
Neale and
Chater,1980)
MK model Extension of the MK model to negative strain paths
(Jalinier and
Schmitt,1982)
MK Model Introduction of damage in the MK model of Hutchinson et al.
(Arrieux et
al.,1982)
MK Model i. New representation: Limit stress curve
ii Takes into account the orthotropy of the sheet metal
(Cordebois and
Ladevèze,1986)
Cordebois i. Formulation in rate and large strain
ii. Searches the maximum of the potential energy function
(Bressan and
Williams,1983)
TTS “Through Thickness Shear Instability Criterion” in order to
take into account the shear fracture mode
(Jones and
Gillis,1984)
Jones and
Gills(J-G)
Localized necking by assuming biaxial stretching of sheet in
three steps i..Homogenous deformation at max load ii. Strain
Concentration under constant load iii. Localized necking from
rapidly load decrease
(Fressengeas and
Molinari,1987)
Perturbations
Technique
i. A model based on homogeneous continuum
ii. Necking is an instability of the mechanical equilibrium state
(Barlat et
al.,1989)
MK model i. Extension of the MK model to the non-quadratic model for
anisotropy of materials
ii. Introduction of the YSSHD (yield surface shape hardening
diagram)
(Hora and
Tong,1994)
(Hora P et
al.,1996)
MMFC Development of Swift model
(Boudeau,1995) Perturbation i. Introduction of the Taylor model in the perturbation technique
ii. An adaptation of the perturbation technique to the prediction
of necking from FE results
(Fromentin,1998) Fromentin i.New representation: equivalent strain at necking with the stain
path
(Zhu X.H,2001) Unified
Bifurcation
Analysis
Include Momentum equilibrium in addition to force equilibrium
in bifurcation analysis
(Brunet,2001) Damage based
FLC
i.Modified form of Gurson’s model to predict and monitor
damage during the forming process
ii. Coupling of the damage model to Swift’s diffused necking
criterion
32
(Smith et
al.,2003)
Modified
Swift
Influence of the through-thickness
stress on the forming limit
(Simha et
al.,2007)
XSFLC Equivalent stress and mean stress at the onset of necking during
in-plane loading
(Stoughton 2008) Generalized
Stress Based
Influence of the stress distribution through the thickness on the
mode of failure
(Bai and
Wierzbicki,2008)
Bai Neck formation in sheets under non-proportional loading based
on the concept of ductile fracture relating accumulated
equivalent plastic strain modified by the stress triaxiality and
Lode angle parameter
(Signorelli et
al.,2009)
Polycrystal
Plasticity
Incorporate a viscoplastic crystal plasticity model of material
behavior into the M-K analysis to allow prediction of various
micro-structural factors on forming limit
(Allwood and
Shouler,2009)
GFLD Modified MK to include six component stresses
(Eyckens et
al.,2009)
TTS-MK Modified MK to include through thickness shear effect
(Safikhani et
al.,2008)
Strain
Gradient
The strain gradient approach is incorporated
into the M–K method for deformation localization
(Stoughton and
Yoon,2011)
Stress based Maximum Shear Stress (MSS) combined with stress based
forming limit to show necking and fracture with post neck
behaviour
(Stoughton and
Yoon,2012)
PEPS FLD Strain-based forming limit criterion based on a polar diagram of
the effective plastic strain
(Volk ,W.,et al.
2012)
Metamodeling
technique
New method for the description of failure behaviour in two-step forming operations by using strain ratio and strain path length.
2.3.1 Sensitivity of MK Model
Many researchers studied the sensitivity of the forming limit curve calculated by M-K
analysis. These mainly include factors including anisotropy, hardening, yield criteria,
texture, microstructure, and strain path. Since these factors imposed some limitation to
use the predicted FLC, enhancement of the original M-K model have been proposed by
many researchers which include study on the effect of yield locus shape by
Banabic(1999, 2001),effect of anisotropy (Aretz,2006; Banabic et al.,2010), influence
of normal pressure by Banabic and Soare(2008), Wu et al,(2008), and Allwood and
Shouler(2009).
Figure 2.3
strain in u
The dashe
with the se
the associa
Figure 2.
uniaxial te
strain of a
transverse
0.13, and
four of the
correspond
3: Changes
uni-axial, p
ed lines ema
econd leg s
ated formin
4: Four fo
ension along
about 0.19, (
e strain of 0
(4) the red
ese curves
ding to uni
33
s to the for
plane-strain
anating from
howing the
ng limit curv
orming limi
g the transv
(2) the tan-c
0.07, (3) the
FLC with a
defines the
iaxial strain
3
ming limit
n, and equi
m the origin
e plane strain
ve
it curves in
verse directi
colored FLC
e blue FLC
a cusp at a t
evolution o
n along the
curves aft
i-biaxial co
n show the p
n path of th
n figure (a
ion (1) bla
C with a cus
C with a cu
transverse st
of the ‘‘sin
e transverse
er pre-stra
nditions.
pre-strain p
he secondary
a) are taken
ack FLC is F
sp close to t
usp close to
train of abo
gle’’ FLC f
e direction.
ain to sever
ath for each
y forming o
n from Figu
FLDo at a l
the horizon
a transvers
out 0.17. Th
for a linear
Figure (b)
ral levels of
h condition,
operation to
ure 2.3 for
ongitudinal
ntal axis at a
se strain of
he set of all
strain path
shows the
f
,
o
r
l
a
f
l
h
e
34
evolution of the stain FLC for a linear strain path corresponding to uniaxial strain along
the rolling direction for the four curves taken from Figure 2.3 (Stoughton and
Yoon,2012)
i. Effect of Strain Path:
A considerable change of strain path occurs during industrial forming operation.
However, industry still uses a conventional forming limit diagram which is only true to
a linear strain path. Limitation of the conventional forming limit discussed by
Nakaziama et al.(1971)and later in several articles (Allwood,2007; Graf and
Hosford,1993 ; Hecker,1973; Hora and Tong,1994; Kleemola and Pelkkikangas,1977).
Graf and Hosford(1993 ) investigated the effect of yield criteria exponent using the MK
Model and showed a significant change of forming limit curve with the pre-strained
samples as shown in Figure 2.3. Zhao et al.(1996) further analysed the effect of strain
path change on the shape and magnitude of forming limit through incorporating
anisotropy, hardening, and strain rate sensitivity. The both works showed that pre-strain
in uniaxial tension raises the limits on the RHS (right hand side) of FLD without much
effect on LHS (left hand side) if the direction of the principal strains is not changed. a
recent article by(Stoughton and Yoon,2012) re-explained the experiment of Graf and
Hosford(1993 )in Figure 2,4 that forming limit curve is dynamic rather than static and it
is evolved even with a linear strain path. Yao and Cao (2002)proposed a methodology
to determine yield surface evolution under large plastic deformation expressed in terms
of change of back stress and yield surface curvature. The predicted forming limit curves
in this approach demonstrated a good improvement in various loading conditions.
ii. Effect of yield surface and anisotropy:
Anisotropy is induced during the cold rolling process. Therefore, sheet metal exhibits
anisotropic behaviours which need to be considered to predict the forming limit.
Different yield functions have been introduced for the MK analysis to assess the
influence of yield surface. For example, Lankford value affects the right side of forming
limit. When a quadratic yield function is used, the influence is remarkable (which is not
correct). Originally the MK analysis was proposed based on Hill’s 1948 criteria (R.
Hill,1948) although later this analysis is found to show overestimated limit strains in
biaxial region and underestimated the limit strain in plain strain regions for the material
with Lankford value less than unity (Painter and Pearce,1974). Sowerby and
35
Duncan(1971) analysis showed that the right hand side of forming limit is increased
with the strain hardening component(n) and forming limit strongly depends on
Lankford value when Hill’s (1948) is used. Later a number of non-quadratic yield
functions have been introduced in forming limit predictions which include Hosford
(1972)’s function(Hosford,1972), Hill’s (1979)yield function(Hill,1979), (Karafillis and
Boyce,1993) and Barlat’s yield functions (1989, 1991, 1997, 2003, 2006) . Graf and
Hosford (1993)’s prediction with Hosford’s non-quadratic yield function shows a better
agreement with the experiment than that of Hill 48. Barlat et al.(1997b) used Yld 94 for
aluminium and steels and the results are found to be reasonably well compared to the
experimental data. More examples to utilize non-quadratic yield functions to predict
forming limit with aluminium alloys cab be found at(Cao et al.,2000; Kuroda and
Tvergaard,2000; Yao and Cao,2002).
iii. Effect of non-planer stresses
A minor level of non-planer stress is generated during sheet forming like punch
stretching and hydraulic bulging. This stress varies from contact to non-contact area
through the thickness; usually minimum (i.e compressive stress) at the concave side and
zero at the convex side. As the nominal stress is much smaller than plane stresses
(especially in case of low sheet curvature), therefore usually out-of-plane stresses are
neglected in sheet forming simulation. However, out of plane stresses can rise to a
considerable effect on the overall formability. Usually it occurs with thicker plate which
undergoes a small bend radius or contact zone with a support tool or die in incremental
sheet forming. The predictions show that more nominal stress i.e. higher compression
delays the onset of the localized necking. The phenomenon was also proved in several
experimental observations. Introducing nominal stresses in forming limit diagram was
attempted to remodel the forming limit as shown in several articles(Assempour and
Nejadkhaki Khakpour 2010; Smith and Averill,2005). Smith and Averill (2005) took
into account the through-thickness normal stress by using strain-to-stress mapping
procedure with the new ratio / and they employed a generalized Stoughton and
Yoon (2005)’s plane stress mapping method. Ahmed and Hamid.(2010)used three
dimensional form of yield function and modified the energy equation in the groove
zone. Eyckens et al.(2009) extended the MK model to consider localized necking with
through thickness shear (TTS). Several case studies (Eyckens,2010; Eyckens et
al.,2009) shows that if TTS(through thickness shear) is present, the critical groove
36
direction changes its strain mode, which delayed necking for all in-plane strain modes
except for equi-biaxial stretching.
2.4 Review of Formability Study in ISF:
Increased formability in incremental sheet forming was observed by early researches
(Filice et al.,2002; Filice et al.,2001; Kim and Yang,2000; Kim and Park,2002;Shim
and Park,2001). The works showed that the conventional forming limit diagram is
underestimated by showing the success in experiment and failure in forming limit. This
contradictory observation gives a strong motivation to develop a new forming limit
diagram that can correctly predict the process for incremental sheet forming. The
overall development of forming limit diagram for incremental sheet forming can be
summarized with the following three phases:
i. Experiment-based observation
ii. Finite Element (FE)-based observation
iii. Non-Conventional approaches for ISF:
2.4.1 Experiment-based observation:
Early formability modelling for incremental forming was mainly based on
experimentally observation of failure for different asymmetric shapes. The motivation
was to study the effects of process parameters, geometry, thickness on formability.
Then, it has been attempted to relate the observations to formability analytically or
statically. Since FE method was not sufficiently developed to accurately simulate the
process considering material characteristics and process mechanism. Although here are
the number of claims on mechanism of incremental sheet forming like bending-
unbending, shear as the contributors to improve forming limit , these could not explain
incremental sheet forming with any theoretical forming limit model,
Kim and Yang(2000) carried out the experiment by ellipsoidal clover cups in single
double pass forming by incremental sheet forming. They observed the improved
formability and better performance was found for double passes based on the
observations of the thickness strain. Felici et al.(2001) developed forming limit
diagram from series of tests with pyramid shape, cross shape (defined as the biaxial
test), cup shape(defined as c-test). They found that forming limit in incremental sheet
37
forming is different from the conventional one, occurs at much higher strain, defined as
a straight line of negative slope in the positive domain of the minor strain in forming
limit diagram. Sim and Park (2001) developed FE-based simulation method with the
verification of different types of part shapes and finally observed a similar straight line
observed by Felici et al. (2001) as shown in Figure 2.5. The failure strains formed
forming limit curve in a straight line with negative slope.
Kim and Park (2002) investigated the effect of anisotropy on formability. They used a
pyramid shape with varying tool diameter and measured the strains along rolling and
transverse directions. Experiment shows that formability along the transverse direction
is greater when smaller diameter tools are utilized, while along the rolling direction the
formability is improved with large diameter tools. Fratini et al.(2004)attempted to find a
correlation between material formability and other mechanical properties of material in
series of tests with a truncated cone shape using copper, brass, DDQ, aluminium alloys
(AA 6114, AA 1050-0). Comparison between the real forming limit in incremental
sheet forming with the initial FLD (FLD0)was conducted through statistical analysis.
The observation(Fratini et al.,2004) concluded that the highest influential factor on
formability in incremental forming processes is hardening coefficient followed by the
strength coefficient and percentage elongation.
