characterization and modeling of the high strain rate

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Characterization and Modeling of the High StrainRate Response of Advanced High Strength SteelsRakan AlturkClemson University, [email protected]

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CHARACTERIZATION AND MODELING OF THE HIGH STRAIN RATE

RESPONSE OF ADVANCED HIGH STRENGTH STEELS

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

Rakan Alturk

December 2018

Accepted by:

Fadi Abu-Farha, Committee Chair

Dr. Nicole Coutris

Dr. Gang Li

Dr. Garrett Pataky

Dr. David Schmuser

ii

ABSTRACT

The push for lightweighting in the automotive industry has motivated metallurgist

and steel manufacturers to produce new generations of steel that provide significant

improvements over the conventional steels to allow them to compete with the

introduction of low-density materials into the industry. To achieve this goal, metallurgist

introduced different (stronger and more ductile) phases into the ferrite-dominant

microstructure of conventional steels. This has led for generations of AHSSs with

significantly improved properties. However, the complex microstructure led to increased

complexities and unpredictability in the behavior of these materials, especially in their

response to variations in the strain rate. This is particularly important, as the materials in

the automotive industry exhibit different strain rates during their lifetime, and the

performance should be predictable especially during high strain rates as these are

encountered during a crash event where performance dictate safety.

This research work aimed at investigating and developing a methodology to allow

for the accurate modeling of multi-phased AHSSs at muli-strain rates. First, a set of

experimental tests were performed on a selected set of AHSSs having a range of

combination of phases in their microstructure at different strain rates. This allowed for

the investigation of the effect of the different phases in the microstructure on the response

of the materials at multi-strain rates, and it gave an insight into the combination of phases

that would result in a material with a favorable response at high strain rates. Then, the

focus was shifted into a particular third-generation AHSS (Medium Mn. steel), the

iii

selection of which was based on its complexity and importance to the future of AHSSs.

Further experiments were performed on this material to characterize its anisotropy at

multi-strain rates. The experiments were used to calibrate an anisotropic yield function at

different strain rates, and the shape and size of the yield locus obtained were observed to

be dependent on the strain rate.

Furthermore, the experimental results of Medium Mn. steel was used to develop a

constitutive relation predicting the strain sensitivity of the anisotropy of the material at

different strain rates. A modification on the Yld2000-2d yield function allowed to

develop a unique strain rate dependent anisotropic yield function that captures the

yielding and anisotropic behavior of the material at multi-strain rates. The model

developed was validated using finite element simulations of the experimental tests.

Finally, the developed model was used to perform crash simulation on a railroad

tank car. The simulations accounted for the strain rate sensitivity of the hardening and

anisotropy of the material. The results highlighted the impact of the current work by

extracting the load-displacement and the energy absorbed from three different

simulations with variations in accounting for the strain rate sensititvity of the material.

The difference in the results emphasizes the impact and importance of utilizing the

methodology proposed in this dissertation for the crashworhiness analysis of AHSSs.

iv

DEDICATION

My PhD journey has been a tough one and wouldn’t have finished without

the unconditional support of my amazing parents who sacrificed everything to get me

where I am, my three wonderful brothers who show me unconditional love, my

wonderful grandmothers who took care of me and pushed me through my life, my

dear friend Nancy who has been my companion and motivator for the past three

years, and the rest of my family and friends. To all, I say thank you, and I love

you

v

ACKNOWLEDGMENTS

First, I would like to express all my gratitude to Dr. Fadi Abu-Farha for

his wise guidance and continued patience during my study and research over

the past four years. He was always willing to help whenever 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. I also thank my committee members,

Dr. Nicole Coutris, Dr. Garett Pataky, Dr. Gang Li, and Dr. David Schmuser,

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.

Louis Hector as well as the entire General Motors for offering me a great

internship opportunity to work in the automotive industry in 2015 and 2016.

During this internship, I accumulated precious working experience, and that

was extremely helpful to my research work.

Also, I cherished the collaboration experience and help of Dr. Jun Hu,

Akshat Agha, Hakan Kazan, Dr. Farbod Niaki, Saeed Farahani, Abdelrahim

Khal, Dr. Bashar elzuwayer 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 educational support.

vi

TABLE OF CONTENTS

Page

ABSTRACT ........................................................................................................... i

DEDICATION ...................................................................................................... iv

ACKNOWLEDGMENTS ..................................................................................... v

TABLE OF CONTENTS ..................................................................................... vi

LIST OF TABLES ................................................................................................ ix

LIST OF FIGURES ............................................................................................... x

CHAPTER

1. INTRODUCTION ........................................................................................ 1

1.1 Lightweighting ................................................................................... 1

1.2 AHSSs ................................................................................................ 3

1.3 DP steels............................................................................................. 8

1.4 Q&P Steel ........................................................................................ 10

1.5 Medium Mn. Steel: .......................................................................... 12

1.6 Motivation ........................................................................................ 13

1.7 Dissertation Overview ..................................................................... 15

1.7 References ........................................................................................ 16

2. EFFECTS OF MICROSTRUCTURE ON THE STRAIN RATE

SENSITIVITY OF ADVANCED STEELS ........................................................ 24

2.1 Introduction ................................................................................... 24

2.2 Experiments ................................................................................... 28

vii

2.3 Results .............................................................................................. 40

2.4 Discussion ........................................................................................ 57

2.5 Summary and Conclusions .............................................................. 65

2.6 References ........................................................................................ 69

3. STRAIN RATE EFFECT ON TENSILE FLOW BEHAVIOR AND

ANISOTROPY OF A MEDIUM MANGANES TRIP STEEL .......................... 80

`3.1 Introduction .................................................................................. 81

3.2 Medium Manganese TRIP Steel ................................................... 85

3.3 Test Specimens .............................................................................. 86

3.4 Instrumentation.............................................................................. 88

3.5 Strain Measurement....................................................................... 89

3.6 Results and Discussion .................................................................. 91

3.7 Conclusions .................................................................................... 101

3.8 References ...................................................................................... 105

4. A RATE DEPENDENT ANISOTROPIC YIELD FUNCTION .............. 111

`4.1 Introduction ................................................................................ 112

4.2 Overview of yield functions........................................................... 113

4.3 Barlat Yld2000-2d ......................................................................... 117

4.4 Calibration of Yld2000-2d ............................................................. 119

4.5 Rate Dependent Yld2000-2d .......................................................... 121

4.6 Numerical Procedure ..................................................................... 126

4.7 FEA Validation .............................................................................. 135

Table of Contents (Continued) Page

viii

4.8 Conclusions and Summary ............................................................ 139

4.9 References ................................................................................... 140

5. CRASH SIMULATIONS OF A RAILROAD TANK CAR .................... 149

5.1 Introduction .................................................................................... 150

5.2 Finite Element Model .................................................................... 153

5.3 Crash simulations using TC128-B ................................................. 159

5.4 Crash simulation using medium Mn (10% wt.) steel ..................... 160

5.5 Summary ........................................................................................ 167

5.6 References ...................................................................................... 167

6. SUMMARY AND CONCLUSIONS ....................................................... 171

6.1 Contributions.................................................................................. 172

6.2 Future Work ................................................................................... 173

Table of Contents (Continued) Page

ix

LIST OF TABLES

Table Page

Table 1.1: Assumed tension properties of steel phases [28] ............................................... 7

Table 2.1 Steel Chemistries (wt. %) ................................................................................. 30

Table 2.2: 0.2% Offset Yield Strength (MPa) at the different strain rates. ...................... 43

Table 2.3: Engineering strain at uniform elongation at the different strain rates. ............ 43

Table 2.4: Ultimate Tensile Strength (MPa) at the different strain rates. ......................... 43

Table 2.5: Engineering Strain at Fracture at the different strain rates. ............................. 44

Table 3.1: Chemical composition (wt.%) of Med. Mn steel. ............................................ 87

Table 3.2: Tensile Properties Associated with RD Flow Curves in Fig. 4.5a. Engineering

stress and strain are reported. ............................................................................................ 92

Table 3.3: Tensile Properties Associated with DD Flow Curves in Fig. 4.5b. Engineering


Table 3.4: Tensile Properties Associated with TD Flow Curves in Fig. 4.5c. Engineering


Table 4.1: Experimental results used to calibrate Yld2000-2d ....................................... 120

Table 4.2: The calibrated 1 8

for each strain rate ..................................................... 121

Table 5.1: the parameters of the simulations .................................................................. 157

x

LIST OF FIGURES

Figure 1.1: The trend in the CO2 and the fuel economy for vehicles in the USA between

1975-2017 as published by the EPA. .................................................................................. 2

Figure 1.2: The “banana” diagram showing the elongation vs. tensile strengths for

different alloys of steels including the three generations of AHSSs [27]. .......................... 6

Figure 1.3: A micrograph showing the typical microstructure of DP steels ....................... 8

Figure 1.4: The iron carbide phase diagram [38] ................................................................ 9

Figure 1.5: Stress-Strain curve showing the effect of martensite on the strength of DP

steels, where the martensite content of CAS>CHCL>CAL [37] ...................................... 10

Figure 1.6: The heat treating processes used to produce (a) DP, (b) TRIP and (c) Q&P

steels [46] .......................................................................................................................... 12

Figure 1.7: Microstructure of QP980 showing the three phases ....................................... 12

Figure 2.1: Micrographs showing the microstructure of the as-received material for (a)

CR5 (b) SS 201 (c) DP590 and (d) DP980 (e) Medium Mn. (10wt%) (f) QP980. .......... 31

Figure 2.2: Experimental setups and test specimens used for (a) quasi-static testing and

(b) high rate testing (Servo-hydraulic) (c) detailed view of the components of the Instron

VHS unit ........................................................................................................................... 34

Figure 2.3: Photron SA1.1 high-speed cameras ................................................................ 36

Figure 2.4: (a) Engineering stress-strain curve of a test ran at a speed of 20m/s showing a

dominant ringing signature and (b) Engineering stress-strain curve of a test ran at a speed

of 20m/s showing the elimination of ringing .................................................................... 37

Page

xi

Figure 2.5: High rate tensile specimen geometry with (a) 20mm gauge length (b) 10mm

gauge length ...................................................................................................................... 39

Figure 2.6: (a) Evolution of strain along the centerline (using a virtual extensometer) of a

10 mm gauge length specimen tested at 500s-1. Zero position denotes the center of gauge

section. (b) Evolution of strain along the centerline (using a virtual extensometer) ........ 40

Figure 2.7: Engineering stress/strain curves at 0.005s-1 and 500s-1 for (a) CR5 (b) SS

201 (c) DP590 (d) DP980 (e) Medium Mn. (10wt%) and (d) QP980. ............................. 45

Figure 2.8: (a) Kolsky bar at NIST (b) Specimen Geometry used for Kolsky testing at

NIST (c) Engineering stress-strain curves with the 500s-1 obtained from the Kolsky

testing. ............................................................................................................................... 46

Figure 2.9: (a) Yield strength (MPa) vs log strain rate (b) Tensile strength (MPa) vs log

strain rate ........................................................................................................................... 52

Figure 2.10: n-value vs log strain rate .............................................................................. 53

Figure 2.11: Percent Change between 0.005s-1 and 500s-1 of (a) Yield Stress (b) Tensile

Strength (c) Total Elongation and (d) Toughness. ............................................................ 56

Figure 2.12: Effect of martensite on the percentage increase of tensile strength ............. 59

Figure 2.13: Volume fraction of martensite measured at different plastic using: (a)

magnetic induction (b) XRD. ............................................................................................ 64

Figure 3.1: Deformation mechanisms in multiphase AHSSs with Mn alloying content .. 82

Figure 3.2: Field emission SEM image of the as-received, medium Mn TRIP steel (two

phases, ferrite (F) and retained austenite (A)) with ~2 μm grain size. Annealing twins

(AT) appear in austenite grains which are indicative of a low stacking fault energy. ...... 87

List of Figures (Continued) Page

xii

Figure 3.3: Tensile specimen geometry for high rate testing. All dimensions are in mm 88

Figure 3.4: INSTRON VHS 65/80-20 (VHS=very high speed) servo-hydraulic test unit 90

Figure 3.5: Engineering stress-strain curves at the five testing strain rates for: (a) 0º (RD

– rolling direction), (b) 45º (DD – diagonal direction), and (c) 90º (TD – transverse

direction). .......................................................................................................................... 94

Figure 3.6: (a) Tensile (2D DIC) strain contours at the ninth image showing the onset of

Lüdering (red contour patch) in the 500 s-1 curve (see Fig. 3.5). (b) Tensile (2D DIC)

strain contours showing Lüders strain at image twelve in the 500 s-1 curve (c) Tensile (2D

DIC) strain contours showing Lüders strain at image twelve in 500 s-1 curve. ................ 95

Figure 3.7: Normalized flow stresses from Eq. (3.1) along the (a) diagonal direction (DD)

(b) transverse direction (TD). ........................................................................................... 97

Figure 3.8: r-value evolution with strain at different strain rates for specimens tested

along the (a) rolling direction (RD) (b) diagonal direction (DD), and (c) transverse

direction (TD). .................................................................................................................. 99

Figure 3.9: Average r-value and normal anisotropy coefficient (rn) variation with the

logarithm of nominal strain rate for the RD, DD and TD. .............................................. 102

Figure 4.1: Yield locus calibrated using data from experiments at different strain rates 122

Figure 4.2: Variation of the Yld2000-2d constants ( 1 8 ) with the strain rate ......... 123

Figure 4.3: Comparison of yield locus plotted using the experimental data at 0.005s-1

(pink), at 0.5s-1 (blue) and using the equation (green), showing an agreement between the

actual and the predicted yield locus. ............................................................................... 124


xiii


(pink), at 5s-1 (blue) and using the equation (green), showing an agreement between the








Figure 4.7:A diagram explaining the process of the cutting-plane algorithm ................ 131

Figure 4.8: Flow chart showing the steps followed in the cutting plane algorithm ........ 136

Figure 4.9: Finite element model used to verify the developed equation ....................... 137

Figure 4.10: Comparison of the experimental results and simulations at 0.005s-1......... 138

Figure 4.11: Load vs. displacement comparison between the experiments and simulations

at 50s-1 using the Yld2000-2d calibration at 0.005s-1 (red curve) and using the developed

equation (green curve) .................................................................................................... 138



equation (green curve) .................................................................................................... 139

Figure 5.1: A schematic showing the collisions associated with a derailment accident [3].

......................................................................................................................................... 151

Figure 5.2: A schematic of a typical North American railroad tank car [18]. ................ 154


xiv

Figure 5.3: The finite element model of the (a) tank (b) impactor ................................. 155

Figure 5.4: The meshed finite element model of the tank .............................................. 155

Figure 5.5: (a) A tank car used for an actual crash test [8] (b) the model developed to

replicate the boundary conditions encountered during a crash test. ............................... 156

Figure 5.6: The model showing the initialization of the fluids inside the tank. Green

(liquid), blue (vapor). ...................................................................................................... 158

Figure 5.7: Comparison of the acceleration profiles of the impactor between (a) the

simulation without lading and (b) Kirkpatrick results [6]. ............................................. 161

Figure 5.8: Comparison of the force-displacement curves of the impactor between (a) the

simulation without lading and (b) Kirkpatrick results [6]. ............................................. 162


simulation with lading and (b) Kirkpatrick results [6]. ................................................... 163

Figure 5.10: Comparison of the three load-displacement profiles of simulations ran using:

1) quasi-static hardening and yielding (green), 2) multi-rate hardening and quasi-static

yielding (blue), and 3) the developed constitutive relation, i.e. multi-rate hardening and

yielding (red) ................................................................................................................... 165

Figure 5.11: Energy absorbed to fracture during a crash simulation of the three different

scenarios described. ........................................................................................................ 166


1

CHAPTER ONE

1. INTRODUCTION

1.1 Lightweighting

In the past decade, research into lightweighting techniques in the automotive

industry has risen significantly. This trend can be mainly attributed to the desire of the

industry to 1) reduce CO2 emissions 2) increase the fuel efficiency of the vehicles 3)

meet the requirement of recent government environmental standards, and 4) reduce the

depletion of energy sources [1, 2]. Assuredly, there are other means by which the goals

above can be achieved, such as: optimizing current technology, or using cleaner fuels.

However, the former has a shallow impact on the overall weight of the vehicle while the

latter has a high initial cost [3]. Whereas, studies have shown that a 10% reduction in

vehicle weight can result in 5-8% improvement in fuel economy [4]. For example, Audi

A2 showed that 60% of fuel consumption is mass-dependent; 10% reduction in vehicle

weight leads to ~6% reduction in fuel consumption [5]. Additionally, lightweighting

leads to overall positive effects on the vehicle such as: mass decompounding and

improving the vehicle handling [6, 7]. Therefore, lightweighting is the most efficient

method to achieve the goals set by the OEMs and the regulations set by the governments.

The first government standard regulating fuel consumption in the USA for

vehicles dates back to the 1970s [8]. Consequently, OEMs began their lightweighting

efforts, cultivating in the shift from body-on-frame (BOF) structure into body-in-white

(BIW) structures, adopting smaller engines and shifting from rear-wheel drive to front-

2

wheel drive systems [9]. These efforts have cultivated in the significant reduction of the

average weight of the vehicles in the USA between 1976 to 1986. However, at the

beginning of the 1990s, there was an opposite trend observed in the average weight of the

vehicle due to an increase in safety and luxury features in vehicles, and the increased

popularity of SUVs and pickup trucks. Nonetheless, the lightweighting drive was again

on the rise in the early 2000s mainly because of the significant increase in fuel prices and

the stringent regulations by governments. Figure 1.1 shows how the lightweighting trend

affected the CO2 emission and the fuel economy of the vehicles in the USA as published

by the EPA in 2018 [10]. Moreover, until today, lightweighting is one of the top priorities

for vehicle manufacturers around the World, although the methods for lightweighting

now has changed.

Figure 1.1: The trend in the CO2 and the fuel economy for vehicles in the USA between

1975-2017 as published by the EPA.

3

The main methods of lightweighting today are material substitution, parts

consolidation and design optimization [1]. Up till today, steel remains the most dominant

material in an automotive BIW [11], with a density of about 7830 kg/m3. Lightweighting

can be achieved with the replacement of steel in the structure with materials with lower

densities such as aluminum (2700 kg/m3) or magnesium (1738 kg/m3). Examples of

material replacements in the automotive industry include the 1994 Audi A8 which had a

BIW entirely made of aluminum [12] and the 2013 BMW i3 which had carbon-fiber

composite structures [13]. Yet, steels are still dominant in the automotive industry body

structures until today, and that status is projected to remain unchanged in the near future

[14]. This dominance of steel is mainly attributed to three points: 1) Mass production: the

automotive manufacturing has been designed to cater to the properties of steels, and the

introduction of a new material (especially magnesium and polymer composites) will

require substantial changes to the infrastructure, 2) Cost: steel is the cheapest metal on

Earth, all of these replacement materials have a much higher cost, and 3) Advanced High

Strength Steels (AHSSs): the steel industry responded to the lightweighting drive with the

introduction of new grades of steels that have improved properties over the conventional

steels and can achieve lightweighting [14].

1.2 AHSSs

In 1975 after the worldwide oil crisis, a sudden increase in the interest of

developing high strength steels for the automotive applications was observed [15, 16].

The first alloys of steel to be introduced were the High Strength Low Alloy (HSLA)

steels that showed a slight improvement in the yield and tensile strengths over the

4

conventional mild steels accompanied by a reduction in the ductility [17]. The unique

combination of strength and ductility required by the industry was found in dual phase

(DP) steels [18], which showed significant improvements on the strength [19, 20].

However, due to the difficulties in adopting the industry to these new steels, the interest

in these materials died out except for some of the conventional High Strength Steels

(HSSs) such as Interstitial Free (IF) steels, Bake Hardenable Interstitial Free (BH-IF) and

HSLA steels [16, 21]. This interest was then revived with the increased competition from

the low-density materials cultivating in the development of the aluminum Audi A8 in

1994, and the demand for improved passenger safety and environment regulations [22].

The response of the steel industry launched a project named Ultra-Light Steel Auto Body

(USLAB) through an international consortium of 35 companies from 18 countries. The

project aimed at developing a lightweight steel BIW comprised mainly of HSS. The

result of this project was the development of a BIW comprised of 90% HSS with a 25%

reduction in weight and improved structural properties [14, 16]. Consequently, there was

an increase in the research for developing new grades of steels to further improve the

lightweighting capabilities, and the interest in DP steels was revived. This led to the

commercialization of DP steels internationally in 1995.

With the more stringent government standards, the increase in fuel prices, and the

infiltration of the low-density materials in the automotive industry provided a push on the

steel industry to seek better combinations of strength and ductility in their steels. These

combinations were achieved through the development of the Advanced High Strength

Steels (AHSSs). The exact definition of AHSSs can be vague sometimes; they can be

5

defined as steels having tensile strengths higher than 600 MPa with “improved”

formability [16]. However, the main feature that differentiates these materials from HSSs

and conventional steels is their microstructure, which contains more than one phase. DP

steels can be considered the first AHSS and belong to what is called the first generation

of AHSSs. The first generation included other materials such as Transformation Induced

Plasticity (TRIP) steels, Ferrite-Bainite (FB) steels, Martensitic Steels (MS) and

Complex-Phase (CP) steels. DP steels, as the name suggests, consist of two phases;

ferrite and martensite, martensite being the phase attributed with the increase in strength.

The TRIP steels show improvement in the strain hardenability and therefore toughness

which can be attributed to the retained austenite in its microstructure [23]. Moreover,

martensitic steels contain more than 80% martensite in their microstructure leading to the

development of very strong yet brittle material. The first generation of AHSSs showed an

increase in the strength with a general decrease of ductility following the trend shown by

the conventional steels in the banana chart shown in Figure 1.2. The first generation of

AHSSs has already been used in the automotive industry. DP and TRIP steels are used

prominently in energy absorbing areas. While the high strength DP and MS steels are

used in areas that require high stiffness and resistance to intrusion [11, 16].

