Procedure and Results for Constitutive Equations For

PROCEDURE AND RESULTS FOR CONSTITUTIVE EQUATIONS FOR

ADVANCED HIGH STRENGTH STEELS INCORPORATING STRAIN, STRAIN

RATE, AND TEMPERATURE

THESIS

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Anthony J. Smith, B.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2012

Master's Examination Committee:

Robert H. Wagoner, Advisor

Rebecca B. Dupaix

Anthony J. Smith

2012

ABSTRACT

A wide range of advanced high strength steel (AHSS) sheets with nominal ultimate tensile strengths (UTS) from 590 to 1180 MPa and engineering strains to failure from 0.09 to 0.51 were tested to obtain the tensile flow stress under various combinations of strain, strain-rate, and temperature. Procedures were developed for selecting suitable constitutive forms corresponding to a generalized framework proposed in the literature by

Sung et al. (Sung et al., 2010). Using the selected forms, the least-squares coefficients were obtained using an efficient optimization procedure that was also developed. The fit accuracy in all cases was similar to the test-to-test experimental scatter. Comparison with results from standard fitting schemes showed that the optimum coefficient values were found using the proposed, more time-efficient, methodology. The resulting constitutive models were compared with novel balanced biaxial bulge test measurements at strains up to several times larger than those accessible to tensile testing. The results show that constitutive models obtained by such procedures from standard tensile data can be extrapolated accurately to strain ranges of interest in bending-affected plastic localization

(“shear fracture”).

This document is dedicated to my loving parents and my brother who have supported and

encouraged me in every endeavor and helped me achieve all of my goals.

iii

ACKNOWLEDGMENTS

First and foremost I would like to thank my advisor, Dr. Robert H. Wagoner, for his constant support and guidance. I would like to thank my fellow research group members and former group members Hojun Lim, Kun Piao, Zhong Chen, and Jason

Ackerman for all of their assistance.

I am grateful for many people who assisted in this work. Dr. Myoung-gyu Lee at

POSTECH provided the balanced biaxial bulge test experimental data. Dr. Jae-bok Nam,

Dr. Hong Woo Lee, and Jae Wook Lee from Posco Global R&D Center and Dr. Dong-Jin

Kim from Posco Technical Research Labs provided the materials tested and technical assistance via many helpful discussions. Ken Kushner and Ross Baldwin provided assistance with mechanical testing at The Ohio State University.

VITA

May 2007 ...... Hilliard Davidson High School

June 2011 ...... Undergraduate Research Assistant,

Department of Materials Science and

Engineering, The Ohio State University

June 2011 ...... B.S. Mechanical Engineering, The Ohio

State University (Magna Cum Laude)

June 2011 to present ...... Graduate Research Associate, Department

of Materials Science and Engineering, The

Ohio State University

FIELDS OF STUDY

Major Field: Mechanical Engineering

TABLE OF CONTENTS

ABSTRACT ...... ii

ACKNOWLEDGMENTS ...... iv

VITA ...... v

LIST OF TABLES ...... vii

LIST OF FIGURES ...... viii

CHAPTER 1: INTRODUCTION ...... 1

CHAPTER 2: EXPERIMENTAL PROCEDURES...... 9

CHAPTER 3: FITTING PROCEDURE ...... 18

CHAPTER 4: RESULTS AND DISCUSSION ...... 29

CHAPTER 5: CONCLUSIONS ...... 44

REFERENCES ...... 46

LIST OF TABLES

Table 1: Standard mechanical properties from tensile testing done by POSCO Steel

(Lee, 2011a) of six AHSS used in this research...... 10

Table 2: Selection of α(T) function between the linear form and the sigmoidal form. ... 22

Table 3: Standard error of fit of plastic constitutive equations determined by H/V methodology for six advanced high strength steels...... 29

Table 4: Overall optimization parameters for the rate-insensitive portion of the plastic constitutive equation for all six materials used in the study...... 31

Table 6: Parameters for strain-rate sensitivity as a function of temperature and strain-rate for all six materials used in the study...... 39

Table 7: Results of comparison between the extrapolation of the plastic constitutive equation and the experimental balanced biaxial bulge test results for each material...... 43

vii

LIST OF FIGURES

Figure 1: Quasi-isothermal tensile test equipment and setup...... 11

Figure 2: Sample results of isothermal tensile tests of DP 780 illustrating the temperature dependence of the stress-strain relationship: (a) Engineering stress-strain curves (b) Strain hardening form...... 12

Figure 3: Typical strain-rate jump test results illustrating the procedure for measuring strain-rate sensitivity: (a) Experimental data and corresponding fit curves (b) Expanded region near the jump strain to illustrate the analysis procedure...... 14

Figure 4: Dimensions of specimens used in balanced biaxial bulge testing (Lee, 2011).

...... 17

Figure 5: Comparison of experimentally-determined values of α at each test temperature with chosen α(T): (a) Linear form from DP 780 (b) Sigmoidal form from DP 590...... 21

Figure 6: Typical comparison of experimentally-measured stresses and their systematic variation with temperature via h(T) illustrating linear thermal softening for DP 780 steel.

...... 23

Figure 7: Linear relationship between coefficients within the Swift constitutive equation and temperature: (a) S1(T) (b) S2(T) (c) S3(T)...... 27

viii

Figure 8: Comparison of the overall constitutive law determined using the proposed methodology with the experimental tensile test data at all test temperatures: (a) DP 590

(b) DP 780 (c) DP980 (d) TRIP 780 (e) CP 1180 (f) TWIP...... 32

Figure 9: Comparison of m values (strain-rate sensitivities) from strain-rate jump tests at three temperatures with their constitutive representation by the rate law: (a) DP 590 (b)

DP 780 (c) DP 980 (d) TRIP 780 (e) CP 1180 (f) TWIP...... 35

Figure 10: Comparison of constitutive equation extrapolation from tensile testing with balanced-biaxial test data: (a) DP 980, 100 °C (b) TWIP, 50 °C...... 42

CHAPTER 1: INTRODUCTION

Advanced high strength steels (AHSS) offer a broad range of strengths (for automotive service requirements, for example) and elongations to failure (for manufacturing via forming). (Kuziak et al., 2008). They achieve these combinations of properties by various deformation and hardening mechanisms that alter the strain hardening behavior. Widespread usage of AHSS has been limited by the practical formability being much lower than predicted values (forming failures occur earlier than predicted). Previous work has shown that these failures are related to “shear fracture”, which is known to occur in AHSS (Huang et al., 2008; Sklad, 2008; Chen et al., 2009). In the current work, we use the typical industrial definition of “shear fracture” to signify only that the fracture occurs in tight-bending regions of a sheet undergoing sheet forming

(i.e. the bending ratio, R/t, where R = tool radius and t = initial sheet thickness, is small).

