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12-2016 Characterization and Modeling of Deformation, Springback, and Failure in Advanced High Strength Steels (AHSSs) Jun Hu Clemson University, [email protected]
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CHARACTERIZATION AND MODELING OF DEFORMATION, SPRINGBACK, AND FAILURE IN ADVANCED HIGH STRENGTH STEELS (AHSSS)
A Dissertation Presented to the Graduate School of Clemson University
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Automotive Engineering
by Jun Hu December 2016
Accepted by: Dr. Fadi Abu-Farha, Committee Chair Dr. Nicole Coutris Dr. Srikanth Pilla Dr. Gang Li
i ABSTRACT
New generations of advanced high strength steels (AHSSs), expected with tension strength exceeding 1000 MPa, sufficient cold-stamping ductility, and low-alloyed microstructure, are being highly sought for automotive lightweighting applications without compromising on neither performance nor cost standards. From the standpoint of metallurgy, the most straightforward solution is to realize complex microstructures in the new AHSSs with significant fractions of the strongest ferrous phase, martensite, in combination with the most ductile phase, austenite. Nevertheless, the preliminary- developed new AHSSs exhibit unique and complex deformation, failure, and springback characteristics compared with the conventional steel grades, which thereby bring significant challenges to the current manufacturing infrastructure in automotive sector.
To thoroughly investigate this uniqueness and complexity, this research work started from a full set of mechanical experimentation on some target AHSSs to comprehensively understand their deformation, springback, and failure. In the selected
AHSSs, in particular, a 980 MPa grade quenched and partitioned (QP980) steel, with tri- phase microstructure of retained austenite, martensite, and ferrite, was primarily investigated, since it would be a baseline material of the developing AHSS grades. Another highlight is that, during the entire experimental work, a stereo digital image correlation
(DIC) system was widely employed to not only in-situ accurately measure the full-field material deformation and displacement, but also control the loading directions when necessary.
i Furthermore, based on the experimental results, a new phenomenological model was proposed to properly characterize not only anisotropy, complex hardening induced by the Bauschinger effect, but also tension-compression asymmetry and the transformation- induced plasticity (TRIP) effect. Then this new model was implemented via user material subroutine in LS-DYNA. Last but not the least, this work eventually wrapped up with a case study of component-level finite element forming and springback predictions compared with the corresponding actual panels, as an implementation and verification of the previous experimentation and modeling.
ii DEDICATION
To my families, especially my parents and passed away grandfather
iii ACKNOWLEDGMENTS
First of all, I would like to express all my gratitude to Dr. Fadi Abu-Farha for his wise guidance and continuous patience during my study and research over the past four years. He was always willing to help when my research encountered bottlenecks. It has been a great pleasure to work with him and I believe that I have learned a lot from him that will affect my future career. I also thank my other committee members, Dr. Nicole Coutris,
Dr. Srikanth Pilla, and Dr. Gang Li, for their patient and inspiring suggestions during my studying experience in their classes and this dissertation writing.
I would particularly take this opportunity to sincerely acknowledge Dr. Lay Knoerr as well as the entire ThyssenKrupp Steel North America group for offering me a great internship opportunity to work in the steel industry in 2015 and 2016. During this internship, I accumulated precious working experience and accepted very professional trainings on many technical software that are extremely helpful to my research work.
In addition, I cherished the collaboration experience with Dr. Youngung Jeong, Dr.
Nan Zhang, Zeren Xu, Rakan Alturk, and all the other group mates. I also appreciate the technical support from the Clemson University library and Palmetto cluster groups.
Furthermore, many thanks to the U.S. Department of Energy for sponsoring my research project, and AK Steel, ThyssenKrupp Steel, General Motors, Fiat-Chrysler Automobiles, and JSOL Corporation for providing the research materials and/or informative support.
Last but not the least, I dedicate this dissertation with the deepest love and gratitude to my families, especially my parents and passed away grandfather, for their unwavering support in the past thirty years.
iv TABLE OF CONTENTS
Chapter Page
ABSTRACT ...... i
DEDICATION ...... iii
ACKNOWLEDGMENTS ...... iv
TABLE OF CONTENTS ...... v
LIST OF TABLES ...... viii
LIST OF FIGURES ...... ix
CHAPTER 1: INTRODUCTION ...... 1
1.1 Overview of AHSSs ...... 2
1.1.1 Advantages of AHSSs ...... 3
1.1.2 Generations of AHSSs ...... 5
1.1.3 Metallurgy of AHSSs ...... 7
1.2 Research Motivations ...... 13
1.2.1 Challenges Brought by Complex Microstructure ...... 14
1.2.2 Research Questions...... 16
1.3 Dissertation Overview ...... 16
CHAPTER 2: MATERIAL CHARACTERIZATION ...... 17
2.1 Tension Properties ...... 18
v 2.1.1 Tension Behavior at Ambient Temperature and Adiabatic Heating ...... 20
2.1.2 Uniaxial Tension Testing between -50 and 300 ºC ...... 32
2.2 Equi-biaxial Loading Testing and Results ...... 50
2.2.1 Experimental Setups ...... 51
2.2.2 Equi-Biaxial Stress-Strain Curve...... 52
2.2.3 Equi-Biaxial Yield Stress and Anisotropy ...... 61
2.3 Failure and Forming Limit ...... 63
2.3.1 Theoretical FLC Models ...... 65
2.3.2 Experimental FLC Determination Techniques ...... 69
2.3.3 Unexpected Failure Due to Microstructure Inhomogeneity ...... 86
2.4 Compression and Cyclic Loading Properties ...... 93
2.4.1 Experimental Setups ...... 97
2.4.2 Compression Properties of QP980 ...... 100
2.4.3 Cyclic Loading Behavior ...... 101
2.4.4 Loading-Unloading Behavior ...... 104
2.4.5 T-C Asymmetry Affected by Multi-Phase Microstructure ...... 107
2.5 Summary ...... 112
CHAPTER 3: CONSTITUTIVE MODEL ...... 114
3.1 Yield Criterion and Function...... 114
vi 3.2 Flow Rule ...... 124
3.3 Hardening Law ...... 128
3.4 Constitutive Model of QP980 ...... 134
3.5 Summary ...... 136
CHAPTER 4: MODEL IMPLEMENTATION AND VERIFICATION ...... 138
4.1 Numerical Implementation ...... 138
4.2 Finite Element Verification ...... 139
CHAPTER 5: CONCLUSIONS ...... 144
5.1 Contributions ...... 144
5.2 Future Plan ...... 145
REFERENCES ...... 146
vii LIST OF TABLES
Table Page
Table 1: Assumed tension properties of the common phases (Matlock and Speer 2006) .. 7
Table 2: Summary of the averaged material properties of the seven orientations ...... 23
Table 3: DIC-based FLC determination time-dependent methods ...... 73
Table 4: Exponential parameters of the T-C asymmetric decaying loading/unloading moduli of QP980 and DP980 ...... 107
Table 5: Calibrated anisotropy coefficients of the Yld2000 yield function (a=8) ...... 120
Table 6: Calibrated coefficients of the CPB2006 yield function (a=2) ...... 120
Table 7: Input parameters that need to be calibrated in the Y-U model ...... 131
Table 8: Flow stress parameters of austenite and martensite (Tsuchida and Tomota 2000)
...... 133
viii LIST OF FIGURES
Figure Page
Figure 1: Diagram of steel grades used in automotive applications, illustrating the elongation and tension strength of conventional steels, HSSs, and AHSSs (Matlock and
Speer 2006, De Moor, Gibbs et al. 2010, Horvath 2010, Thomas, Matlock et al. 2015) ... 6
Figure 2: Microstructure of (a) DP590, (b) DP780, (c) DP980, and (d) DP1180: ferrite in white and martensite in black ...... 8
Figure 3: The Iron-Carbide phase diagram (Callister and Rethwisch 2009) ...... 9
Figure 4: The complete isothermal Fe-C alloy TTT diagram (A: austenite, B: bainite, M: martensite, P: pearlite) (Callister and Rethwisch 2009) ...... 10
Figure 5: Heat treating processes of producing (a) DP, (b) TRIP, and (c) Q&P steels (De
Moor, Matlock et al. 2012) ...... 13
Figure 6: EBSD map of as-received QP980 (red: BCC ferrite; black: BCT martensite; blue:
FCC austenite; green: unindexed phases) ...... 13
Figure 7: Martensitic transformation in carbon steel microstructure, two neighboring FCC austenite (in solid black) lattices to a BCT martensite (in dash red) lattice ...... 15
Figure 8: Experimental setup for uniaxial tension testing and adiabatic heating evaluation
...... 21
Figure 9: Engineering stress-strain curves of QP980 at seven directions ...... 22
Figure 10: n-value relative to true strain curves of QP980 at seven directions ...... 24
Figure 11: r-values of QP980 are the slope of the ɛ2- ɛ3 curves ...... 25
Figure 12: The verification testing of the thermal camera and results ...... 26
ix Figure 13: (A) The evolution trends of full-field true strain (upper) and surface temperature
(lower) of the same specimen during a complete quasi-static uniaxial tensile testing process at room temperature; (B) the average temperature, maximum temperature, and engineering stress curves based on the testing time ...... 28
Figure 14: Mechanical properties of the selected steels: (A) true stress-strain curves
(elongation and strength), (B) toughness ...... 29
Figure 15: Adiabatic heating induced temperature change of the selected steels: (A) maximum temperature (near the cracking position) with engineering strain curves; (B) the overall maximum temperature (near the cracking position) and average temperature (of the entire gauge area) ...... 30
Figure 16: (A) Although the strength and ductility of QP980 are not apparently strain-rate- dependent from 0.001/s to 0.1/s, (B) the highest surface temperature induced by the adiabatic heating is approximately doubled ...... 31
Figure 17: The experimental setup showing a 3D DIC system monitoring the uniaxial tensile deformation of the test specimen through the window of an environmental chamber
...... 33
Figure 18: The stress-strain curves of QP980 at quasi-static strain rate from -50 to 300 ºC
...... 34
Figure 19: The (A) strength and (B) elongation of QP980 vary with the environmental temperature ...... 35
x Figure 20: The temperature variation of retained austenite (RA) measured by EBSD and
XRD techniques and grain average misorientation (GAM-degrees) measured using EBSD
(reproduced with the permission by (Coryell, Savic et al. 2013)) ...... 38
Figure 21: The strength of QP980 varies with the environmental temperature and strain rate...... 41
Figure 22: The (A) TE and (B) UE of QP980 vary with the environmental temperature and strain rate ...... 42
Figure 23: The PLC banding occurrence map based on various strain rate and temperature conditions (red solid dots represent that the PLC bands are observed in the DIC results, while black circles indicate no PLC bands observed) ...... 42
Figure 24: Stress-strain curve of QP980 at 150 ºC, 0.0022/s ...... 45
Figure 25: Evolution of the PLC bands at 150 ºC: (A) serrations on the stress-time curve and five critical moments are selected for analysis; (B) the PLC bands exhibited on the strain and strain-rate propagation maps from 30 to 45 second testing time based on the DIC results; (C) the strain rate with three section lines diagrams from 37.5 to 41.5 second reveal the regeneration and reorientation process of the PLC bands ...... 46
Figure 26: Stress-strain curve of QP980 at 200 ºC, 0.0022/s ...... 49
Figure 27: Evolution of the PLC bands at 200 ºC: (A) serrations on the stress-time curve and four critical moments are selected for analysis; (B) the PLC bands exhibited on the strain and strain-rate propagation maps from 30.5 to 60 second testing time based on the
DIC results; (C) the strain rate with three section lines diagrams from 56.5 to 58 second reveal the regeneration and reorientation process of the PLC bands ...... 50
xi Figure 28: Hydraulic bulge testing setups and equi-biaxial stretching schematic ...... 52
Figure 29: Work principle of the bulge testing ...... 53
Figure 30: Effective stress-strain curves of QP980 based on the bulge testing (BT) and the uniaxial tension testing (UT) ...... 55
Figure 31: Mean radius evolution curves of the three candidate primitives fitted to the actual bulge surface...... 56
Figure 32: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.02 ...... 57
Figure 33: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.04 ...... 58
Figure 34: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.06 ...... 59
Figure 35: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.08 ...... 60
Figure 36: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.1 ...... 61
Figure 37: The stress-strain curves of QP980 at low strain level (up to 0.02) ...... 62
Figure 38: Biaxial anisotropy coefficient derivation via fitting ɛ1-ɛ2 curve of the bulge apex area ...... 63
Figure 39: Schematics of Forming limit diagram, curves, and safety margin ...... 65
Figure 40: Family of theoretical FLC models...... 67
Figure 41: Family of experimental FLC determination techniques ...... 70
xii Figure 42: (a) Formability testing setups, specimen geometry, and (b) schematics of the
Nakajima and Marciniak testing ...... 76
Figure 43: Testing-technique compatibility of the selected FLC determination methods under (a) uniaxial tension strain path, (b) plane strain path, and (c) equi-biaxial tension strain path (each point is a representative of three repeated testing results) ...... 78
Figure 44: Comparison of FLCs of QP980 determined by the selected methods based on
(a) the Marciniak testing and (b) the Nakajima testing...... 80
Figure 45: Full strain field FLDs of QP980 based on the LBF method results ...... 81
Figure 46: Strain-section curves in the last stage before cracking in the DIC measurements on the 40 mm width Marciniak testing specimens of DP980, QP980, and SS201 from left to right ...... 82
Figure 47: In the last stage before cracking, (a) the QP980 specimen behaves multiple localized necking, while (b) the SS201 specimen behaves almost diffuse necking ...... 83
Figure 48: Comparison of FLCs of (a) DP980 and (b) SS201 determined by the selected methods based on the Marciniak testing ...... 84
Figure 49: Schematic of the new time-dependent method to determine safety margin lower boundary in FLD ...... 85
Figure 50: The safety margin determined using the new method in the FLD of QP980 based on (a) the Marciniak and (b) the Nakajima testing results ...... 86
Figure 51: Full strain paths of QP980 based on (a) the Marciniak and (b) the Nakajima testing results ...... 88
xiii Figure 52: Comparison of repetition of the picked strain paths based on (a) the Marciniak and (b) the Nakajima testing results...... 89
Figure 53: The tested four QP980 specimens of 140 mm gauge width exhibited very scattered forming limit points in the FLD ...... 90
Figure 54: Inconsistent failure in the four QP980 specimens of 140 mm gauge width: (a)
No.G0401-140RD-1 specimen and (b) No.G0402-140RD-2 specimen cracked due to localized ɛxx strain bands, while (c) No.G0201-140RD-3 specimen and (d) No.G0403-
140RD-4 specimen failed due to ɛyy strain localization ...... 92
Figure 55: Unexpected band-like strain distribution can also be observed in (a) the uniaxial tension specimens that are not cut along the rolling direction, (b) optical microstructure examination reveals that such a phenomenon is due to microstructure inhomogeneity in multi-phase AHSSs ...... 93
Figure 56: Reversal tension-compression loading (red for the tension stress) during passing over a tool radius (left) or through a draw-bead (right) ...... 96
Figure 57: Customized in-plane cyclic loading experimental setup and specimen geometry.
The setup can realize self-alignment, in-situ strain-controlled testing, and in-situ side-force controlling and recording ...... 98
Figure 58: Uniaxial tension stress-strain curves of DP980 are not significantly affected by the side pressure from the anti-buckling mechanisms ...... 98
Figure 59: Flow stress curve correction via compensating the friction and the biaxial stress induced by the side force ...... 99
xiv Figure 60: Comparison of true stress-strain curves of QP980 in uniaxial tension and compression ...... 101
Figure 61: The Lankford parameters (R-values) under uniaxial compression are the slope values of the best-fit lines of the ɛ2-ɛ3 curves ...... 101
Figure 62: Cyclic loading responses of (a) QP980 and (b) DP980...... 103
Figure 63: T-C asymmetric loading/unloading stress-strain curves of (a) QP980 and (b)
DP980 ...... 105
Figure 64: Linear fitting middle segment of a loading/unloading stress-strain curve at each loading/unloading loop and the slope of the fitting line is the instantaneous loading/unloading modulus at that strain level ...... 106
Figure 65: Exponential fitting the decaying tendency of loading/unloading modulus at all the strain levels ...... 107
Figure 66: The monotonic loading and cyclic loading testing results of (a) DP590, (b)
DP780, (c) DP980, and (d) DP1180 ...... 110
Figure 67: Asymmetric yielding and hardening of the tested steels ...... 111
Figure 68: The cyclic loading testing results of (a) CR4 and (b) PH1500 ...... 112
Figure 69: Family tree of some common yield criteria in sheet metal modeling ...... 116
Figure 70: Summary of experimental results as inputs for yield Functions ...... 121
Figure 71: Comparison of the yield surfaces at ɛ=0.005 (initial yielding) with the experimental results in the normalized stress coordinate system; the selected yield functions including (a) the von Mises, the Hill1948-3r, the Hill1948-3σ, the Hill1948- asymmetric, (b) the Yld1989, the Yld2000, the Yoshida2011, and the CPB2006 ...... 122
xv Figure 72: Comparison of (a) the yield stress and (b) the r-values in different rolling directions prescribed by the selected yield functions with the experimental results in Table
2 (the Yoshida 2011 is excluded due to lack of necessary data in the derivation) ...... 124
Figure 73: Schematics of associated and non-associated flow rules ...... 126
Figure 74: Comparison of tri-component yield surface (blue) and plastic potential surface
(red) of QP980 prescribed by (a) the CPB2006 function, (b) the Hill1948 function, (c) the
Hill1948-asymmetric function with the experimental results ...... 128
Figure 75: Isotropic and kinematic hardening and some representative hardening laws in the literature ...... 129
Figure 76: The Yoshida-Uemori model (Yoshida and Uemori 2002, 2003) and its four parts
...... 131
Figure 77: Retained austenite volume fraction (RAVF) of QP980 decreasing curve during uniaxial tension ...... 133
Figure 78: The multi-surface model for QP980 (or any TRIP-like AHSSs) is similar to the original Y-U model ...... 135
Figure 79: Schematic of numerical implementation procedure based on the semi-implicit algorithm ...... 139
Figure 80: A typical procedure of cold stamping an AHSS panel in FE prediction ...... 140
Figure 81: Four material cards implemented in the FE simulations, with their corresponding required material parameters, represented material characteristics, and needed experiments
...... 141
Figure 82: Compare the FE predicted to the scanned actual panel before trimming ...... 142
xvi Figure 83: Compare the FE predicted to the scanned actual panel after trimming...... 143
xvii CHAPTER 1: INTRODUCTION
Lightweighting is nowadays a major trend in the automotive manufacturing sector.
In general, there are four drivers boosting it: 1) reducing energy consumption, 2) decreasing gas emissions, 3) satisfying upcoming strict government regulations, such as the 2025
Corporate Average Fuel Economy (CAFE) standards in the U.S., and 4) more functional features added in the modern and future vehicles (Rosato 2012). The strategies to realize the lightweighting objectives can be divergent, but the main efforts by the Original
Equipment Manufacturers (OEMs) are towards three primary directions: 1) optimal structure design, 2) efficient manufacturing processes, and 3) lightweight materials (Rosato
2012). From only the standpoint of material, the most straightforward way of lightweighting is to adopt the low-density materials, such as aluminum, magnesium, and composites, rather than the conventional steels, in manufacturing the automobile bodies, and this indeed is a popular strategy considered by the automakers. For example, the Audi
A8 was launched into the market in 1994 as the first mass-production all-aluminum body- in-white (BIW) car (Ulrich 2010), while the BMW i3 in 2013 was the first mass-production model featuring carbon-fiber polymer structures (Hahn 2010, Davies 2012).
However, the low-density materials still do not threat the overall dominant role of steels that has been established for a century in the automotive manufacturing sector.
According to the statistics and predictions in the 2010 U.S. Department of Energy (DOE) material technologies report, steel was, is, and will still be the majority material (at least
40%) in the automotive compositions from the year 1977 until 2035. The main reason of
1 this situation is because of the appearance and development of advanced high-strength steels (AHSSs) (Keeler and Kimchi 2015).
