Materials Science and Engineering A 423 (2006) 253–261

Recrystallisation driving forces against pinning forces inhot rolling of Ti-microalloyed steels

M.I. Vega a, S.F. Medina b,∗, A. Quispe c, M. Gomez b, P.P. Gomez b

a FCC Construction S.A., Research & Development Department, c/Acanto 22, 28045 Madrid, Spainb National Centre for Metallurgical Research (CENIM-CSIC), Av. Gregorio del Amo 8, 28040 Madrid, Spain

c National University Jorge Basadre, University City, Av. Miraflores s/n, Tacna, Peru

Received 1 May 2005; received in revised form 1 November 2005; accepted 1 February 2006

Abstract

In this work the pinning forces exerted by TiN particles in the austenitic phase in four Ti-microalloyed steels have been determined and comparedwith the driving forces for recrystallisation determined in each rolling pass. The thermomechanical simulation has been carried out in the laboratoryby means of torsion tests in a sequence of 20 passes with the final passes in a mixed austenitic/ferritic phase. The driving forces were found tobe approximately two orders of magnitude higher than the pinning forces, which explains why the austenite in these steels barely experiencesh©

K

1

avodtaio

ttartaimp

0d

ardening during rolling and why the accumulated stress prior to the austenite → ferrite transformation is insufficient to refine the ferritic grain.2006 Elsevier B.V. All rights reserved.

eywords: Ti/N ratio; Recrystallisation; Precipitation; Driving and pinning forces; Hot rolling simulation

. Introduction

During hot deformation, the behaviour of steels with titaniums the only microalloying element forming nitrides or carbides isery different to that of other microalloyed steels with niobiumr vanadium. The reason for this is that while niobium or vana-ium precipitates are dissolved at the reheating temperatures,itanium nitrides (TiN) have very little solubility in austenitend their complete dissolution is not possible unless the reheat-ng temperature is very high, close to the melting point of steelr even at temperatures corresponding to the liquid state [1–5].

This characteristic makes Ti and N ideal elements for con-rolling the austenite grain size in heating processes, such ashe heat-affected zone in welding [6,7]. For this reason, theirddition is indispensable in structural steels with weldabilityequirements. Besides possessing good weldability characteris-ics, structural steels must also present good levels of strengthnd toughness in parent plates. In order to achieve this objectivet is necessary to obtain a fine microstructure, which is the only

echanism that contributes to increasing these two mechanicalroperties.

The results reported in the literature agree that a Ti/N ratioof between 1 and 3, less than the stoichiometric ratio (3.42),provides good results in austenite grain size control, especiallywhen precipitation takes place in the solid state, since fine pre-cipitates are obtained [8–10].

The role of TiN particles in austenite grain size control iswell known and the mechanism by which they exert this influ-ence may be explained by Zener and Gladman’s expressions forpinning forces [11,12], which relate the precipitate size, precipi-tated fraction and austenite grain size. When the grain boundaryintersects a particle such as a titanium nitride, pinning is exertedand unpinning requires the supply of additional energy by thedriving force for grain boundary movement. If the pinning forcesare greater than the driving forces for recrystallisation, the pro-cess is halted.

Previous studies [13,14] have shown that the rolling of low-alloy steels containing Ti as a microalloying element barelyproduces any hardening of the austenite. However, growth ofthe austenitic grain at the reheating temperatures has been com-pletely controlled in steels with a Ti/N ratio lower than thestoichiometric ratio [15].

∗ Corresponding author.E-mail address: smedina@cenim.csic.es (S.F. Medina).

The object of this work has been to determine the pinningforces exerted by TiN precipitates and the driving forces forinterpass recrystallisation, establishing a comparison betweenthe two. In this way it has been possible to explain the results

921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2006.02.037

254 M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261

Table 1Chemical composition (wt.%) of the steels used

Steel C Mn Si P S Ti Al N

S2 0.13 1.23 0.32 0.011 0.013 0.021 0.045 0.0046S3 0.13 1.16 0.31 0.011 0.013 0.047 0.045 0.0043S5 0.12 1.18 0.29 0.007 0.012 0.018 0.034 0.0080S6 0.11 1.09 0.25 0.005 0.012 0.031 0.035 0.0083

found in the rolling simulation regarding the poor hardeningachieved in the austenite and the difficulty of achieving a suitablyrefined ferritic microstructure.