This type of fracture is contrasted with typical failures with traditional steels, i.e. where the onset of localization occurs in flat or gently-curve areas of the sheet part, and is generally unaffected by the bending. The formability of AHSS are significantly affected by the high strain-rates and high strains which are encountered during practical industrial applications with sheet metals (Kim, 2011).

The high strains and strain-rates (typically about 10/s (Fekete, 2009) are normally found in low R/t regions as the sheet is pulled over a die radius. Because of the high rates, local deformation-induced heating plays a significant role in the strain hardening behavior (Wagoner et al., 2009 a,b; Wagoner and Chenot, 1997; Ghosh, 2006, Kim,

2011; Sung, 2012), and thus in the failure characteristics. Plastic deformation in high strain regions results in high temperatures that lead to local softening and reduced strain hardening of the AHSS. This causes reduced formability and promotes fracture in forming processes (Kleemola and Ranta-Eskola, 1979; Ayres, 1985; Lin and Wagoner,

1986; Gao and Wagoner, 1987; Lin and Wagoner, 1987; Wagoner et al., 1990; Ohwue et al., 1992).

These factors motivate the need for reliable plastic constitutive equations in the ranges of strain, strain-rate, and temperature encountered during these forming operations. Particular emphasis is on the critical ranges of these variables for shear fracture: strains up to 0.5, strain-rates up to 10/s, and temperatures up to 100-120 °C

(Kim, 2011; Sung, 2012). In order to use these new materials and capabilities effectively and efficiently, the plastic constitutive response must be known accurately, often at much larger strains, strain-rates, and temperatures than are accessible using typical tensile testing. It is therefore crucial to have available accurate constitutive models for AHSS, such that they can be accurately extrapolated outside of their measured domain. A standard, reliable, efficient, and accurate methodology is needed to obtain constitutive equations incorporating strain, strain-rate, and temperature in the ranges of interest. This methodology should be versatile enough to effectively fit a wide range of the most

2 common and the most advanced AHSS, both in terms of the forms of the components of the integrated equations, and also in terms of the values of the final coefficients.

A wide range of plastic constitutive equations describing strain hardening at a constant strain and temperature exist in literature. A central division among strain- hardening forms is based on the behavior of the model at large strains. Saturation or

Voce-type models approach a saturation stress at large strains (Voce, 1948; Follansbee and Kocks, 1988), while power-law or Hollomon-like models are unbounded at large strains (Hollomon, 1945). A typical variation of the Hollomon approach incorporates a pre-strain coefficient and is known as a Swift form (Swift, 1952; Chung et al., 2011). The

Voce, Hollomon, and Swift constitutive equations respectively are:

( - e p(- )) (1)

(2)

s ( ) (3)

Where V1, V2, V3, H1, H2, S1, S2, and S3 are material constants.

Sung et al. reviewed many of the constitutive equations in the literature (Sung et al., 2011). Such breadth will not be repeated here. When combinations of effects have been considered (i.e. strain, strain rate, and temperature) either complicated integral forms have been introduced (e.g. Johnson and Cook, 1983; Khan, 1999) or composite functions which depend on one or more variables which appear together via a multiplicative combination have been used. Choosing a multiplicative combination

(which typically eliminate cross-terms between strain and strain-rate sensitivity, for example) provides conceptual simplicity and has shown to be effective in many cases

(Hutchison, 1963; Kleemola and Ranta-Eskola, 1979; Johnson and Cook, 1983; Lin and

Wagoner, 1986; Wagoner, 1981 b; Ghosh, 2006; Zerilli and Ronald, 1987). One extension of this multiplicative structure involves using standard constitutive equations at a fixed temperature, then allowing the coefficients to vary in a simple manner with temperature. The concept of allowing the strain hardening parameters within one of these models to vary in a prescribed manner with temperature has been studied for both power- law strain hardening behavior and saturation strain hardening behavior (Rusinek and

Klepaczko, 2001; Lin and Wagoner, 1987).

Measuring and characterizing strain rate sensitivity is particularly difficult with

AHSS, which typically are much stronger than traditional steels and have commensurately smaller strain-rate sensitivity. In fact, the effect of rate-sensitivity on flow stress for typical changes of strain-rate (10x, for example) is often smaller than the normal test-to-test scatter of flow stress inherent in material and specimen variability.

This effectively eliminates the possibility of obtaining accurate strain-rate sensitivity forms from multiple tensile tests.

In the current work, the strain-rate sensitivity forms and coefficients were obtained from strain-rate jump tests among various strain-rates and at a range of temperatures. While differences have been observed between strain-rate sensitivity measurements from multiple tensile tests and jump tests (Wagoner, 1981b), utilizing a suitable method to reduce the effect of transient behavior immediately following a jump may serve to produce better agreement. Previous literature (Wagoner, 1981a; Saxena and

Chatfield, 1976) has shown that strain-rate sensitivity is nearly independent of strain for

4 steels, so jump tests at a single intermediate strain were presumed to represent the behavior over the tensile strain range.

Sung et al. introduced a framework for constitutive equations that was shown to work well for modeling the plastic behavior of dual-phase (DP) steels. That work proposed a model that used a special modification of the multiplicative structure with three functions: f(ε, T), h(T), and g( ̇) as follows:

( , , ̇) f( , ) g( ̇) h( ) (4)

The framework is not purely multiplicative as strain and temperature appear in a complex relationship in f(ε,T) (Sung et al., 2010), while a typical separate scaling of flow stress with temperature is accommodated by h(T). This structure takes into account the possibility of significant thermal softening on the behavior of the steel (via h(T)), as well as the possibility of temperature dependence on the material’s strain hardening (via f( ,T)). The special “H/V” form of the function f( ,T) introduced by Sung builds on the common observation that steels and other metals often exhibit Hollomon-like behavior at low temperatures and more Voce-like behavior at higher temperatures (Sung et al., 2010).

The H/V strain-hardening form at a fixed temperature consists of a linear combination of the Hollomon (Equation 2) and Voce (Equation 1) hardening laws, with linear combination coefficient α, as follows (Sung et al., 2010).

α H ( -α) V α ( -α) ( - e p(- )) (5)

Where it is apparent the α is a measure of the contribution of the Hollomon constitutive equation to the final form, with α = 1 being pure Hollomon behavior and α = 0 being pure

Voce behavior. In the special H/V form proposed by Sung, the only dependence of f( ,T) on temperature occurs in the linear combination coefficient α, i.e.

α α( ) (6)

This equation represents the transition between Hollomon-like and Voce-like strain- hardening behavior as the temperature is varied. Sung found that a linear form of α(T) was adequate to represent the DP steels tested there:

α( ) α ( - ) α (7)

Where α1 and α2 are constants and T0 = 25°C is arbitrarily chosen as a room-temperature baseline (typically where other functions in the composite model are best known).