1.1 Overview of AHSSs
Actually back to the 1970s, to meet the automotive lightweighting requirements brought by the first and second world oil crises, steel producers developed multiple grades of high-strength steels (HSSs) for the automotive industry (Horvath 2010). Unfortunately, these HSSs, especially high-strength low alloyed (HSLA) and solid solution strengthened
(SSS) steels, turned out to be largely unsuccessful in the market due to their limited improvement of strength but unacceptable decrease of ductility compared with the conventional low-carbon steels (Horvath 2010). Later on, a real threat from the low-density materials came in 1994, when Audi A8, the first mass-production all-aluminum BIW car, was launched into the market. As a response, in the same year, Ultra-Light Steel Auto Body
(ULSAB) program was commenced by a consortium of 35 steel producers from 18 countries (Fonstein 2015, Keeler and Kimchi 2015). In the next year, 1995, the first step of ULSAB program was commercializing dual-phase steels in Japan, the U.S., and Europe, and in the following decade, ULSAB program achieved the development and applications of more AHSSs, which eventually won considerable attention back in the lightweight materials competition and have become the fastest growing materials for automotive applications (Fonstein 2015, Keeler and Kimchi 2015). In addition, in 2008, World Auto
Steel, a branch of World Steel Association, proposed another program called Future Steel
Vehicle (FSV), aiming at 29% mass reduction, 5-star crash safety rating, and 70% life cycle emission reduction at no cost penalty in the future vehicles via developing and promoting
2 Giga-Pascal-strength AHSSs with the OEMs (WorldAutoSteel 2011). Furthermore, from
2013 until now, U.S. Automotive Material Partnership, collaborating with multiple academia and industry partners, led a DOE project of developing third generation AHSSs
(3G-AHSSs) for future lightweight vehicles (Hector and Krupitzer 2014), and this dissertation work was sponsored by this project. Meanwhile, there are some other 3GAHSS programs going on independently, such as the projects of Nanosteel (Branagan 2014) and
Baosteel (Baosteel 2015a, 2015b).
1.1.1 Advantages of AHSSs
The blooming development of the AHSSs is due to the overwhelming advantages compared with those low-density materials in automotive applications. First of all, the cost of steels is always the most competitive in the market mainly due to the unbeatable gigantic raw steel production worldwide. For example, in 2014, the global overall steel production exceeded 1,670 million metric tonnages, completely suppressing the aluminum production
(49 million metric tonnages), which was the second largest production material, let alone magnesium and carbon-fiber composites (Bray 2015, WorldSteelAssociation 2015).
Therefore, it was estimated that compared with steel, the unit cost of aluminum is about five times higher, magnesium is ten times higher, and carbon-fiber composites is fifteen times higher (Fine and Roth 2010). This is why most of the low-density materials are tend to be applied in supper-mid-class cars. Taking Audi cars for example, with the class level and price from low to high, A4 is made of mostly steel, A6 is steel-aluminum hybrid, A8 is mostly aluminum, and Audi R8 is aluminum-carbon-fiber-composites hybrid (Davies
2012).
3 In addition, unlike the low-density materials, lightweighting of AHSSs components is realized via increasing material strength and decreasing component thickness, without compromising any safety and functional standards. Meanwhile, due to the high strength (>
700 MPa), AHSSs are widely applied to manufacture vehicle strategic structures, especially the safety cage, to protect inside passengers from fatal inquiry during impact.
This is why even Audi A8, begun to adopt AHSS, no longer aluminum, as its B-pillar material in the third generation structure since 2009 (Davies 2012). Furthermore, AHSSs belong to the great steel family, which includes a wide range of grades, from high-strength up to 2GPa (martensitic steels) to high-ductility up to 80% (annealed austenitic stainless steels). This not only provides more options to car body designers, but also resists galvanic corrosion when joining and assembling different steel components (Keeler and Kimchi
2015).
Another remarkable advantage of AHSSs is more environment-friendly than those low-density materials. First of all, the carbon dioxide emission during steel production, no matter AHSS or conventional steels, is about eight times lower than that of producing aluminum, ten times lower than carbon-fiber composites, and over twenty times lower than magnesium (WorldAutoSteel 2011). Furthermore, unlike carbon-fiber composites, over
92% automotive steels can be recycled without compromising any properties
(SteelRecycleInstitute 2013). In addition, in sheet metal forming, in contrast to many aluminum and magnesium grades, AHSS panels are typically cold-stamped to automotive components, which further decreases the overall cost of using AHSSs in automotive applications.
4 1.1.2 Generations of AHSSs
In the past two decades, the initial two generations of AHSSs have been widely applied in car bodies such as pillars, bumper beams, door rims, and floor cross-members
(Keeler and Kimchi 2015). As shown in Figure 1, the first generation AHSSs (1G-AHSSs) include dual-phase (DP), transformation-induced plasticity (TRIP), and complex-phase
(CP) steels, while the second generation grades (2G-AHSSs) are mostly austenitic-based steels, such as twinning-induced plasticity (TWIP) and lightweight steel with induced plasticity (L-IP). The mechanical properties of the 1G-AHSSs still follow the ‘banana’ trend formed by the conventional steels and HSSs, which is, the higher strength, the lower ductility, and vice versa. In contrast, the 2G-AHSSs possess both high ductility and strength, which seem to be attractive in applications. However, these remarkable properties are realized based on retaining enough austenite during metallurgic production via adding a large amount of alloys, such as manganese (Mn), nickel (Ni), chromium (Cr), and/or molybdenum (Mo), which unfortunately causes high cost and poor weldability of the 2G-
AHSSs (De Moor, Gibbs et al. 2010, Keeler and Kimchi 2015).
Therefore, 3G-AHSSs are expected to combine the advantages of both the initial two generations and meanwhile avoid the drawbacks. To achieve this objective, Matlock et al. (Matlock and Speer 2006, 2009) suggested low-alloyed austenite-martensite dominant microstructure in 3G-AHSSs, rather than the conventional ferrite-martensite dominant microstructure in HSSs and 1G-AHSSs, since austenite is the most ductile micro- constituent and martensite the strongest as shown in Table 1. To follow up, De Moor et al.
(2010) summarized five possible strategies to achieve 3G-AHSSs, including 1) enhancing
5 DP or 2) TRIP steels, 3) developing ultrafine bainitic or 4) quenched-and-partitioned
(Q&P) steels, and 5) decreasing Mn content in TWIP or adding Mn into TRIP steels. Based on the latest development, Fonstein (2015) further emphasized three types, including carbide-free bainitic (CFB) (also called TRIP with bainitic ferrite (TBF)) (Sugimoto,
Kanda et al. 2002), medium Mn (Merwin 2007, Aydin, Essadiqi et al. 2013), and Q&P steels (Speer, Matlock et al. 2003), as the prospective baseline materials that are very potential to be improved to 3G-AHSSs in the near future. Some of their preliminary grades have already been commercialized nowadays, such as TBF1180 by Kobe Steel
(KOBELCO 2015), QP980 and 1180 by Baosteel (Baosteel 2015).
Figure 1: Diagram of steel grades used in automotive applications, illustrating the elongation and tension strength of conventional steels, HSSs, and AHSSs (Matlock and Speer 2006, De Moor, Gibbs et al. 2010, Horvath 2010, Thomas, Matlock et al. 2015)
6 The undergoing DOE 3G-AHSSs project targets at achieving a grade of at least
1200 MPa tension strength with 30% total elongation and/or a grade of at least 1500 MPa tension strength with 25% total elongation steels (Thomas, Matlock et al. 2015). The baseline material is QP980, which is also the target material in this dissertation.
Table 1: Assumed tension properties of the common phases (Matlock and Speer 2006)
Constituent Tension Strength (MPa) Uniform True Strain
Ferrite (α-Fe) 300 0.3 Austenite (γ-Fe) 640 0.6 Martensite (α’-Fe) 2000 0.08
1.1.3 Metallurgy of AHSSs
From the standpoint of microstructure, most AHSSs were developed based on adjusting combinations of iron-carbon (Fe-C) alloy phases, different from HSSs based on changing strengthening mechanisms (Horvath 2010). There are four common phases in microstructure of steels, including austenite, martensite, ferrite, and bainite (which is basically dislocation-rich ferrite with cementite precipitation (Bhadeshia 2015)). Most of mild steels and HSSs have ferrite dominant microstructure. For AHSSs, DP steel consists of ferrite and martensite, TRIP steel consists of austenite, bainite, ferrite, and martensite,
TWIP steel is fully austenitic, and future 3G-AHSSs are expected to contain partial austenite and partial martensite (Matlock and Speer 2006). Since these phases are of apparently divergent mechanical properties as shown in Table 1, theoretically by means of combining them with different volume fractions, various steel grades, either ductile or strong or even both (austenite with martensite), can be produced. For example, as shown
7 in Figure 2, the strength grade of the four selected DP steels increases with the increase of martensite volume fraction (illustrated in black) as 19.7% in DP590, 54.5% in DP780,
62.2% in DP980, and 70.9% in DP1180.
(a) (b)
(c) (d) Figure 2: Microstructure of (a) DP590, (b) DP780, (c) DP980, and (d) DP1180: ferrite in white and martensite in black
1.1.3.1 Retaining Austenite
In actual steel production, however, the ideal combination of austenite and martensite is very difficult. One of the biggest challenges is to retain enough austenite in microstructure after production processes. This is because according to the iron-carbide phase diagram as shown in Figure 3, austenite in steel of C content between 0.02% and
0.77% (the eutectoid point) is stable only when temperature is above 727°C (the eutectoid
8 temperature); when below this temperature, to reach an energy equilibrium state, austenite tends to release extra free energy and transforms to comparatively more stable phases
(Verhoeven 2007). As for which exact phases, it highly depends on the cooling path in the temperature-time-transformation (TTT) diagram as shown in Figure 4: generally speaking, slow cooling results in pearlite (mixture of ferrite and cementite), moderate cooling mainly produces bainite (due to cementite precipitation in ferrite in a proper temperature range), and rapid quenching results in martensite (due to less time for C atoms to diffuse out of ferritic matrices and generate cementite) (Callister and Rethwisch 2009).
Figure 3: The Iron-Carbide phase diagram (Callister and Rethwisch 2009)
Metallurgists have developed two ways to retain austenite in steel microstructure.
The first one is alloying third-party elements, such as Mn (Hong and Lee 2004), Ni, Al
(Shun, Wan et al. 1991, Shun, Wan et al. 1992, Jin and Lee 2012), or Cu (Lee, Kim et al.
9 2011) in steels to increase the stacking fault energy (SFE) out of the favored range of martensitic transformation (<16 mJ/m²) and thereby stabilize the austenite (Frommeyer,
Brux et al. 2003, Allain, Chateau et al. 2004, Curtze and Kuokkala 2010, Peng, Zhu et al.
2013). Both austenitic stainless steels and TWIP steels are developed in this way, but the difference between the two is that the SFE in TWIP steels is much higher (typically between 18 and 45 mJ/m²) and consequently even under plastic deformation, the austenite in TWIP steels behaves only twinning (Dastur and Leslie 1981, Scavino, D'Aiuto et al.
2010, Peng, Zhu et al. 2013), rather than transforming to mixed two types of martensite
(hexagonal close-packed (HCP) (ε-Fe) and body-centered tetragonal (BCT) (α’-Fe)) in austenitic steels (Hsu 1999, Xu and Zhao 2004, Nishiyama 2012, Chen, Rana et al. 2014).
Figure 4: The complete isothermal Fe-C alloy TTT diagram (A: austenite, B: bainite, M: martensite, P: pearlite) (Callister and Rethwisch 2009)
10 The second way of retaining austenite is to utilize enriched C atoms to stabilize austenite via adjusting heat treatment processes as well as adding Al, Si, P, and/or Mo to suppress the generation and growth of cementite (De Moor, Speer et al. 2009, Chen, Rana et al. 2014). Most TRIP-like steels, including Q&P (De Moor, Lacroix et al. 2008) and
TRIP-aided-DP, -ferritic, -bainitic, -annealed martensitic steels (Sugimoto, Kanda et al.
2002, Caballero, Garcia-Mateo et al. 2008) were developed based on this technique. In addition, this way is considered more promising than the first way due to the cost and welding issues introduced above.
1.1.3.2 Advantages of Q&P
In multi-phase TRIP-aided steels, retained austenite can improve the ductility, while other phase(s), especially martensite determines the overall strength grade.
Therefore, how to optimize the heat treatment processes in steel production to guarantee as much as both retained austenite and martensite is another major challenge. As a very prospective solution, Q&P processes were developed at Colorado School of Mines (Speer,
Matlock et al. 2003, Speer, Rizzo Assuncao et al. 2005) and continuously discussed and improved in the following decade in metallurgy (De Moor, Lacroix et al. 2008, De Moor,
Penning et al. 2008, Jin, Park et al. 2009, Li, Lu et al. 2010, Pastore, De Negri et al. 2011,
Speer, De Moor et al. 2011, Zhuang, Jiang et al. 2012, Choi, Hu et al. 2014, Speer, De
Moor et al. 2015). Figure 5 shows the comparison of heat treating processes to produce (a)
DP, (b) TRIP, and (c) Q&P steels. (a) As a reference, for the DP processing, all the austenite transforms to martensite during the first quenching (illustrated in red). (b) For the TRIP processing, the first quenching (in red) stops above the martensitic transformation starting
11 temperature (Ms ~357 °C) but in the bainitic transformation temperature range. During the following tempering step (in green), some austenite may transform to bainite/ferrite and some can be stabilized by the surrounding C atoms when Si, Al, and/or P suppress cementite. Then in the second quenching to room temperature (in blue), some unstable austenite transforms to martensite as the desired strong phase. (c) As for the Q&P processing, the first quenching step (in red) drops temperature between the Ms and Mf
(martensitic transformation finish temperature) and thereby transforms some austenite to martensite. Then, the second step (in green) is to isothermally reheat to a higher temperature in order to transport C atoms from martensite to partition and stabilize the remaining austenite islands to prevent secondary martensitic transformation during the final quenching step to room temperature (in blue) (Bigg, Edmonds et al. 2013).
Apparently, the advantage of the Q&P processing compared with the TRIP processing is that more martensite, rather than bainite/ferrite, in the microstructure can be guaranteed when the retain austenite amount of the two is equivalent (De Cooman and Speer 2006).
Furthermore, the optimal Q&P processing in the near future is very likely to even achieve the ideal austenite plus martensite dual-phase microstructure.
12 Figure 5: Heat treating processes of producing (a) DP, (b) TRIP, and (c) Q&P steels (De Moor, Matlock et al. 2012)
So far, as the first commercial grade produced based on the Q&P processes, QP980 contains tri-phase microstructure of ~10% retained austenite, ~50% martensite, and ~40% ferrite as shown in the electron-backscattered-diffraction (EBSD) map in Figure 6 and processes good combination of strength and ductility. Therefore, QP980 was even categorized into the 3G-AHSSs category by Coryell, Campbell et al. (2012), Coryell, Savic et al. (2012), (2013) and Sun and Wagoner (2013), or at least archived by De Moor, Gibbs et al. (2010) as a candidate grade for developing qualified 3G-AHSSs in the near future.
Figure 6: EBSD map of as-received QP980 (red: BCC ferrite; black: BCT martensite; blue: FCC austenite; green: unindexed phases)
1.2 Research Motivations
Despite the improved mechanical properties, the complex microstructure of the new developed AHSSs also bring unexpected macroscopic deformation characteristics,
13 which may greatly challenge the current material categorization and modeling infrastructure.
1.2.1 Challenges Brought by Complex Microstructure
First of all, the retained austenite in TRIP-like AHSSs is still metastable. At a proper temperature and/or strain rate, either stress or strain can induce martensitic transformation. Austenite has face-centered cubic (FCC) crystal structure, which has close- packed {1 1 1} slip planes as well as twelve slip systems and thereby is more ductile than either body-centered (BCC) or hexagonal close-packed (HCP) structures (Askeland, Fulay et al. 2010). In low-alloyed carbon steels, under external shear stress, the Fe atoms from two neighboring lattices tend to generate a BCT structure (α’-Fe) as shown in Figure 7.
Due to some C atoms occupying the octahedral interstitial sites in the FCC lattice, the BCT structure cannot further transform to a fully symmetric BCC structure (unless the martensite is tempered so that extra external energy can boost the interstitial C atoms diffusing out of the octahedral sites) and thereby has less slip systems than BCC structure, this is why martensite has very limited ductility; meanwhile, the interstitial C atoms lead to microscopic strain hardening and high-dislocation density inside the BCT lattices, which explains the high strength of BCT martensite (Callister and Rethwisch 2009).
From a macroscopic point of view, this martensitic transformation, on the one hand, can effectively prevent early strain localization in sheet metals during plastic deformation.
The austenite continuously transforms to martensite at different locations of a sample, actually transferring local deformation areas dynamically (Hsu 1999), and thereby extend uniform deformation and postpone necking onset (Yu 2008). This phenomenon is called
14 the TRIP effect. On the other hand, however, this TRIP effect is very sensitive to either temperature (Tamura 1982, Berrahmoune, Berveiller et al. 2004) or strain rates (Choi, Son et al. 2006, Oliver, Jones et al. 2007, He, He et al. 2012), and hence greatly affects the deformation behavior. The mechanism behind this effect can be explained by either the
‘accommodation process’ proposed by Greenwood and Johnson (1965) or the ‘orientation process’ proposed by Magee (1966) or even both (Cherkaoui 2002). Furthermore, under different strain paths, the martensitic transformation rate is different (Radu, Valy et al.
2005). Therefore, the TRIP effect on deformation under complex loading conditions cannot be simply extrapolated from the basic uniaxial tension testing; instead, more advanced experimental work needs to conduct.
Figure 7: Martensitic transformation in carbon steel microstructure, two neighboring FCC austenite (in solid black) lattices to a BCT martensite (in dash red) lattice
The TRIP effect is a primary, but not the only factor manifested by new AHSSs that increases the research complexity. Other factors, including anisotropy, tension-
15 compression asymmetry, static and dynamic strain aging, and the Bauschinger effect, all bring challenges but on the other hand motivations, to improve the research infrastructure.
1.2.2 Research Questions
In this dissertation, the target material is mainly on QP980, but also including various grades of DP, TRIP, austenitic stainless, medium Mn, and mild steels for comparison when necessary. Two questions need to be answered:
How does the complex microstructure of QP980 affect the material Question 1 mechanical properties?
How to adjust phenomenological modeling for QP980 to reflect the Question 2 unique behavior of deformation, springback, and failure?
1.3 Dissertation Overview
Therefore, to explore the answer of the first question, a series of experimental material characterization work on QP980 will be presented in Chapter 2, with various highlighted core experimental setup, results, and unique phenomena. Then, in Chapter 3, based on the experimental results in Chapter 2, a dedicated continuum model of QP980 will be proposed and calibrated. This new model can reflect the unique manifestations observed in the experiments. Furthermore, in Chapter 4, the new model will be implemented via user material subroutine in finite element (FE) simulation to predict the deformation and springback results of forming QP980 sheets. The prediction results compared with the actual experimental results can verify the proposed model and indicate the importance of different parameters in the model. To wrap up, the main contributions of this dissertation work is summarized as well as a brief future plan in Chapter 5.
16 CHAPTER 2: MATERIAL CHARACTERIZATION
In this Chapter, the target material QP980, as well as other reference steels, will be thoroughly investigated experimentally on its mechanical properties based on different aspects, including temperature, strain rate, loading direction, and microstructure.
Before proceeding to the details of the experiments, it is necessary to highlight the use of a stereo digital image correlation (DIC) system, which was the most important experimental device during the entire experimentation in this research. The DIC system can not only in-situ measure the full-field material deformation and displacement, but also feedback signals of instantaneous strain measurement results to the load frame controller to control loading directions when necessary. Developed in the early 1980s (Peters and
Ranson 1982, Sutton, Wolters et al. 1983), DIC progressed rapidly and has become a widely-used technique in support of all types of mechanical material testing due to its unique capabilities of measuring surface strains in-situ of material deformation (Tong
1998, Tong and Zhang 2001). In traditional uniaxial tensile testing, extensometers are clipped on certain sections of a specimen’s gauge length (and/or width) to measure the elongation or contraction within the sections in order to compute the strains. However, this application has many limitations. The strains measured by extensometers are “nominal”, since they simply represent the global deformation captured between two points. DIC on the other hand provides the data for closer examination at different areas of interest as deemed needed for the tested material (including areas, lines and points). Tong, Tao et al.
(2005) and Coryell, Campbell et al. (2012) demonstrated this advantage of DIC and showed examples of analyzing DIC results in various ways and the corresponding impact on the
17 extracted material deformation behavior (such as stress-strain curves) from global to local deformation. Beyond that, the true value of DIC can be exploited in special cases of capturing the localization of material deformation, such as the Portevin Le-Châtelier (PLC) bands, Lüders bands, and static bands due to microstructure inhomogeneity, where no clip- on extensometer can offer an alternative (Hu, Zhang et al. 2016). More details of the advantages of using DIC will be presented in this chapter.
2.1 Tension Properties
In the literature, Coryell, Campbell et al. (2012), Coryell, Savic et al. (2012) investigated the basic tension properties of QP980, including the strength, ductility, anisotropy, martensitic transformation, and facture. De Moor, Matlock et al. (2012) compared the strain hardening and stretch-flangeability of QP980 with DP and TRIP steels.
