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Nisr United States Department of CommerceTechnology AdministrationNational Institute of Standards and Technology

NIST Technical Note 1500-6Materials Reliability Series

Constitutive Behavior Modeling of Steels

Under Hot-Rolling Conditions

Y.W. ChengR.L. Tobler

B.J. Filla

K.J. Coakley

QC100.U575310.1500-999

NIST Technical Note 1500-6Materials Reliability Series

Constitutive Behavior IVIodeling of Steels

Under Hot-Rolling Conditions

Y.W. ChengR.L. Tobler

B.J. Filla

K.J. Coakley

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

325 BroadwayBoulder, Colorado 80303-3328

April 1999

''4TES O*

U.S. DEPARTMENT OF COMMERCE, William M. Daley, SecretaryTECHNOLOGY ADMINISTRATION, Gary R. Bachula, Acting Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Raymond G. Kammer, Director

National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1500-6, 112 pages (April 1999)CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICEWASHINGTON: 1999

For sale by the Superintendent of Documents, U.S. Government Printing Office, Wasfiington, DC 20402-9325

Contents

Foreword v

1. Introduction 2

2. Background 2

3. Constitutive Equations and Factors Affecting Flow Stress 5

3.1 Temperature and Strain Rate 5

3.1.1 Apparent Activation Energy Q 6

3.1.2 Constants A and a 6

3.1.3 Value for /w 6

3.2 Dynamic Recrystall ization 7

3.3 Chemical Composition : 7

3.4 Austenite Grain Size 7

3.5 Precipitation 8

3.6 Adiabatic Heating 8

4. Experimental Approach 9

4.1 Materials 10

4.2 Test Specimen 10

4.3 Apparatus 10

4.3.1 Computer and Peripherals for Machine Control and Data Acquisition .. 11

4.3.2 Load Frames 11

4.3.3 Heating, Cooling, and Vacuum 12

4.3.4 Machine Performance and Reproducibility 13

4.3.4.1 Alignment 13

4.3.4.2 Reproducibility 13

4.3.4.3 Strain Rates 13

4.3.4.4 Specimen Strength 14

4.3.4.5 Adiabatic Heating 14

5. Results and Discussion 14

5.1 Raw Data and Data Analysis 14

5.2 Round-Robin Test Program with UBC 15

5.3 Machine Compliance 17

5.4 Mathematical Description of Stress-Strain Curves 18

5.5 Determination ofModel Parameters 20

5.5.1 Parameter Estimation 21

5.5.2 Uncertainty Analysis 22

5.6 Experimental Stress-Strain Curves and Model Predictions 23

5.6.1 A36 23

5.6.2 DQSK and HSLA-V 23

5.6.3 HSLA-Nb, HSLA-50/Ti-Nb, and IF Steels 24

5.6.4 HSLA-80/Ti-Nb 25

5.7 Strain at Peak Stress 25

in

Contents (continued)

5.8 Boundary Separating Dynamic Recrystallization and

No-Dynamic Recrystallization 26

5.9 Effects of Prior Heat Treatments: HSLA-Nb 26

5.10 Effects of Reheating Temperature and Model Adjustments: HSLA-80/Ti-Nb . 27

5.11 Effects of Retained Strain: HSLA-Nb 28

5.12 Comparison with Data from CANMET 28

5.13 Calculation of Roll-Separating Forces 29

6. Summary and Conclusions 31

7. References 33

Figures 37

IV

Foreword

The Materials Reliability Series ofNIST Technical Notes are reports covering significant research

accomplishments of the Materials Reliability Division. The Division develops measurement

technologies that enable the producers and users of materials to improve the quality and reliability of

their products. Measurement technologies are developed for process control to improve the quality and

consistency of materials, for nondestructive evaluation to assure quality of finished materials and

products, and for materials evaluation to assure reliable performance. Within these broad areas of

measurement technology, the Division has focused its resources on three research themes:

• Intelligent Processing of Materials—To develop on-line sensors for measuring the materials'

characteristics and/or processing conditions needed for real-time process control.

• Ultrasonic Characterization of Materials—To develop ultrasonic measurements for

characterizing internal geometries of materials, such as defects, microstructures, and lattice

distortions.

• Micrometer-Scale Measurements for Materials Evaluation—To develop measurement techniques

for evaluating the mechanical, thermal, and magnetic behavior of thin films and coatings at the

appropriate size scale.

This report is the sixth in the series. It covers research on constitutive behavior of steels under hot-rolling

conditions, one of the projects on Intelligent Processing of Materials. Previous reports in this series are:

Technical Note 1500-1 Tensile Testing of Thin Films: Techniques and Results,

byD.T. Read, 1997

Technical Note 1500-2 Procedures for the Electron-Beam Moire Technique,

by E.S. Drexler, 1998

Technical Note 1500-3 High-Energy, Transmission X-ray Diffraction for Monitoring

Turbine-Blade Solidification, by D.W. Fitting, W.P. Dube,

and T.A. Siewert, 1998

Technical Note 1500-4 Nondestructive Characterization of Reactor Pressure Vessel Steels:

A Feasibility Study, by H.I. McHenry and G.A. Alers, 1 998

Technical Note 1500-5 Electron-Beam Moire Technique: Advances, Verification, Application,

by E.L. Drexler, 1998

CONSTITUTIVE BEHAVIOR MODELING OF STEELS

UNDER HOT-ROLLING CONDITIONS

Y. W. Cheng, R. L. Tobler, and B. J. Filla

Materials Reliability Division

National Institute of Standards and Technology

Boulder, CO 80303

K. J. Coakley

Statistical Engineering Division

National Institute of Standards and Technology

Boulder, CO 80303

The purpose of this work is to develop constitutive equations that predict the

stress-strain curves of steels as functions of hot-rolling process variables, such as

temperature, strain rate, and microstructural features. Eight steels, A36, DQSK,HSLA-V, HSLA-Nb, HSLA-50/Ti-Nb, HSLA-80/Ti-Nb and two interstitial-free

(IF) grades, were selectively tested at temperatures in the range from 900 to 1200

°C, strain rates from 1 to 50 s'\ and austenite grain sizes from 0.012 to 0.29 mm.The data were analyzed to support the development of constitutive models.

To cover a wider range of strain rates, a new computer-controlled servo-

hydraulic laboratory apparatus was designed and built for high strain-rate

experiments. The machine has a load capacity of 100 kN and controllable

constant strain rate of 50 s\ It has been used to generate all the data with the

strain rate of 50 s'' in this project. Data for lower strain rates were generated with

existing equipment.

Models for predicting stress-strain behaviors for the eight steel grades listed

above have been developed. Model predictions correlate well with the

experimental data, indicating that the models adequately describe the stress-strain

behaviors of various steel grades under hot-rolling conditions. Model predictions

for A36 and DQSK are further validated with high strain-rate (up to 150 s"') data

obtained at CANMET.

The models have been used in the Sims equations to calculate the roll-

separating forces which agree well with mill measurements. The models are

currently used in the Hot-Strip Mill Model developed at UBC, Canada.

Key words: Constitutive behavior; dynamic recrystallization; elevated-

temperature mechanical properties of steels; hot-strip rolling; IF

steels; microalloyed steels; plain-carbon steels; thermomechanical

process simulation.

1. Introduction

In 1993, the American Iron and Steel Institute (AISI) sponsored a research project on

microstructural engineering in hot-strip mills. The objective was to develop a predictive tool that

quantitatively links the properties of hot-rolled products to mill processing parameters. The

predictive tool will be a user-friendly computer model that incorporates knowledge of heat flow,

microstructural phenomena (recrystallization, grain growth, precipitation, and austenite

decomposition), and structure-composition-property relationships. It will compute the thermal

and microstructural evolution of steel during hot rolling, as well as the final product properties,

as a function of hot-strip mill design and operating practices.

The AISI project is a cooperative effort involving the National Institute of Standards and

Technology (NIST), the University of British Columbia (UBC-Canada), and U.S. Steel. Major

portions of the research have been performed at NIST and UBC. The University of British

Columbia conducts heat-transfer studies, quantitative characterization of kinetics of

microstructural evolution, and model verification with plant and pilot-plant measurements.

NIST focuses on constitutive behavior and structure-composition-property relationships. The

goal of the constitutive-behavior studies is to model the flow behavior of steels as functions of

chemistry, temperature, strain rate, and other processing variables. This report presents the

results of the constitutive-behavior studies, including experiments to provide stress-strain data

and to understand the effects of different parameters and models to predict stress-strain curves

under hot-rolling conditions.

2. Background

Characterizing the plastic flow of steels is essential to improve industrial hot-forming

operations. During forging or hot rolling, the plastic-flow properties determine the power

requirements of industrial equipment, as well as the mechanical properties and dimensional

accuracy of finished products.

Forming at high temperatures and high strain rates involves a sizable number of

variables. Material response is affected by changes in temperature and strain rate, as well as

steel chemistry and microstructural factors including grain size, grain growth, recrystallization,

and precipitation during processing. The demand for accurate flow-property predictions under

hot-forming conditions has increased in recent years due to research pursuits in the field of

deformation simulation and to commercial needs to manufacture products with ever increasing

efficiency.

Figure 1 presents the block diagrams (rectangular blocks) of a typical hot-strip rolling

operation and the various computer models for different stages (oval-shape blocks) of the

operation. Steel experiences the following conditions during its conversion from slab to hot

strip;

• Temperature may reach 1250 °C at the first stand of the roughing mill, and 900 °C or

lower at the last stand of the finishing mill.

• Strain rate may reach 1 s'^ at the first stand of the roughing mill, and 100-200 s'' or

higher at the last stand of the finishing mill.

• Grain size may exceed 0.25 mm at the first stand of the roughing mill, but may be as

small as 0.015 mm at the last stand of the finishing mill.

Obviously, the material's flow stress varies considerably during different stages of

processing. The research objective is to develop and validate constitutive equations for use in

strip-rolling simulations using numerical techniques, such as the finite element method.

Constitutive equations describe the nonlinear relationships that exist among the process variables

and the internal structures of materials, and they permit us to predict or estimate the steel's flow

properties for specific strip-rolling conditions.

Figure 2 shows some experimentally determined flow-stress curves for DQSK steel. The

relationship between true stress and true strain is called the flow-stress curve because it plots the

stress required for plastic flow at a given strain. The shape of the flow-stress curve relates to

different structural changes going on in the steel as it undergoes hot deformation in the austenite

phase. The three shapes of these curves indicate three possible responses to hot deformation at

high temperatures and high strain rates. The Type I flow-stress curve (at 900 °C) rises

continuously, owing to strain hardening in the austenite. The Type II curve (at 1000 °C) rises at

first, and then reaches a plateau. This plateau is caused by the occurrence of dynamic recovery

in amounts sufficient to counterbalance the effect of strain hardening in austenite. Finally, the

Type III curve (at 1 100 °C) rises to a peak and then subsequently decreases. The decline of the

flow stress in a Type III curve is caused by dynamic recrystallization, which begins when the

critical strain ec is reached.