To address strain-rate sensitivity, Sung et al. examined several g( ̇) functions. The functions examined in the current work incorporate strain-rate sensitivity into the constitutive equations in a multiplicative manner as follows (Wagoner, 1981b):

̇ ( ) ( ) (8) ̇

Where f( ,T) is the strain-hardening at a constant base strain rate, ̇ , and m is the strain- rate sensitivity index and is defined in terms of derivatives. It appears as a constant in this equation. At constant strain and temperature (and any other significant variables), the standard definition of strain-rate sensitivity is as follows:

d ln ( ) ̇ d m (9) d ln( ̇) d ̇

Two of the models referenced by Sung, the power-law model (Zener and

Hollomon, 1944) and the Wagoner model (Wagoner, 1981a) respectively are:

̇ g( ̇) ( ) (10) ̇

m ̇ m √ ̇ ̇ g( ̇) ( ) (11) ̇

Where m , m0, and m1 are material constants and ̇ is a base strain rate where f( ) is measured. Equations 10 and 11 are approximate forms; they assume that only the first order variation of σ with ̇ is retained (m is assumed to be constant over the very small range of ̇ sampled). Thus, in terms of the quantity m, which may vary with strain, strain- rate, and temperature (and other significant variables, if any), the underlying laws for

Equations 10 and 11 are:

m m (Power-law model) (12)

m m m ̇ (Wagoner model) (13)

To complete a specific example of the H/V framework, Sung et al. examined several thermal softening functions, h(T), to fit the dual-phase steels examined in that work. The simplest of these was a linear model (Hutchison, 1963).

h( ) ( - (T °C-T0)) (14)

⁄

Where k and σ0 are material constants used to normalize the thermal softening function by the stress at the arbitrarily chosen room-temperature baseline, T0 = 25 °C.

The current work is focused on extending the framework developed by Sung et al. as necessary to apply to other AHSS, beyond DP steels. It is also aimed at developing and verifying a standard, efficient, effective methodology for choosing forms and obtaining optimal coefficients from a broad array of choices. In order to accomplish these goals, a

7 wide range of AHSS, with nominal ultimate tensile strengths ranging from 590 to 1180

MPa, and tensile elongations (engineering strains to fracture) ranging from 0.09 to 0.51 were tested for this purpose, constitutive models and procedures were developed, and extrapolations were compared with balanced biaxial experiments.

CHAPTER 2: EXPERIMENTAL PROCEDURES

Six typical advanced high strength steels in use and being considered for use were acquired from POSCO Steel. Standard mechanical properties given by the steel provider are shown in Table 1.TRIP refers to a transformation-induced plasticity steel, CP refers to a complex phase steel, and TWIP refers to a twinning-induced plasticity steel, part of a special experimental lot provided for this research. The numbers in the steel designations refer to the nominal σUTS (Ultimate Tensile Strength) of the material. Tensile testing was performed with a strain rate of 10-3 /s before material yielding and 0.008 /s after material yielding. The strain hardening index was calculated over the strain range from 6% to the uniform elongation for all materials except DP 980 and CP 1180. For DP 980 and CP

1180 the strain hardening index was calculated over the strain range from 4% to the uniform elongation. The r-value was calculated at an engineering strain of 4%for DP 980 and CP 1180 and at a strain of 5% for all other steels (Lee, 2011a).

Table 1: Standard mechanical properties from tensile testing done by POSCO Steel (Lee, 2011a) of six AHSS used in this research.

Material Thickness σy σUTS eu et n r (mm) (Mpa) (Mpa) (%) (%) DP 590 1.4 372 606 15.3 25.5 0.16 1.18 DP 780 1.2 527 830 13.0 20.2 0.14 0.79 DP 980 1.2 800 1030 4.8 10.2 0.06 0.79 TRIP 780 1.4 489 804 20.4 27.1 0.24 0.86 TWIP 1.4 543 977 49.2 51.0 0.34 0.67 CP 1180 1.2 939 1200 4.7 9.2 0.06 0.84 Key: σy – 0.2% offset yield strength σUTS – material ultimate tensile strength eu – uniform elongation (engineering strain at maximum tensile load) et – total elongation (engineering strain at failure) n – strain hardening index r – ratio of width to thickness strain in a tensile specimen at fixed extension

In order to obtain the necessary stress-strain data for subsequent analysis, a series of tensile tests was performed at various temperatures between room temperature and 125

°C. For each of the materials in the study, 2% tapered, ASTM E8-08 specimens oriented in the rolling direction of the material were used. This taper ensured failure at the center of the specimen for all tests at the cost of a reduction in total elongation, but with no significant effect on stress or strain measured in the uniform strain region (Sung et al.,

2010). Each specimen had a gage length of 80 mm. Testing was done using an MTS 810 tensile frame, equipped with a 100 kN capacity MTS Model 318.10 force transducer

(MTS, 2006). Strain measurements were taken using an Electronic Instrument Research

Model LE-01 laser extensometer (EIR, 2010). Data was collected at a sampling frequency of 1000 Hz. Three repeat tests were performed at room temperature, 50 °C, 75

°C, 100 °C, and 125°C. These tests were done using the equipment and setup developed by Piao and shown in Figure 1 to obtain quasi-isothermal conditions (Piao, 2011). Each 10 tensile test was performed with a crosshead velocity of 0.08 mm/s, providing a nominal strain-rate of 10-3 /s.

Pressurized Heating (Compressed Plates Air) Clamping Cylinder

Voltage Variable Auto- Transformer Electronic Temperature Control

Figure 1: Quasi-isothermal tensile test equipment and setup.

A typical set of the tensile results, one at each temperature, are shown in Figure 2.

Examination shows that the behavior of DP 780 conforms to the main principle of the

H/V framework, i.e. that the strain hardening varies with temperature (as well as the flow stress varying with temperature) (Sung et al., 2010). Figure 2(a) shows the engineering stress-strain curves and their variation with temperature including any effect on total elongation, while Figure 2(b) emphasizes the constitutive response in terms of true stress and true plastic strain. 11

900 50 °C 800

700 125 °C 100 °C 75 °C 28 °C 600 DP 780 t = 1.2 mm

500 Isothermal, RD EngineeringStress (MPa) Strain Rate = 10-3 /s 400 0 0.05 0.1 0.15 0.2 0.25 Engineering Strain (a)

1000 50 °C 28 °C 800 75 °C 100 °C 125 °C 600 DP 780

t = 1.2 mm True Stress True (MPa) Isothermal, RD Strain Rate = 10-3 /s 400 0 0.05 0.1 0.15 True Plastic Strain (b)

To obtain the strain-rate sensitivity data for each material, a series of strain-rate jump tests were completed using the same procedures as for the standard tensile tests.

These tests were conducted at six pairs of down-jump rate changes, with one jump per test at an engineering strain near ½ εu, where u is the true stress at uniform elongation.