Ham and Jesweit(2007) carried out a large scale experimental campaign (46
experimental runs with 500 samples) using Box-Behnken design to determine the effect
of material type, material thickness, shape, step size and tool size on the maximum
forming angle, effective strain, and major and minor strains. Material type had the
greatest effect on formability followed by the shape, which was varied with a forming
angle. Hussain et al.(2007) defined the forming limits in terms of two formability
parameters by conducting the experiments with a truncated funnel shape: (1) thinning
limit, (2) forming angle limit. . The research reached to an analogy to the sine law (Kim
and Yang, 2000), i.e., the formability of a sheet-metal depends upon the slope along the
depth of a part, or in other words it depends upon the slope along the depth of a part. A
higher slope produces a less formability ( . sin , where initial thickness t0 is
deformed to thickness t while incremental forming is processed at slope angle θ). When
θ approaches to zero, the final thickness is converged to zero, which is not formable.
38
2.4.2 Finite Element (FE)-based observations:
Finite Element (FE) approaches in incremental forming were attempted popularly
during the last decade including the early attempts by Iseki(2001),Kim and Park (2002).
In the early stage, FE analysis for incremental sheet forming mainly focused on the
solution schemes, FE elements, process conditions etc. Recently, FE approaches started
to explore the process mechanism including constitutive modelling and the limitation of
conventional forming limit diagram. Hirtet al.(2004) implemented GTN(Gurson-
Tveergard-Needleman) constitutive law available in Abaqus/explicit (Bambach et
al.,2003b)with a pyramid shape. The observation by Hirt et al.(2004) emphasised the
stress state as an important factor with respect to formation and growth of voids that
eventually lead to the fracture in ISF. Using GTN, Hirt et al.(2004) also analysed the
effect of the tool and step size on forming limit in a qualitative manner. The result from
the damage mechanics confirms qualitatively the fact that a higher forming limit can be
achieved with a smaller forming head and larger step size. Han and Han and Mo(2008)
formed a truncated cone by Abaqus/Explicit using Hill’s (1948)yield function with 4-
node shell element and showed good prediction of thickness profile. They claimed
stepwise increase of strain path as a reason for increase of forming limit. Han and
Mo(2008) also mentioned about increasing hydrostatic stress accompanied with
increased plastic strain .
He et al.(2005) simulated a cone shape using a sectional FE model using
Abaqus/Standard with Von Mises yield function and Swift hardening. It is also found
the presence of serrated strain path. The model later used by Bael et al.(Bael et
al.,2007)attributed serrated strain path to a major reason to improve formability and
proposed a strain-based FLD based on an modified M-K approach. The proposed
modified MK model considered the initial texture based anisotropy and its hardening.
The observation shows that forming limits are considerably higher for a monotonic
loading, but underestimates the experimentally observed formability during incremental
forming. The research also proposed to consider bending and reverse bending effects to
overcome the discrepancy occurred.
Wilko et al.(2007) theoretically related the improvement of forming limit to shear strain
in incremental sheet forming. Later in a review paper by Emmens et al. (2009), the
claim is extended by including bending under tension, cyclic loading, and nominal
stresses. A
experimen
between
membrane
failure. Th
previous a
governing
Recently
utilizing th
of local b
good pred
finally pro
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experimen
transverse
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Figure 2.5
2.4.1 Non
The sectio
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Although E
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Malhotra e
he stress-ba
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o explain i
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5 :FLC obt
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on discusse
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39
Emmens e
ations, they
and theor
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ased damag
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9
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DPIF(doub
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PIF and cam
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The author
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ming by the
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the effects
me up with
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incremental
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formability
under non-
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l
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-
40
Based on the discovery of the process mechanism for ISF, it became quite obvious to
introduce an alternate of conventional forming limit approach that can incorporate strain
path change, nominal stress, thickness gradient effect etc. An overview the FLDs
developed in strain space shows the development of the representation space developed
in two ways. (i) Researches attempted to incorporate strain path change in the principal
strain space, (ii) Replace the principal space by an alternative strain spaces including
equivalent plastic strain verses triaxiality. Among non-conventional forming limits,
Muschenborn and Sonne(1975) first employed equivalent plastic strain and strain-rate
ratio from the transformation of the conventional FLD to take the account of strain path
change. Yoshida and Kuwabara(2007) proposed the space of effective plastic strain and
principle stress ratio , .Zeng et al.(2008) proposed effective plastic strain with the
principle strain ratio , . The work has been extended to a path independent forming
limitin the polar space ( cinθ, cosθ called Polar Effective Plastic Strain (PEPS)
diagram by Stoughton and Yoon (2012). Another recent article by Volk et al(2012)
came up with anew model to describe failure behavior in two-step forming operations
by using a metamodeling technique. In this phenomenological approach, forming limit
strain is parameterized with function of strain ratio and strain path length ,
for a bilinear strain path . The model shows acceptable agreement with
experimental observation although does not provide a generalized solution for arbitrary
nonlinear path.
In incremental sheet forming, necking behaviour should be considered through the
thickness contrary to a conventional forming limit constructed with membrane strains.
To incorporate through thickness effect, Allwood and Shouler(2009) proposed a new
forming limit represented by the full components of strain tensor to display strain space
in-plane strain ( , , , and transverse shear& nominal
strains( , ,, , ,defined as Generalized Forming Limit(GFLD). They developed a
theoretical MK-based model. In GFLD improvement of forming limit attributed to
nominal and transverse shear strains. Major drawback of the proposed model is
extensive experiments for proportional loading to create the 3D diagram with the
experimental difficulty to handle the contact side of the specimen.
Figure 2.6
materials.
Inspired b
forming, E
effect of t
limit occu
of bending
control fo
bending u
using set
bending t
aluminium
convention
mechanism
details are
from a rec
in experim
tensile typ
6: Maximum
The dashed
by the role
Eyckens et
through thi
urs by incorp
g under the
ormability.
under Tensio
t of three r
test. The t
m (AA 601
nal tensile
m includes
e explained
cent draw be
ment for AH
pe strain loc41
m observed
d line shows
of transve
al.(2009, 2
ickness she
porating TT
e tension m
They(Emm
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16, 1.3mm)
e test. The
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in Emmen
ending test
HSS. Acc
calization o
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s the 1:1 rel
rse shear t
2011) the
ar (TTS) c
TS. Emmen
mechanism in
mens and B
ed on the or
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2.6) show
) which is
CBT test c
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ns and Boog
by Kim et
ording to t
occurs at a l
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lation. (Emm
o increase
modified M
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ns and Boog
ncremental
Boogaard,20
riginal test
down the
ws around
much high
can be linke
bending und
gaard(2011)
al.(2011) w
the observa
large R/t (ra
h tensile tes
mens and B
formability
MK model
-MK. Signi
gaard(2009)
sheet form
011)carried
developed b
specimen,
90% elong
her than th
ed to ISF m
der tension,
). Similar o
which observ
ation of dra
adius of rol
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Boogaard,20
y in increm
by incorp
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)investigated
ming as a ke
out CBT(C
by Benedyk
similar to
gation at f
he one ach
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, and cyclic
observation
ved three fa
aw bending
ller/thicknes
test for all
011)
mental sheet
porating the
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d the effect
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Continuous
k (1971) by
three point
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hieved with
as the test
c effect The
is obtained
ailure zones
type tests,
ss of sheet)
l
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2.5 Stres
The path
interest fo
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Figure 2.
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ing under te
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and the ben
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g to the dev
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major challe
entifying th
eveloping e
odel.
ss Based F
dependence
or the use o
s curves (F
ntally observ
ressures is
ependent of
n multistage
7 :Represe
limit curve
t al.,2007)
42
ension type
em of draw
nding angle
h cause mu
elopment o
s in increm
nges in the
e major fac
experimenta
FLD:
e of the co
of forming l
LSCs).The
ved by Mar
applied for
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, and the
2
e strain loca
bending tes
that sheet u
ultiple fract
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ental sheet
field of form
tors to stabi
al test to de
onventional
limit stress
path indepe
rtin et al.(19
r aluminium
ng path. F
rocess for fa
f strain bas
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sts is that th
undergoes. S
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mability for
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Sometimes
ften a fractu
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rming limit
mit diagram
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fracture at
are much co
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iated experi
mmarized th
al sheet form
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ontrolled by
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imental and
hat there are
ming :
ure.
theoretical
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ith forming
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l forces and
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be a more
tress-based
e (XSFLC)
.
y
l
d
d
e
l
t
g
t
d
s
e
d
)
43
Figure 2.8:Path Independency of experimental strain-FLC if plotted as stress-FLC
: Experimental forming limit curves for linear strain paths and for a bilinear strain path
after 0.07 strain in equal biaxial tension in (a) strain and (b) stress spaces. The green
dashed lines with arrows in both figures show the corresponding strain and calculated
stress increments due to pre-strain and three blue dashed lines show selected strains
and corresponding calculated stress increments to the final point on the strain FLC.
Note that the overlay of the two experimental stress FLCs is proof that stress-based
FLCs are independent of the loading history. (Stoughton and Yoon,2012)
44
According to the review of Stoughton and Zhu (2004), Stress based FLD was first
introduced by Kleemola and Pelkkikangas(1977)and rediscovered by Arrieux et
al.(1982). In 80s and 90s, this invention remains less explored by most sheet metal
researchers except for Gronostajski(1984)and Zhao et al.(1996), who experimentally
demonstrated FLDs and FLSDs for various loading conditions involving two
proportional paths and found that FLSDs are almost identical. In the last decades,
Stoughton (2000) demonstrated the path independence of stress-based forming limit by
using the data by Graf and Hosford(1993 ) as explained in Figure 2.8.
Continuous update of FLSD has been carried out with contemporary anisotropic yield
criteria, experiments and FE approaches(Stoughton 2008; Stoughton and
Yoon,2005;Stoughton and Zhu,2004). Stoughton and Yoon (2005) proposed a clear
guideline to asses stress-based formability in a complex loading condition using planar
anisotropic material model. The claim is supported by the simulation of multistage
deep drawing processes with Balrat’s Yld2000-2d material model(Barlat et
al.,2003)under plane stress condition. Plane stress FLSDs are extended to consider 3D
stress state by including through thickness compressive component by Simha et
al.(2007) and introduced an extended Stress-Based Forming Limit Curve(XSFLC).
XSFLC represents the stress space with equivalent stress and mean stress during in-
plane loading(Figure 2.7). Yoshida et al.(2005)performed the biaxial tension test
utilizing tension-internal pressure testing machine to investigate the path dependency
behaviour of forming limit stresses with aluminium tubes. The result confirmed the
limited path independency of forming limit stresses. In the later experiments, Yoshida
subsequently calculated the forming limit stresses for two stages paths using the MK
model to clarify the mechanism behind the path independency of stress-based forming
limit(Yoshida et al.,2007).
The investigation to find the path dependency of stress-based FLC is extended to the
hardening models in the later work of Yoshida and Suzuki (2008). These investigations
concluded that the path dependency of stress-FLC depends on stress-strain behavior
during the subsequent loading stages. So the stress-based FLC can be claimed path
independent if the work hardening behavior remains unchanged with a change of strain
path. However, the experiment by Yoshida et al. (2005, 2007)was carried out in the
45
combined loading conditions. Stoughton and Zhu (2004)earlier mentioned the
impracticality of calibration of forming limit through stress measurement especially due
to unloading. A more practical solution is to calculate stress-based FLC from the
measured strain path. Stress-based FLC calculated from the measured strain-based FLC
also ensures that the stress-based approach leads to an identical assessment of
formability as the strain-based approach isa special case of linear strain paths.
Stoughton and Zhu (2004) shows very insignificant difference in the calculated stress
for linear and nonlinear strain paths for the mapping from experimental strain FLCs to
stress-FLC(Figure 2.8). In finite element approach, this is much simpler as stresses can
be directly obtained and mapped to the equivalent stress through the hardening
relationship. A detail mapping process to generate FE (finite element) stresses is
developed by Stoughton and Yoon (2011) which extended the stress-based FLC by
incorporating fracture criterion and considering the stress distribution through the
thickness of the sheet metal to identify the mode of failure.
2.6 Forming Limit Curve at Fracture (FLC-F):
Prediction of fracture limit (FLC-F) along with necking limit (FLC-N) during the sheet
metal forming process is very important in order to identify the conditions that
deformed sheet leads to ductile fracture. This is obvious for incremental forming of
aluminium alloy sheets, where some grades show large post necking or fracture often
occurs without any obvious necking phenomenon preceding the forming limit as shown
in Figure 2.9. Further, necking suppression is commonly observed in incremental sheet
forming, which also causes fracture without necking. Hence forming limit needs to be
determined based on FLC-F. Conventional forming limit curves at fracture can be also
obtained in the strain space. But, it does not consider non- proportional loading
condition. In order to observe both necking and fracture, stress-based fracture criteria
can be more reliable and accurate, although an accurate selection and validation of the
fracture curve is required to complete FLC-N and FLC-F.
46
Figure 2.9: Typical evolution of the FLD at necking and at fracture: (left) high-
ductility materials; (right) low-ductility materials.