Ding, Lin et al. (2012) tested QP980 via uniaxial tension testing to indicate the acceleration from diffuse necking to localized necking with the increase of strain rates (from 1.3E-4/s to 8.3E-2/s). Focusing on a wider range of testing velocity, Liu, Wang et al. (2013) investigated the rate effect of testing QP980 in uniaxial tension deformation, and found that this type of steel actually exhibited heterogeneous rate dependency: its strength increased with strain rate, while its elongation decreased from 10E-4/s to 10/s, then suddenly rose to the peak at 80/s, and fell again from 102/s to 103/s. Such behavior was also observed by He, He et al. (2012) on two grades of TRIP steels and they explained that the ductility initially decreasing with strain rate increasing was due to not enough time for the martensitic transformation, while then the following enhancement was because of the large amount of adiabatic heat emitted during the high-speed testing. Coryell, Savic et al.
18 (2013) published their tension testing results on QP980 from -100 to 200 ºC, showing the significantly unique ductility- and strength-temperature relationships.
Such a popular research topic on the unique sensitive temperature and strain rate dependency of the martensitic transformation in QP980 (and other TRIP-like) steel sheets is reasonable and critical for further investigation on either manufacturing or crashworthiness of the similar AHSSs. From the manufacturing perspective, like the conventional automotive steels, AHSS panels are stamped at ambient temperature, i.e.
‘cold’ stamped, to automotive components. However, since high material toughness can release significant high adiabatic heat during deformation (Gao and Wagoner 1987, Kim and Wagoner 1987, Mason, Rosakis et al. 1994, Kapoor and Nemat-Nasser 1998,
Macdougall 2000), AHSSs can release a large amount of heat during plastic deformation.
Without effective thermal dissipation in a typical cold stamping setup, such heat energy, as well as friction-induced heating, can gradually propagate in a large-lot rapid production, and thereby significantly increase the actual stamping temperature between stamping tools.
Such an implicit phenomenon has been observed in the actual field. For example, Pereira and Rolfe (2014) evaluated the actual stamping temperature of DP780 to be ~181 ºC. We also measured ~100 ºC die temperature in stamping mild steels at a local plant. QP980 is expected to emit more heat than the same strength-level ferrite-martensite based steels at the same strain rate due to the TRIP effect induced by plastic strain. It is of great importance to quantify this amount of energy for two reasons. First, the temperature change on the tooling dimensions and lubrication may affect the forming accuracy of QP980 components.
19 Second, the deformation characteristics of QP980 is also sensitive to the surrounding temperature.
Therefore, in Section 2.1, this research work sheds some light on evaluation of the fundamental tension properties and the adiabatic heating of QP980 as well as other steel sheets for comparison. In the first part (Section 2.1.1), a DIC system and an infrared thermography system (thermal camera) were applied to measure respectively the temperature and strain evolution on uniaxial tension specimens at ambient temperature.
Based on the DIC measurement results only, the tensile deformation properties of the target materials can be categorized. These results will be further implemented for model calibration in Chapter 3. As for the DIC and the thermal camera together, since the two measurements proceeded synchronically, the relationship between the strain and temperature evolutions can also be revealed. In the second part (Section 2.1.2), the temperature effects on the deformation characteristics of Q&P steel were evaluated based on a series of isothermal uniaxial tension tests at various temperature and strain rate conditions.
2.1.1 Tension Behavior at Ambient Temperature and Adiabatic Heating
In this section, the objective is to evaluate the mechanical properties and the adiabatic heating of QP980 during uniaxial tension deformation.
2.1.1.1 Experimental Setups
As seen in Figure 8, the testing setup was centered around an electro-mechanical
INSTRON 5985 universal load frame, fitted with a 250 kN load cell. A stereo DIC system was used to monitor the deformation of the test specimens. Its two five-megapixel CCD
20 cameras were set at ~24º viewing angle, and at ~700 mm away from the target (test specimen). The field of view was approximately 100 × 125 mm², or about 0.05 mm per pixel. QP980 sheets (~1.4 mm thickness) were mainly tested in this work and tension specimens were prepared by water-jet cutting according to standard ASTM E8 (gauge area
= 50 × 12.7 mm²). The specimens were cut towards seven orientations relative to the sheet metal rolling direction, including 0º, 15º, 30º, 45º, 60º, 75º, and 90º, and were tested until fracture; to guarantee repeatability, of each orientation, five specimens were tested in the same conditions. In addition, to meet the requirements of DIC measurements, after cleaning the specimens, a random black-white speckle pattern was sprayed onto the surface of each specimen using flat high-heat paints rated to ~650 ºC, for either the ambient or temperature- varying testing. The universal load frame and the DIC system were triggered simultaneously at the beginning of each test, thus all images were tagged by the corresponding force during testing.
Figure 8: Experimental setup for uniaxial tension testing and adiabatic heating evaluation
21
In addition, to monitor the temperature change induced by the adiabatic heating, a
FLIR A40 thermal camera was deployed between the DIC and the specimens as seen in
Figure 8. Considering the distance and angle between the two DIC cameras, the thermal camera was placed closer to the clamped specimens without blocking the DIC measurements. Last but not the least, the initial strain rate was set to be 0.001/s, 0.01/s and
0.1/s, and other tested steels for comparison included DP-T980, DF-140T, SS201,
TRIP780, and CR4 mild steel.
2.1.1.2 Mechanical Properties of QP980 under Uniaxial Tension
The stress-strain curves of QP980 at the seven orientations are shown in Figure 9.
Based on these curves, the fundamental tensile properties of QP980 can be derived in Table
2, including Young’s modulus (E), 0.2% yield strength (YS), ultimate tensile strength
(UTS), uniform elongation (UE), total elongation (TE), strain hardening exponents (n- values), and Lankford coefficients (r-values, a measurement of plastic anisotropy).
Figure 9: Engineering stress-strain curves of QP980 at seven directions
22
Table 2: Summary of the averaged material properties of the seven orientations E 0.2%YS UTS UE TE n- r-
(GPa) (MPa) (MPa) (%) (%) value* value** Avg. 199.24 813.35 1022.50 9.84 14.19 0.112 0.826 0º Std. Dev. 4.00 12.09 4.76 0.38 0.34 0.004 0.007 Avg. 192.07 812.60 1015.77 9.25 13.14 0.109 0.842 15º Std. Dev. 6.49 11.59 3.30 0.22 0.52 0.001 0.006 Avg. 193.97 826.51 1012.91 8.83 12.98 0.103 0.877 30º Std. Dev. 8.68 2.93 6.35 0.28 0.77 0.001 0.006 Avg. 202.83 799.22 1010.00 9.31 12.79 0.110 0.912 45º Std. Dev. 1.48 10.16 5.50 0.21 0.29 0.002 0.004 Avg. 200.38 774.38 990.35 8.72 12.61 0.112 0.912 60º Std. Dev. 2.83 10.45 5.18 0.20 0.51 0.003 0.005 Avg. 201.67 763.27 985.45 8.90 12.67 0.114 0.902 75º Std. Dev. 2.84 8.16 2.24 0.16 0.61 0.001 0.013 Avg. 205.72 783.58 1003.05 8.50 12.24 0.111 0.899 90º Std. Dev. 1.42 14.21 8.17 0.26 0.34 0.003 0.007 * The n-values are calculated based on 0.04 to 0.08 true strain region. ** The R-values are fitted based on the uniform plastic deformation.
Among these parameters, the E, YS, UTS, UE, and TE are very straightforward to represent the material ductility and strength. As for the n-value, it is the measure of the maximum allowable stretch-ability of metals (Keeler and Kimchi 2015). Generally speaking, the higher n-value, the higher strain hardening rate and the better formability of the material, because the higher deformed area becomes stronger and thereby has less tendency of further localized thinning. Therefore, the n-value was used to predict the
23 forming limits in some preliminary criteria (Considère 1885, Keeler and Brazier 1977). It can be defined by the power law Equation (1) (ISO 10275 and ASTM E646).
𝜎 = 퐾휀푛 (1)
Other than the calculation as a constant value based on a certain strain region in
Table 2, n-value can also be calculated instantaneously with major strain as shown in
Figure 10. Note that the constant n-values do not show obvious dependency with the anisotropy, in contrast to the instantaneous n-value curves.
Figure 10: n-value relative to true strain curves of QP980 at seven directions
Regarding the r-value, or the Lankford coefficient, it reveals the directionality of plastic deformation of sheet metals (Lankford, Synder et al. 1950), which is induced by statistical orientation tendency of grains produced in rolling (Hosford and Caddell). It is expressed as the ratio of the true plastic width strain to the true plastic thickness strain (2)
(ISO 10113 and ASTM E517).
휀푤 푟 = (2) 휀푡
24
From the expression of the r-value, it can be deduced that the greater r-value, the better thinning resistance of the sheet metal, and thereby the better deep draw-ability and stretch-flangeability (Keeler and Kimchi 2015). The r-values can also be calculated instantaneously based on the evolving plastic strain, but not really necessary for QP980, since as shown in Figure 11, all the ɛ2- ɛ3 curves are very close to linear. Also, according to ISO 10113, for QP980, the weighted average r-value that represents the normal anisotropy can be calculated using Equation (3) and the degree of planar anisotropy using
Equation (4).
푟 + 2푟 + 푟 푟̅ = 0 45 90 = 0.888 ± 0.006 (3) 4 푟 + 푟 − 2푟 ∆푟 = 0 90 45 = −0.049 ± 0.005 (4) 2
Figure 11: r-values of QP980 are the slope of the ɛ2- ɛ3 curves
2.1.1.3 Adiabatic Heating in Uniaxial Tension
25 The tensile properties based on the DIC measurement have been presented; in this section, they will be coupled with the adiabatic heating evaluation based on the thermal camera measurement to reveal more information of the target material.
Before the tests, the start-up was to verify the thermal camera measurement. As seen in Figure 12, five thermal couples were soldered in the gauge area on the specimen in a column with 10 mm spacing between each two neighboring soldering points. The thermal couples were connected to five testing channels of a signal reader. In the verification test, the signal reader was set to synchronize with the thermal camera, the load frame, and the
DIC, so that in the post-processing, signals from different devices can be coupled together according to the same on/off testing time. In Figure 12, the average temperature-time locus
(curve in red) of the thermal camera measurement is close to the trend of other curves that represent the measurements of the thermal couples. This indicates the positive verification of the thermal camera measurement.
Figure 12: The verification testing of the thermal camera and results
26 Both the DIC and thermal camera can monitor the evolving history as well as the full field mapping of their respective measurements. As shown in Figure 13(A), the adiabatic heating induced temperature change on a specimen surface during a process of uniaxial tension test was captured stage by stage by the thermal camera, and so was the strain evolution by the DIC. With the dynamic legend setting, the localization of temperature and strain can be shown and the two trends are very close. Also as shown in
Figure 13(B), the adiabatic heating induced temperature change (average and maximum) and engineering stress are plotted together based on the synchronized timeline. It reveals that at quasi-static strain rate the average temperature on the entire gauge area can rise to around 40 ºC, while the maximum temperature near the final cracking position can rise up to about 85 ºC. In addition, the diffuse necking onset is approximately the moment of the maximum temperature starts to increase at a more rapid rate, which reveals the relationship between the adiabatic heating and the deformation instability. This is similar to the observation in TRIP800 (Rusinek, Klepaczko et al. 2005) and SS304 (Rodriguez-Martinez,
Pesci et al. 2011).
27
Figure 13: (A) The evolution trends of full-field true strain (upper) and surface temperature (lower) of the same specimen during a complete quasi-static uniaxial tensile testing process at room temperature; (B) the average temperature, maximum temperature, and engineering stress curves based on the testing time
Adiabatic heating is determined primarily by material toughness, which was defined as the ability of a material to absorb energy and plastically deform resisting to fracture (Hertzberg, Vinci et al. 2012). Toughness can be calculated by integrating the stress-strain curve using Equation (5), and its unit is thereby MPa × mm/mm = 10² × J/m³.
휀푓 𝜎푖 + 𝜎푗 푈 = ∫ 𝜎 푑휀 ≈ ∑ (휀 − 휀 ) (5) 푇 2 푖 푗 휀푒
Therefore, the target steel, QP980 is calculated to possess about 99.96E2 J/m³ toughness, comparatively, higher than 68.74E8 J/m³ of the mild steel due to the higher strength, higher than 74.46E2 J/m³ of DP-T980 and 88.22E2 J/m³ of DF-140T due to the higher ductility, but lower than 116.10E2 J/m³ of TRIP780 and 342.26E8 J/m³ of SS201 due to the less ductility. Figure 14 shows the comparison of the ultimate tensile strength, uniform elongation, and toughness of QP980 with the other selected steels. Generally, the
28 higher toughness, the more adiabatic heating can be emitted. Figure 15 shows such a tendency.
(A)
(B) Figure 14: Mechanical properties of the selected steels: (A) true stress-strain curves (elongation and strength), (B) toughness
29 (A)
(B) Figure 15: Adiabatic heating induced temperature change of the selected steels: (A) maximum temperature (near the cracking position) with engineering strain curves; (B) the overall maximum temperature (near the cracking position) and average temperature (of the entire gauge area)
Strain rate effects on adiabatic heating was also investigated. The strain rate of the sheet metal in the actual stamping circumstance is about 0.1/s to 1/s, much higher than the quasi-static rates. In this work, due to the limitation of the load frame, the highest strain rate is set to be 0.1/s. Three repeated tensile tests were conducted on QP980 respectively
30 at 0.001/s, 0.01/s, and 0.1/s. It is shown in Figure 16(A) that the strength and ductility of
QP980 is not apparently strain-rate-dependent. However, Figure 16(B) shows that from
0.001/s to 0.1/s, the highest temperature induced by the adiabatic heating increases from
~80 ºC to ~150 ºC. This is due to that at higher strain rate, the adiabatic heat has less time to dissipate, and thus accumulates more on the specimen and induces higher temperature.
(A)
(B) Figure 16: (A) Although the strength and ductility of QP980 are not apparently strain-rate-dependent from 0.001/s to 0.1/s, (B) the highest surface temperature induced by the adiabatic heating is approximately doubled
31
2.1.2 Uniaxial Tension Testing between -50 and 300 ºC
In this section, the objective is to specify the temperature effects on the deformation characteristics of QP980, in order to emphasize the importance of understanding the adiabatic heating in the practical manufacturing. The target temperature range is expanded to -50 to 300 ºC in this research so that the particular temperature sensitivity of QP980 can be thoroughly represented.
2.1.2.1 Experimental Setups
Testing procedure starts with setting an environmental chamber fitted with the load frame to the desired test temperature, and allowing the chamber to reach and equilibrate at that temperature. The chamber can be either heated up as high as 500 ºC via convection heating or cooled down by connecting to a supply of liquid nitrogen for testing at sub- ambient temperatures. The equilibrium state was not based on the chamber’s internal thermocouple measurement, rather, by monitoring the temperature of additional thermocouples embedded into the grips. On average, heating time before testing varied between 2-4 hours. Once ready, the furnace door is open and a specimen is inserted into the grips within 5-10 seconds, and the furnace is allowed 5 minutes to re-equilibrate before testing (initiating material deformation). The grips are designed for fast loading in extreme temperature testing, and exert the tension load on the specimens through shoulders; more information about the design of these grips and the compatible specimen geometries can be found in Abu-Farha and Curtis (2009). The setup overview is seen in Figure 17.
32
Figure 17: The experimental setup showing a 3D DIC system monitoring the uniaxial tensile deformation of the test specimen through the window of an environmental chamber
Last but not the least, the DIC system monitored the uniaxial tension deformation of the test specimen through a ~100 × 150 mm² window on the furnace door. Furnace window (glass) and black-body radiation emitted at elevated temperatures are expected to distort the captured images in the DIC measurements, and thus a detailed research on analyzing and offsetting the corresponding errors has been conducted by Hu, Zhang et al.
(2016).
2.1.2.2 Temperature Effects on Tensile Properties
In this work, the quasi-static tension tests were all conducted in the immersive isothermal condition. Figure 18 shows the stress-strain curves of QP980 between -50 and
300 ºC, and Figure 19 highlights the strength and elongation of QP980 varying with the environmental temperature. Different deformation mechanisms lead to the divergent deformation responses at different temperature ranges. Below are the detailed analyses based on the strength and elongation in each temperature range.
33
Figure 18: The stress-strain curves of QP980 at quasi-static strain rate from -50 to 300 ºC
(A)
34 (B) Figure 19: The (A) strength and (B) elongation of QP980 vary with the environmental temperature
From -50 to 0 ºC, the 0.2%YS does not show a consistent trend, the UTS decreases, while both the UE and TE slowly increase to peaks. This is due to the transition from the stress-assisted martensitic transformation to the strain-induced martensitic transformation around Ms(σ) temperature (~0 ºC in this case). Below the Ms(σ) temperature, the retained austenite in QP980 starts to transform to martensite with the aid of the applied external stress that is lower than the regular austenite yield strength. In other words, the yielding of
QP980 is determined by the initiation of the stress-assisted martensitic transformation rather than the regular slip mechanism. The early appearance of martensite can effectively improve both the 0.2%YS and the UTS. However, with the temperature closer to Ms(σ), such a mechanism becomes weaker and the 0.2%YS and the UTS decrease gradually.
Above the Ms(σ) temperature, the yielding is initiated by regular slip processes in the parent phases, while the applied external stress must exceed the yield strength, i.e. in the plastic deformation regime, to trigger the martensitic transformation. The relative theory
35 of distinguishing two types of martensitic transformation and their impacts on material behavior were first proposed by Olson and Cohen (1972), (1976), and has been supported by later on experimental observations, such as Tamura (1982) on Fe-Ni-C, Fe-Cr-Ni, and
Fe-Mn-C austenitic alloys, Berrahmoune, Berveiller et al. (2004) on TRIP600, Wang,
Huang et al. (2006) on TRIP700, and Coryell, Savic et al. (2013) on QP980. Nevertheless, it shall also be pointed out that from Ms(σ) temperature on, the 0.2%YS keeps a general decreasing trend as shown in Figure 19. This is because austenite yields first in the three phases in QP980 and its yield strength decreases with temperature Olson and Cohen (1972) reported.
The martensitic transformation can effectively suppress strain localization and extend uniform elongation, namely the TRIP effect. However, such an effect becomes apparently weaker and weaker with the temperature increases from 25 to 150 ºC, and thus both the elongation and strength gradually drop to valleys. According to Coryell, Savic et al. (2013), for QP980, 150 ºC is roughly the Md temperature, which stands for the stop limit for retained austenite transforming to martensite induced by plastic deformation.
Similar observations were also reported in the literature. For example, Tamura (1982) reported Md = ~25 ºC for Fe-Ni-C alloy, Berrahmoune, Berveiller et al. (2004) obtained
Md = ~100 ºC for TRIP600, Wang, Huang et al. (2006) tested Md = ~80 ºC for TRIP700.
In other words, from Ms(σ) to Md, retain austenite transforming to martensite becomes more and more inactive during the plastic deformation. To verify this deduction, Coryell,
Savic et al. (2013) examined the retained austenite volume of some pre-strained QP980 specimens using both XRD and EBSD. Figure 20 is reproduced based on their plot. It
36 clearly shows that from 25 to 150 ºC, more and more retained austenite can be detected in the pre-strained specimens, which reveals the suppression of the temperature to the martensitic transformation in this regime. In addition, the peak grain average misorientation degree (GAM-degree) value at 150 ºC also indicates that the primary plasticity and strain hardening mechanism is the regular dislocation slip rather than the phase transformation. As Coryell, Savic et al. (2013) proposed, this might be caused by additional solute atoms (C or N-atoms) that diffuse actively due to the temperature to the boundary between the retained austenite and other phases, and thereby effectively stabilize the austenite from transforming to martensite. It may be also due to the SFE increasing with the temperature (Rémy and Pineau 1978). To be specific, as Peng, Zhu et al. (2013) summarized, the martensitic transformation occurs at SFE ≤ 16 mJ/m² (Frommeyer, Brux et al. 2003), while dislocation slip happens at SFE ≥ 45 mJ/m² (Curtze and Kuokkala 2010).
Either explanation will need further microstructure-based verification.
37 Figure 20: The temperature variation of retained austenite (RA) measured by EBSD and XRD techniques and grain average misorientation (GAM-degrees) measured using EBSD (reproduced with the permission by (Coryell, Savic et al. 2013))
From 150 to 250 ºC, the trends of the UTS, TE, and UE go uphill again. For the increase of the UTS, it can be explained that the dynamic strain aging (DSA) is activated and manifests both blue brittleness and PLC banding (from 150 to 200 ºC). Regarding the blue brittleness, it was defined and confirmed by Dolzhenkov (1971), Yushkevich,
Manankova et al. (1974), and Ohmori, Harada et al. (1996) that at a certain temperature range (typically 150 to 450 ºC) a blue oxide film will form on steels and increase the UTS.
