A New Model for Interpass Softening Based on the Strain Hardening Rate Prior to Unloading

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1 Materials Science Forum Online: ISSN: , Vols. 5-51, pp doi:1.428/ 25 Trans Tech Publications, Switzerland A New Model for Interpass Softening Based on the Strain Hardening Rate Prior to Unloading J.J. Jonas and E.I. Poliak* McGill University, Dept. Metallurgical Eng., 361 University St., Montreal, PQ, H3A 2B2, Canada *Mittal Steel, Research Laboratories, 31 E. Columbus Dr., East Chicago, IN, 46312, USA john.jonas@mcgill.ca, *evgueni.poliak@mittalsteel.com Keywords: interpass softening, modeling, strain hardening rate. Abstract. It is shown that the kinetics of softening between mill passes can be modeled more simply when the normalized strain (reduction) per pass is employed rather than the conventional strain. This method requires a second important input, namely the strain hardening rate at the end of preloading. Using this approach, the number of input parameters and experiments required for their determination are drastically reduced. The use of the Law of Mixtures to describe the behaviors of the recrystallized and unrecrystallized volume fractions is then illustrated. Finally, the approach required for quantifying the precipitation kinetics (in microalloyed steels) is described. Introduction In multipass hot rolling, the kinetics of softening during unloading for a time t between the passes are traditionally quantified by the JMAK equation for the volume fraction softened X n t X () t = 1 exp.693. (1) t5 Based on constant strain rate simulations, the time t 5 required to attain 5 % fractional softening has been found to depend on the conditions during and after deformation as [1]: t 5 b p ε q ( Q RT ) = CD Z exp. (2) In Eqs. 1 and 2 n is the Avrami exponent, ε the strain or accumulated strain, Z the Zener-Hollomon parameter pertaining to the preceding loading, Z = ε& exp( Q def RT def ), ε& the strain rate, Q def and Q s the activation energies for deformation and softening, respectively, and D the initial grain size prior to loading. T def and T s are the absolute temperatures of deformation and softening, respectively; these are determined mostly by the mill configuration and rolling speed. For convenience, the sensitivity exponents p and q are taken as positive here. The major difficulty associated with the use of the model of Eqs. 1 and 2 is that the parameters of post-deformation softening p, q, n and Q s vary depending on the strain, strain rate and temperature of the preceding deformation, as well as on the controlling softening mechanism, viz., static recovery, static recrystallization (SRX) or metadynamic recrystallization (MDRX). For this reason, the model of Eqs. 1 and 2 is an oversimplification and determination of the model parameters requires considerable experimentation. Furthermore, values established, for example, in laboratory simulations cannot be applied directly to industrial conditions since the latter can differ markedly from the former and, consequently, the softening model itself must be complemented by a procedure that permits the extrapolation of laboratory results to industrial mills. Further difficulties are faced when interpass softening is incomplete and the concept of the retained strain accumulated from pass to pass is employed. The problems associated with the use of the retained strain to predict the amount of softening that takes place during a given interpass interval are described in a companion paper [2]. These involve the lack of a rigorous definition of fractional softening on the one hand as well as the s s All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-1/5/16,23:28:15)

