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Materials Science and Engineering A 528 (2011) 2843–2858

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

onstitutive description of a low C-Mn steel deformed underot-working conditions

.S. Puchi-Cabreraa,b,∗, Mariana H. Staiaa

School of Metallurgical Engineering and Materials Science, Faculty of Engineering, Universidad Central de Venezuela, Postal address 47885, Los Chaguaramos, Caracas, VenezuelaVenezuelan National Academy for Engineering and Habitat, Palacio de las Academias, Bolsa a San Francisco, Caracas 1010, Postal address 1723, Venezuela

r t i c l e i n f o

rticle history:eceived 10 November 2010eceived in revised form1 December 2010ccepted 13 December 2010vailable online 21 December 2010

eywords:onstitutive descriptionlastic deformation

a b s t r a c t

The present investigation has been conducted in order to develop a constitutive description for a lowC-Mn steel employed in structural applications, when this material is deformed under hot-working con-ditions. The set of equations that encompass the constitutive description of the steel are based on asimplified form of the Mechanical Threshold Stress Model. Accordingly, it has been shown that the flowstress of the material can be considered to arise from two different contributions: solid solution andprecipitation hardening, which do not evolve in the course of plastic deformation and work-hardening,the only evolving component of the flow stress. The evolution of work-hardening has been introducedin the model by means of the work-hardening law proposed by Estrin and Mecking, in differential form.Also, the temperature and strain rate dependence of the contributions of solid solution and precipita-

-Mn steelsot-working conditionsechanical Threshold Stress Model

tion hardening to the flow stress, as well as that of the saturation stress in the work-hardening law, aredescribed by means of the model earlier advanced by Kocks. The constitutive formulation encompassesseven material parameters which have been determined employing the experimental data correspond-ing to the strain interval where the material only exhibits positive work-hardening. It is able to predictaccurately both work-hardening and work-softening transients when changes in deformation conditionsoccur. Furthermore, it can be easily implemented in any advanced computational tool employed for the

roces

analysis of hot-working p

. Introduction

Industrial hot rolling operations constitute very complex pro-esses which involve the extensive plastic deformation of theaterial at temperatures usually greater than 0.6 TM and mean

train rates above 0.1 s−1. In the preceding, TM stands for the abso-ute melting temperature of the alloy. During the hot rolling of botherrous and non-ferrous alloys, the total deformation applied to the

aterial is imparted in a sequential manner, under conditions ofecaying temperatures and increasing strain rates. Such changes ineformation conditions give rise in turn to a complex microstruc-ural evolution which has a significant effect on the alloy strengthnd therefore, on the hot rolling parameters and final geometry ofhe rolled product.

In the past few years, important research efforts have been made

n order to improve the prediction of the changes in rolling load,orque and power consumption with deformation conditions, asell as the microstructural evolution of the material that takeslace throughout the schedule. This, however, is a very complex

∗ Corresponding author. Fax: +58 212 7539017.E-mail addresses: epuchi@reacciun.ve, epuchi@cantv.net (E.S. Puchi-Cabrera).

921-5093/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2010.12.042

ses of this material.© 2010 Elsevier B.V. All rights reserved.

task which requires the use of advanced computational tools sincethe intrinsic heterogeneities in the strain, strain rate and tempera-ture distributions throughout the cross section of the slab, as wellas the dynamic and static microstructural changes that take placeduring deformation and in the interpass periods under hot workingconditions, give rise to marked differences in microstructure andmechanical properties from the surface to the center of the materialunder processing.

Just to highlight some of the complexities involved in theindustrial hot rolling of C-Mn steels, Fig. 1 illustrates a defor-mation profile of a program typically applied to steels employedin structural applications. In this case, the slab of initial dimen-sions 250 × 1000 × 11,000 mm3, is reheated at 1250 ◦C and rolledwith a Steckel mill. The rolling operation takes about 180 s and isconducted in 14 passes. The first seven passes correspond to theroughing stage, which is carried out employing rolls of 525 mm indiameter. The last seven passes correspond to the finishing standand are conducted with rolls of 370 mm in diameter. As can be

seen, the effective strain applied to the material in each pass dur-ing the roughing stage increases from about 0.2 to approximately0.5. The strain applied continues to increase during the first twopasses of the finishing stand up to about 0.75 and then decreasesin the five remaining passes to approximately 0.16. In this way, the

2844 E.S. Puchi-Cabrera, M.H. Staia / Materials Science

Nomenclature

Greek symbols˛ angular coordinate along the rolling arc of contact,

rad˛0 bite angle, radε effective strainε̇ effective strain rate, s−1

ε̇′ plane strain rate, s−1

ε̇Ki material constant, s−1

ε̇KεS material constant, s−1

εP peak strainεss effective strain required to achieve 95% of the value

(�s − �0)�, �(T) temperature-dependent shear modulus, MPa�0 temperature-dependent shear modulus at 0 K, MPa� flow stress as a function of strain, strain rate and

temperature, MPa�0 flow stress at the beginning of plastic deformation,

MPa�a athermal stress, MPa�Comp. computed value of the flow stress, MPa�Exp. experimental value of the flow stress, MPa�S apparent saturation stress, MPa� ′

S true saturation stress, MPa�(T, ε̇) current flow stress, MPa�ε(T, ε̇) �ε, contribution of work-hardening to the current

flow stress, MPa�i(T, ε̇) contribution of solid solution and precipitation

hardening to the current flow stress, MPa�̂ Mechanical Threshold Stress, MPa�̂i contribution of solid solution and precipitation

hardening to the Mechanical Threshold Stress, MPa�̄ ′ mean plane flow stress in Sims theory, MPa�̄ ′

g mean plane flow stress for the computation ofrolling torques in Sims theory, MPa