Although experimental fracture study on incremental sheet forming was carried out
from the early stage of ISF development, but modelling of a suitable failure criterion is
challenging. In fact, forming limit curves presented in earlier and recent articles
actually used the fracture limit derived from the crack or fracture(Filice et al.,2002;
Shim and Park,2001; Silva et al.,2011). Early stage of fracture predictions in
incremental sheet forming were based on the conditions:(i)Experimental crack study on
the parts, (ii) Maximum force required, (iii) Maximum thinning with drawing angle.
The first case was already discussed in the prior section. Thinning and slope relation to
fracture were confirmed by the studies(Kawai et al.,2001b; Kim,2000; Strano et
al.,2004;Young and Jeswiet,2004).Simultaneously tremendous effort has been made in
deriving an empirical relation to predict the force curve(Aerens et al.,2010). These
approaches were derived based on the plane strain condition or fitting experimental
curve and so, it does not comply with the process variations. So it is a challenge to
define a fracture limit that can precisely define the fracture location & timing by
considering nonlinear path and cyclic loading condition in incremental sheet forming.
Limitation of the strain space in defining forming limit and merits of stress-based
forming limit for ISF was already discussed in the previous sections. So the aim of this
section is to incorporate a suitable and accurate fracture model for incremental sheet
forming in the stress space.
47
2.6.1 Ductile Fracture Criteria (DFC) and Shear Fracture Criteria
To describe the forming limit diagram at fracture, various ductile fracture criteria have
been used in forming process. These criteria had been related to the macroscopic
variables. The hypothesis of the ductile fracture criterion is that the ductile fracture
occurs when the maximum damage value of the workpiece exceeds a critical damage
value (CDV). Ductile failure criteria are usually expressed as an integral form,
representing the effect of the deformation history of the process parameters.
(2-5)
Where the effective strain at fracture and F is is a function of the process parameters.
That means that ductile fracture depends on plastic deformation. The integral criteria
often take an integral form of a stress function over the effective strain field.
Fracture can be also governed by the condition of maximum shear stress as,
= (2-6)
If explained with the principle stress components, it becomes,
12max , , min , , (2-7)
where , are the principle stresses. In quantity, this equation resembles
Tresca yield criterion.
Several researchers reviewed ductile fracture criteria for sheet metal in the last decade,
mainly emphasized the calibration of different materials. Lee(2005) presented
remarkable studies which include experimental, numerical, and analytical approaches
and the work presented various facture criteria to investigate the fracture of thin plates
subjected to localized static and impulsive loading. Wierzbicki et al.(2005)
experimentally investigated the fracture of AA 2024-T 351 with 15 different bulk and
sheet specimens shown in figure 2.10. They proved that the fracture is not geometry
dependent, but depends only on the state of local triaxial stress. The work investigated
the major fracture criteria including constant equivalent strain, the Xue–Wierzbicki
(called X–W), the Johnson–Cook, the crash FEM, and the maximum shear (MS) in
terms of accuracy compared to experimental data.
Figure 2.
Al 2024-T
and flat gr
(5–9), she
2.7 Form
Gradien
Thickness
controvers
and Hecke
on the she
for thicke
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forces, too
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10: Wierzb
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rooved spec
ar (10; 11),
mability
nt:
s effect on
sial. The inf
er(1974). K
eet thicknes
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These incl
ol contact p
various punc
rain with th
xpressed as
48
bicki’s exp
carried on
cimens used
and tensile
Analysis
formabilit
fluence of th
Keeler and B
s with hot a
Marciniak(
lude the gr
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eriments o
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arpentier(19
e ratio of sh
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en fracture
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)
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rst addressed
endence of
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Based on
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s (2 and 3),
. Upsetting
ess/Strain
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formability
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s on sheet
ess, friction
n Nakazima
he increase
ss to punch
kness/punch
r
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g
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y
s
t
n
a
e
h
h
49
radius), the limit strain increase of a thicker sheet at a large punch radius is rapid
compared to a thinner sheet stretched by a smaller punch radius (Fromentin,1998).
Figure 2.11:For different starting geometries, Nakazima strips are deformed in a
hemispherical punch test to generate the different strain paths in the canter of the
test specimens.
However, there are a few articles to study the effect of the stress / strain gradient on the
failure of metal sheets. The number of publications is very limited compared to the
overall research in the field of formability. The early works include Aifantis (1984,
1987)followed by Zbib and Aifantis (1989). The work of Aifantis (1984) developed the
constitutive relations for formability models by incorporating higher-order strain
gradient terms into hardening or yield condition. Based on the formulation of Aifantis
(1987), (Assempour et al.,2009) developed the modified MK prediction with strain
gradient using boundary value problem. Other significant research with the gradient
effect includes El-Domiaty et al.(1996), Tharett and Stoughton (2003), Col et al(2007),
and (Domingo et al.,2013).
Figure 2.12 :Non-linear behaviour of strain paths in stretch-bending with a
cylindrical punch
50
Figure 2.13 :Sum of the principal strains for a 50 wide strip of 1008 AK steel
stretch-bent over a punch wedge with a ¼ inch radius to the depth at which onset
of necking occurs, as reported(Tharrett and Stoughton,2003). The forming limit is
characterized as a simple limit on the sum of the principals because the minor
strain was less than or equal to zero at all points along the strip in a region of the
FLD characterized by a limit on thinning strain for this metal. The FLC and FLDo
was obtained from standard FLD tests independent of the stretch-bend test
(Stoughton and Yoon, 2011)
The effect of the strain gradient through the sheet thickness can be easily explained by
assuming the metal sheet as a superposition of each layer in thickness, all having the
properties of the base material, with each one sustaining its corresponding strain/stress
(Col and Balan,2007). With a ductile material like aluminium alloy (6xxx, 5xxx) failure
is observed through necking which is usually suppressed in low ductile materials. If the
strain gradient is considered, for example, pure bending does not allow necking where
moderate strain gradient is present through the thickness and necking before failure is
common. Given that necking is a kind of plastic instability, it is reasonable to assume
that the plastic instability of the sheet occurs when all layers through the sheet thickness
51
become plastically unstable. Due to the presence of the gradient, the outer layer
observed(Uko et al.,1977) to be more strained than the inner one in a stretch bending
test (Figure 2.12). Thus, the material on the inner side delays the onset of necking
because this layer will be the last to reach the plastic instability(Col and Balan,2007).
Tharret and Stoughton(Tharrett M.R and T.B.,2003) at General Motors conducted a
series of simple bending under tension tests for steel, aluminium, and brass with
different punch tip radii to determine the behaviour of necking through the thickness
direction. The results followed the idea of concave side rule (CSR) which necking is not
initiated when the membrane strains exceed the strain FLC, as was previously thought,
but much later in the forming process, when the strains on the concave side of the sheet
rise to the level of the FLC. While the tests were limited to plane strain conditions, the
results were confirmed in all materials and tooling geometry. As shown in Figure
2.13we see the two necks observed in this specimen on either side of the punch-tip
radius at the location where the strains on the concave side, shown by the enlarged
circles, rise to the level of the FLC for this material. However, the CSR in the strain
space has two major drawbacks :(a) As the rules are expressed in terms of strains, this
approach is valid only for proportional loading conditions,(b) It assumes that the failure
is controlled by the evolution of the stress/strain state just at the sheet surface, which is
rather a restrictive view. That's why later an alternative approach explained by
(Stoughton,2011) and more recently by (Domingo et al.,2013) to evaluate the
formability of sheet metal combining stress-based description with the forming limit
stress-based diagram (FLSD),which appears to be more insensitive to the strain path.
The stress-based approach (Stoughton and Yoon,2011) explained why it is not possible
to form any neck under pure bending, since the stress on the concave side is in
compression, and consequently is always below the limit for initiation of through-
thickness necking instability as shown in the figures2.12 and 2.13. On the other hand,
necking can be formed at high tension under bending when the stresses on all surfaces
exceed the stability limit. Consequently, a reliable necking criterion requires that all
layers through the thickness exceed the instability limit that applies to in-plane tension,
before any insipient through-thickness neck can be develop or be predicted
(Stoughton,2011). One of the most convenient ways to investigate stress change can be
achieved through FEA analysis, where a forming limit or failure criterion is applied for
52
every integration points and, then failure can only be confirmed when all layers exceed
the limit. This idea also explains the limitation of the membrane strains in predicting
failure. Since incremental sheet forming involves cyclic bending/unbending and
combined bending/stretch, stress gradient analysis imposes a real picture of the
deformation mode to predict failure accurately.
53
Chapter 3
Mapping of FLC-N and FLC-F between Stress and Strain
Space
3.1 Introduction
Path independent behavior of forming limit stresses is discussed in the previous section.
The modeling of stress-based FLC requires to incorporate a constitute relations with
flow rule, yield criterion and hardening rule. Strain hardening parameters and
anisotropic coefficient can be experimentally measured. As reliable forming limit
stresses cannot be measured directly from experiments, we start with a conventional
forming limit curve in the strain space obtained from either experiment or MK model.
This chapter explains the mapping process of forming limit curve to stress space starting
from a strain-based forming limit predicted from MK model. Basic theoretical formulas
are reviewed, but, we focus on the framework to build stress-based forming limit. Based
on the material properties of AA6022-T43,the effects of yield criteria and anisotropy on
forming limit is investigated to select the most reliable yield function and hardening
law, which can also be used for FEA in the next section.
3.2 Review of the M-K model
Figure 3.1: Schematic View of MK model
In MK model, it has been assumed that there is a narrow groove in the surface. Thus the
sheet is composed of a safe zone and a groove zone which are denoted by “a” and “b”,
respectively. This groove leads to localized necking in the sheet (Figure 3.1) . For
54
modelling the groove, an imperfection factor is introduced which represents the
thickness ratio f = tb/ta, where, “t” denotes material thickness. The safe area is subjected
to proportional strains. Also it is assumed that strains at the groove direction in the two
areas are equal for compatibility condition. In deformation process, strain ratio
(minimum strain to maximum strain) outside the groove is constant. This ratio decreases
in the groove zone. In practice, this type of groove can be caused by surface roughness
or local thickness variation which could be formed before the process.
Because of plane stress assumption, strain and stress increments in the groove can
directly be solved with respect to the strain increments in safe zone. All strains are zero
in the beginning and then, a small value for ad is assumed to start the analysis. The
calculation is performed in the safe region first and then to the groove region. For a
small value assumed for ad , the equivalent strain, a , is calculated. Equivalent stress
is calculated by substituting a in the hardening law. Thus, all strain and stress
components can be calculated. To calculate stress components, yield function is used,
which incorporates the anisotropy. It is often convenient to use the principal stress ratio,
11
22
and equivalent stress is calculated from hardening law along rolling direction
11( )a Then, yield function is replaced by ,1 for 11,a 22
a respectively. Then, α is
calculated. Thus 11,a is obtained. Stress at TD ( 22
a ) is also calculated by use of α and
calculated 11,a . Unit vectors for this tensor are at xyz or safe system of coordinates. Unit
vectors for the stress tensor can be changed to the groove system by using a rotation
matrix.
For area a
(3-1)
.
55
For the area b 2 (3-2)
.
2
Where‘n’and‘t’are the normal and tangential directions of the groove, respectively.
According to the force equilibrium of the section between the areas a and b, we
have
(3-3)
In other
words,
(3-4)
and / / (3-5)
(3-6)
and
where and the initial thickness of the normal and groove areas before deformation,
respectively; ta and tb denotes the true thickness during deformation of the normal and
groove areas, respectively; ε3 is the strain in the direction of thickness.
Setting where f0 is the initial thickness imperfection we can get from equation:
.
2
.
(3-7)
The angle of the groove, φ, is updated during deformation as:
tan
11
(3-8)
Where and are the principal strain increments in the plane of the area a.
56
The compatibility equation between the areas a and b is
2 . (3-9)
Various methods are available to determine an instability point. In general the
instability point is determined when the ratio of the strain increments outside the groove
to those inside the groove is smaller than a certain value (Cao et al.,2000). The
nonlinear equations are solved by Newton-Raphson iteration method.
Detail method of calculation for MK-based FLC can be found at the references(Barata
et al.,1984; Barlat and Jalinier,1985) for isotropic and quadratic yield criteria with
isotropic hardening models. (Butuc et al.,2010) also explained a method to develop MK
based forming limit with advanced constitutive yield criteria including Yld2000-2d
(Barlat et al.,2003).
3.3 Constitutive Modeling of Stress and Strain forming limits:
The governing equation to compute strain-based forming limit through MKmodel can
be represented as follows:
1 D
1
(3-10)
where D is the value to characterize and quantify the thickness imperfection. For typical
commercial alloys, the studies from damage microscopic observations and probability
57
calculations have shown that D is about 0.4% for aluminium alloy (Hong et al.,2008).
Therefore, D = 0.004 is used in this work.