2. Experimental procedure

Four steels were manufactured in a vacuum induction furnacefrom high purity electrolytic iron. On the basis of their nitrogencontent, the steels can be classified in two groups (Table 1). Thefirst group includes two steels with 40 ppm nitrogen (S2 and S3)and the second group the other two steels with 80 ppm nitrogen(S5 and S6). On the other hand, the Ti and N contents of thesteels used guarantee that the precipitation of titanium nitride(TiN) particles takes place in the solid state, since their solubilitytemperatures are below 1500 ◦C, which is approximately themelting temperature of the studied steels.

Thermomechanical simulation has been carried out by mul-tipass torsion tests. The test specimens had a gauge length of50 mm and a radius of 3 mm, and were protected during test-ing with an argon stream in order to prevent their oxidation.The torque and number of revolutions have been transformedinto equivalent stress and strain values using Von Mises crite-rion [16]. The reheating temperature was 1300 ◦C × 10 min andsubsequently the temperature was lowered to 1175 ◦C, corre-sponding to the temperature of the first pass. The simulationconsisted of 20 passes, with a temperature drop between eachpTrT0ts

e(tac

3

3

1i

Table 2Austenite grain size at reheating temperature

Steel Ti/N Reheatingtemperature(◦C)

Linear interceptaustenite grainsize, l (�m)

Austenite grainsize, d (�m)

S2 4.56 1300 269 302S3 10.93 1300 190 213S5 2.25 1300 34 38S6 3.73 1300 57 64

pressurised water stream flowing through the interior of the tube,achieving a high quenching severity and thus guaranteeing thecomplete transformation of the austenite into martensite. Thesespecimens were also used for the analysis of precipitates, as willbe described below.

The austenite grain size was determined according to ASTMstandard E-112. The mean linear intercept and the average grainsize were noted, both being deduced from the ASTM number.The results obtained are shown in Table 2. Among the steelsused, steel S5 gave the smallest austenite grain size, with a Ti/Nratio of 2.25.

3.2. Analysis of precipitates and calculation of volumefraction

The studied precipitation state was that corresponding to thereheating temperature of 1300 ◦C in all the steels. Fig. 1(a andb) shows two TEM micrographs of fine TiN precipitates corre-sponding to the temperature of 1300 ◦C × 10 min for steels S3and S6, respectively.

The analysis of precipitates showed a size distribution thatvaried from just a few nanometres up to micrometres. For thisreason it was necessary to establish a division of the precipitatesinto two groups: fine (size < 100 nm) and coarse (size > 100 nm).The size selected to separate the fine precipitates from the coarsepfI(X

ffoosIb[

ahwtsf

ass of 25 ◦C, so the temperature of the final pass was 700 ◦C.wo interpass times (�t) of 30 s and 60 s were tested, which cor-esponded to cooling rates of 0.83 K/s and 0.42 K/s, respectively.he effect of the strain (ε) was also tested, applying values of.20 and 0.35 in two simulation series, respectively. Similarly,wo strain rates (ε) of 3.63 s−1 and 1.45 s−1 were applied in twoimulation series, respectively.

The analysis of precipitates was carried out by transmissionlectron microscopy (TEM) and scanning electron microscopySEM). TEM was used to analyse the finer precipitates withhe carbon replica extraction technique, while SEM was used tonalyse the coarser precipitates. The studied precipitation statesorresponded to the austenitisation temperature of 1300 ◦C.

. Results and discussion

.1. Austenite grain size

Firstly, the austenite grain size was determined at 1300 ◦C for0 min, in the same conditions as the reheating of the specimensn the hot rolling simulation. The specimens were quenched by a

recipitates was established bearing in mind that the pinningorces are very weak for precipitates larger than 100 nm [12].dentification of the observed precipitates as titanium nitridesTiN) was achieved by electron diffraction and energy dispersive-ray analysis.The sizes of the fine and coarse precipitates were measured

or each steel. In the first case the population of precipitates usedor the size measurement was approximately 300, and in the sec-nd case was at least 40. In this way a frequency diagram wasbtained for each steel, such as that shown in Fig. 2, referring toteel S6, for an austenitisation temperature of 1300 ◦C × 10 min.t can be seen that the distribution is lognormal. The size distri-ution obtained for the other steels was also of a lognormal type15].