The true jump strain, εj, for each material was as follows: for DP 590, εj = 0.075; for DP

780, εj = 0.065; for DP 980, εj = 0.025; for TRIP 780, εj = 0.1; for CP 1180, εj = 0.025; for TWIP, εj = 0.25. These jump strains were chosen to provide a range of plastic strain data within the uniform strain region both above and below the jump strain in each test to facilitate extrapolation for analysis. The tests were done using the following pairs of strain-rates: .5 /s → .1 /s, .1 /s → . /s, . 3 /s → . 3 /s, . /s → . /s, . 3 /s →

. 3 /s, and . /s → . /s. hree repeated tests were performed at the . /s → . /s and the . 3 /s → . 3 /s pairs of rates to establish the experimental scatter for the materials. An approximate strain-rate sensitivity value, m , and a logarithmic average strain-rate, ̇avg, were determined as follows, where σ1 is the stress at the higher first rate,

̇ , and σ2 is the stress at the lower second rate, ̇ (Sung et al., 2010).

ln ( ⁄ ) m (15) ln ( ̇ ⁄ ̇ )

̇avg √ ̇ ̇ (16)

Data from a typical down-jump test and the procedure used for analysis is illustrated in Figure 3.

1200 TRIP 780 t = 1.4 mm Fit - Curve 2

Fit - Curve 1

800 Test Data True Stress (MPa) True

400 0 0.05 0.1 0.15 0.2 0.25 True Strain (a)

900 . TRIP 780   2 t = 1.4 mm 1 m' = ln( / ) / ln(. /. ) 2 1 2 1 880 Fit - Curve 2 Fit - Curve 1 Data . 1 

860 2 True Stress (MPa) True

   j+0.005 j+0.01 840 j 0.09 0.1 0.11 0.12 True Strain (b)

To obtain the value of σ1, the data from = 0.02 to j was fit to the form of the constitutive equation determined at that temperature from the tensile tests then used to extrapolate: σ1 = σfit ( = εj+0.005).

To obtain the value of σ2, the process is the same except that the data range used is = εj+0.005 to u. Using these stress values and the strain-rates tested, the approximate strain-rate sensitivity was calculated for each test according to Equation 15. Plotting m vs. ̇avg in a semi-logarithmic plot shows the variation, if any, of the strain-rate sensitivity of the material with strain-rate (Sung et al., 2010). Error! Reference source not found.(a) shows a typical result for DP 780, along with the constitutive equation fit to each specified range of data within the test (identified as Fit – Curve 1 and Fit – Curve 2).

Error! Reference source not found.(b) shows the area of interest surrounding the jump strain, the extrapolations of each curve fit, and how the values of σ1 and σ2 were determined.

The balanced biaxial bulge test data used in this research was provided by

JeongYeon Lee and Myoung-Gyu Lee of the Graduate Institute of Ferrous Technology at the Pohang University of Science and Technology (POSTECH). The tests were performed using a specially modified Bulge/FLC Tester Model 161 machine

(ERICHSEN, 2012), using a “drawing speed”1 of 0.20 mm/s for DP 590 and DP 780,

0.22 mm/s for DP 980, and 0.30 mm/s for TRIP 780 and TWIP. The testing rates were chosen such that the strain rate was kept on the order of 10-3 /s while the total testing time remained reasonable. For all materials, a blank holding force of 1000 kN was used during

1Manufacturer specified “drawing speed” controlled via proportional distributing valve with pressure balance (ERICHSEN, 2012). 15 the test. The bulge diameter for these tests was 200 mm. The specimen dimensions used for this testing for all materials except for TWIP are shown in Error! Reference source not found.. The TWIP material was tested using a 300 mm x 300 mm square specimen.

Elevated temperature testing was done through the use of bulge oil heating equipment which supplied heated oil to the bulge test cylinder. For this testing, a special silicon oil

(UNISILKON TK 017/200 THERM, KLUBER, Germany) was used (Lee, 2011b).

POSTECH provided stress-strain results from three repeated tests each at temperatures of room temperature, 50 °C, and 100 °C for all steels except CP 1180.

90 mm

300 mm

Figure 4: Dimensions of specimens used in balanced biaxial bulge testing (Lee, 2011).

The effective stresses and strains were based on Hill’s non-quadratic yield function and were calculated, respectively, using the equations shown (Hill, 1979;

Wagoner, 1980).

ia ial ̅ (17) ( ( r̅))M

M ̅ ia ial( ( r)̅ ) (18)

Where r ̅ is the normal plastic anisotropy parameter and M is the power in Hill’s non- quadratic yield function.

CHAPTER 3: FITTING PROCEDURE

Three repeat isothermal tensile tests were performed for each material at each temperature. The stresses and strains from these tests were averaged to obtain a single set of “composite” stress-strain data points at each temperature and a standard deviation value representing experiment-to-experiment scatter in the following manner.

From each data set, twenty equally spaced strain values were chosen for the strain range ε = 0.02 to the uniform limit (at UTS). Around each of these chosen strain values, the five stress-strain data points both above and below the strain value were used to obtain an average stress value and an average strain value (over a strain range of approximately 0.0002). This resulted in a set of twenty stress-strain data points describing each tensile test. This stress-strain data point at each strain value was then averaged together with the stress-strain data points at the corresponding strain value from the two repeated tests performed at the same conditions. This resulted in a set of twenty stress- strain data points describing three repeated tensile tests for each material at each temperature. Experimental scatter, as determined in this process, was obtained by calculating the standard deviation of each set of stress data points at a given strain value from their average over three repeated tests, then averaging these standard deviations

18 over the full strain range and the full temperature range to determine a value for experimental scatter representing the material as a whole.

These thirty composite data sets (six materials, five temperatures each) were then used for all subsequent analysis and curve fitting. All data analysis was done using

Microsoft Excel 2010 (Microsoft, 2010), and all curve fitting was done using least- squares fitting in KaleidaGraph 4.0 (Synergy, 2005). For all steels except TWIP, the following steps describe the procedure used to generate the plastic constitutive equations reported in this work. The procedure used to generate equations for the TWIP steel will be presented separately.

First, choose a general form of the strain hardening portion of the constitutive equation, i.e. f(ε), that fits at a single temperature. For each material, the Swift constitutive equation (Equation 3) and the combined “H/V” equation (Sung et al., 2010)

(Equation 5) were examined. As will be shown later, the combined “H/V” equations fit the composite stress-strain data within the experimental scatter at each temperature and no further forms were pursued.

Once a suitable form for f( ) was determined, the next step was aimed at determining the form of f( ,T). For the five alloys that follow the H/V form, this consisted of finding the form of α(T). To do this, Hollomon (Equation 2) and Voce

(Equation 1) forms were fit to each composite tensile test data set, thus obtaining explicit values of H1 and H2 (Hollomon) and V1, V2, and V3 (Voce). Initial trial values were chosen based on Sung’s resultant best-fit coefficients for fitting the Hollomon and Voce equations separately. The values used were as follows: H1: 1250 MPa, H2: 0.15, V1: 1000

MPa, V2: 0.4, V3: 20 (Sung et al., 2010). To better ensure the uniqueness of each fit, each trial value was adjusted by a factor of ½ and by a factor of 2. In every case, the initial best-fit coefficients remained unchanged or the adjusted trial values resulted in a curve fit with a smaller correlation coefficient, R2.