� ′ plane flow stress, MPa�εS(T, ε̇) saturation stress, MPa�̂εS saturation stress at 0 K, MPa�y(T, ε̇) yield stress of the material, MPa�� work-softening due to dynamic recrystallization,

MPa�0 initial work-hardening rate, MPa� sum of squares, MPa2

˝ sum of squares, MPa2

Arabic symbolsA material constantb magnitude of the Burgers vector, me′ plane engineering straine′

fplane engineering strain corresponding to therolling pass

fP fraction of the peak strain at which dynamic recrys-tallization starts

g0i normalized activation energyg0εS normalized activation energyk Boltzmann constantk′ material constantm′ material constantN number of experimental stress-strain curvesNTot total number of experimental data pointsT absolute temperature, KTM absolute melting temperature, K

and Engineering A 528 (2011) 2843–2858

slab achieves a final thickness of approximately 1.5 mm, whereasthe final width only increases just to about 1010 mm.

On the other hand, Fig. 2 illustrates that the effective strain rateof the material increases progressively in each pass in a significantmanner, from approximately 3 s−1 at the beginning of the first passto about 200 s−1 at the end of the finishing stand. On the contrary,the mean temperature of the slab, as shown in Fig. 3, decreasesmoderately from approximately 1220 ◦C at the first pass and toabout 850 ◦C at the end of the rolling schedule.

These changes in deformation conditions also determine thechanges in rolling load, torque and power consumption of themill, which are presented in Figs. 4–6. As can be seen in Fig. 4,the rolling load exhibits a profile similar to that displayed by theeffective strain, in the sense that it increases progressively fromabout 1140 ton at the first pass to a maximum of approximately1720 ton at the ninth pass, decreasing to a minimum of 1020 tonat the end of the rolling program. On the contrary, the maximumvalues of the rolling torque, as shown in Fig. 5, are achieved duringthe first four passes of the roughing stand, attaining a magnitude ofup to 180 ton-m and then decreasing continuously to about 6 ton-m and the end of the schedule. On the other hand, the powerconsumption of the mill follows a similar trend to that of therolling load, as shown in Fig. 6. A maximum value of approximately13,700 kW is achieved at the ninth pass, which decreases to a mini-mum value of approximately 1790 kW at the last pass of the rollingprogram.

One of the essential aspects of any computational tool employedfor the prediction of the mechanical response of the material andits microstructural evolution during a processing operation of thisnature is the availability of a sound constitutive description of thealloy as close as possible to the actual deformation conditions. Asindicated by Abbod and co-workers [1–3], such a description shouldbe able to provide the value of the flow stress of the material interms of the deformation conditions (temperature and strain rate)as well as its microstructure given in terms of a number of internalstate variables which include dislocation density, grain structureand crystallographic texture, among others.

The literature devoted to the description of the constitutivebehavior of both ferrous and non-ferrous materials deformed underhot-working conditions is quite extensive and includes differentsound approaches able to provide the information required infinite element codes employed for modeling industrial hot-workingoperations, as that previously described. In some of these con-stitutive models, the flow stress of the material is given as anintegrated equation, as a function of the strain applied, strainrate and temperature [1–6]. In the case of steels deformed underhot-working conditions, this approach allows the description ofthe work-softening transient in the stress–strain curve, associatedwith the occurrence of dynamic recrystallization. A disadvantageof these models is that it is not possible to update the changes indeformation temperature and strain rate that occur in the courseof plastic deformation.

Other approaches employed for the constitutive description ofmetals and alloys make use of physically based models whichseparate the contribution of athermal and thermal barriers to dis-location motion. Some of these formulations also provide the flowstress of the material as an integrated equation [7,8], whereas otherapproaches consider the evolution of the flow stress in the courseof plastic deformation in differential form [9–21]. Consequently, inthe latter case it is possible to update changes in deformation condi-tions throughout the deformation process. Among the latter group

of models, one of the most widely used is the so called MechanicalThreshold Stress (MTS) Model. This methodology for describing theconstitutive behavior of a material deformed in a wide range of tem-peratures and strain rates was developed at Los Alamos NationalLaboratory and it is based on one internal state variable called the

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2845

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Eff

ecti

ve S

trai

n

Low Carbon Steel

Finishing

R = 370 mm

Roughing

R = 525 mm

ing pr

M[

tmifMuts

eiao

Fig. 1. Typical deformation profile of an industrial hot roll

echanical Threshold Stress or flow stress at a temperature of 0 K11–16].

The MTS model has been employed quite successfully forhe description of a wide range of materials, including pure

etals, ferrous and non-ferrous alloys [11–21]. The differencesn the approach employed in each case arise mainly from theorm of the evolution function that describes the change of the

TS with strain and that of the extrapolation functions that aresed to compute the current flow stress from the MTS, whereashe main concepts which provide its robustness remains theame.

In spite of its sound physical basis, the MTS model is not

mployed extensively in finite element simulation due in part tots apparent complexity regarding its instrumentation within thevailable computational tools. On the contrary, as has been pointedut by Abbod et al. [3], most finite element codes employed for

1

10

100

1000

7654321

Pass

Eff

ecti

ve S

trai

n R

ate,

s-1

Low Carbon Steel

Roughing

Fig. 2. Changes in the effective strain rat

Pass Number

ogram of C-Mn steels employed in structural applications.

modeling of hot rolling processes assume the validity of mechani-cal equations of state and therefore, under changing conditions oftemperature and strain rate, the flow stress of the material at anyapplied strain is assumed to be a function only of the instantaneousvalue of these two variables.