2 16 Microalloying for New Steel Processes and Applications extensive experimentation required to characterize the effects of the rolling parameters on the amount of softening. However, it has recently been shown [3, 4] that laboratory simulations can be considerably simplified if the normalized strain w i = ε i /ε p is kept constant instead of the pass strain ε i, where ε p is the peak (or steady state) strain. Under these conditions, although the pass strains change with strain rate and temperature, all the log t 5 w plots have similar shapes (here t 5 is the time of half softening). These differences can be better understood by referring to Figs. 1 and 2. In Figs. 1a [5] and 1b [6], the conventional time of half softening, t 5, is plotted against the conventional strain. It is readily apparent that the softening behavior is represented by a series of curves. Such curves are intrinsically difficult to model as strain, strain rate and temperature all play a role. When the observations are plotted against the normalized strain, w i = ε i /ε p, Figs. 2a and 2b, the common features of the various curves are seen more clearly.. a b Fig. 1. Strain dependence of t 5 : (a).85 C - 1 Mn.45 Nb steel [5]; (b).7 C 1.5 Mn.3 Nb.4 V.1 Ti steel, strain rate 3 sec -1 [6] /sec.5/sec 5/sec C 9 C 875 C 85 C log t5 1 log t a Normalized strain b Normalized strain Fig. 2. Plots of log t 5 against the normalized strain: (a) for Fig. 1a; (b) for Fig. 1b. Note that, at small w, log t 5 decreases rapidly with w, while at high w, log t 5 attains a sort of a steady state. The transition from the strong w-dependence of t 5, actually characteristic of SRX, to the strain rate dependent steady state (characteristic of MDRX) occurs when the preloading

3 Materials Science Forum Vols normalized strain lies in the vicinity of the DRX critical strain εc (w c = ε c /ε p ). The change in t 5 dependence within the transition range is attributable to the heterogeneous microstructure that develops during the initial stages of DRX: due to the limited extent of DRX associated with preloading to strains just above w c, two post-deformation softening mechanisms, SRX and MDRX, operate concurrently. The balance of this paper is concerned with this normalized strain approach and with how it can be employed to model interpass softening. Temperature-Compensated Time The observations exemplified in Figs. 1 and 2 prompted the proposal that the temperaturecompensated time for half-softening [7] be expressed as a function of the normalized strain w [3]: Q p p q 5 exp w ε p Z. (3) τ = s b p q b t5 = CD ε Z = CD RT s The advantage of employing τ 5 is that it is related only to the parameters of the prestrain. Although the initial grain size D has a sensitive effect on the softening kinetics, it also affects the deformation behavior and can be incorporated into the prestrain term. The strain dependence of t 5, and also the w-dependence of τ 5 vanish completely when ε i is equal to ε p (that is, at w = 1), cf. Figs. 1 and 2. Consequently, it is possible to introduce the temperature-compensated time for softening at the stress peak (or similarly at the steady state) as Q τ s b q 5 = t5 exp = CD Z RT, (4) s where t 5, Q s, q, in general, differ from the respective quantities that pertain to smaller values of the normalized strain. This allows further unification of the strain dependencies of t 5 because the normalized softening curves actually overlap, considerably simplifying the curve fitting required in this way, as illustrated in Fig. 3 for the plots shown above in Figs. 1 and log (t5/t 5) Normalized strain Fig. 3. Combination of the plots in Fig. 2; the time for half-softening is normalized by t 5. Since the Zener-Hollomon parameter Z is uniquely related to the peak stress and to the peak strain, the temperature-compensated time τ 5 must also depend solely on the peak stress and/or the steady state stress.