As for the PLC banding, it is the most common and pronounced macroscopic manifestation of the DSA effect (Robinson and Shaw 1994), and the detailed analysis will be presented in Section 2.1.2.4. However, no matter which DSA manifestations, they are typically reported to decrease the material ductility, which conflicts with the experimental result in
Figure 19(B). Therefore, it is very possible that the suppression effect from the DSA on the elongation is overcame by another elongation-aiding effect. Figure 20 shows a sharp consuming rate of the retained austenite between 150 and 200 ºC, which indicates that the martensitic transformation was reactivated. Since no relative research has been found in the literature so far, the explanation for this phenomenon can be proposed that when the deformation temperature is risen in the DSA range, the diffusivity of the C-atoms is highly boosted. On the one hand, as the ‘arrested model’ (proposed by McCormick (1972) and developed by van den Beukel (1975) and Mulford and Kocks (1979)) describes, the diffusing C-atoms can gather around the moving dislocations due to the opposite energy
38 polarity to pin their motion until the external stress releases the dislocations from the pinning. Then since the dislocations move slower than the C-atoms, they will be caught up and pinned again, until more external stress release them again. Such a cyclic pinning- unpinning process results in the increased strain hardening rate and the PLC banding. On the other hand, since the C-atoms play a significant role of partitioning the retained austenite with other phases, when the temperature boosts the C-atoms to diffuse to the entire microstructure, the retained austenite can no longer be stabilized and thereby massively transform to the martensite. Consequently, the reactivated TRIP effect results in the improvement of both the elongation and the strength.
From 250 to 300 ºC, the UTS keeps on a certain level but the elongation drops. It indicates that the DSA still affects the QP980 deformation, even though the PLC banding regime has passed, while the TRIP effect slowly fades again as shown in Figure 20, probably due to the continuous increase of the SFE.
To summarize, in the sub-zero temperature range (below Ms(σ)), the stress-assisted martensitic transformation mainly affects the deformation characteristics of QP980. Since the actual stamping is not likely conducted in this temperature range, the further discussion will not be presented in this work. As for the environmental temperature between Ms(σ) and Md, the strain-induced TRIP effect on QP980 is decaying with the increase of the temperature. Therefore, the stamping performance of QP980 actually degrades in this range. With the further increase of the temperature, the UTS and ductility of QP980 rise again, probably due to the combined influence of the DSA and TRIP effects.
2.1.2.3 Temperature-Strain Rate Effects on Tensile Properties
39 In the actual stamping, the metal sheets deform at a much wider strain rate range than the quasi-static. Therefore, in this section, the strain rate effect is added in the investigation with the temperature effect. Four strain rate levels, including 0.0001/s,
0.001/s, 0.01/s, and 0.1/s were conducted at 25 to 300 ºC. Figure 21 and Figure 22 show the variation of the strength and the elongation respectively with the strain rates as well as the temperature. For the 0.2%YS, no matter how the strain rate changes, the general trends are similarly decreasing with the temperature rising. This again confirms that the strain- induced martensitic transformation can only affect the plasticity deformation of the material. For the UTS, TE, and UE, from 25 to 75 ºC, the effect from the strain-induced martensitic transformation is decaying at all the selected strain rates. However, the divergence begins from 75 ºC since the deformation of the specimens at 0.0001/s starts to be affected by the DSA effect, and then gradually becomes more and more pronounced when the deformation of the specimens at 0.001/s, 0.01/s, and 0.1/s enters the DSA regime respectively at 125 ºC, 175 ºC, and 250 ºC. This also confirms that the DSA behaves positive temperature sensitivity and negative strain rate sensitivity (Robinson and Shaw
1994). According to the ‘arrested model’ (McCormick 1972), the slower strain rate, the easier for the C-atoms to catch up and pin the dislocations. In this work, the DSA regime is recognized based on the appearance of the PLC bands in the DIC results as shown in
Figure 23, even though theoretically the DSA range can be wider than this recognition and is not necessarily manifested by the PLC banding (Kim, Ryu et al. 1998). Notice that the
PLC bands of QP980 are observed in the temperature ranges that the actual stamping circumstances can very likely reach due to the adiabatic heating as well as the friction-
40 induced heating. Meanwhile, the PLC banding is reported to cause some practical problems in the actual application, such as rough surface rough surface (Robinson and Shaw 1992), decreased fatigue life (Kim, Ryu et al. 1998), and early necking (Kang, Wilkinson et al.
2006). Therefore, in the following section, the observation and analysis on the PLC banding phenomena of QP980 will be presented in detail.
Figure 21: The strength of QP980 varies with the environmental temperature and strain rate
(A)
41 (B) Figure 22: The (A) TE and (B) UE of QP980 vary with the environmental temperature and strain rate
Figure 23: The PLC banding occurrence map based on various strain rate and temperature conditions (red solid dots represent that the PLC bands are observed in the DIC results, while black circles indicate no PLC bands observed)
2.1.2.4 The Portevin Le-Châtelier Effect
42 The PLC banding, named after the two pioneer researchers and their work (Portevin and le Chatelier 1923), is the most commonly observed macroscopic manifestation of the
DSA (Robinson and Shaw 1994). This phenomenon typically appears by means of repeated serrated stress-strain curves and mobile band-like strain and strain rate localization on the specimen surface. The PLC banding was first (and usually) observed on either flat or cylindrical uniaxial tensile specimens (Portevin and le Chatelier 1923, Benallal, Berstad et al. 2008), but also reported in uniaxial compression (Wang, Guo et al. 2015) and multi- axial loading tests (Romhanji, Glisic et al. 1998, Cieslar, Karimi et al. 2002, Cieslar,
Fressengeas et al. 2003, Swaminathan, Abuzaid et al. 2015). Rodriguez (1984) categorized the PLC bands to five types (Type A, B, C, D, and E, but only A, B, C are common) based on different serration appearances on the stress-strain curves. However, since the actual experimental results usually do not show differentiable serration characteristics as in the schematics, Yilmaz (2011) proposed that the categorization as well as all the other relative analysis on the PLC banding should be comprehensively based on band motion, orientation, spatiotemporal appearances, and serration characteristics. In this case, the DIC technique is a qualified way for the PLC-related research due to its in-situ full-field displacement and distortion measurement capability. Examples of some efforts where DIC was used to capture such “banding behaviors” of several materials can be found in the review in Yilmaz (2011). Some representative efforts include Benallal, Berstad et al.
(2008) and Ait-Amokhtar and Fressengeas (2010) on characterizing the PLC bands in 5xxx aluminum alloys, Zavattieri, Savic et al. (2009) and Renard, Ryelandt et al. (2010) for PLC bands in TWIP steels, and Swaminathan, Abuzaid et al. (2015) on Ni alloys. Nevertheless,
43 very few efforts in the literature, such as Manach, Thuillier et al. (2014), utilized DIC to study PLC banding behavior of Al alloys at higher-than-ambient temperatures.
In this work, the PLC banding were captured by the DIC system at different deformation conditions as shown in Figure 23. Among them, the results at 150 ºC and 200
ºC at 0.0022/s strain rate are picked out for detailed analysis in this section for their different but representative characteristics.
First of all, the tension behavior of QP980 between 150 and 175 ºC at 0.0022/s is significantly unique. Since the PLC banding appearance at 175 ºC is very similar to that at
150 ºC, this paper presents the 150 ºC results only. In this temperature region, the stress- strain curves exhibit apparent serrations around the diffuse necking onset in Figure 24. This is because the environmental temperature in this case is still not high enough to provide enough energy for the interstitial atoms (such as C and N atoms) to diffuse, catch up, and
‘arrest’ the moving dislocations, until in the large strain region, the dislocation density is high enough to slow down the moving dislocations. Figure 25(A) shows the timeline of the stress evolution. Five moments, including 32, 37.5, 39, 39.5, 41.5 second, are selected for analysis. Figure 25(B) exhibits the strain and strain rate propagation maps from 30 to 45 second based on the DIC measurements. Since the PLC banding is a strain rate softening instability manifestation (Ananthakrishna 2007), the strain rate maps can highlight the PLC bands much clearer than the strain maps. The five selected moments are also marked in the two propagation maps. Figure 25(C) exhibits at 37.5, 39, 39.5, 41.5 second, the four diagrams of the strain rate with three section lines which are virtually sketched in the DIC post-processing software.
44
Figure 24: Stress-strain curve of QP980 at 150 ºC, 0.0022/s
(A)
(B)
45 (C) Figure 25: Evolution of the PLC bands at 150 ºC: (A) serrations on the stress-time curve and five critical moments are selected for analysis; (B) the PLC bands exhibited on the strain and strain-rate propagation maps from 30 to 45 second testing time based on the DIC results; (C) the strain rate with three section lines diagrams from 37.5 to 41.5 second reveal the regeneration and reorientation process of the PLC bands
The three parts in Figure 25 need to be analyzed together based on the five selected critical moments. At 32 second, the stress reaches the ultimate value, which indicates the beginning of the diffuse necking, meanwhile two type-A PLC bands nucleate at a certain location in the gauge area but towards two crossing directions. Notice that the nucleation position of the bands is not around the gripping area. Similar phenomenon was also reported by Zavattieri, Savic et al. (2009) on a TWIP steel. This is discrepant with some
DSA theories like the proposal by Mesarovic (1995) and the experimental observations by
46 Lebedkina, Lebyodkin et al. (2009) on a TWIP steel and Ranc and Wagner (2005) on an aluminum-copper alloy. Lee, Kim et al. (2011) summarized the experimental results in the literature and proposed that the contents of nitrogen and aluminum in the microstructure may be responsible for shifting the PLC banding nucleation positions from the gripping ends to the gauge area. However, the detailed mechanisms still need to be further investigated.
After the initiation of the bands, at 37.5 second, Figure 25(A) shows an apparent stress valley, Figure 25(B) shows a backslash-like (‘\’) band in the DIC strain rate map, and Figure 25(C) shows a peak of strain rate (about 0.05/s) in the band. In the following stress peak at 39 second in Figure 25(A), the band moves about 3 mm according to the sketch gauge sections in Figure 25(C). Meanwhile, a secondary slash-like (‘/’) band starts to nucleate as shown in the DIC strain rate map in Figure 25(B) and the peak strain rate in this band is about 0.007/s as shown in Figure 25(C). Then the stress quickly drops to another valley at 39.5 second in Figure 25(A). At the same time, the secondary slash-like band becomes the primary band in Figure 25(B) and the peak strain rate in it reaches about
0.05/s, while the previous backslash-like band fades out with the peak strain rate in it at only about 0.01/s in Figure 25(C). In the last selected moment at 41.5 second, the stress rises to another peak in Figure 25(A), the slash-like band continues the movement, and the backslash-like band fades away in Figure 25(C). After 41.5 second, similar phenomena keep appearing as a cyclical process until the fracture occurs.
As for the second picked case, when the environmental temperature is increased to
200 ºC, the entire plastic deformation region of the stress-strain curve becomes serrated as
47 shown in Figure 26. Different from the case at 150 ºC, when the plastic deformation just starts, the interstitial atoms can obtain enough energy from the environment to cyclically diffuse, catch up, and ‘arrest’ the moving dislocations. Figure 27(A) shows the timeline of the stress evolution. Similar to the previous case, four moments, 56.5, 57, 57.5, 58 second, are selected for analysis in this case. At these moments, the stress waves respectively to peak, valley, peak, and valley as shown in Figure 27(A). Notice that in this case the time interval between neighboring stress peaks and valleys is less than that at 150 ºC, which also indicates that the diffusivity of the interstitial atoms is higher at 200 ºC and thus accelerates the ‘arrest’ cycles. Meanwhile in this period, a PLC band moves to the gripping end, and then a new band nucleates at another position in a different orientation as shown in the strain rate map in Figure 27(B). This process can also be observed in detail in Figure 27(C).
At 56.5 second when the stress is at a peak, the band moves to the end of the gauge area.
The strain rate peak in this band is about 0.011/s. At the same time, a new band is nucleating at a position about 15 mm away from the old band. From 57 to 57.5 second when the stress waves to a valley and another peak, the old band fades out, while the new band grows to become the primary. At 58 second when the stress reaches to another valley, the old band is about to diminish at the gripping end, while the new band finally grows up, with the peak strain rate in it increases to about 0.008/s. After 58 second, the new band seems to be stuck at its nucleation position and eventually develops to the crack there as shown in Figure
27(B).
48
Figure 26: Stress-strain curve of QP980 at 200 ºC, 0.0022/s
(A)
(B)
49 (C) Figure 27: Evolution of the PLC bands at 200 ºC: (A) serrations on the stress-time curve and four critical moments are selected for analysis; (B) the PLC bands exhibited on the strain and strain-rate propagation maps from 30.5 to 60 second testing time based on the DIC results; (C) the strain rate with three section lines diagrams from 56.5 to 58 second reveal the regeneration and reorientation process of the PLC bands
2.2 Equi-biaxial Loading Testing and Results
Uniaxial tension properties are straightforward to obtain experimentally but not sufficient enough, especially for anisotropic materials, to represent material behavior under biaxial loading in actual sheet metal forming. Therefore, as a necessary complement, biaxial loading testing was introduced in material characterization first by Hill (1948).
There are basically six methods of biaxial loading testing, including 1) hydraulic/pneumatic bulge (Gutscher, Wu et al. 2004, Abu-Farha, Hector et al. 2010), 2)
50 multiaxial tube expansion (Kuwabara and Sugawara 2013), 3) combined tension-torsion
(Naghdi, Essenburg et al. 1957), 4) biaxial compression (Hosford 1966, Barlat, Maeda et al. 1997), 5) disk compression (Barlat, Brem et al. 2003), and 6) cruciform biaxial testing
(Abu-Farha, Hector et al. 2010, Hanabusa, Takizawa et al. 2013, Iadicola, Creuziger et al.
2014, Deng, Kuwabara et al. 2015). However, all of these methods have some drawbacks in experimental implementation. For example, 1) stress in bulge testing is indirectly measured but rather calculated based on a membrane theory, which may cause inaccuracy especially for thick sheets, 2) tube expansion and 3) combined tension-torsion testing is not suitable for sheet metal, 4) stress measurement in biaxial compression is not accurate due to the interactive forces, such as friction, among the stacked metal sheets, 5) disk compression does not allow stress measurement, and 6) cruciform biaxial testing needs a robust specimen geometry that not only allow direct stress measurement, but also guarantee the desired strain path consistently at different strain levels. Therefore, these methods are still undergoing improvement, even though 1) hydraulic bulge and 6) cruciform biaxial testing being standardized (ISO 16808 and 168442, respectively).
2.2.1 Experimental Setups
In this research, hydraulic bulge testing was adopted to characterize the target material under the equi-biaxial loading condition. As shown in Figure 28, the base device is an Interlaken servo-hydraulic press. A pair of hydraulic bulge form dies are centered and bolted onto the platforms. Before testing, the center hole in the lower die is fulfilled with liquid and covered by a round QP980 specimen (diameter >180 mm). During testing, the two dies are first clamped to each other with an 800 kN force and then a piston below the
51 liquid pushes up continuously until the bulged specimen bursts by the liquid eventually. A sensor connected into the lower die can detect the liquid pressure during testing.
Furthermore, a 3D DIC system is setup on top of the press and measure the deformation and displacement of the specimen upper surface through a hole in the center.
Figure 28: Hydraulic bulge testing setups and equi-biaxial stretching schematic
2.2.2 Equi-Biaxial Stress-Strain Curve
The biaxial stress in hydraulic bulge testing can be derived based on a membrane theory first proposed by Hill (1948). As seen in Figure 29, it is assumed that when the specimen is bulged, its isotropic internal surface can fit a sphere of radius ρ. According to
ISO 16808, with the detected liquid pressure signal p and the derived actual thickness t, at the apex of the bulge, the stress in the x and y directions are considered equivalent and can be expressed in Equation (6).
52 푝𝜌̅ 𝜎 = 𝜎 = 𝜎 = (6) 퐵 푥 푦 2푡
The actual thickness t is calculated based on the true thickness strain ɛ3 at the bulge apex in Equation (7). Since thickness deformation cannot be directly measured by DIC, the volume consistency Equation (8) in ISO 16808 is suggested to calculate ɛ3, assuming an ideal plane stress condition at the bulge apex and excluding the bending strain.
휀3 푡 = 푡0푒 (7)
휀3 ≈ −휀1 − 휀2 (8)
Figure 29: Work principle of the bulge testing
So far it can be seen that the default ISO 16808 equations are based on several ideal assumptions, especially the isotropic biaxial deformation and the plane stress condition, which may lead to errors. To improve the algorithm, Atkinson (1997) argued that the material planar anisotropy (Δr in Equation (4)) significantly affected the biaxial deformation; therefore, the fitting primitive radius ρ in the x and y directions should be differentiated. To realize this difference numerically, the fitting primitive should be no longer a sphere but an ellipsoid (or a 2nd degree surface in general) or rather a higher degree free surface (up to 4th degree). Furthermore, the bending strain at the bulge apex
53 should also be added in thickness strain derivation. The corresponding new formula have many versions, such as the proposals by Young, Bird et al. (1981), Yoshida (2013), Mulder,
Vegter et al. (2015), and Min, Stoughton et al. (2016).
In this work, since the thickness of QP980 specimens is only 1 mm, comparatively very small with the curvature radius of the fitting primitives (shown in ), the thickness strain and actual thickness derivations are still base on the equations (7) and (8). Yet, the fitting primitive is picked from on a sphere (Equation (9)) (ISO 16808), a 2nd degree free surface (Equation (10)) (ISO 16808: Annex C-1), and a 4th degree free surface (Equation
(11)) (GOM 2012), based on which one is the best-fit to actual. In addition, for fitting primitives having two different curvature radius in the x and y directions, the mean curvature radius is calculated based on Equation (12).
2 2 2 2 𝜌 = (푥 − 푥0) + (푦 − 푦0) + (푧 − 푧0) (9)
2 2 푧 = 푎0푥 + 푎1푦 + 푎2푥푦 + 푎3푥 + 푎4푦 + 푎5 (10)
4 4 2 2 2 2 푧 = 푎0푥 + 푎1푦 + 푎2푥 + 푎3푦 + 푎4푥 푦 + 푎5푥푦 + 푎6푥 + 푎7푦 + 푎8 (11)
−1 1 1 1 𝜌̅ = ( ( + )) (12) 2 𝜌푥 𝜌푦
Effective stress-strain curves based on the three candidate fitting primitives, compared with the uniaxial tension stress-strain curve in Section 2.1.1, are shown in Figure
30. Comparatively, the effective stress-strain curves based on bulge testing can reach much higher uniform strain level than that of uniaxial tension testing. This is because the effective strain in equi-biaxial stretching (ɛ1+ ɛ2) is theoretically twice the stain in uniaxial tension
(ɛ1) (Nasser, Yadav et al. 2010). In addition, unlike in uniaxial tension testing, during equi-
54 biaxial stretching, the specimens do not have early strain location issues induced by width necking. As for the higher flow stress in the equi-biaxial curves than the uniaxial, it is due to the material planar anisotropy (Nasser, Yadav et al. 2010).
Figure 30: Effective stress-strain curves of QP980 based on the bulge testing (BT) and the uniaxial tension testing (UT)
Among the three fitting primitives, the effective stress-strain curves based on the sphere and the 2nd degree free surface are very close to each other over the entire strain, while the stress-strain curve based on the 4th degree free surface shows a higher stress level between the 0 to ~0.15 strain region and a slightly lower stress level above ~0.2 strain.
Since the strain region between 0 and 0.15 will be a critical reference for model parameter calibration in Chapter 3, here it is very necessary to clarify which stress-strain curve can better represent the real material behavior.
Also, based on the comparison of the fitting curvature radius in Figure 31, the most obvious discrepancy between the 4th degree free surface and the other two primitives appears at ~0.02 strain, approximately the transition region from elastic to plastic
55 deformation. The most likely reason for this discrepancy is that the fitting derivations of the three primitives to the actual specimen surface are actually different even within the fitting tolerance.
Figure 31: Mean radius evolution curves of the three candidate primitives fitted to the actual bulge surface
Therefore, five effective strain levels (0.02, 0.04, 0.06, 0.08, and 0.10) are targeted, and at each strain, the deviation distribution DIC maps and the normal distribution curves are shown and compared respectively from Figure 32 to Figure 36.
56
Figure 32: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.02
57
Figure 33: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.04
58
Figure 34: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.06
59
Figure 35: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.08
60
Figure 36: Deviation contours and normal distribution plot of fitting the three candidate primitives to the actual bulge surface at effective strain equals to 0.1
Obviously, on the five targeted strain levels, the highest deviation distribution probability peaks at 0 mm indicate that the 4th degree free surface can always fit the actual surface better than the other two primitives. In contrast, with involving the planar anisotropy, the sphere is comparatively the most inaccurate fitting primitive. Nevertheless, the discrepancy among the three fitting primitives continuously decreases from 0.02 to 0.10 strain. This trend matches the comparison of the stress-strain curves in Figure 30.
Therefore, the effective stress-strain curve of the 4th degree free surface fitting is considered the most accurate and will be used in the following sections.