Thus, the present investigation has been conducted in orderto employ a simplified form of the MTS model for the descrip-tion of the constitutive behavior of a C-Mn steel deformed underhot working conditions, specifically in the strain interval whereit exhibits positive work-hardening. Such a simplified formulationkeeps the fundamental aspects of the original model and therefore,it constitutes a robust description of the change in flow stress with

deformation temperature, rate of straining and microstructure,with the additional advantage that it can be easily implementedin such finite element codes for modeling hot rolling operations,such as those described earlier.

98 10 11 12 13 14

Number

Finishing

e throughout the rolling schedule.

2846 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

500

600

700

800

900

1000

1100

1200

1300

1400

1500

7

Pa

Tem

per

atu

re, °

C

Finishing

Low Carbon Steel

Roughing

tempe

2

tttwa

ttmpsrao

1 65432

Fig. 3. Changes in the slab deformation

. Essential aspects of the model employed

The fundamental basis of the MTS model is the considerationhat plastic deformation in a polycrystalline aggregate occurs byhe accumulation and motion of dislocations and that the rate con-rolling deformation mechanism is the interaction of dislocationsith defects, such as grain boundaries, forest dislocations, solute

toms, etc. [11–16].In the present case, the formulation of the constitutive descrip-

ion of the steel was carried out employing the concepts behindhe MTS model. It has been considered that the flow stress of the

aterial arises basically from three different microstructural com-

onents: (1) athermal barriers, (2) solid solution and precipitationtrengthening, which are able to modify the yield stress of the mate-ial, but which do not evolve in the course of plastic deformationnd (3) work-hardening (dislocation–dislocation interactions), thenly component that evolves in the course of plastic deformation.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

1 765432

Pass

Ro

llin

g L

oad

, to

n

Low Carbon Steel

Roughing

Fig. 4. Changes in rolling load thro

98 10 11 12 13 14

ss Number

rature throughout the rolling schedule.

Therefore, the flow stress during plastic deformation at any tem-perature (T) and strain rate (ε̇), that is to say, the current flow stress,can be expressed as [11–16]:

�(T, ε̇) = �a + �

�0�i(T, ε̇) + �

�0�ε(T, ε̇) (1)

where � represents the temperature-dependent shear modulus ofthe material, which for plain carbon steels can be expressed as [22]:

�(T) MPa = 88, 885 − 37.3T (2)

In Eq. (1), �a represents the athermal component of the flowstress, which characterizes the rate-independent interaction of dis-

locations with long-range barriers (e.g., grain boundaries), �i (T, ε̇)the contribution of solid solution and precipitation hardening to thecurrent flow stress, �0 the value of � at 0 K (�0 = 88,885 MPa) and�ε (T, ε̇) the contribution of work-hardening to the current flowstress. In the present work, �i (T, ε̇) has been expressed as func-

98 10 11 12 13 14

Number

Finishing

ughout the rolling schedule.

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2847

0

20

40

60

80

100

120

140

160

180

200

87654321 109 11 12 13 14Pass Number

Ro

llin

g T

orq

ue,

to

n-m

Low Carbon SteelRoughing

Finishing

ue thr

ta

�

wtkvc

l�d

Fig. 5. Changes in rolling torq

ion of temperature and strain rate by means of the model earlierdvanced by Kocks [23]:

i(T, ε̇) = �̂i

(ε̇

ε̇Ki

)kT/�b3g0i

(3)

here �̂i represents the contribution of solid solution and precipi-ation hardening to the flow stress at 0 K, that is to say, to the MTS,is the Boltzmann constant, g0i an experimental normalized acti-ation energy, b the Burgers vector (0.286 nm) and, ε̇Ki a material

onstant.

As indicated above, the component �i does not exhibit any evo-ution in the course of plastic deformation, whereas the componentε does evolve. In the present work, such an evolution has beenescribed in differential form by means of the work-hardening law

0

2000

4000

6000

8000

10000

12000

14000

16000

7654321Pas

Ro

llin

g P

ow

er, k

W

Low Carbon Steel

Roughing

Fig. 6. Changes in rolling power thr

oughout the rolling schedule.

advanced by Estrin and Mecking [24], according to which:

d�ε

dε= �2

0�ε

[1 −

(�ε

�εS(T, ε̇)

)2]

(4)

where

�0 = �

A(5)

A represents a material constant and �εs (T, ε̇) is the saturationstress, achieved at large strains when work-hardening is balancedby dynamic recovery. This work-hardening law has been derived by

specifying the evolution of the dislocation density, employing theconcept that hardening is controlled by the competition of storageand annihilation or rearrangement of dislocations, processes thatare considered to be additive. Also, it is assumed, that the storageof dislocations is determined by the spacing between impenetrable

8 109 11 12 13 14s Number

Finishing

oughout the rolling schedule.

2848 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

0

50

100

150

200

250

300

350

400

0.20 0.4 0.6 0.8 1 1.2 1.4 1.6

ffect

Eff

ecti

ve S

tres

s, M

Pa 850°C - 25 s-1

950°C - 2.5 s-1

1000°C - 2.5 s-1

1050°C - 0.25 s-1

C-Mn Steel

obtain

omgi

oo

�

wma

poRvcesata

3

ns0mtrrsaPa

E

Fig. 7. Typical effective stress–strain curves

bstacles or dislocation sinks and, therefore, that the dislocationean free path is a constant prescribed either by particle spacing or

rain size. Additionally, it is considered that the annihilation terms proportional to the dislocation density.