Strain-based forming limit is defined as :
1(Strain-based FLC)
(3-11)
The principal strain ratio (ρ) in equation(3-11 can be converted into the principal stress ratio
( ) in the stress space as
1 (Stress-based FLC) (3-12)
A detailed mapping procedure for general non-quadratic yield criteria is presented by
Stoughton and Yoon (2005) which will be reviewed below :
Under proportional loading, the plastic strain tensor components at any point on the
strain-based FLC are proportional to the gradients of the plastic potential, at the
corresponding stress thatresulted in the plastic strain,
(3-13)
Equation (3-13, can be written alternately as
(3-14)
Where is the effective plastic strain at the forming limit, which varies from point to
pointalong the FLC. An equivalent nomenclature for the gradient of the plastic
potential is a equivalent nomenclature of gradient of plastic potential and a known
explicit dimensionless function of the stress tensor where , . The principal
strains on the forming limit are given by
58
2
(3-15)
Substituting equation (3-15 to equation (3-14 :
1
2
(3-16)
Equation (3-9) gives a definition of strain ratio as a function of gradient of plastic
potential as follows:
(3-17)
A representation of general plane-stress state in terms of major principle stress and
the major stress ratio , can simplify the derivation of stress state corresponding
to a given plastic strain state obtained under linear loading condition. The stress tensor
for plane-stress condition can be written explicitly in terms of the two principal stresses,
and the orientation angle θ of the major stress component to the rolling direction of the
sheet as follows,
Alternately it can be written
(3-18)
=1
=
(3-19)
Using Eq. (3-19with the selected material model, we can define the yield function with
a constant ,
59
, , , , (3-20)
Using the power law and Eq. (3-20, we can finally calculate the
major principal stress as
, ,
(3-21)
α
Barlat’s Yld 2000-2d is used to derive the stress independent normalized yield function,
, , at equation (3-20)and associated gradient of plastic potentials
, , at equation (3-14) which is required to calculate effective plastic strain
in Eq.(3-16).
3.3.1 Hardening law description:
Material is defined in the model macroscopically by its Yield surface and its work
hardening law. Usually this law can be defined by Swift or Voce for isotopic hardening:
i. Swift Law:
(3-22)
where, K, n, are material constants and are effective stress and
effective plastic strain, respectively.
ii. Voce Law
Voce function defines hardening as :
(3-23)
where A, B, C are material constants. These constants are calculated by fitting
experimental stress/strain data.
60
3.3.2 Yield Criteria Description:
i. Hill ’(1948) quadratic function:
Hill’48 yield function (Hill,1948) is a quadratic function considering with six
coefficients. The function is expressed as:
2 2
2 2
(3-24)
Under the plane stress condition, it reduced to four parameter model
2 2 2 (3-25)
Where F,G,H, N values are calculated based on three Lankford values along 0,45, 90
degrees from the rolling and one yield stress ( along the reference direction (rolling or
biaxial):
12 ;
11
2 ;1
2
12 1
2
(3-26)
ii. Yld89 (Barlat and Lian, 1989) :
The yield function proposed by Barlat et al.(1989) is a non-quadratic yield criterion
with four coefficients under plane stress condition. Yld89(will be mentioned asBarlat-
89 from here on)can be represented as
, ,2| |
2| |
2|2 | / (3-27)
where
, ,h2
, ,h2
a; c; p and h are material parameters. For FCC materials, usually the exponent “m”
equals to 8. Four material parameters can be determined from either r-values or yield
stresses.
61
iii. BarlatYld 2000-2d:
Non-quadratic yield function by Barlat et al. (2003) (called Yld2000-2d) is widely used
in order to describe the anisotropic material behaviour of aluminium alloys due to the
balance in accuracy and computational time.
The plastic potential is written for the model as
, , 1/2 /
(3-28)
where “a” is material coefficient and
| | (3-29)
|2 | |2 |
In Eq,(3-28, ′, ′′ 1,2 are defined by two linear transformations as
s= (3-30)
s=
The 2Dlinear transformations can be expressed for plane stress condition as follows:
00
0 0,
00
0 0 (3-31)
These anisotropy coefficients can be connect to eight independent constants
1 8 as
2 /3
/3
/3
2 /3
(3-32)
8 2 2 2 /9
4 4 4 /9
4 4 4 /9
8 4 4 /9
62
The eight parameters in Eq. (3-18) can be calculated by an iterative procedure from
eight independent testing.
3.4 Forming limit Representation in Strain and Stress Spaces with
Different Yield Criteria: In the present work the forming limit prediction can cover various hardening laws and
yield functions. Strain-based forming limit is calculated using M-K theory where rigid
plasticity, plane stress condition, and isotropic hardening are assumed. Program
structure for forming limit prediction in strain space is shown in Figure 3.2 . The
detailed mapping procedure to the stress space is shown in Figure 3.3(Von
Mises),Figure 3.4(Hill’s quadratic function) and Figure 3.5(General non-quadratic
function).
Figure 3.2:Structure of FLD code for strain and stress FLC : The Subroutine
Structure for Forming Limit Curve Prediction in Strain Space : a.Hardening law
b. Yield Function c. Flow Rule .
63
Figure 3.3: Stress –Strain relation for Isotropic Yield Criteria (Von Mises)
Figure 3.4: Stress-Strain relation for Quadratic Model(Hill Normal Anisotropy):
Figure 3.5: Stress-Strain Relation for Non Quadratic Model(Yld2000-2d)
64
3.4 Forming Limit Modeling for AA 6022-T4E32 with Different Yield
Criteria :
The data from hydraulic bulge test, and uniaxial test at seven different directions (00,
150, 300, 450, 600, 750, 900) are available from Numisheet benchmark data for AA 6022-
T4E32(Brem et al.,2005). To check the validity of these data for the selected sample,
directional uniaxial tensile tests along 0, 45 ad 90 degrees from the rolling were carried
out and the results are found to fit well the experimental hardening curve (Fig 3.7a).
Hardening curve is generated using Swift law using the material constant for K and n
(Swift fits better than Voce up to fracture for this material). The MK modelling code
requires input of plastic strain ratio (Lankford coefficient) and normalized yield stresses
at the angles of 00, 450, 900from the rolling and the biaxial direction. The data are used
to calculate coefficients of selected yield criteria as mentioned in section 3.3.2. Input
stress ratios are calculated from minimum plastic workto fracture. In this method, stress
values are taken as a function of minimum plastic work (which is the area under
hardening curve).
(3-33)
Where is the plastic strain corresponding to plastic work (W) to fracture.
Plastic work up to fracture strain ( ) is calculated along 00, 450, 900directions from the
rolling and the biaxial direction using Swift law. Among the four plastic works to
fracture (W0, W45, W90, Wb), the minimum plastic work is identified as W(min) . For AA
6022-T4E32,the minimum plastic work to fracture strain is found at 00 hardening curve
W(min =W0).Now plastic strain and effective stress for each directional curve can be
found when W(min) is given (Figure 3.6).Then, effective stresses obtained from the four
curves are normalized with respect to the rolling. The calculated values are presented in
Table 3.1.
Tensile test along 45 degrees has been conducted up to fracture. A local strains was
measured by ARAMIS system. To model appropriate hardening curves, both Voce and
Swift laws are modelled. Although Voce shows good prediction up to 20 % of strain,
we obtained a less square fit error with Swift law for the overall behaviour up to fracture
(400Mpa at 0.56) as can be seen in Fig.3.7(b). It is remarkable to keep a good hardening
to the fracture. It is because of the saturation behaviour with Voce curve and the work
hardening
strain lim
much high
element si
Figu
Table 3.1Angle
0
45
90
Biaxial
Table 3.2
Calculated
Table 3.2:M
α1
0.949
Material C
a
1.076
Material C
F
1.5033
with Swif
it does not
her maximu
imulation, w
ure 3.6:Un
: Material UTS R
136.0
131.2
127.6
: Material
Material Con
Material Cons
α 2
1.081
onstants for B
c
0.924
onstants for H
G
1.063
65
ft near fract
t guarantee
um plastic
which can av
ique plastic
Properties R value
1.029 0
0.532 0
0.728 0
1.149 0
Constants
nstants to plot
stants for Yield
α 3
0.943
Barlat Yld89( b
h
1.097
Hill 48 (based
H
1.094
5
ture. Also,
any reliabi
strain is m
void seriou
c work for
for AA602n
0.258 47
0.254 45
0.258 44
0.255 44
for Differe
Yield Functio
d Function Yl
α 4 α 5
1.055 0.9
based on R-va
p
1.000
on R-value)
N
2.649
extrapolatin
ility. Equi-
more reliable
s extrapolat
different h
22-T4E32 K M
79.92
55.00
42.45
48.58
ent Yield C
ons for AA602
ld2000-2d (Ex
5 α 6
993 0.952
alue)
ng the unia
biaxial bu
e as the ref
tion.
hardeningc
Max plastic
strain )
0.196
0.217
0.210
0.516
riteria
22 T43
xponent m=8)
α 7
2 0.971
axial plots t
ulge test wh
ference curv
urves (W1=
Plast
4
5
5
15
)
α8
1 1.17
to a higher
hich allows
ve in finite
=W2)
tic work
49.94
53.37
50.12
58.31
72
r
s
e
66
Figure 3.7:( a) Hardening curves are plotted for uniaxial tension and biaxial data using
Swift law and compared with experiment uniaxial tension test for 450curve. (b) Fitting
Swift and Voce hardenings curve with experimental curve upto fracture stress.
67
Normalized stresses and R-values predicted from different yield functions including
quadratic and non-quadratic models are compared in Fig.3.8 and 3.9. In Fig. 3.8,
predictions of stress directionality from Hill 48 and Barlat89 fail if r-value based
coefficients are used. On the other hand, stress directionality is coincided well with the
experiment if stress-based coefficients are used for Hill 48 and Barlat 89. The similar
analogy can be found in Fig. 3.9 for r-value directionality. If r-values are used for the
calibration of yield function, prediction is accurate. Predictions from Yld2000-2d are
found to be excellent for both stress and r-value anisotropies. It is because the function
uses eight coefficients which are sufficient to consider both directionalities, Yield
surfaces are also plotted in Fig.3.10. It is found that Barlat 89 (Yld89) is located at the
most inside in plane strain region.
Strain-based FLD is generated by MK approach incorporating various yield criteria
already discussed earlier. In Fig. 3.11, the results are compared with the experimental
data shown in Progress Report published by US department of Energy(Esteban et
al.,2008) which determined the necking limit using the limit-dome test apparatus at the
Advanced Materials Processing Laboratory at NWU. It is found that the left side of
forming limit curve is less sensitive in response to yield criteria, although the right hand
side displays the large variations especially for the biaxial forming direction, which is
one of the most vital zone for incremental sheet forming. Quadratic functions
overestimates forming limit. Hill or von Mises models are not obviously a good choice
for incremental sheet forming. Yld89 slightly over estimates the forming limit, although
the prediction is reasonable limit. Yld2000-2d is found to match pretty closely the
experimental forming limit curve with good coincidence on the cracks occurred in both
plane strain and biaxial and directions. This observation confirmed Yld2000-2d model
as the most acceptable criteria to be incorporated. Strain-based forming limit is
converted to stress-based forming limit based on the procedure explained in section 3.4
(See Fig.3.12). Predicted stress-based forming limit model shows that the material
possess higher forming limit in plane strain and biaxial tension for Yld 2000-2d model
compared to Yld89. .
68
Figure 3.8: Stress directionality predicted from various yield functions for AA
6022-T4E32
Figure 3.9:r-value directionality predicted from various yield functions for AA
6022-T4E32
69
Figure 3.10:Yield locus predicted from various yield functions for AA 6022-T4E32
Figure 3.11:Predicted strain-based forming limit curves for AA 6022-T4E32
70
Figure 3.12:Predicted stress-based forming limit curves for AA 6022-T4E32
3.5 Modeling of Ductile and Shear Fracture Criteria:
In this section several ductile and shear fracture criteria are studied and modelling
procedure is demonstrated to integrate these fracture criteria in the stress space. By
presenting both forming and fracture limits, post necking behaviour can be traced. Five
ductile fracture criteria and maximum shear stress criterion are studied for the purpose.
Fracture criteria used in the work include (i)Cockcroft-Latham(CL) (Cockcroft and
Latham,1968), (ii) Brozzo(Brozzo et al.,1972.) (iii). Oh (Oh et al.,1979), (iv) Ko(Ko et
al.,2007), (v) Maximum Shear Stress (Stoughton and Yoon, 2011). The equations are
summarized in Table 3.3
For the convenience of fracture modelling,von Mises function is used with Swift
hardening law. To determine the fracture strain, uniaxial tensile test data are fitted with
the left hand side of experimental forming limit ( . from Fig,3.7. The material
71
constants calculated from uniaxial tensile test are used for the hardening in Swift Law.