An important factor that determines which type of the flow-stress curve will manifest is

the Zener-Hollomon parameter [1]:

Z- e expQ_RT (1)

Here, 8 is strain rate, T'\s temperature, R is the universal gas constant, and Q is the apparent

activation energy of the rate-controlling dynamic softening process. High Z promotes strain

hardening, whereas low Z favors dynamic recrystallization, which fosters the decline of flow

stresses following a peak stress. Deformation at intermediate Z corresponds to a balance

between strain hardening and dynamic recovery, which is characterized by a constant flow stress

after initial hardening. If strain rate is held constant in eq (1), then Z increases with decreasing

temperature and the flow stress at low temperature increases. Conversely, the flow stress will

decrease at low strain rates.

The flow-stress curve is fundamental to materials engineering. From it, the important

properties of a steel are interpolated, and the mechanisms of deformation can be determined.

Many attempts have been made to fit mathematical expressions to empirical flow curves at

selected temperatures and strain rates. Exponential equations or power laws and their derivatives

Table 1 . Stress-strain relationships cited in the literature.

Equations References

(2) 2

(3) 3

(4) 4

(5) 5

(6) 6

(7) 7

(8) 7

(9) 8

(10) 7,9

(11) 10

a = Ks"

(T = C7o + Ks"

a - K[£q + £)

a - K£"' + exp(/r + n^£)

0" = cro + (5+ £)"

O = 0ro + 51og(f)

o- = o-,[l-exp(-5f)]"

cr = cTq + ^1 - exp(-C£')]"

cr =(7o

- (/^ - 5) • exp(-C£')

o^B l-exp(-C^"")]" (12)

o- = 1.64-o-^^''

Gp = peak stress; n = 0.22 + 0.001 -ap(13) 11

are commonly used. The parametric constants that relate true stress a, and true strain £, in these

equations must be determined experimentally.

Table 1 lists the various equations that are cited in the literature [2-11] and have been

proposed for use in modeling. These equations were applied to describe Type I or Type II

behaviors. Most equations in table 1 were developed to fit stress-strain relationships at relatively

low temperatures; they may fit the experimental data well, but only for certain materials, and for

limited experimental ranges. For steels at elevated temperatures, eq (4), which is a modified

form of Ludwik's power-law equation of eq (3), has been shown to fit some experimental data

rather well [4]. At NIST, we successfully used eq (9) to describe the stress-strain behavior of bar

steels [8]. The parameters B and n are nearly constant for medium-C microalloyed steels when

as is taken as the flow stress in the steady state. Regarding Type III flow curves, no single

governing equation has been found. Different approaches [9,12-14] have been used to account

for the amount and kinetics of softening due to dynamic recrystallization.

3. Constitutive Equations and Factors Affecting Flow Stress

Constitutive equations describe nonlinear relationships that exist among the process

variables such as effective stress, effective strain rate, and temperature at different strains. Such

equations are unique for each material and processing condition. Usually, for practical

applications they are developed using data obtained under simplified experimental conditions.

The constitutive equations are usually described using a rate expression

e = f{<jJ,S), (14)

where e is strain rate, cr is stress, Tis temperature, and S is the structural state of the material. In

general, S is affected by deformation (strain and strain rate), and temperature.

One of the goals in constitutive-behavior modeling is to predict the flow stress of a given

steel under various processing conditions. For this purpose, rate expressions, such as eq (14), are

rearranged to express the flow stress explicitly as a function of strain, temperature, and strain

rate.

3.1 Temperature and Strain Rate

The rate at which strain is applied can have an important influence on the flow stress. In

general, increasing strain rate increases flow stress. Also, the shape of the stress-strain curve

depends strongly on temperature. Decreasing temperature, in general, increases flow stress.

Typically, the temperature and strain-rate dependence of the rate expression is described by the

hyperbolic sine equation [15]

s - yl[sinh(«o-)]'" expQRT (15)

where .4, a, and m are experimentally determined constants. At low stresses (Qro-<0.8), the

equation is reduced to a power relation such as the equation used to describe creep behavior,

£ = A^g"^ expQRT (16)

and at high stresses {a<j>\.2), it reduces to an exponential relation.

£ = A2 expyfimjcr) expRT (17)

Equation (15) was derived based on results of steady-state creep experiments. It has been

observed that the parameters in eq (15) decrease with strain, especially at small strains, where

steady state has not been reached. After the steady-state stress is reached, these parameters

approach a constant with little variation. It has been shown that eq (15) describes the hot-

deformation behavior of steels better than eqs (16) and (17) [16]. Rearranging and expressing

the steady-state stress as a function of Z, we find from eq (1 5) that

a- ^ 08)

Low Z promotes dynamic recrystallization (Type III behavior in figure 2). As Zincreases, the peak in the curve broadens and the strain that corresponds to the peak stress also

increases. With further increase of Z, Type III behavior approaches Type II behavior and the

peak disappears.

Many hot-deformation studies are devoted to determining the constants A, a, m, and Q in

eq (15). Once these constants are determined for a given steel, eq (15) can be used to interpolate

or extrapolate to conditions for which data are unavailable.

3.1.1 Apparent Activation Energy Q

For C-Mn steels the values ofO fall in the range 262 to 366 kJmol"' [1,4, 17-20]. The

values ofO tend to increase with additions of certain alloying elements including Nb. Values

reported for some Nb microalloyed steels fall in the range 325 to 461 kJ-mof' [12,20,21]. The

apparent activation energy is frequently related to the activation energy for self-diffusion of iron

in austenite, and the reported values are similar. However, some materials show distinct

differences between activation energies determined under hot-working conditions and those

measured during creep [22]. Such materials include steels [22]. In such cases, one should

beware not to extrapolate to conditions beyond the range of actual experimental observations

since Q might vary with strain rate. Abnormal behavior of (7 has been reported for Nb steels [4].

3.1.2 Constants /i and a

The unit for the constant A is s"', and values are usually expressed in \xhA. Some reported

values for In^^ are in the range 25 to 32 [4,12]. The values of a are usually fixed at 0.012 [12] or

0.013 MPa' [4].

3.1.3 Value for m

The values of /w reported in the literature range from 3.2 to 5.0 [4,12,20,22]. This

dimensionless parameter relates to the strain-rate sensitivity of the material, and this parameter is

a measure of the dependence of flow stress on strain rate. The higher the strain rate, the higher

the flow stress. It is widely reported that the strain-rate sensitivity increases with increasing

temperature [23,24]. However, for hot-worked steels, the value of /w remains constant after the

steady-state condition is reached [16]. Note that m, as expressed in eq (15), decreases with an

increase of the strain-rate sensitivity.

3.2 Dynamic Recrystallization

The occurrence of dynamic recrystallization may profoundly affect the flow stress, as

shown in figure 2. Although dynamic recrystallization and its effects have been widely studied,

it has been argued that prospects for the occurrence of dynamic recrystallization during rolling in

a single pass are remote since the critical strain Sc for dynamic recrystallization is seldom

achieved [20,25]. However, the attainment of this critical strain is more probable during multi-

pass rolling in the finishing stands. The interpass time in that case is short, and the rolling

temperature is low; therefore grains deformed in a previous pass do not have sufficient time to

statically recrystallize after deformation, and the strain is retained to the next pass. Thus, strain

increments accumulated in the process of several passes can reach e^ and induce dynamic

recrystallization. Although the occurrence of dynamic recrystallization in an industrial rolling

process seems likely, there has been no report in the literature to substantiate the hypothesis.

A model that considers the occurrence of dynamic recrystallization on stress-strain

behavior has been reported [13,14,24]. This model derives from the ftindamental theories of

hardening and softening (recovery and recrystallization), and it applies to a wide range of

materials and test conditions. As a consequence of its broad applicability, this model contains a

large number of constants that must be determined experimentally. This necessitates some creep

test experiments.

A simpler model that involves four constants has also been proposed for hot-worked

steels [12]. Two of these four constants are material properties: the steady-state flow stress after

dynamic recrystallization and Sc, which is a ftinction of the initial grain size and Z. The steady-

state flow stress after dynamic recrystallization Orexd for a Nb-B steel is expressed as

^rexd = 84.34 sinh-1

5.2x10"

0.2

(19)

3.3 Chemical Composition

In general, the effects of chemical composition on flow properties are embedded in the

constants of the stress-strain equations. For example, additions ofNb increase Q in eq (15), as

noted previously. Few attempts have been made to express the flow stress in terms of

composition [26]. But systematic research quantifying the effects of individual alloying

elements is scarce.

3.4 Austenite Grain Size

Grain size is an important parameter governing the low-temperature mechanical

properties of steels, such as strength and toughness. However, the existing literature for steels

contains little information describing the effects of grain size on flow stress under hot-working

conditions, aside from its effects on the occurrence of dynamic recrystallization.

It is stated that equilibrium subgrain structures are established early during deformation at

high temperatures [22,27]. These subgrain structures determine the flow stress under hot-

working conditions, except at the very early stage of deformation before the subgrain structures

are established. Thus, the initial austenite grain size plays little role in determining the flow

stress of steels. Nevertheless, the austenite grain size does change from one pass to another

during rolling. And the initial grain size influences the value of £c, which increases with an

increase of grain size [1]. Therefore, more research is needed to arrive at conclusive results

about the effects of grain size on the flow behavior of steels.

3.5 Precipitation

Precipitation is another factor that has potent effects on low-temperature mechanical

properties but is not widely discussed in the field of high-temperature flow properties. During

hot rolling, precipitation in C-Mn steels is insignificant, but it does occur in microalloyed steels.

Dynamic precipitation is not likely to take place in single-pass rolling because insufficient time

is available [12,25]. However, the conditions for strain-induced precipitation seem to be

sufficient in the finishing stands of a multipass rolling operation, especially in the case of Nb-

alloyed steels. Temperature is relatively low during rolling in the finishing stands, and strain is

accumulated over several passes. However, even if precipitates do form during rolling, their

potency remains at issue.

3.6 Adiabatic Heating

Adiabatic heating is a factor in tests at high strain rates. Typically, at strain rates

exceeding 1 s'\ the temperature of a test specimen will increase continuously while the test is

progressing. The heat generated in a specimen during deformation at such rates cannot be

dissipated quickly enough to maintain an isothermal condition. If we assume that all the

mechanical energy is converted into heating the specimen, then the temperature increase AT due

to deformation can be estimated [4]:

^T-^. (20)

{am 5e) is the area under the stress-strain curve without temperature correction, p is the steel's

density, and Cp is its heat capacity. The incremental stress decrease due to the adiabatic heating

is

Ao- =\maR)\T T + d^T.'

'

^^^

Depending on test conditions, the specimen's temperature rise can be as high as 30 °C at

a strain of 0.7 [4]. An increase in temperature of 10 °C will generally decrease the flow stress by

2 to 5 MPa. According to one report, the measured and calculated temperature increases due to

adiabatic deformation agree well [4].