The saturation stress, �εs(T, ε̇) can also be expressed as a functionf deformation temperature and strain rate by means of an equationf the same form as Eq. (3):

εS(T, ε̇) = �̂εS

(ε̇

ε̇KεS

)kT/�b3g0εS

(6)

here �̂εS represents the saturation stress at 0 K, that is to say, theechanical threshold saturation stress, ε̇KεS a material parameter

nd g0εS an experimental normalized activation energy.Therefore, the component �ε is computed recursively from its

revious value, at each temperature and strain rate, by meansf the numerical integration of Eq. (4) employing a fourth orderunge–Kutta method. The integration procedure is started from thealue of the constant �a (Eq. (1)). Thus, Eqs. (1)–(6) comprise theonstitutive description of the steel under investigation. It involvesight parameters to be determined from the experimental strain,tress, temperature and strain rate data, namely: �a (Eq. (1)), �̂i, ε̇Ki

nd g0i (Eq. (3)), A (Eq. (5)), �̂εS , ε̇KεS and g0εS (Eq. (6)). These parame-ers can be determined simultaneously from the experimental data,s explained in the forthcoming.

. Experimental techniques

The present investigation has been carried out employing aumber of effective stress–effective strain curves obtained fromamples of a C-Mn steel of the following composition (wt.%):.11 C, 1.11 Mn, 0.26 Si, 0.03 P, 0.002 S and bal. Fe. The speci-ens were deformed under plane strain compression conditions in

he temperature range 850–1150 ◦C, at nominally constant strainates of approximately 0.25, 2.5 and 25 s−1. Effective strains in the

ange of 1.5–2.0 were achieved in each test. Prior to testing, theamples were lubricated with Dag 2626 (Acheson Colloids Ltd.)nd the deformation temperature was measured by means of ayrotenax chromel–alumel thermocouple with the bead locatedt mid-length, mid-breadth and mid-thickness. These experiments

ive Strain

ed under different deformation conditions.

were conducted at the Department of Engineering Materials of theUniversity of Sheffield, U.K. and details of the plane strain compres-sion testing equipment and testing techniques have been reportedelsewhere [25,26].

4. Experimental results

Fig. 7 illustrates a number of representative stress–strain curvesobtained after deformation at different nominally constant tem-peratures and strain rates. As can be observed, an appropriatelubrication of the specimens prior to testing allowed the achieve-ment of effective strains in the range of 1.4. This, in turn, allowedthe identification of the strain interval where the main dominantmicrostructural mechanisms, which take place during deforma-tion under hot working conditions of this material, occur. At thebeginning of the deformation up to the maximum flow stress (alsoknown as peak stress), indicated by the vertical broken lines drawnon each curve, strain hardening prevails over dynamic recovery,which leads to a steady increase in flow stress and a decrease in thework-hardening rate as deformation proceeds.

At strains close to the peak stress, before its attainment, dynamicrecrystallization takes place, which gives rise to a significant work-softening of the material. This phenomenon leads to a decreasein flow stress, until a balance between dislocation annihilationand dislocation generation by work-hardening is accomplished,which gives rise to the achievement of a true steady-state con-dition. Under these circumstances, the flow stress would remainconstant as deformation continues to be applied. The above fea-tures are more clearly observed when deformation takes place atrelatively elevated temperatures and low strain rates, that is tosay, temperatures above 950 ◦C and strain rates less than 2.5 s−1.As deformation temperature decreases and strain rate increases,the strain interval where work-hardening prevails is significantlylarger than for other deformation conditions, as shown in the curve

corresponding to 850 ◦C and a strain rate of 25 s−1.

During plane strain compression testing, the strain rate is main-tained relatively constant throughout the tests, particularly afterthe initial stages of the deformation process, as shown in Fig. 8.Here, it can be observed that, particularly in the range of strain

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2849

0.1

1

10

100

0.20 0.4 0.6 0.8 1 1.2 1.4 1.6

ffect

Eff

ecti

ve S

trai

n r

ate,

s-1

850°C

950°C

1000°C

1050°C

C-Mn Steels

durin

rt00

mciimdri

esmdwi

Ft

E

Fig. 8. Changes in effective strain rate

ates between 2.5 and 25 s−1, such a parameter is more readily con-rolled, whereas during tests at lower strain rates, in the range of.25 s−1, a steady increase from values of approximately 0.14 up to.9 s−1 are observed in the course of deformation.

On the other hand, Fig. 9 shows the evolution of the specimenean temperature during testing, as a function of the deformation

onditions. When deformation occurs at elevated temperatures,n the range of 1050 ◦C and nominal strain rates of 0.25 s−1, thencrease observed in the temperature of the specimen due to defor-

ational heating is just of approximately 3 ◦C. However, as theeformation temperature of the specimen decreases and the strainate increases, e.g., 850 ◦C and 25 s−1, the increase in temperatures observed to achieve approximately 35 ◦C.