Table 3 also indicates the weighted mean size of the finend coarse precipitates, respectively, for each steel. On the otherand, the Ti and N contents in solution and in precipitated stateere calculated, along with the precipitated volumetric frac-

ions. The calculations were performed bearing in mind theolubility product of the TiN particles [1] and the expressionsor determining the precipitated volume [17], respectively.

M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261 255

Fig. 1. TEM images of fine TiN particles in austenite at 1300 ◦C × 10 min. (a) Steel S3; (b) Steel S6.

Fig. 2. Relative frequency of size for fine TiN precipitate in steel S6 reheated at1300 ◦C × 10 min.

The Ti and N contents in solution and in precipitated statewere calculated, along with the precipitated volumetric frac-tions, at 1300 ◦C (1573 K) for steels used (Table 4). Steel S3shows the highest Ti content (0.035 wt.%) in solution at 1300 ◦C,

Table 3Weighted mean size of fine and coarse precipitates, respectively, at 1300 ◦C

Steel S2 S3 S5 S6

Ratio Ti/N 4.56 10.93 2.25 3.73Fine precipitates (nm) 23 65.2 13.8 14.1Coarse precipitates (�m) 2.18 2.57 1.26 2.32

but the N content in solution is very small (0.007 wt.%) andthe possible strain induced precipitation in continuous cooling,which takes place in the simulations performed, precipitatingboth TiN and TiC, would be weak (lower proportion than inisothermal conditions). The other steels yield an even smallersolubility product (TiN)s than steel S3 and the possibility ofinduced precipitation in a broad range of temperatures belowthe no-recrystallisation temperature (Tnr) is lower [13].

3.3. Determination of pinning forces

The pinning forces opposing grain growth and the progressof recrystallisation were determined. It is known that the finerprecipitates are those that act on grain growth control, and forthis reason the value of “r” in the expressions of pinning forcescorresponds to the distribution of fine precipitates. On the otherhand, the different expressions used to calculate the pinningforces assume the particles to be spherical, and for this rea-son it was necessary to convert the average TiN particle size,corresponding to the side of a square (�), to the radius of a cir-cle (r), by means of the approximate equality of areas, so thatr = �/

√π. Given that most TiN particles present a rectangular

shape in the micrographs obtained by TEM, the average side of

Table 4Titanium and nitrogen (wt.%) as dissolved (s) and as precipitated (p), at 1300 ◦C,d(

S

SSSS

etermined using Turkdogan’s solubility products [1], and precipitated volumeVp)

teel Tis Tip Ns Np Vp

2 0.012 0.009 0.0019 0.0027 1.75 × 10−4

3 0.035 0.012 0.0007 0.0036 2.50 × 10−4

5 0.005 0.013 0.0043 0.0037 2.42 × 10−4

6 0.010 0.021 0.0022 0.0061 4.03 × 10−4

256 M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261

Table 5Pinning forces exerted by fine TiN particles for steels used

Steel Reheatingtemperature (◦C)

Particle radius(nm)

Precipitatedvolume 50%

Gladman, 3γf/2r(MN/m2)

Rigid boundary model,6γf/πr (MN/m2)

Flexible boundary model,3γf2/3/πr (MN/m2)

S2 1300 12.98 0.88 × 10−4 0.008 0.010 0.116S3 1300 36.79 1.25 × 10−4 0.004 0.005 0.052S5 1300 7.79 1.21 × 10−4 0.019 0.024 0.240S6 1300 7.96 2.02 × 10−4 0.030 0.039 0.330

their equivalent square was obtained as � = √�1�2, where �1

and �2 are the lengths of the sides, respectively.The pinning force exerted by a number of particles per unit

of area (Na) on the grain boundary is given by [12]:

Fp = πrγNa (1)

where γ is the interfacial energy per unit of area (0.8 J/m2) andr is the average radius of the second phase particles.