4. Experimental Approach

The goal of constitutive modeling is to predict material flow properties. Knowledge of

the stress-strain curve is needed for efficient processing, along with the yield- and flow-strength

data. Experiments provide an accurate mechanical-property database for each steel composition.

Models must then be developed, based on equations that relate the deformation parameters.

In this study, we used uniaxial compression tests and right-circular cylindrical specimens

to characterize the mechanical behavior of selected steels. There are no standard test procedures

for determining flow-stress curves at the relatively high temperature and strain-rate conditions

encountered in industrial processing and metalworking. Compression testing offers advantages

over other test methodologies, especially in simplicity and economy of experimental apparatus.

Tension and torsion tests are viable alternatives. Tension test specimens are not desirable

for hot-deformation studies because they typically neck at strains in the range 10 to 30 %. Muchlarger strains without plastic instability can be achieved using torsion or compression tests.

However, the strain and strain rate in torsion tests will vary from a maximum at the surface of

the cylindrical specimen to at the center. The measured flow stress is then an average stress

over a certain volume of material at the outer layer of the specimen. The strain and strain rate in

torsion tests are likewise averages. It is assumed that the outer layer of the torsion specimen

provides the major contribution to the flow stress because of the stress distribution and work

hardening [28]. Torsion test specimens are usually rigidly fixed in the drive system. Therefore,

the strain-rate transient depends on the time needed to attain the operating speed. The strain rate

in this transient period is usually limited to less than 10 s"' [29],

The right-circular cylinder is often used in tests of metals to simulate thermomechanical

processing effects. Cylindrical specimens are small and readily machinable compared to

specimens used in other test methodologies. The cylindrical specimen is also recognized in the

ASTM Standard Practice for Compression Tests of Metallic Materials at Elevated Temperatures

with Conventional or Rapid Heating Rates and Strain Rates (ASTM E 209 - 65).

A disadvantage of the cylindrical specimen is that it barrels during compression testing.

Barreling in axisymmetrical compression testing is caused by (1) the friction at the platen-

specimen interface, and (2) the temperature gradient between the platen-specimen interface and

the middle of the specimen.

Barreling in compression tests causes nonuniform strain and strain rates. Nevertheless,

reports show that barreling does not affect the measured flow stress if the specimen aspect ratio

(the specimen diameter divided by its height) is kept between 0.5 and 0.8 [30,3 1]. Higher loads

are needed to deform a specimen when friction is present. However, barreling is usually

accompanied by shear instability, which reduces the load. In certain specimen geometries the

load increase due to friction is possibly compensated by the load reduction due to shear

instability. For this reason, we used no lubricants for the experiments described in this study.

Lubricants at the platen-specimen interface may reduce barreling, but they introduce other

complications.

Nonuniform deformation from barreling does cause problems in the interpretation of

microstructures that are developed during deformation [32,33]. Strain imposed on the specimen

during deformation is a major factor influencing the recrystallized austenite grain size after

deformation. The nominal strain that is measured during deformation differs significantly from

the actual strain of the specimen [33]. Therefore, barreling may be ignored if the testing is solely

for the measurement of flow stress, but it must be taken into account if microstructural

evaluation is involved.

4.1 Materials

Two plain C steels, two interstitial-free (IF) steels, and four microalloyed steels were

tested in this study. Table 2 identifies the eight steels and their chemical compositions.

Mechanical properties of the steels at room temperature are reported in a different report [34].

4.2 Test Specimen

The cylindrical test specimen was 10 mm in diameter and 15 mm high. The cylindrical

specimens were machined from crop samples such that the cylindrical axis coincided with the

through-thickness direction. The steel types were identified in table 2.

4.3 Apparatus

Two servohydraulic test machines are available in our laboratory: one has a load capacity

of 250 kN and the other 100 kN. In addition to this difference in load capacity, the two machines

differ in attainable strain rates, and in heating and cooling techniques. The 250 kN machine is an

existing machine, and the 100 kN machine was developed and commissioned during the

Table 2. Chemical compositions (mass %).

Elem.HSLA-V

HSLA-Nb

HSLA-50

Ti-Nb

HSLA-80

Ti-NbA36 DQSK

IF

Nb rich

IF

Nb lean

C 0.045 0.082 0.071 0.07 0.17 0.038 0.0028 0.002

Mn 0.45 0.48 0.758 1.35 0.74 0.30 0.17 0.106

S 0.005 0.005 0.013 0.003 0.008 0.008 0.006 0.008

Si 0.13

V 0.080

Ti 0.013 0.042 0.035 0.059

Nb 0.036 0.023 0.085 0.035 0.009

AI 0.078 0.024 0.053 0.039 0.040 0.040 0.027 0.033

N 0.0072 0.0054 0.0067 0.0075 0,0047 0.0052 0.0029 0.0041

10

investigation of this project. Using these machines, we compressed specimens between quartz or

ceramic platens located inside an evacuated test chamber at elevated temperatures.

Fused quartz or ceramic rods are used for platens. The ceramic platens are

transformation-toughened zirconia (TTZ) in the form of solid white rods. The TTZ rods are

more expensive than quartz rods, but are preferable in our experience since quartz rods are more

susceptible to crack after repeated use.

The apparatus are described elsewhere [35,36], and only brief descriptions of the

apparatus are given here. The subcomponents of the apparatus include a computer-controlled

servohydraulic load frame, a test chamber providing vacuum environment, a furnace equipped

with a programmable temperature controller, a microcomputer, and a storage oscilloscope. The

axisymmetric compression test methodology offers benefits since the required specimen size is

relatively small. This facilitates system design by minimizing the volume and mass of test

fixtures.

4.3.1 Computer and Peripherals for Machine Control and Data Acquisition

Figure 3 is a flowchart of the plan for machine control and data acquisition. The

computer is a 25 MHz 80486 personal computer (PC), equipped with a board-level analog-digital

(A/D) plug-in board having a 45 kHz throughput and an IEEE GPIB board. The A/D board

contains eight differential A/D channels (input) and two digital-analog (D/A) channels (output).

Peripherals include a line printer, a 340 MB hard disk, a 1 .2 MB floppy disk drive, and a 360 KBfloppy disk drive.

The apparatus is equipped with a programmable digital storage oscilloscope for high-

speed data acquisition. The oscilloscope includes four 15 bit, 100 kHz digitizers. Each channel

stores up to 4000 data points when all four channels are operating simultaneously. The same

computer and the storage oscilloscope are used interchangeably with the two test machines.

The computer components and accessories described here are existing pieces of

equipment that serve multiple uses in our laboratory. They do not represent up-to-date

technology. Different types of components may perform the same functions. For example, the

storage oscilloscope is used for data acquisition simply because it is readily available and has

been used for such purposes at NIST for the past 15 years [37,38]. More affordable board-level

A/D plug-in boards are available and may be substituted for the oscilloscope described here.

Also, the capabilities of today's personal computer and its peripherals, such as disk drives, are

much superior to those described in this investigation.

4.3.2 Load Frames

Each machine is equipped with a closed-loop servohydraulic load frame. One machine

has a load capacity of 250 kN and the other 100 kN. These machines are not conventional in

terms of actuator travel speed. The machines have been modified by using high flow-rate

servovalves and large accumulators. The modification gives attainable actuator speeds up to 500

11

mms"'. With the currently used specimen geometry, this travel speed produces controllable

strain rates up to 25 s*' for the 250 kN machine, and 50 s'' for the 100 kN machine.

The most desirable deformation mode for rolling simulation is a constant true-strain-rate

compression, which greatly eases data analysis and interpretation. Constant true-stain-rate

compression is accomplished through displacement control of the load train by a PC using the

equation

h{t)^ h,exp{-i£). (22)

Here, h{t) is the specimen's instantaneous height at time /, h, is the initial height, and ^is the

desired constant true strain rate.

In a high strain-rate experiment, the actuator's travel starts from a rest position and then

accelerates to the desired speed that results in slow rates at the very beginning of the experiment.

To overcome the problem, we retract the actuator at the beginning of a test so that the specimen

is compressed using only that portion of the actuator travel that produces constant rate. Then, the

value of ^is calculated using the actual signals recorded by the data acquisition system [39].

4.3.3 Heating, Cooling, and Vacuum

For the 100 kN machine, a vertically mounted electrical tube furnace (power rating is 6.0

kW at 240 V and 21 .6 A) is used for heating. The heating elements are four tubular tungsten-

filament quartz lamps, each backed by an elliptical reflecting wall. The vacuum chamber is a

quartz tube that encapsulates the ceramic rods and the specimen. The tube is 50.8 mm in

diameter and 264 mm long. The elliptical reflectors, which surround the quartz tube, focus

radiant energy from the tungsten lamps to the outer surfaces of the specimen in the quartz tube at

the center of the heating chamber. The physical sizes of this furnace and its power supply are

small, which contributes substantially to the compact design and efficiency of the entire test

facility.

Heating of a specimen is achieved through radiation and the heating rate is limited to

about 2 °C-s''. There is no forced cooling device for fast cooling of a heated specimen. The

cooling is achieved through radiation to the surrounding and through conduction to the loading

rods. Thus, cooling of a heated specimen after a test is slow, and it takes 2 to 3 h to furnace-cool

the specimen. The vacuum system consists simply of a pressure gage and a mechanical pump by

which the chamber is evacuated during a test to minimize oxidation of the specimen at high

temperatures.

The 250 kN apparatus uses an induction furnace and a controlled gas-flow cooling

process. A 10 kW, 450 kHz radio-frequency (rf) induction generator generates the heat

demanded by the temperature controller. The heating coil is a Cu tube 4.76 mm in diameter. Astainless-steel vacuum chamber encloses the specimen, and the heating and cooling coils. Avacuum is obtained with a 100 mm diffusion pump which can achieve a vacuum of 1 .33 x 10

Pa (1 X 10'^ Torr) within 20 min. At a pressure of about 1.33 x 10'' Pa (lO"' Torr), gasses are

12

-3

ionized between the specimen and the heating coils. At this pressure, it is impossible to heat the

specimen because the rf field arcs and shuts off the rf generator.

Cooling of a specimen for the 250 kN machine is controlled with a PC that regulates gas

flow onto the specimen through a Cu coil 6.35 mm in diameter and concentric with the heating

coil. The side of the cooling coil facing the specimen is perforated to provide a gas flow that

cools the specimen uniformly. Either He or N gas is used for cooling. Because of its higher

specific heat (5230 versus 1030 J-kg'^K"' at 25 °C) and greater mobility, He is more efficient

than N in rapid cooling. For a cylindrical steel specimen 10 mm in diameter by 18 mm in height,

the cooling rate is controllable to a maximum rate of 25 °Cs"\

Both furnaces are operated automatically by means of a programmable temperature

controller that has ramp-and-soak and multiple set-point capabilities. Before testing, the desired

thermal profile is selected and preprogrammed into the controller. The temperature at any phase

ofthe experiment is then monitored by means of an S-type thermocouple (Pt and Pt-10%Rh) that

is spot-welded to the surface of the cylindrical specimen at the midpoint of height.