The changes in deformation temperature and strain rate that arexperimented in the course of plane strain compression testing, ashown in Figs. 8 and 9, represent intrinsic features of the testing

ethod which should be taken into consideration regarding the

erivation of a constitutive description of the material. Thus, theork-hardening law employed for the description of the change

n flow stress in the course of deformation must necessarily be

800

850

900

950

1000

1050

1100

0.20 0.4 0.6 0.8 1 1.2 1.4 1.6

Effective Strain

Def

orm

atio

n T

emp

erat

ure

, K

C-Mn Steel

ig. 9. Changes in deformation temperature during plane the strain compressionsests.

ive Strain

g the plane strain compression tests.

expressed in differential form and the flow stress computed in arecursive manner, from its previous value as indicated in Section2. Only in this way, it would be possible to update the values ofdeformation temperature and strain rate at each step of the calcu-lation, prior to the computation of the new magnitude of the flowstress, for the corresponding strain interval. If the flow stress ofthe material were expressed in the form of an integrated equationas a function of the strain applied, deformation temperature andstrain rate, a prior temperature and strain rate iterative correctionprocedure of the experimental flow stress data would be requiredbefore determining the actual value of the parameters involved inthe constitutive equation.

As indicated above, the purpose of the present investigationis the development of a constitutive description of the materialspecifically in the strain interval in which it exhibits positive work-hardening, which is the relevant part of the stress–strain curveas far as the computation of rolling loads, torques and powerconsumption is concerned. Thus, Figs. 10–15 illustrate the set ofexperimental stress–strain curves that were employed in the anal-ysis, which encompass nominal deformation temperatures in therange of 850–1100 ◦C, effective strain rates nominally between0.25 and 25 s−1 and effective strains in the range of approximately0.18–0.70, depending on the deformation conditions.

Such an experimental framework partially encompasses theconditions found in the industrial hot rolling schedule described inFigs. 1–3. Particularly, in relation to the effective strain rate appliedto the material, it can be observed that in the industrial practice,such a parameter achieves much higher values than those attainedat experimental level.

The first step in the development of the constitutive descriptionof the material is the determination of a first approximation of thecontributions to the current flow stress of both the athermal com-ponent, �a, and that arising from solid solution and precipitationhardening, �i(T, ε̇). For this purpose, the values of the yield stressof the material, �y(T, ε̇), obtained from the stress–strain curves at

different temperatures and strain rates, were fitted to the equation:

�y(T, ε̇) = �a + �̂i

(ε̇

ε̇Ki

)kT/�b3g0i

(7)

2850 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

0

50

100

150

200

250

300

350

0

Eff

Eff

ecti

ve S

tres

s, M

Pa

0.25 s-1

2.50 s-1

25 s-1

T = 850°C

C-Mn Steel

EXPERIMENTAL

COMPUTED

curve

p

˝

at

g�voo

0.10 0.2 0.3

Fig. 10. Experimental and computed stress–strain

Thus, the value of the constants �a, �̂i, ε̇Ki, and g0i were com-uted simultaneously by defining the sum of squares, ˝, as:

=N∑

j=1

{�y(Tj, ε̇j) − �a − �̂i

(ε̇j

ε̇Ki

)kTj/�jb3g0i

}2

(8)

nd solving the system of equations that results from the minimiza-ion condition:

∂˝

∂�a= 0;

∂˝

∂�̂i= 0;

∂˝

∂ε̇Ki= 0;

∂˝

∂g0i= 0 (9)

The later step was accomplished by means of the conjugate

radient method. The results of the above analysis indicates thata ∼= 0, �̂i = 7.66 MPa, ε̇Ki = 2.94 × 104 s−1 and g0i = 0.16. Suchalues were employed subsequently for the global computationf all the parameters that encompass the constitutive descriptionf the material. However, accordingly these first results, it would

0

50

100

150

200

250

300

0.10 0.2 0.3

E

Eff

ecti

ve S

tres

s, M

Pa

0.25 s-1

2.50 s-1

C-Mn Steel

Fig. 11. Experimental and computed stress–strain curve

.4 0.5 0.6 0.7 0.8

ective Strain

s determined at 850 ◦C and different strain rates.

seem that the contribution to the current flow stress which arisesfrom long-range barriers to dislocation motion were largely super-seded by the contribution provided by the short-range barriersrepresented by solid solution and precipitation hardening.

Fig. 16 illustrates the comparison between the experimental val-ues of the yield stress and those provided by the second term of Eq.(7), which is observed to provide a satisfactory description of theexperimental data. The fact that the contribution of long-range bar-riers to dislocation motion is apparently negligible in comparisonwith the contribution of solid solution and precipitation strength-ening would lead to a simplification of the constitutive descriptionof the material, which allows the expression of the current flow

stress simply as:

�(T, ε̇) = �

�0

{�̂i

(ε̇

ε̇Ki

)kT/�b3g0i

+ �ε(T, ε̇)

}(10)

0.4 0.5 0.6 0.7 0.8

ffective Strain

25 s-1

T = 900°C

EXPERIMENTAL

COMPUTED

s determined at 900 ◦C and different strain rates.

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2851

0

50

100

150

200

250

300

0.050 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

E

Eff

ecti

ve S

tres

s, M

Pa

2.5 s-1

T = 950°C

25 s-1

0.25 s-1

C-Mn Steel

EXPERIMENTAL

COMPUTED

curve

dittbtti

o

�

Fig. 12. Experimental and computed stress–strain

As indicated above, the second term of the above equation isetermined from the numerical integration of Eq. (4), which also

ncludes Eqs. (5) and (6). Thus, the complete set of parametershat allow the constitutive description of the material are reducedo: �̂i, ε̇Ki, and g0i, for which an initial approximation has alreadyeen determined, together with constants A, �̂KεS , ε̇KεS , and g0εS. Allhese parameters should be computed simultaneously, employinghe stress, strain rate and temperature data recorded in each strainnterval during every plane strain compression test.