For the calculation of pinning forces consideration was madeof the expressions of Gladman’s rigid boundary and flexibleboundary models, respectively, which are differentiated in theway of calculating Na [18–20]:

(a) Gladman expression:

Fp = 3fγ

2r(2)

(b) Rigid boundary model:

Fp = 6γf

πr(3)

(c) Flexible boundary model:

Fp = 3γf 2/3

πr(4)

iatrai

d

lap

3

tcc

S

Fig. 3. Hot torsion simulation: stress–strain curves corresponding to 20 passtorsion sequence for steel, Steel S3.

the temperature decreases, which supposes a greater tendencyto strengthening. In the final passes there is a reduction in theflow stress as a consequence of the � → � transformation.

To determine the hot rolling critical temperatures, such as theno-recrystallisation temperature (Tnr) and phase transformationtemperature (Ar3), the mean flow stress (MFS) is determined ineach pass, given by the area of the corresponding stress–straincurve divided by the applied strain. Figs. 5 and 6 show examplesof MFS versus the inverse of the temperature corresponding tosome of the studied steels. In these figures it is possible to seein greater detail the four stages mentioned above and marked inFig. 5:

(1) Stage I corresponds to an austenite which recrystallises com-pletely between passes.

(2) Stage II corresponds to an austenite whose recrystallisationis partially inhibited between passes.

Ft

Given that only the fine precipitate fraction has been takennto account in the calculation of pinning forces, it has beenssumed, in accordance with Gao and Baker [21], that this frac-ion is approximately 50% of the total precipitated volume. Theesults (Table 5) show that the rigid boundary and flexible bound-ry models yielded greater pinning forces than those calculatedn accordance with Gladman.

The greatest pinning forces corresponded to steels S5 and S6,ue especially to the smaller average size of the precipitates.

The driving forces for grain growth at 1300 ◦C were calcu-ated for each steel using Zener and Gladman’s expressions [22],nd their values were smaller than those corresponding to theinning forces.

.4. Hot rolling simulations

Bearing in mind that two values were applied for each ofhe three deformation variables (�t, ε, ε), six simulations werearried out for each steel. Thus a total of 36 simulations werearried out by means of torsion tests.

Figs. 3 and 4 show the simulation of 20 passes for steel S3 and5, respectively. At first sight an increase is seen in the stress as

ig. 4. Hot torsion simulation: stress–strain curves corresponding to 20 passorsion sequence for steel, Steel S5.

M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261 257

Fig. 5. Dependence of mean flow stress on inverse absolute temperature, accord-ing to given schedule, illustrating as the residual stress (�σ) is measured. Fourdifferent phases (I–III) can be seen. Steel S2.

(3) Stages III and IV correspond to the austenite/ferrite+ pearlite transformation.

The intersection of the regression lines of phases I and IIdefines the value of Tnr and the intersection of lines II and IIIdetermines the value of Ar3. The real critical temperature Ar1corresponds to a point after the minimum where the curve startsto change slope to become an increasingly straight line [23].

The values found for the two critical temperatures are shownin Table 6. It is interesting to see how in certain deformationconditions Tnr does not appear defined, which means that thesteel recrystallises completely between passes until the austen-ite starts to transform into ferrite (Fig. 6). This occurs when thestrain applied to each pass is high (0.35), which notably accel-erates the recrystallisation kinetics. In general, and as in otherconditions, an increase in the strain causes a decrease in the Tnr,or this magnitude does not appear.

An increase in both the strain rate and the interpass timealso has the consequence of causing a decrease in Tnr, as

Fi

Table 6Tnr, Ar3, Ar1 and �σr values for steel S1 determined by 20 passes hot rollingsimulation from 1175 to 700 ◦C and interpass temperature of 25 ◦C

Steel �t (s) vc (K/s) εpass ε (s−1) Tnr (◦C) Ar3 (◦C) ��r (MPa)

S2 30 0.83 0.20 1.45 887 769 2230 0.83 0.20 3.63 876 755 2530 0.83 0.35 1.45 a 759 a

30 0.83 0.35 3.63 860 760 1260 0.42 0.20 3.63 865 747 1460 0.42 0.35 3.63 a 747 a