4 18 Microalloying for New Steel Processes and Applications The role of ε p (and of w) is confirmed in Fig. 4, where values of τ 5 are plotted against the preloading stress σ i. All the values of τ 5 that correspond to preloading strains close to stress peaks fall on the same straight line (note the logarithmic scale in Fig. 4). Since the peak stresses for the steel in Fig. 4 obey the power law, σ p Z 1/m, the straight line can be described as ln 5 τ = K Plnσ p = K Pln A mpln Z, (5) b where P is a constant, m is the strain rate sensitivity of the stress and ln( CD ) K = C 95 C 9 C 1 sec log τ Fig. 4. Stress dependence of the temperature-compensated time for softening,.6 C 1.4 Mn.3 Si.3 Nb steel. The straight line represents the locus of logτ 5 -values Role of the Strain Hardening Rate It is implicit in the plots of Figs. 3 and 4 that the strain dependence of τ 5 vanishes when the strain θ = σ ε & drops to zero during preloading. This generalization remains hardening rate ( ) ε, T def, D valid, regardless of the operating softening mechanism, and is therefore independent of whether DRX has taken place prior to unloading or not. It can further be assumed that τ 5 (and t 5 ) should depend, not only on T, ε&, ε (or w) and D, but should also be related to the strain hardening rate during preloading, since the latter is also a function of T, ε, ε& and D. Indeed, the strain dependence of logτ 5 displays the same trend as the strain dependence of the strain hardening rate θ, as exemplified in Fig. 5. Both logτ 5 and θ decrease with strain, this decrease being rapid at small strains but becoming quite slow at higher strains. These considerations are further illustrated in Fig. 6, in which the dependence of (logτ 5 /τ 5 ) on strain hardening rate θ i at the end of preloading is displayed. The ratio log(τ 5 /τ 5 ) exhibits a near linear dependence on strain hardening rate, as can be described by: ln τ5 = ln τ + aθ i, (6) where a is an independent constant. The combination of Eqs. 5 and 6 leads to ln τ5 = K q ln Z + aθ and therefore to τ = Z q q 5 exp( K + aθ) and τ = A σ m 5 p exp( K + aθ). (7) log σ i 5

5 Materials Science Forum Vols Eq. 7 allows for a radical reduction in the number of experiments required to characterize the rate of post-deformation softening, provided data regarding the Z-dependence of the peak stress are available and the strain hardening behavior is obtained beforehand by the analytical or numerical differentiation of continuous (uninterrupted) flow curves. 4 Strain hardening rate θ, MPa o C 95 o C 1 sec True strain Fig. 5. Typical strain dependencies of strain hardening rate θ,.6 C 1.4 Mn.3 Si.3 Nb steel log τ5/τ Strain hardening rate θ, MPa Fig. 6. Dependence of the temperature-compensated time for softening on strain hardening rate,.6 C 1.4 Mn.3 Si.3 Nb steel. These results make it clear that the initiation of recrystallization after hot deformation is determined, not only by critical amount of the stored energy associated with the flow stress and expressed through Z, but also by the rate of energy accumulation during deformation as specified by θ. The rate of post-deformation recrystallization therefore depends on some function of the stress and strain hardening rate. This is somewhat similar to the initiation of dynamic recrystallization [8] but not that of static recrystallization after cold deformation. At low deformation temperatures, the rate of energy accumulation (and hence the strain hardening rate) is almost insensitive to strain rate and temperature, so that only the stored energy is important for the initiation of recrystallization.

6 2 Microalloying for New Steel Processes and Applications It is now possible to evaluate the critical strain hardening rate θ c required to initiate postdeformation recrystallization. Conventionally, SRX begins once a certain time (the incubation period) has elapsed after deformation. This is the time required for SRX to progress to a detectable extent equivalent to, say, a volume fraction of 5 % (X =.5). If the interpass time t ip is equal to the incubation period for SRX, t X=.5, then, solving Eqs. 1 and 7 for θ one can get θ c 1 = ln a 2 ( ) 1 n t τ ip 5 Q art SRX s s. (8) The higher θ c, the lower the preloading strain after which SRX can start. The temperature compensated time τ 5 is uniquely related to the peak stress and so is the critical strain hardening rate for a given strain rate and temperature. This is illustrated in Fig. 7, where the flow curves are shown in θ σ form. Note that the longer the allowable incubation period, the higher the critical strain hardening rate for SRX and hence the lower is the critical strain sec -1 9 C Strain hardening rate θ, MPa C 95 C θ c SRX, t ip = 1 sec θ c SRX, t ip = 5 sec θ c DRX True stress, MPa Fig. 7. The θ σ plots for.6 C 1.4 Mn.3 Si.3 Nb steel; t ip = t X=.5. In contrast to SRX, softening by MDRX is considered not to have an incubation period. MDRX starts if the material has been preloaded beyond the DRX critical strain corresponding to the inflection point in the θ σ plot [8], Fig. 7. Nonetheless, a certain time is still necessary for MDRX to progress to a detectable extent. Using an equation analogous to Eq. 8 and substituting Q s MDRX for Q s SRX, it is possible to evaluate the minimum interpass interval after which softening by MDRX can be detected after preloading to θ i < θ c DRX. Eqs. 6 and 7 are significantly more convenient for hot mill control because the strain hardening rate can be computed from the flow stress subroutine used for rolling force calculations in the mill model so that the deformation and softening computations can be combined in one block. As a result, it is no longer necessary to specify the values of p and q in Eq. 2 and their dependencies on the deformation conditions. Instead, only the two independent constants a and q are required. Evaluation of Incomplete Softening and the Law of Mixtures. It is readily evident that the external dimensions and shape of a bar do not change during unloading. As a result, the retained strain defined in the literature is not a strain in the conventional sense (i.e., it is not a measure of local displacements or changes in the workpiece dimensions and shape).