2.2.3 Equi-Biaxial Yield Stress and Anisotropy
Based on the effective stress-strain curve from bulge testing, the biaxial yield stress of QP980 can be derived. However, as shown in Figure 37, the elastic segment of the biaxial curves is very scatter and steep, and hence is believed inaccurate to use the same yield stress derivation method as in uniaxial tension testing (Barlat, Maeda et al. 1997).
61 Therefore, Barlat, Maeda et al. (1997) proposed to use the amount of plastic work at yield point on the uniaxial tension stress-strain curve as a reference to calculate the biaxial yield stress. In other word, in Figure 37, the area in red is equal to the area in orange, and the peak point of the area in orange is considered the biaxial yield stress. Therefore, since the tension yield stress of QP980 (0°) is 813.35 MPa (Table 2), corresponding to 3.197 MPa plastic work (integrated based on Equation (5)), the biaxial yield stress point on the biaxial effective stress-strain curve (4th degree free surface) is 854.68 MPa. As comparison, the yield stress points on the other two biaxial stress-strain curves are respectively 841.98 MPa
(sphere) and 814.60 MPa (2nd degree free surface).
Figure 37: The stress-strain curves of QP980 at low strain level (up to 0.02)
Besides the yield stress, the anisotropy coefficient, denoted as rb, in equi-biaxial loading condition is also derived differently from the uniaxial tension. Barlat, Brem et al.
(2003) first proposed this parameter as a ratio of the principal strain in rolling direction and in transverse direction (expressed in Equation) based on disk compression testing results,
62 and Pöhlandt, Banabic et al. (2002) also used it in processing cruciform biaxial testing results. As shown in Figure 38, the slope of ɛ1-ɛ2 curve of the bulge apex area is the rb of
QP980 and its value (1.150) is consider constant during the entire deformation.
휀2 푟푏 = (13) 휀1
Figure 38: Biaxial anisotropy coefficient derivation via fitting ɛ1-ɛ2 curve of the bulge apex area
Alternatively, as Barlat, Brem et al. (2003), Barlat, Aretz et al. (2005) suggested, rb can also be derived based on the Yld96 model (Barlat, Maeda et al. 1997) using some uniaxial tension parameters (yield stress and r-values along and transverse to rolling direction) and equi-biaxial tension yield stress. In this case, the rb of QP980 is derived as
0.930. More details about this model will be included in Chapter 3.
2.3 Failure and Forming Limit
No matter undergoing which forming conditions, metal sheets will reach forming limits, beyond which failure is anticipated (Keeler 1965). Forming limit is actually a very
63 broad concept, since it refers to plastic deformation bounding of ‘a given shape without defects,’ in which the ‘shape defects’ can be wrinkling, tearing, rough surface, and any other manifestations that violate certain manufacturing quality criteria (Banabic 2010).
Nevertheless, in sheet metal forming, the primary defect concern is tearing
(cracking/fracture), which means a sheet has lost its load carrying capacity. Typically, tearing is preceded by localized necking (also termed as thickness/local necking (Keeler and Kimchi 2015)), a manifestation of critical strain localization and a certain stage of component strength reduction (Keeler and Backofen 1963, Backofen 1972). Therefore, onset of localized necking is usually considered the forming limit in sheet metal stamping
(Keeler 2003). In some cases, such as very ductile materials formed at high temperature and do not behave apparent localized necking, facture or onset of diffuse necking can hence be the forming limit (Banabic 2010, Li, Carsley et al. 2013).
To represent this kind of forming limits, originally Keeler (1965) just plotted pre- fracture maximum localized principal major and minor strain as points in a ε1- ε2 coordinate system and called this point cloud as ‘formability limits’ in stampings. Since
Keeler’s experimental work was mainly based on biaxial stretched HSLA parts (ε1 > 0, ε2
> 0), later on Goodwin (1968) extended this point cloud to uniaxial stretching states (ε1 >
0, ε2 <0) based on various cup tests as well as actual stampings. Also, Goodwin (1968) specified that the ‘formability limits’ were actually ‘onsets of necking.’ Agreed with
Goodwin, Keeler and Brazier (1977) started to use ‘onsets of localized thinning’ instead of
‘formability limits,’ and first named the strain point cloud in the ε1- ε2 coordinate system as forming limit diagram (FLD) with the best-fit curve of the point cloud as forming limit
64 curve (FLC). The red line in Figure 39 is a FLC obtained experimentally or predicted numerically, which splits forming failure and marginal/safe zones. The marginal band is generated via shifting down the red line by 10% (default recommendation by Keeler
(2003)), 10~30% (an empirical number, varying with material types and stamping workshops), or a calculated number based on dimensions of draw-beads and material thinning after flowing through draw-beads (Keeler 2003). The existence of the marginal band is to allow not only experimental data variance but also any possible bias of FLC due to some factors that affect in component-level manufacturing but are not or not fully included during coupon-level testing, such as pre-straining, temperature effect, strain rate effect, stamping lubrication, and microstructure inhomogeneity (Banabic 2010). Therefore, the green line in Figure 39 shifted from the red is the actual FLC implemented in industrial applications.
Figure 39: Schematics of Forming limit diagram, curves, and safety margin
2.3.1 Theoretical FLC Models
65 There are many theoretical models that can predict or derive a material’s FLC.
Generally speaking, in this research community, even the conventional principal-strain- based FLC (ɛ-FLC) (Figure 39) is only one of the four common FLC types (Figure 40). A major drawback of ɛ-FLC is its strain-path sensitivity, in other words, a ɛ-FLC generated based on pre-strained specimens shifts from that based on as-received specimens, and the shifting magnitude and direction vary with the pre-straining paths (Ghosh and Laukonis
1976, Basak, Panda et al. 2015). This may result in overestimation of forming limits based on experimental results but unexpected early cracking in actual stampings. To avoid this drawback, Stoughton (2000) proposed principal-stress-based FLC (σ-FLC) and proved its advantage of strain-path independency even for anisotropic materials under non- proportional loading (Stoughton and Zhu 2004, Wu, Graf et al. 2005, Basak, Panda et al.
2015). Simha, Grantab et al. (2007) further improved σ-FLC to extended stress-based FLC
(XSFLC) for thick sheets to predict necking under three dimensional loading. Yet, σ-FLC is highly dependent on adopted hardening laws; even switching between Voce’s and
Swift’s laws (both can fit the same material’s hardening behavior), the resulted σ-FLCs are different (Basak, Panda et al. 2015). Therefore, Stoughton and Yoon (2012) returned to ɛ-
FLC but introduced another FLC type, polar-effective-plastic-strain-FLC (PEPS-FLC), by means of representing movement of ɛ-FLC in a polar coordinate system.
66
Figure 40: Family of theoretical FLC models
These new alternative FLC types proposed in recent years still need further development and evaluation based on more different materials. In addition, a very apparent obstacle that prevents the wide acceptance of the new FLC types in industrial applications is their lack of use-friendliness: they cannot be directly compared with experimental results, which are still obtained from standardized strain-based measurements. Last but not the least, the potential risk of unexpected early cracking cause by the pre-strain-induced shift of ɛ-FLC can be neutralized by the safety margin in Figure 39.
Back to the ɛ-FLC models category, according to the summary in Banabic (2010), there have already been various models based on different theories developed in the past half a century. As seen in Figure 40, first of all, regarding the semi-empirical models,
Keeler and Brazier (1977) investigated the relationship between FLC and sheet metal geometry as well as properties, and figured out a semi-empirical equation (14) of the valley point of FLC (denoted as FLC0) represented by sheet nominal thickness (0 < t ≤ 3 mm) and terminal strain hardening exponent (n-value at UTS) (0 < n ≤ 0.21).
67 푛 퐹퐿퐶 = (23.3 + 14.13푡)/100 (14) 0 0.21
Later, Equation (14) was further extended to a more complete piecewise equation
(15), so that an entire FLC curve can be sketched based on the only two input parameters.
Therefore, even though the Keeler-Brazier (K-B) equation was developed based on HSLA steel data, it is now widely used in North American stamping workshops to preliminarily predict FLCs for all types of mild steels, HSSs, and AHSSs (Sriram, Huang et al. 2009). In addition, the K-B equation was also added with strain-rate sensitivity coefficient, internal damage parameters, and the Lankford coefficients by Cayssials (1999), Cayssials and
Lemoine (2005) to become new dedicated models for predicting FLCs of AHSSs and
UHSSs.
퐹표푟 휀2 ≤ 0: 휀1 = 푙푛(퐹퐿퐶0 + 1) − 휀2 (15) 휀2 휀2 퐹표푟 휀2 > 0: 휀1 = 푙푛 (퐹퐿퐶0 + (푒 − 1)(−0.86(푒 − 1) + 0.78) + 1)
Besides the semi-empirical models, FLCs can also be predicted based on different theories. For example, based on necking/instability theories, the representative models include Swift’s and Hora’s diffuse necking models (Swift 1952, Hora, Tong et al. 2003) and Hill’s localized necking model (Hill 1952). Also, Dudzinski and Molinari (1991) and
Storen and Rice (1975) proposed their FLC models based on respectively linear perturbation and bifurcation theories. These models are independent with materials’ yield criteria but more relative to hardening laws and sheet thickness. Since these theoretical models are all based on ideal deformation of homogeneous metal sheets, Marciniak and
Kuczynski (1967), (1979) argued that in reality even a random piece of metal sheet could always have either geometrical variance through thickness or inhomogeneous
68 microstructure. Therefore, their model, namely the M-K model, assumes a unit sheet with a pre-existing groove deforms at different strain paths; when strain ratios of areas inside and outside the groove reach a critical level, it is considered the onset of localized necking, and hence the corresponding strain on the FLC is obtained. The original M-K model could only generate half of a FLC in the biaxial strain domain (ɛ1 > 0, ɛ2 > 0), while Hutchinson and Neale (1978) (H-N model) further extended it to the entire strain domain.
The theoretical FLC models have been just briefly reviewed in this section since they are not the focus in this research. Only the FLC generated based on the Keeler-Brazier equation will be included in the following section as a reference.
2.3.2 Experimental FLC Determination Techniques
Although the Keeler-Brazier equation has been successfully implemented to predict
FLCs of conventional automotive steel sheets in North American stamping workshops during the past decades, it has also been found not applicable for TRIP-like steels due to these steels’ continuous high strain hardening rate even at higher strain levels that effectively retards strain localization (Keeler and Kimchi 2015). Therefore, to guarantee the stamping quality, the reliable FLCs can only be determined from experimental results.
The representative methods can be overviewed in Figure 41.
69 Figure 41: Family of experimental FLC determination techniques
2.3.2.1 Position-dependent Methods
FLC experiments are actually based on various strip and cup forming methods with different punch and/or specimen geometry (Banabic 2010). Today, the most common methods are Nakajima and Marciniak testing, both having been standardized in ISO 12004-
2. Nevertheless, the main research attention is on how to determine FLCs from forming limit tests. Conventionally, in North America, Keeler (1968) first proposed to imprint small circle grids on blanks by photographic or electrochemical marking techniques before stamping; after stamping, by means of analyzing distortion of the closest circles around the first crack, the critical major and minor strain can be tape-measured. This is the so-called circle grid analysis (CGA) method, and has been standardized in ASTM E2218. Similarly, in Europe, small grids need to be etched onto specimen surface, while an obvious difference from the CGA method is, rather than round circles, these grids are square and actually formed by intersected section lines along or transverse to sheet metal rolling direction; when the first crack just appears during deformation, longitudinal strain of the square grids on the section lines crossing perpendicularly to the crack are measured, and by means of fitting the curve of the longitudinal strain with the section length with a parabola, the strain corresponding to the parabola peak is considered a point on FLC. This is called position-dependent method, and has been standardized in ISO 12004-2.
Both of the conventional methods are very costly and low efficient. First of all, a large amount of specimens need to be repeated since each test can only generate one single sample (a cracked cup) and very limited data can be extracted from it. The worse thing is,
70 if the grids need to be adjusted due to dimension, geometry, or crack position issues, the complete testing process must be reworked. In addition, experimenters have to spend a large amount of time on tape-measure and strain calculation, grid by grid on each specimen.
Furthermore, the conventional methods are incapable of determining FLCs from materials behaving inhomogeneous strain distribution (ISO 12004-2: 2008 – Annex G). Therefore, even though these methods have been applied on a large scale due to their simplicity, new techniques are still expected to avoid the drawbacks.
The introduction of DIC to sheet metal forming satisfied such an expectation. On the one hand, DIC-measure is far more accurate and efficient than manual tape-measure.
For example, in DIC-aided ISO 12004-2 method, etched grids are no longer needed; any virtual grids can be added or modified freely in DIC-recorded images, without repeating the whole testing. Also, the grid distortion measurements and strain calculation can be conducted automatically, swiftly, and accurately (for example, pixel resolution is ~0.04 mm/pixel in this research work) in DIC-associated data-processing software. On the other hand, in each test, a DIC system can continuously record complete deformation history of a patterned specimen surface (for example, at least 100 images were recorded in each test in this research work). Therefore, the post-cracking measurement of the conventional methods is overshadowed by the in-situ measurement of new DIC-aided methods. For example, Geiger and Merklein (2003) improved the ISO 12004-2 method by calculating arithmetic middle values of major strain gradient in last multiple pre-cracking stages of
DIC measurements. Similarly, Zhang, Lin et al. (2013) proposed to trace distortion history of circle grids at different positions in DIC measurements and derive strain of necking
71 onset. They both claimed to successfully improve the conventional methods to be more efficient and accurate.
2.3.2.2 Time-dependent Methods
Nevertheless, other than aiding the position-dependent techniques, DIC-measure also inspired new development of a series of time-dependent methods. Indeed, the time- dependent methods fully take advantage of the two most unique functions of DIC: in-situ and full-field measurement. Due to the in-situ measurement, the most common idea in all the time-dependent methods is to trace a certain deformation attribute evolving with testing time and eventually detect a critical moment of the target attribute exhibiting a sharp change as onset of necking. Such an attribute can be major strain ɛ1 (including major strain rate and acceleration), for example in Vacher, Haddad et al. (1999), Huang, Sriram et al.
(2008), Högstrom, Ringsberg et al. (2009), Merklein, Kuppert et al. (2010), Hotz, Merklein et al. (2013), and Li, Carsley et al. (2013), or thickness strain ɛ3 (or thickness strain rate), such as Hotz, Merklein et al. (2013) and Volk and Hora (2011), or even specimen surface topography variance (Wang, Carsley et al. 2014). Also, since the entire deformation evolution (from undeformed until fracture) is recorded, the critical moment of the attribute can be detected from either pre-necking or post-necking DIC stages (or both), dependent on different methods. Furthermore, due to the full-field measurement, not only the first crack area, but also any inhomogeneous local deformation areas can be traced and analyzed if necessary. A summary of these timed-dependent methods is shown in Table 3.
72 Table 3: DIC-based FLC determination time-dependent methods
Techniques Traced Attribute Traced DIC Stages
Vacher et al.’s method Major strain & strain rate Pre-necking Huang et al.’s method Major strain acceleration Pre-necking Hogstrӧm et al.’s method Major strain acceleration Pre-necking Correlation coefficient (CC) Major strain acceleration Pre-necking method (Merklein et al., 2010) Gliding correlation coefficient Major strain acceleration Pre-necking (GCC) method (Hotz et al., 2012) Gliding difference of mean to median (GDMM) method (Hotz Thickness strain rate Pre-necking et al., 2012) Linear best fit (LBF) method Thickness strain rate Pre- & post-necking (Volk & Hora, 2011) Li et al.’s method Major strain Pre-cracking Wang et al.’s method Surface height difference Pre-necking
Among these methods, the LBF method was highly recommended by Hotz,
Merklein et al. (2013) based on the comparison work of the CC, GCC, GDMM, LBF in
Table 3 and the ISO 12004-2 methods after processing FLCs of over thirty different grades of steels and aluminums. Comparatively, the LBF method is concluded to have not only the most robust algorithm and the widest material applicability, but also the best physical basis (Hotz, Merklein et al. 2013). The general algorithm of the LBF method is to trace thickness strain rate evolution of a set of filtered representative points in DIC measurements and identify the most nonlinearity moment as onset of necking; this moment is detected by intersected two lines respectively fitting stable and unstable deformation regimes of the thickness strain rate evolution curve. It makes more physical sense to trace thickness strain rate, rather than major or minor strain (including major or minor strain rate
73 and acceleration), because the thickness strain rate represents the deviatoric character of the two plane principle strain of the deformation rate tensor, so it indicates deformation instability in either major or minor strain with time; the thinning acceleration may also be an instability detector, but the 2nd derivation with time also increases noise in it, which makes it very inaccurate (Volk and Hora 2011).
In the following two sections, FLCs of the target and comparative AHSSs will be processed and compared based on different formability testing techniques and FLC determination methods. A new time-dependent method of safety margin in FLD will also be proposed in the end.
2.3.2.3 Experimental Setups
The formability testing setups were still based on the Interlaken servo-hydraulic press with the roof DIC system that has been introduced in Section 2.2.1, while on the internal platform, the central mechanical mechanisms were changed to either a Marciniak or a Nakajima testing punch (both with a diameter of four inches) with the same associated dies. Since in the industry both the Marciniak and Nakajima testing are widely implemented, so are in this research work. As seen in Figure 42(a), the specimen geometry is the same (~180 mm diameter) in both of the testing methods. With the gauge width varying from 20 to 180 mm, the geometric effect contributes to the specimen strain paths covering from uniaxial to equi-biaxial stretching in FLD shown in Figure 39. Nevertheless, the two testing methods also have some differences, not only the punch geometry, but also the ways to generate stress conditions on the specimens. As seen in Figure 42(b), in the
Nakajima forming, a hemispherical punch directly contacts with the specimens (excluding
74 a lubricant layer) and creates a bend-stretching loading state concentrating around the apex of the dome. As for the Marciniak testing, a flat punch indirectly forms the specimens with carrier blanks between them (also excluding the lubricant layer). The carrier blanks are made of a very ductile CR5 mild steel of 0.8 mm thickness in this research work. The high ductility of the carrier blank material can cause the carrier blanks later rupture than the specimens. Each carrier blank was cut with the same outer diameter as the specimen geometry and a 1.25-inch-diameter hole in the center to ensure the specimen having a homogeneous strain distribution and a final crack in the center hole area. Last but not the least, all the tests were conducted at a punch speed of 0.5 mm/s and the DIC captured at least 150 images per test.
(a)
75 (b) Figure 42: (a) Formability testing setups, specimen geometry, and (b) schematics of the Nakajima and Marciniak testing
2.3.2.4 FLCs of Tested AHSSs
In this section, four selected FLC determination methods, including the ISO 12004-
2 (abbreviated as ISO in this section), the LBF, the CC, and the K-B equation, are compared based on mainly the QP980 formability testing results. The first step is to explore the testing-technique dependency of the selected methods for comparing the compatibility of these methods. As seen in Figure 43, along each strain path, three types of testing techniques are able to provide forming limit data. To be specific, (a) very close uniaxial tension strain paths can be generated in the uniaxial tensile test and both the Nakajima and the Marciniak tests on 20 mm width specimens, and the Marciniak test on 20 mm width specimens; yet based on these strain paths, the forming limit points determined by the FLC determination methods are scattered. For example, in Figure 43(a), the LBF determines the most scattered forming limit points (in blue), while the CC methods determines the closest results (in red). (b) In Figure 43(b), plane strain paths can be generated in the uniaxial tensile test on 40 mm width specimens, the Nakajima test on 100 mm width specimens, and the Marciniak test on 120 mm width specimens. Due to the initial bending, the
76 Nakajima testing generates a plane strain path on a different side of the ɛ2 axis with the other two testing techniques. The LBF determination results are still scattered, similar to the CC (in yellow), but worse than the ISO (in red). (c) Equi-biaxial tension strain paths can be generated in the hydraulic bulge testing and both the Nakajima and the Marciniak tests on 180 mm width specimens. As seen in Figure 43(c), in this case, the ISO determination results (in red) are the most scattered, while the CC determination results (in yellow) are comparatively more concentrated. Therefore, to summarize, the CC method is more compatible in the different testing techniques along either the uniaxial tension or equi-biaxial tension strain paths, the ISO method is more compatible along either the uniaxial tension or plane strain paths, while the LBF method is only compatible along the equi-biaxial strain path.
(a)
77 (b)
(c)
Figure 43: Testing-technique compatibility of the selected FLC determination methods under (a) uniaxial tension strain path, (b) plane strain path, and (c) equi- biaxial tension strain path (each point is a representative of three repeated testing results)
78 Other than the testing techniques, the second more important step is to investigate the different tendencies of the FLC determination/prediction methods. In the literature,
Hotz, Merklein et al. (2013) compared four time dependent methods and the ISO method mainly based on the Nakajima testing of over thirty grades of steels and aluminums. Their conclusion was that the CC method tended to generate the most optimistic predictions (too close to fracture), the ISO method typically generated the most conservative predictions
(too far from fracture), while the LBF method gave the best FLCs right between the previous two. In addition, Sriram, Huang et al. (2009) compared the K-B equation results with the ISO method and the Huang, Sriram et al. (2008) method (similar to the CC method) based on DP and TRIP steels using both the Marciniak and Nakajima testing.