The second generation of AHSSs can be shown to break the trend observed with

the conventional steels and the first generation of AHSSs by improving both the ductility

and strength (Figure 1.2). The main characteristic of the second generation of AHSSs is

that their microstructure is austenite-based and they have high alloy content. The most

known of the second generation AHSSs is the Twinning Induced Plasticity (TWIP) steels

6

which show strengths reaching 1000MPa and elongations of about 60% [24, 25]. The

properties of the second generation are desirable; however, these materials have a high

cost (alloying), and poor manufacturability and welding. Consequently, their applications

in the automotive industry are limited as compared to the first generation [26].

The difficulties shown with the second generation of AHSSs have led the industry

to focus the research on developing the new third-generation AHSSs that can fill the gap

between the first and second generations on the banana chart. Studying the

microstructure of the first and second generation of AHSSs, it was suggested that to

Figure 1.2: The “banana” diagram showing the elongation vs. tensile strengths for

different alloys of steels including the three generations of AHSSs [27].

7

achieve the desired properties, a microstructure that dominantly contains austenite and

martensite with low alloying is required [28]. The reasons for this combination can be

explained by looking at table 1.1, which shows the different properties associated with

the three main phases of steels; ferrite, martensite, and austenite. Martensite is the phase

that provides the highest strength while austenite provides the highest ductility, therefore,

a combination of these two phases should provide the desired properties. The research

conducted for the development of the third generation AHSSs was summarized by

Fonstein [16] in the following materials. 1) Bainitic ferrite TRIP steel which improves the

strength and ductility of the conventional TRIP steels [29], 2) Quenched and Partitioned

(Q&P) steels, which are produced through a modification on the production process of

TRIP steels [30, 31] and 3) Medium Mn. steels (4-10% wt.) which contain a combination

of ferrite and austenite with a medium Mn. content, that improves the ductility and

strength significantly [32], these steels are discussed in detail in this thesis.

Table 1.1: Assumed tension properties of steel phases [28]

Phase Tensile Strength (MPa) Uniform True Strain

Ferrite 300 0.3

Austenite 640 0.6

Martensite 2000 0.08

8

1.3 DP steels

As mentioned earlier, DP steels were among the first steels to be introduced and

classified as AHSSs. They provide a right combination of strength and ductility and have

been widely used in the automotive industry. They can provide these properties because

of their microstructure, shown in Figure 1.3, which consists of a ferrite matrix with

martensitic islands dispersed in it. DP steels are typically produced by first heating the

material to the inter-critical temperature region, soaking to allow austenite to nucleate

and grow followed by slow cooling to the quench temperature and finally rapid cooling to

allow the austenite to transform into martensite [33, 34]. The different phases and the

temperatures they exist at, are shown in the iron carbide phase diagram shown in Figure

1.4. Note that martensite cannot be seen on the diagram, and that is because martensite

can only be achieved through the transformation of austenite. Austenite is a very unstable

phase at room temperatures and will transform to martensite. It is only stable at high

temperatures and through specific alloying techniques.

Figure 1.3: A micrograph showing the typical microstructure of DP steels

9

The austenite to martensite transformation which occurs during cooling results in

an increase in the density of mobile dislocations within the ferrite phase, which

accommodates the majority of the plastic deformation and causes the excellent

continuous yielding observed in the material [35, 36]. On the other hand, the presence of

the martensite phase causes the high strength shown by the DP steels. As can be seen

with the different DP steels produced, the higher the martensite content, the higher the

yield and the ultimate strength of the material (Figure 1.5) [35, 37]. The volume fraction

of martensite in DP steels is controlled by varying the annealing temperature and the

cooling rate in the quenching process.

Figure 1.4: The iron carbide phase diagram [38]

10

Figure 1.5: Stress-Strain curve showing the effect of martensite on the strength of DP

steels, where the martensite content of CAS>CHCL>CAL [37]

1.4 Q&P Steel

As discussed earlier, austenite is an unstable phase that will transform into

martensite at room temperature (see Figure 1.3). However, austenite has desirable

properties, and its presence in the microstructure will provide improvements on the

material. Therefore, research was conducted to find methods to stabilize austenite at room

temperatures. Two methods are generally used by the industry; the first method is mainly

using specifying alloying element (Mn, Ni, Al, and Cu) to increase the stacking fault

energy of the austenite and thus stabilizing it [39-43]. This method is used to produce

most austenitic stainless steels and TWIP steels. However, the increased allying content

causes an increase in the cost and difficulties in welding. The second method uses a

modification on the heat treatment process used to produce the steels with a little help of

11

Al, Si, P, and Mo to suppress the growth of cementite [44]. This method gave birth to

TRIP steels and Q&P steels.

Q&P steels were first developed at the Colorado School of Mines [30, 45], with a

modification to the process used to produce TRIP steels. TRIP steels are produced [46],

by heating to the inter-critical temperature, similar to DP steels, then quenching the

material to a temperature above the that at which martensite begins forming (~357 oC),

followed by a tempering step at which some austenite is stabilized by the carbon atoms

which are suppressed to form cementite by the alloys. Finally, the material is quenched to

room temperature, during which the unstable austenite transforms to martensite. The

process is visualized in Figure 1.6, which shows a comparison between the three different

processes for forming DP, TRIP and Q&P steels. The figure shows that for Q&P, the first

quenching step does not stop before the martensite starts temperature but instead stops at

a temperature between the martensite start and the finish temperatures. This means that

some of the austenite transforms to martensite during that step. Then the material is

reheated again above the temperature martensite starts forming and held at that

temperature allowing carbon atoms to partition from martensite and work to stabilize the

remaining austenite, preventing its transformation back to martensite during the final

quenching phase [47]. This method allows for a better control over the amount of

martensite the material has and thus can produce materials with higher strengths than

TRIP steels [48]. Figure 1.7 shows a micrograph of the microstructure of QP980,

showing the three phases: ferrite, austenite, and martensite.

12

Figure 1.6: The heat-treating processes used to produce (a) DP, (b) TRIP and (c) Q&P

steels [46]

Figure 1.7: Microstructure of QP980 showing the three phases

1.5 Medium Mn. Steel

One of the methods to stabilize austenite mentioned earlier was using an alloying

element with Mn being one of the main alloying elements that can achieve that. TWIP

steels have high Mn content (~20% wt.) to accomplish that. However, that high content

13

causes an increase in the cost and problems with the manufacturing processes and

welding [49]. Therefore, research was done to develop a steel with an austenite content

higher than TRIP steels and lower than TWIP steel to maintain the cost and weldability

[50]. Aydin et al. [32] developed a steel alloy with medium Mn. content based on having

a stacking fault energy between that of TRIP and TWIP steels. The study showed the

importance of having the Mn content in the microstructure to about 10% to obtain the

desired austenite content and stability to improve over TRIP steels. A medium Mn. steel

is the main steel focused on in this thesis and will be described in more detail in chapters

2 and 3.

1.6 Motivation

The enhancement of the mechanical properties of AHSSs over the conventional

steels comes at the cost of increased complexities in the microstructure. The complex

multi-phased microstructure of AHSSs contains various micro-elements that can change

the macro behavior of the material significantly. This will translate into increased

complexities and difficulties in modeling and predicting the behavior of these materials.

Taking austenite as one of those micro-elements. It remains a metastable phase

even when the material contains enough alloys to stabilize it at room temperature, loading

the material will cause it to transform into martensite (TRIP effect) [51, 52]. While that is

a preferred property that improves on the strength and ductility of the material, it is

highly unpredictable due to its dependence on multiple variables, such as temperature

[53, 54] and strain rate [55, 56].

14

Strain rate, in particular, is a very important factor especially for the automotive

industry, as the materials used in the automotive industry undergo loading at strain rates

ranging from 0.001s-1 to 1000s-1. It is well established in the literature that the mechanical

properties of materials are highly dependent on the strain rate [57-60]. It is, therefore,

crucial to understanding how the strain rate affects the behavior of AHSSs if they were to

be used in an automobile. As mentioned earlier, AHSSs are made of multi-phase

microstructure, each having its own independent properties and mechanisms involved

with deformation. Therefore, each will have its own response to changes in strain rate and

that will provide a challenge to model and predict.

Yet, the aforementioned properties do not represent the full spectrum of

challenges involved with modeling the new AHSSs. Anisotropy is another property that

can influence the accuracy of these model if it is not considered. Anisotropy is the

dependence of material properties on the crystallographic direction at which they are

measured. The anisotropy in a sheet metal can be caused by the manufacturing processes

that impart a crystallographic direction or texture on the material and due to the variation

of atomic spacing in different crystallographic directions [38]. This will cause further

complications in modeling AHSSs because as is the case with strain sensitivity and other

properties, the different phases in the microstructure will have its own degree of

anisotropy.

All these complications will manifest themselves when trying to model the

behavior and performance of these AHSSs in crash simulations. During these

simulations, the strain sensitivity, the anisotropy and the combination of both will

15

significantly affect the behavior of these materials. Therefore, despite the number of

attractive properties AHSSs offer, there is a need for these challenges to be addressed and

studied and that what motivated this work. The focus will be on improving the

understanding of the AHSSs, their microstructure, and its effect on the various properties

of interest for the crashworthiness of these materials. Different grades of DP, QP980 and

Medium Mn. steels will be studied and presented. However, the focus will be on Medium

Mn. steel as it is still a development third-generation AHSS that present great

complexities in its modeling, especially regarding its anisotropy. The thesis aims to

answer three questions:

1) What is the effect of the phases in the microstructure of the material on its macro

properties at multiple strain rates?

2) How does the anisotropy of Medium Mn. steel gets affected at high strain rates?

3) How to develop a phenomenological model to capture unique behavior exhibited

by Medium Mn. steel, with regards to its strain rate sensitivity and anisotropy at

multiple strain rates?

1.7 Dissertation Overview

Chapter 2 in this dissertation will aim to explore the answer to the first question

by presenting a series of experimental techniques and results on multiple AHSSs with

varying microstructures. The unique experimental setups of multi-strain rate testing are

presented and discussed, followed by a detailed discussion of the contribution of each of

the three main phases in the different microstructure on the experimental observations.

16

Then, in chapter 3 the focus will be drawn to Medium Mn. steel, when its

anisotropy at multiple strain rates is explored and discussed. The anisotropy is studied

through different properties at different strain rates to show how strain rate sensitive is

the anisotropy. Following that, chapter 4 will be a detailed presentation and derivation of

a constitutive model for Medium Mn. steel aiming to capture the unique strain rate

sensitive anisotropy of the material. Furthermore, its compilation and the numerical

procedure of incorporating it for use in a finite element software will be discussed. The

model developed will be verified through simulations of experimental test and

comparison with the results. Finally, in chapter 5, the model developed will be used to

perform simulations on a railroad tank car to quantify the effect of incorporating the

different variables in a crash simulation.

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19

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[32] H. Aydin, E. Essadiqi, I.-H. Jung, and S. Yue, "Development of 3rd generation

AHSS with medium Mn content alloying compositions," Materials Science and

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initial stages of continuous annealing of cold rolled dual-phase steel," Materials Science

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[35] U. Liedl, S. Traint, and E. Werner, "An unexpected feature of the stress–strain

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AA5182-O," University of Waterloo, 2015.

[37] M. Singh, A. Das, T. Venugopalan, K. Mukherjee, M. Walunj, T. Nanda, et al.,

"Impact of Martensite Spatial Distribution on Quasi-Static and Dynamic Deformation

Behavior of Dual-Phase Steel," Metallurgical and Materials Transactions A, vol. 49, pp.

463-475, 2018.

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worked 316L stainless steel: influence of dynamic strain aging," International Journal of

Fatigue, vol. 26, pp. 899-910, 2004.

[40] S. Allain, J.-P. Chateau, O. Bouaziz, S. Migot, and N. Guelton, "Correlations

between the calculated stacking fault energy and the plasticity mechanisms in Fe–Mn–C

alloys," Materials Science and Engineering: A, vol. 387, pp. 158-162, 2004.

[41] X. Peng, D. Zhu, Z. Hu, W. Yi, H. Liu, and M. Wang, "Stacking fault energy and

tensile deformation behavior of high-carbon twinning-induced plasticity steels: Effect of

Cu addition," Materials & Design, vol. 45, pp. 518-523, 2013.

[42] T. Shun, C. Wan, and J. Byrne, "A study of work hardening in austenitic Fe

Mn C and Fe Mn Al C alloys," Acta metallurgica et materialia, vol. 40, pp. 3407-

3412, 1992.

[43] J. Hu, "Characterization and Modeling of Deformation, Springback, and Failure

in Advanced High Strength Steels (AHSSs)," 2016.

[44] E. De Moor, J. Speer, D. Matlock, C. Föjer, and J. Penning, "Effect of Si, Al and

Mo alloying on tensile properties obtained by quenching and partitioning," MS&T 2009,

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[45] J. G. Speer, F. C. R. Assunção, D. K. Matlock, and D. V. Edmonds, "The"

quenching and partitioning" process: background and recent progress," Materials

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Hole Expansion Properties of Quench & Partitioned, Quench & Tempered and

Austempered Steels," SAE Technical Paper 0148-7191, 2012.

[47] T. D. Bigg, D. V. Edmonds, and E. S. Eardley, "Real-time structural analysis of

quenching and partitioning (Q&P) in an experimental martensitic steel," Journal of

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[48] B. C. De Cooman and J. G. Speer, "Quench and Partitioning Steel: A New AHSS

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transformation plasticity in steels," 서울대학교 대학원, 2016.

[50] D. K. Matlock and J. G. Speer, "Third generation of AHSS: microstructure design

concepts," in Microstructure and texture in steels, ed: Springer, 2009, pp. 185-205.

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deformation," Materials Science and Engineering: A, vol. 479, pp. 333-338, 2008.

[53] I. Tamura, "Deformation-induced martensitic transformation and transformation-

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the martensitic transformation at various scales in TRIP steel," Materials Science and

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23

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24

CHAPTER TWO

2. EFFECTS OF MICROSTRUCTURE ON THE STRAIN RATE SENSITIVITY

OF ADVANCED STEELS

ABSTRACT

The dependence of the strain rate sensitivity of advanced ~1GPa tensile strength

steels on the phases present in their microstructures was studied by testing different steels

at multi-strain rates, namely 0.005s-1, 0.5s-1, 5s-1, 50s-1, and 500s-1. The intermediate and

high strain rate tests were performed using a servo-hydraulic machine, while the quasi-

static tests were performed using a universal testing machine. The four main steels of

interest were the Ferrite-Martensite DP590 and DP980, the Ferrite-Austenite Medium

Manganese steel (10% wt.), and the Ferrite-Martensite-Austenite QP980; the latter being

a transformation induced plasticity (TRIP) assisted steel. For comparison, ferritic CR5

mild steel and austenitic stainless steel 201 were also tested under the same conditions.

Though the differences in the steel chemistries were not taken into account, the results

obtained here suggest a strong relationship between the phase-content of the steel and its

response to the changes in the loading rate. The relationships between the observed

mechanical behavior and the phases present in the microstructure are discussed.

2.1 Introduction

With the ever-increasing push for lightweighting in the automotive sector,

Advanced High Strength Steels (AHSSs) continue to infiltrate the industry due to their

unique properties and merits [1]. The uniqueness of AHSSs can be primarily attributed to

25

their complex multi-phase microstructures, as compared to conventional single-phase

steels, which enable achieving particular blends of strength and ductility. While strength

drives the lightweighting potential of the material, ductility defines the practical limits of

its stamping. However, beyond forming materials into the various automotive

components, materials need to pass specific tests and meet certain criteria to be used in

vehicles; some of the most critical ones are crashworthiness-related [2] Finite element

(FE) analyses are typically performed to simulate a crash event; the accuracy of these

simulations relies heavily on how well the constitutive equations describe the material

behavior.

Automotive materials undergo loading at different strain rates during their

lifetime. From quasi-static (< 0.1 s-1) to intermediate (around 1 s-1) during stamping, to

high rates (>100 s-1) during a crash event. It is well-established in the literature that the

deformation behavior and the properties of steels are highly influenced by strain rate.

Thus, the governing equations used in the various analyses should be able to incorporate

rate effects on the behavior of the material. In order for the equations to achieve that,

accurate experimental material data must be obtained at different strain rates.

Special test instrumentation must be used to reliably generate material data over

the nominal high strain rate range since screw-driven load frames commonly used for

measurement of material data at quasi-static rates (e.g. <0.5s-1) are not applicable. Servo-

hydraulic load frames, in which an actuator, accelerated to the desired speed, engages a

slack adaptor assembly to dynamically deform a test specimen in tension, are the most

commonly employed to test automotive body structure steels at high strain rates [3-6]. A

26

common upper testing limit on strain rate for servo-hydraulic test machines is 500s-1. The

split-Hopkinson bar [7-10], also known as the Kolsky bar [11, 12], is commonly applied

in dynamic material tests at strain rates in the vicinity of (or above) 1000s-1 (there are

some notable exceptions where tests have been conducted near 500s-1). Here, the

specimen is plastically deformed by an elastic wave pulse initiated by a striker through

the load train consisting of incident and transmission bars and the specimen. Unlike a

servo-hydraulic load frame, the test specimen is often not deformed to fracture in a split-

Hopkinson bar test. Another test used to measure the deformation response of automotive

materials at high strain rates is the instrumented falling weight impact tester [13, 14]. In

general, high rate tests are more involved and therefore often more costly than quasi-

static tests for a variety of reasons. Among the more significant of these are the

application of strain gauges to address load cell ringing; tensile specimen design to

eliminate non-uniform strain evolution during elongation; and test system artifacts that

produce spurious specimen bending during elongation. Load cell ringing, which is

common in servo-hydraulic units, appears as a spurious oscillatory signal superimposed

onto the “true” material response (e.g. load vs. time or load vs. displacement) [15]. This

effect results from stress wave propagation, the inertia of the load train components, and

the frequency response of the load cell. Ringing is most severe near the resonant

frequency of the load train (~9kHz for many servo-hydraulic testers) which obscures the

desired material response leading to inaccurate flow data. Most test frame manufacturers

provide the means for reducing ringing through data filtering or fitting routines.

However, uncertainties remain regarding the accuracy of the material data since filtering

27

is in and of itself an intentional distortion. Other techniques for reducing ringing, such as

tuning a test specimen geometry via load train modal analysis for a specific test

instrument, have also appeared in the literature [3, 16]. Strain gauges are commonly

applied to one of the test specimen grip regions. These essentially replace the signal from

the test unit load cell and provide voltage signals that can be converted to force assuming

that Young’s modulus of the material is known within reasonable accuracy and that a

uniform force is transmitted through the test specimen during loading. Strain gauge

measurement of plastic deformation in the test specimen gauge region is often

problematic and digital image correlation (DIC), a non-contact whole field optical strain

mapping technique using high-speed photography is now the preferred method for both

quasi-static and high strain rate testing [17, 18]. For example, issues with adhesion of the

strain gauge at large strains can arise, maximum measurable strains are dependent upon

the strain gauge design, and test artifacts such as specimen bending cannot be easily

revealed. At present, only two European standards for dynamic tensile testing are

available, viz., ISO 26203-1,2 [19] and SEP1230 [20].

Strain rate effects on the mechanical properties of steels have been studied by

many researchers over the past decade; examples include these references [5, 21-28].

However, the majority of these studies did not focus on the microstructure and the role it

plays in dictating the strain rate sensitivity of the steel grade. Few studies considered the

microstructure and its correlation to rate sensitivity, particularly in advanced steel grades.

Hwang et. al [29] studied the effects of ferrite grain size on the strain rate sensitivity in

DP steels. Oliver et al. [6] tested DP and TRIP steels at 0.001 s-1 and 200 s-1 to compare

28

the changes in the microstructure at different strain rates. Tarigopula et. al [17] studied

the effect of strain rate on the mechanical properties and strain localization of DP800. Li

et al. [30] studied the effects of strain rate on the deformation-induced transformation of

austenite in quenched and partitioned steels, up to strain rates of 0.1s-1.

This chapter investigates the response of selected advanced steels with complex

multiphase microstructures to external loading at multi-rates ranging from quasi-static to

rates close to those encountered during a crash event. The responses at the intermediate

and high rates are compared to that at quasi-static rates to evaluate the rate sensitivity of

each material. Moreover, the differences in their responses are correlated to the types of

constituent phases present in their microstructures. This would be the first step to develop

models that can incorporate rate effects in each of the various phases within the

microstructure of advanced high strength steels. The main four materials of interest are

the Ferrite-Martensite DP590 and DP980, the Ferrite-Austenite Medium Manganese steel

(10% wt.), and the Ferrite-Martensite-Austenite QP980. Nevertheless, the deformation of

two single-phase steels, ferritic CR5 mild steel and austenitic stainless steel 201, are also

investigated to provide some basis for comparison and establish the rate dependency of

the individual phases.