Using the constitutive relation shown in Figure 3.13with the fracture criteria listed in
Table 3.3, fracture limit (FLC-F) is generated in both stress and strain spaces in
Figs.3.14 and 3.15
Figure 3.13: Mapping procedure for fracture criteria between strain and stress
spaces
Table 3.3 : Selecetd fracture criteria
72
Figure 3.14: Comparison of fracture limit curve presented in strain spacefor AA
6022 T4E32(reference maximum fracture strain . .
Figure 3.15: Comparison of fracture limit curve presented in stress space for AA
6022 T4E32.(reference maximum fracture strain . .
73
3.5.1 Shear Fracture Modeling using Advanced Constitutive Equations :
Maximum shear stress criterion has a similarity with Tresca surface. To complete the
plot, shear stress at compression and tension is required. Alternatively, it can be
extended to Coloumb-Mohr fracture model. However, for incremental sheet forming,
failure is unlikely to occur in the compression zone, so only the first quadrant with
tension and biaxial direction is considered. The fracture polygon is drawn with the
available maximum shear stress data in uniaxial tension direction. For the comparison
of FE solutions later, the stress fracture curve is also transformed to the strain space
using Yld2000-2d and Swift hardening law. The calculation procedure is shown in
Fig.3.16. Integrated plot of maximum shear fracture criterion with necking limit is
shown for both stress and strain space (Figs. 3.17 and 3.18).
Figure 3.16:Mappingprocedure of fracture surface from stress space to strain
space using Yld2000-2d :
74
.
Figure 3.17: Strain space presentation of forming limit (FLC-N) and fracture limit
(FLC-F) with Yld 2000-2d
Figure 3.18: Stress space presentation of forming limit (FLC-N) and fracture limit
(FLC-F) with Yld 2000-2d
3.6 Summary:
The chapter gives a complete procedure to generate stress-based forming and fracture
limits using advanced anisotropic yield function. The theory has been implemented for
Hill 48, Barlat 89, and Yld2000-2d. Yld2000-2d is found to be most reliable to be used
by confirming the experimental data.
75
Chapter 4
Experimental Observations and Data Analysis
4.1 Introduction:
Two different types of experiments are carried out using Robot and CNC machine
available in the AusAMRC (Australian Advanced Manufacturing Research Centre)at
Swinburne University of Technology. The experimental setup for incremental sheet
forming is although simple, the process involved many complexities depending on the
parameters which include tool size, machine capacity, lubrication, tool path, feed rate,
punch speed etc. A good number of literatures are available in this regards, hence the
author did not orient his work to investigate the effects of process parameters. In order
to ensure reliable experimental data, stable and repeatable experiments are required.
That way the experimental parts can be utilized for the comparison with the result from
finite element method for the analysis of formability. For this reason experimental
parameters which were formed successfully are only adopted for measurement and
analysis purpose.
4.2 Design of Experiment:
4.2.1 CAD System and Tool Path Design:
To develop CAM based approach, G-Code is generated from the CAD data (IGES file)
using a commercial software called Power Mill v10. Power mill generates the tool path
that finally used in CNC machine to form the blank accordingly. The tool path is kept to
an inclination angle of 450to avoid any edge contact of the tool with the formed surface.
Another important factor is to control the step down mode of the tool path. Step down
motion can be done applying a skim mode, sliding over the surface along a single line.
Skim mode for downward motion sometimes causes a crack as observed in a few tests
with a pyramid and cone shape forming According to FE analysis, skim mode generates
the reaction force rapidly. Other observations show that step down motion along the
same string line causes stretching, so step down motion was carried out along the
different lines. Noticeably a formed zone like corner can be avoided for the step change
path in order to escape a possible biaxial stretch.
76
Fig (a)
Fig(b)
Fig(c)
Figure 4.1: (a)Tool Path generated for different shapes(b) Straight and Skim
modes downward step.(c)Reference dimensionsfor pyramid and cone Shape
4.2.2 Forming Tool Design:
A solid hemispherical head is generally used for asymmetric single point incremental
forming. For very steep wall it is necessary to design a smaller tool shank than the
77
sphere diameter to avoid the contact between shank and sheet. The tool-head shape must
take into account the tool path. Two important factors to design tools can be
summarized as:(i) Minimum friction,(ii) Optimum surface quality. The ball-head
diameter is chosen properly according to the steps required considering machine
capacity. A wide range of tool diameters is used, starting from small diameter of 6 mm
to a large tool diameter of 100 mm for the manufacturing of a large part(Jeswiet et
al.,2005). Tool diameter is usually selected based on the smallest concave radius
required in the part. Tool diameter also influences the surface quality and/or the
manufacturing time. An early work by Kim and Park(2002)found good formability for
tool head range within of 5 to 10 mm diameter. The effect of tool diameter is presented
with various experimental evidences by Silva(Maria B. Silva,2011) which utilized 5 sets
of punch tool ranging from 4 to25 mm to construct a cone and a pyramid shape.He
found that a larger tool ensures safer forming. The most commonly used diameters are
located between 12 and 12.5 mm(Bambach et al.,2003a; Jeswiet et al.,2005; Kim and
Yang,2000)
In present work, three types of tools have been manufactured with the tip diameter of
12.66 mm:
i. Punch tool with a brazed ball at top.
ii. Tool with a free rolling ball at the top((a). with a fluid channel,(b) without a fluid
channel)
iii. Tool with a spherical roller.
Fig (a) Fig(b)
Figure 4.2 : (a) Design of tool with lubrication channel, (b) Complete assembly of
tool with force sensor mountings.
78
In first kind the ball is placed on a cone groove. The shaft end diameter is smaller than
the one of the ball so that any angle of slope and curvature can be formed without
interface. For the experiment Second kind of tool is used which used a steel ball placed
in a machined grooved followed by locking at edges. Then, the ball can roll in any axis
or rotation. A channel is machined inside the tool in order to hold and supply lubricants
from inside. Figure 4.2 shows the detail design of tool and designed mounting to hold
force sensor with it.
4.2.3 Fixture Plate, Die Design:
Fixture plate is designed by placing the holding plate on a square shape box. The base is
kept heavy to avoid any vibration. Two sets of support plates are implemented to form
pyramids and cone shape. The blank size of the fixture is selected: 222 X 222 mm.
Figure 4.3: Fixture for experiment
4.2.4 Forming Machines:
Experimental test was carried out with a 5 axis CNC machine and in an ABB robot.
Five axis machine is suitable to produce parts with high accuracy but has a limitation of
79
part size due to its limited workspace. On the other hand, ABB robot can produce quite
large parts with all six degrees of freedom. However, accuracy is in issue due to its
limited rigidity of robot arm. Advantage of using a robot for incremental sheet forming
is that forming process can be observed clearly and the process can be interrupted at any
time. A constant speed of 1000 mm/min and anticlockwise motion is maintained for
both CNC and ABB robot. Downward step size is kept constant with 0.5 mm at the
completion of each incremental cycle. Oil film is introduced onthe sheet surface and in
the hollow channel of the tool to provide ample lubrication. Sometimes the parts formed
by robot shows slight wrinkles because of the vibration of the tool head. Configuration
of arm in a closer space (making the robot arm shorter)improves this chattering. The
complete experimental process is presented in Figure 4.6
Figure 4.4: Robot(left) and CNC machine in ISF
Figure 4.5: Wrinkling occurred while forming using Robot (Pyramid (left) and
cone (right)).
80
Figure 4.6: The complete CAD/CAM-Robot setup for the experiment and FE
analysis.
Table 4.1:Experimental Conditions for Pyramid and Cone Shape Machine i) DACKEL‐MAHO CNC 5 AXIS
ii) ABB (IRB6640 ;180 Kg,2.55)
Tool Path i. G‐Code by Power Mill 10.1 ii. Rapid code
Feed 1000mm/min
Lubrication Holcut 807 ( Metal Cutting Fluid Concentrate, Sp. Gravity (1.03 at 150 C)
i. Thin Film Applied on Sheet Surface ii. Applied in Channel inside the tool to flow on
gravity.
Force Sensor Schaevitz Engineering , Pennsauken,
New Jersey , Model: FTD -1U-1000, S/N: 2160Capacity: 1000 lbs , calibrated for : -15v-+15v for maximum load,
81
4.3 Measurement of Strain:
4.3.1 Strain Measurement by CMM, GPA, and ASAME Target Model :
Three techniques are used to measure strains for the formed parts.
i. Coordinate Measuring Equipment (CMM): Sheffield CMM with software PC-
DMIS EMS.
ii. GPA (Grid Point Analyser) Model : GPA-100 and software GPA-V3
iii. ASAME Target Model (Model : TRM-25) with software ASAME Version 4
i. CMM : Blank is marked with laser for a rectangle grid (2.5 mm x 2.5mm) and
measured the strains in CMM (Nodal coordinates are measured for the selected area
after the shape is formed). Nodal coordinates are converted to strain data using a in-
house code developed with FORTRAN. The code follows the 3D membrane/ shell
method. Accuracy of CMM depends upon the accurate positioning of the stylus tip at
each grid node. Therefore, it is found that error is higher. For this reason, the
measurement from CMM is not used for the comparison with FE results.
Figure 4.7 in-house 3D membrane/ shell strain measurement ofpyramid part by
CMM(Yoon et al., 2002)
ii. Grid Point Analyser(GPA):As shown in Figure 4.8 , it is a hand holding system that
uses a standard USB video camera with a close up lens and viewing one grid element
such as circles or squares. From the original undeformed grid size, the GPA-100 can
measure the strain within ±2. 0% (±1. 5% under good grid conditions).Strains are
measured with laser grid marked pyramid part with this system. Major and minor strains
82
are displaced in Figs.4.9 and 4.10. Also, the strains on forming limit diagram are
presented in Figure 4.11.
Figure 4.8 Thicknessmeasurement of laser marked pyramid with GPA system
Figure 4.9: Major Stain distribution for pyramid as measured with GPA
83
Figure 4.10: Minor Strain Distribution for Pyramid as measured by GPA
Figure 4.11: Experimental Strain FLD plot for Pyramid shape part.
84
iii. ASAME Target Model:It is an accurate and reliable method used for the strain
measurement shown in Figure 4.12. The system measures the surface geometry and
calculates the strains on the part using digital image correlation, where two or more
views of an area on the part are photographed at the different positions. These offset
view are digitized with two dimensional space and photo geometry principle is applied
to determine the three-dimensional map of the area. Based on the undeformed grid size
and the three-dimensional data for each deformed grid, the surface strains are
calculated. The experimentally measured major and minor strains are presented with
forming limit curve in Figure 4.15.
Figure 4.12 Strain measurment of a cone shape using the ASAME.
85
Figure 4.13: Major strain distribution for a cone shape part.
Figure 4.14: Minor strain distribution for a cone shape measured in ASAME.
86
Figure 4.15: Strain-based forming limit plot for a cone shape measured in ASAME
4.4 Conclusions:
The strain measurements using CMM, GPA, and ASAME are carried out in this work.
It is found that GPA and ASAME show a similar pattern of forming limit prediction.
Experimental results in forming limit curve shows that the strain distributions are much
above the necking limit, although the part is formable. This clearly invalidates the
conventional forming limit curve to justify the formability in incremental sheet forming.
This also complies with acceptable experimental observations from many researchers.
87
Chapter 5
Development of Stress FLD through FE Approaches
5.1 Introduction
Finite element analysis of incremental sheet forming is challenging, since it requires a
large number of increments to be performed. Secondly, the small contact area between
tool and blank undergoes relatively large deformation per step. Implicit simulation often
encounters a convergence problem due to a large deformation and also explicit method
needs extremely long computational time due to many incremental steps. An implicit
simulation of single point incremental forming provides a very good agreement with
experimental data (Bambach et al.,2004), although convergence is still in issue. In FE
analysis for a conventional sheet forming, Newton–Raphson method per iteration is
cheaper with a linear convergence behavior (Zienkiewicz and Taylor,2005) but for
incremental sheet forming it requires more computing time as it takes more iterations to
achieve convergence at each step (Hadoush and Boogaard, 2009). But, it is still efficient
compare to explicit solution.
Regarding yield criteria for incremental sheet forming, Barlat 89 (Yld89)reached 89 %
of accuracy (compared to experimental data) which is higher than 64% accuracy from
Von Mises(Malhotra et al.,2010). In the present work, the author carried out intensive
research with element types and material models in order to build a reliable model
justifying a stress-based approach for incremental sheet forming. The finite element
results are verified through thinning and punch force data. The predicted strains and
stresses are also mapped into strain and stress spaces for necking and fracture analyses.
5.2 Implicit FE analysis of Incremental Sheet Forming :
A number of software attempted to implement incremental forming process. In finite
element model, element type and size, and material model need to be selected carefully
for accuracy and convergence. For the present work, ABAQUS/Standard encountered
convergence issue to form a pyramid shape. MSC. Marc and LS-DYNA3Dwere found
to be more stable in terms of convergence. All the results presented in this work are
based on the implicit simulation using LS-DYNA 3D.