4.3.4 Machine Performance and Reproducibility

4.3.4.1 Alignment

Both test machines are rigidly constructed. The specimen dimensions were measured

before and after compression using a micrometer to confirm the final value of strain, and to

confirm that good axial symmetry is retained after testing. The fact that the top and bottom

surfaces of specimens remain parallel after compression testing is a qualitative indication of

good mechanical alignment.

4.3.4.2 Reproducibility

Figure 4 shows the results of triplicate tests ofDQSK specimens using the 250 kNmachine. The results demonstrate excellent reproducibility of the machine. Figure 5 shows a

pair of stress-strain curves for DQSK specimens that were tested using the 100 and 250 kNmachines at nominally identical test conditions. In this example, the grain size of the steel is

0.114 mm, the test temperature is 1000 °C, and the strain rate is 25 s'' (the maximum possible

strain rate of the 250 kN apparatus). The near equivalence of the stress-strain curves presented

in this figure demonstrates acceptable reproducibility.

4.3.4.3 Strain Rates

At low strain rates, the actuator of the machine follows the command signals quite well.

As a result, the nominal constant strain rate can be achieved and maintained. Figure 6 shows the

results of tests at nominal constant strain rates of 1 and 10 s'\ Here, the strain rates were

calculated using the actuator's displacement-versus-time signals.

If the command signals call for instant high speeds, the actuator is not capable of

following the demands instantaneously due to inherent characteristics of servohydraulic test

13

machines. The actuator's speed will be slower than the command signals at the beginning and at

the end of a command cycle. After receiving the commands at the onset of a test, the actuator

needs time to accelerate from a rest position to the desired speed. Similarly, it needs time to

decelerate from a high-speed travel to rest as called for by the command signals at the end of the

test. This behavior is clearly demonstrated in figure 7.

As mentioned in section 4.3.2, the problem of slower speed at the beginning of a test has

been remedied by actuator retraction. However, the problem at the end of a test has not been

corrected in our laboratory. The problem limits the valid strain range during a high strain-rate

test. Thus, the actual strain rate during a high-speed test varies instantaneously, so specified

values are necessarily quoted in nominal terms.

4.3.4.4 Specimen Strength

The machine's performance is slightly affected by secondary factors such as the

specimen's size and flow strength. The data in figure 8 show implicitly that the machine's

performance is influenced by the properties of the test material. Here, machine performance is

gaged in terms of the ratio ///////„,. The measured and commanded final heights are Hfm and Hfc,

respectively. A ratio of 1 means the commanded strain value is precisely achieved. Specimens

in this series were tested at the same nominal strain rate, using identical machine settings (span

of ram travel fixed at a constant value). Larger grain size and higher temperature implies softer

steel; so the trend of resuhs in figure 8 reflects the 100 kN machine's ability to meet a

commanded value of compressive strain at high strain rate, and how that is affected by the

specimen response.

4.3.4.5 Adiabatic Heating

Figure 9 is an example of specimen self-heating during a typical compression test at a

strain rate of 50 s"\ Experiments indicate that the temperature rise zlTat a strain of 0.7 ranges

from 5 to 10 °C, with modest variation depending on test conditions. Normally, zlT values less

than 10 °C may be considered negligible for engineering purposes. On the other hand, the

observed values here are certainly smaller than expected. Based on predictions from

mechanical-equivalence equations assuming all deformation energy is converted into heat as

described in section 3.6, the calculated zlT^ is about 20 °C. The disparity may be due to

nonuniform heating throughout the specimen. Therefore, additional analysis is needed to clarify

the role of adiabatic heating in hot-deformation studies. Because of this uncertainty, the stress-

strain results presented in this report are not adjusted for the adiabatic heating correction.

5. Results and Discussion

5.1 Raw Data and Data Analysis

During each test, the load, actuator's displacement, temperature, and time were measured,

recorded by the storage oscilloscope, and saved onto floppy disks for processing. Outputs of

interest are the curves of true stress as a function of true strain, the true strain rate as a function

14

Table 3. Summary of test results from UBC and NIST.

Steel ^,s-^ T, °C(Tat f == 0.5, MPa

UBC NIST

900 135 154

1

1000 98 108

1100 71 66

DQSK1200 43 44

900 171 174

101000

1100

129

103

138

102

1200 75 74

900 133 133

1

1000 95 89

1100 66 57

A361200 46 44

900 171 165

101000

1100

120

104

131

104

1200 80 76

of true strain, and the temperature change during deformation. Information on the software

programs for the data reduction and analysis is given elsewhere [39].

5.2 Round-Robin Test Program with UBC

A round-robin test program with UBC was initiated to compare the test results using the

respective laboratory facilities and analyses. Two steels, the A3 6 and DQSK grades, from the

same slabs after rough rolling were used for the program. The round-robin test program is

important because there are no published test standards for the high-temperature and high-strain-

rate compression tests, and test procedures vary widely from laboratory to laboratory. Consistent

and agreeable stress-strain data generated from different laboratories are essential for any

modeling effort. The cylindrical specimens used were 15 mm in height and 10 mm in diameter.

The testing parameters included 3 strain rates, 0.1, 1, and 10 s"\ and 4 temperatures, 900, 1000,

1100, and 1200 °C.

The test results are presented in figures 10 through 13. The flow stresses at a strain of 0.5

are tabulated in table 3. In general, the agreement between the results generated in the two

laboratories is good. At a strain of 0.5, the difference in flow stress is within 10 MPa, except that

15

Table 4. Constants in hyperbolic sine equation for DQSK and A36 steels.

Steel Laboratory a, MPa"^ Q. kJmor' m \r^A

DQSKUBCNIST

0.013

0.013

293

326

4.07

4.41

25.6

28.4

A3 6UBC

NIST

0.013

0.013

314

321

4.34

4.39

27.6

28.4

C-Mn Literature 0.013 l(il to 366 3.2 to 5.0 25 to 32

in two cases, where one difference is 19 and the other is 1 1 MPa. This is within the variability

observed for the same steel tested with the same machine.

The flow stresses at a strain of 0.5 were used to fit eq (15). The results are presented in

table 4. Here again the agreement between the results from the two laboratories is good. The

results also agree well with those published in the literature.

However, there is a consistent difference observed in the results from the two

laboratories; that is the hardening behavior at strains below about 0. 1 . This is due to the

difference in the way strain has been calculated. At UBC, the uniaxial strain was inferred from

the direct measurement of diametrical changes during testing. The measurement was performed

directly on the specimen at its mid-section. At NIST, the axial strain was calculated from the

displacement signals of load train with the assumption of incompressibility of steel during plastic

deformation (constant-volume assumption). Implicit in this strain calculation is the assumption

of uniform deformation.

As mentioned previously, at NIST the displacement signals were not measured directly

on the specimen. The displacement signals also contained the displacement of the load train in

addition to the displacement due to specimen. With rigid machines such as those at NIST, the

displacement due to the load train was negligible. However, even this usually negligible

displacement due to the machine becomes noticeable at small strains. This results in a slight

overestimation of strain in the results produced at NIST. The correction for displacement due to

machine compliance will be discussed in section 5.3, "Machine Compliance."

In addition to the difference in strain calculations, there is another significant difference

between the machines at UBC and at NIST. That is the heating method. At UBC, resistance

heating was used, and the jaws were water-cooled. This produced a temperature gradient along

the specimen axis from the mid-section of the specimen to the interface between the specimen

and the anvils. The temperature difference was between 50 and 70 °C. At NIST, induction

heating was used, and the specimen was placed on a quartz platen. The upper platen was

retracted; thus it was not in contact with the specimen before testing. This specimen-platen

arrangement produced a slight temperature gradient with higher temperature at the top of the

specimen and lower temperature at the bottom. However, because of the low thermal

conductivity of quartz platen plus poor contact between the specimen and the platen before

testing, the temperature gradient should be small.

16

Induction heating is inherently slower than resistance heating, such as used at UBC. For

tests at NIST, with a strain rate less than 0.5 s'\ heat loss from the specimen to the platen

through conduction was significant during deformation. The heater control cannot respond to

this heat loss until about halfway through the test at a strain of about 0.5. Therefore, heat loss in

tests of a strain rate of 0. 1 s'^ is significant. The temperature of the test specimen decreased

slowly from the start of the test and then slowly increased again after a strain of about 0.5 whenthe heater control started to respond to the temperature decrease. The isothermal test condition

had not been maintained. Flow stress from these tests (strain rate of 0. 1 s"') showed higher

values due to temperature drop during test. Due to this effect, the results of testing at the strain

rate of 0. 1 s'^ are not presented in this report. Under hot-rolling conditions, the strain rate is

expected to be much higher than 0. 1 s'\ so the results at this strain rate are much less important

than those at high strain rates.

Another point worth mentioning is the friction between the specimen and the anvil. At

UBC, because heating is resistive, the contact between specimen and anvil has to be good in

order for the electrical current to go through. This specimen-anvil arrangement increases the

friction between the specimen and the anvils. Large friction between specimen and anvil

(platen) causes barreling and increases the deformation loads. This is evidenced in most of the

UBC's results at large strains; the flow stress increases at strains greater than about 0.7. This

increase in flow stress at strains greater than 0.7 is not real, but due to large friction between

specimen and anvil. However, barreling is usually accompanied by shear instability that reduces

the load. Specimens tested at both laboratories did show barreling after deformation. Reports in

the literature [4,31] have shown that with the specimen dimension used in this round-robin

program, barreling will not affect the measured flow stresses, at least at strains less than 0.7, in

both laboratories. This is probably because the increase in load due to friction is cancelled out

by reduction in load due to shear instability.

Despite all these differences between the test machines at UBC and at NIST, the

agreement between the test results is reasonably good. Although detailed comparison between

the results obtained from this program with those published in the literature is impossible

because of the differences in steel grades and testing details, computed results using Shida's

equation [26] with comparable C contents and testing conditions agree well with present results.

This confirmation seems to exclude the possibility that both laboratories produce erroneous data.

5.3 Machine Compliance

The accurate way to measure strain is to measure the displacement, from which strain is

calculated, directly on specimen with strain gages, extensometers, or optical devices. In our

experiments, however, the strain has been calculated using displacement signals that are not

taken directly on the specimen (due to experimental difficulties), but away from the specimen.

The experimental difficulties arise because of space constraints resulting from heating furnace,

high temperature, and large strain. The displacement signals recorded away from the specimen

consist of the displacement due to the specimen plus the displacement of the load train.

Our experimental setup measures the displacement at a location about one meter away

from the specimen with a linear variable displacement transducer (LVDT). The displacement of

17

the load train typically consists of two parts. First, it comes from the loose contact between the

anvils and the specimen at the beginning of test. This portion of the displacement will vary from

specimen to specimen and is relatively large. The other is the elastic deformation of the load

train. This portion of the displacement is proportional to the applied load and is reproducible

from test to test.