These explicit values of H1, H2, V1, V2, and V3 were constrained to be constant and an H/V fit was then performed to find the optimum value of α (under the constraint of constant Hollomon and Voce coefficients). Alpha was then plotted vs. T (see, for example, Figure 5) to suggest suitable forms for α( ). Trial values were chosen for this fit based on Sung’s resultant best-fit coefficient for fitting the combined H/V equation assuming a constant value of α. Sung’s results pointed towards a trial value of α .8. o ensure that the optimum value of α was found for each case, trial values of α , .6, .4 and 0.2 were also considered. In every case, the initial best-fit α coefficient remained unchanged or the adjusted trial value resulted in an H/V curve fit with smaller R2 value.

he two functions which best described the behavior of α(T) were a linear form introduced by Sung (Sung et al., 2010) and a new sigmoidal form motivated by the concept that a pure Voce law (α ) would be obeyed at sufficiently high temperature and a pure Hollomon law (α ) would be obeyed at sufficiently low temperature. The linear form of α( ) is given by Equation 7 (Sung et al., 2010) and the sigmoidal form of

α( ) is as follows. In both cases, a reference temperature of 25 °C was used to normalize to room temperature.

α( ) (19) α3 e p(α4 ( - ))

1 DP 780 t = 1.2 mm 0.8 Data

 0.6 Best Fit Line Hollomon-like

Alpha, 0.4 (T) =  *(T-T )+ 1 0 2  = -0.0009,  = 0.7379 1 2 0.2 Voce-like

0 0 25 50 75 100 125 Adjusted Temperature, T-25 (°C) (a)

1 (T) = 1/(1+ *exp( *(T-T ))) 3 4 0 0.8  = 0.1553,  = 0.2273 3 4

Best Fit Line  0.6

Hollomon-like Data

Alpha, 0.4

0.2 DP 590 Voce-like t = 1.4 mm

0 -100 0 100 200 300 Adjusted Temperature, T-25 (°C) (b)

Figure 5: Comparison of experimentally-determined values of α at each test temperature with chosen α(T): (a) Linear form from DP 780 (b) Sigmoidal form from DP 590.

The set of best-fit α values for each temperature were fit with both the linear and the sigmoidal forms of the α(T) function. For fitting the linear α(T) function, trial values 21 were again motivated by the resultant best-fit coefficients from Sung. In this case, the values chosen were as follows: α1: 0, 0.0025, 0.005; α2: 0.1, 0.5, 0.9 (Sung et al., 2010).

For fitting the sigmoidal α(T) function, the trial values chosen were as follows: α3: 0,

0.25, 0.5; α4: 0, 0.25, 0.5. Based on these starting values, the set representing the maximum R2 value was chosen as optimal for the linear form and for the sigmoidal form.

Whichever form represented a lower error of fit between the curve fit and the α values was chosen to represent that material. Table 2 shows the five materials using the

Hollomon/Voce equations, the error of fit for each form of the α(T) function, and the choice of function for each material (designated by the value appearing in bold).

Table 2: Selection of α(T) function between the linear form and the sigmoidal form.

Material Linear Error of Fit Sigmoidal Error of Fit <α> <α> DP 590 0.0568 0.0394 DP 780 0.0252 0.0255 DP 980 0.0162 0.0128 TRIP 780 0.1139 0.0622 CP 1180 0.0359 0.0424

Error! Reference source not found.(a) shows the linear form of the α(T) function fit to the DP 780 data and Error! Reference source not found.(b) shows the sigmoidal form of the α(T) function fit to the DP 590 data as well as the best-fit coefficients for each curve. In all cases, the material exhibited more Hollomon-like behavior at lower temperatures and more Voce-like behavior at higher temperatures. The sigmoidal form showed more consistent Hollomon-like and Voce-like regions at low and high temperatures respectively with a steep transition region in between the two regions. 22

The next step was to determine the form of the thermal softening function, h(T).

For each material, the stress at a true strain of ½ εu was plotted vs. temperature. In each case a linear function with a negative slope, Equation 14 (Hutchison, 1963), represented h(T) within the estimated experimental scatter, an example of which is given in Figure 6.

Also shown are the values necessary for obtaining T1 (σ0, k).

850 DP 780 Data Best Fit Line t = 1.2 mm

/2 (MPa) /2 825 u  = 848 MPa

 0 = =  k = -0.562

800 h(T) = (1-T *(T-T )) 1 0 T = k/ = -0.00066 1 o

True Stress at True T = 25 °C 0 775 0 25 50 75 100 Adjusted Temperature, T-25 (°C)

Figure 6: Typical comparison of experimentally-measured stresses and their systematic variation with temperature via h(T) illustrating linear thermal softening for DP 780 steel.

Once the forms of all parts of the f(ε,T) and h(T) functions were established, two initial optimization methods were used to find the best-fit values for the rate-independent portion of the constitutive equation. For the first method, the best-fit coefficients from the

α(T) and h(T) functions already determined (i.e. α1, α2, and T1 or α3, α4, and T1) were assumed to be constant. The least-squares method was then used to find the best-fit

23 remaining parameters (H1, H2, V1, V2, and V3) in Equation 5 using all of the composite data. To obtain initial trial values for the remaining parameters, the values of H1, H2, V1,

V2, and V3 obtained from each individual temperature fitting of the combined

Hollomon/Voce equation were averaged to obtain ̅̅̅ ̅, ̅̅̅ ̅, ̅ , ̅ , and ̅ . These average values were used as the initial trial values for the first fitting method. The eight values obtained from this optimization method (five from curve fitting and three from the already determined best-fit coefficients from the α(T) and h(T) functions) were referred to as Optimization 1.

For the second method, the values of ̅̅̅ ̅, ̅̅̅ ̅, ̅ , ̅ , and ̅ were assumed to be constant while the least-squares fitting method was then used to find the best-fit values for the chosen forms of α(T) and h(T) (i.e. α1, α2, and T1 if the linear α(T) was chosen or

α3, α4, and T1 if the sigmoidal α( ) was chosen) in Equation 5 using all of the composite data. The initial trial values for this fitting were chosen to be the previously found coefficients for α(T) and h(T) (i.e. the coefficients assumed to be constant in

Optimization 1). The eight values obtained from this optimization method (five from the averaged H/V coefficients and three from curve fitting) were referred to as Optimization

2. In each method, the final set of parameters fit a rate-independent equation of the form shown as follows.

H ( , ) (α( ) (H ) (( -α( )) V ( -V e p(-V3 ))) h( )) (20)

These two methods were used to obtain trial values for use in an Overall

Optimization, which let all parameters vary freely in an attempt to minimize the standard error of fit between the optimized equation and the full set of composite stress-strain data. 24

The starting trial parametric values consist of the values H1, H2, V1, V2, and V3 obtained in the first optimization method and the values of α1, α2, (or α3, α4) and h(T) obtained in the second optimization method.