88
5.2.1 Tool Path Generation:
For a reliable simulation, the tool path in simulation should be compatible with that of
experiment. In the present work, the tool path used in experimental test is directly
implemented into simulation to mimic the real process. Several researches claimed to
achieve a stable numerical simulation with the direction change at each step
(anticlockwiseclockwiseanticlockwise),which is not compatible with the
experimental tool path. So, it is not implemented. The tool path generated in G-code is
converted to APT file, then to a rapid file. Rapid file consists of the coordinates along
the defined path. As tool motion in the rapid file is controlled by Cartesian coordinate, it
poses tessellation in generating a circular profile. It can be reduced by generating the
toolpath points close to small incremental step. The coordinates are extracted and
implemented in LS-DYNA3D as the required motion of the tool. In a similar way, a tool
path for a complex shape is also generated for finite element simulation. Tool is
modeled as a rigid ball with the restricted to self-rotation. Small amount of
friction(0.01) is introduced for contact considering very well lubricated experimental
condition.
5.2.2 Element Selection:
Element selection is also very challenging for incremental sheet forming to ensure
economic, accurate solution throughout the whole process. Shell element is found to be
more suitable compared to solid elements in incremental sheet forming. Malhotra et
al.(2010) carried out experiment and simulation for a cone shape (70 Degree) AA 5052
with 1 mm thickness for the comparison with continuum element solution. The research
found that solid element generates an excessive punch force than the experimental
observation while shell element gives a bit less force but predicts thinning closer to the
experiment.
Belytschko-Tsay shell element called BTL (Belytschko and Tsay,1981) is widely used
in research and industry for its great computational efficiency. The shell element is
constructed by combining flat 4-nodes with a plane quadrilateral bending and, so
warping is not considered, since the co-rotational system located at the center of an
element is used for the entire element. This incapability of warping causes severe plastic
strain undergoing a local deformation. The simulation result for a pyramid shape
confirms high effective plastic strain with BTL element (Figure5.1) and also shows
89
irregular thickness changes with a cone shape simulation. Mesh refinement with solid
and shell element is also attempted. It is observed that mesh refinement does not
improve the result in incremental sheet forming simulation rather increasing the
simulation time significantly. For same reason, solid element is not an attractive
solution for incremental sheet forming.
A remarkable improvement can be achieved by using the Belytschko, Wong and Chang
element called BWC(Belytschko,1989) with competitive computational power. The
element is based on continuum-based shell which includes the projection operators for
transverse shear and bending and then, it is suitable for dealing with warping. Figure5.1
shows the capability for compensating severe warping with BWC element. Fig. 5.2
shows the comparisons of effective plastic strain along an inner circle. It can be shown
that BTL and thick shell predict unrealistic results. Fig.5.3 shows thickness strain
distribution along x-direction. In a cone shape, the draw angle is kept the same
throughout the process, which should give uniform thickness strain along the section.
This is also in accordance with the spinning sine law(Kobayashi et al.,1961)and the
paper by (Jeswiet et al.,2005). However, in Figure 5.3, BTL element without warping
stiffness shows non-uniformity in thickness distribution, which is not observed
fromBWC 4 or 8 nodes element. In a thick shell, hourglass modes may cause the severe
distortions.
Figure5.1:(left) Increase in displacement of BTL element due absence of warping
stiffness (right): Warping occurred in a pyramid corner with BWC element
90
Figure5.2: Comparison of effective plastic strain in a critical segment along an
inner circle.
Figure 5.3: Comparison of thickness strain distribution along a side wall of x-axis .
Cone and pyramid shapesare variefied with Yld 2000-2dmodel (Barlat et al., 2003)
using the coefficients determinded from AA 6022-TE43 material properties. FE
91
modelusedselected shell elements with implicit solution scheme. The outcomes are
listed inFigure5.1
Table 5.1: Effect of shell element on thickness predictionwith element size of 2.5 x
2.5 for a cone shape (Yld 2000-2d)
Element Type Max Predicted
Z-force(N)
Thickness
Reduction (%)
BWC(4N) 1445 35.24
BWC(8N) 1505 35.695
BTL(4N) 1510 37.686
5.2.3 Mesh Sensitivity Analysis:
Finite Element analysis for incremental sheet forming is mesh sensitive uptoa certain
range. Different size of meshes have been investigated for the accuracy of predicting the
principle strains and thinning. Results also compared for simulation time and data size
requirements. Keeping the same input parameters and material models, simulation with
a cone shape is carried out with the mesh sizes of 2.5 x 2.5 mm, 2x 2 mm, 1 x 1 mm,
0.5 x 0.5 mm. In the third and fourth cases, adaptive remeshing is used starting from the
original mesh sizes of 4 and 2 mm, respectively. One section for each geometry (cone or
pyramid) shapes is selected to compared the strain distributions with experimental test
(Figs. 5.4 and 5.5). FE results obtained from the four selected mesh sizes does not show
noticeable differences and all three strains (major, minor, thickness)are reasonably close
to that of experimental results. The observation confirms the mesh insensitivity even
with remeshing. Implementing very fine mesh size which is less than the thickness leads
to worse results. In present work, a selection of 2.5x2.5 mm size for 1 mm thickness is
found to be an optimum considering that a little decrease of mesh size causes a
significant increase of computational time.
92
Table 5.2: Effect of mesh size with a cone shape Mat Model Shape Element
type
Element Size Max Thickness
Reduction (%)
Max
EPS(Ref
Mid)
CPU Time
4CPU
(Seconds)
Barlat 2000-2d, Cone BWC(4N) 2.5x2.5 35.695 0.4657 34481
Barlat 2000-2d, Cone BWC(4N) 2 x2 36.314 0.4955 50635
Barlat 2000-2d, Cone BWC(4N) 1 x 1(RM) 36.0121 0.4735 55396
Barlat 2000-2d Cone BWC(4N) 0.5x0.5(RM) 37.472 0.576 156952(1cpu)
Figure 5.4: Comparisons of the two principle strains and thickness strain predicted
from different mesh sizes with experimental result for a cone shape
93
Figure 5.5: Comparisons of the two principle strains and thickness strain predicted
from different mesh sizes with experimental result for a pyramid shape.
5.2.4 :Effect of yield criterion:
In this section, several yield functions have been investigated using the material models
available in LS-DYNA3D under the same condition.. Hill’s 48 and Yld 89 (Barlat and Lian,
1989), and Yld 2000-2d (Barlat et al., 2003) are selected for the evaluation. Maximum
effective plastic strain and thinning are compared in Table 5.3. Yld 2000-2d shows the
closest prediction of thickness strain with experiment. The results will be further compared
for the analyses of forming limits in strain and stress spaces later.
Table 5.3: Effect of different Yield criteria on performance of FE of ISF
Shape Yield Cr Element Size Element
Type
Thickness
Reduction
(%)
Max EPS
(Ref.Mid)
Cone Yld2000-2d 2.5x2.5 BWC(4N) 35.24 0.45934
Cone Yld 89 2.5x2.5 BWC(4N) 35.699 0.46442
Cone Hill 48 2.5x2.5 BWC(4N) 39.945 0.47179
Cone vonMises 2.5x2.5 BWC(4N) 32.765 0.49597
94
5.2.5 Prediction of Punch (Tool) Force:
Tool force can be used a representative value of the process. Tool force predicted from
FE analysis is investigated in this section. Cerro et al.(2006) used ABAQUS/Explicit
with shell elements and obtained a 5% difference of the maximum punch force between
simulation and experimental data. Influence of the tool speed is also investigated by
Yamashita et al.(2008)who suggested introducing a right mass scaling in simulation.
Duflou et al.(2007a) assumed that a considerable drop in the tool force after reaching a
pronounced peak value is an indicator for failure. In spite of a significant amount of
applications, verification of tool force curve is sometimes misleading. First of all,
predicted force curve differs based on the element used(Malhotra et al.,2010).
Experimental test accompanies the distortions as and frictional resistance which makes
difficult the measurement for z-directional reaction, while the results in simulation are
purely dedicated to z-directional reactions. Figs. 5.6 and 5.7 present the simulation
results of the predicted z forces with different yield criteria and element types. FE
prediction from all selected criteria seems to concede well with the experimental data,
although Yld 2000 gives the best fit. Element types are also found to influence the z-
force to a large extent. As shown in Fig.5.7, four node or eight-node thin shell predicts
z-force close to experiment but, solid-shell and thick shell produce over the
overestimated results.
Figure 5.6: Tool force sensitivity from yield criteria
95
Tool force along normal direction shows oscillation that usually occurs during
incremental step change. The tool undergoes release of contact followed bysudden
contact and stretching at tool tipduring the change of downward step. This causes
instant drop and rise in force curve. The mechanism is further explained at section 6.2.1.
Figure 5.7: Tool force sensitivity from element types.
5.3 Forming Limit Analysis from FE Results
From the observations so far in Chapter 4 and 5, FE prediction from mesh size of 2.5
mm with BWC shell element and Yld2000-2d with Swift hardening curve is found to be
the most reliable and accurate. Therefore, Strain-based and stress-based forming limits
are verified with the FE results in incremental sheet forming. Fig.5.8 shows thickness
strain distribution at the final step.
96
i. Cone Shape Results(Yld2000-2d):
Figure 5.8: Thickness strain distribution for a cone shape simulation with Yld
2000-2d
Table:5.4 Properties of FE model for Cone Shape Number of elements: 6400
Type of elements: THIN SHELL 4 NODE , Formulation : Belytschko, Wong and Chang (BWC)
Contact property model Surface to Surface
Friction formulation Low Static Friction (0.01)
CPU clock speed 2.1 GHtx
Number of cores per CPU 4
Main memory 4 GB
Operating system Windows 7(WINX64)
Total CPU time 1 1hours 36 minutes 43 seconds
The principal plastic strains extracted from the integration points in the top, middle and
bottom layers are presented in the strain space with the associated necking and failure
limits in Fig.5.9. It is found that the results are located above the necking limit. The final
strain data are located near plane strain tension direction for all the layers. The bottom
layer(non-contact) has a higher value of plastic strain than any other integration points,
while the top layer(tool contact) has a lower strain. According to the shear fracture curve in
the strain space, the bottom plane exceeded the maximum fracture limit.
97
A different observation is noticed in the stress space plotted with the projected stress values.
Because of continuous loading and unloading by the tool, FE data need to be compensated.
A projection method based on the final stresses was introduced to compensate the current
unloaded stress by projecting it back to the current yield surface or hardening (called the
projected stress). The calculation method for the projection is described in section 3.3.
Projected final stress data in top, middle, and bottom integration points of the sheet are
plotted in the stress space incorporating necking fracture limits in Fig.5.10. Onthe bottom
surface, stresses are evolved mainly toward the minor tension direction, where onthe top
surface, stresses are dominant toward the major tension and minor compression directions.
Stress distributions show necking in all layers but no exceeding fracture limit.
Figure 5.9: Predicted strains from Yld2000-2d for a cone shape in strain-based
forming limit: mid layer(left);Bottom layer (middle) and top layer (right)
98
Figure 5.10: Stress plots from Yld2000-2d for a cone shape(top, mid and bottom
layers)
99
5.3.1 Pyramid Shape Results(Yld2000):
Similar to a cone shape, incremental sheet forming simulation for a pyramid shape was
conducted. Fig.5.11 shows the thickness strain contour at the final stage. The critical
thinning occurs uniformly along the wall. Fig, 5.12 shows the principal plastic strains
extracted from the integration points in the top, middle and bottom layers. Strains for a
pyramid shape show the strain distributions between biaxial and plane strain directions
which also varies through the thickness. Predicted results show necking along all three
layers and fracture at top and mid-layers. In Fig.5.13, stress distributions at three layers
are over the necking limit but avoid shear fracture, which is compatible with
experiment.
Figure 5.11: Thickness strain distribution for a pyramid shape simulation with Yld
2000-2d
100
Table 5.5: Properties of FE model for Pyramid shape Number of elements: 6400
Type of elements: THIN SHELL 4 NODE , Formulation : Belytschko, Wong and Chang (BWC)
Contact property model Surface to Surface
Friction formulation Low Static Friction (0.01)
CPU clock speed 2.1 GHtx
Number of cores per CPU 4
Main memory 4 GB
Operating system Windows 7(WINX64)
Total CPU time 35 hours 34 minutes 44 seconds
Figure 5.12: Predicted strains from Yld2000-2d for a pyramid shape in strain-
based forming limit: mid layer(left);Bottom layer (middle) and top layer (right)
101
Figure 5.13: Stress plots from Yld2000-2d for a pyramid shape(top, mid and
bottom layers)
102
5.3.2 Cone shape results from Yld 89 and Hill 48(mid plane only) :
For the comparison purpose, the stress distributions from Yld89 and Hill’s (1948) are
plotted for the mid-layer. Much lower level of stress distributions is observed from
Hill’s 1948, while the result from Yld89 is compatible with the previous prediction
from Yld2000-2d. It means that Yld89 may start the plastic deformation earlier than
Hill’s 1948 as shown in Figs.3.10 and 3.12.
(a)
(b)
Figure 5.14:Stress plots for a cone shape (mid layer): (a) Yld89 (b) Hill48
103
5.4 Non-planer stress analysis based on stress-based FLC:
Figure 5.15:Effect of nominal and transverse shear stresses presented in von Mises
yield surface.