The computation of the above constants requires the definition

f a second sum of squares function, of the form:

=NTot.∑k=1

{�Comp.

k(Tk, ε̇k, �̂i, ε̇Ki, g0i, A, �̂KεS, ε̇KεS, g0εS) − �Exp.

k

}2(11)

0

50

100

150

200

250

0.050 0.1 0.15 0.2

Effectiv

Eff

ecti

ve S

tres

s, M

Pa

0.25 s-1

C-Mn Steel

Fig. 13. Experimental and computed stress–strain curve

ffective Strain

s determined at 950 ◦C and different strain rates.

In this expression, NTot. represents the total number of experi-mental data points, whereas �Comp. and �Exp. refer to the computedand experimental values of the current flow stress. As indicatedabove, the determination of the different parameters involved inthe above relationship is carried out by solving simultaneously theset equations that results from the minimization of function � :

∂˝

∂�̂i= 0;

∂˝

∂ε̇Ki= 0;

∂˝

∂g0i= 0;

∂˝

∂A= 0

∂˝

∂�̂KεS= 0;

∂˝

∂ε̇KεS= 0;

∂˝

∂g0εS= 0 (12)

Again, this can be accomplished numerically, by means of theconjugate gradient method. Table 1 summarizes the values of the

0.25 0.3 0.35 0.4 0.45

e Strain

2.50 s-1

T = 1000°C

25 s-1

EXPERIMENTAL

COMPUTED

s determined at 1000 ◦C and different strain rates.

2852 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

0

50

100

150

200

250

0.050 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Effective Strain

Eff

ecti

ve S

tres

s, M

Pa

2.50 s-1

T = 1050°C

0.25 s-1

25 s-1

C-Mn Steel

EXPERIMENTAL

COMPUTED

Fig. 14. Experimental and computed stress–strain curves determined at 1050 ◦C and different strain rates.

0

20

40

60

80

100

120

140

160

180

00 .050 .1 0.15 0.20 .250 .3 0.35 0.4

fectiv

Eff

ecti

ve S

tres

s, M

Pa

T = 1100°C

2.50 s-1

25 s-1C- Mn Steel

EXPERIM ENTA L

CO MP UTE D

curve

pm

tsu

TP

Ef

Fig. 15. Experimental and computed stress–strain

arameters that encompass the constitutive description of the

aterial.Figs. 10–15 also illustrate the computed stress–strain curves

hat were obtained for each nominal deformation temperature andtrain rate employing Eqs. (4), (6) and (10), together with the val-es presented in Table 1. In general, it can be observed that the

able 1arameters involved in the constitutive description of the investigated alloy.

�̂i , MPa ε̇Ki , s−1 g0i A, MPa �̂KεS , MPa ε̇KεS , s−1 g0εS

389 2.16 × 103 0.104 77 1278 1.39 × 105 0.106

e Strain

s determined at 1100 ◦C and different strain rates.

predicted stress–strain curves compare quite satisfactorily with theexperimental ones, both in terms of the magnitude of the flow stressand work-hardening rate of the material. This result can be furtherconfirmed in Figs. 17 and 18, which show both the comparison ofthe experimental and computed values of the flow stress, as wellas the change in the magnitude of the relative error between suchvalues, as a function of the experimental flow stress.

The comparison between the experimental and computedvalues of the flow stress indicates that for some deformation condi-tions, particularly at relatively low deformation temperatures andelevated strain rates, the largest differences occur at the first strainintervals and effective strain values less than approximately 0.005.

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2853

50

100

150

200

250

50 100 150 200 250

Exp

erim

enta

l Y

ield

Str

ess,

MP

a

850°C

900°C

950°C

1000°C

1050°C

1100°C

C-Mn Steel

r2 = 0.97

kT

iσε

ε

µb3g0i

ental

Afafe

p

Fig. 16. Comparison between the experim

lso, relatively large deviations, in the range of 10–20% can be foundor some of the data obtained at 1273 K at 2.5 s−1, 1323 K at 0.25 s−1

nd 25 s−1, as well as 1373 K at 2.5 s−1. However, as shown in Fig. 18,

or the vast majority of the experimental data points the relativerror is within the range of ±10% or less.

Therefore, it can concluded that the set of equations that encom-ass the constitutive description of the material, together with the

0

50

100

150

200

250

300

350

500 100 150

Experimental F

Co

mp

ute

d F

low

Str

ess,

MP

a

850°C

900°C

950°C

1000°C

1050°C

1100°C

r2 = 0.986

Fig. 17. Comparison between the experimenta

Ki

and computed values of the yield stress.

appropriate parameters, provide a satisfactory prediction of theflow stress during tests conducted under nominal constant tem-peratures and strain rates. However, most hot working processes

are carried out under conditions in which deformation temperatureand strain rate undergo significant changes. Therefore, the capabil-ity of the present constitutive description for predicting flow stresstransients, which represent the microstructural evolution of the

200 250 300 350

low Stress, MPa

C-Mn Steel

l and computed values of the flow stress.

2854 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

-20

-15

-10

-5

0

5

10

15

20

05 0 100 150 200 250 300 350

Flow

% R

elat

ive

Err

or

850°C

900°C

950°C

1000°C

1050°C

1100°C

C- Mn Steel

d and

mf

5

idiorr

Experimental

Fig. 18. Variation of the % of relative error between the compute

aterial when changes in deformation conditions occur, has to beurther analyzed.