To test the validity and uniqueness of the set of coefficients found by this procedure, new trial parametric sets were created by multiplying each initial parameter at a time by a factor of ½ and a factor of 2 while keeping the remaining trial coefficients at the previously determined optimum values. Other values within this range were also tried, utilizing at least 16 sets of trial coefficients starting alternate optimization computations for each material. None of these alternate sets of starting values resulted in final parameters representing a lower standard error of fit than the Overall Optimization.

A final graphical check was then done for each material to show that there were no systematic deviations with the fit. This completes the fitting procedures for 5 of the 6

AHSS tested here.

As noted earlier, the H/V model did not fit the composite tensile tests within the experimental scatter for one of the tested materials: TWIP. The Swift equation, Equation

5, was used to visit TWIP behavior in the literature (Chung et al., 2011) and therefore was pursued further here. As will be shown later, it represents the composite tensile tests within the experimental scatter.

Each composite curve from the TWIP stress-strain data was fit to Equation 5, resulting in a set of Swift coefficients (i.e. S1, S2, S3) at each temperature. These coefficients were then plotted vs. temperature to obtain a curve describing each as a

25 function of temperature. Each of these curves showed a linear relationship between the coefficient and temperature, as shown in Figure 7.

2400 TWIP t = 1.4 mm Data 2300

Fit

(MPa) 1

S 2200 S =S +S (T-T ) 1 1_1 1_2 0 S =2344.4, S =-1.8837 1_1 1_2 2100 0 40 80 120 Adjusted Temperature, T-25 (°C) (a) 0.125 Data TWIP t = 1.4 mm Fit

2 0.1 S

S =S +S (T-T ) 2 2_1 2_2 0 S =0.12, S =-0.0004 2_1 2_2 0.075 0 40 80 120 Adjusted Temperature, T-25 (°C) (b) 0.7 TWIP Data t = 1.4 mm Fit

3 0.6 S

S =S +S (T-T ) 3 3_1 3_2 0 S =0.65, S =-0.0007 3_1 3_2 0.5 0 40 80 120 Adjusted Temperature, T-25 (°C) (c)

Figure 7: Linear relationship between coefficients within the Swift constitutive equation and temperature: (a) S1(T) (b) S2(T) (c) S3(T).

Using the form of these individual functions provided a temperature-dependent

Swift model, given as follows, that was used to describe the TWIP strain hardening.

( - ) ( ) ( ( - ))( ( - ) ) (21)

Where S1 = S1_1 + S1_2(T-T0), S2 = S2_1 + S2_2(T-T0), and S3 = S3_1 + S3_2(T-T0).

The trial values for the TWIP curve fitting were the coefficients obtained from the initial functions (i.e. S1_1, S1_2 from S1(T), S2_1, S2_2 from S2(T) etc.). Using these values, the best-fit values for the Overall Optimization were found by leaving all variables free to adjust during least-squares fitting to all of the composite data. To test the validity and uniqueness of the set of coefficients found by this procedure, new trial parametric sets were created by multiplying each initial parameter at a time by a factor of ½ and a factor of 2 while keeping the remaining trial coefficients at the previously determined optimum values. Using this method, 12 sets of trial coefficients were used to start alternate optimization computations. None of these alternate sets of starting values resulted in final parameters representing a lower standard error of fit than the Overall Optimization. A final graphical check ensured that there were no systematic deviations with the fit.

CHAPTER 4: RESULTS AND DISCUSSION

The results of the form selection, curve fitting, and optimization are shown in

Table 3.

Table 3: Standard error of fit of plastic constitutive equations determined by H/V methodology for six advanced high strength steels.

Optimization Optimization Alt. Overall Experiment 1 2 Sets Optimization % % % <σ> % <σ> <σ> <σ> <σ> Material of of of Range of (MPa) (MPa) (MPa) (MPa) UTS UTS UTS (MPa) UTS DP 590 6 1.1 30 4.7 16 2.6 7-33 7 1.1 DP 780 6 0.7 19 2.3 16 2 6-38 6 0.7 DP 980 12 1.2 14 1.4 9 0.9 5-19 5 0.5 TR 780 3 0.3 39 4.9 49 6.1 17-41 17 2.2 TWIP 17 1.8 8-44 8 0.8 CP 1180 6 0.5 9 0.7 11 0.9 4-15 4 0.4

Table 3 summarizes the quality of the fits following the procedures introduced in the previous section. In general, the experimental scatter among repeated tests at a single temperature represents 0.7 to 1.0% of the flow stress, approximately the same standard error of fit for the full (non-rate dependent terms) of the final constitutive model

(“Overall Optimization”): .5-2.2%. The only material for which the final constitutive model exhibited significantly more deviation was TRIP 780. As will be shown later, this 29 appears to be related to the extreme temperature sensitivity of this alloy as compared with the others (range of flow stress is approximately 200 MPa for TRIP 780 versus 20-50

MPa for the other alloys). Thus, comparing experimental scatter at a single temperature is not a good comparison.

Table 3 also shows that Optimizations 1 and 2 do not achieve nearly the overall minimum standard error of fit (as is expected by the constraints imposed). Nonetheless, using these intermediate procedures to obtain trial values for the Overall Optimization allowed determination of the best-fit parametric values in a single optimization execution, as is demonstrated by comparing the standard errors of fit for the optimizations starting from the alternate starting sets. In general, the curve fitting done using the alternate trial values produced one of two results. The first result was a set of parameters that was very similar to the Overall Optimization parameters, resulting in a standard error of fit that was often 5-10 MPa greater than that of the Overall Optimization. The second result was a set of parameters that introduced a systematic deviation from the experimental data, typically resulting in a standard error of fit that was 3-5 times greater than that of the

Overall Optimization. CP 1180 showed the least variation in standard error of fit over the entire range of alternate trial values (4 MPa – 15 MPa), while TWIP showed the most variation (8 MPa – 44 MPa). In all cases, the minimum standard error of fit in this range is represented by the results of the Overall Optimization.

The final parameters used to define the overall optimization of the constitutive equation for each material are given in Table 4.

Table 4: Overall optimization parameters for the rate-insensitive portion of the plastic constitutive equation for all six materials used in the study.