Using aluminum specimens at room temperature (investigation at high temperature is

difficult to conduct, if not impossible), we determined the displacement of the load train as a

function of load experimentally. In this experiment, a clip-on gage was attached to measure the

gap between the two anvil surfaces that compress the specimen. This device measured the actual

specimen displacement. At the same time the displacement was recorded away from the

specimen just like those experiments conducted at high temperatures. The difference between

these two quantities is the displacement due to machine. Except for those results in section 5.2,

the stress-strain results presented in this report have been corrected for the machine

displacement.

5.4 Mathematical Description of Stress-Strain Curves

The goal of this study is to provide models that estimate the stress-strain curves of steels

under hot-rolling conditions. These models are then used to calculate the roll-separating forces

during rolling and to provide data as input to the numerical tools, such as finite-element method,

for detailed calculations of stress, strain, and related properties. The focus has been on modeling

the stress-strain behavior as functions of temperature, strain rate, and austenite grain size. Little

has been performed to detail the effects of individual parameters.

Besides chemical composition, three other major parameters have significant influence

on the flow behavior of steels: temperature, strain rate, and austenite grain size. For models to

be useful, they have to consider these parameters explicitly. In this study, there are not enough

systematic variations of chemical compositions to provide enough data for modeling. Thus,

there is no general modeling effort in this area and each grade of steel will have its own model

that considers the effects of temperature, strain rate, and austenite grain size.

The effects of temperature, strain rate, and austenite grain size on flow behavior are given

in figures 14 through 16. As expected, decreasing temperature increases flow stress (figure 14).

Also, increasing strain rate increases flow stress (figure 15).

The influence of austenite grain size is more complicated. As indicated in figure 16,

reduction of grain size promotes dynamic recrystallization. Grain-size reduction also increases

the flow stress and the rate of hardening in the early stage of deformation. As for the effects of

grain size on the steady-state flow stresses, either with or without dynamic recrystallization, the

results are not very conclusive. However, observations support the theory that grain size has

little effect on the steady-state flow stresses [22,27]. Equilibrium subgrain structures are

established in the early stage of deformation. The development of subgrain structure controls the

early stage of deformation and the equilibrium subgrain structure controls the steady-state

stresses. So, the initial grain size has little influence on the steady-state flow stresses.

18

To describe the stress-strain curve mathematically, our initial choice was eq (9), which

fitted the elevated-temperature experimental data well for the stress-strain portion that was prior

to dynamic recrystallization. Later, UBC suggested that we use the rigid-plastic model to

describe the flow behavior and incorporate yield stress at the start of the stress-strain curve.

Thus, eq (9) was modified and the following equation was used.

0-, = o-o + {(T^^ - o-o)[l - exp(-^f)]".

(23)

Here, oo is the 0.2 % offset yield stress in MPa, oisi is the steady-state stress without dynamic

recrystallization in MPa, /9is a dimensionless constant related to the hardening behavior, and n is

a constant related to the relationship between applied stress and dislocation density.

The hyperbolic sine equation (eq (15)) is used to describe the combined effects of

temperature and strain rate on the steady-state stresses. Grain size is considered to have no

influence on the steady-state stresses. The effects of temperature, strain rate, and grain size on

Ob and y9 are treated separately. The constant n is considered to be independent of temperature,

strain rate, and grain size.

Different equations have been proposed to treat the effects of temperature, strain rate, and

grain size on oo and /?. The task has proven to be difficult because the equations must describe

the effects that follow some physical principles and still cover a wide range of conditions. The

equations are required to meet the following conditions:

• ao increases with decreasing temperature, increasing strain rate, and decreasing grain

size.

• Within the range of interest, /? increases with increasing temperature, decreasing strain

rate, and decreasing grain size.

• Within the experimental range, where the machines are capable of generating reliable

data, the equations should be able to reproduce the stress-strain curves.

• Outside the experimental range, where no reliable experimental data exist, the equations

should be able to estimate the stress-strain curves reasonably well. The operational

ranges are temperature from 800 to 1300 °C, strain rate from 1 to 400 s'', and grain size

from 0.02 to 0.5 mm.

After some trial and error, we found that the power law performs well and used the

following equations.

£70=^1 -V' (24)

^ = b,

f TXH £\^( d^^'

\Tr,J

£

Wo.(25)

19

where a\ and b\ are constants to be determined experimentally, and To = 1273, £q= 10, and do

= 0.08 are scaling factors. T, e, and d are temperature, strain rate, and grain size in K, s'*, and

mm, respectively.

To account for the softening due to dynamic recrystallization, we introduced,

Act - max 0,(c^.„-o-„,)

// (8\

\m^^

1-- exp -*

V V \£ J ^ J

(26)

where Zlcris the amount of softening due to dynamic recrystallization after the peak stress in the

stress-strain curve has been reached, £•* is a scaling factor depending on temperature, strain rate,

and grain size, and m^ is a constant to be determined experimentally. A simple power law

f ^\

kT^j

ni( • N '?3 / ,\')i

yd., J(27)

is used to describe the dependence of £:* on temperature, strain rate, and grain size. o'ss2 is the

steady-state stress after the occurrence of dynamic recrystallization. The dependence of crss2 on

temperature and strain rate is treated with the hyperbolic sine equation (eq (15)). Grain size is

assumed to have no influence on crss2-

Equation (27) is substracted from eq (23) after the peak stress in the stress-strain curve is

reached, and the overall stress component becomes

(7 = a, - Act. (28)

The determinations of the strain at peak stress and the boundary separating dynamic

recrystallization and no dynamic recrystallization are treated in sections 5.7 and 5.8 later in this

report.

The combination of eq (18) and eqs (23) through (28) ftilly describes the stress-strain

curve. The subsequent task is to determine the coefficients in these equations.

5.5 Determination of Model Parameters

At the beginning of the modeling effort in this project, experimental data were fitted to

obtain the coefficients in eq (23) for each individual specimen. Then, coefficients in other

equations were obtained individually and separately. This procedure has the advantage for

detailed analyses of the effects of individual parameters on the overall stress-strain curve, and of

the deformation mechanisms. But it is very tedious and time-consuming. For the purpose of this

study, which is to provide models that estimate the stress-strain curve for hot-rolling simulations,

some statistical techniques that estimate all the coefficients in all the equations with one

computer run are available.

20

This latter approach, which we adopted, is fast in obtaining the coefficients that describe

the stress-strain behavior adequately. It also tends to smooth out the experimental abnormalities.

However, it loses the details of the effects of individual parameters. Also, the original meanings

of some parameters are lost. These include the steady-state stresses without or after dynamic

recrystallization. And the significance of the value of £* , which was originally proposed to be

the critical strain for dynamic recrystallization [9], no longer exists. Therefore the reader is

cautioned that, as presented in this report, the parameters should not be interpreted as necessarily

having their intended original meanings. They must be regarded only as fitting parameters.

5.5.1 Parameter Estimation

The empirical mathematical model just described has 20 adjustable parameters, with oisi

and (Tss2 expressed as

^ssl=

sinh-'(^,Z,^^)

a(29)

and

^ss2=

sinh-'(^3Z/^)

a(30)

Here, Zi (i = 1 and 2) is the Zener-Hollomon parameter

Z, = e expRT

(31)

For each material, there are K experimental stress-strain curves. The empirical mode! was fitted

to the data in two stages. Yield stress, as described in eq (24), was modeled and fitted separately.

To estimate the constants ai, a2, 03, and ^4, we minimized

L = Y,(ao{k)-aM)'' (32)

k=\

In this expression, <Jo(k) and ao(k) denote the predicated and observed true stress for the ^-th

curve (at true strain equal to 0). We minimized L using quasi-Nev4on method, where the

Hessian matrix and gradient are approximated numerically [40].

Once the fitted values of ^i, i= 1, 2, 3, 4, were obtained, we determined the value of the

other 16 parameters by a "penalized" least-squares fit. Specifically, we minimized

k=\ 1=1

(33)

21

where, oik,i) and 6-(k,i) denote the measured and predicted true strain for the /-th sample in the

k-xh experimental curve. For the k-th curve, there are A^k samples in the stress-strain curve. The

first term on the right side is the sum of the squared prediction errors. The second term is a

penalty function which increases L when the predicted asymptotic stress d-ss2, varies from a

reasonable guess a ss2 of the asymptotic stress. The relative contribution of the penalty function

was controlled by the value of the (positive) parameter /^. Estimates of d-ss2 are not available for

all experimental curves. For curves where there was no prior information, we set /Ik - 0. Whenwe have prior information, we set Ak = 0. 1. For the cases studied, for ?tk ^ 0. 1, the agreement

between the observed and predicted data is about the same. However, when extrapolating the

model to higher strains, our judgement is that a nonzero value of Xk improves the quality of the

prediction.

In the fitting procedure, the parameters were constrained to fall between selected lower

and upper bounds. For the /-th parameter, if the bounds are L, and f/i, we expressed the

constrained parameter a, in terms of an unconstrained parameter cj^:

a,=L, +/^'~f' y (34)l + exp(<r/,j

We minimized L as a fianction of the unconstrained parameters using a quasi-Newton method

where the Hessian matrix and gradient are approximated numerically [40]. In a complicated

optimization problem such as ours, results may depend on the initial guess for the model

parameters and on the type of algorithm used to estimate the parameters. As a check, for each

material, we refitted the model at least 20 additional times. In each additional run, the initial

parameter values were randomly perturbed. In all cases, additional runs did not yield a lower

value of Z. However, sometimes a large value ofL was obtained.

This analysis generated values for all the coefficients: a\-a4, bx-b^, kx-k^, a, Qx, Q2, t]i-

774, n, and /W4. They were generated separately for grades HSLAVTi-Nb, HSLA-Nb, DQSK, and

A36. The stress-strain behaviors of grades HSLA-50/Ti-Nb, and the two IF steels were found to

be similar to that ofHSLA-Nb. As a result, we used the model for HSLA-Nb for these steels.

For the same reason, the DQSK model was used for the HSLA-V grade. The units for the

parameters in the model are a in MPa, £ in s"\ Tin K, t/in mm, and Q in J-mol''. R = 8.314

J-mor'-K'' is the universal gas constant. The range of experimental conditions that were used for

the determination of model parameters is tabulated in table 5. It should be noted that in DQSK,the data of strain rate higher than 50 s'' are from CANMET, and those lower than 0. 1 s'' in A36are from UBC. All other data were generated at NIST.

5.5.2 Uncertainty Analysis

We estimated the random statistical error in the parameter estimates by a bootstrap

resampling procedure [41,42]. For each material, we have data from K experiments. Weindexed the experimental curves as / = (1,2, ...K) and simulated synthetic data by bootstrap

resampling. First, we simulated a new index set /* = (/i, /2, ../k) In this resampled index set, /j

is a random integer which is uniformly distributed between 1 and K. All of the /j were simulated

22

Table 5. Range of experimental conditions used for statistical regression.