Their conclusion was that the K-B equation predicted the most optimistic FLCs, while the
ISO method determined the most conservative results. In contrast, Vysochinskiy, Coudert et al. (2012) argued that the ISO method determined a more optimistic FLC of AA6061 than the CC method in the Marciniak testing.
In this research, a similar comparison work was conducted using both of the testing techniques on QP980. As shown in Figure 44(a), based on the Marciniak testing results, the K-B equation is comparatively the most optimistic method in the left FLD regime, while the LBF method is the most optimistic in the right FLD regime; the ISO method generates the most conservative FLC, especially in the right FLD regime. In Figure 44(b), based on the Nakajima testing results, the forming limit points seem to be less scattered than in Figure 44(a). The ISO method still determines the most conservative FLC, while
79 the CC method turns out to be the most optimistic, both matching the observation in (Hotz,
Merklein et al. 2013).
(a)
(b) Figure 44: Comparison of FLCs of QP980 determined by the selected methods based on (a) the Marciniak testing and (b) the Nakajima testing
In addition, if added with the forming limit results (based on the LBF methods) in the transverse rolling direction, the FLD of QP980 can be extended to full strain field as
80 shown in Figure 45. Consistent with the tension properties in Section 2.1, the forming limits of QP980, no matter based on which testing techniques, do not exhibit obvious anisotropy.
Figure 45: Full strain field FLDs of QP980 based on the LBF method results To further investigate the different tendencies of the selected methods, two comparative steels, DP980 and SS201, were also tested using the Marciniak testing technique. DP980 was selected due to its strength grade same as QP980 but less ductility due to its lack of retained austenite that cannot retard any early strain localization. In contrast, regarding SS201, its high ductility attributes to its fully austenitic microstructure.
Therefore, the onset of localized necking of these three steel grades shall be ranked from early to late as DP980, QP980, and SS201. This can be verified from a comparison of strain-section curves in the last stage before cracking in the DIC measurements on the specimens of 40 mm gauge width in Figure 46. It indicates that right before cracking, the localized necking area in DP980 has already developed with a higher strain peak than in
QP980 and SS201. In other words, these three steels do not even behave the same types of necking: as shown in Figure 47, right before cracking, the QP980 exhibited multiple
81 localized strain bands, while the SS201 specimen formed an approximately single blunt strain peak, reveling a more likely diffuse necking. In this case, a comparison based on these materials will be necessary and reveal certain material-sensitivity of the selected FLC determination/prediction methods.
Figure 46: Strain-section curves in the last stage before cracking in the DIC measurements on the 40 mm width Marciniak testing specimens of DP980, QP980, and SS201 from left to right
(a)
82 (b) Figure 47: In the last stage before cracking, (a) the QP980 specimen behaves multiple localized necking, while (b) the SS201 specimen behaves almost diffuse necking
The determined/predicted FLCs of DP980 and SS201 are shown in Figure 48.
Comparatively, the K-B equation prediction is the most unstable method: it predicts the most conservation FLC of DP980, but the most optimistic FLC of SS201 (even higher than the cracking strain). After all, this is understandable since the K-B equation needs only two input parameters from a uniaxial tension test and was developed mainly based on conventional steel testing. The ISO method, a representative of the position-based methods, is in general the most conservative but not sensitivity to testing techniques nor materials.
In contrast, the CC method, representing the ɛ1-based time-dependent methods, is very sensitive to either testing techniques or materials. As for the LBF method, a representative of the ɛ3-based time-dependent methods, is indeed quite stable and not as conservative as the ISO method as concluded in (Hotz, Merklein et al. 2013).
83 (a)
(b) Figure 48: Comparison of FLCs of (a) DP980 and (b) SS201 determined by the selected methods based on the Marciniak testing
The final step in this section is to propose a new method of determining safety margin in FLD. As mentioned above, the safety margin was conventionally determined based on empirical or semi-empirical methods. This determination way, however, may need a large quantity of try-outs to verify, especially for new materials. Therefore, as shown in Figure 49, when trace a curve of evolving Pearson product-moment correlate coefficient
84 (Equation (16) (Rodgers and Nicewander 1988)) of in-situ thickness strain rate of the first cracking area in DIC measurements with time (or stage or punch displacement), a nonlinearity peak can be detected before onset of necking, and this peak moment is proposed to be the lower limit of safety margin in FLD.
∑푛 (푋 − 푋̅)(푌 − 푌̅) 푟 = 푖=1 푖 푖 (16) 푛 ̅ 2 푛 ̅ 2 √∑푖=1(푋푖 − 푋) √∑푖=1(푌푖 − 푌)
Figure 49: Schematic of the new time-dependent method to determine safety margin lower boundary in FLD
Applying this new method to the FLD of QP980, the safety margin can be determined as seen in Figure 50. Other than avoiding the empirical try-outs in applications, the primary advantage of this method is that the determined safety margin is not simply shifted based on the FLD0 but varies with strain paths, which guarantees the fully use of material formability as well as avoiding potential cracks according to different loading conditions.
85 (a)
(b) Figure 50: The safety margin determined using the new method in the FLD of QP980 based on (a) the Marciniak and (b) the Nakajima testing results
2.3.3 Unexpected Failure Due to Microstructure Inhomogeneity
For QP980 particularly, it can be seen in Figure 44(a) that the determined forming limit points based on the Marciniak testing results between the plane strain and equi-biaxial stretching strain paths are very scattered, compared with that in Figure 44(b) that are based on the Nakajima testing results. In Figure 51, full strain paths of QP980 on the first failure
86 areas are plotted based on the two testing techniques. Unlike the gradual spinning of the strain paths from left to right with the specimen gauge width from narrow to wide in the
Nakajima testing results (Figure 51(b)), development of the strain paths in the Marciniak testing results (Figure 51(a)) becomes irregular in the specimens of gauge width between
140 mm and 145 mm. Furthermore, focusing on five picked strain paths as shown in Figure
52, the QP980 specimens of 40 mm, 100 mm, and 180 mm gauge width can generate highly repetitive strain paths using no matter which testing technique, while the specimens of 140 mm and 145 mm can generate close strain paths in the Nakajima testing but very divergent strain paths in the Marciniak testing.
(a)
87 (b) Figure 51: Full strain paths of QP980 based on (a) the Marciniak and (b) the Nakajima testing results
(a)
88 (b) Figure 52: Comparison of repetition of the picked strain paths based on (a) the Marciniak and (b) the Nakajima testing results
Actually, in this research work, totally four specimens of 140 mm width have been tested using the Marciniak technique, and their necking onsets distribute far from each other as seen in Figure 53. To further investigate the reasons behind this discrepancy, for each 140 mm width specimen, the DIC images in the two stages right before and after cracking happening with the associated strain maps are summarized in Figure 54.
89 Figure 53: The tested four QP980 specimens of 140 mm gauge width exhibited very scattered forming limit points in the FLD
Four images (a), (b), (c), and (d) in Figure 54 are corresponding to No.G0401-
140RD-1, G0402-140RD-2, G0201-140RD-3, and G0403-140RD-4 in Figure 53.
Apparently, G0401-140RD-1 and G0402-140RD-2 exhibited cracking in the local y- direction (along the rolling direction) due to the unexpected localized ɛxx strain bands, while the other two cracked horizontally in the x-direction caused by the ɛyy strain localization. Actually, from the standpoint of specimen geometry design, the failure transverse to the rolling direction is expected due to the geometric effect, just like those specimens with less gauge width. Therefore, since the geometric effect, a macroscopic factor, is not the only determination factor of the crack along the x-direction, there is a certain microscopic effect, which results in a band-like strain distribution along the rolling direction as seen in Figure 54, can compete with the geometric effect and trigger the failure occurring in the y-direction.
(a)
90 (b)
(c)
91 (d) Figure 54: Inconsistent failure in the four QP980 specimens of 140 mm gauge width: (a) No.G0401-140RD-1 specimen and (b) No.G0402-140RD-2 specimen cracked due to localized ɛxx strain bands, while (c) No.G0201-140RD-3 specimen and (d) No.G0403-140RD-4 specimen failed due to ɛyy strain localization
Accordingly, in the same sheet where one of the 140 mm width specimens was cut from, two specimens of respectively 45° and 90° orientations to the rolling direction were cut as ASTM E8 geometry and tested in the uniaxial tension setups in Section 2.1.1. As seen in Figure 55(a), the DIC images of these two specimens also reveal some band-like strain distribution along the rolling direction in their uniform deformation stages.
Furthermore, in Figure 55(b), based on optical microstructure examination, in a strain valley, there are more needle-like martensite grains, which are strong but not ductile, while in a strain peak, there are more island-like ferrite grains, which are comparatively more ductile. Such a microstructure inhomogeneity was primarily caused in sheet metal rolling processes. It has also been reported in the literature by Yan (2009) and observed in many other sheet metals, especially those multi-phase AHSSs, according to our unpublished
92 results. Although further research on this topic is not included in this research work, it is of great importance to investigation the physical reasons and practical influence of this phenomenon in the future.
(a)
(b) Figure 55: Unexpected band-like strain distribution can also be observed in (a) the uniaxial tension specimens that are not cut along the rolling direction, (b) optical microstructure examination reveals that such a phenomenon is due to microstructure inhomogeneity in multi-phase AHSSs
2.4 Compression and Cyclic Loading Properties
93 Until this section, the formability of QP980 has already been characterized. For conventional steels, such a procedure is sufficient to calibrate material models and process simulations to aid morph stamping dies and guarantee steel panels do not fail. However, nowadays especially for AHSSs due to their high strength, other than failure, springback is becoming another primary concern in sheet metal forming (Yan 2009, Keeler and
Kimchi 2015). Therefore, FE simulation results are also highly expected to be predictive of deformation induced by springback, even can be directly relied as critical references for tool design to prevent potential geometric and dimensional inaccuracy and inconsistency problems in the actual operations. Nevertheless, due to the complex loading states during stamping, the main challenge of springback prediction comes from the Bauschinger effect
(Bauschinger 1886), which manifests in material plastic deformation as 1) early re-yielding and smooth elastic-plastic transition in the re-yielding stages, 2) work-hardening stagnation, 3) permanent softening, and 4) decaying loading/unloading modulus
(Hasegawa, Yakou et al. 1975, Dieter and Bacon 1988, Yoshida and Uemori 2002). One of the prominent material models that can predict the Bauschinger effect is the Yoshida and Uemori’s (Y-U) model (Yoshida and Uemori 2002). This model has been evaluated to well predict the springback of metal sheets compared to the actual stamping results
(Yoshida and Uemori 2003, Ghaei, Green et al. 2015a, 2015b). More details about the Y-
U model will be presented in Chapter 3.
Like other commonly used material models in stamping simulation, the Y-U model is calibrated based on only the tension-initiate/-based stress-strain relationships. This process implicitly assumes that 1) the material deformation characteristics under the
94 tension and compression loading conditions are symmetric and/or 2) the metal sheets experience only the tension-initiate/-based loading conditions during the entire stamping process. However, the experimental results inconsistent with the first assumption have been frequently reported in the literature. Sometimes termed as the ‘anisosensitive’ (Życzkowski
1981) or the strength differential (S-D) effect (Singh, Padmanabhan et al. 2000, Callister and Rethwisch 2009), the tension-compression (T-C) asymmetry was mostly researched on hexagonal close-packed (HCP) metals due to their significantly different T-C deformation mechanisms: directional slip in tension but twinning in compression (Hosford
1966, Kelley and Hosford 1968, Teodosiu and Hu 1998, Cazacu, Plunkett et al. 2006). In recent years, especially when more and more new steel grades are developed, their pronounced T-C asymmetry, different from the conventional steel grades, commences to attract more research interests. For example, Singh, Padmanabhan et al. (2000) investigated the correlation between the T-C asymmetry and the steel microstructure and concluded that the higher volume fraction of untempered martensite, the more pronounced T-C asymmetry can be exhibited. Moreover, Kuwabara, Saito et al. (2009) investigated the T-C asymmetry in 304 stainless steels based on the comparison of the cyclic loading results between the tension followed by compression (tension-initiate) and compression followed by tension
(compression-initiate) tests. They also reported the effect of anisotropy on the T-C asymmetry and summarized three possible principles of the T-C asymmetry: 1) twinning
(Kocks and Westlake 1967), 2) hydrostatic pressure dependent plastic flow (Lowden and
Hutchinson 1975), and 3) directionality and rearrangement of dislocation structures
(Barlat, Duarte et al. 2003, Yapici, Beyerlein et al. 2007). Last but not the least, to involve
95 the T-C asymmetry in the simulation, Verma, Kuwabara et al. (2011) modified the Hill-48 yield function so it can reflect the T-C asymmetry of a high-strength steel and then Noma and Kuwabara (2014) further enhanced this model based on the testing data of DP780.
As for the second assumption, it can be directly overthrown by the FE simulation results. As seen in Figure 56, the upper and lower layers of the metal sheet are undertaking different tension-compression loading conditions when the sheet flows through a die radius
(Case I) or a draw-bead (Case II). For Case I, the upper layer is only in tension while the lower layer is loaded successively by tension, compression, and tension. Similarly, for Case
II, the upper layer undertakes successively compression, tension, compression, and tension; meanwhile, the lower layer undertakes successively tension, compression, and tension.
Even though in either case only local areas are under the complex cyclic loading, the global dimensions change significantly in the springback driven by these local areas.
Figure 56: Reversal tension-compression loading (red for the tension stress) during passing over a tool radius (left) or through a draw-bead (right)
96 Therefore, no matter from the standpoints of either the material itself or the loading conditions, the T-C asymmetry needs to be emphasized together with the Bauschinger effect in both the model calibration and the simulation inputs. This section will start from experimentally evaluating the compression properties of QP980, and then compare with a series of DP (ferrite and martensite) steels, including DP590, DP780, DP980, and DP1180, with a ferritic mild steel (CR4) and a martensitic steel (PH1500) to explore any microstructure influences on the Bauschinger effect and the T-C asymmetry.
2.4.1 Experimental Setups
A customized setup was built for the in-plane cyclic loading experiments. As seen in Figure 57, the whole system was also centered on the electro-mechanical INSTRON
5985 universal load frame, fitted with the 250 kN load cell. Two grips installed on the load frame can mount a specimen via four bolts. Each specimen was machined to 20 mm by 10 mm gauge length and width, which referred to the optimal anti-buckling specimen design suggestions in Boger, Wagoner et al. (2005). Also, based on such a gauge geometry, the experimental uniaxial tension stress-strain curves are very close those based on ASTM E8 specimens, as seen in Figure 58. The thickness side of each specimen was patterned with random black-white speckles and turned towards the DIC system, which can not only in- situ measure the strain but also real-time control the load cell to change loading directions according to the pre-settings. The highlight feature of this design is on the anti-buckling mechanisms: to prevent the occurrence of thickness buckling during compression, the left pneumatic cylinder provides a side force to press the mounted specimen tightly between two plates to the right load cell, which is used to verify the side force magnitude and
97 provide in-situ side force data for friction force evaluation; the whole frames can automatically self-center itself via a mechanical mechanism on the bottom to fit various materials of different thickness as well as thickness change of specimens during testing.
Figure 57: Customized in-plane cyclic loading experimental setup and specimen geometry. The setup can realize self-alignment, in-situ strain-controlled testing, and in-situ side-force controlling and recording
Figure 58: Uniaxial tension stress-strain curves of DP980 are not significantly affected by the side pressure from the anti-buckling mechanisms
98
In addition, to reduce errors caused by friction forces induced by the side anti- buckling force (~5 kN, suggested by Boger, Wagoner et al. (2005) and verified experimentally), two layers of Teflon (0.1 mm thickness) were attached on the surfaces of each anti-buckling block as well as some lubricant sprayed on them. The side anti-buckling force may also result in extra biaxial stress in the longitudinal direction, particularly for comparatively low strength materials. Therefore, a correction equation (Boger, Wagoner et al. 2005, Knoerr, Sever et al. 2013) is used to compensate the total errors in the stress- strain curves. As shown in Figure 59, the red stress-strain curve was based on a uniaxial tension testing with the side force, which reached a higher stress level than the blue curve that was based on a similar testing but without the side force. By deducting the friction and biaxial stress, a corrected curve (in green) was shifted from the red curve and matched the blue one, which indicates the correction of experimental stress.
Figure 59: Flow stress curve correction via compensating the friction and the biaxial stress induced by the side force
99 Furthermore, in this research work, such experimental setups are used to conduct three types of loading conditions: 1) monotonic loading (tension or compression), 2) tension-compression cyclic loading, and 3) multi-cycle loading-unloading. To investigate the T-C asymmetry, these three types of testing can be conducted separately as tension- initiate and compression-initiate loading types. Last but not the least, in this work, all the specimens were tested at an initial strain rate of 0.002/s at ambient temperature. Therefore, the effects of anisotropy, strain rate, and temperature are not discussed here.
2.4.2 Compression Properties of QP980
The first step in this section is to characterize the compression properties of QP980.
Similar to the equi-biaxial tension properties in Section 2.2, the yield stress and the
Lankford parameters (R-values) of QP980 are also different in uniaxial compression from those in uniaxial tension. As seen in Figure 60, the compression stress-strain curves along and transverse to the rolling direction can reach higher stress levels than the tension stress- strain curves. Using the same method in Section 2.1.1, the yield stress of QP980 under uniaxial compression can be derived based on the stress-strain curves to be 865.06 MPa and 865.43 MPa, respectively along and transverse to the rolling direction.
100
Figure 60: Comparison of true stress-strain curves of QP980 in uniaxial tension and compression
Similarly, by fitting the ɛ2-ɛ3 curves in Figure 61, the derived R-values are 1.011 and 1.098 respectively along and transverse to the rolling direction.
Figure 61: The Lankford parameters (R-values) under uniaxial compression are the slope values of the best-fit lines of the ɛ2-ɛ3 curves
2.4.3 Cyclic Loading Behavior
101 To follow up, Figure 62 shows multiple stress-strain curves of QP980 and DP980 based on the monotonic and cyclic loading experiments, including uniaxial tension, compression, and three cyclic loading conditions. Both of the materials exhibit some typical manifestations of the Bauschinger effect. First of all, comparing with the circled area 1, the circle area 2 shows apparent early re-yielding and smooth elastic-to-plastic transition. From the microstructure point of view, as deformation occurs, dislocations increase and start to slip gradually. Around some obstacles (for example, grain boundaries, dislocation forests, or any other defects), dislocations can easily accumulate as clusters, which generate microscopic back stress and thus bounce dislocations moving in the reverse direction. The re-yield strength therefore becomes lower (Banabic 2010). However, both of the AHSSs do not show any work hardening stagnation in the circled area 2 nor obvious permanent softening in the circled area 3 as observed by Yoshida and Uemori (2002),
Banabic (2010). A very possible reason is due to the higher strength in compression than in tension. Such a difference can be seen in Figure 62 that the compression stress-strain curve (flipped to positive) (in green) is higher than the tension curve (in blue). This is understandable from the microscopic standpoint since breaking atomic bonds (tension) is definitely easier than pushing close atoms (compression) (Callister and Rethwisch 2009).
More discussions about the T-C asymmetry will be included in Section 2.4.5.
102 (a)
(b) Figure 62: Cyclic loading responses of (a) QP980 and (b) DP980
On the other hand, the differences between the two 980 MPa steels are also obvious.
First of all, in the circled area 1, the initial yield stress in the compression stress-strain curve of DP980 is lower than in its tension stress-strain curve; the compression flow stress exceeds the tension flow stress at ~0.01 true strain. Yet in QP980 the compression stress- strain curve is always higher than the compression curve after yielding. In addition, in the circled area 3, the flow stress of QP980 does not reach saturation in the higher strain level
103 cycle, which indicates the continuous isotropic hardening characteristics of this material.
In contrast, at the same area, the flow stress of DP980 is consistent in both the low and high strain cycles, which reveals the dominant kinematic hardening in this steel. Such a tendency was also reported in Stoudt, Levine et al. (2016).
2.4.4 Loading-Unloading Behavior
The cyclic loading tests can reveal stress evolution in complex loading conditions during stamping, while after the stamping, residual stress in the stamped sheets releases out partially and the remained forms a new stress balance. To predict the consequent deformation, namely the springback, it is of critical importance to understand the stress- strain constitutive relation during the stress release. Conventionally, it is believed that such a relation is linear, with its slope equivalent to the Young’ modulus (Keeler and Kimchi
2015). However, Lems (1963) first pointed out that this constitutive relation was actually a non-linear elastic-plastic type, and its slope value, termed as loading/unloading modulus, presenting material stiffness, was decaying with plastic strain from the Young’s modulus at no strain to a saturated value at large strain. Also, Callister and Rethwisch (2009) explained this phenomenon from the material science point of view that during unloading, the sheared away atoms could move close back to each other but always with a shift.