Para- DP DP TRIP CP meter Eq. Unit 590 DP 780 980 780 1180 TWIP H1 2 MPa 1001 1200 1440 1446 2024 H2 2 0.1753 0.1965 0.0831 0.2200 0.0624 V1 1 MPa 589.0 1377 1109 810.6 1361 V2 1 0.2230 0.2831 0.3435 0.4126 0.2463 V3 1 0.0910 13.67 76.90 4.907 54.14 α1 7 -0.0005 0.0956 α2 7 K -1 0.7190 0.0013 α3 19 0.0273 0.4231 0.0401 α4 19 K -1 0.0134 0.0180 -0.0156 -1 T1 14 K 0.0004 0.0008 0.0002 0.0016 0.0004

S1_1 21 MPa 2353

S1_2 21 MPa/K -2.072

S2_1 21 0.1195 -1 S2_2 21 K -0.0004

S3_1 21 0.6627 -1 S3_2 21 K -0.0008 Key: Eq: The equation number in the current work where the parameter is introduced Unit: The proper units for each parameter when applicable

The plastic constitutive equations found using the proposed methodology and the experimental composite data is shown in Figure 8. Also included is the standard error of fit, denoted by <σ>, between the constitutive equation and the data for each material.

750 DP 590 20 °C 50 °C t = 1.4 mm, RD 75 °C 675 Strain Rate = 10-3 /s 100 °C 600 125 °C

<  > = 7 MPa (1%)

True Stress True (MPa) 525

450 0 0.05 0.1 0.15 0.2 True Strain (a)

1000 DP 780 28 °C 50 °C t = 1.2 mm 900 RD, Strain Rate = 10-3 /s 75 °C 100 °C 800 125 °C

<  > = 6 MPa (1%)

True Stress True (MPa) 700

600 0 0.05 0.1 0.15 True Strain (b)

Figure 8: Comparison of the overall constitutive law determined using the proposed methodology with the experimental tensile test data at all test temperatures: (a) DP 590 (b) DP 780 (c) DP980 (d) TRIP 780 (e) CP 1180 (f) TWIP.

continued

Figure 8 continued

1150 25 °C DP 980 50 °C t = 1.2 mm RD, Strain Rate = 10-3 /s 1100 75 °C

100 °C 1050 125 °C

True Stress (MPa) True <  > = 5 MPa (1%)

1000 0 0.02 0.04 0.06 True Strain (c)

1100 50 °C 75 °C <  > = 17MPa (2%) 1000 25 °C 900 100 °C 800 125 °C 700 Scatter < Marker TRIP 780 True Stress True (MPa) Size t = 1.4 mm 600 RD, Strain Rate = 10-3 /s 500 0 0.1 0.2 0.3 True Strain (d)

continued

Figure 8 continued

1400 CP 1180 50 °C t = 1.2 mm 25 °C RD, Strain Rate = 10-3 /s 1350 75 °C <  > = 4 MPa (1%) 100 °C 125 °C

1300 True Stress (MPa) True

1250 0 0.01 0.02 0.03 0.04 0.05 True Strain (e)

1500 TWIP 50 °C t = 1.4 mm -3 1250 RD, Strain Rate = 10 /s 75 °C

27 °C 100 °C 1000 125 °C

<  > = 8 MPa (1%)

True Stress (MPa) True 750

500 0 0.1 0.2 0.3 0.4 0.5 True Strain (f)

Following the procedure put forth in the Experimental Procedures chapter, the approximate strain-rate sensitivity, m , was calculated at six different pairs of jump-down 34 rates and three different temperatures (25 °C, 50 °C, and 100 °C) for all six materials.

The strain-rate sensitivity data at all three temperatures along with the parameters for the function representing strain-rate sensitivity as a function of rate and temperature appears for all six materials in Figure 9. Strain-rate sensitivity at all three temperatures was plotted on a semi-logarithmic scale vs. the logarithmic average strain-rate to show the variation of the strain-rate sensitivity of the material with strain-rate and with temperature. Experimental scatter is shown, representing three repeated tests where it is available.

0.015 DP 590 t = 1.4 mm 0.01 25 °C

0.005 50 °C 100 °C

0 StrainRate Sensitivity, m

-0.005 0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (a)

Figure 9: Comparison of m values (strain-rate sensitivities) from strain-rate jump tests at three temperatures with their constitutive representation by the rate law: (a) DP 590 (b) DP 780 (c) DP 980 (d) TRIP 780 (e) CP 1180 (f) TWIP.

Figure 9 continued

0.01 DP 780 0.008 t = 1.2 mm 25 °C

0.006

0.004 50 °C 0.002 100 °C

0 Strain Rate Sensitivity, Rate Strain m

-0.002 0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (b)

0.004 DP 980 25 °C t = 1.2 mm

0.002

50 °C 0

-0.002 100 °C StrainRate Sensitivity, m -0.004

0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (c)

continued 36

Figure 9 continued

0.025 TRIP 780 0.02 t = 1.4 mm 50 °C 0.015 25 °C 0.01

0.005

0 Strain Rate Sensitivity, Rate Strain m 100 °C -0.005 0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (d)

0.004 CP 1180 t = 1.2 mm 25 °C 0.002

0 50 °C -0.002

100 °C StrainRate Sensitivity, m

-0.004 0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (e)

continued 37

Figure 9 continued

0.005 TWIP t = 1.4 mm 25 °C

50 °C -0.005

100 °C Strain Rate Sensitivity, Rate Strain m

-0.01 0.0001 0.001 0.01 0.1 1 Logarithmic Average Strain Rate (f)

All six materials followed one of two functional forms for rate sensitivity as a function of rate and temperature: the “linear” law or the “Wagoner” law, given as follows (Sung et al., 2010; Wagoner, 1981a).

̇ m m log ( ⁄ ) m m3 (22) ̇

̇ m6 m m (m4 m5) ( ⁄ ) -(m m ) (23) ̇ 8

The strain-rate sensitivity data for each material was fit to both equations, and the form which produced a smaller error of fit was chosen to represent the strain-rate sensitivity of the material. All steels except TRIP 780 followed the linear law with a smaller error of fit while the TRIP 780 steel followed the Wagoner law with a smaller

38 error of fit. Table 5 shows the parameters used to define the strain-rate sensitivity of each material.

Table 5: Parameters for strain-rate sensitivity as a function of temperature and strain-rate for all six materials used in the study.