Steel r, °c ^,s-' d, mm

HSLA-80/Ti-Nb 900 to 1200 lto25 0.012 to 0.2

HSLA-Nb 900 to 1200 1 to50 0.015 to 0.249

DQSK 884 to 1200 1 to 163 0.04 to 0.29

A36 900 to 1200 0.1 to 50 0.014 to 0.244

independently. For instance, \iK= 5, a simulated index set might be (3,1,2,1,4). For this case,

data from experiment 1 are represented twice and data from experiments 2, 3, and 4 are

represented once. Data from experiment 5 are not represented. To the w-th synthetic data, wefitted the model and got a bootstrap replication of the model parameters. The fitting procedure

was the same as used for the observed data. That is, we got a new set of parameter estimates (20

in all) for each synthetic data set. From this set of parameter estimates, we estimated the

standard deviation of each of the 20 model parameters. We divided the bootstrap estimate of the

standard deviation of the model parameter by the model parameter to get a fractional standard

deviation. When this fractional error is high, it means the model parameter is not well

determined. In general, except in a few cases, the parameters were well determined.

5.6. Experimental Stress-Strain Curves and Model Predictions

This section presents the results of model predictions versus experimental data for each

individual grade of steel. The experimental conditions are given on the graphs.

5.6.1 A36

The results for A36 are presented in figures 17 through 24. In general, the agreement

between the experimental data and the model predictions is excellent.

5.6.2 DQSK and HSLA-V

The results for DQSK are presented in figures 25 through 36. In general, the agreement

between the experimental data and the model predictions is excellent.

For HSLA-V grade, the test matrix covers a range of temperature from 900 to 1200 °C,

strain rate from 1 to 50 s'\ and grain size from 0.03 to 0.3 mm. After completing the

experiments, we compared the data ofHSLA-V specimens with predictions obtained using the

DQSK model. Figure 37 presents some of the typical results showing that the DQSK model

predictions agree well with the HSLA-V experimental data.

We believe that V has little effect on the flow response of the steel in the process

variable ranges of interest and that the model developed for the DQSK grade steel is suitable for

the HSLA-V steel because the two steels have similar chemical compositions. Figure 37

23

Table 6. Dynamic recrystallization ofDQSK and HSLA-V steels.

DQSK HSLA-V

Temp,

°C

Strain rate. Grain size,

mm fcp

Grain size,

mm 8,*

1200 10 0.29 0.3 0.3 0.33

1200 1 0.29 0.17 0.3 0.185

1100 25 0.114 0.44 0.1 0.42

1100 10 0.29 0.33 0.3 0.35

1100 10 0.114 0.337 0.1 0.335

1100 10 0.114 0.342 0.11 0.335

1100 10 0.114 0.34 0.1 0.335

1100 10 0.114 0.336 0.1 0.335

1100 0.29 0.22 0.3 0.235

1100 0.114 0.195 0.1 0.225

1000 0.29 0.321 0.3 0.33

1000 0.114 0.26 0.1 0.3

900 0.04 0.331 0.03 0.335

* Strain at which peak stress occurs in the stress-strain curves. This table lists only those

specimens that indicate the presence of the occurrence of dynamic recrystallization.

presents data that support these expectations. As shown in table 6, detailed analysis of dynamic

recrystallization in the two grades further demonstrates the similarities in flow behavior for these

two grades. Therefore, we used exactly the same model for HSLA-V and DQSK.

5.6.3 HSLA-Nb, HSLA-50/Ti-Nb, and IF steels

The results for HSLA-Nb grade are presented in figures 38 through 46. In general, the

agreement between the experimental data and model predictions is excellent.

Selected experiments with a scaled-down test matrix were completed for HSLA-50/Ti-Nb

grade. The experimental conditions span temperatures from 900 to 1 150 °C, strain rates from 1

to 25 s'\ and grain sizes from 0.013 to 0.061 mm. We expected the flow behavior of the HSLA-50 grade to be similar to that ofHSLA-Nb because the compositions of the two grades are

similar. Therefore, the series of experiments performed using this grade was less extensive than

those for other grades.

24

Some flow stress measurements for HSLA-50 were compared with predictions using the

model developed for HSLA-Nb, and the results are presented in figure 47. As expected, the flow

curves agree well with predictions from the HSLA-Nb model. On this basis, we conclude that

the model developed for the HSLA-Nb grade is in fact suitable for HSLA-50 also.

Two IF steel grades, Nb-rich and Nb-lean, were tested. Identical experimental conditions

were used for both grades. Our results lead to the conclusion that the flow behaviors of both

grades of IF steel are similar to that of the HSLA-Nb grade within the range of experimental

variables studied.

For the Nb-rich IF grade, figure 48 shows some of the results, wherein the experimental

stress-strain curves for the EF grade are compared with predictions made using the model for the

HSLA-Nb grade. The flow behavior of this IF grade is very similar to that of the HSLA-Nbgrade. The same conclusion is obtained from torsion experiments conducted at UBC. The

results of the torsion experiments are presented in figure 49. We conclude that the model

developed for the HSLA-Nb grade also predicts the flow behavior of the Nb-rich IF steel.

For the Nb-lean grade, figure 50 shows some of the experimental data compared with

predictions made using the HSLA-Nb model. We find that the flow behavior of this grade is also

similar to that of the HSLA-Nb. However, results obtained in multiple-hit torsion tests at UBCshow that the flow stresses of this grade are lower than those of the Nb-rich IF grade when

temperatures are below 980 °C and total true strains are larger than 1 .2.

Scrutiny of the torsion test results has indicated that little softening takes place in Nb-rich

grade between the hits which produce higher flow stresses, whereas substantial softening occurs

in the Nb-lean grade. This means that there is little static recrystallization between hits for the

Nb-rich grade, whereas the Nb-lean grade almost fully recrystallized. This is probably due to Nbin retarding the static recrystallization for the Nb-rich grade and explains why the Nb-rich grade

has higher flow stresses in later stages of deformation in multiple-hit experiments and at low

temperatures.

5.6.4 eSLA-SOATi-Nb

The results for HSLA-80/Ti-Nb grade are presented in figures 51 through 62. In

general, the agreement between the experimental data and model predictions is excellent.

5.7 Strain at Peak Stress

Dynamic recrystallization is an important phenomenon in hot deformation. It not only

softens material's resistance to deformation, but also refines the microstructure. It is useful to

estimate the conditions under which it will occur.

Based on the experimental data generated in this study, we developed models correlating

the strain at the peak stress in the stress-strain curve with temperature, strain rate, and grain size

for the A36, DQSK, HSLA-Nb, and HSLA-80 grades. Two different forms of equations were

used to make the correlation. One used the power law, and the other used the conventional

25

approach involving the Zener-Hollomon parameter. For the power-law approach, we used the

equation

£p = AV^o-

^4

(35)

where £p is the strain at which the peak stress appears in the stress-strain curve. Aj, A2, A3, and

A4 are constants. Other parameters were previously defined in section 5.4. The results are

plotted in figure 63.

For the conventional approach incorporating the Zener-Hollomon parameter, we used the

equation

e^ = B,Z'^d'\ (36)

where Bj, B2, and B3 are constants. Other parameters were previously defined. The results are

plotted in figure 64. It should be noted that the reported value of .^2 in the literature is 0. 1 5,

while the values for ^3 range from 0. 18 to 0.5 [29,43]. As shown in figures 63 and 64, the two

approaches have a similar statistical correlation.

5.8 Boundary Separating Dynamic Recrystallization and No-Dynamic Recrystallization

As described in section 5.7, if strains exceed Sp as described by eq (35) or (36) the steel

will undergo dynamic recrystallization. However, there is experimental evidence suggesting that

under certain conditions, dynamic recrystallization will not occur with a single deformation even

though strains exceed those described by eq (35) or (36). To determine whether dynamic

recrystallization will occur during a single deformation regardless of the conditions set forth by

eq (35) or (36), we used the Zener-Hollomon parameter as a criterion. IfZ is greater than a

critical value Zc, then dynamic recrystallization will not occur in a single deformation. The

model, as described by the equation

Z^ = AQxp{Bd) + C, (37)

also takes grain size into consideration, where ^, B, and C are experimentally determined

constants. Z and d are defined previously.

5.9 Effects of Prior Heat Treatments: HSLA-Nb

We also conducted experiments to evaluate the effects of precipitates on the flow

behavior ofHSLA-Nb steel. In earlier experiments we obtained small austenite grain sizes using

low reheating temperatures. Under these conditions, the Nb precipitates were not dissolved in

the matrix when the experiments were conducted. This represents a departure from mill

conditions, where the reheafing temperature is high and the Nb precipitates are dissolved.

26

Specimens used in this series of experiments were reheated to 1200 °C , held for 30 min,

and water-quenched to the ambient temperature. The majority of the precipitates would be

dissolved after this treatment. The specimens were then reheated to 1000 °C with a heating rate

of 5 °C-s"\ held at that temperature for 150 s, cooled to 900 °C at a rate of 1 °Cs"', and held for

30 s before testing. In tests at 1000 °C, the specimens were tested immediately after the 150 s

hold at that temperature.

Figures 65 through 68 present the experimental results. For some test conditions, there is

no discernible difference in the flow behavior with or without prior heat treatment. However, at

the test condition of 900 °C and 1 s"' (figure 65), there is a significant difference in flow

behavior after a strain of 0.4. In this case, the specimen without prior heat treatment shows

evidence of dynamic recrystallization, whereas the specimen with prior heat treatment shows

continuous hardening, with no indication of dynamic recrystallization. It is likely that the

dissolved Nb retards the process of dynamic recrystallization in the specimen with heat

treatment.

Although mill conditions will not produce a strain rate of 1 s'^ at 900 °C, the observed

experimental effect deserves closer attention, particularly in cases where dynamic

recrystallization could occur. The effect vanishes at higher strain rates or higher temperatures.

5.10 Effects of Reheating Temperature and Model Adjustments: HSLA-80/Ti-Nb

In comparison with those samples that were reheated to 1200 °C, UBC has observed that

in torsion experiments, the flow stresses are lower when samples were reheated to 1250 °C and

held for 30 min. To confirm this observation, we conducted experiments with the following two

different reheating conditions:

1. Reheated at 1250 °C and held 10 min to obtain a grain size of 0.3 mm2. Reheated at 1250 °C and held 10 s to obtain a grain size of 0. 1 mm

Typical results of the experiments are presented in figures 69 through 72. As shown in

figures 69 and 70, under condition 2 there is no significant difference between the experimental

data and the model predictions, which were based on the experiments using lower reheating

temperatures. However, as figures 71 and 72 indicate, a higher reheating temperature and a

longer holding time (condition 1), appears to lower the flow stress.

The effects of reheating temperature on the flow behavior ofHSLA-80 were examined

fiirther in experiments using different test temperatures. The flow stresses for specimens with

reheating temperature of 1250 °C and holding time of 10 min appear to be approximately 9 %lower than those for specimens reheated at a lower reheat temperature. This 9 % drop is

consistent throughout the whole stress-strain curve and has been observed for specimens tested at

various temperatures. Thus, the model described in section 5.5 should be adjusted for this effect

ifthe reheating temperature and holding time are high and long.