Although incremental sheet forming is mainly plane stress operation, the nominal stress
occurs locally from the tool contact force. This force constrains the propagation of the
plastic flow. The deeper the tool propagates, the contact forces continue to increase. The
phenomenon is more critical if the tool radius is large, which causes higher deformation
force. Fig.5.15 shows the effect of nominal and transverse shear stresses with von Mises
yield surface. The plot shows that nominal stress moves yield surface to compressive
area and transverse shear shrinks yield surface.
Nominal stresses was incorporated in forming limit to remodel the strain-based FLC by
several researchers (Assempour and Nejadkhaki Khakpour 2010; Smith and
Averill,2005). Smith and Averill (2005) took into account the nominal stress by using a
strain-to-stress mapping procedure by introducing a new ratio / . Eyckenset
al.(Eyckens et al.,2009; Eyckens et al.,2011) extended the MK model to consider
localized necking through thickness shear (TTS). Stoughton and Yoon (2011)proposed
to use (sigm_1-sigma_3) and (sigma_2-sigma_3) space to remove the effect of
hydrostatic pressure.
104
5.4.1 :Nominalstress analysis using Hill’s (1948) quadratic function
Hill’s 48 yield model in 3D stress space can be expressed as:
2 2
2 2
(5-1)
Where F,G,H, N are the anisotropic coefficients. .
Hill’s 48 yield criterion is modified by replacing , , with , , and
as follows:
. . 2 . 2 . 2. . . 2. . .
. . . 2 2 2 2
2 0
or
0
If we take the real root of the above equation and ignore transverse shear parts, we get
4. /
2
α
where
. . 2 . 2 .
2. . . 2. . .
. . . 2
And iscalculatedfromSwiftlawandα can be calculated from the principle stress
ratio / .
Simulation is carried out with LS-DYNA element type 25 (shell element with full stress
components). In the element, in order to make the element accommodate nominal stress,
modification was made to decouple the thickness degrees of freedom (Bischoff and
105
Ramm,1997.). Projected FE stresses at the mid layer are presented in Fig.5.16. The
stress space clearly shows the overall reduction of stress distribution i.e., increase of
formability in the plane strain direction.
Figure 5.6(a)Stress-based FLC for plane stress (bottom) Stress-based FLC
considering nominal stress.
106
5.5 Conclusions
This chapter discussed effect of different governing factors to select a reliable FE model
for ISF. FE analysis with Yld 2000-2d with swift hardening found to be most accurate
and accepted for analysis. The results are presented in strain FLD and stress FLD , when
later gives the results within fracture in compliance with experimental observations. The
stress plot at different planes also presents the directional changes of stresses explaining
the deformation mechanism. The effect of through thickness stress is further explained
with stress plot introducing non planer stresses in FE analysis and stress mapping
process .
107
Chapter 6
Mechanism to Suppress Necking in Incremental Sheet Forming
6.1 Introduction
Stretching perpendicular to the tool direction is the major deformation mechanism in
single point incremental sheet forming (Filice,2001; Kim,2000). This deformation
always occurs irrespective of the material, thickness and wall angle. The phenomenon is
narrowly defined by the change of wall thickness for a wall angle and described by the
sine law:
. sin (6-1)
In Eq.(6-1), Initial thickness t0 is deformed to thickness t, when a cone shape is formed
with the slope angle θ from the vertical axis (the tool movement direction).According to
the sine law, the major strain increases with the angle, and at a certain angle, this leads
to failure, since thickness converges to zero when the angle is zero. Another observation
includes in-plane shear parallel to the tool direction (Allwood et al.,2007; Kathryn
Jackson,2009). Allwood(2007), with a specialized experimental test, demonstrated that
incremental sheet forming induces significant transverse shear strains and bending
under tension (BUT) which delays necking behaviour (Emmens,2006). Emmens and
Van den Boogaard(2009.)also discussed the factors including contact stress, shear,
cyclic loading effect, geometrical inability, and hydrostatic pressure which delays the
overall necking by imposing stability in the process.
Figure 6.1: Stress states occurring in incremental sheet forming.
Figure 6.1
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109
In the chapter 2,the overall mechanism of incremental sheet forming is reviewed by
referring both experimental and numerical approaches. Also, in the chapter 4, the stress
gradient through the sheet thickness was investigated through simulation. In this
chapter,the dynamic change of strain and stress states through the thickness is analysed.
6.2.1 Effect of Tool Force on Deformation:
Change of loading conditions during incremental forming can be experimentally
observed through the change of z-force in each cycle. Figure 6.3: Z-Force evolution in
time for a cone shapeFig.6.3 presents the behaviour of z-force over a period of time for
a cone shape forming. In a cone shape, z-force shows a rapid rise while step changes
downwards and it remains fairly constant during in-plane motion although it follows a
ratchet-like path. This ratchet path may be attributed to tool vibration undergoing
frictional resistance during the ball rolling motion (Fig.6.4). A continuous tensile force
acting on the sheet causes BUT (bending under tension) and a stretch force to act under
the tool contact, resulting in a step up of the z-force. During the in-plane motion (RD)
this effect is unchanged but during the tool downward step motion this BUT plus stretch
is significant, resulting in a rapid increase of the z-force .
Figure 6.3: Z-Force evolution in time for a cone shape.
110
Figure 6.4: Working directions in incremental sheet forming
6.3 Mechanics of Necking Suppression
Stress analysis carried over by other authors were mainly based on membrane
analysis(Martins et al.,2008; Silva et al.,2008). Although this type of analysis can relate
stress to deformation mechanisms, it does not provide sufficient information about the
governing factors for necking and failure during the process. A more thorough
assessment can be achieved with a gradient analysis through the sheet thickness, as
discussed in detail in the Chapter 4. For simplicity, only three layers through thickness
were considered for the study.
6.3.1 Strain-Based Analysis:
Plasticity holds volume constancy for plastic strain part as
PPP332211
(6-2)
For a selected element, thickness strain is monitored for the three integration points
through thickness. The lowest thickness strain variationwas observed on upper plane
(tool contact), while the highest thickness strain occurred on the bottom plane(no tool
contact). This indicates that the membrane strain is higher at a non-contacting plane
than that of a contacting plane. Based on the strain gradient theory, this phenomenon
suggests the presence of a simple bending in the element and anticipates early necking
in the non-contacting plane. From Figure 6.3it can be seen that the upper plane
experiences a rapid increase in effective plastic strain with the decrease of thickness
strain (indicated by a red circle).
111
Figure 6.3: Evolution of thickness strain and effective plastic strain for a selected
element.
Incremental sheet forming before and after the tool contact are intensively analysed to
understand the deformation mechanics. For convenience the tool motion is classified
as:
i. In-plane motion along a defined path.
ii. Step down motion with a defined step size.
i. In-Plane Tool Motion:
The change of effective plastic strain is studied at top, bottom and mid layers for a
selected element before and after the tool contact: (ball positions A (before contact), B
(during contact) and C (after contact) in Figure 6.4). Biaxial stretching is dominant
during the tool contact. The highest major strain is observed at the bottom layer (non-
contact surface) and the lowest at the top layer (contact surface), which explains why
the sheet undergoes bending along TD direction(right angle to tool motion). The
increase of the minor strain along the thickness layers explains the presence of
transverse shear strain. Effective plastic strain plot is displayed in Fig.6.7. Effective
plastic strain increases during the tool contact through a sequence of elastic (A: before
contact) elasto-plastic (B: during contact) elastic (C: after contact). The increase is
higher at the top and bottom planes compared to the mid plane.
Figure
Figure 6.
contact fo
6.4 : Strain
.5: Change
or in-plane
11
n state chan
e of effect
tool motio
2
nge of cont
ive plastic
on.
tacting elem
c strain on
ment for too
n an eleme
ol in-plane
ent before
motion.
and after
r
113
Figure 6.6: Change of effective plastic strain for three consecutive downward steps.
ii. Tool Downwards Motion:
At each downward step the effective strain increased during the tool contact through a
sequence of changes in the form of: elastic (before contact) elastic-plastic (during
contact) elastic (after contact), resulting in an overall increase of the effective plastic
strain as shown in Fig.6.5. This rise is sharper on the top and bottom plane than that
observed for the mid plane. The strain path is now extended to a number of consecutive
downwards steps and observed the change. Similar consequence of in-plane and step
down motion is continuously repeated as shown in Figs. 6.6. Increase in the effective
strain level during progressive steps results in the distribution of the overall thinning
uniformly over the forming area, hence ,the transition of material localization(necking)
is delayed. This phenomenon resembles the “Noodle Theory of Fracture” by (Malhotra
et al.,2012). Strain path at the top and bottom layers undergoes the continuous changes
of direction at each step, while the mid layer seems fairly insensitive. Consequently, if
the mid plane strain or average strain is used to predict failure, error is significant as the
strain path change does not taken into account by this layer. At each consecutive step,
the minor strain increases as shown in Fig.6.7 which illustrates transverse shear during
deformation as discussed earlier. This phenomenon may increase formability by
avoiding the short cut to necking toward plane strain tension.
114
Figure 6.7: Strain path change in three consecutive steps.
Figure 6.8: Overall strain path change for selected element.
ii. Overall Strain Path :
The following three distinct behaviours can be observed for a complete strain path
throughout the process:Phase1: elastic loading-unloading; Phase2: plastic loading;
Phasse3: unloading.
The first phase occurs when an element is not in contact with the tool (ball) but its
adjacent elements are undergoing plastic deformation. During this phase, loading occurs
as the ball passes over the adjacent elements and unloading occurs as the ball leaves the
elements. The elastic loading and unloading occurs mainly along RD direction(ball
rolling dir
ball along
phase is t
deformatio
ball. The p
only a par
as the stra
third phas
occurs. Th
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Figure6.9
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he strain p
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he latter can
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mid-plane w
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Deformatio
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116
6.3.2 Stress-Based Analysis:
In this section, the mechanism of incremental sheet forming is analysed by studying the
stress state change at different thickness layers, which provides more realistic prediction
of the actual process.
Step i. In-Plane motion :
Figure6.9 (i) and (ii) describe the local phenomenon occurring during in-plane tool
motion along the defined tool path. At contact, the tool forms a spherical deformation
profile, which turns into a quadro-cylindrical path as the tool moves forward
(Figure6.9(ii)). In the figure element ”b” experiences the biaxial tension followed by the
plane strain tension later at the position of element ‘c’.On the contrary, element
“d”(located at the next to element “b”) experiences the tension from bending-stretching
along the radial direction. Figure 6.10 shows the stress states at the top and bottom
planes of the selected element for the three positions (1: before contact, 2: during
contact, 3: after contact) within one step. The steps indicated in Figure 6.10also
represent three global consecutive downward steps. As can be seen in the figure, the
stresses at the top and bottom surfaces do not exceed the necking limit simultaneously
and these are well balanced. Considering that necking only occurs when the both
surfaces are over the necking limit, the balanced stresses of the two surface stresses
suppresses necking.
Formability can be improved by ensuring a higher stability in the local stresses, which
can be controlled by selecting appropriate tool speeds, friction, material thickness, tool
radius, tool path, among others. For example, smaller tool radii cause an increase in
stress to deform at the tip (due to localized contact surface area for the applied force),
hence, tensile stress rises critically on the bottom layer (non-contact). The phenomenon
also explains the initial observation of Kim and Park (2002), who showed that strains
along the transverse direction are greater when small diameter tools are utilized.
117
Figure 6.10: Stress path change before and after contact (1: before contact , 2:
during contact, 3: after contact) for three consecutive downward steps.
118
Step ii: Tool step down motion:
As shown in Figure6.9, the sheet moves through the locations of “A”, “B”,”C” and ”D”.
When the tool takes the downward steps, element “A” undergoes bending under
tension(BUT) followed by unbending and plain strain tension, element “B” experiences
stretch to bending under tension(BUT), “C” undergoes stretch-bending and “D”
undergoes bending. A schematic diagram for thickness strain change is drawn in
Figure6.9(iii) and (iv) based on punch stretching concept, which depicts the strain
evolution as the element undergoes stretch-bending to bending under tension and finally
unbending and tension.
The process can be explained more clearly with Figure 6.10. The three steps are
explained with the evolution of element B in the sketch in Figure6.9, (iii). As the tool
comes in contact with the element, it undergoes a critical change of stress while shifting
the overall stress state to a higher value for both top and bottom planes. This behaviour
changes as the tool moves downwards, at different steps, as shown in Figure 6.10. At
step 1, minor compression is observed at the bottom layer indicating bending of the
selected element. At step 2, the element is deformed through the contact of the ball,
hence, the material undergoes stretch under bending. At the same step, the bottom plane
is biaxial stretched while the top plane undergoes a major drop of stresses. In step 3, the
stress at top and bottom planes increases towards opposite directions, indicating
unbending under tension. The mechanism predominantly causes increases of the stress
level in the both planes.
iii.Complete Strain Path:
The complete deformation path illustrated Figure 6.11shows the repeated in-plane and
downward tool motions. The stress behaviour is quite different before and after the tool
contact, which starts to deform plastically the element. Therefore, the stress states for
the overall deformation are presented in three consecutive phases using the simulation
process of a cone shape.