Param DP DP DP TRIP CP -eter Law Unit 590 780 980 780 1180 TWIP

m1 Lin 0.0028 0.0022 0.0007 0.0011 0.0020 -1 m2 Lin K -4E-5 -2E-5 -5E-5 -2E-5 -7E-5

m3 Lin 0.0132 0.0091 0.0052 0.0031 0.0070 -1 m4 Wag K -2E-4

m5 Wag 0.0249 -1 m6 Wag K -5E-3

m7 Wag 0.6912 -1 m8 Wag K 4E-5

m9 Wag -1E-3 - 1 ̇ Rate s 0.001 0.001 0.001 0.001 0.001 0.001

Once the form and coefficients for the strain-rate sensitivity were determined for each material, the g( ̇, ) function was added to the composite constitutive equation, for example:

( ̇) ( ( )( ) (( - ( )) ( - (- ))) ( )) ( ̇, ) (24)

The forms of this function, g( ̇, ), are given as follows. Equation 25 is used when the strain-rate sensitivity of the material is represented by the function shown in

Equation 22. Equation 26 is used when the strain-rate sensitivity of the material is

represented by the function shown in Equation 23. The new coefficient, m , is defined by

Equation 27 and serves to change the base of the logarithm in Equation 22 and Equation

25.

m m ̇ ̇ 3 g( ̇, ) e p (m ln ( ) ln (( ) )) (Linear) (25) ̇ ̇

m m ̇ m6 m ̇ m8 m g( ̇, ) e p ( 4 5 ( ) - ln (( ) )) (Wagoner) (26) m6 m ̇ ̇

m m (27) .86858

To confirm the accuracy of the constitutive equations extrapolated at high strains as encountered in forming applications or in draw-bend testing, they were compared with the results of new and novel, elevated-temperature, balanced biaxial bulge tests conducted at POSTECH. In order to compare the tensile test results with those from a bulge test, an effective stress and strain were calculated from the bulge test results. The effective stress and strains were based on Hill’s non-quadratic yield function (Hill, 1979).

The explicit equations for calculating this effective stress and strain are given as Equation

17 and Equation 18 respectively (Wagoner, 1980).

For all of the analysis done in this study, r ̅ was taken to be . . Hill’s M value was fit for each material and temperature such that ( ̅, ̅) ( ̅, ̅) ensile at the tensile UTS. This equality allowed for comparison of the predicted high-strain behavior of the material from the constitutive plastic equation with the experimental balanced biaxial bulge test experimental results. A typical result (DP 980 steel at 100 °C) as well as an atypical result (TWIP steel at 50 °C) are shown in Figure 10. The experimental balanced biaxial bulge test results show high scatter at low strains, but at high strain values the scatter was typically between 3-6 MPa. For the DP 980, the standard error of fit between the

40 constitutive equation extrapolated to high strains and the experimental results was 2 MPa

(0.2% of UTS), an insignificant difference when compared with experimental scatter in tensile testing or balanced biaxial bulge testing. The constitutive equation developed using the H/V methodology provides a very accurate prediction of the stress-strain behavior at strains up to six times larger than the tested range. For the TWIP steel, the standard error of fit between the constitutive equation extrapolated to high strains and the experimental results was 4.2% of the UTS.

Table 6 shows a summary of all of the comparisons between the constitutive equation e trapolations and the balanced bia ial bulge test results. he Hill’s M values varied from 2.05 in the DP 590 steel to 2.5 in the TWIP steel, but show small variation with temperature. The reported errors of fit were calculated over an effective strain range from 0.15 to the maximum recorded strain in the balanced biaxial bulge test. The TRIP

780 steel showed the closest overall agreement between the constitutive equation and the bulge test results (0.6% of UTS to 0.7% of UTS), while the TWIP steel showed the largest difference between the two results (3.7% of UTS to 5.7% of UTS).

1400 DP 980 <  > = 2 MPa t = 1.2 mm H/V-Biaxial T = 100 °C, M = 2.1 1200 H/V Biaxial Data Extrapolation

1000 Balanced Biaxial Bulge Test Data Effective True Stress True (MPa) Effective Tensile Data H/V Extrapolation 800 0 0.2 0.4 0.6 Effective True Strain (a)

2500 TWIP H/V Extrapolation t = 1.4 mm T = 50 °C, M = 2.5 2000 <  > = 67 MPa H/V-Biaxial 1500 Biaxial Data (Scatter < Marker Size)

Biaxial Bulge Test Data

1000 EffectiveTrue Stress (MPa) Tensile Data H/V Extrapolation 500 0 0.2 0.4 0.6 0.8 1 Effective True Strain (b)

Figure 10: Comparison of constitutive equation extrapolation from tensile testing with balanced-biaxial test data: (a) DP 980, 100 °C (b) TWIP, 50 °C.

Table 6: Results of comparison between the extrapolation of the plastic constitutive equation and the experimental balanced biaxial bulge test results for each material.

Material Temp. Exp. Standard <σ>0.15-max % of M (Hill) Deviation UTS (°C) (MPa) (MPa) RT 3.9 16 1.9 2.05 DP 590 50 3.2 11 1.4 2.10 100 2.1 13 1.6 2.08 RT 1.9 6.0 0.50 2.09 DP 780 50 3.2 9.0 0.80 2.09 100 2.0 12 1.1 2.12 RT 6.8 11 0.80 2.09 DP 980 50 8.0 28 2.1 2.18 100 7.5 2.0 0.20 2.10 RT 2.0 7.0 0.60 2.12 TRIP 780 50 3.2 6.0 0.60 2.19 100 7.0 7.0 0.70 2.19 RT 7.0 95 5.7 2.50 TWIP 50 2.4 67 4.2 2.50 100 4.2 60 3.7 2.50

CHAPTER 5: CONCLUSIONS

The framework proposed by Sung (Sung et al., 2010) has been extended to include new forms of the α(T) and g( ̇) functions, and a function of the form g( ̇, T) has been introduced. It has been extended to incorporate a range of AHSS beyond DP steels.

Based on the experiments, fitting, and analysis in this work the following conclusions were reached:

1. The proposed methodology for fitting constitutive equations provides unique and

optimal coefficients without requiring a large number of optimization runs or trial

value sets. That is, the methodology is efficient.

2. The constitutive forms and corresponding optimized coefficients fit the

experimental data within the scatter of the data for the DP, TWIP, and CP steels

in this study. That is, the methodology is effective.

3. The constitutive equation as determined for the DP and TRIP steels can be

extrapolated to strains several times larger than the measured domain, and they

agree with experiments within the experimental scatter.

a. For CP 1180 steels this agreement was not tested, as the specimens broke

when clamped in the balanced biaxial bulge test machine.

b. For TWIP steel, uniaxial and biaxial results are not equivalent as the

proportions of the different mechanisms involved in deformation, slip and

twinning, are dependent on the strain state (Lee et al., 2009).

4. Introduction of a new strain, strain rate, and temperature dependent constitutive

model for TWIP steel. For the measured domain, the model resulted in a standard

error of fit of 8 MPa (1% of material UTS), which showed an agreement with

experiments within the experimental scatter.

5. The H/V constitutive equation form (Sung et al., 2010) was extended through the

introduction of a new sigmoidal form of α(T) (Equation 19).

6. Strain-rate sensitivity was measured and presented as a function of temperature

and strain-rate for AHSS. New forms of g( ̇,T) were introduced to describe strain-

rate sensitivity.

a. Strain-rate sensitivity of DP, CP, and TWIP steels decrease linearly with

temperature over the tested range while increasing with strain-rate.

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