S3 30 0.83 0.20 1.45 927 784 4330 0.83 0.20 3.63 915 788 4830 0.83 0.35 1.45 913 785 4430 0.83 0.35 3.63 905 799 4560 0.42 0.20 3.63 917 791 4360 0.42 0.35 3.63 890 788 25

S5 30 0.83 0.20 1.45 865 779 1130 0.83 0.20 3.63 862 766 1830 0.83 0.35 1.45 a 777 a

30 0.83 0.35 3.63 838 769 660 0.42 0.20 3.63 879 760 1160 0.42 0.35 3.63 850 758 7

S6 30 0.83 0.20 1.45 893 780 2130 0.83 0.20 3.63 894 779 1230 0.83 0.35 1.45 870 782 830 0.83 0.35 3.63 a 774 a

60 0.42 0.20 3.63 864 779 860 0.42 0.35 3.63 a 785 a

Reheating conditions: 1300 ◦C × 10 min.a Does not exist.

can be seen by comparing the results obtained with the samedeformation conditions except for the aforementioned variables.This is due to the fact that both variables favour the kineticsof the static recrystallisation that takes place between passes.These results are similar to those obtained by other authors[24].

It seems clear that the size of the initial precipitates at 1300 ◦Chas not influenced the value of Tnr, because if this were thecase, steel S5 would have yielded higher values than steel S2.According to Table 4, steel S3 notably differs from the others interms of its Ti content in solution at 1300 ◦C. It is known thatTi in solution delays static recrystallisation in a similar way toNb, and this could be the reason why the Tnr values have beengreater in this steel than in the others [25]. Furthermore, steelS3 was the only one to always show a value for Tnr, even whenthe deformation conditions were the most favourable for staticrecrystallisation to be completed between consecutive passes,i.e. when the strain was 0.35, the interpass time 60 s, and thestrain rate 3.63 s−1.

With regard to the transformation temperature Ar3, the valuesdetermined were also similar in almost all the steels, even whenthe deformation conditions were modified. It is seen that steelS3 also presents somewhat higher Ar3 values than the others,despite the fact that its carbon content is somewhat higher. It isknown that Ti tends to increase the value of Ar3 [26], but thereason for this increase in steel S3 has rather been due to thefd

ig. 6. Dependence of mean flow stress on inverse absolute temperature, accord-ng to given schedule, Steel S6.

act that the accumulated stress (�σr) in the austenite, imme-iately before the start of the � → � transformation, has been

258 M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261

Fig. 7. Microstructure obtained in steel S2 by hot rolling simulation: ε = 0.35;ε = 3.63 (s−1); �t = 30 s; final pass temperature (Tf) = 800 ◦C; T(800−500) ◦C = 5(K−1); D� = 13 �m.

considerably greater in this steel than in the others. This accu-mulated stress can be measured on the graph of MFS versus theinverse of the temperature and will be given by the length of thevertical segment drawn by Ar3 and limited by the phase I straightregression line and the intersection with the prolongation of thephase II straight line, as is illustrated in Fig. 5.

The values of �σr, determined in this way, are shown in thefinal column of Table 6, it being observed, as has been mentionedabove, that these are greater for steel S3. This means that theaustenite in this steel has undergone a greater strengthening andtherefore has reached a greater content of dislocations, whichare possible nucleation sites for ferrite.

However, the stress accumulated in each pass up to Ar3 hasbeen relatively low, even in the case of steel S3, and is clearlyinsufficient to achieve a highly strengthened austenite that helpsto notably refine the ferritic grain size [27]. This is due to thefact that in those cases where Tnr was determined, the austeniteonly experienced partial inhibition of recrystallisation betweensuccessive passes. In other words, the precipitation state at thereheating temperatures (RT), plus the possible strain-inducedprecipitation occurring at temperatures below Tnr in steel S3,with free Ti and N at RT, was not sufficient to strongly inhibitinterpass recrystallisation [22].

Microalloyed steels are normally rolled at temperatures aboveAr3 and the final pass, e.g. in the hot strip mill, is performed ata temperature close to A . In this sense, simulations were car-rpatiapatt(rs

Fig. 8. Microstructure obtained in steel S3 by hot rolling simulation: ε = 0.35;ε = 1.14 (s−1); �t = 30 s; final pass temperature (Tf) = 800 ◦C; T(800−500) ◦C = 5(K−1); D� = 9 �m.