2.2 Experiments

2.2.1 Materials and Initial Microstructures

Table 2.1 summarizes the chemical compositions of each of the six materials

tested. The initial microstructures of the six materials were examined by optical

29

microscopy and scanning electron microscopy; the obtained micrographs are shown in

Figure 2.1. The CR5, a very ductile grade of mild steel which is most commonly used for

high formability applications including exterior panels and closures. Its microstructure

(Figure 2.1a) primarily consists of moderate-grained ferrite (~5-25 µm equivalent grain

size), and the removal of interstitials such as C and N from the ferritic matrix is achieved

via microalloying additions. The SS 201 is an austenitic stainless steel, the microstructure

of which exhibits a similar heterogeneous distribution of grains, though larger than the

CR5. The microstructure of this chromium-nickel-manganese alloy exhibits complex

deformation behavior [31]. The DP590 and DP980 microstructures, shown in Figures

2.1c and d, consist of martensite islands/particles dispersed in a matrix of ferrite grains

[32-34]. The area fractions of the two phases (both are very fine) were measured to be

~60% ferrite and ~40% martensite (DP590) and ~30% ferrite and ~70% martensite

(DP980). As part of an ongoing DOE project aimed at the development of an ICME

model for third generation advanced high strength steels (3GAHSS), an experimental

medium Mn. steel was produced through inter-critical annealing in the ferrite-austenite

region [44]. Austenite is stabilized through enrichment with Mn and C, and the initial

microstructure contains no martensite. Its initial microstructure, shown in Figure 2.1e,

consists of ferrite and austenite. Annealing twins appear on the austenite grains and are

suggestive of a low stacking fault energy. Plastic deformation and phase transformation

in tension are highly inhomogeneous and occur in propagating bands as confirmed by

synchrotron X-ray measurements [35]. The QP980 microstructure is a three-phased

TRIP steel subject to a two-step quenching and partitioning heat treatment [36]. Xiong et

30

al. [37] identified both film-like and blocky forms of carbon-enriched austenite in QP980.

The result is a microstructure composed of ~40% ferrite, ~50% martensite and ~10%

retained austenite. The ferrite and martensite grains are fine (similar to the DP980), while

the austenite islands are particularly small (~0.5-1 µm); an example is indicated in the

micrograph in Figure 2.1f. As noted, phase distinction in this material is challenging;

thus, the reported phase fractions were obtained from both optical microscopy and EBSD

maps (not presented here).

Table 2.1 Steel Chemistries (wt. %)

Steel C Mn Si Al Cr Ni Mo P N S Ti CR5

0.0018 0.14 0.005 0.038 0.029 0.035 0.005 0.008 0.006 0.007 0.077

SS 201 0.045 6.5 0.48 0.002 16.7 4.5 0.17 0.036 0.13 0.001 0.002

DP 590 0.06 1.5 0.32 0.023 0.78 0.039 0.10 0.010 0.009 0.002 0.003

DP980 0.09 2.3 0.65 0.040 0.022 0.007 0.11 0.007 0.004 0.001 0.029

Medium

Mn. 0.16 10.56 0.22 1.56 <0.002 - - 0.0023 0.0027 0.0016 -

QP980 0.20 1.79 1.52 0.039 0.03 - - 0.014 0.004 0.003 -

31

(a) (b)

(c) (d)

(e) (f)

Figure 2.1: Micrographs showing the microstructure of the as-received material for (a)

CR5 (b) SS 201 (c) DP590 and (d) DP980 (e) Medium Mn. (10wt%) (f) QP980.

32

2.2.2 Experimental Setups and Procedure

The six materials were tested in tension under both quasi-static, intermediate and

high rate conditions using the setups shown in Figure 2.2. The quasi-static tension tests

were performed on a universal electromechanical load frame, at a constant crosshead

speed of 18mm/min. A standard ASTM E8 specimen geometry was used, and the

corresponding strain rate of deformation was ~0.005s-1. For accurate strain

measurements, a 3D digital image correlation (DIC) system was used to monitor material

deformation during testing. High strain rate testing, on the other hand, was performed at

the General Motors R&D department using the servo-hydraulic machine shown in Figure

2.2b. The Instron VHS (very high speed) unit, shown in Figure 2.2b, is a two column

servo-hydraulic tester with static and dynamic load ratings of 80 kN and 50 kN,

respectively. The load frame has a 250 kN rating. Deformation of a test specimen occurs

via the motion of an actuator, which can achieve a maximum speed of 20m/s, connected

to the lower grip of the gripping mechanism or slack adaptor. The actuator retracts into

the bottom portion of the tester and is hidden from view in Figure 2.2b by the black base.

Actuator speed is controlled using oil pressure which is regulated using a remote pump

and servo-valves on the back side of the unit (not shown). The VHS unit runs in two

modes: a low-pressure mode (pressure = 185bar), which is used to move the actuator

manually when gripping or un-gripping a test specimen, and a high-pressure mode

(pressure = 210bar), which is needed to perform the high-speed testing. The upper grip is

connected to a Kistler 9071A piezoelectric load washer (400 kN rating, 5 MHz sampling)

which is connected to a movable yoke (shown in Figure 2.2c). Also shown in Figure 2.2c

33

are two high-speed digital cameras for strain measurement. These are attached to a

custom tripod and placed on the backside of the tester to capture images of a test

specimen as it is deformed.

As mentioned earlier, the lower grip of the VHS unit is connected to the slack

adaptor which is very crucial for the operation of the servo-hydraulic machine at high

speeds. A slack adaptor is a mechanism for gripping a test specimen that consists of a

stationary upper grip, a lower grip that is connected to the actuator via an aluminum

collar and damping washers. It allows the actuator to attain a set constant speed before

the load is applied to the specimen, and it can be used in both open (high speeds) and

closed-loop control (low speeds). The slack adaptor unit shown in Figure 2.2c is a

redesigned version of the original Instron unit which was found to be unsuitable for high

strain rate testing of AHSS. To minimize inertia and hence load train ringing, the grips

and slack rod were designed with Ti-6Al-4V. To further reduce load train ringing, two

polyurethanes and two aluminum impact washers were placed between the lower grip and

the aluminum collar that screws into the top the actuator (the lower grip screwing into the

collar). The aluminum collar was redesigned to enable a reciprocating fit for the slack rod

that runs through the collar into the actuator. The slack rod (also Ti-6Al-4V) is a

cylindrical rod with one end as a conical (i.e., tapered) section that is engaged by the

aluminum collar. The gripping of the specimens to the grips was done using smooth M2

tool steel pins (see Figure 2.2c) as opposed to the original design of the unit which

involved bolt/nut combinations which resulted in spurious twisting of a test specimen

(detected by DIC) since the specimen would ride up and down the bolt threads during

34

(a) (b)

(c)

Figure 2.2: Experimental setups and test specimens used for (a) quasi-static testing and

(b) high rate testing (Servo-hydraulic) (c) detailed view of the components of the Instron

VHS unit

35

loading. To reduce vibrations that result from the engagement of the slack adaptor rod

with the collar/actuator combination, a damping material was wrapped around the conical

section of the slack rod.

Strain measurement was conducted with a high-speed 3D DIC system. A

significant advantage of 3D DIC is that it provides full-field strain maps of the entire

gauge section while eliminating any load train compliance effects. This is especially

useful in quantifying the inhomogeneous deformation at high strain rates such as local

plastic deformation in propagative instabilities (e.g., these were noted at quasi-static rates

during tensile testing of the Med. Mn. (10wt.%) steel). The 3D DIC system also provides

a careful check on instrumentation artifacts (e.g., spurious bending and twisting of the

test specimen) during high strain rate deformations that otherwise go undetected with

other means. The 3D DIC system consisted of two Photron SA1.1 black and white

cameras (Figure 2.3), multi-channel data acquisition (DAQ) box for image capture and

data logging (strain gage voltages and load washer loads).

The Photron SA1.1 cameras have a maximum frame rate of 675,000 frames/s at a

64x16 pixel resolution. To capture the gauge area of the specimens until failure, a

minimum resolution of 512x512 pixels is needed which limits the maximum camera

frame rate to 20,000 frames/s. At this frame rate, the optimal exposure time is 1/95000 s.

If the longer exposure time is used, then the images will contain motion blur which

prevents strain measurement. This problem occurs when there is a significant motion of

the specimen between the period at which the camera aperture opens and closes.

36

The maximum strain rate required was 500s-1, which was achievable with the

maximum frame rate of 20,000 frames/s. The other nominal strain rates used in the tests

were: 0.5s-1, 5s-1, and 50s-1. These rates were deemed reliable for adequate sampling of

high strain rate behavior of the various steels tested. The captured images are then loaded

in the post-processing software to obtain a point-by-point mapping, on a user-chosen

grid, of displacement and strain field components as well as velocities and strain rates.

A significant challenge with high-speed testing using servo-hydraulic units is load

signal ringing due to vibrations in the load train. The piezoelectric load cell used in the

VHS Instron tester, Kistler 9071A (Fig. 3.2b), provides a ringing-free load signal up to

actuator speeds near 1.0m/s. Beyond this speed, ringing obscures the material behavior

Figure 2.3: Photron SA1.1 high-speed cameras

37

since the load signal swamps that from the actual material response, as shown in Figure

2.4a. To reduce ringing and make sure any minor specimen bending was eliminated, one

linear strain gauge was adhered to both sides of the two gripper ends on each specimen

tested at 500s-1 and specimens tested at 50s-1. Linear strain gauges, with a resistance of

350 Ω and gage factor of 2.04 were used since elastic strains are measured in the grip

region in one direction only. Each strain gauge can measure up to 5% strain. The two

terminals of each strain gauge are connected to a signal conditioning box. The signal

conditioner completes the quarter bridge circuit using in-built resistances and outputs the

voltage signal to the DAQ box. Figure 2.4b shows a flow curve in which ringing has been

substantially minimized.

(a) (b)

Figure 2.4: (a) Engineering stress-strain curve of a test ran at a speed of 20m/s showing a

dominant ringing signature and (b) Engineering stress-strain curve of a test run at a speed

of 20m/s showing the elimination of ringing

38

2.2.3 Specimen Geometry

An extensive search of the literature revealed a wide variety of different servo-

hydraulic test specimen geometries. Yan et al. [15] showed four examples used in their

round-robin study on dynamic testing while ten different geometries are shown in

Appendix IV of Ref. [38]. It was determined that the most important considerations for

the test specimen geometry are that it must take into account the gripping mechanism,

load train ringing [3], strain gauge application, and it must not promote non-uniform

strain evolution. A careful review of literature references [39], resulted in the two test

specimen geometries (Figure 2.5). Each is a dogbone geometry. A fundamental

difference between the two is the gauge length, i.e. one has a gauge length of 20mm

(Figure 2.5a), and the other has a gauge length of 10mm (Figure 2.5b). One gripper end

was designed to be longer than the other to accommodate linear strain gauges. Shorter

gauge lengths allow for higher average strain rates at lower actuator speeds with reduced

ringing. However, inferences about the effect of gauge section length of strain evolution

uniformity from ISO 26203-1,2 [19], SEP1230 [20], and the modal analysis of servo-

hydraulic load train ringing by Yang et al. [3] suggest that shorter gauge lengths can

adversely affect the strain field homogeneity in the gauge area.

The DIC methodology was used to examine the effect of the gauge length on the

strain homogeneity. Repeated tests of AHSS specimens that neck using both geometries

showed anomalous strain accumulation artifacts during plastic deformation. Figure 2.6a

provides the evolution of strains along the centerline of a 10mm gauge length specimen

which is distinctly non-uniform. Figure 2.6b shows uniform strain accumulation in a 20

39

mm gauge length specimen. While Figure 2.6c shows the strain countors after yielding.

Based upon the results in Figure 2.6 and the inferences regarding specimen selection in

ISO 26203-1,2 [19], SEP1230 [20], the specimen with 20 mm gauge length was selected

to perform the test with (Figure 2.5b).

(a)

(b)

Figure 2.5: High rate tensile specimen geometry with (a) 20mm gauge length (b) 10mm

gauge length

40

(a) (b) (c)

Figure 2.6: (a) Evolution of strain along the centerline (using a virtual extensometer) of a

10 mm gauge length specimen tested at 500s-1. Zero position denotes the center of gauge

section. (b) Evolution of strain along the centerline (using a virtual extensometer)

2.3 Results

2.3.1 Flow Behavior

Yield Strength, strain at uniform elongation, tensile strength, and strain at fracture

per steel at each strain rate are listed in Tables 2.2-2.5. Engineering stress-strain curves

obtained at the five strain rates are shown for all six materials in Figure 2.7. Part (a)

displays typical tensile flow curves generated for the CR5 steel. These show a monotonic

increase in flow stress, both yield, and tensile stress, as strain rate is increased. A

significant increase in the yield strength and the ultimate strength is noted for the higher

strain rate, accompanied by a reduction in the uniform and total elongation of the

material; this is indicative of strong strain rate sensitivity. This is typical behavior of mild

steels, and several researchers have reported the same behavior in previous studies [22,

41

26, 40]. The red curve at 500s-1 shows a single hump just beyond 0.1 engineering strain

which is unlikely to be related to ringing since it lacks the periodic waviness

characteristic of ringing. Instead, this results from a stiffness issue pertaining to the 0.75

mm initial thickness of the CR5 steel. Repeated tests of the CR5 at 500s-1 showed the

same behavior which for some tests was more extreme, and for others, it was less

extreme. This was the only material that exhibited this behavior even with dual strain

gauges that functioned as the load cell. A search through the literature revealed very little

information on dynamic tensile testing results for sheet steels with initial thicknesses at or

close to the 0.75 mm thickness of the CR5 material using a servo-hydraulic tester. In the

course of searching for alternatives to servo-hydraulic testing of a thin sheet, only two

notable references were located, and both describe custom high strain testing

instrumentation to deform metallic foils in tension. Specifically, Paul and Kimberly [41]

describe a desktop Kolsky bar apparatus for testing metallic foils (99% Magnesium, 0.2

mm thick) at strain rates up to 1000s-1. Also, Kwon et al. [42] developed a “High-Speed

Micro Material Testing Machine” and tested 0.096 mm thick AA1100 foils over the 1s-1 -

100s-1 range. As a check on the servo-hydraulic instrumentation and test methodology

detailed previously, the CR5 material was tested at 500s-1 on a Kolsky bar at NIST,

Gaithersburg. The NIST Kolsky bar is shown in Figure 2.8a, and the associated specimen

geometry is shown in Figure 2.8b. Here, the specimen is plastically deformed by an

elastic wave pulse initiated by a striker through the load train consisting of incident and

transmission bars and the test specimen. Mates and Abu-Farha [12] provide additional

details on the NIST Kolsky bar. Using strain fields computed from stereo DIC (see the

42

two cameras called out in Figure 2.8a), the 500s-1 flow curve (red) in Figure 2.8c does not

have the hump in the red curve at 500s-1 in Figure 2.7a. Repeated tests of the CR5

material with the Kolsky bar produced results essentially identical to the 500s-1 curve in

Figure 2.7a. An overlay of the 500s-1 curves in Figures 2.7a and 2.8c provided confidence

that a “fit” through the red curve at 500s-1 in Figure 2.7a, so as to eliminate the hump

from the servo-hydraulic test, is reasonable. Servo-hydraulic testing can, therefore, be

used in future tests with these thin sheet materials provided that any filtering or alteration

of the data is validated by other means, such as a Kolsky bar test.

Figure 2.7b shows flow curves for the 201LN (fully austenitic) stainless steel.

Here, there is a reduction in both strain hardening rate and tensile strength at the 0.5s-1

and 5s-1 strain rates relative to 0.005s-1 (see Table 2.3). This is presumably due to

reductions in the transformation hardening (TRIP) effect. Tensile strengths for the 201LN

steel, however, begin to increase with strain rate after an initial decrease and exceed that

of the quasi-static rate at the highest tested rates (see Table 2.4). Inspection of Figures

2.7a, b and Table 2.3 reveals that both the CR5 and 201LN steels exhibit significant

reductions in uniform elongation with increased strain rate. The significant reduction in

work hardening rates between quasi-static and high rate testing is likely an effect of the

reduction of strain hardening due to reduced austenite transformation at increased strain

rates in the 201LN steel. This is attributed to increased adiabatic heating as discussed by

Isakov et al. [43] but a definitive demonstration would require the addition of an infrared

camera that is coupled with the DIC image capture process. The increased yield strength

for CR5, which is most probably due to the dislocation slip in ferrite with little increase

43

in strain hardening rate, is in-line with behavior reported in the literature [44]. Note that

the 500s-1 (red) flow curve in Figure 2.7a shows a very modest ringing artifact (i.e., small

amplitude oscillations) which for all practical purposes can be ignored.

Table 2.2: 0.2% Offset Yield Strength (MPa) at the different strain rates.

Table 2.3: Engineering strain at uniform elongation at the different strain rates.

Table 2.4: Ultimate Tensile Strength (MPa) at the different strain rates.

Steel 0.2% Offset Yield Strength (MPa)

0.005s-1 0.5s-1 5s-1 50s-1 500s-1

CR5 160 226 265 307 322

201LN 320 370 395 416 580

DP590 Steel 365 380 417 447 480

DP980 715 763 776 832 900

Med. Mn TRIP 690 743 765 782 815

QP980 680 720 760 780 870

Steel Engineering strain at uniform elongation

0.005s-1 0.5s-1 5s-1 50s-1 500s-1

CR5 0.27 0.22 0.21 0.122 0.11

201LN 0.59 0.45 0.454 0.422 0.34

DP590 Steel 0.17 0.15 0.15 0.154 0.19

DP980 0.09 0.08 0.077 0.082 0.097

Med. Mn TRIP 0.33 0.30 0.31 0.29 0.237

QP980 0.111 0.113 0.114 0.11 0.136

Steel Ultimate Tensile Strength (MPa)

0.005s-1 0.5s-1 5s-1 50s-1 500s-1

CR5 287 311 335 372 440

201LN 743 700 740 755 870

DP590 Steel 630 640 680 714 800

DP980 1052 1080 1090 1110 1158

Med. Mn TRIP 1115 821 826 867 967

QP980 1017 1020 1050 1065 1130

44

Table 2.5: Engineering Strain at Fracture at the different strain rates.

(a) (b)

Steel Engineering Strain at Fracture

0.005s-1 0.5s-1 5s-1 50s-1 500s-1

CR5 0.495 0.41 0.433 0.48 0.33

201LN 0.63 0.59 0.623 0.63 0.59

DP590 Steel 0.26 0.28 0.292 0.315 0.34

DP980 0.143 0.147 0.17 0.188 0.211

Med. Mn TRIP 0.37 0.39 0.40 0.407 0.44

QP980 0.158 0.185 0.20 0.23 0.245

45

(c) (d)

(e) (f)

Figure 2.7: Engineering stress/strain curves at 0.005s-1 and 500s-1 for (a) CR5 (b) SS 201 (c)

DP590 (d) DP980 (e) Medium Mn. (10wt%) and (d) QP980.

46

(a) (b)

(c)

Figure 2.8: (a) Kolsky bar at NIST (b) Specimen Geometry used for Kolsky testing at

NIST (c) Engineering stress-strain curves with the 500s-1 obtained from the Kolsky

testing.

47

A different behavior is observed in the dual-phase steels; the DP590 and DP980

(Figure 2.7c and d); overall, there is a monotonic increase in flow stress, both yield, and

tensile stress, as strain rate is increased. Total elongation is notably larger at high strain

rates in both cases; the change in the uniform elongation, however, is not as notable and

stays almost the same for both materials. The stress-strain curves do not show the change

in shape observed in CR5 and SS 201; i.e., a significant drop in strain hardenability at

higher rates. On the contrary, the DP980, in particular, seems to strain to harden more

during high rate deformation, while DP590 does not show any notable change between

the different strain rates.

Flow curves in Figure 2.7e for the Medium Mn (10 wt. %) TRIP steel, like the

201LN steel, also exhibit significant differences when the strain rate is increased from the

quasi-static rate. Here, the hardening rate associated with quasi-static deformation is

significantly higher than that at the dynamic strain rates. As strain rate is increased, the

ultimate tensile strength of the Medium Mn (10 wt. %) TRIP steel decreases significantly

between 0.005s-1 and 0.5s-1 and 5s-1 (see Table 2.4). Yield point elongation is clearly

indicated in flow curves at all strain rates except that for 500s-1. An issue that was noted

pertained to results with and without dual strain gauges for strain rates ≥50s-1. If strain

gauges were not used, ringing with a frequency close to that of the resonant frequency of

the load train (~9.5 kHz) was observed only during yield point elongation at the dynamic

rates. This effect vanished when strain gauges were applied to the Medium Mn (10 wt.

%) TRIP test specimens. It is currently thought that the ringing effect within the yield

point elongation regime (which is minimal to nonexistent during subsequent work

48

hardening) results from the reduced damping of the tensile specimen during lateral

deflection of the tensile specimen during high rate testing. That is, the lack of stress

increase with specimen deflection (strain) during yield point elongation results in reduced

specimen damping and notable ringing at the resonant frequency of the load train. Again,

an expansive look through the literature on dynamic materials testing revealed no

dynamic tensile testing results for sheet steels (or any other metal alloys) that exhibit a

protracted yield point elongation up to 500s-1. Plastic instability, denoted by a single

Lüders band during yield point elongation [45] followed by one or more bands that

propagate along the gauge length during hardening, was observed at all strain rates at

which the Medium Mn (10 wt. %) TRIP steel was tested. However, the number of bands

and band propagation speed decreased significantly with increasing strain rate as

indicated above for the 500s-1 rate. The propagative instabilities resembled Type A

Portevin-Le Châtelier bands that are associated with dynamic strain aging and a negative

strain rate sensitivity of the flow stress. This behavior has been observed in other AHSS

such as fully austenitic TWIP steels [46]. Highly inhomogeneous deformation in the

Medium Mn (10 wt. %) TRIP steel has been associated with austenite transformation

localized to the propagating bands [47] using synchrotron X-ray diffraction. Not only is

plastic deformation inhomogeneous, but austenite transformation is inhomogeneous as

well.

Figure 2.7f shows the flow curves for the triple-phased QP980. The response of

the flow curves to strain rate in QP980 is similar to that of DP980. The notable difference

is the drop in the strain hardenability at higher strain rates of QP980. This can be mainly

49

attributed to the presence of austenite in the microstructure and the effect higher strain

rates has on its deformation-induced transformation.