The effect of reheating temperature has not been studied with other grades. The effect

might be due to larger grains resulting from a higher reheating temperature and longer holding

27

Table 7. Test matrix for double-hit experiments: HSLA-Nb.

Case

#

Soaking

temp., °C

Soaking

time, s

Grain size,

mmTesting

temp., °C

Strain for

the first hit

Strain

rate, s''

Inter-hit

time, s

1 1200 600 0.249 900 0.2 10 20

2 1200 600 0.249 900 0.6 10 10

3 1200 600 0.249 900 0.6 25 5

4 1200 600 0.249 1000 0.2 10 5

5 1000 150 0.015 900 0.2 10 5

time. If this is the cause for lowering the flow stress, then the effect should also be observed for

other grades of steels. The other potential cause for the effect is due to the dissolution of

precipitates of microalloying elements, also resulting from a higher reheating temperature and

longer holding time. If this is the factor, then the effect will be observed in the Ti-Nb- and Nb-

bearing grades because the precipitates of these grades will not dissolve unless the reheating

temperature is high and the holding time is long. The effect on plain-C or V-bearing grades

should be minimal because precipitates in these grades usually dissolve at lower temperatures

during reheating.

5.11 Effects of Retained Strain: HSLA-Nb

Using a double-hit experiment, we evaluated the effects of retained strain on the flow

behavior ofHSLA-Nb steel. The strain retained from one pass to the next not only significantly

increases the rolling forces but also affects microstructural development. The retained strain is

usually small, and the effects are negligible in plain-C (A36 or DQSK grades) and V-treated

steels because static recrystallization is rapid in these steels. Retained strain is important for the

HSLA-Nb grade in the last few finishing passes because the interpass time is short and the

process of static recrystallization is comparatively sluggish.

The test matrix of the double-hit experiments is given in table 7. Figure 73 presents

results which suggest that, under the experimental conditions described in table 7, little softening

takes place between the two hits. As indicated in these figures, the established model predicts

the flow behavior well once the retained strain is taken into consideration.

5.12 Comparison with Data from CANMET

We received from CANMET sets of high-strain-rate stress-strain data for three steels:

A36, DQSK, and HSLA-Nb. These experimental data covered temperatures from 900 to 1 100

°C and strain rates from 10 to 150 s'\ Austenite grain sizes ranged from 0.014 to 0.25 mm for

A36, from 0.038 to 0.155 mm for DQSK, and from 0.015 to 0.249 mm for HSLA-Nb.

We compared CANMET's data with our own model predictions for A36 and DQSK.Figure 74 compares some CANMET's data with our model predictions for the DQSK grade.

28

The agreement is generally good for stresses in regions of high strain (larger than 0.6).

However, poor agreement is observed in the regions of small strain (less than 0.4).

A similar trend is found for the A36 grade, as shown in figure 75. In general, the

agreement between CANMET's data and our model predictions for A36 is not as good as that for

DQSK, and in a few cases the agreement is poor for the entire stress-strain curve, as shown for

example in figure 76.

Two reasons are advanced to explain the disagreement between the observed trend at

small strains. First, CANMET's strain values include displacements due to machine compliance,

which are accounted for in NIST's analyses. As discussed earlier, including displacements due

to machine compliance in strain calculations promotes large errors in the stress-strain curve at

small strains. Second, we have assumed rigid-plastic behavior in our models, so the flow stress

begins at the yield stress when strain is 0.

In view of these differences in data treatment, we performed a second comparison. Wecompared our experimental data (without machine compliance correction) with CANMET's data

at similar test conditions. The results in figures 77 through 79 show fair, but better, agreement.

CANMET's data tend to exhibit lower strain hardening at the onset of deformation, and the

hardening continues to large deformations (large strains). In contrast, NIST's data tend to show

a rapid increase in stress due to a high strain hardening rate at the onset of deformation, and

thereafter the stress levels off at large deformations. Thus, CANMET's data exhibit lower

stresses at small strain regions and higher stresses at large strain regions. The cause of this

difference in hardening behaviors between results obtained at CANMET and NIST remains

unclear. It should be noted that the hardening behavior observed at NIST is more consistent with

some findings reported at UBC.

Efforts have been made to correct for the effect of machine compliance in CANMET'sdata. When machine compliance is accounted for, the stresses in the early portion of the flow

curve increase; thus the agreement between NIST's model predictions (or data) and CANMET'sdata improves. Figure 79 is a typical example showing good agreement between NIST's model

prediction and CANMET's data for A36 after correction for compliance.

5.13 Calculation of Roll-Separating Forces

One of the important applications of constitutive equations in hot rolling is the roll-force

calculation, which requires other parameters, such as roll diameter during rolling, thickness of

slab at roll entry and roll exit, and the friction between the workpiece and the rolls. One of the

commonly used models for roll-force calculations is the Sims method [44], which simplified the

slab analysis [45] and assumed that only sticking friction exists in the roll contact zone. The roll-

separating force per unit width is given by Sims as

F=2kR — - In —;— + —4 h, 2

( 1 ^

Kh^j(38)

29

where k is defined as the yield strength of the workpiece in pure shear, R'\s the radius of

deformed roll, hj and h2 are the entry and the exit slab thicknesses, and A/? is the draft, which

equals to {h] - hi), (p is the contact angle, which is approximated as

(j) = yJRAh, (39)

and h((f)n) is the slab thickness at the neutral point, which is the point along the surface of contact

between the roll and the workpiece where the surface velocity of the roll is equal to the velocity

of the workpiece.

According to the von Mises yielding criterion, k is equal to 57.7 % of the yield strength

of the workpiece in uniaxial tension or compression; it is only 50 % if the Tresca criterion is

applied. In this investigation, the von Mises criterion is adopted and the yield strength under

uniaxial compression is taken to be the stress calculated with models presented in section 5.5.

The strain used to calculate the stress is the true strain computed from hj and h2 or the

cumulative strain if there are retained strains from previous passes.

The value of 7? 'depends on the roll material, the original roll radius R, the roll-separating

force, and the draft. The relationship among these parameters is [26]

R CF— = 1 +R Ah

(40)

For steel rolls, the value of C is between 2. 1 x lO""* and 2.5 x 10"'* mm'^N'\ and is taken to be

2.21 X 10"'* in the present investigation [46].

The angle at the neutral point is calculated from

— In4 \hj

fR'

\

[}k J

tan (41)

The thickness at the neutral point is

h[(t>,) = 2R[\-cos[<l>„))^h^. (42)

The results of roll-force calculations along with actual mill measurements for a 13-pass

rolling on HSLA-80 steel are presented in table 8 and figure 80. Based on precipitation

calculation using the model developed by Dutta and Sellars [47], precipitation ofNb(CN)

particles occurred after the first pass at 965 °C. No static recrystallization between passes in all

13 passes was assumed and strain was accumulated from one pass to the subsequent passes. Thepredicted results agree reasonably well with the actual measurements. Due to the assumption of

sticking friction, however, the Sims equations are expected to slightly overestimate the rolling

loads.

30

Table 8. Comparison of calculated roll forces with actual mill measurements.

Roll forces, ton

Pass# Measured Calculated % difference

1 3324 3608 8.54

2 3456 3326 -3.76

3 3690 3646 -1.19

4 3113 3226 3.63

5 2678 2957 10.42

6 2137 2370 10.90

7 1827 2035 11.38

8 2990 3046 1.87

9 3080 3182 3.31

10 2720 3153 15.92

11 2610 2587 -0.88

12 2428 2754 13.43

13 2250 2393 6.36

6. Summary and Conclusions

The purpose of this work is to develop constitutive equations that predict the stress-strain

curves of steels as functions of hot-rolling process variables, such as temperature, strain rate, and

microstructural features. Eight steels: DQSK, A36, HSLA-V, HSLA-Nb, HSLA-50/Ti-Nb,

HSLA-80/Ti-Nb, and two interstitial-free (IF) grades, were selectively tested at temperatures in

the range from 900 to 1200 °C, strain rates from 1 to 50 s\ and austenite grain sizes from 0.012

to 0.29 mm. The data were analyzed to support the development of constitutive models.

To cover a wider range of strain rates, a new computer-controlled servo-hydraulic

laboratory apparatus was designed and built for experiments at high strain rate. A constant strain

rate of 65 s' was obtained with the machine. The usable range of strain at this rate is about 0.4.

The strain rate drops off substantially at strains larger than 0.4. At a constant strain rate of 50 s'',

the usable range is about 0.6. The strain rate of 50 s'' was used for the majority of tests in this

project. The machine has a load capacity of 100 kN and its radiant furnace can heat a specimen

(a cylinder with 10 mm diameter and 15 mm long) to 1200 °C. The maximum heating rate is

about 2 °C-s"\ The machine has muUiple-strike capabilities and is equipped with a rough-

vacuum pump to prevent oxidation during test at high temperatures. The machine and the data-

acquisition system are computer controlled.

Models for predicting stress-strain behaviors for A36, DQSK, HSLA-Nb, and HSLA-80/Ti-Nb were developed. The model developed for DQSK is also adequate for HSLA-V, since

31

we find that V has little influence on the flow behavior of low-C steels, and the chemical

compositions of the two grades are similar (except for the V addition in the HSLA-V grade).

The model developed for HSLA-Nb was also found to be suitable for predicting the stress-strain

behaviors of HSLA-50/Ti-Nb, and two Nb-treated IF steels, which all contain Nb. Model

predictions correlate well with the experimental data, indicating that the models adequately

describe the stress-strain behaviors of various steel grades under hot-rolling conditions.

Models were also developed for correlating the strain at peak stress in the stress-strain

curve with temperature, strain rate, and austenite grain size for the following four steels: A36,

DQSK, HSLA-Nb, and HSLA-80/Ti-Nb. In addition, we proposed models for predicting the

conditions under which there will be no dynamic recrystallization for the same steels. This will

precede the former conditions.

We also conducted experiments to evaluate the effects of precipitates on the flow

behavior ofHSLA-Nb grade. It was found that under one condition with low strain rate (1 s'*)

and low temperature (900 °C), the specimen with undissolved precipitates displayed occurrence

of dynamic recrystallization, while the specimen with prior high-temperature soaking heat

treatment (precipitates dissolved at the time of experiment) did not show the same behavior. It is

likely that the dissolved Nb retards the process of dynamic recrystallization under this condition.

At high temperatures and higher strain rates, there is no difference in flow behavior between the

specimens (with or without the high-temperature soaking heat treatment). It should be noted that

mill conditions rarely encounter a strain rate as low as 1 s'' at 900 °C.

For the HSLA-80/Ti-Nb grade, it was found that specimens with treatment of reheating at

1250 °C and holding for 10 min (0.3 mm in grain size) reduced the flow stress about 9 % whencompared with those specimens that were reheated at lower temperatures. This behavior has

been observed for various experiments with testing at different temperatures and strain rates. It

is, therefore, suggested that the 9 % reduction in flow stress be corrected if the actual conditions

are close to our experimental conditions (reheated at 1250 °C and held for 10 min).