Yoshida, Uemori et al. (2002) suggested an exponential equation to fit the decaying loading-unloading modulus. Also, Sun and Wagoner (2013) proposed a quasi-plastic- elastic (QPE) model to predict the nonlinear loading-unloading curves. In addition, for steels, Zhu, Huang et al. (2004), Levy, Van Tyne et al. (2006) both observed that the DP and TRIP steels behaved more decayed loading/unloading modulus in large plastic strain.
104 In this work, first of all, loading/unloading stress-strain curves of QP980 and
DP980 were obtained experimentally as shown in Figure 63. To emphasize the T-C asymmetry, the loading/unloading tests were conducted separately via tension-based and compression-based loading means, denoted as T-LU and C-LU.
(a)
(b) Figure 63: T-C asymmetric loading/unloading stress-strain curves of (a) QP980 and (b) DP980
105 An instantaneous loading/unloading modulus is equivalent to the slope value of linear fitting the middle segment of each loading/unloading loop as shown in Figure 64.
After that, a full set of instantaneous loading/unloading moduli is fitted using the exponential equation (17) (Yoshida, Uemori et al. 2002), where E0 is the Young’s modulus, Ea the saturation value, and COE represents the decaying rate. The best-fit curves are shown in Figure 65 and the corresponding exponential equation parameters of QP980 and DP980 at tension and compression conditions are listed in Table 4. Based on the comparison, in general, both of the two 980 MPa AHSSs exhibit different decaying trends in tension and compression. DP980 has the deepest decaying tendency in tension, while
QP980 behaves the slowest decaying trend to a similar saturation level with DP980 in tension. Their decaying curves under compression are very close to each other and both saturate at a higher level than the tension-based curves.
Figure 64: Linear fitting middle segment of a loading/unloading stress-strain curve at each loading/unloading loop and the slope of the fitting line is the instantaneous loading/unloading modulus at that strain level
106
퐸 = 퐸0 − (퐸0 − 퐸푎)(1 − 푒−퐶푂퐸∙휀̅̅̅푝̅) (17)
Figure 65: Exponential fitting the decaying tendency of loading/unloading modulus at all the strain levels
Table 4: Exponential parameters of the T-C asymmetric decaying loading/unloading moduli of QP980 and DP980
Material Loading E0 (GPa) Ea (GPa) COE
Tension 200.04 176.55 28.04 QP980 Compression 199.06 179.88 143.27 Tension 199.16 176.22 68.17 DP980 Compression 193.83 181.31 141.32
2.4.5 T-C Asymmetry Affected by Multi-Phase Microstructure
The T-C asymmetry of QP980 and DP980 has been compared in Section 2.4.3. In this section, further investigation on the effect of multi-phase microstructure of AHSSs continues. The target materials in this section include four DP steels of increasing
107 martensite fraction and strength grade (Figure 2), as well as a ferritic mild steel (CR4) and a martensitic steel (PH1500). The experimental results of the four DP steels are shown in
Figure 66. In the monotonic testing results, the T-C asymmetry of DP steels manifests as different yielding and hardening behavior. For all the DP steels, the results show that the initial yield strength (elastic limit) under compression is always lower than that under tension; nevertheless, due to the higher strain hardening rate under compression, after about
0.01 strain, the strength under compression exceeds that under tension. Similar results on
DP780 were also reported in Noma and Kuwabara (2014).
(a)
108 (b)
(c)
109 (d) Figure 66: The monotonic loading and cyclic loading testing results of (a) DP590, (b) DP780, (c) DP980, and (d) DP1180
However, such a trend cannot be found in the testing results of the reference single- phase steels, neither CR4 (fully ferritic) nor PH1500 (fully martensitic), as seen in Figure
67. Therefore, it is unlikely that this T-C asymmetry manifestation is due to a certain deformation mechanism of a steel phase. More microstructure examinations are needed to help explain this phenomenon.
110 Figure 67: Asymmetric yielding and hardening of the tested steels
Moreover, for CR4 (0% martensite) in Figure 68(a), DP590 (19.7% martensite), and DP780 (54.6% martensite), the strain hardening continuously increases with the loading cycles. For example, at 0.02 strain in either tension or compression, the strength level in the second loading cycle (between -0.05 and 0.05 strain) is higher than that in the first cycle (between -0.025 and 0.025 strain). In contrast, for DP980 (62.2% martensite),
DP1180 (70.9% martensite), and PH1500 (100% martensite) in Figure 68(b), such a cyclic hardening trend is hardly noticeable. This indicates that under the cyclic loading conditions, with the increase of martensite, the dominant hardening mechanism gradually shifts from isotropic to kinematic hardening. Nevertheless, except for CR4, other steels do not show obvious cyclic T-C asymmetry after the first reversal of loading direction, since the tension-initiate and compression-initiate stress-strain curves tend to overlap to each other as shown in Figure 67 and Figure 68(b). This implies that the Bauschinger effect may be able to compensate the T-C asymmetry, but this deduction as well as why CR4 is an exception needs further microstructure examinations as verification and explanation evidences.
111 (a)
(b) Figure 68: The cyclic loading testing results of (a) CR4 and (b) PH1500
2.5 Summary
To sum up, the deformation characteristics under various loading conditions, including uniaxial tension, compression, equi-biaxial tension, and cyclic loading conditions, and also the failure behavior of QP980, as well as some other comparative materials, have been categorized via experimentation in this chapter. Although some of the experimental techniques have been implemented or even standardized for many years, such
112 a comprehensive experimental investigation on the target material is still of vital importance, not only because of the improvements and modifications of these testing, but also for the adjustments for particularly QP980 and incoming 3GAHSSs when they exhibit unique behavior that cannot be thoroughly characterized using the current experimental infrastructure. To further verify this point of view, the extracted parameters will be implemented and evaluated in the following chapters of modeling and simulation.
113 CHAPTER 3: CONSTITUTIVE MODEL
Nowadays in the industry, FE simulation is commonly implemented in sheet metal forming processes, not only before actual try-outs, but also during entire stamping tooling design and morphing procedure. FE simulation is expected to be predictive enough to foresee possible failure and simulate sheet metal deformation during and after stamping.
The determinant of the FE prediction precision is material constitutive model. In sheet metal forming, it represents continuum relation between strain and stress. There are various types of constitutive models, but a phenomenological model is in most cases the chosen one, mainly due to its straightforward applicability and comparative simplicity. Therefore, in this chapter, such a model for QP980 will be proposed ultimately, based on the experimental data in Chapter 2, and will be verified via FE simulations compared to actual results in Chapter 4. Essentially, a phenomenological model consists of three components:
1) a yield function, 2) a flow rule, and 3) a hardening law. The yield criterion determines not only elastic deformation limit, but also stress state after yielding occurs, the flow rule represents relationship between plastic strain increment and stress, and the hardening rule describes how the material is strain hardened with plastic strain increases. In the following three sections, these three components are briefly reviewed and calibrated for QP980 in detail.
3.1 Yield Criterion and Function
Yield criterion theory in metallic materials is actual a branch of generalized strength theories developed from the late eighteenth century. Yu (2002) summarized three mainstream categories of strength theories, from narrow to wide, including single shear
114 stress series (SSS) (such as the Tresca criterion (Tresca 1864)), octahedral shear stress series (OSS) (such as the von Mises criterion (Mises 1928)), and twin shear stress series
(TSS) series (such as the Hill1948 criterion (Hill 1948)). Before 1990s, most of the common yield criteria, such as the Hershey (1954), the Hosford (1972), the Yld1989
(Barlat and Lian 1989) took the SSS criteria as the lower bound and the OSS criteria as the upper bound, while later on, more yield criteria were proposed lying between the SSS and the TSS criteria to fit more complex yielding behavior (such as the anisotropy and T-C asymmetry) of new metallic materials (Yu 2002, Yu, Ma et al. 2005).
To specify, the development of the yield criteria has passed through several critical stages and gradually evolved towards divergent directions. As seen in Figure 69, first of all, the initial isotropic criteria, such as the Tresca criterion represented by linear functions and the von Mises criterion represented by the quadratic function of the second deviatoric stress invariant, was soon replaced by some isotropic criteria with non-quadratic formulations (Hershey 1954). However, with more anisotropic effects observed in experiments, new anisotropic yield criteria were proposed. In 1948, Hill simply added different parameters in the von Mises quadratic yield function and proposed the well- known Hill1948 model (Hill 1948) as Equation (18) (plane stress version). These new parameters can be derived based on three Lankford parameters (r-values) or three yield stress in different rolling directions. The former version is called the Hill1948-3r model as
Equation (19), and the latter one is the Hill1948-3σ model.
2 2 2 2 푓푦 = 퐴1𝜎푥 − 퐴2𝜎푥𝜎푦 + 퐴3𝜎푦 + 퐴4휏푥푦 = 𝜎̅ (18)
2푟0 푟0(1 + 푟90) (푟0 + 푟90)(1 + 2푟45) 퐴1 = 1, 퐴2 = , 퐴3 = , 퐴4 = (19) 1 + 푟0 푟90(1 + 푟0) 푟0(1 + 푟0)
115
2 2 2 𝜎0 2 𝜎0 𝜎0 2𝜎0 2 𝜎0 퐴1 = 1, 퐴2 = 1 + ( ) − ( ) , 퐴3 = ( ) , 퐴4 = ( ) − ( ) (20) 𝜎90 𝜎푏 𝜎90 𝜎45 𝜎푏
Figure 69: Family tree of some common yield criteria in sheet metal modeling
The Hill1948 model (especially the 3r version) is still being commonly used in the industry, mainly due to its user-friendly function and capability of representing the basic anisotropy. The parameters in Hill1948-3r need only uniaxial tension tests along three directions. However, this yield criterion cannot represent materials with severe anisotropy.
Therefore, there were several improvements based on it, such as Hill (1979), Hill (1990),
Hill (1993), Lin and Ding (1996), Leacock (2006), which, however, also meant more parameters and/or more complex formulations, and thereby lost the user-friendly advantage.
Another mutation branch of the Hill1948 yield criterion based on high order polynomial equations was started by Gotoh (1977a), (1977b). The basic idea is to use a
116 fourth order polynomial function to represent the complex anisotropic yielding. Moreover, some fourth, sixth, and even eighth order polynomial formulations (Pan and Wang 2006,
Hu 2007, Soare, Yoon et al. 2007, Yoshida, Tamura et al. 2011) followed up. For example,
Yoshida (2011) proposed a sixth order polynomial function as Equation (21) (plane stress version) with sixteen parameters. These functions can represent complex yield behavior well, but are also limitedly used in the industry due to their numerical complexity.
6 5 4 2 3 3 2 4 푓푦 = 퐶1𝜎푥 − 3퐶2𝜎푥 𝜎푦 + 6퐶3𝜎푥 𝜎푦 − 7퐶4𝜎푥 𝜎푦 + 6퐶5𝜎푥 𝜎푦 5 6 − 3퐶6𝜎푥𝜎푦 + 퐶7𝜎푦 4 3 2 2 3 + 9(퐶8𝜎푥 − 2퐶9𝜎푥 𝜎푦 + 3퐶10𝜎푥 𝜎푦 − 2퐶11𝜎푥𝜎푦 (21) 4 2 2 2 4 + 퐶12𝜎푦 )휏푥푦 + 27(퐶13𝜎푥 − 퐶14𝜎푥𝜎푦 + 퐶15𝜎푦 )휏푥푦 6 6 + 27퐶16휏푥푦 = 𝜎̅
Also based on the von Mises criterion, Hershey introduced a non-quadratic yield function (Hershey 1954), and later was extended to be able to represent anisotropy by
Hosford (1972). The main advantage of Hosford’s model is fitting the function order exponent based on crystal plasticity theory (Logan and Hosford 1980, Hosford and Caddell
2011). Furthermore, Balart and his coauthors proposed a series of ‘Yld’ criteria via substituting the stress components in the Hosford’s model by different stress transformation deviators (Barlat, Brem et al. 2003, Barlat, Aretz et al. 2005). The representative ones are the Yld1989 as Equation (22) and the Yld2000 as Equation (23) for plane stress deformation. A non-monotonic loading deformation version of the Yld2000 was also developed as the Yld2004 (Barlat, Duarte et al. 2003).
1 푓 = (|퐾 + 퐾 |푎 + |퐾 − 퐾 |푎 + |2퐾 |푎) = 𝜎̅푎 (22) 푦 2 1 2 1 2 2
1 푓 = (|푋′ + 푋′ |푎 + |푋′′ + 2푋′′|푎 + |2푋′′ + 푋′′|푎) = 𝜎̅푎 (23) 푦 2 1 2 1 2 1 2
117
In addition, Banabic and his coauthors adopted the classical yield function development method as adding weight coefficients and extended the Herskey’s formulation to a series of Banabic-Balan-Comsa (BBC) yield criteria (Banabic, Kuwabara et al. 2003, Banabic, Kuwabara et al. 2004, Comsa and Banabic 2007). Mathematically, the BBC2005 is equivalent to the Aretz model (Aretz 2004) and the Yld2000 (Barlat, Brem et al. 2003). Also based on the Hersey’s formulation, Vegter and his coauthors first proposed an isotropic yield criterion but later challenged the usual way of using a function to define a yield criterion, instead, they proposed to directly use the Bezier curve to interpolate the yield locus based on the experimental results (Vegter and Van Den
Boogaard 2006). This proposal, however, was also not widely adopted due to its unfriendly computation process as well as the mathematical difficulty to be extended from a 2D version with three stress components for plane stress states to a 3D version with six stress components for non-monotonic loading conditions.
In the new century, the strong T-C asymmetry of newly developed materials became another critical factor influencing FE prediction as introduced in Section 2.4.
Correspondingly, Cazacu and her coauthors rediscovered Drucker’s isotropic yield criterion and developed it to be orthotropic, namely the series of Cazacu-Plunkett-Barlat
(CPB) criteria (Cazacu and Barlat 2004, Cazacu, Plunkett et al. 2006, Plunkett, Cazacu et al. 2008). A representative in this series is the CPB06 model as Equation (24), where the parameter k includes the asymmetric T-C yield stress ratio ((σT/ σC)). Like in the Yld2000, in the CPB2006 the Σi are also stress transformation deviators and need to be calibrated based uniaxial tension/compression, shear, and biaxial tension/compression results. The
118 3D version of the CPB06 was also published as the CPB06-ex2, -ex3, and -ex4 (Plunkett,
Cazacu et al. 2008).
푎 푎 푎 푎 푓푦 = ||훴1| − 푘훴1| + ||훴2| − 푘훴2| + ||훴3| − 푘훴3| = 𝜎̅
𝜎 1 − ℎ( 푇) (24) 𝜎퐶 푤ℎ푒푟푒 푘 = 𝜎 1 + ℎ( 푇) 𝜎퐶
In addition, Verma, Kuwabara et al. (2011) modified the Hill1948-3σ by incorporating additional linear terms to be an asymmetric yield function, namely the
Hill1948-asymmetric, as Equation (25), where all the parameters contain asymmetric T-C yield stress ratio (σT/ σC).
2 2 2 0.5 푓푦 = 푎(𝜎푥 − 퐴𝜎푥𝜎푦 + 퐵𝜎푦 + 퐶휏푥푦) + (푘1𝜎푥 + 푘2𝜎푦) = 𝜎̅
푐 푇 푇 푐 푇 𝜎0 − 𝜎0 𝜎0 (𝜎90 − 𝜎90) 푤ℎ푒푟푒 푘1 = 푐 , 푘2 = 푐 푇 , 2𝜎0 2𝜎90𝜎90 푐 푇 푐 푇 푐 푇 2 𝜎0 + 𝜎0 𝜎0 𝜎0 (𝜎90 + 𝜎90) 푎 = 푐 , 퐵 = { 푐 푇 푐 푇 } (25) 2𝜎0 𝜎90𝜎90(𝜎0 + 𝜎0 ) 푇 2 1 𝜎0 퐴 = 1 + 퐵 − [ { − (푘1 + 푘2)}] 푎 𝜎푏 푇 2 1 2𝜎0 퐶 = [ { − (푘1 + 푘2)}] − (1 − 퐴 + 퐵) 푎 𝜎45
Therefore, in this research work, the yield stress and r-values of QP980 in Error! R eference source not found. are used to calibrate eight selected yield functions, and then the comparison of these yield functions back with the experimental results can reveal which one can better describe QP980 based on three comparison criteria: 1) the yield stress in different rolling directions, 2) the anisotropy, and 3) the T-C asymmetry. The eight yield functions include the von Mises yield function (as the benchmark), the Hill48-3r (19), the
119 Hill1948-3σ (20), the Hill1948-asymmetric (25), the Yld1989 (22), the Yld2000 (23), the
CPB2006 (24), and the Yoshida2011 (21), almost covering all the development branches in Figure 69. For parameter calibration, their required input experimental data are summarized in Figure 70.
Both of the Yld2000 and the CPB2006 need extra calibration for anisotropy coefficients in the stress linear transformation deviators. The coefficients in the Yld2000 are calculated based on the Newton-Raphson algorithm (Table 5). The exponent a is set to
8 due to the FCC structure austenite in QP980. In the CPB2006, the coefficients are calibrated based on the constrained nonlinear minimum optimization using the sequential quadratic programming (SQP) algorithm in MATLAB (Table 6).
Table 5: Calibrated anisotropy coefficients of the Yld2000 yield function (a=8)
α1 α2 α3 α4 α5 α6 α7 α8 YS 0.9081 1.0826 0.8638 1.0037 0.9862 0.8632 0.9989 1.1168
Table 6: Calibrated coefficients of the CPB2006 yield function (a=2)
C11 C12 C13 C22 C23 C33 C66 YS 0.9434 2.0036 1.8332 0.7322 1.7659 0.4529 1.2030
120
Figure 70: Summary of experimental results as inputs for yield Functions
Figure 71 shows comparison of the yield surfaces at ɛ=0.005 (initial yielding) with the experimental results (black circles) in the normalized stress coordinate system.
Different yield functions need different experimental data for calibration, and thus there are always some discrepancy between the yield surface with the unused experimental results. For example, the Yld2000 function is calibrated based on four r-values and four yield stress in 0°/45°/90°/biaxial orientations to the rolling direction only under tension, so apparently its yield surface cannot fit the two points of the experimental results under compression in Figure 71(b). Although the CPB2006 is calibrated using all the data in
Figure 70, its calibration algorithm based on constrained optimization cannot guarantee
100% fitting the yield surface with the experimental results. In this case, the yield functions that do not include the biaxial stress in their parameters show some discrepancy in the biaxial tension direction. These yield criteria include the von Mises, the Hill48-3r, and the
Yld1989. Similarly, only the Hill1948-asymmetric and the CPB2006 include the T-C
121 asymmetry parameters, so only their yield surfaces fit the corresponding experimental results.
(a)
(b) Figure 71: Comparison of the yield surfaces at ɛ=0.005 (initial yielding) with the experimental results in the normalized stress coordinate system; the selected yield functions including (a) the von Mises, the Hill1948-3r, the Hill1948-3σ, the Hill1948- asymmetric, (b) the Yld1989, the Yld2000, the Yoshida2011, and the CPB2006
As a further evaluation of the applicability of the selected yield functions to the target material, Figure 72 shows comparison of (a) the yield stress and (b) the r-values in
122 different rolling directions (from 0° to 90° and 15° increment) prescribed by the selected yield functions with the experimental results in Figure 70. Unfortunately, the Yoshida 2011 is excluded in this comparison due to lack of necessary data in 22.5° and 67.5° in the derivation of its r-value and yield stress curves.
The two diagrams in Figure 72 further indicate the effects on the prescribed results from how many and what types of experimental results are used in function parameters.
For example, the Hill1948-3r function has three r-values (in the 0°/45°/90° orientations) in its anisotropy parameters, so it can approximately fit the experimental r-value tendency but not the experimental yield stress curve. In contrast, the Hill1948-3σ and the Hill1948- asymmetric functions can fit the yield stress but not the r-value tendency. The only yield function that can fit both is the Yld2000.
(a)
123 (b) Figure 72: Comparison of (a) the yield stress and (b) the r-values in different rolling directions prescribed by the selected yield functions with the experimental results in Table 2 (the Yoshida 2011 is excluded due to lack of necessary data in the derivation)
Therefore, based on the three comparison criteria, none of these selected yield functions can completely represent the material behavior of QP980. However, since the
Hill1948-asymmetry can fit both the T-C asymmetry and the yield stress of different orientations, as well as the very limited anisotropy of QP980, it will represent QP980 in the constitutive model and used in the FE-aided verification in Chapter 4.