Fig. 9. Microstructure obtained in steel S5 by hot rolling simulation: ε = 0.35;ε = 3.63 (s−1); �t = 30 s; final pass temperature (Tf) = 800 ◦C; T(800−500) ◦C = 5(K−1); D� = 11 �m.

Fig. 10. Microstructure obtained in steel S6 by hot rolling simulation: ε = 0.2;ε = 3.63 (s−1); �t = 60 s; final pass temperature (Tf) = 800 ◦C; T(800−500) ◦C = 5(K−1); D� = 15 �m.

r3ied out in the same conditions as the previous simulations buterforming the last pass at 800 ◦C and cooling the specimen inn argon stream at a rate of approximately (5 K/s) (800–500)◦Co room temperature. The microstructures obtained were sim-lar and the ferritic grain size was similar from one steel tonother irrespective of the conditions, except for steel S3 whichresented a somewhat finer microstructure due to its greaterccumulated stress. Figs. 7–10 show microstructures of each ofhe studied steels, obtained in the conditions noted in the respec-ive captions with regard to strain (ε), strain rate (ε), cooling rateT ) between 800 ◦C and 500 ◦C, and interpass time (�t). The fer-itic grain sizes measured (D�) are very approximate, except forteel S3 which in the indicated deformation conditions yielded

M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261 259

a smaller ferritic grain size due to the greater accumulated strainin the austenite.

In practice, when the objective is to obtain a much finermicrostructure, other microalloying elements such as Nb areadded which help to achieve higher �σr values [28,29], althoughaustenitic grain control at high temperatures, e.g. in welding,would be worse [17].

3.5. Driving forces for interpass recrystallisation

In hot rolling, austenite recrystallises statically betweenpasses and dynamic recrystallisation (DRX) takes place onlywhen the temperatures is high and the applied strain is greaterthan the peak strain (εp), i.e. the strain corresponding to the max-imum stress reached on the flow curve for that temperature. Asthe temperature decreases, so dynamic recrystallisation becomesmore difficult, since the peak strain rises with the temperature.In Fig. 3 it can be seen that a strain of 0.20 did not reach thepeak stress in the first passes. However, in Fig. 4, correspond-ing to steel S5, it is observed that the strain of 0.35 applied ineach pass exceeded the peak strain in the first passes and thusan important dynamically recrystallised volume was produced.This was observed only with steels S5 and S6, which presentedthe smallest austenitic grain sizes, a fact that produces a decreasein the peak strain and thus a reduction in the critical strain (εc)nre

fiiicu

F

wvdtdc

r

σ

w

lsipifs

Fig. 11. Driving forces for recrystallisation between passes against inverse abso-lute temperature, Steel S3.

Figs. 11 and 12 show examples of Fr calculated in each passfor the steels used, in accordance with Eqs. (5) and (6), versusthe pass temperature. As the temperature decreases, the driv-ing forces also increase as a consequence of the increase in thestress and thus in the density of dislocations. In all cases, Fr hasbeen represented up to a temperature somewhat higher than Ar3,always in the austenitic phase.

The values reached for Fr in all cases exceed the pinningforces by more than two orders of magnitude, which demon-strates that in these steels, where most of the precipitation isnot strain-induced, recrystallisation always progresses and is nothindered by the presence of precipitates. However, in microal-loyed steels with Nb or V, where the precipitation is completelystrain-induced, very fine precipitate distributions are reachedwhich may cause the delay or even the complete inhibition ofstatic recrystallisation.

The representation of Fr versus 1/T also defines a Tnr valuewhich is practically the same as that determined with MFS ver-sus 1/T, although the latter is preferred since it yields a bettercorrelation index.

Fl

ecessary for DRX to start. Between the two strains there is aelationship of εc = aεp which varies from one author to another,stimating a value of “a” of between 0.4 and 0.95 [30,31].