In HSLA-80/Ti-Nb, we found the static recrystallization to be sluggish. Our double-hit

experiments show that at testing temperatures of 900 and 1000 °C and inter-hit times of 5 to 20 s,

little softening occurred between hits. If retained strain is properly considered, the model based

on single-hit deformation is adequate to describe the stress-strain behaviors under multiple-hit

conditions.

Our model predictions for A36 and DQSK were compared with Camplastometer's high

strain-rate data obtained at CANMET. CANMET's data cover the strain-rate range from 10 to

150 s'\ which are beyond NIST's experimental capability. The agreement between CANMET'sdata and NIST's model predictions is good at large strains (> 0.6) but poor at small strains (<

0.4). The discrepancy mainly comes from the different treatment of data in the two laboratories.

NIST corrected the data for displacements due to machine compliance and used rigid-plastic

model to describe the stress-strain behavior, whereas CANMET did not correct for machine

compliance and reported the data as generated (no stress-strain model had been used). After

correction for machine compliance in CANMET's data, the agreement was improved, and the

results indicated that NIST's models can be extrapolated beyond the experimental range that

32

were used to obtain the coefficients in the constitutive equations. However, caution should be

exercised in extrapolation because the results might not be reliable.

Finally, we used the Sims equation with NIST's constitutive model to calculate the roll-

separating forces for rolling an HSLA-80 slab. The calculated results were compared with the

actual mill measurements. The results are encouraging. For 8 of the 13 passes, the calculated

loads are within 10 % of the measured loads. The worst discrepancy is an overestimation by

15.9 %. For the other 4 passes, the discrepancy is slightly over 10 %. For 10 of the 13 passes,

the calculated loads are higher than the measured. This is to be expected since the Sims equation

assumes the presence of sticking friction (maximum friction) between the rolls and the

workpiece.

This work represents one subtask of the Microstructural Engineering in Hot-Strip Mill

Project, a cooperative research project sponsored by the AISI Advanced Process Control

Program. Other tasks of this project have been performed at the University of British Columbia

and at the U.S. Steel Technical Center. We gratefully acknowledge the support and sponsorship

of the American Iron and Steel Institute, particularly the two directors of the Project: L. W.Kavanagh and J. R. Vehec.

The actual values for the coefficients for the constitutive equations are not published in

this report, as these are considered proprietary data of AISI.

This work was performed under the supervision of G. Alers and H. I. McHenry. We also

acknowledge J. R. Orth (Rocket Seals Corporation, Denver, CO), and C. McFadeen (Coors

Ceramics Division, Golden, CO) for providing advice and information about the high-

temperature polymer seals and ceramic materials used in the construction of the hot-deformation

test apparatus.

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33

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34

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[28] McQueen, H. J. Deformation mechanisms in hot working. J. Metals: 31-38; 1968 April.

[29] Sellars, C. M. Physical simulation to determine hot forming behavior. Proc, Physical

simulation of welding, hot forming, and continuous casting; 1988 May 2-4; Ottawa,

Ontario, Canada; 1988. in-1-14.

[30] Baragar, D. L.; Crawley, A. F. Frictional effects on flow stress determination at high

temperatures and strain rates. J. Mech. Working Technol. 9: 291-299; 1984.

[31] Gelin, J. C; Oudin, J.; Ravaland, Y. Determination of flow stress-strain curves for

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[33] Kuziak, R. M.; Cheng, Y. W. Microstructural evolution in microalloyed medium-carbon

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strength low alloy steels. ISS, Warrendale, PA.; 1992. 51-64.

[34] Purtscher, P. T; Cheng,Y. W. Structure-properties relations in steels. To be published as

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[35] Cheng, Y. W.; Rosenthal, Y.; McHenry, H. I. Development of a computer-controlled

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35

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206; 1987.

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250 H

£200

2 100

50-

900 °C. 1 s-"", 0.015 mm

Case 1

Case 2

0.0

-I—I—I r

0.2 0.4 0.6

True strain

I I I II

I I I I

0.8 1.0

Figure 65. Effect of prior heat treatment on flow behavior: HSLA-Nb. There is no

discernible differences in the flow behavior below strain of 0.4 between the two cases.

Above 0.4, the behavior of Case 2 shows dynamic recrystallization while Case 1 shows

continuous hardening. Case 1 : with prior heat treatment; Case 2: without prior heat

treatment.

300

250-

900 X, 25 s-'', 0.015 mmCase 2

50-

0.0 0.2 0.4 0.6

True strain

0.8 1.0

Figure 66. Effect of prior heat treatment on flow behavior: HSLA-Nb. The differences

in the two cases are within the scattering band of experimental data. Case 1 : with prior

heat treatment; Case 2: without prior heat treatment.

96

ouu--

1000 °C, 1 s-l, 0.015 mm250-

2 200-

^ 150-Case 2

str ^^'^0)

2 100- Af^f Case 1

50-1

0-

0.0 0.2 0.4 0.6

True strain

0.8 1.0

Figure 67. Effect of prior heat treatment on flow behavior: HSLA-Nb. The differences

in the two cases are within the scattering band of experimental data. Case 1 : with prior

heat treatment; Case 2: without prior heat treatment.

300

250-1

; 1000 °C, 25 s-'', 0.015 mm

2 200H

S 150H

50-

I I I I

0.0 0.2

T 1I

I I 1 II

I I I II

I I I I

0.4 0.6

True strain

0.8 1.0

Figure 68. Effect of prior heat treatment on flow behavior; HSLA-Nb. There is no

difference between the two cases.

97

(0Q-

w(U

w

:3

200-

IIOOX. 25s'\ 0.1 mm

Experimental data

150-

^^^...^.^===1=—

100-/^ Model predictions

50-

0-

f0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain

Figure 69. Effect of reheating temperature on flow behavior: HSLA-80. The differences

between the experimental data and model predictions are insignificant.

250

200-

(0Q_

150-M

a?

CO

0)13

-1

1000 "C, 25 s", 0.1 mm

Experimental data

Model predictions

100

50-

I1

1 I I I I I I I 1 1I

I I I1

1 I I I I

I

I I I1

1 I I I1

1 I I I I

I

I I I I

I

1 1 I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain

Figure 70. Effect of reheating temperature on flow behavior: HSLA-80. There is no

significant difference between the experimental data and model predictions.

98

250

200

(0CL

150

V)

(D3100-

1200X, 25s-\0.3mm

Model predictions

Experimental data

I 1 1

I

I I 1 1

I

1 1 I

11 I 1 1 I

1

1 I 1 1

I

I I I

11 I I I I

I

I 1

1

I

I

I I II

I

1 1 I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain

Figure 71. Effect of reheating temperature on flow behavior: HSLA-80. Experimental

data are about 9 % lower than the model predictions.

250

200-

(0Q.

150toto(Ut_

w0)3

100-

1000 °C, 25 s-''. 0.3 mm

Model predictions

Experimental data

I 1 1 1 I I I

II { I

11 I I

I '

' ' '

I

'

'

' '

I' ' ' '

I

' ' ' 'I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain

Figure 72. Effect of reheating temperature on flow behavior: HSLA-80. Experimental

data are about 9 % lower than the model predictions.

99

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100

300

: 878°C, 146s'\0.04mm

250-

(?200

.2 100

50-

INIST's model predictions

- CANMETs data

0.2

1

0.4 0.6

True strain

I I II

I I 1 1 1 1 I I 1 1 1 I 1

1

0.8 1

(a)

300- 986 X, 140.5 s-\ 0.114 mm

250

2 200

(A

$ 150

100-

50-

NIST's model predictions5==?^^

CANMET'sdata

I 1 1 1 1 I 1 1 1 1 1 1 1

1

1 1 1 I

0.2 0.4 0.6

True strain

0.8 1

(b)

Figure 74. Examples of comparison between CANMET's high strain-rate results and

NIST's model predictions: DQSK.

101

300

250

£^200

(A

0) 150i_

a>

.5 100

50

/

-^^^S^00""^ ^ ^^^ 4^

..^-' NISTs model predictions

•A/

•' - CANMET's data

H

>

880 °C, 140 s'\ 0.014 mm1

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1

0.2 0.4 0.6

True strain

0.8

(a)

300

250

t^200

50-

992 °C, 134.6 s'V 0.25 mm

NIST's model predictions

CANMET's data

0.2 0.4 0.6

True strain

0.8

I I I I I I I—I I II I

(b)

Figure 75. Examples of comparison between CANMET's high strain-rate results and

NIST's model predictions: A36.

102

300

250

2 200

50-

997 °C, 96.1 s-"", 0.25 mm

NIST's model predictions

CANMETs data

I 1 1 I I I II

I I I I I I I I I I I II

I I I I I I I I I I I I I I I I I I I

0.2

1

0.4 0.6

True strain

0.8

Figure 76. Example of comparison between CANMET's high strain-rate results and

NIST's model predictions: A36. The agreement is poor.

300

250

1^200

v>

$ 150

N 1ST: 1 000 X, 50 s'"" , 0. 1 05 mmCANMET: 1017 °C, 46.5 s-\ 0.105 mm

NIST's data

CANMET's data

I I I I I I I

1

1 I

0.2

' '

I

' '

' I I I I I I

II I I I I I I I I

0.4 0.6

True strain

0.8

Figure 77. Example of comparison between CANMET's high strain-rate data and NIST's

data: A36. Neither set of data corrected for machine compliance. The agreement is fair.

103

owu

NIST: 1000 °C, 10 s-\ 0.114 mm250- CANMET: 985 °C, 9.5 s-\ 0.1 14 mm

2 200-CANMET's data

^ 150^- ^.«

jft"^^^^ A0)

2 100-NIST's data

/^

50- (Oh

0.2 0.4 0.6

True strain

0.8

(a)

300

250-

2 200H

$ 150-1

3 100

50

NIST: 1000 °C, 50 s-\ 0.114 mmCANMET: 987 X, 49.3 s-\ 0.1 14 mm

oi 1 1

11 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1

11 1 1 1 1 1 1 1 1

1

r

0.2 0.4 0.6

True strain

0.8

(b)

Figure 78. Examples of comparison between CANMET's iiigh strain-rate data and

NIST's data: DQSK. Neither set of data corrected for machine compliance. The

agreement is fair.

104

200

150(0a.

^ 100 H

50-

1097 °C, 50.1 s-\0.25mm

No compliance correction

(CANMETdata)

Compliance corrected

(CANMETdata)

NISTs model predictions

I I I I

II I I I I I I I I

II I I I I I I I I

II I I I I I I I I

II I I I I I I I I

0.2 0.4 0.6

True strain

0.8

Figure 79. Comparison ofCANMET's data (A36) with NIST's model prediction.

Agreement is improved after correction for machine compliance.

4000

3500 H

3000

2500 -

(D

^ 2000 ^

1 1500-1

1ooo^

500Open symbol: predicted values

Solid symbol: measured values

-11 1 \ 1 \ 1 \ \ \ \ \ T"

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pass number

Figure 80. Results of roll-separating-force calculations. The agreement between the

measurements and predictions is very good.

105

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