. Discussion

One of the main limitations of the present constitutive analysiss the fact that it has been developed employing the stress–strain

ata where the material exhibits only positive and decreas-

ng work-hardening. Therefore, it is assumed that in the coursef deformation work-hardening is counterbalanced by dynamicecovery which would eventually lead to the achievement of a satu-ation stress at large strains. Fig. 6 illustrates that the true behavior

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4

Effectiv

Eff

ecti

ve s

tres

s, M

Pa

C-Mn Steels

Fig. 19. Hypothetical instantaneous change in deformation conditions from 850 ◦

Stress, MPa

experimental values of the flow stress, as a function of the latter.

of the material is somewhat more complex due to the presence ofa work-softening transient after the attainment of the peak stress,which has been associated with the occurrence of dynamic recrys-tallization.

The decrease in flow stress, ��, at strains greater than that asso-ciated with the peak stress, has been commonly described by asoftening term of the form [2,3]:

�� = (�S − � ′S)

{1 − exp

[−k′

(ε − fPεP

εss − εP

)m′]}(13)

0.5 0.6 0.7 0.8 0.9

e strain

850°C - 25 s-1

1100°C - 0.25 s-1

C and 25 s−1, to 1100 ◦C and 0.25 s−1. The change occurred at a strain of 0.2.

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2855

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

fecti

Eff

ecti

ve s

tres

s, M

Pa

850°C - 25 s-1

1100°C - 0.25 s-1

C-Mn Steels

850 ◦

wbuessiaaAs

Ef

Fig. 20. Hypothetical instantaneous change in deformation conditions from

In the above equation, �S represents the saturation stress thatould be hypothetically achieved as a consequence of the balance

etween work-hardening and dynamic recovery, � ′S the actual sat-

ration stress achieved after the work-softening transient, εP, theffective strain associated with the peak stress, also known as peaktrain, fP the fraction of the εP at which dynamic recrystallizationtarts and εss the strain at which (� − �0) = 0.95 (�S − �0) undersothermal conditions. In this case, � represents the flow stress

t any strain, temperature and strain rate and �0 the flow stresst the beginning of plastic deformation. k′ and m′ represent thevrami constants in the kinetic equation. The amount of work-oftening given by Eq. (13) is considered to be zero in the region

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4

Effecti

Eff

ecti

ve s

tres

s, M

Pa

C-Mn Steels

Fig. 21. Hypothetical instantaneous change in deformation conditions from 850 ◦

ve strain

C and 25 s−1, to 1100 ◦C and 0.25 s−1. The change occurred at a strain of 0.4.

of positive work-hardening, whereas at strains greater than εP itis subtracted from the value of the flow stress. However, such anapproach requires that the flow stress is computed from an inte-grated form of the stress-strain equation that describes the firstpart of the curve. In the present work, the region of positive work-hardening is described in differential form and therefore the aboveprocedure cannot be employed directly.

According to Figs. 1, 3 and 7 the effective strain applied in

most passes of the rolling schedule, without taking into consid-eration possible effects of strain accumulation in between passes,are greater than the corresponding peak strains of the stress–straincurves determined in the temperature range of 950–1100 ◦C. This

0.5 0.6 0.7 0.8 0.9

ve strain

850°C - 25 s-1

1100°C - 0.25 s -1

C and 25 s−1, to 1100 ◦C and 0.25 s−1. The change occurred at a strain of 0.6.

2856 E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

fecti

Eff

ecti

ve s

tres

s, M

Pa

850°C - 25 s-1

1100°C - 0.25 s-1

C-Mn Steels

1100

fpstc

aptt

�

Ef

Fig. 22. Hypothetical instantaneous change in deformation conditions from

act somehow highlights the limitation of considering only the firstart of the stress–strain curve for which the occurrence of work-oftening is not taken into account. However, in most hot-rollingheories, rolling loads and torques are determined on the basis ofomputing the mean flow stresses of the material.

For example, in the analytical theory earlier advanced by Sims,s described by Richardson et al. [27] for the calculation of sucharameters, the mean plane flow stress employed in the computa-

ion of rolling loads is determined over the arc of contact betweenhe rolls and the material, as:

¯ ′(�̂, ε̇′, T) = 1˛0

∫ ˛0

0

� ′(�̂, ε̇′, T)d˛ (14)

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4

Effecti

Eff

ecti

ve s

tres

s, M

Pa

C-Mn Steels

Fig. 23. Hypothetical instantaneous change in deformation conditions from 1100

ve strain

◦C and 0.25 s−1, to 850 ◦C and 25 s−1. The change occurred at a strain of 0.2.

where ˛0 represents the bite angle. On the other hand, for the com-putation of the rolling torque the mean flow stress is calculated onthe basis of the engineering plane strain applied to the materialduring the rolling pass, as:

�̄ ′g(�̂, ε̇′, T) = 1

e′f

∫ e′f

0

� ′(�̂, ε̇′, T)de (15)

In the above equations, the term �̂ represents the MTS and it

has been introduced instead of the effective strain, ε, in order toemphasize the microstructural dependence of the material flowstress rather than its strain dependence.

Thus, although the present constitutive description overesti-mates the flow stress of the material by assuming that dynamic

0.5 0.6 0.7 0.8 0.9

ve strain

850°C - 25 s-1

1100°C - 0.25 s-1

◦C and 0.25 s−1, to 850 ◦C and 25 s−1. The change occurred at a strain of 0.4.