2.3.2 Strain Rate Sensitivity

Figure 2.9a shows the yield strength variation with the strain rate for all the

materials tested. An approximately equal and logarithmic increase in yield strength is

shown by CR5 and Medium Mn (10 wt. %) in Figure 2.9 as measured by the slope of the

linear plots. That is, the relationship between yield strength and logarithm of the strain

rate is linear and of equal slope. The same is not true for other materials; the three other

materials show a variation in the slope of the yield strength increase. 201LN for instance

exhibit an increase in the slope at 500s-1, while DP590 exhibits an increase in the slope at

0.5s-1 and finally DP980 shows the increase of slope at 50s-1.

The same trend of variation in yield strength is observed in all the materials

except 201LN and Medium Mn. (10 wt. %) steels, in other words, materials without any

significant presence of austenite in the microstructure (Figure 2.9b). For 201LN and

Medium Mn. (10 wt. %) steels, there is an initial drop in the ultimate tensile strength

when the strain rate increases from 0.005s-1 to 0.5s-1 and then the tensile strength

increases with strain rate. The initial drop in the tensile strength is significantly larger for

Medium Mn. (10 wt. %) steel as compared to 201LN and the reason for the drop and the

variation between the two materials is discussed in section 2.4.

A differential (or instantaneous) strain hardening coefficient, pn n , which is

independent of any hardening law, and can be defined by [48]:

50

ln

(ln )p

dn

d

(2.1)

where is true plastic stress and p is a true plastic strain. n can be computed from a

true stress-true strain curve using the following algorithm

5 5 4 4 3 3 2 2 1 1

5 5 4 4 3 3 2 2 1 1

(ln ln ) (ln ln ) (ln ln ) (ln ln )(ln ln )

(ln ln ) (ln ln ) (ln ln ) (ln ln ) (ln ln )

i i i i i i i i i ii

i i i i i i i i i i

n

(2.2)

Note that there are other approaches to computing ( )pn n in the literature [ref].

Among these, the method proposed by Krauss and Matlock [49] is of interest:

( )p

p

dn

d

(2.3)

where d

d

is the true strain hardening rate. Equation 2.3 is derived from the Holloman

equation:

n

pK (2.4)

The predicted n from both approaches are essentially identical.

Equation 2.2 was used to calculate the strain hardening rate for each material at

all strain rates, and the results are plotted in Figure 2.10. The strain hardening exponent

was calculated over the 5-10% true strain range and plotted. A decrease in strain

hardening index is observed with increased strain rate in Medium Mn. (10 wt. %) steel,

201LN, and CR5 steels. This reduction is especially pronounced in the Medium Mn (10

wt. %) steel for strain rates > 0.005s-1, and this is again attributed to the effect of strain

rate on the austenite in the microstructure. The DP steels, in contrast, display minimal

51

sensitivity to the strain rate and this insensitivity is likely attributable to the large

population of mobile dislocations in ferrite before the onset of deformation during testing

and to the significant presence of martensite in the microstructure. As for QP980, the

strain hardening exponent remains almost constant up until 5s-1, and then the strain

hardening exponent increases, this behavior will be discussed in section 2.4.

Energy absorbed until the fracture is critical to the performance of the material

during a vehicle crash incident, and this property can be represented by toughness.

Toughness can be calculated by calculating the area under the stress-strain curve. This

means that the increase in strength with strain rates does not necessarily mean that the

material can absorb more energy during a crash, both strength and ductility must be taken

into consideration. To take a closer quantitative look at the effect of strain rate on the

energy absorbing capabilities of the six materials, the percentage change in four

mechanical properties between 0.005 s-1 and 500 s-1 was calculated and plotted in Figure

2.11. Part (a) shows how strain rate affects the yield strength in the materials. As

expected, the largest increase in the yield strength was observed in CR5, which showed

an increase of ~105%, followed by SS 201, which showed an increase of ~79%. As for

the multi-phased materials, the increase is not that significant, starting with DP590 which

showed an increase of 28% while its counterpart DP980 showed only a 2% increase.

While Medium Mn. (10 wt. %) and QP980 showed 14% and 9% respectively. The same

trend was almost observed when comparing the change of tensile strength in Figure

2.11b, but the values were different. The two materials that don’t follow the same trend

52

(a)

(b)

Figure 2.9: (a) Yield strength (MPa) vs log strain rate (b) Tensile strength (MPa) vs log

strain rate

53

Figure 2.10: n-value vs log strain rate

was SS201 and Medium Mn. (10 wt. %). Both showed a significant drop in the

percentage change of the tensile strength when comparing them to the percentage change

in the yield strength. Medium Mn. (10 wt. %) even showed a drop in its tensile strength.

This apparent change in the trend is due to the austenite phase in the microstructure that

is hugely influenced by the strain rate that affects the strain hardening behavior and

subsequently the tensile strength. This will be discussed further for SS201 in the next

section and Medium Mn. (10 wt. %) in the next chapter. However; for the other

materials, the strain hardening is not significantly changed at different strain rates, and

thus, the percentage change in the tensile strength is similar to that of yield strength. As a

result, the percentage increase in ultimate tensile strength for CR5 was ~57%, and that of

54

SS 201 was ~21%. For DP980 and QP980, strain hardenability was not significantly

affected at high rates, confirmed by the similar percentage increase of both the ultimate

tensile strength and the yield strength for the two materials (~7%).

Total elongation is an indication of material ductility and has a strong influence

on the energy absorption capabilities of the materials. As in the case of tensile strength,

the phases in the microstructure affect the response of the material to changes in strain

rate. Figure 2.11c shows the percentage change in total elongation for the six materials.

The total elongation of both CR5 and SS 201 decreased at high rates, agreeing with the

classic theory of strain rate sensitivity, which dictates that ductility decrease with the

increase in strain rate (for materials with a positive strain rate sensitivity). DP590 and

DP980 both showed an increase in the total elongation, more significantly with DP980

(due to its higher martensite content, as discussed in the next section). Medium Mn. (10

wt. %) steel and QP980 also showed an increase but to a very different magnitude.

Medium Mn. (10 wt. %) showed an increase of only ~19%, while QP980 showed the

most significant percentage increase of all materials at ~56%. The results obtained from

the multi-phase materials do not conform to the definition of positive or negative strain

rate sensitivity, as those materials showed a disconnect between the effect of strain on the

strength and ductility, and some of them even showed a disconnect between the response

of yield and ultimate strength to different strain rates.

Finally, the combined effects of all these properties will dictate the toughness of

the materials at different strain rate and therefore, their energy absorbing capabilities at

different strain rates. Figure 2.11d compares the effect of strain rate on the toughness of

55

the four materials. As noted, the percentage increase of toughness for DP590 was the

highest (~69%), which can be easily predicted due to the increase in the strengths (both

yield and tensile) and ductility observed in the material. Followed by both DP980 and

QP980 with ~28% and ~30% respectively, this would not be very easy to predict if you

only look at the stress-strain curves as it shows that the material is barely affected by the

change in strain rates. Then the austenite-based materials follow with almost identical

percentages increase of ~20%, and finally, CR5 showed a negligible increase of only

~5%. In summary, it is clear here that the six materials (driven by their different

microstructures) responded differently to increasing strain rate, leading to a net change in

toughness observed in Figure 2.11d. Total ductility reduction in CR5 at high rates

counter-acted the increase in the strength, leading to a small increase in toughness.

Toughness increase in SS 201 was strength-driven, since total ductility remained

unchanged, while toughness increases in DP980, Medium Mn. (10 wt. %) and QP980

was primarily ductility-driven. Moreover, of course, DP590’s increase in toughness was

both strength and ductility driven.

56

(a) (b)

(c) (d)

Figure 2.11: Percent Change between 0.005s-1 and 500s-1 of (a) Yield Stress (b) Tensile

Strength (c) Total Elongation and (d) Toughness.

57

2.4 Discussion

All the results above suggest that the phases present in the microstructure of steel

greatly affects its response to increased loading rates. Based on Orowan’s principle [50],

the strain rate is directly proportional to dislocation velocity and other dislocation

parameters. When materials are loaded at high rates, the density and the speed of

dislocations are higher. The relationship between dislocation velocity and the critical

resolved shear stress have been studied extensively in the literature, starting with the

work of Gilman and Johnston [51, 52] and followed by many others; such as the work of

Parameswaran et al. on aluminum [53]. Different relations were derived, but all assume

the speed of dislocation movement is directly proportional to the critical resolved shear

stress. Therefore, when materials are loaded at higher strain rates, higher stress is needed

to overcome the critical resolved shear stress to move the dislocations at the speed

required for the strain rate. This, in turn, implies higher yield and flow stresses during

deformation at high rates, and hence reduced ductility.

The above coincides with the behavior of CR5 at high rates, where the ultimate

tensile strength increased significantly; this is mainly due to the ferrite phase that

constitutes 100% of the microstructure of CR5. However, when looking at the results of

the other materials, it is noticeable that the strain rate sensitivity showed different trends.

To explain this, it is important to look closely at what happens during high strain rate

testing. During any kind of a mechanical test, where work is used to produce deformation

on the materials, heat will be generated. Under quasi-static conditions, test duration is

long giving enough time for the heat generated during the test to be dissipated. Thus a

58

small increase in temperature is observed; it is therefore well established in the literature

that isothermal conditions take place during quasi-static testing. This is not the case

during high rate testing where the test occurs in a very short period; the 500s-1 tests

carried out in this study were completed within 0.4ms. In this short period, the generated

heat does not have enough time to be dissipated, and thus the specimen’s temperature

increases significantly, changing the conditions from isothermal to adiabatic. For CR5

mild steel, which consists of soft and ductile ferrite only, the effect of temperature rise

softening is not as significant as the work hardening due to the higher loading rate. Thus

the net effect is a significant increase in strength at higher strain rates. Moving to the

dual-phase steels, which contain both ferrite and martensite in their microstructure which

has different properties and responses to different strain rates. Martensite is a hard and a

brittle phase, and high temperature (at high strain rates) causes significant softening in the

behavior of the material. This leads to two opposing effects taking place when testing

DP590 and DP980 at high strain rates: work hardening which occurs primarily in the

ferrite; and softening due to temperature rise, which occurs primarily in the martensite.

The latter was reported in the work of Hwang et al. [29], which showed a slight decrease

in strength and an appreciable increase in ductility in a Martensitic steel as the result of

an increase in deformation rate. These opposing effects cause the reduction in the

strength increase observed with the dual-phase steels as opposed to CR5. The strength

increase reduced from 58% (CR5) to 30% (DP590) to 6% (DP980). This also confirms

the effect of martensite as the percentage of martensite increase from 0% to 40% to 70%

(see Figure 2.12). Regarding ductility, a similar but opposite trend can be observed, the

59

higher the martensite content, the higher is the increase of ductility with strain rate. This

is because the drop in ductility of ferrite is outbalanced by the increase in ductility in the

martensite.

Another point to add is that the post-necking elongation portion increases (this is

the case with all six steels regardless of the microstructure). Note the post-necking

elongation of ~12-20% strains in all six materials at 500 s-1 is significantly higher than

the quasi-static post-necking strains. This is believed to be primarily driven by the

localized temperature increase, shifting the necking from localized to diffused, and thus

prolonging the deformation.

Figure 2.12: Effect of martensite on the percentage increase of tensile strength

Moving on to the third main phase in steels and the most complex of all; austenite.

Austenite is also a ductile phase like CR5, but it has a unique property that causes its

60

behavior and modeling to be unique and challenging. That unique property is that it

undergoes a transformation to the martensite phase when deformed. The stability of the

austenite phase dictates the rate of formation and the amount of martensite formed during

deformation. The stability mainly depends on the steel’s composition, the rate of

deformation and temperature, as well as the amount of plastic strain, stress state during

deformation, grain size and its morphology [54]. Of the prime three factors, the effects of

composition and temperature on the rate of transformation have been studied extensively

in the literature [55, 56]. It is well-documented, based on many studies on 300 series

austenitic stainless steels (SS300), that temperature rise due to deformation heating

reduces the rate of transformation, thus giving the effect of stabilizing the austenite during

deformation, yet the degree of stabilizing depends on the composition of the material

[57]. The third factor and the factor most relevant to automotive applications and this

study is the strain rate. The changes in the transformation kinetics at high strain rates can

be related to temperature. As mentioned earlier, high deformation rates are considered to

occur under adiabatic conditions, as opposed to isothermal conditions during quasi-static

deformation; the significant increase in temperature via adiabatic heating affects the

kinetics of phase transformation [58]. The latter was studied by some researchers such as

Talonen et al. [57], Lichtenfeld et al. [54] and Isakov et al. [43] who showed that

austenite-to-martensite transformation in SS3xx was generally inhibited at high strain

rates, leading to lower ductility.

Studying austenite-to-martensite transformation kinetics requires accurate

measurements of phase volume fractions and their changes in response to plastic strain.

61

Of the different methods used for such measurements, the most common is magnetic

induction, which takes advantage of the ferromagnetic properties of α’ martensite to

determine its volume content within a specimen [59]. X-ray diffraction (XRD), which

takes advantage of the fact that α’ martensite and austenite have different crystal

structures, is another common method for measuring the austenite/martensite volume

fraction. The intensity of the measured diffraction peaks is proportional to the volume

fraction present. Other methods of phase fraction measurements include neutron

diffraction and electron backscatter diffraction (EBSD); while the former is probably the

least used due to availability or accessibility limitations and the fact that it is relatively

slow, the latter is two dimensional (area fraction measurement technique) and lacks the

statistical representation of the material being analyzed. Comparisons between the

different measurement methods can be found in this reference [60].

Though the majority of efforts in the literature have focused on 300 series

stainless steels and reached similar conclusions regarding the adiabatic heating effects on

inhibiting transformation at high deformation rates, in almost all cases, different alloys

were affected by different degrees, and thus their high rate tensile behaviors and

transformation kinetics were different. To our knowledge, there have been no reports in

the literature on the high rate deformation and transformation kinetics in 200 series

stainless steels (SS2xx).

Beginning with the single-phased austenitic stainless steel 201LN, positive strain-

rate sensitivity is clearly inferred when looking at the yield strength; however, the quasi-

static tensile strength is higher than that observed at the intermediate rate. This is a direct

62

result of the significant reduction in strain-hardening rate (strain hardening exponent n)

with increasing strain rate. Total elongation, on the other hand, is noted to decrease as the

strain rate increases, while there is hardly any change in the total elongation with strain

rate. While the Orowan’s principle mentioned earlier explains the general systematic

increase in yield strength with loading rate, it does not explain the drop in hardening

behavior, and as a result, the unique trend of changes observed in the tensile strength. For

that, a closer look at the phase fraction changes with plastic strain is needed.

The two methods for measuring the volume fraction of martensite produced

similar trends, as shown in the two parts of Figure 2.13, but different values, especially

for the quasi-static condition. It is out of the scope of this paper to discuss the differences

and relative accuracies of the two methods; previous researchers [61, 62] concluded that

there is significant variability in the measurements when using different methods, with no

robust way of identifying which method is the best. The results in Figure 2.13 indicate a

strong strain rate effect on the transformation of austenite to martensite in SS201. The

volume fraction of martensite formed at quasi-static rates at around 50 % strain was

measured to be 13 % using magnetic induction and 24 % using the XRD. These values

dropped significantly for the higher strain rates with no significant difference between the

three strain rates (1, 10 and 500 s-1). This behavior is different from what has been noted

in the literature for SS3xx alloys, which have been studied extensively at different rates.

Though increasing the strain rate has been reported to slow down the austenite-to-

martensite transformation in SS3xx alloys, the transformation is not entirely inhibited; in

fact, over 50% transformation is reported at ~200 s-1 [54, 57]. In the SS201 studied here,

63

the transformation is almost entirely inhibited at low-intermediate rates (as low as 1 s-1).

This can be related to alloy content; the SS201 has a lower nickel content and higher

manganese content compared to SS301, and this was shown to affect the stability of the

austenite phase during quasi-static deformation in the work of Tavares et al. 2009 [63].

The results shown here when compared to the results shown in the literature for SS3xx,

lead to the same conclusion for high strain rates; that is the greater stability of austenite in

SS2xx compared to that in SS3xx. For instance, the plots of martensite formation in [57]

and [54] show a gradual drop at different strain rates, and not as abrupt as the ones sown

here in Figure 2.13.

Moreover, while more than 80% of the austenite transforms in SS3xx, a smaller

fraction of the austenite transforms into martensite in SS201 (up to ~25% at quasi-static

conditions). These disparities in the transformation behavior are responsible for the

significant difference in strain rate sensitivity of the two grades; rate effects on the

stress/strain curves of SS201 (Figure 2.7b) are strong compared to those reported in the

literature for SS3xx [54, 57]. The less stable austenite in the SS3xx results in the

formation of larger fractions of martensite; the latter is known to have low/negative rate

sensitivity, which results in the lower impact on the stress/strain behavior [64].

For QP980, the small increase in strength at higher strain rates is similar to that

observed in DP980, which can be attributed to the similarities in the microstructure

regarding the presence of martensite and ferrite. The slightly lower martensite and

slightly higher ferrite volume fractions in QP980 (compared to DP980) results in the

64

(a) (b)

Figure 2.13: Volume fraction of martensite measured at different plastic using: (a)

magnetic induction (b) XRD.

slightly higher rate sensitivity (Figure 2.11a and b). However, regarding ductility (Figure

2.11c), QP980 showed an increase of ~55% in total elongation at higher strain rates,

which is higher than that exhibited by DP980 and by all of the other materials. Since this

cannot be driven by the ferrite/martensite portion of the microstructure, it is believed to

be associated with the austenite present in QP980. However, the effects and exact

mechanisms are not apparent. While austenite is known to exhibit deformation-induced

transformation into martensite, causing additional resistance to necking and thus

enhanced tensile ductility and strain hardening, it was shown earlier that in SS201 the

austenite-to-martensite transformation is inhibited at high rates since temperature

increase reduces the chemistry driving force and increases the stacking fault energy,

which has the combined effect of inhibiting the transformation. So, this does not agree

nor explain why QP980 exhibits improved total elongation, on the level of ~55%, at high

65

strain rates. The effects of high loading rates on the austenite-to-martensite

transformation in TRIP-assisted steels (including QP and Medium Mn. steels) is not fully

understood. While some researchers claim that transformation should be inhibited at high

rates (based on the results in fully austenitic stainless steels), others indicate the opposite.

For instance, Liu et al. [65] showed that at strain rates higher than 100s-1, QP980 shows

increased ductility (which is similar to the results shown here in Figure 2.11c). Liu et al.

also measured the austenite content in the material after deformation and showed that the

transformation occurs at higher rates when the material is tested at high strain rates.

Current understanding based on data presented in this and a previous study [66] suggests

that the increased ductility with strain rates associated with QP980 and Medium Mn. (10

wt. %) are a result of combined and potentially counteracting effects. Thermal

stabilization of austenite to transformation via adiabatic heating may delay transformation

to greater strains, thereby extending the onset of necking. Additionally, modified

dislocation slip (increasingly planar with increased strain rate and increasingly non-planar

with increased temperature) may compete concerning the creation of dislocation

interactions necessary to induce austenite transformation to martensite.

2.5 Summary and Conclusions

The effects of the phase content on the strain rate sensitivity of six materials with

different microstructures were investigated. The materials were: CR5 (100% ferrite),

SS201 (100% austenite), DP590 (60% ferrite, 40% martensite), DP980 (30% ferrite, 70%

martensite), Medium Mn. (10 wt. %) Steel (34% ferrite, 66% austenite) and QP980 (40%

66

ferrite, 50% martensite, 10% austenite). The materials were deformed at 0.005, 0.5, 5, 50

and 500 s-1, and their mechanical response was examined. The ferrite phase was shown to

have a strong “classical” positive strain rate sensitivity, exhibited by an increase of

strength and a decrease of ductility with strain rate. When introducing martensite in the

mixture, the behavior is changed. This is because martensite in its nature is a hard and a

brittle material and has good sensitivity to the high-temperature conditions of high strain

rate testing. The softening effect of temperature on martensite counteract the effect of

ferrite and causes the increase in strength experienced by DP590 to be lower than that of

CR5 and caused the ductility of DP590 to increase with an increase in strain rate. When

increasing the martensite volume fraction in the microstructure as is the with DP980, it

magnifies the effects that were observed in DP590, and the material showed almost no

rate sensitivity regarding strength and a significant increase of ductility at high strain

rates. The third phase that was studied was the austenite, which was present in three

materials: 1) single-phased SS201, 2) double-phased Medium Mn. (10 wt. %), and 3)

triple-phase QP980. Austenite is known to exhibit a deformation-induced transformation

to martensite (a harder phase), causing significant strain hardening in the plastic phase of

deformation. The effect of strain rate on this phenomenon is what mainly dictates the

behavior of austenite-containing materials. The effect of strain rate on this phenomenon

was measured by two techniques. The results show that the volume fraction of formed

martensite indicate that austenite-to-martensite transformation is only evident at quasi-

static rates (to a maximum of ~25%), and it is almost entirely inhibited at rates above

~0.5s-1. The overall impact of this on the tensile behavior is a completely diminished

67

strain hardening behavior at high strain rates, leading to a reduction in the increase of

tensile strength of the material as compared to CR5. Also, this contributed to the

reduction of the total elongation exhibited by this material at high strain rates. QP980

had a similar martensitic content as DP980 which explains the negligible rate sensitivity

regarding strength shown by both materials; however, QP980 showed notably enhanced

ductility, which cannot be explained based on the known responses of the individual

phases. Even austenite, the highly ductile phase, is known to exhibit a drop in uniform

ductility at high rates due to the inhibition of austenite-to-martensite transformation,

which was also confirmed by the results obtained here for the SS 201. Further

investigation of austenite-to-martensite transformation kinetics and its rate dependence

from the literature and a separate study helped us to understand that is most likely the

result of two counteracting phenomena: the decrease in the transformation rate of the rate

due to adiabatic heating at high strain rates delays the transformation to higher strains

causing an extension to the necking, on the other hand, dislocation slips are affected by

high strain rates and can affect the creation of dislocation interactions that are necessary

to induce austenite transformation to martensite. The combined effect of all these

properties leads to a variation in the property that is most important to the automotive

industry and plays a crucial role in deciding whether this material is deemed viable. The

change of the toughness of these materials with strain rate highlights the importance of

looking at the combined effect of these properties and just the individual effect.