3.2 Flow Rule
After yielding, materials are continuously hardening during plastic deformation until failure. In the corresponding model component, how to specify the continuum relation between the stress and the plastic strain increment is a necessary step. For the increasing stress, if the r-values do not change, the stress components keep a consistent relationship on an instantaneous subsequent yield surface, still represented by the original yield function but with different size and center. As for the increasing plastic strain, in most cases it is
124 formulated based on the normality principle (Drucker 1951), which is, as shown in Figure
73(a), the plastic strain increment is normal to the instantaneous subsequent yield surface and relative to the instantaneous r-value (Hosford and Caddell 2011) in Equation (26). In this case, the plastic potential function (fp) defining the plastic strain increment is identical with the subsequent yield stress function (fy) defining the continuum relations of the stress components, and thereby the plastic strain increment is ‘associated’ with the increasing stress. Therefore, this is called ‘associated flow rule.’ Even though the normality principle does not have a valid theoretical background (Hill 1948), the associated flow rule is not only mathematically simple, but also satisfies Drucker’s postulate (Drucker 1951) of material deformation stability. Based on the associated flow rule and the work conjugate formulation, the plastic strain increment, or approximately the plastic strain rate, can be derived to an expression of the effective strain and stress as well as the stress tensor in
Equation (27).
휀̇ 푑휀 휕𝜎 푑푟 2 ≈ 2 = − 1 = − (26) 휀1̇ 푑휀1 휕𝜎2 푑푟 + 1
휕푓푝 휕푓푦 푫풑 = 휆̇ = 휆̇ ; 푊̇ 푝 = 𝜎̅휀̇̅ = 흈: 푫풑 휕흈 휕흈 휕푓푦 휕𝜎̅ 풑 ̇ ̇ 흈: 푫 흈: 휆 흈: 휆 (27) ⟹ 휀̇̅ = = 휕흈 = 휕흈 ≈ 휆̇ 𝜎̅ 𝜎̅ 𝜎̅ 휕𝜎̅ ⟹ 푫풑 = 휀̇̅ 휕흈
However, for materials behaving very strong anisotropy, the limited number of anisotropy coefficients in a classical yield function is insufficient to precisely represent both the anisotropy and the yield stress. Therefore, if fp ≠ fy, the number of anisotropic coefficients is doubled and more likely to represent the material deformation. In this case,
125 the plastic strain increment is no longer associated with the stress, and thereby it is called
‘non-associated flow rule.’ In the non-associated flow rule, since the plastic potential function is different from the instantaneous subsequent yield function in principle, the normality principle is only valid for the plastic potential function as shown in Figure 73(b).
In addition, two sets of anisotropic coefficients dedicated for each function need to be calibrated separately.
Figure 73: Schematics of associated and non-associated flow rules
In this research work, the non-associated flow rule is actually not necessary for
QP980 due to the limited anisotropy shown in Table 2. So it is only for references in this section to calculate the plastic potential functions based on the CPB2006, the Hill1948, and the Hill1948-asymmetric. In the non-associated flow rule, the yield surfaces are plotted based on only the yield stress values from the experiments, while the plastic potential surfaces based on only the r-values from the experiments.
126 Figure 74 shows at ɛ=0.005 the tri-component yield surfaces (blue) and plastic potential surfaces (red) plotted based on (a) the CPB2006, (b) the Hill1948, and (c) the
Hill1948-asymmetric functions. It indicates that both the CPB2006 and the Hill1948 functions generate very close yield and plastic potential surfaces, while the Hill1948- asymmetry function prescribes very discrepant two surfaces in the third quadrant of the diagram.
(a)
(b)
127 (c) Figure 74: Comparison of tri-component yield surface (blue) and plastic potential surface (red) of QP980 prescribed by (a) the CPB2006 function, (b) the Hill1948 function, (c) the Hill1948-asymmetric function with the experimental results
3.3 Hardening Law
The last essential component of a constitutive model is a strain hardening law. As introduced in Section 3.2, based on the classical plasticity theory, when a material is deforming plastically, the stress components are flowing on a certain subsequent yield surface prescribed by the initial yield function but with a growing size and/or a shifted center as shown in Figure 75. Such a subsequent yield surface expansion and translation are termed as isotropic hardening and kinematic hardening, respectively. They can also be expressed as Equation (28). In a uniaxial tension and then compression case, as shown in
Figure 75, the re-yield stress based on an isotropic hardening law is equivalent to the stress in value when the loading direction is reversed, while if based on a kinematic hardening law, the re-yield stress is less in value. In practical analysis, especially when the re-yield stress needs to be considered, typically a mixed isotropic-kinematic hardening law is used.
128 In some alternative hardening models, the yield surface can expand and distort (Barlat,
Gracio et al. 2011), rotate and distort (Francois 2001), translate and rotate (Choi, Han et al.
2006, Choi, Han et al. 2006), or even all the four types of change (Suprun 2006). In this research work, the model is still based on a classical isotropic-kinematic hardening law.
Figure 75: Isotropic and kinematic hardening and some representative hardening laws in the literature
Isotropic hardening: 푓 = ∅(흈) − (푌 + 푅) = 0 Kinematic hardening: 푓 = ∅(흈 − 휶) − 푌 = 0 (28) Isotropic-kinematic hardening: 푓 = ∅(흈 − 휶) − (푌 + 푅) = 0
For the classical isotropic hardening laws, except for Odqvist’s linear law (1933), two basic forms are Hollomon’s equation and Voce’s equation as shown in Figure 75, and others are mutations or combined forms of these two. Larour (2010) investigated the extrapolation ranges of the common isotropic hardening laws, and ranked them from high to low as Ludwik (1909), Swift (1952), Hollomon (1945), Ghosh (1977), Hockett-Sherby
129 (1975), and Voce (1948). Typically, to guarantee a wide representation range of hardening, the Swift’s law and the Voce’s law are combined with a weight ratio for adjustment as
Equation (29).
푅 = 휈푅푆푤푖푓푡 + (1 − 휈)푅푉표푐푒 (29)
As for the classical kinematic hardening laws, the fundamental theory was proposed by Mises (1928), while Prager (1957) first proposed a linear kinematic hardening law and identified the key parameter as the backstress α. The Prager’s law was found unable to represent the curvature in the elastic-plastic transition segment (circled area 2 in Figure 62), therefore a series of nonlinear kinematic hardening laws were proposed, such as Chaboche
(1986), Ohno and Wang (1993), Frederick and Armstrong (2007). Furthermore, Krieg
(1975), Dafalias and Popov (1976) introduced a virtual ‘bounding surface’ outside the yield surface and indirectly guides the yield surface expanding and moving inside it. In this way, not only the nonlinear transition, but also some large-strain scale cyclic loading behavior, such as permanent softening and hardening saturation, can be well represented. In the new century, Yoshida and Uemori (2002) further developed the two-surface kinematic hardening law to a more comprehensive model, the Yoshida-Uemori (Y-U) model as mentioned in Section 2.4. As illustrated in Figure 76, this Y-U model consists of four parts:
1) a yield surface that can perform only the kinematic hardening during the plastic deformation, 2) a bounding surface that can perform both the isotropic and kinematic hardening, 3) a stagnation surface that can also perform the mixed hardening rules, and 4) decaying loading-unloading modulus (Yoshida and Uemori 2002, 2003). The parameters that need calibrations are described in Table 7.
130
Figure 76: The Yoshida-Uemori model (Yoshida and Uemori 2002, 2003) and its four parts
Table 7: Input parameters that need to be calibrated in the Y-U model
Notations Descriptions
Y Initial size of the yield surface
B Initial size of the bounding surface
SC Kinematic hardening coefficient of the yield surface
b Kinematic hardening coefficient of the bounding surface Kinematic hardening coefficient of the bounding surface and isotropic K hardening coefficient of the yield surface if use the Voce’s law RSAT Isotropic hardening coefficient of the bounding surface
h Coefficient of the stagnation surface Isotropic hardening coefficient of the bounding surface if use the C1 Swift’s law Isotropic hardening coefficient of the bounding surface if use the C2 Swift’s law Isotropic hardening coefficient of the bounding surface if use the К Swift’s law
131 Ea Saturation value of the decaying loading/unloading modulus
COE Coefficient in the decaying loading/unloading modulus equation
This Y-U model has been evaluated to be very predictive in practical case studies
(Yoshida and Uemori 2003, Ghaei, Green et al. 2015a, 2015b). However, since this model implicitly assumes four conditions: 1) isotropic yield function (von Mises), 2) associated flow rule, 3) T-C asymmetry, and 4) no isotropic hardening of the yield surface (Yoshida,
Uemori et al. 2002), its prediction potential can still be further explored. Some researchers have already noticed this and started to propose the new modifications. For example, Kano,
Hiramoto et al. (2013) investigated the effects of using anisotropic yield functions in the
Y-U model. Also, Uemori, Sumikawa et al. (2013) realized that during reversal loading, the yield function could expand (isotropic hardening), so they added an extra parameter to the original Y-U model but did not specify the physical meaning of this parameter.
Furthermore, Ghaei and Taherizadeh (2015) integrated the non-associated flow rule into the Y-U model and successfully predicted the springback of a severely anisotropic aluminum.
In this research work, an improvement of the original Y-U model is dedicated for not only the target material QP980, but also any TRIP-like steels. Due to occurrence of the martensitic transformation as illustrated in Figure 7, the ~10% austenite in QP980 gradually decreases to ~2% during a uniaxial tension test according to the synchrotron examination results in Figure 77. Since martensite possesses a much higher strength than austenite as shown in Table 1, the re-yield stress of QP980 (or any TRIP-like steels) should increase, rather than keeping as a constant number. Such a yield stress increasing rate can
132 be calculated in Equation (30), dependent on the austenite volume fraction decreasing rate and the flow stress of martensite (M) and austenite (A), which can be expressed in Swift’s law as Equation (31). The flow stress parameters of austenite and martensite are based on the data in Table 8 published by Tsuchida and Tomota (2000). As for the plastic strain of each phase, it can be estimated based on a ratio of their uniform elongation in Table 1, pF:pM:pA=0.3:0.6:0.08.
Figure 77: Retained austenite volume fraction (RAVF) of QP980 decreasing curve during uniaxial tension ̇ 푌̇ (휀)̅ = 푓퐴 ∙ (푌푀 − 푌퐴) (30)
푝 퐶3 푌푖 = 퐶1푖(퐶2푖 + 휀 푖) , 𝑖 = 푀, 퐴 (31)
Table 8: Flow stress parameters of austenite and martensite (Tsuchida and Tomota 2000)
Austenite (α-Fe) Martensite (α’-Fe)
C1 C2 C3 C1 C2 C3 880 2E-2 0.39 2500 1E-7 0.29
133 Another unknown in Equation (30) is the austenite volume fraction decrement function (dfA). The point cloud in Figure 77 can only reveal the martensitic transformation tendency in uniaxial tension but not along other strain paths. Radu, Valy et al. (2005) investigated the kinetics of martensitic transformation under different loading conditions and proposed a universal transformation model as Equation (32). Under an isothermal condition, the temperature-dependence parameter g(T) is 1. In addition, Shan, Li et al.
(2008) determined the stress triaxiality (η) dependent function φ(η) based on their experimental results as Equation (33). The stress triaxiality is a ratio of hydrostatic stress
(σm) and von Mises effective stress in Equation (34).
휑(휂) 푓 = 푓 ∙ 푒푥푝 (− ∙ (휀)̅ 푎) (32) 퐴 퐴0 푔(푇)
휑(휂) = 1.45휂 + 2.89 (33)
𝜎 휂 = 푚 (34) 𝜎̅
Therefore, since in uniaxial tension the stress triaxiality is 1/3, the exponent a can be derived based on Figure 77 to be a=0.5 for QP980. Then since the initial retained austenite volume fraction of QP980 is 0.1, the austenite decreasing rate (as well as the yield surface expanding rate) can be expressed in Equation (35).
̇ −0.5 0.5 푓퐴 = −0.05휑 ∙ (휀)̅ 휀̇̅ ∙ 푒푥푝 (−휑 ∙ (휀)̅ ) (35)
3.4 Constitutive Model of QP980
Based on the calibrated yield function, flow rule, and hardening law, the new model for QP980 (or any TRIP-like AHSSs) can be established in this section. As shown in Figure
78, the new model is based on the multi-surface strategy like the original Y-U model.
134 However, the yield function, as well as the bounding surface function and the stagnation surface function, is based on the Hill1948-asymmetric model as in Equation (25).
Moreover, the yield surface also conducts isotropic hardening due to the martensitic transformation expressed in Equation (30).
Figure 78: The multi-surface model for QP980 (or any TRIP-like AHSSs) is similar to the original Y-U model
To be specific, the relative kinematic motion rate of the yield surface with respect to the bounding surface can be expressed in Equation (36), where a and R can be derived in Equation (29) and (37). Similar to the Y-U model, the kinematic hardening rate is still represented as in Equation (38). Then the absolute kinematic hardening rate of the yield surface can be expressed in Equation (39).
푎 푎 휽̇ = 푆퐶( 휹 − √ 휽)휀̇ ̅ (36) 푌(휀)̅ 휃̅
푎 = 퐵 + 푅 − 푌(휀)̅ (37)
푏 휷̇ = 퐾( 휹 − 휷)휀 ̇̅ (38) 푌(휀)̅
135
흈̇ = 휶̇ + 휹̇ = 휷̇ + 휽̇ + 휹̇ (39)
On the other hand, based on the plastic work conjugate and associated flow rule, the plastic strain rate tensor has been derived in Equation (27). The Jaumann’s objective
Cauchy stress rate is thereby expressed in Equation (40).
휕𝜎̅ 흈̇ = 푪: 푫풆 = 푪: (푫 − 푫풑) = 푪: (푫 − 휀̇̅ ) (40) 휕휹
Furthermore, based on the plastic consistency principle, the derivative of the yield function to time is zero. Therefore, with the substitutions of Equations (36)-(40), the effective strain rate can be derived as in Equation (41).
휕푓푦 휕𝜎̅ 휕𝜎̅ 푓̇ = : 휹̇ = : 휹̇ = : (흈̇ − 휷̇ − 휽̇ ) = 0 푦 휕휹 휕휹 휕휹 휕𝜎̅ : 푪: 푫 ⟹ 휀̇̅ = 휕휹 (41) 휑 휕𝜎̅ 휕𝜎̅ 휕𝜎̅ 푏 휕𝜎̅ 푎 푎 푤ℎ푒푟푒 휑 = : 푪: + : 푲 ( 휹 − 휷) + 푆퐶 : ( 휹 − √ 휽) 휕휹 휕휹 휕휹 푌 휕휹 푌 휃̅
Eventually, the objective stress rate can be derived as Equation (42), via substituting Equation (41) to (40).
휕𝜎̅ 휕𝜎̅ 휕𝜎̅ : 푪: 푫 휕𝜎̅ 푪: ⊗ : 푪 흈̇ = 푪: (푫 − 휕휹 : ) = (푪 − 휕휹 휕휹 ) : 푫 = 푪풆풑: 푫 (42) 휑 휕휹 휑
3.5 Summary
In this chapter, the constitutive model of QP980 was proposed based on calibrating a proper yield function, using associated flow rule, and adding martensitic transformation functions to the original Y-U model. The new model should thereby precisely represent the anisotropic deformation, the T-C asymmetric stress, the TRIP effect, and the Bauschinger
136 effect of QP980. Such an anticipation of the new model will be implemented in FE subroutine and verified in a cold stamping case study.
137 CHAPTER 4: MODEL IMPLEMENTATION AND VERIFICATION
In this chapter, the constitutive model of QP980 will be numerically implemented via LS-DYNA user subroutine as customized material cards. Then, in the second section, the customized material cards will be verified via a component level case study.
4.1 Numerical Implementation
In order to implement the constitutive model in LS-DYNA, it is required to incrementally update all variables including stress and internal variables. As shown in
Figure 79, 1) the total strain tensor is first prescribed for an increment, and hence the equivalent plastic strain, plastic strain tensor, internal variables, and stress tensor can be calculated consequently. Obviously, all of these variables need to be assumed before this strain increment. It is first assumed that the total strain increment as fully elastic. If the deformation in the increment is still elastic (fy < Y), the calculated stress is updated. 2)
However, if plastic deformation is detected after an increment (fy ≥ Y) and the calculated stress is beyond the yield surface, the equivalent plastic strain rate as well as other variables needs to be iterated implicitly until the loading point returns back to the yield surface according to the return mapping algorithm. 3) To follow up, all the stress tensors are updated explicitly based on the corresponding stress rate tensors. 4) In addition, if the bounding surface center is calculated to be beyond the stagnation surface, the isotropic hardening rate of the bounding surface is updated to zero and the effective strain rate needs to be re-iterated until the bounding surface center is back to the range of the stagnation surface again. This is namely the semi-implicit algorithm based on the proposals by (Ghaei and Taherizadeh 2015).
138
Figure 79: Schematic of numerical implementation procedure based on the semi- implicit algorithm
4.2 Finite Element Verification
The implemented material cards are verified in a cold stamping case study in this section. Details of the scanned actual panel data and the stamping procedure parameters were provided by an OEM. As an overview, in Figure 80, the stamping consists of five stages: gravity, draw form, first springback, trimming, and secondary springback. The occurrences of twice springback is due to two times of residual stress releasing in the panel after unclamping from the tools. Accordingly, the comparison of the predicted results with the actual are performed twice.
139
Figure 80: A typical procedure of cold stamping an AHSS panel in FE prediction
As shown in Figure 81, there are four material cards implemented in this case study.
MAT37 is based on the Hill1948-r ̅ yield function with tabulated flow stress hardening curve and MAT287_Voce is based on the Hill1948-3r yield function with the original Y-
U model. Both of them are provided by the LS-DYNA distributors. MAT41 is based on the Hill1948-3r yield function with the new model and MAT43 is based on the Hill1948- asymmetric yield function with the new model. Therefore, all the four material cards are based on the Hill1948 yield function, while the compared targets are on the differences of
1) with or without the kinematic hardening (between MAT37 and MAT287), 2) with or without the martensitic transformation component in the hardening model (between
MAT287 and MAT41), and 3) with or without the T-C asymmetric yielding (between
140 MAT41 and MAT43). Correspondingly, these material cards require different material parameters as inputs, with various experiments from simple to complex.
Figure 81: Four material cards implemented in the FE simulations, with their corresponding required material parameters, represented material characteristics, and needed experiments
All the four material cards were implemented in the LS-DYNA user subroutine.
The FE simulations proceeded without changing other input parameters. The prediction results needed to be offset half of the panel thickness from the middle layer to the top layer, and then were overlapped with the laser-scanned actual panels based on the minimum 3-2-
1 alignment according to the provided checking fixture constraint coordinates. The comparison results at the two springback stages are shown in Figure 82 and Figure 83, respectively, with the deviation tolerance ±1 mm in the first comparison and ±0.5 mm in the second comparison. Apparently, in both cases, MAT43 is the most predictive one since its prediction results show the highest within-tolerance percentages with the actual panel.
141 This is due to its most comprehensive capability of representing the anisotropy, the T-C asymmetry, the Bauschinger effect, and the TRIP effect of the target material within these material cards. In contrast, even though MAT37 is still widely implemented in the industry due to its simplicity, its predictions in both cases show the largest out-of-tolerance percentages compared to the actual panels. This again implies the inadequacy of the current simple material characterization and simulation infrastructure in dealing with the new generations of AHSSs.
Figure 82: Compare the FE predicted to the scanned actual panel before trimming
142
Figure 83: Compare the FE predicted to the scanned actual panel after trimming
143 CHAPTER 5: CONCLUSIONS
To sum up, this research work characterized QP980, a representative new grade
AHSS developed in recent years, on its deformation, failure, and springback behavior based on a comprehensive set of experiments. This target material, compared with other conventional AHSSs, did exhibit unique properties during the experimental investigations on the anisotropy, the T-C asymmetry, the Bauschinger effect, the TRIP effect, as well as the strain rate and temperature effects, mainly due to its complex multi-phase microstructure. These unique properties greatly affected the modeling ways and implementation accuracy in the FE predictions.
5.1 Contributions
The first main contribution in this research work is the comprehensive and robust experimentation. For conventional steels, a stamping workshop only needs to conduct a few uniaxial tension tests along different orientations and then can to some extent predict the material’s forming limit and springback angle and magnitude. However, the experimental results of QP980 indicate that for these new AHSSs, especially those with complex microstructure like QP980, a comprehensive experimental investigation is necessary. This is the answer to the first research question.
As the answer to the second research question, the second main contribution is the specification of enhancing the current phenomenological models by adding or adjusting some necessary components, such as a proper yield function with parameters representing
T-C asymmetry and anisotropy and some parameters that can represent the TRIP effect in an isotropic-kinematic hardening law.
144 5.2 Future Plan
As emphasized in Section 2.1.2, the actual cold stamping of QP980 will be under
>150°C non-isothermal condition at strain rate >0.1/s, and the material behavior, such as the martensitic transformation, the Bauschinger effect, and the failure of QP980 and any
TRIP-like AHSSs will greatly shift from those at the ambient temperature and quasi-static strain rate. Therefore, the future model needs to properly include some temperature and strain rate sensitivity parameters to represent such complexity.
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