E.S. Puchi-Cabrera, M.H. Staia / Materials Science and Engineering A 528 (2011) 2843–2858 2857

0

50

100

150

200

250

300

350

400

0.10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Effective strain

Eff

ectiv

e st

ress

, MP

a

850°C - 25 s -1

1100°C - 0.25 s -1

C-Mn Steels

1100

raitsasrcws

tdtdssi2spo

sdrnpctdaTdsag

Fig. 24. Hypothetical instantaneous change in deformation conditions from

ecovery is the only operative microstructural restoration mech-nism, such overestimation is not expected to have a significantnfluence on the computation of the rolling parameters due mainlyo two reasons. Firstly, the fact that average values of the flowtress are employed in the calculation. Secondly, the temper-ture and strain rate dependence of both the peak strain andtress, which leads to an increase in these parameters at lowerolling temperatures and higher strain rates. These deformationonditions are typical of the last passes of the rolling schedule,hich limits the strain that can be applied to the material, as

hown in Fig. 1.The second aspect of the proposed constitutive description

hat should be discussed is that related to its capability for pre-icting strain transients when changes in deformation conditionsake place. Figs. 19–24 illustrate several hypothetical changes ineformation conditions which clearly show that the proposed con-titutive description is able to provide a satisfactory prediction ofuch transients. In Figs. 19–21, it can be observed that by chang-ng instantaneously the deformation conditions, from 850 ◦C and5 s−1 to 1100 ◦C and 0.25 s−1, an instantaneous drop in the flowtress value in the range of approximately 123–132 MPa takeslace, depending on the strain at which the change in conditionsccurs.

Such a drop is basically associated with the contribution of solidolution and precipitation hardening to the current flow stress andepends mainly on the final deformation temperature and strainate achieved during the change. The results indicate that the mag-itude of the flow stress drop tends to increase slightly as the strainrior to the change in deformation conditions increases. On theontrary, the hypothetical changes shown in Figs. 22–24 indicatehat the instantaneous increase in flow stress that takes place wheneformation conditions change from 1100 ◦C and 0.25 s−1 to 850 ◦Cnd 25 s−1, remains constant at a value of approximately 98 MPa.

hese results can be readily explained by the fact that under con-itions of 1100 ◦C and 0.25 s−1, the stress–strain curve achievesaturation at effective strains of less than 0.1, whereas at 850 ◦Cnd 25 s−1, the curve exhibits positive work-hardening up to strainsreater than approximately 0.8.

◦C and 0.25 s−1, to 850 ◦C and 25 s−1. The change occurred at a strain of 0.6.

Regarding the magnitude of the strain transients, the resultsalso agree with the expected behavior. When changes in defor-mation conditions occur from low deformation temperatures andelevated strain rates to higher temperatures and lower strain rates,strain transients are relatively short, in the range of approximately0.14–0.15. Once the transient is completed, the flow stress attainsthe saturation stress value corresponding to the lower stress–straincurve, as shown in Figs. 19–21.

On the contrary, when changes in deformation conditions occurin the opposite sense, the strain transients are observed to be muchgreater than in the previous case, as shown in Figs. 22–24. In fact,in these hypothetical examples, after the change, the flow stressdoes not achieve the saturation stress corresponding to the uppercurve, even after effective strain greater than 0.6. These resultscan be readily explained on the basis of the pronounced work-hardening behavior observed in the upper curve at the beginning ofdeformation, which are reflected after the change in deformationconditions.

At the present time, more analytical work is being conductedin order to incorporate in a rational manner the work softeningtransient, associated with dynamic recrystallization, into the dif-ferential form of the hardening law. In this way, the flow stresscould still be computed recursively from its previous value, as in thepresent case, but without the need of assuming that the dynamicrecovery constitutes the only operative restoration mechanism ofthe material.

6. Conclusions

The constitutive description of a C-Mn steel deformed underhot working conditions has been carried out on the basis of asimplified form of the Mechanical Threshold Stress (MTS) Model.The evolution of the flow stress during plastic deformation due

to work-hardening has been introduced in the model by meansof the work-hardening law proposed by Estrin and Mecking. Thetemperature and strain rate dependence of the contributions ofsolid solution and precipitation hardening to the flow stress, aswell as that of the saturation stress in the work-hardening law, are

2 cience

dpitsbtaowptoeticttadct

A

cop

R

[[[[

[

[

[

[

[[

[

[

[[[[

[van der Winden, “Measuring Flow Stress in Hot Plane Strain Compres-

858 E.S. Puchi-Cabrera, M.H. Staia / Materials S

escribed by means of the model earlier advanced by Kocks. Thearameters involved in the model have been determined employ-

ng experimental data corresponding to the strain interval wherehe material only exhibits positive work-hardening. It has beenhown that, in the present case, the flow stress of the material cane considered to arise essentially from two different microstruc-ural components, which include the contribution of solid solutionnd precipitation strengthening, which do not evolve in the coursef plastic deformation and, on the other hand, work-hardening,hich on the contrary evolves during such a phenomenon. The pro-osed constitutive equation is able to predict an accurate value ofhe flow stress and work-hardening rate of the material in termsf deformation temperature, rate of straining and microstructure,mploying only seven material parameters that are computed fromhe experimental data. Moreover, the formulation of the harden-ng law in differential form allows the updating of the deformationonditions prior to its numerical integration in order to determinehe corresponding value of the flow stress. Therefore, the constitu-ive formulation is able to predict accurately both work-hardeningnd work-softening transients when changes in deformation con-itions occur. The constitutive equation that has been developedan be easily implemented in finite elements codes employed forhe analysis of hot-working processes of this material.

cknowledgements

The present investigation has been carried out with the finan-ial support of the Scientific and Humanistic Development Councilf the Universidad Central de Venezuela (CDCH-UCV) through theroject No. PG 08-7775-2009.

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