Toughness in both DP980 and QP980 was shown to improve significantly at rates close

to crash conditions, which compared favorably to the conventional CR5 mild steel that

68

showed no increase in toughness. When in fact CR5 showed an increase of 105% in yield

strength as opposed to a negligible increase in the yield strength of DP980 and QP980.

As was mentioned in the introduction, the automotive sector is looking for new

alloys of steel that have the advantage in performing over the current materials that are

being used and thus can lead to a reduction in the weight of the vehicles. The state-of-the-

art research in the steel alloys is AHSSs; which utilize a complex microstructure that can

produce superior properties compared to the conventional steels. In this chapter, four

AHSSs were shown and tested with a range of combinations of phases in the

microstructure, all have shown superior properties when comparing them to the

conventional steel (CR5) which is used heavily in the automotive industry. Then the

question becomes which of these materials must be used, what is the criteria for deciding

the choice of the material, and ultimately is there a way to reverse engineer the

microstructure that can provide the optimum properties for a material to achieve

lightweighting. As for the criteria, there are multiple aspects to keep in mind, such as

cost, formability, corrosion resistance, energy absorption… In this dissertation, our

primary focus was on the performance of these materials when loaded at high strain rates

(i.e., during a crash event) and the criteria we took into account was the energy

absorption capability. We have shown that these new materials do not follow most

classical theorems and it is of the utmost importance to be able to obtain high quality,

accurate experimental data at multiple strain rates to be able to judge the materials, as

some of these materials’ performance enhance at high strain rates while others did not.

We have also shown that the microstructure has a significant influence on the behavior of

69

these materials and their strain rate sensitivity and it is a troublesome task to accurately

quantitively predict how the addition of a new phase or the removal of an already existent

phase would change the properties. Martensite, in particular, showed very promising and

improved energy absorption capabilities despite its negative strain rate sensitivity

regarding strength. Austenite, on the other hand, is a very complex material to predict and

its influence on energy absorption at high rates was not as promising as martensite. Yet, it

is still a desirable phase due to its unique transformation induced plasticity (TRIP) effect

that can substantially improve its ductility and thus its performance. Yet, high strain rates

seem to be working oppositely regarding the TRIP effect and thus more work is required

to fully understand and predict the effects strain rate has on the TRIP effect in austenite.

In summary, the goal of this chapter was to provide a comprehensive overview of

the effect of the three main phases in AHSSs (ferrite, martensite and austenite) on the

performance of the material at high strain rates and provide a basis and a starting point to

allow for designing the microstructure of the steel to have specific properties.

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80

CHAPTER THREE

3. STRAIN RATE EFFECT ON TENSILE FLOW BEHAVIOR AND

ANISOTROPY OF A MEDIUM MANGANESE TRIP STEEL

ABSTRACT

Plastic anisotropy dependence on nominal strain rate of a medium manganese (10

wt. % Mn) TRIP steel consisting of an initial 66% austenite volume fraction (balance

ferrite) is investigated. This material exhibits yield point elongation, propagative

instabilities during hardening, and austenite transformation to α'-martensite either directly

or through ε-martensite. Uniaxial strain rates within the 0.005-500 s-1 range along the 0º,

45º, and 90 º orientations were selected based upon their relevance to automotive

applications. The plastic anisotropy (r) and normal anisotropy (rn) indices corresponding

to each direction and strain rate were determined using the strain fields from stereo digital

image correlation (DIC) systems that enabled both quasi-static and dynamic

measurements. The results provide evidence of significant, orientation-dependent strain

rate effects on both the flow stress and the evolution of r and rn with strain. This has

implications not only for material performance during forming but also for the

development of future strain rate dependent anisotropic yield criteria. Since tensile data

alone for the subject medium manganese TRIP steel does not satisfactorily connect

microstructural mechanisms responsible for the observed macroscopic-scale behavior

from tensile testing, a discussion of additional tests that must supplement the mechanical

test results presented herein is provided.

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`3.1 Introduction

Usage of advanced high strength steel (AHSS) continues to increase in

automotive structural applications due to their favorable combinations of strength and

ductility [1]. However, fundamental challenges remain to implementation in vehicle body

structures. Examples are flow behavior dependence on strain rate and temperature,

martensitic transformation dependence upon deformation mode and strain path,

springback, and liquid metal embrittlement in spot welds [2].

Many AHSS grades are characterized, in part, by their multi-constituent

microstructures relative to the primarily single-constituent steels of common use. The

role of manganese (Mn) is especially important in the 1st, 2nd and 3rd generations of

AHSSs and is considered in Figure 3.1 regarding three main deformation mechanisms,

viz. dislocation-mediated plasticity, transformation induced plasticity or the TRIP effect,

and twinning induced plasticity (TWIP) observed in so-called TWIP steels. Manganese

alloy content is denoted by the large open arrow wherein 0% is associated with stable

ferrite in traditional steels and many 1st Gen AHSSs, such as dual-phase steels [3]. As Mn

content increases, increased austenite stabilization is made possible. Application of strain

to austenite bearing steels may result in the TRIP effect which involves the diffusionless

shear transformation of austenite to martensite (denoted by the blue region in Figure 3.1)

[4]. The TRIP mechanism is especially crucial in 3rd generation AHSSs [5]. These steels

can be produced with heat treatments such as quenching and partitioning (Q&P) [6], or

inter-critical annealing wherein ferrite rejects both C and Mn which partition to austenite

[7]. Enhanced work hardening results along with a delay in necking under tension, and

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greater elongations to fracture for a given steel strength [6, 8]. As Mn content is further

increased, austenite to martensite transformation may occur, in part, through the

formation of ε-martensite (denoted by the red region), the hexagonal close-packed form

of iron [9, 10]. This was recently reported for a so-called medium Mn TRIP steel with 10

wt.% Mn, which is a 3rd generation AHSS [11]. The ε-martensite phase is nucleated at

intersecting shear bands [12]. Steels that exhibit both the TRIP and TWIP mechanisms

have been associated with Mn content as low as 10 wt.% in bulk [13]. Fully austenitic

steels that exhibit the TWIP effect (2nd Gen AHSSs) have the highest Mn content and

lowest stacking fault energies [14].

Figure 3.1: Deformation mechanisms in multiphase AHSSs with Mn alloying content

Phases are as follows: ferrite (, body-centered cubic), austenite (, face-centered cubic),

epsilon martensite (, hexagonal close-packed), martensite (', body-centered tetragonal).

Note that denotes shear stress. The solid red arrows indicate that yield stress decreases,

in general, with increasing Mn content.

83

Automotive body structure materials are subject to different deformation modes,

strain paths and strain rates both during manufacturing and when integrated into a vehicle

body structure [15]. It is well-established that many steels exhibit strain rate dependent

flow properties, such as tensile yield and ultimate strengths as well as ductility [16-25].

Strain rates in forming of automotive components are typically in the range of 0.1-10 s-1,

and the various stress modes and strain paths require extensive materials characterization

to support sufficient formability prediction. Strain rates during vehicle impact, however,

can reach as high as ~1000 s-1. Increased temperatures during stamping, both due to the

effects of friction and adiabatic heating at elevated strain rates, may reduce the rate of

retained austenite transformation, thereby lowering the work hardening rate and tensile

strengths of TRIP steels, in particular [26-28].

Although AHSSs offer enhanced combinations of strength and ductility, practical

limits exist on their formability, and various tests have been devised to characterize and

predict in-vehicle performance [29]. Advanced fracture models, either microstructure

based [30] or based upon a phenomenological model, such as GISSMO (Generalized

Incremental Stress-State Dependent Damage model) [31], require additional tests ranging

from compression of small to ligament-type specimens that probe specific stress

triaxialities. Perhaps the most common test instrument for material data generation is the

screw-driven tensile load frame, which is limited to quasi-static strain rates of

approximately 0.1 s-1 or less. For higher strain rates, a servo-hydraulic load frame is

commonly employed to test automotive sheet steels [17, 23, 32, 33]. Here, a test

specimen is accelerated to a target speed with an actuator that engages a slack adaptor

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assembly to rapidly deform a test specimen in tension. While the upper limit on nominal

strain rate for servo-hydraulic test machines can achieve 1000 s-1, the 500 s-1 rate is

commonly employed as a useful upper limit. One of several persistent challenges to high

strain rate testing is ringing which appears as a spurious oscillation superimposed onto

the “true” material response (e.g., load vs. displacement) [23]. Ringing is most severe

near the resonant frequency of the load train, and it obscures the desired material

response leading to inaccurate flow data. The split-Hopkinson bar [34-37], also known as

the Kolsky bar [38, 39], is commonly applied in dynamic material tests at strain rates at

or above 1000 s-1, but there are notable exceptions utilizing this technique in which tests

have been conducted at or near 500 s-1. Here, the specimen is plastically deformed by an

elastic wave pulse initiated by a striker through the load train consisting of incident and

transmission bars and the test specimen. Unlike a servo-hydraulic load frame, the test

specimen is often not deformed to fracture in a split-Hopkinson bar test thereby

precluding determination of fracture strains.

The majority of mechanical property studies of AHSS do not consider strain rate

effects on plastic anisotropy, notably the plastic anisotropy index, alternatively termed r

or r-value. Recently, there have been some reports in which plastic anisotropy has been

investigated at different strain rates. For example, Huh et al. [40] showed that the plastic

anisotropy of DP780 and TRIP590 tends to diminish at high strain rates. They also

constructed yield loci at different strain rates and noted changes in the general shape of

yield loci. Rahmaan [41] studied strain rate effects on the anisotropy of DP600 and

TRIP780; however, strain rate had a negligible effect on anisotropy. Li et al. [42] showed

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that anisotropy changes with strain rate in high strength low alloy (HSLA) steels. These

effects were then characterized by calibrating different anisotropic yield functions and

observing the strain rate sensitivity of the coefficients in those yield functions.

Strain rate effects on the plastic anisotropy of AHSS, 3rd generation AHSS in particular,

require additional investigation. In this study, strain rate effects on the plastic anisotropy

of a developmental Medium Mn (10 wt.%), duplex (ferrite+ austenite) TRIP sheet steel is

investigated. Of particular interest is the effect of strain rate on the tensile flow stresses

along different in-plane loading directions and on the r-value and its evolution with

plastic strain. For this purpose, a servo-hydraulic load frame was used to apply nominal

strain rates in the 0.5 – 500 s-1 range. The extent to which the r-value changes at selected

nominal strain rates are quantified. The value of generated results for calibration and

validation of material constitutive models for CAE analyses is discussed regarding the

need for a methodology that incorporates strain rate effects on sheet anisotropy effects in

plasticity models. Additional experimentation required to connect important

microstructural mechanisms to anisotropy dependence on strain rate under alternative

strain paths is suggested.

3.2 Medium Manganese TRIP Steel

The chemical composition in weight percentage (wt.%) for the uncoated medium

manganese (Fe-0.16C-10.0Mn-1.56Al-0.2Si, wt.%) TRIP steel, from now on Med. Mn

TRIP is listed in Table 3.1 [43]. Processing involved inter-critical batch annealing at

600-700 °C for 96 hours. According to Gibbs et al. [7], this facilitates the enrichment of

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retained austenite with both C and Mn. The material was then hot rolled in billet form,

and cold rolled into 1500 mm long strips each with a nominal 1.1 mm thickness and 200

mm width. The as-manufactured material contains 66% austenite volume fraction with

the balance of ferrite. Figure 3.2 is a field emission SEM image of the as-manufactured

microstructure showing the morphology of ferrite and austenite phases. The combined

average grain size of ferrite and austenite phases, similar in size and morphology, is ~2

μm. The presence of annealing twins in the austenite grains is noted and is suggestive of a

low austenite stacking fault energy as a result of alloy enrichment in the austenite phase.

Martensitic transformation in the Med. Mn TRIP under quasi-static tension as well as

deformation and strain path dependence in a T-shaped stamping have been detailed in

Abu Farha et al. [11] and Wu et al. [15], respectively. Texture evolution was noted in

synchrotron X-ray data for in-situ quasi-static tests of specimens with the 0º orientation,

but no similar tests were performed on specimens with the 45º and 90º orientations [11].

3.3 Test Specimens

Test specimen geometries are shown in Figure 3.3. All quasi-static tests were

conducted with the ASTM E8/ISO 6892 Type 1 [44] test specimen geometry. A search of

the literature revealed a wide variety of sheet specimen tensile geometries for high strain

rate testing with servo-hydraulic load frames. For example, Yan et al. [45] show four

examples used in their round-robin study on dynamic testing while Borsutzki et al. [46]

show ten different tensile geometries. The high rate test specimen geometry must take

into account the gripping mechanism, facilitate strain gauge application, and not

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contribute itself to load train ringing and non-uniform strain evolution. After a careful

literature review, the dog-bone test specimen geometry in Figure 3.3 was selected. This

geometry is based upon that in Yang et al. [47]. One gripper end was designed to be

longer than the other to accommodate linear strain gauges used as load cells at strain rates

Table 3.1: Chemical composition (wt.%) of Med. Mn steel.

Elements Mn C Al Si P S Fe

wt.% 10.0 0.16 1.37 0.19 0.002 0.0018 Balance

Figure 3.2: Field emission SEM image of the as-received, medium Mn TRIP steel (two

phases, ferrite (F) and retained austenite (A)) with ~2 μm grain size. Annealing twins

(AT) appear in austenite grains which are indicative of a low stacking fault energy.

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in excess of 5 s-1. It guarantees strain homogeneity in the gauge section length as

suggested by ISO 29203-1,2 [48] and SEP1230 [49]. The second version with a 10mm

gauge length was tested, but non-uniform strain accumulation along the gauge length

resulted. All test specimens were cut with wire EDM and cleaned before application of

the DIC contrast pattern.

Figure 3.3: Tensile specimen geometry for high rate testing. All dimensions are in mm

3.4 Instrumentation

Quasi-static tension tests were conducted on an Instron 5985 universal

electromechanical load frame. The load frame has a 250 kN static load capacity with an

actuator that can achieve a maximum speed of 17 mm/s. The upper grip is attached to a

precision force transducer (with 250 kN rating) that is connected to the moving

crosshead. The tests were performed at a constant crosshead speed of ~0.3 mm/s with

quasi-static strain rate 0.005 s-1.

Nominal strain rates from 0.5 s-1 to 500 s-1 were achieved with an INSTRON

VHS 65/80-20 servo-hydraulic test unit shown in Figure 3.4. This is a two-column load

frame with an 80 kN dynamic load rating. Deformation of a test specimen occurs via

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linear motion of an actuator, which can achieve a ~20m/s maximum speed, connected to

the lower grip of a slack adaptor unit. The upper grip is connected to a Kistler 9071A

piezoelectric load washer which is between the upper (stationary) grip and a manually

adjustable yoke. A significant challenge with servo-hydraulic units during high rate

testing is ringing from the sudden engagement of the lower grip, discussed in detail in

Yang et al. [47]. To substantially minimize ringing and to make sure that any minor

specimen bending was eliminated, a linear strain gauge connected to a ¼ bridge circuit

adhered to both sides of the longer grip end on each high rate test specimen. The strain

gauges functioned as the load cell for strains > 5 s-1.

3.5 Strain Measurement

For accurate strain measurements, stereo digital image correlation (DIC) systems

were used to monitor material deformation during testing. A black and white speckle

contrast pattern was applied to the gauge section of each test specimen. This facilitated

strain measurement in DIC post-processing of images captured during a test. For the

quasi-static tests at the nominal 0.005 s-1 rate, the GOM 3D DIC system was used, which

consisted of two black and white 5M cameras with 50mm lenses, two high-intensity

lights, and the ARAMIS software for post-processing. A frame rate of 5 f/s was selected

for the quasi-static tests.

For the high rate tests in the 0.5 – 500 s-1 range, digital images of a specimen

gauge section were captured at a fixed 20,000 f/s rate with a high-speed stereo DIC

system from Correlated Solutions. This system, which is shown in part in Figure 3.4,

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consisted of the following components: two Photron SA1.1 black and white cameras,

high intensity movie lights, a National Instruments multi-channel data acquisition (DAQ)

box for image capture and data logging (e.g. image number tagged with corresponding

strain gage voltages and load washer loads), and the Vic-Snap image capture software

from Correlated Solutions. After testing, all DIC images were processed consistently to

obtain the full-field history of strains using algorithms detailed in Sutton et al. [50].

Figure 3.4: INSTRON VHS 65/80-20 (VHS=very high speed) servo-hydraulic test unit

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3.6 Results and Discussion

3.6.1 Tensile Flow Behavior

Figures 3.5(a)-(c) show engineering stress-strain curves for the Med. Mn TRIP

steel along the rolling (RD), diagonal (DD), and transverse (TD) directions, respectively,

at 0.005, 0.5, 5, 50 and 500 s-1. All specimens were tested to fracture. Each curve is

characterized by a protracted yield point elongation (YPE), denoted by the plateaus in the

stress/strain curves which span approximately 7% engineering strain and followed by

hardening before fracture. The YPE is not obvious in the 500 s-1 curves due to an artifact

in the load signal that the strain gauges were unable to eliminate. However, confirmation

that the yield point elongation persists to 500 s-1 was acquired through examination of

DIC strain rate contours in Figures 3.6(a)-(c). Relevant mechanical properties as a

function of strain rate and orientation are listed in Tables 3.2-3.4. An initial negative

strain rate sensitivity regarding the flow stress and ultimate strength occurs for all three

directions. This is observed by the high strain hardening at 0.005 s-1 followed by the

generally significant drop in strength at 0.5 s-1. Beyond 0.5 s-1, flow stress once again

increases indicating a positive strain rate sensitivity, but strength levels observed at 0.005

s-1 are not achieved in the 0.5 s-1 to 500 s-1 range. The reduction in strain hardening is a

result of suppressed martensite transformation from austenite due to adiabatic heating and

austenite stabilization during deformation at high strain rates [24]. Multiple studies have

shown that the increase in temperature inhibits the deformation-induced transformation of

austenite and thus leads to lower strain hardening at higher rates in fully austenitic,

duplex ferrite-austenite, and Q&P steels [51-54]. It is also worth noting that the strain

92

Table 3.2: Tensile Properties Associated with RD Flow Curves in Fig. 4.5a. Engineering

stress and strain are reported.

Nominal Strain

Rate (s-1)

0.2% Yield

Strength (MPa)

Ultimate

Tensile

Strength (MPa)

Strain at

Uniform

Elongation

Strain at

Fracture

0.005 690 1115 0.33 0.41

0.5 743 821 0.30 0.40

5.0 765 826 0.31 0.39

50.0 782 867 0.29 0.41

500.0 815 967 0.24 0.44

Table 3.3: Tensile Properties Associated with DD Flow Curves in Fig. 4.5b. Engineering


Nominal Strain

Rate (s-1)

0.2% Yield

Strength (MPa)

Ultimate

Tensile

Strength (MPa)

Strain at

Uniform

Elongation

Strain at

Fracture

0.005 702 1030 0.39 0.45

0.5 712 805 0.36 0.51

5.0 736 819 0.35 0.49

50.0 770 840 0.34 0.52

500.0 795 865 0.32 0.53

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Table 3.4: Tensile Properties Associated with TD Flow Curves in Fig. 4.5c. Engineering


Nominal Strain

Rate (s-1)

0.2% Yield

Strength (MPa)

Ultimate

Tensile

Strength (MPa)

Strain at

Uniform

Elongation

Strain at

Fracture

0.005 700 1160 0.33 0.37

0.5 770 828 0.28 0.40

5.0 805 852 0.28 0.43

50.0 845 865 0.28 0.42

500.0 890 902 0.25 0.40

(a) (b)

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(c)

Figure 3.5: Engineering stress-strain curves at the five testing strain rates for: (a) 0º (RD

– rolling direction), (b) 45º (DD – diagonal direction), and (c) 90º (TD – transverse

direction).

hardening varies with tensile loading orientation relative to the sheet rolling direction,

which is due to an orientation dependence of both the texture evolution and martensite

transformation. Previous studies indicate that martensite formation from austenite is

suppressed when the tensile axis is oriented along <001> in both austenitic stainless

steels and high Mn steels that exhibit two stage TRIP behaviors (γ→ε→α’) [55-57]. The

elevated work hardening rates in both the RD and TD orientations relative to the DD

orientation suggest that a considerable <001> austenite component exists along the DD

orientation. Serrations beyond yield point elongation, which are clearly evident in

hardening regions of the 0.005 s-1 curves, are indicative of dynamic strain aging. This is

95

(a) (b) (c)

Figure 3.6: (a) Tensile (2D DIC) strain contours at the ninth image showing the onset of

Lüdering (red contour patch) in the 500 s-1 curve (see Fig. 3.5). (b) Tensile (2D DIC)

strain contours showing Lüders strain at image twelve in the 500 s-1 curve (c) Tensile (2D

DIC) strain contours showing Lüders strain at image twelve in 500 s-1 curve.

manifested by propagative instabilities resembling the highly spatially correlated Type A

Portevin-Le Châtelier bands reported in tensile tests of other AHSS [58, 59]. Band

propagation beyond yield point elongation for all orientations ceases at 500 s-1 as noted

via examination of strain rate contours computed in DIC post-processing. Total

elongation tends to increase for strain rates higher than 0.005 s-1 along the DD and TD.