During the first phase (before contact), bending and unbending within the elastic limit
occurs. As shown in Figure 6.11, element ‘c’ is in the brown zone where the thickness
change is not initiated. During the second phase (during contact),major plastic
deformation occurs, as discussed in previous sections. During the third phase (after
contact), an interesting phenomenon is observed, in which a dramatic change of the
stress direction takes place. As the tool travels downwards and ceases element contact,
119
plastic deformation from the tool becomes inactive and then, the element undergoes
bending recovery followed by minor compression or spring back. This results in a stress
path change on the top and bottom planes and allows the element to escape a critical
necking value, which could have occurred due to the progressive stretching and tension.
The overall stress path for the selected element for three thickness locations is
represented in Figure 6.12.The progressive evolution of yielding can be visualized. This
explains the suppression of the necking phenomenon at the top and bottom planes with
the change in the deformation path from the critical direction.
Figure 6.11:Stresspath for top (left) and bottom (right) planes for a selected
element C for the overall process of forming to show the complete stress change.
120
Figure 6.12: Step wise yield surface evolution of a selected element C.
6.4 Summary:
This chapter discovered the mechanism of incremental sheet forming from the
viewpoint of strain and stress paths through thickness. For the sake of simplicity only
three plane were analyzed. The strain and stress states change over the selected element
is distinguished for three cases.It has been found that incremental sheet forming has the
mechanism to suppress necking by changing its deformation path to avoid the instability
during the repeated unloading processes.
121
Chapter 7
Optimization of Process Parameters for Incremental Sheet
Forming
7.1 Introduction:
Selection of optimum process parameters determines the accuracy of the designed parts
to be manufactured in incremental sheet forming. Therefore, major process parameters
have been extensively investigated by various researchers; these include: feed rate,
vertical part slope, part geometry, tool radius, tool path, sheet material, sheet thickness,
tool geometry, etc( Hamilton,2010; G. Ambrogio,2008; Elisabetta Ceretti,2004; Kim
and Park,2002; Matthieu Raucha,2009). Among them, part slope has been confirmed to
be directly related to thinning and fracture phenomena (Kawai et al., 2001; Kim and
Yang, 2000; Strano, 2003). Decreasing the feed rate positively contributes to improve
formability (Kim and Park,2002; Strano,2004;Wong,2003),however, it leads to an
economically unviable process for industrial applications. The effect of the part
curvature on accuracy was investigated by Strano (2005).The author finds that the effect
of the part curvature is minor in sharp corners, where the local stretch from the tool
radius is the major deformation following the biaxial path. An empirical formula was
proposed in relation to sine law to show effect of curvature was proposed by Strano
(2005), i.e.,
. 1 (7-1)
Where , is the actual strain and is the nominal strain from sine law(
ln sin , z is the part depth and r is the curvature radius from the symmetry center
point. In Eq. (7.1), the deformed sheet thickness decreases by increasing the curvature.
Various studies on the process parameters have been looking for clear correlations
between the process parameters and the material performance to find out an optimum
manufacturing process (Figure 7.1). Incremental sheet forming appears to be as a
superior option to other forming techniques in terms of flexibility. However, the process
contains a higher number of manufacturing parameters to be optimized. Based on the
observations from this thesis work and the information available in the literatures, the
122
author proposed a process table on the contributions from manufacturing and material
parameters according to the typical deformation modes, as illustrated in Figure 7.2. This
table in Fg.7.2 can be used to control the process parameters and materials elections for
a certain deformation path.
Figure 7.1:Factors that need to be considered for design incremental sheet forming
process.
Figure 7.2: a process table on the contributions from manufacturing and material
parameters according to the typical deformation modes.
123
7.2 Advanced ISF Process Development
7.2.1 Numerical Simulation of a Complex Shape in ISF
Manufacture of complex shapes in incremental forming is feasible if CNC or Robot
technologies are used with the tool path generated from CAM software. However,
numerical simulation of the forming process is of higher difficulty as commercial FE
software require either coordinates input or parametric relations to generate the
appropriate motions. As a result, most researchers choose a cone, pyramid, or other
asymmetric shapes, which can be geometrically represented using analytical formulas.
For the current study, a novel approach is proposed to investigate several patches within
a part to identify the critical forming zones and adjust the process parameters in these
regions. Process optimization can be done by altering the process parameters such as
tool, process step, curvature, inclination angle, and others. Stress-based forming limit is
used a distinguished tool for this analysis as it can reliably predict the necking and
failure as discussed in the previous chapters. The strategy is that the process parameters
can be iteratively modified based on the stress limits toward succeeding the forming of
a given part. For the demonstration purpose ,the tool path of a complex shape for finite
element simulation is developed from experimental data. A complex shape with three
type of patches are modelled as(See Fig.7.3)
A: Funnel shape with increasing slope angle.
B. Plane wall with the fixed slope of 450 .
C. Cone shape with the fixed slope 450.
To build a finite element model of the shape, a three-axis tool path is generated with the
depth of 35 mm. Yld2000-2d (Barlat et al., 2003) is used with the projection to the
proposed stress-based forming and fracture limits. The predicted results in Fig.7.4 are
investigated to identify the most critical zones. As can be seen in the figure, the stress
analysis clearly identifies the patch A as the most critical zone, most likely to fail. The
results also show a high effective plastic strain at this zone in the final step.
Additionally, the change of effective plastic strain along the depth (from top to probe,
marked as piece of three lines) is investigated. The plot also confirmed that patch-A
shows the highest effective plastic strain. In the patch-C (a cone shape wall) the stress
level is the lowest. A constant slope allows the bending with stretch which prevent the
pure stretch and also, the presence of shear deformation delays necking. In the patch-B,
124
critical stress level is observed In the patch, plain strain tension is dominant through the
profile, except at interference with other profile, where biaxial deformation is more
dominant. Hence, a risk of failure remains in the plane wall from plane strain tension.
Figure 7.3: Three-dimensional representation of the different patches analyzed.
Figure 7.4: Strain and stress-based analyses for a complex shape forming
composed of three patches.
125
7.2.2 Multistage Forming
One of the limitations in incremental sheet forming is that material flow occurs locally
without draw-in. For this reason, critical stretch predominantly leads to excessive
thinning at a high slope. Stretching in incremental sheet forming can be suppressed by
the presence of bending. Hence, selection of a moderate slope is recommended.
However, this is a major limitation in part design.
Several process developments to improve the formability have been proposed which
include: tool path development (Matthieu Raucha,2009), heating of the blank (J.R.
Duflou,2007; Hino,2008; Tong,2010),flexible support, multipoint toolpath and
backdrawing (Micari 2007), and multistage process development (Duflou,2008;
Skjoedt,2008). In spite of high formability, heating adversely affects the surface quality.
Tool path improvement by changing the step-down strategy enhances the performance
and accuracy, but does not overcome the limitation of deep angle forming. However, the
tool path improvement incorporating multistage processes was found to be a good
choice to augment formability and deep angle forming. When multi-stage process is
used in incremental sheet forming, it is important to allow sufficient bending to avoid a
local stretching. Additionally, in multi-step process ,the tensile force on the blank can
rise significantly, which can usually be depicted from the rise in the axial tool force,
finally resulting in the tearing of the sheet. Therefore, both step size and step angle need
to be controlled carefully along the tool path.
7.2.3 Developing process plan for a cup forming:
Usually a cup forming with 900angle is not possible if a simple tool path is used in
incremental sheet forming. Although several strategies can be followed to design the
process plan, investigation is restricted to three types of strategies for tool path plan as
shown in Fig.7.5. In all the three strategies, it is important to distribute the downward
steps so that stretch or tension cannot build up in a particular area. Z-displacement is
distributed in such a way that repetition of steps can be avoided in a particular zones.
Strategy one consists of a simple two-step process. Forming of a cone at 450 and then
forming a 900 cup. Strategy two follow a similar strategy but the cone is formed with
incremental process throughout four steps, i.e., 450 is followed by 600 ,750 and finally
900 . In the third strategy, the 450cone and 900 cup are formed alternatively at each
126
downward step. Analytical tool path is developed from an analytical formula
considering depth, height and slope:
/ /180
cos /180 sin /180
Figure 7.5 Different process strategies for cup forming :Strategy1: Two step
incremental forming; Strategy2: Four step incremental forming ; Strategy 3.
Progressive step incremental forming
127
Figure 7.6: (left) Stress-based necking and fracture limits; (right) Thickness strain
distribution
128
Figure 7.7: Z-force for the different strategies.
Finite element simulation is performed following the above mentioned three strategies.
The FE results, although not experimentally verified, give an idea of the process design
with stress-based analysis. In Fig.7.6, strategy-2 (4 Step) exhibits the lowest level of
stress than that of the other strategies although it is predicted to be failed. Strategy-3
incurs the very early failure due to the rapid stress increase in both uniaxial tension and
biaxial stretch directions. The thickness-strain distributions are also presented in
Fig.7.6.It can be seen that the 4-step process has a fairly lower rise in thickness strain
compared to the other processes, which support the stress analysis The Force curve in
Fig.7.7clearly shows the maximum forces predicted from three different processes. In
the cure, strategy-3exhibits of the lowest maximum force. Continuous loading and
unloading does not allow the increase of the maximum force, although local stress level
increases. A proposed method can be possibly adopted to form a part with a deep slope.
7.3 Summary:
This chapter is presented to promote the practical application of stress-based forming
and fracture limits in incremental sheet forming. It has been proposed a process table on
the contributions from manufacturing and material parameters according to the typical
deformation modes. Stress-based analysis for three patch shapes is found that stress
analysis can be successfully used for a complex shape forming. In further investigation,
a process path improvement strategy is investigated with FE analysis, for three distinct
types of tool paths. To achieve the desired slope in forming, multiple step incremental
forming can be a good strategy.
129
Chapter 8
Conclusions and Recommendations
8.1 Overview and Conclusions
In this thesis a new approach to predict necking and failure for incremental sheet
forming was introduced by using stress-based forming and fracture limits. The
limitation of the conventional forming limit diagram to handle non-proportional
loadings occurred in incremental sheet forming motivated this research. The reliability
of this new stress-based approach for necking and failure were successfully verified
through material modeling, experimental testing, and finite element simulation.
For an appropriate modeling of the forming and fracture limits, advanced constitutive
models including Yld2000-2d were introduced in both strain and stress analyses. Strain-
based forming limit curve was predicted from MK model combined with Yld2000-2d
model and then, the curve was mapped to the principal stress space. Fracture limit was
modeled by Maximum Shear Stress (MSS) criterion based on fracture testing. The
asymmetrical parts with pyramid and cone shapes were successfully formed with CNC
and ABB robot. The strains are measured through ASAME system. The experimental
tool path data was imported directly to finite element simulation.
Reliable finite element solution was obtained from Yld2000-2d material model and
BTL element with the mesh size of 2.5mm. Both FE results and experimentally
measured strains predicted failure in the conventional strain space, although it was
successfully formed experimentally. However, it is found that FE results mapped to the
stress-space are successfully located within the fracture limit, which is compatible with
experimental observation.
Furthermore, finite element results are analyzed through the element thickness (top,
middle, and bottom layers) and the stresses in all layers are located below the failure
limit in all cases. In addition, the finite element simulation considering the nominal
stress was conducted. It showed that nominal stress delays necking. The mechanism of
incremental sheet forming has been discovered from the viewpoint of strain and stress
paths. It has been found that incremental sheet forming has the mechanism to suppress
necking by changing its deformation path to avoid the instability during the repeated
unloading processes.
130
To promote the practical application of stress-based forming and fracture limits in
incremental sheet forming, it was proposed a process table on the contributions from
manufacturing and material parameters according to the typical deformation modes.
Stress-based analysis for three patch shapes showed that stress analysis can be
successfully used for a complex shape forming.
8.2 Recommendations for Future Study
The literature referenced in Chapter 2 describes two major challenges in the accurate
construction of the forming and fracture limits for incremental forming: (i) identifying
the major factor to govern necking and failure, (ii) developing experimental tests to
determine the forming limit for incremental sheet forming. This research provides a
small step forward to overcome these challenges.
A conventional necking and fracture limits in the strain space is not appropriate for
incremental sheet forming. A new methodology to handle non-proportional loadings in
the strain space will be an effective tool for FE analysis.
It has been found that nominal stress delays necking from the verification using a
quadratic yield function. Implementation of an advanced non-quadratic anisotropic
model is expected to predict more accurate results. Also, directional hardening is
recommended to be included for highly anisotropic materials.
Modeling of incremental sheet forming for difficult-to-form materials including
titanium is an interesting research topic to be investigated considering that the process is
commercially available.
Successful incremental forming is being achieved by the control of a local deformation
behavior through the adjustments of the process parameters. The proposed stress-based
approach can be incorporated in the design cycle as a feedback control tool.
131
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