Disregarding the possibility that DRX can take place in therst passes, or even accepting this possibility, which would

mply that metadynamic recrystallisation (MDRX) would occurn the first passes, the driving forces for recrystallisation werealculated for the four steels and all the deformation conditionssing the following expression [21]:

r = µb2�p

2(5)

here µ is the shear modulus, (4 × 104 MN/m2), b Burger’sector (2 × 10−10 m) and �p is the variation in the density ofislocations associated with the movement of the recrystallisa-ion front in the deformed zone. Although this equation does notirectly reflect the influence of the temperature, its influence isontemplated in �p.

On the other hand, the value of �p during hot working iselated with the increase in flow stress [32]:

m = 0.2µb√

�p (6)

here σm is the stress reached in the flow curve for each pass.The driving forces for recrystallisation (Fr) were calcu-

ated for each steel according to the above expressions fromtress–strain curves and for each of the tested conditions. Anncrease in the pass strain, while the other conditions (pass tem-erature and strain rate) remain unchanged, supposes an increasen the density of dislocations, which implies a greater drivingorce and thus a reduction in the time necessary for recrystalli-ation to be completed.

ig. 12. Driving forces for recrystallisation between passes against inverse abso-ute temperature, Steel S5.

260 M.I. Vega et al. / Materials Science and Engineering A 423 (2006) 253–261

Fig. 13. Driving forces for recrystallisation between passes against inverse abso-lute temperature, calculated when the stress σm and MFS are used, respectively,Steel S6.

It should be noted that the comparison between pinning forcesand driving forces has been carried out in conditions that yieldminimum values for the former and maximum values for thelatter. If all the precipitated volume had been used in the caseof the pinning forces, their value would have been double butwould continue to be lower than the driving forces. In the caseof the driving forces, if the yield stress had been subtracted atthe maximum stress reached in each step, which accounts forapproximately one third of the latter, Fr values of approximatelyhalf those represented in the preceding figures would have beenobtained. In any case, a new comparison of both forces wouldshow that the driving forces would be more than one order ofmagnitude greater than the pinning forces.

On the other hand, if MFS had been used as the stress in Eq.(6), the result would be similar to the previous case, since theMFS/σm ratio acquires a value of between 0.6 and 0.7, dependingon the pass temperature. By way of example, Fig. 13 shows Frversus the inverse of the absolute temperature using both σm andMFS, respectively, as the stress in Eq. (6). It is seen that the rela-tionship between the values of Fr is a little lower than the ratioof stresses and its value is observed to be between 0.5 and 0.65.In short, if MFS is taken as the dislocation-generating stress,the driving forces would be approximately half those obtainedif σm were taken, but would continue to be much greater thanthe pinning forces. The regression lines define both Tnr valueswith very similar magnitudes.

ssiap

4

1

precipitates and the pinning forces exerted by them werecapable of controlling grain growth at the reheating temper-ature of 1300 ◦C.

2. The hot rolling simulation of the studied steels has shownthat the stress accumulated (�σr) in the austenitic phase forthe deformation conditions applied in this work is very smalland in certain conditions is nil. This means that the fractionrecrystallised between passes has been high and in some caseswas complete.

3. The driving forces for interpass recrystallisation were alwaysmuch greater than the pinning forces exerted by the TiN par-ticles, irrespective of the Ti and N contents and irrespectiveof the deformation conditions.

4. This superiority of the driving forces over the pinning forces,by almost two orders of magnitude, shows why the values of�σr have been so low and why it is difficult to notably refinethe ferritic grain after hot rolling in these steels.

References

[1] J. Kunze, Met. Sci. 16 (1982) 217.[2] J. Kunze, B. Beyer, S. Oswald, W. Gruner, Steel Res. 66 (1995) 161.[3] E.T. Turkdogan, Iron Steelmaker 3 (1989) 61.[4] K. Inque, I. Onhuma, H. Ontani, K. Ishida, T. Nishizawa, ISIJ Int. 38

(1998) 991.

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[

[

Finally, the comparison between pinning and driving forceshows that interpass recrystallisation is thermodynamically pos-ible, although from the point of view of kinetics completenterpass recrystallisation would depend on the variables thatffect it, such as the interpass time, temperature, strain in eachass and strain rate.

. Conclusions

. Steels with a Ti/N ratio lower than the stoichiometric ratio,such as steels S5 and S6, showed a finer distribution of TiN

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