The data in Tables 3.2-3.4 show that flow properties along the DD experience the

lowest magnitude of strain rate sensitivity of yield strength, wherein the yield stress

increases by 13% between 0.005 s-1 and 500 s-1. By comparison, tests along the RD and

TD exhibit yield strength increases of 18% and 27%, respectively. The same trend by

96

orientation is observed in the change in uniform elongation with strain rate from 0.005 s-1

to 500 s-1. Tests along the DD show a change of 17% in ductility, while the change in

uniform elongation observed along the RD and TD is 27 % and 25%, respectively.

The anisotropy in the flow behavior may be characterized by normalized true flow stress,

σn, calculated as a function of true strain via [40, 42]:

0

o

n

(3.1)

where is the flow stress in either the DD or the TD relative to the RD flow stress, σ0. The

material is isotropic when 1n . Figure 3.7 shows n vs. ε in both the DD and the TD

for all five strain rates. The normalized true flow stress was calculated in the true strain

range of 0.05-0.25 to avoid both YPE and the region of non-uniform deformation (post

necking). Although n at all strain rates varies as a function of total strain in both the DD

and TD, its reduction is especially prominent with the DD at 0.005 s-1. Again, this is

attributed to the reduction in work hardening as a result of martensite transformation

suppression, presumably along <001> fibers in austenite, as a result of as-produced

texture. As strain is increased, the much smaller changes in n suggest that the material

becomes increasingly isotropic. The same trend toward isotropy with increasing strain

rate is also observed, as the plots at higher strain rates tend to move closer towards 1n

in Figure 3.7.

97

(a) (b)

Figure 3.7: Normalized flow stresses from Eq. (3.1) along the (a) diagonal direction (DD)

(b) transverse direction (TD).

3.6.2 Anisotropy

A well-known and important parameter that both denotes the level of anisotropy

and quantifies the resistance to thinning under tension in sheet metals is the Lankford

coefficient or r-value [60], also termed the plastic anisotropy ratio. The r-value, which is

the ratio of the in-plane (transverse) to thickness strains, is used to gauge anisotropy in

stamping as well as calibrate anisotropic yield criteria for finite element simulations [61].

Strain fields parsed in DIC post-processing of image data recorded during the tests

provided a useful alternative to extensometers for computing r-values [62]. Figure 3.8

shows the r-value evolution with true strain at different strain rates for each of the tested

orientations. The measured r-value variation with orientation changes with both strain

rate and total strain. The r-value at 0.005 s-1 in the three loading directions has a high

98

dependency on the strain, which is a strong indicator of the texture evolution with strain

[11, 63]. The curves corresponding to 0.005 s-1 tend to be outliers with those for the

remaining strain rates grouping together. The change in all three directions during quasi-

static loading is towards perfect isotropy (r=1); along the RD the r-value changes from

0.54 to 0.63, along the DD the r-value changes from 1.15 to 1.1 and along the TD the r-

value changes from 0.72 to 0.8. Figure 3.8 shows, in summary, that the plastic anisotropy

index tends toward a common value with strain for all tested orientations due to tensile

texture evolution, although the common value at a given orientation is not equal to that of

an alternate orientation. Additionally, the plastic anisotropy index tends toward isotropy

(r=1) with increased strain rate, and this indicates that material texture effects on plastic

anisotropy are mitigated at increased strain rate. Of particular note in Figure 3.8 are the

relative differences of quasi-static r-values and elevated strain rate r-values in each

orientation. A clear decrease in r-value is observed above quasi-static test rates in the DD

orientation, but both the RD and TD orientations reveal an increased r-value with

increased strain rate. As previously noted, this is presumably an effect of both

transformation kinetics related to as-produced austenite texture and tensile texture

evolution in the ferrite and austenite phases during deformation. Additional studies are

required to characterize the structural evolution during deformation fully and definitively

account for the observed behavior. The change of r-value with strain is, in part, due to

the non- homogenous yielding observed with the Med. Mn. TRIP, i.e., yield point

elongation [40, 63, 64]. This behavior was recently confirmed by coupled stereo digital

99

(a) (b)

(c)

Figure 3.8: r-value evolution with strain at different strain rates for specimens tested

along the (a) rolling direction (RD) (b) diagonal direction (DD), and (c) transverse

direction (TD).

100

image correlation and synchrotron X-ray diffraction where plastic deformation and

austenite transformation were found to be highly inhomogeneous and occur in

propagating bands [11].

Figure 3.9 shows the average r-value as a function of true strain for thee RD, DD,

and TD, calculated across a true strain range of 0.05-0.25. Also shown in Figure 3.9 is the

coefficient of normal anisotropy (nr )

0 45 902

4n

r r rr

(3.2)

which is a useful indicator of formability. The highest r-value is observed along the DD,

which corresponds with the ductility results shown in tables 3.2-3.4 (the highest ductility

is observed when testing along the DD). Similar to Figure 3.8, the anisotropy exhibited in

the three directions is most significant (furthest away from an isotropic value of 1.0) at

the quasi-static rate (r0avg=0.6, r45avg=1.1 and r90avg=0.78). The r-values are closest to 1

(thus approaching isotropic behavior) at the highest rate (500 s-1). These results are in

qualitative agreement with Huh et al. [40] who identified significant strain rate effects on

the r-value of TRIP590 and DP780. The reported r-values of Huh et al. also tended

toward unity as the strain rate was increased. A similar conclusion was reached by Li et

al. [42] who were able to quantify the strain rate effect on the anisotropy for an HSLA

steel tested at different rates. For the Med. Mn. TRIP of interest here, the normal

anisotropy increases with strain rate indicates better resistance to thinning at higher strain

rates; and thus, improved formability.

101

The aforementioned results suggest a significant influence of strain rate on the

texture evolution and austenite transformation of Med. Mn TRIP, leading to a change in

the normalized flow stress and the r-value observed along different loading directions.

This effect, in turn, raises the question of the accuracy of using quasi-static flow data to

model plastic anisotropy. Modeling of plastic anisotropy is often accomplished via an

anisotropic yield function, a critical part of the constitutive relations needed to accurately

describe the behavior of metal sheet plasticity. The strain rate effects shown here on the

flow stress and the r-values suggest that the calibration of the yield function using quasi-

static data is only accurate at strain rates close to quasi-static. As with the different

magnitudes of flow stresses and r-values at different strain and strain rates, comes yield

loci with different shapes and sizes. Therefore, there is a need for the development of a

methodology that will allow for modeling plastic anisotropy during high rate deformation

for steels such as the Med. Mn TRIP considered herein.

3.7 Conclusions

The effect of strain rate on the plastic anisotropy of a development Med. Mn.

TRIP steel with an initial 66% volume fraction of austenite (balance ferrite), was

investigated. During tensile straining, austenite transforms to martensite either directly or

through -martensite, the transient hexagonal close-packed form of martensite. Tests

were conducted in three different orientations, viz, 0º or RD, 45 º or DD, 90 º or TD, in

uniaxial tension at selected nominal strain rates within the 0.005 s-1 to 500 s-1 range. The

change in anisotropy with strain rate and the total strain was characterized using three

102

Figure 3.9: Average r-value and normal anisotropy coefficient (rn) variation with the

logarithm of nominal strain rate for the RD, DD, and TD.

parameters: normalized flow stress, the r-value, and the normal anisotropy ( nr ). Strain

rate significantly affected all parameters; the normalized flow stress was observed to

approach unity as the strain rate increases, and a similar trend was observed with the r-

value. Moreover, the r-value was also shown to be dependent on total true strain. This

behavior has been previously noted in many other steels; however, we believe that is

further accentuated in this material due to the non-homogeneous strain accumulation

associated with the yield-point-elongation (YPE) and dynamic strain aging as well as

austenite transformation localized to propagative instabilities reported in Abu-Farha et al.

[11].

While results from the tensile tests are instructive for identifying strain rate

dependence on anisotropy, they do not establish the role of underlying microstructural

103

mechanisms responsible for the observed macro-scale behavior. Indeed, there are

observations from the extant literature that are suggestive of some of these mechanisms

which likely are at work in the Med. Mn. TRIP steel of interest in this paper. Some of

these mechanisms, such as the effect of strain rate on texture evolution and austenite

transformation to martensite were alluded to in the course of discussing the observed

anisotropy dependence on strain rate. Additional data from electron backscatter

diffraction (EBSD) and X-ray (or neutron) diffraction will be required to address this

issue. For example, variations of austenite volume fraction along with texture evolution

in the RD, DD, and TD over the range of strain rates relevant for automotive applications

would provide a more substantive basis for interpretation of the tensile results presented

herein. A clearer picture of how strain rate affects -martensite formation and subsequent

'-martensite formation would also be forthcoming. A repeat of these measurements for

different deformation modes (e.g., biaxial, plane strain, shear) would provide insights into

how anisotropy evolves in forming processes wherein uniaxial tension is not the

predominant deformation mode. The addition of a high rate infrared camera coupled to

the DIC system would give needed temperature information that can be coupled to

anisotropy and the martensitic transformation. Further investigations should focus on

quantifying the effect of strain rates on the shape and size of the yield locus requiring

calibration of different anisotropic yield functions using uniaxial and biaxial experimental

data. This should be the initial step in developing a rate-dependent anisotropic yield

function.

104

Major conclusions from this study are as follows:

[1] The flow response of the Med. Mn (10 wt.%) TRIP over the 0.005 s-1 to 500 s-1 range

of nominal strain rates is characterized by a protracted yield point elongation followed by

a hardening stage that precedes fracture. Propagative instabilities resembling Type A

Portevin-Le Châtelier bands are suggestive of dynamic strain aging and appear at all

tested strain rates except 500 s-1.

[2] During uniaxial tensile testing of the Med. Mn. TRIP, strain hardening decreased with

increasing strain rate due to the suppression of the austenite to martensite transformation

which was likely a result of adiabatic heating. This resulted in an orientation dependent

drop ranging ~16% to 30% of ultimate tensile strength between 0.005 s-1 the elevated

strain rates.

[3] At high strain rates (0.5 s-1 and greater) the anisotropy of the Med. Mn TRIP steel was

diminished, as demonstrated by the change in the normalized flow stresses and r-values

with increased strain rate, i.e., both approached unity indicative of perfect isotropy.

Additionally, the r-values at strain rates other than the quasi-static rates tend to group

together; these two effects are a result of the mitigation of the material texture effect at

high strain rates (> 0.5 s-1).

[4] The increase in the coefficient of normal anisotropy ( nr ) of the Med. Mn. (10 wt.%)

TRIP sheet at elevated strain rates indicates that the resistance to thinning of the material

is improved at elevated strain rates, which suggest that this aspect of formability is

improved relative to quasi-static strain rates.

105

[5] Anisotropy observed in this study has the following implications for the calibration of

anisotropic yield criteria: the calibration of yield criteria using quasi-static data is valid

only at quasi-static deformation; the changes in the normalized flow stress and the r-value

directly affect the shape and size of the yield locus; to accurately model the anisotropy

during forming, rate-dependent anisotropic yield criteria are necessary.

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111

CHAPTER FOUR

4. A RATE DEPENDENT ANISOTROPIC YIELD FUNCTION

ABSTRACT

The results of the previous chapter confirmed the strain rate dependence of the

anisotropic properties of Medium Mn. (10 wt. %) steel. To quantify this dependence, the

calibration of an anisotropic yield function is needed. First, a comprehensive review of

the different yield functions in the literature is provided, showing the strength and

weaknesses associated with each of them. The result of that is a systematic choice of an

anisotropic yield function that can accurately model the strength and strain anisotropic

nature of Medium Mn. (10 wt. %) steel. The experimental results obtained from the

previous chapter are used to calibrate at each strain rate, allowing for plotting the yield

locus at each strain rate. The shape and size of the yield loci were observed to change

with the varying strain rate; thus showing a strong dependence on the strain rate. This

provided the motivation to develop a methodology to model the strain rate dependence of

the yield locus. As mentioned in Chapter 3, there are currently no anisotropic functions

in the literature that can predict the effect of strain rate on anisotropy. Therefore, a

methodology to model the rate dependent anisotropy of materials is proposed at the end

of this chapter. Followed by the detailed numerical procedure of a material card that can

capture the strain rate in the hardening and anisotropy of Medium Mn. (10 wt. %) steel.

112

`4.1 Introduction

Accurate modeling of the behavior of the materials in FEA simulations is very

crucial, especially when the materials in question are AHSSs. This is mainly due to: 1)

the desire to optimize the amount of material needed (and therefore the cost) by

increasing the accuracy of the simulations, and 2) the complex nature of these materials

and the difficulty in predicting their responses to different loading conditions.

Models to predict the response of materials (constitutive models) can be divided

into two types: physical models and phenomenological models. While physical models

are more accurate, yet they have a very complex nature and are difficult and costly to

implement in a finite element code. Therefore, phenomenological models are more

widely used in the automotive industry as they require calibration of a fewer number of

parameters and are relatively more straightforward to implement in a finite element code.

A phenomenological model consists of three components [1]: 1) a yield function, 2) a

flow rule and 3) a hardening law. The yield function is an expression to define the

relationship between the different stress component at the onset of yielding, and through

plastic deformation. The flow rule is an expression which defines the relationship

between the components of strain rate and stress. Finally, the hardening law defines how

the initial yield stress evolves during plastic deformation [2].

In chapter 2, it was shown that the range of different microstructure in the

different AHSSs affects their strain rate sensitivity at speeds close to that encountered

during a crash event (hardening). In chapter 3, it was also shown that the anisotropy of

the material changes at different strain rates (yield function). In this chapter, a modeling

113

approach will be devised to incorporate the strain rate sensitivity in both the hardening

and anisotropy of Medium Mn. (10 wt. %) steels during crash simulations.

4.2 Overview of yield functions

Yield functions define the stress state at which the elastic behavior of the material

ends, and plastic flow begins. During a uniaxial tensile test, this can be easily found by

looking at the stress-strain curve and finding the stress at which plastic flow begins.

However, during a multi-axial loading condition, such as that encountered during a crash

scenario, it is not as straightforward as it is in the case of a uniaxial test. Therefore, an

implicit relationship between the principal stresses is needed to define the onset of plastic

flow, this is known as the yield function. The function will represent a three-dimensional

surface of the principal stresses. The elastic state of the material includes all the points

inside the surface and the points on the surface itself present the plastic state. Points are

not allowed to be outside the surface. One of the oldest and most used yield functions is

the “maximum shear stress criterion” proposed by Tresca [3] in 1864 and the “maximum

distortion energy criterion” by Von Mises [4] in 1913 [5]. The former defines the onset of

plastic flow as when the maximum shear stress reaches a critical value, equation 4.1

1 2 2 3 3 1max , , y (4.1)

where 1 2 3, , are the stresses in the three principal directions and y is the yield stress.

Von Mises function, on the other hand, is a quadratic relation (instead of linear with

Tresca) of the second deviatoric stress invariant, equation 4.2. The basis of the function is

114

that the plastic flow of the material begins when the elastic energy of distortion reaches a

critical value.

2 2 2 2

1 2 3 2 1 3 2 y (4.2)

These functions were followed by other formulations such as Drucker [6] in 1949

which tried to find a formulation to model the material with results lying between Tresca

and von Mises. Followed by Hershey [7] in 1954 which suggested a generalized

formulation of the von Mises equation. Nonetheless, all these equations were proposed on

the assumption of isotropy in the material properties. However, experimental results

showed that some materials exhibited a strong anisotropy in their behavior and therefore

there was a need for an anisotropic yield function. In 1948, Hill [8] made an

improvement on the generalized von Mises equation by adding parameters that depend on

anisotropic properties that can be derived using experimental methods (the yield stresses

in rolling, transverse and diagonal directions, or the r-values in the three directions) as

shown in equation 4.3

2 2 2 2 2 2

22 33 11 22 33 11 23 31 122 ( ) 2 2 2 1ijf F H G L M N

(4.3)

where F, H, G, L, M, N are the new parameters that introduce anisotropy to the equation.

These parameters can be calculated by using either the three r-values or the yield stresses

in the three directions relative to the rolling directions. That means that this equation

assumes an inherent relation between the r-values and the yield stress. Although this is

true for some materials, it is not a universal rule. This is a significant drawback, as some

materials show anisotropy in the yield stresses and the r-values in varying magnitudes

115

(such as Medium Mn. (10 wt. %) shown in chapter 3), and Hill 1948 definitely can’t

capture that behavior. Other drawbacks were highlighted by another researcher when they

used Hill 1948 for the calibration of their materials, such as Pearce [9], Woodthorpe and

Pearce [10] and Banabic et al. [11]. Due to these reasons and others that are detailed in

the references, Hill made improvements to his earlier equation and developed the Hill

1979 yield criterion [12]. The new Hill yield criterion followed a non-quadratic

formulation, and it improved on the original yield criterion while maintaining some of the

simplicity. Nonetheless, there were still drawbacks about this equation, mainly its

assumption that 0 90r r and

0 90 . Due to this, Hill proposed two other yield criteria in

1990 [13] and 1993 [14]. Some of the problems mentioned were resolved by one of these

models, but it came at the cost of increased complications in the formulation (and

therefore higher computational cost) as in the case of Hill 1990 or with the introduction

of more assumptions as in the case of Hill 1993 [2].

The Hill family of yield criteria used the von Mises yield criterion as a basis for

their formulations. There exist another family of yield criteria that used Hershey’s model

[7] as a basis for their formulations. The first person to do this was Hosford [15] in 1972,

who proposed a function that was a generalization of Hershey’s criteria (equation 4.4)

22 33 33 11 11 22

a a a aF G H (4.4)

Looking at this equation and Hill’s equation, one can notice some similarities

between them, yet the main difference is the identification of the parameter “a” in

Hosford’s and “m” in Hill 1979 and beyond. “a” is determined based on the

crystallographic structure of the material [16-18]. Leading to a connection between the

116

microstructure and the yield function, which was considered a significant advantage of

this yield criterion and all Barlat yield criteria as they used equation 4.4 as a basis for

their formulations.

The first criterion developed by Batlat was published in 1987 with Richmond

[19], which used Hershey’s model as a basis with a modification in using generalized

coordinates rather than material coordinates. This criterion was simple and suffered

multiple drawbacks, most importantly is its limitations in predicting the onset of yield for

other than uniaxial conditions [2]. In 1991, Barlat [20] proposed a new yield criterion

(Yld91) formulated in terms of the deviatoric principal stresses (equation 4.5)

1 2 2 3 1 3 2m m m m

yS S S S S S (4.5)

This model was simple, general and easy to manipulate to fit varying conditions,

yet it could not predict the yielding of different kinds of materials. Therefore, Barlat tried

to improve on the 1991 model and proposed a model in 1997 (Yld94) [21] by adding

weighting parameters, in front of the deviatoric principal stresses, that are a function of

the anisotropic properties. These parameters introduced a right amount of flexibility to

the model to allow it to predict the anisotropic yielding of a wide range of materials. Yet,

the model suffered from complexities in application to numerical codes that limited its

use as compared to Yld91 [22]. Other models that were proposed in the literature include

BBC [23], Karafillis-Boyce [24], Budiansky [25], Gotoh [26, 27] and CPB family [28-

30].

117

4.3 Barlat Yld2000-2d

Even with the good applicability of Yld91 and its common use in finite element

simulations of anisotropic materials, there were still some drawbacks associated with it.

The most important of those are the lack of proof of convexity and the inaccuracy in

describing yielding for complex loading conditions [31]. Therefore in 2003, Barlat et al.

[31] developed a new yield function (Yld2000-2d) that can address these drawbacks.

Barlat et al. wanted the yield function to satisfy two primary conditions: 1) the function

must have at least seven parameters to take into account the three uniaxial yield stresses

in the three directions, the r-values in the three directions and the biaxial yield stress. 2)

the function must satisfy the convexity condition.

To ensure convexity, Barlat et al. used an isotropic yield function that was

proposed by Hershey in 1954 [7] and Hosford 1972 [32] which was made up of the sum

of two functions (equation 4.6)

2 a (4.6)

where

1 2 2 1 1 2........ 2 2a a a

s s s s s s (4.7)

This function was made to account for anisotropy by using a linear transformation on the

deviatoric stress tensors, which can be written in terms of the Cauchy stress tensors. This

would lead into two linearly transformed stress tensors ,X X , where

. . . .

. . . .

X C s C T L

X C s C T L

(4.8)

118

where T is the transformation matrix for the Cauchy stress tensor to the deviatoric stress

tensor. Consequentely, this will lead to the replacement of 1s ,

2s with X and

X respectively. With certain conditions imposed on the coefficients of the ,C C , the

,L L matrices reduce to:

11

12 1

21 2

22 7

66

2 / 3 0 0

1/ 3 0 0

0 1/ 3 0

0 2 / 3 0

0 0 1

L

L

L

L

L

(4.9)

311

12 4

21 5

22 6

66 8

2 2 8 2 0

1 4 4 4 01

4 4 4 1 09

2 8 2 2 0

0 0 0 0 9

L

L

L

L

L

(4.10)

The coefficients 1 8 can be found using the seven experimental values

mentioned above, and the extra coefficient can be determined using the biaxial r-value.

The yield function now reduces to the same form as equation (4.6) with the replacement

of ,X X in place of 1 2,s s in equation (4.7). This final form of the yield equation

satisfies the two conditions stated above and on top of that provides an equation that is

relatively easier to derive than Yld91 [31].

Yld2000-2d have proved to be one of the most accurate models that can predict

anisotropy in different materials showing varying degrees of anisotropy in different

properties while keeping the application in a numerical code to a practical level for

industrial applications. Nonetheless, the numerical complexity of implementing this

119

function is still higher than other equations. In regards to the anisotropy observed in

Medium Mn. (10 wt. %), it was observed that it showed anisotropy with both the flow

stress (strength) and the r-value (strain). For that reason, Yld2000-2d was chosen to be

the basis for the modification that will allow for the development of a rate-dependent

anisotropic yield function.

4.4 Calibration of Yld2000-2d

In order to fully calibrate the Yld2000-2d, eight experimental values are needed.

Namely, the yield stresses in the three rolling directions, the r-values in three rolling

directions, the biaxial yield stress and the biaxial r-value. The first six parameters were

obtained at the different strain rates in the previous chapter, while the biaxial yield stress

and r-value at 0.005s-1 were obtained using a hydraulic-bulge test. There are no

experimental capabilities available to allow biaxial testing at high strain rates. Therefore

the biaxial yield stress at strain rates higher than 0.005s-1 was estimated based on how

45 and 90 changed with strain rates with respect to 0 . This necessarily induces some

inaccuracies in the results [33, 34], yet that is a limitation that future research will be

focused on. As for the r-value at the high strain rates, it was calculated analytically using

Yld96 [35]. The calculation was done for the 0.005s-1, and it was very close to the

experimental value. The values that are used in the calibration of the Yld2000-2d are

shown in table 4.1.

120

Table 4.1: Experimental results used to calibrate Yld2000-2d

Strain Rate (s-1) 0

0

90

0

45

0

0r 90r

45r 0

b

br

0.005 1 1.014 1.017 0.47 0.55 0.78 1.240 0.85

0.5 1 1.036 0.958 0.58 0.69 0.70 1.211 0.88

5 1 1.052 0.962 0.58 1.03 1.05 1.210 0.92

50 1 1.080 0.985 0.52 0.85 1.25 1.229 0.95

500 1 1.092 0.975 0.64 0.90 0.93 1.115 0.98

Using a numerical procedure that was developed based on the work done in the

paper published by Barlat et al. [31, 36], the eight parameters (

1 8

) were found for

each strain rate, and the values are shown in table 4.2. It can be deduced that the strain

rate has a significant effect on the value of these eight parameters in varying magnitudes,

with the maximum effect being observed with 3 (~46% increase). This suggests a

powerful influence of strain rate on the anisotropy and the shape of the yield locus, and

therefore ultimately on the accuracy of using the quasi-static parameters to model the

behavior at high strain rates. To further quantify this, the parameters obtained at each

strain rate were used to plot the yield locus on the normalized stress axes as shown in

Figure 4.1. The dependency of the yield locus on the strain rate is significantly apparent

in Figure 4.1. The yield locus moves closer towards isotropy with the increase in strain

rate as the results of the previous chapter led to conclude. This further asserts the

inaccuracies of calibrating yield functions using quasi-static experimental results to

model material behavior at high strain rates as in the case with crash simulations.

121

Table 4.2: The calibrated

1 8

for each strain rate

Strain Rate (s-1) 1 2

3 4

5 6

7 8

0.005 0.7412 0.9234 0.5559 0.9264 0.9442 0.5415 0.9156 1.2494

0.5 0.8586 0.9206 0.6428 0.9173 0.9446 0.5866 0.9379 1.3832

5 0.7774 1.0321 0.7159 0.8940 0.9426 0.5659 1.0001 1.3142

50 1.0080 0.9559 0.8098 0.9214 0.9447 0.7527 1.0341 1.1488

500 0.9241 0.9841 0.8414 0.9645 0.9642 0.7367 1.0144 1.1686

4.5 Rate Dependent Yld2000-2d

The experimental testing results shown in chapter 3 and the calibration results of

the Yld2000-2d both exemplifies the motivation behind developing the methodology to

numerically model the anisotropic nature of materials at multiple strain rates. The

introduction of a strain rate dependent parameter in the formulation of an anisotropic

yield function is a complex task, especially when the function in question is Yld2000-2d.

Yet the observation made by Li et al. [34, 37] regarding the rate dependency of

the parameters of Hill48 when used to model an HSLA steel, inspired the investigation of

this phenomenon for state-of-the-art developmental steel with a complex microstructure

and behavior (Medium Mn. steel). The results shown in table 4.2 and visualized below in

Figure 4.2 showed that indeed the parameters of Yld2000-2d are strain rate dependent.

The next question that comes to mind is whether that dependency can be represented by a

mathermatical relation. After different mathematical forms of equations were tried, the

optimum form that was selected to best model the trend of the parameters is the cubic

form shown below

3 2

0

(log ) (log ) (log )a b c d

(4.11)

122

where 0 is the parameter calibrated with quasi-static results.

The equation developed for the parameters of Yld2000-2d were used to calculate

the

1 8

at the different strain rates and used these values to plot the yield locus at

different strain rates. The results are shown in Figures 4.3-4.6, wherein each figure three

yield loci are shown. A yield locus plotted using the 0.005s-1 results, a yield locus plotted

Figure 4.1: Yield locus calibrated using data from experiments at different strain rates

123

using the results from the strain rate shown, and a yield locus plotted using equation

(4.11) for the strain rate shown. Figure 4.3 shows the comparison of the three yield loci

between the 0.005s-1 and the 0.5s-1. It can be observed that the yield locus plotted using

the parameters obtained by the equation suggested is in a good agreement with the yield

locus plotted using the experimental results performed at 0.5s-1. While the yield locus

plotted using the experimental results obtained at 0.005s-1 varies in shape and size from

the yiled locus plotted using the experimental results performed at 0.5s-1. The variation is

significant especially where the arrows are drawn.

Figure 4.2: Variation of the Yld2000-2d constants (

1 8

) with the strain rate

124

As the strain rate increases to 5, 50 and 500s-1, the difference between the loci

plotted using the experimental results at the specified strain rates and the locus plotted

using the experimental results obtained at 0.005s-1 starts to increase, while the yield loci


(pink), at 0.5s-1 (blue) and using the equation (green).

125

plotted with the parameters obtained using equation (4.11) maintains an excellent

agreement at all strain rates.

Figure 4.4: Comparison of yield loci plotted using the experimental data at 0.005s-1

(pink), at 5s-1 (blue) and using the equation (green).

126



4.6 Numerical Procedure

4.6.1 Flow Rule

As the materials yield and begin plastic deformation, hardening begins, and a

relation is required to describe how the stress respond to plastic strain increments. This

relation, or a flow rule, usually takes the form shown in equation (4.12) [38]

127

( )ij

ij

dgd d

d

(4.12)

where g is called the plastic potential and d is a scalar function. This general form of

the flow rule describes what is known as a non-associated flow rule. In the particular case

where the plastic potential is the yield function, this becomes an associated flow rule. The

associated flow rule is based on the normality principle which was proposed by Drucker

in 1951 [39]. The principle states that the increment in plastic strain is outward normal to



128

the yield locus surface at any given point. This allows for the replacement of the plastic

potential with the yield function and thus simplifying the mathematical formulation. The

use of associated flow rule with different materials have been researched and verified for

metals but not for materials like rock and soil [40].

Another relation is required to adequately describe the behavior of the material,

which is the hardening law. There are a significant amount of work in the literature that

studies the effectiveness of different hardening laws to capture the strain rate sensitivity

exhibited by different materials, such as Johnson Cook (JC) [41-43], Reyes [44, 45],

Zerilli and Armstrong [46], Combined Holloman and Voce equation [47] and many

others. The selection of the equation depends mainly on the material chosen as it was

shown in Chapter 2 that the microstructure, significantly influences the material’s

behavior at multi-strain rates. There is no widely acceptable equation for AHSSs at multi-

strain rates conditions. That is mainly because of the inhomogeneity of the response of

the material’s hardening properties with strain, as was strongly exemplified with the

chosen material. With the results observed for Medium Mn. (10 % wt.) steel, it was

decided to code the material using the hardening curves at different strain rates as the

input to obtain the values otherwise obtained using a hardening equation. The decision

was made based on the high accuracy of the experimental results obtained, and the range

of strain rates at which these experiments were performed. The LS-DYNA user manual

describes the procedure to code a material card using hardening curves at multi-strain

rates instead of a rate dependent hardening law.

129

4.6.2 Stress Integration

The strain rate dependent Yld2000-2d developed and the associated flow rule will

be used to predict the material properties and develop a numerical code that can be used

to code a user-defined material card in LS-DYNA (UMAT) to model the behavior of

Medium Mn. steel during crash simulations. The code must have an algorithm defined to

integrate the constitutive relation at each integration point in the elements in the finite

element model [48, 49]. Therefore, a stress integration algorithm must be selected and

used to develop the code [50, 51]. The two most well-known algorithms for stress

integration are: 1) the closest point projection iterative method, and 2) the cutting-plane

algorithm [52, 53]. The closest point projection iterative method is usually applied to

relatively simple constitutive relations as otherwise, it will lead to computationally

expensive codes and complex mathematical relations to develop [54]. While the cutting-

plane algorithm (shown in Figure 4.7) was proposed to avoid the complex codes and

mathematical operations by abandoning the need to calculate the gradient of the flow rule

and the yield function in the formulation [55]. Therefore, the cutting-plane algorithm

based on the incremental deformation theory of plasticity was selected to write the code

as described by Abedrabbo et al. [53].

The incremental deformation theory of plasticity was selected as it provides

significant advantages over the total deformation theory by avoiding some unnecessary

steps that might affect the accuracy of the formulation as observed by Abedrabbo et al.

[53] following the work of Ortiz and Simo [56, 57]. The incremental theory of plasticity

was applied by researchers starting with Chung and Richmond [58] that used the

130

minimum plastic work path to integrate the constitutive relations, and Yoon et al. [36, 59]

which used the same theory to integrate Yld96 and its associated flow rule. In the

incremental deformation theory, the material’s strain increments are equal to the true

strain increments, and the rotation of the material is also found at each step through polar

decomposition to find the current incremental angle [53].

In this work, the cutting plane algorithm was used to numerically integrate the

constitutive equation with the associated flow rule. In LS-Dyna, the strain increment n ,

the previous stress state value 1n are provided at the beginning of each time step. The

strain is assumed to be elastic, and therefore the stress is calculated through the elastic

relations. That trial stress and the strain rate are used to calculate the stress state using the

newly developed strain rate dependent Yld2000-2d, and the size of the yield function is

calculated using the hardening flow rule. A check is performed to test whether the

calculated trial stress lies inside the yield surface, if the condition is met, then the trial

stress is the actual true stress, however, if it did not then the material has yielded, and the

stress state is plastic. The Newton-Raphson method is then used to iteratively return the

trial stress to the yield surface by calculating the normality parameter (λ), which tries to

“cut” the stress back to the yield surface, and thus the name ‘the cutting plane algorithm’.

Figure 4.8 is a flowchart explaining the flow of calculations used in the stress

integration process. The first step is to read the history variable which stores the values

for the current strain increment n and the previous stress

1n . It is assumed that the

current strain increment is elastic and therefore, the current stress can be calculated based

on the elasticity equation (4.13)

131

1 :n n nC (4.13)

where C is the elasticity matrix. The strain rate is calculated using the incremental strain,

and the strain rate is used to calculate the coefficients (

1 8

) using the equations

developed in this chapter. Then the stress calculated, which will be called “initial stress”,

and the strain rate dependent (

1 8

) will be used to calculate the value of the yield

function using equations (4.6-4.10) and the derivative of the yield function using the

steps below:

1

1 2 1 2 1 2

1( )( 2 2 )2

aa a aX X X X X X

(4.14)

where is the effective stress. The next step is calculating the prime values for ,X X

matrices.

Figure 4.7: A diagram explaining the process of the cutting-plane algorithm

132

2

2

1

222

( )

2 2[ ]

( )

2 2

xx yy xx yy

xy

xx yy xx yy

xy

X X X XX

X

XX X X X

X

(4.15)

2

2

1

22

2

( )

2 2[ ]

( )

2 2

xx yy xx yy

xy

xx yy xx yy

xy

X X X XX

X

X X X X XX

(4.16)

Equation (4.14) can then be differentiated using the chain rule

1 2 1 2 1 2

1

( 2 2 )1

2

a a a

a

X X X X X X

a

(4.17)

2 3

11 1

1( )

2

nc d c d

ac d i k c d k c d k

X X

a X X

(4.18)

where d is xx, yy, and xy.

Once these two values are calculated, the code is instructed to find the value of

the flow stress and the derivative using on the input hardening stress-strain curve based

on the current strain and strain rate. The first check is now performed to determine if the

effective stress calculated by the yield function is elastic or not (i.e. if the stress state falls

within or on the yield locus) by comparing it with the stress obtained from the hardening

curves:

( , , ) ( , ) 0p p

n n n n n nH (4.19)

133

where ( , )p

n nH is the hardening stress obtained from the input curves. If the condition

does not hold, then the material is in the plastic region now, the “initial stress” calculated

is wrong and has to be adjusted and the strain increment is not totally elastic. In order to

achieve that, the normality principle and the associated flow rule are used. As a reminder

the flow rule is:

p F

(4.20)

where F is the yield function or in other words the effective stress and is what is

called the normality parameter [54]. It is required to find the value of to be able to

calculate the increment of the plastic strain. It should be noted that according to the

incremental deformation theory of plasticity, the normality parameter is equal to the

equivalent plastic strain [53]

:: p

p

(4.21)

Once is found, the new stress value can be calculated as follows:

: ( )p

n n nC (4.22)

moreover, it is known that:

1

p p p

n n (4.23)

substituting that into equation (4.22) leads to

1: ( ) :p p

n n nC C (4.24)

substituting the flow rule in place of p , and using the initial stress:

1 :i

i i nn n C

(4.25)

134

where the “i” donates the trial number, so i

n represents the trial stress calculated based

on elasticity, and the “i+1” is the new trial stress that has been “cut back” using equation

(4.25). The next step would be to again to repeat the procedure shown between equation

(4.14) to (4.19) and see if the new stress state is on the yield locus. If not, then the

process must be repeated again, starting with updating the normality parameter, which is

explained next.

Before the steps mentioned in the previous paragraph can be performed, the

normality parameter must be calculated. The method to calculate the normality parameter

will take advantage of linearization using Taylor’s expansion [53, 56, 57] of equation

(4.19)

( ) 1 ( 1) ( )0 ( , ) ( ) ( )i i

i i p i i i p i p i

n n n n n n

(4.26)

from equation (4.25):

1 :i

i i nn n C

(4.27)

and

( 1) ( )p i p i

n n (4.28)

substituting both back into equation (4.26)

( )0 ( , ) ( : )ii i

i i p i nn n C

(4.29)

Therefore, the normality parameter can be defined as:

( )( , )

: :

i i p i

n n

i i i

n n

p

HC

(4.30)

135

The linearized Taylor’s expansion on the yield function represents the tangent

lines shown in Figure 4.7. As mentioned before, this value of the normality parameter is

fed back into equation (4.25) to calculate the new stress value, which is again verified

using equation (4.19), however this time replacing the zero with a small number like (10-

9) which is the convergence criteria. Once that is met, the stress and the equivalent plastic

strain are returned to the FEM code to continue the simulation.

4.7 FEA Validation

The developed UMAT card is compiled and implemented in LS-DYNA. The first

step is to validate the accuracy of this model by running simulations of the experiments

performed at different strain rates and comparing the simulation results with the

experimental results. Figure 4.9 shows the finite element model of the tensile test

specimen used to run the simulations. The model consisted of fully integrated shell

elements with five integration point across the thickness. The elements had a size of

0.5mm, giving enough elements across the width (10) to ensure the good accuracy of the

results.

The first simulation was performed at a strain rate of 0.005s-1 using the Mat133 as

the material card, which is an already developed material card of Yld2000-2d in LS-

DYNA. Another simulation was performed at the same strain rate using the newly

developed material card, and load-displacement curves of both simulations were

compared with the experimental load-displacement curves shown in Figure 4.10.

136

Figure 4.8: Flowchart showing the steps followed in the cutting plane algorithm

The results shown confirm that the formulation and coding of the material card

are accurate and matches the coding of the LS-DYNA material card at quasi-static rates.

137

Figure 4.9: Finite element model used to verify the developed equation

The next simulation aimed at testing the validity of the strain-rate dependent Yld2000-2d

model developed in this chapter. Therefore, tensile test simulations were performed at the

strain rates at which the experimental data were performed, and again the load-

displacement of the simulations and the experiments were compared in Figure 4.11 and

Figure 4.12. In addition to that, load-displacement curves from the simulation run using

Mat133 (rate-independent Yld2000-2d) are also shown in the figure.

138

Figure 4.10: Comparison of the experimental results and simulations at 0.005s-1



equation (green curve)

139



equation (green curve)

The results shown in the figures validate that the material card was coded and

formulated correctly and that the equation developed provide a significant improvement

in the accuracy of predicting the behavior of the material at multi-strain rates

4.8 Conclusions and Summary

In this chapter, a new rate-dependent yield criterion was proposed to model the

anisotropy and hardening of Medium Mn. (10 % wt.) at multi-strain rates. In the

beginning, a comprehensive review of constitutive relations was performed, explaining

their different types and the different components. Then different yield criteria were

140

discussed, and the Yld2000-2d was chosen due to its accuracy and flexibility in capturing

different modes of anisotropy. Yld2000-2d was calibrated using experimental data from

multi-strain rates, and the results showed that the yield locus’ shape and size is strain rate

dependent and thus it is necessary to capture this dependency to improve the accuracy of

constitutive relations that are used to model similar anisotropic material at multi-strain

rates.

With that motivation, it was concluded that the parameters of the Yld2000-2d are

strain rate dependent and can be mathematically modeled in relation to the strain rate.

These relations were developed and used to develop a User Defined Material (UMAT)

card to be used in a commercial FEM software (LS-DYNA). The numerical procedure to

develop the code was described in detail with most of the mathematical manipulation

involved, and then the material card was compiled and implemented in LS-DYNA.

Several simulations were performed using the newly developed material card, the

results of which were compared with the experimental results showing an excellent

agreement between the two. The results highilighted the inaccuracies of using a yield

function calibrated using the quasi-static results to model the behavior of anisotropic

materials, in this case a third-generation AHSS, during loading at multi-strain rates.

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elevated temperatures–Part 1: Material characterization," International Journal of

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149

CHAPTER FIVE

5. CRASH SIMULATIONS OF A RAILROAD TANK CAR

ABSTRACT

Following the development and the validation of the user-defined material card

(UMAT), that includes the constitutive relations describing the behavior (hardening and

anisotropy) of Medium Mn. (10 % wt.) at multi-strain rates, the next step is to use the

newly developed material card to perform crash simulations and analyze the significance

and impact on the modeling accuracy. The material card was to perform finite element

crash simulations on a railroad tank car. The simulations were performed in accordance

with the regulations of the FRA regarding crash tests of tank cars. First, the finite element

model was developed based on the information provided by a commercial tank car

manufacturer. The accuracy of the model was verified by running simulations of an

actual crash test performed by a tank manufacturer, ensuring to correctly define the initial

and boundary conditions, and the material properties of TC128-B, which is the material

used in railroad tank cars. The results of the simulation were compared with the results of

the actual crash test allowing for the validation of the finite element model developed.

Following that, a different set of simulations were performed using the Medium Mn. (10

% wt.) steel as the material for the tank. The simulations were run using three types of

material cards: 1) A material card based on the quasi-static results, 2) a material card

predicting the strain rate sensitivity of the material (hardening only) and 3) the material

card coded in chapter 4 which can predict the strain sensitivity of the material (hardening

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and anisotropy). The results of these simulations were compared and analyzed based on

the force-displacement history and energy absorbed until fracture. The impact and

significance of the work done are highlighted by comparing the results of these

simulations.

5.1 Introduction

Due to the great difficulty in obtaining automotive crash models and crash test

data from an OEM, and the availability of both from a tank car manufacturer, it was

decided to use the material card developed to perform crash simulations on railroad tank

cars. The simulations will follow the standards and regulations instated by the

Association of American Railroads (AAR) and the Federal Railroad Administration

(FRA) regarding crash tests and simulations [1].

The interest in studying the crashworthiness of tank cars is not recent, as there are

studies that date back to the 1970s that were performed with the association of the AAR

[2]. However, there has been an increased interest in recent years to improve the

modeling capabilities due to the increase in derailment accidents that led to the spilling of

hazardous materials into the environment [3-5]. The accidents have shown that there is a

need for more research to be done to improve the structural integrity of the tank cars.

Consequently, steps were taken by the government and the industry, manifested by the

Next Generation Rail Tank Car (NGRTC) project [6], which aimed to research and study

the different crash scenarios that can lead to structural failure, and consequent spilling, to

develop an accurate test methodology to replicate the scenarios provide solutions. The

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studies focused on the prevention of the release of hazardous materials from the tank

during a crash.

Studying the recent crash incidents, two accident scenarios were highlighted and

were considered most critical. The first scenario involves a derailment accident, which

causes the tank car to derail from the track, causing different track cars to collide together

(Figure 5.1). The other scenario involves a train on a train collision. These two scenarios

have led to the development of two cross-sectional profiles for the impactor to be used in

tests and simulations [7, 8]. The cross sections correspond to two different parts of the

tank car: 1) the coupler shank, and 2) the draft sill. The structure of the tank must be able

to maintain its integrity when collided by either of these profiles. The structural integrity

of the tank is characterized mainly using two parameters: 1) the impact energy absorbed

by the structure before puncture, and 2) the minimum velocity of the impactor that causes

tank puncture (impact velocity) [8].

Figure 5.1: A schematic showing the collisions associated with a derailment accident [3].

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The finite element model development has also been a significant point of interest

for governmental and industrial agencies due to the complex nature of the model. The

complexity associated with the simulations mainly comes from the necessity of

accurately modeling the response of the tank material at multi-strain rates and the fact

that tank cars are actually carrying fluids inside of them. The first complexity has been

the goal of this research and the previous chapter has led to the development of a material

card that is capable of accurately modeling the multi-rate behavior. However, to tackle

the second complexity, it is crucial to accurately define the fluid and the fluid-structure

interaction (FSI) using appropriate material models and equations of state. This is

complex because, in finite element analysis, the methodology used to model solids is

different from the one used with fluids. Numerical discretization in finite element

simulation can be performed using different methods [9], such as Lagrangian, arbitrary

Lagrangian Eulerian (ALE), smooth particle hydrodynamic (SPH), element-free Galerkin

(EFG), and discrete element. Lagrangian Methods are the most commonly used in

forming and crash simulations as they can model the behavior of solid materials under

small deformations accurately at a low computational cost. Lagrangian methods are

characterized by a deformable mesh, with every material point coinciding with every

node (i.e., the mesh moves and deforms) [10]. While Eulerian methods, on the other

hand, consist of a fixed non-deformable geometrical mesh within which the material can

move freely. A combination of both methods is known as an Arbitrary Lagrangian

Eulerian formulation which allows the mesh translational and rotational motions while

still maintaining the non-deformability and the separation between the material and the

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mesh [11, 12]. Fluids flow when being loaded, and the flowing would cause a

considerable distortion in the element which would cause the simulation to fail (negative

Jacobian); therefore, fluids can only be modeled using Eulerian or ALE elements.

Solving a problem using ALE formulation requires an additional time step (advection

time step) and requires 3D-Solid elements. A model containing both element

formulations need special care to correctly define the interaction between the ALE mesh

and the Lagrangian mesh, and this was studied in the literature regarding railroad tank

cars [8, 13, 14].

This chapter will focus on the development of the finite element model of the

railroad tank car. Beginning with some details about the parameters in the model and the

procedure followed to verify the accuracy of the base model developed. Moreover, the

model will then be used to run crash simulations using Medium Mn. (10% wt.) steel,

using the constitutive relation developed in the previous chapter. The simulations will

aim to quantify the impact and importance of using the constitutive relation on the results

of an impact.

5.2 Finite Element Model

Figure 5.2 shows a schematic of a typical railroad tank car, describing its different

components. The particular tank car modeled in LS-DYNA is based on the

DOT117J100W design as per the definition and standards of the AAR [1]. The auxiliary

parts on the tank such as (manways, pressure relief devices, bolsters, etc…) are not

included in the model to reduce computation times, and their inclusion was proven to not

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affect the results of the simulations [15-17]. The tank heads were designed based on the

FRA regulations as an ellipsoid of revolution with the major axis equaling the diameter of

the tank and the minor axis to be half of the major axis. The tank was designed with an

outer diameter of 120.5” and a thickness of 9/16”, while the jacket had a thickness of

11GA (~3mm). The developed model is shown in Figure 5.3a. The model was meshed

using shell elements of a length of 200mm, while the center area of the tank where the

impact will occur, had a refined mesh of size of 20mm. The final model with mesh is

shown in Figure 5.4. The impactor chosen was based on the standards of the AAR and

FRA for impact testing. It had face dimensions of 12”x12” with an edge radius of 0.5”

(Figure 5.3b). The impactor was also meshed wih similar size to the impact area mesh

despite the fact that no computation is required on those elements. However, this is

crucial to ensure a smooth accurate contact algorithm.

Figure 5.2: A schematic of a typical North American railroad tank car [18].

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(a)

(b)

Figure 5.3: The geometry of the (a) tank (b) impactor

Figure 5.4: The meshed finite element model of the tank

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The boundary and initial conditions in the crash simulations were defined with the

goal to replicate the real scenarios encountered during an actual crash test [19]. The

actual test includes impacting the tank on one side with a ram car, while a wall rests on

the other side to prevent the tank from rolling. A similar setup was used for our model,

with the wall on the left side of the tank car (Figure 5.5). Another rigid wall is added on

the lower side of the tank to act as the floor and prevent any vertical motion.

Figure 5.5: (a) A tank car used for an actual crash test [8] (b) the model developed to

replicate the boundary conditions encountered during a crash test.

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Removing the wheels from the model mandates adding boundary conditions that

can correctly represent their effect. Failing to do so will give extra degrees of freedom for

the elements of the tank leading to an inaccurate deformation behavior in the impact area

and the lack of the actual structural integrity. The impactor was assigned an initial

velocity of 18 mph toward the longitudinal and vertical center of the tank.

The material used in the simulation is TC128-B, which is a typical carbon steel

material used for plate materials. The material has a yield strength of about 400MPa and

ultimate tensile strength of about 700MPa. The TC128-B used in tank materials

undergoes a heat treatment process known as normalizing, which is described in detail by

Hayashi et al. [20]. With the material and boundary conditions of the simulations well

defined, the simulations are ready to be performed. A summary of the parameters of the

simulations are summarized in table 5.1

Table 5.1: the parameters of the simulations

Lading Chlorine

Gross Weight 286000lbs

Tank Inside Diameter 119 3/8”

Thickness 9/16”

Internal Pressure 100psi

Outage 5%

Weight of Impactor 286000lbs

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As mentioned earlier, the tank cars modeled transport liquified goods, the fluid, in

this case, was chlorine. Under the normal conditions, the material being transported exist

in two phases: liquid and vapor. In order to correctly define the pressure in the fluids and

the interaction between the fluid, the vapor, and the tank, it is necessary to model the

coexisting of the two phases inside the tank in addition to the atmospheric air surrounding

the tank. Therefore, the fluid was modeled using three regions: the liquid chlorine in the

tank, the vapor chlorine in the tank, and the air surrounding the tank from outside. The

three fluids were modeled using solid elements. The interaction between the fluids and

the tank was defined using the FSI capabilities of LS-DYNA. The Mie-Gruneisen

equation of state (EOS) [21] already defined in LS-DYNA was used to model the state of

the chlorine, while the ideal gas EOS was used for the air.

Figure 5.6: The model showing the initialization of the fluids inside the tank. Green

(liquid), blue (vapor).

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5.3 Crash simulations using TC128-B

The initial crash simulations were performed without modeling fluids using ALE

formulation, but rather as just an additional mass on the tank. The objective of the initial

run was to ensure the structural part of the model was correctly defined and leads to

accurate results. To verify that, the results of the initial run of simulations were compared

to results published in the literature specifically the results published by Kirkpatrick in

2010 whose performed crash tests and simulations on a tank car with similar design and

parameters. First, the acceleration profile of the impactor was compared to that obtained

in the literature by Kirkpatrick in 2010. The results are shown in Figure 5.7. Second, the

force-displacement curves extracted from the impactor were compared with the results

obtained from Kirkpatrick (Figure 5.8). The results of the simulations performed in this

study agree to a certain degree of accuracy (in terms of the magnitude and overall trend)

with the results from Kirkpatrick publication. The difference can be mainly attributed to

the lading modeling.

The results obtained from the first set of simulations ensured the validity and

accuracy of the structural part of the model. Therefore, the next set of simulations

involved modeling lading using the ALE formulation as described earlier. Other than the

lading modeling, the parameters used in the initial run were kept the same, and the

simulations were run at the same initial impactor speed. The load-displacement curve was

extracted from these simulations and compared with the same results by Kirkpatrick in

Figure 5.9. The circle is drawn on the maximum load value in the second set of

simulation to be compared with the actual crash test results presented by Kirkpatrick

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rather than with his simulation results. To better visualize the accuracy and improvement

over the first run, the load-displacement curve from the crash test published by

Kirkpatrick was digitized and plotted with the results of the second set of simulation

(Figure 5.9c). Our simulation predicted a maximum force of 0.98 Mlbs, while during the

actual crash test, the force recorded was 0.96 Mlbs, which is accurate to about 2%. This

means that the finite element model developed is accurate and can capture the actual

deformation of the tank during a crash event.

5.4 Crash simulation using medium Mn (10% wt.) steel

With the results of the previous section validating the accuracy of the finite

element model developed, the material of the tank was now replaced with Medium Mn.

(10% wt.) steel and the developed material card described in chapter 4 was used. Three

different simulations were performed to quantify the impact and importance of the newly

(a)

161

(b)

Figure 5.7: Comparison of the acceleration profiles of the impactor between (a) the

simulation without lading and (b) Kirkpatrick results [6].

(a)

162

(b)


simulation without lading and (b) Kirkpatrick results [6].

(a)

163

(b)

(c)


simulation with lading, (b) Kirkpatrick results [6], and (c) direct comparison between the

simulation results and the actual test results published by Kirkpatrick.

164

developed constitutive relation. The first simulation was performed using the

experimental results obtained using the quasi-static experimental results, i.e., that only

one hardening curve was used in the simulation (quasi-static curve) and the anisotropic

Yld2000-2d was used as the yield function calibrated using the quasi-static data. The

second simulation was performed using the experimental results obtained using the multi-

rate experimental results for hardening and the quasi-static experimental results for the

yield function, i.e., that the hardening curves at the different strain rate were used in the

simulation and the anisotropic Yld2000-2d was used as the yield function calibrated

using the quasi-static data. Finally, the last simulation was performed using the

experimental results obtained using the multi-rate experimental results for hardening and

multi-rate experimental results for the yield function, i.e. that the hardening curves at the

different strain rate were used in the simulation, and the newly developed rate-dependent

anisotropic Yld2000-2d (chapter 4) was used as the yield function calibrated using the

multi-rate data (the material card coded in chapter 4 was used). Figure 5.10 shows the

force-displacement curves obtained from the three simulations. The figure clearly shows

that the three simulations have distinct load-displacement profiles from the three different

input cards and the dependency of the simulation results on capturing the anisotropy

(notice the difference between the red and the green curves). The results show that

simplifying the input card by modeling only the quasi-static response of the material lead

to overestimation of the puncture resistance of the tank, whereas when improving the

accuracy of the input card by modeling the multi-rate response in terms of hardening and

anisotropy the load-displacement curve tend to significantly drop in terms of the area.

165

Figure 5.10 represents a qualitative comparison between the three scenarios,

however, to further prove the necessity and impact of accurate modeling, the energy

absorbed to fracture was calculated in the simulation for the three different scenarios.

Figure 5.11 shows a bar graph representation of the energy. Scenario 1 corresponds to the

Figure 5.10: Comparison of the three load-displacement profiles of simulations ran using:

1) quasi-static hardening and yielding (green), 2) multi-rate hardening and quasi-static

yielding (blue), and 3) the developed constitutive relation, i.e., multi-rate hardening and

yielding (red)

166

first simulation where only quasi-static experimental results were used. Scenario 2

corresponds to capturing multi-rate hardening while only modeling quasi-static yielding,

and finally, scenario 3 corresponds to using the rate-dependent anisotropic Yld2000-2d

developed in the previous chapter. The results show that capturing the multi-rate

hardening behavior of Medium Mn. (10 % wt.) steel reduces the energy absorbed by the

tank before puncture by 13%. Furthermore, accurately modeling the full spectrum of the

multi-rate response of the material (hardening and anisotropy) manifested by the material

card coded in chapter 4 reduces the energy absorbed by 27% as compared to the

conventional modeling technique (quasi-static only). The significant drop in energy

shown here emphasize the necessity and impact of the work performed, as the errors

introduced by following the conventional approach for AHSSs can prove to be fatal.

Figure 5.11: Energy absorbed to fracture during a crash simulation of the three different

scenarios described.

167

5.5 Summary

A finite element model of a railroad tank car was built to perform crash

simulations. The model was verified by running crash simulations using TC128-B (the

material used in the industry) and the results were compared to the results of simulations

and crash tests in the literature. Following that, crash simulations were performed on the

tank with Medium Mn. (10% wt.) steel as the material of the tank. Several variations of

the constitutive relations were used, and three scenarios of simulations were run. The first

scenario involved running the simulations while modeling only the quasi-static behavior

of the material; the second scenario involved modeling the multi-rate hardening behavior

and quasi-static yielding and finally, the third scenario involved using the developed rate-

dependent anisotropic yield function with multi-rate hardening (coded in a material card).

When comparing the energy absorbed before tank puncture in the three simulations, it

was apparent that the difference between the three scenarios is significant, emphasizing

the necessity and impact of accurately modeling the different properties of the materials

at multi-strain rates and not only the hardening.

5.6 References

[1] A. o. A. Railroads, "MANUAL OF STANDARDS AND RECOMMENDED

PRACTICES: SECTION C—PART II," in DESIGN, FABRICATION, AND

CONSTRUCTION OF FREIGHT CARS, ed. Washington, D.C. 20024: The Association

of American Railroads, 2015, p. 497.

168

[2] E. Phillips and L. Olsen, "Final Phase 05 Report on Tank Car Head Study," RPI-

AAR Tank Car Safety Research and Test Project, RA-05-17, 1972.

[3] S. W. Kirkpatrick, "Detailed Impact Analyses for Development of the Next

Generation Rail Tank Car: Part 1—Model Development and Assessment of Existing

Tank Car Designs," in ASME 2009 Rail Transportation Division Fall Technical

Conference, 2009, pp. 43-51.

[4] NTSB, "Collision of Norfolk Southern Freight Train 192 With Standing Norfolk

Southern Local Train P22 With Subsequent Hazardous Materials Release at Graniteville,

South Carolina," Railroad Accident Report NTSB/RAR-05/04 PB2005-916304. Notation

7710A, 2005.

[5] D. Jeong, Y. Tang, H. Yu, and A. B. Perlman, "Engineering analyses for railroad

tank car head puncture resistance," in ASME 2006 International Mechanical Engineering

Congress and Exposition, 2006, pp. 1-7.

[6] S. Kirkpatrick, "Detailed Puncture Analyses of Various Tank Car Designs,"

Applied Research Associates, Mountain View, CA, 2009.

[7] D. Tyrell, D. Jeong, K. Jacobsen, and E. Martinez, "Improved tank car safety

research," in ASME 2007 Rail Transportation Division Fall Technical Conference, 2007,

pp. 43-52.

[8] H. Yu and D. Y. Jeong, "Impact dynamics and puncture failure of pressurized

tank cars with fluid–structure interaction: A multiphase modeling approach,"

International Journal of Impact Engineering, vol. 90, pp. 12-25, 2016.

169

[9] L.-D. K. U. s. Manual and I. Volume, "Version 971," Livermore Software

Technology Corporation, vol. 7374, p. 354, 2007.

[10] T. J. Hughes, W. K. Liu, and T. K. Zimmermann, "Lagrangian-Eulerian finite

element formulation for incompressible viscous flows," Computer methods in applied

mechanics and engineering, vol. 29, pp. 329-349, 1981.

[11] J. Donea, S. Giuliani, and J.-P. Halleux, "An arbitrary Lagrangian-Eulerian finite

element method for transient dynamic fluid-structure interactions," Computer methods in

applied mechanics and engineering, vol. 33, pp. 689-723, 1982.

[12] N. Takashi and T. J. Hughes, "An arbitrary Lagrangian-Eulerian finite element

method for interaction of fluid and a rigid body," Computer methods in applied

mechanics and engineering, vol. 95, pp. 115-138, 1992.

[13] H. Yu, Y. H. Tang, J. E. Gordon, and D. Y. Jeong, "Modeling the effect of fluid-

structure interaction on the impact dynamics of pressurized tank cars," in ASME 2009

International Mechanical Engineering Congress and Exposition, 2009, pp. 59-69.

[14] H. Yu, D. Jeong, J. Gordon, and Y. Tang, "Analysis of impact energy to fracture

unnotched charpy specimens made from railroad tank car steel," in ASME 2007 Rail

Transportation Division Fall Technical Conference, 2007, pp. 123-130.

[15] D. Jeong, J. Gordon, Y. Tang, A. Perlman, and H. Yu, "Analysis of railroad tank

car shell impacts using finite element method," in IEEE/ASME/ASCE 2008 Joint Rail


170

[16] Y. Tang, H. Yu, J. Gordon, and D. Jeong, "Finite element analyses of railroad

tank car head impacts," in ASME 2008 Rail Transportation Division Fall Technical


[17] Y. Tang, H. Yu, J. Gordon, M. Priante, D. Jeong, D. Tyrell, et al., "Analyses of

full-scale tank car shell impact tests," in ASME 2007 rail transportation division fall

technical conference, 2007, pp. 29-37.

[18] C. P. Barkan, S. V. Ukkusuri, and S. T. Waller, "Optimizing the design of railway

tank cars to minimize accident-caused releases," Computers & operations research, vol.

34, pp. 1266-1286, 2007.

[19] D. Tyrell, K. Jacobsen, B. Talamini, and M. Carolan, "Developing strategies for

maintaining tank car integrity during train accidents," in ASME 2007 Rail Transportation

Division Fall Technical Conference, 2007, pp. 59-68.

[20] K. Hayashi, A. Nagao, and Y. Matsuda, "550 and 610 MPa class high-strength

steel plates with excellent toughness for tanks and penstocks produced using carbide

morphology controlling technology," JFE Technical Report, vol. 11, p. 20, 2008.

[21] A. I. Burshtein, Introduction to thermodynamics and kinetic theory of matter:

John Wiley & Sons, 2008.

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CHAPTER SIX

6. SUMMARY AND CONCLUSIONS

In summary, this research focused on developing a methodology to characterize

the behavior of new grades of AHSSs at strain rates equivalent to that encountered during

a crash event. First, the effects of microstructure on the strain sensitivity of different

AHSSs was studied by performing experiments at multiple strain rates on a variety of

AHSSs with a wide array of phases compositions in their microstructure. The

microstructures of these different materials contained a variety of different combination

of the three main phases of steel. The results showed the significant impact the

microstructure has on the behavior of the steels at multi-strain rates. Following that, the

focus of the work shifted to a specific third-generation AHSS, which was Medium Mn.

(10% wt.) steel. This steel is considered one of the most promising third-generation

AHSSs, due to its unique properties and microstructure.

Further experiments were performed on this material to study its multi-rate

behavior deeply. The results showed that the material exhibit anisotropy at quasi-static

strain rates and at high strain rates, however, the anisotropy varied at different strain

rates. This strain rate sensitive anisotropy was significant and therefore had to be

accurately modeled in the crash simulations performed on this material. A survey of the

literature showed that to the best knowledge of the author, there is no strain rate

dependent anisotropic yield function capable of capturing this kind of behavior.

Therefore, after a thorough literature review of the different anisotropic yield functions,

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Yld2000-2d was selected to be the basis to develop a new rate-dependent anisotropic

yield function. The developed yield function was numerically coded in a material card

to be used in the commercial FEA software, LS-DYNA. The material card was first

validated by running simulations of tensile tests and comparing it with the experimental

results. Once this was accomplished, the material card was used to perform crash

simulations on a railroad tank car. The results of the simulations showed that using the

material card developed had a significant impact on the results, which was quantified by

calculating the energy absorbed by the tank before puncture occurred. The energy

absorbed dropped by 27% when using the newly developed model as compared to the

conventional method of modeling (quasi-static only). The significant difference in the

energies highlights the impact of capturing the full spectrum of the multi-rate behavior of

AHSSs (hardening and anisotropy) when performing crash simulations. The significance

is manifested in the fatal implications of having errors in that magnitude when running

crash simulations, most importantly the safety of the humans and the environment.

6.1 Contributions

The main contributions of this work are:

1) Developed the experimental capabilities and approach to enable robust material

testing at a wide range of strain rates (up to very high strain rates).

2) Performed a comprehensive and detailed experimental study on the effects of the

different steel phases (ferrite, austenite and martensite) on the macro-mechanical

173

response of AHSSs at different rates up to the highest strain rates encountered

during a crash event.

3) Investigated and captured hardening and anisotropy in one of the latest

generations of AHSSs (Med. Mn Steel Grade 10%wt); a promising material with

very complex behavior.

4) Developing a new rate-dependent anisotropic yield function based onYLD2000-

2d that can capture the change in anisotropy observed in medium Mn. (10% wt.)

steel at different strain rates. This presents a significant modification to the

methodology of modeling and characterizing the behavior of material especially

AHSSs at multi-rates.

5) A UMAT card, for the modified constitutive relations for Med. Mn Steel, was

developed for use in LS-DYNA; the card was validated through a series of

simulations of tension testing that show the ability to accurately capture hardening

and anisotropy in the material at different strain rates.

6) Full impact simulations of rail tank cars were performed as a final overall

validation of the developed material card; the simulations provide a quantitative

measure of the importance of accurately modeling the hardening and anisotropy at

multi-strain rates in complex materials.

6.2 Future Work

The future work will focus on developing a relation that can predict the behavior

of the materials at high strain rates based on the initial phase composition of the

microstructure. This is particularly important in minimizing the costs associated with the

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development of the new generations of AHSSs. Also, further work needs to be done

regarding the new methodology manifested by the new anisotropic yield function.

Several experiments need to be performed to improve the accuracy of the model at

different plastic work. Finally, a fracture model must be added to the constitutive relation

to increase the accuracy of crash simulations. The research will focus on developing a

rate-dependent fracture model.

It is also worth noting that the current rate dependent anisotropic yield function

includes a parameter that depends on the crystallographic configuration of the

microstructure. When austenite, which has an FCC crystallographic structure, transforms

into martensite, which has a BCT crystallographic structure, the yield function needs to

be updated in accordance with the new structure. In the current work, it was shown that

the transformation is inhibited at high strain rates, and therefore even with the

transformation taking place, the microstructure will be majorly an FCC structure and thus

justifies the choice for the parameter. However, it is worth investigating the effect of the

change in the structure on the yield function in the future work.

characterization and modeling of the high strain rate

Documents