7 Materials Science Forum Vols Instead, it is a quantity related to the internal state of the material, which must be associated with the stored energy (flow stress) and rate of energy accumulation (strain hardening rate). The complications involved in determining the conventional retained strain stem from the fact that, during loading, the stored energy and the rate of its variation are non-linear functions of the strain and so is the stored energy retained in the material after partial softening. Furthermore, the notion of retained strain is itself controversial since its definition is based, on one hand, on the assumption that the strain distribution is uniform, but, on the other hand, on the additivity of the strain that implicitly assumes strain non-uniformity within the material. The approach based on the strain hardening rate prior to unloading can be used to overcome these problems and to account for the non-linearity since it does not require determination of the retained strain to evaluate the amount of interpass softening during multipass deformation. For this purpose, it should be recalled that the Taylor model allows for description of the flow stress using the rule of mixtures σ, + 1 = σ, X + σ, ( 1 X ). (9) i i This expression is applicable to both laboratory simulations and to rolling mills, with σ i and σ,i+1 being the offset (yield) stresses at the start of the ith and (i + 1)th passes, respectively, σ f,i is the flow (yield) stress at the end of the ith pass and X i is the fractional softening between the ith and (i + 1)th passes. Within the framework of the Taylor model, the yield stress can be differentiated with respect to the strain. In particular, if the reloading offset strain (i.e. strain at yielding at the beginning of the (i + 1)th pass) is assumed equal to the strain at the moment of unloading at the end of the ith pass, then ε σ = σ ε + ε i f i ( X ) = θ = θ X + θ ( X ), i+ 1, i Xi σ f, i 1 i, i+ 1, i i f, i 1 i. (1) That is, the strain hardening rate at the beginning of each reloading stage can be described with the aid of the rule of mixtures using the fractional softening achieved during the previous unloading. Furthermore, since the fractional softening X i during prior unloading is independent of the strain during subsequent reloading, differentiation of the flow stress at any uniform reloading strain is admissible within the framework of the Taylor model as long as neither the stress state nor the strain path has changed upon reloading. Therefore, using the value of X i in Eqn. 1 pertaining to the previous pass (which is already known), both the strain hardening rate at the end of the next pass and hence the softening after the next pass can be predicted. The computational procedure is straightforward. Let the material entering the first pass (pass strain ε 1 ) be homogeneous and fully recrystallized; it then undergoes partial softening X 1 after the first pass. The strain hardening rate attained at the end of the first pass (which is the hardening rate pertaining to the fully recrystallized state) defines the temperature-compensated time τ 5,1 for softening after this pass. When the partially softened material is deformed during the second pass to a strain ε 2, the strain hardening rate at the end of the second pass, θ f2, is given by an equation similar to Eq. 1, as illustrated schematically in Fig. 8: ( ε ) X + θ ( ε )( ) f, 2 = θr 2 1 nr 2 1 X1 i θ. (11) This results from the strain hardening contributions of both the recrystallized (softened) θ r (ε 2 ) and unrecrystallized θ nr (ε 2 ) portions of the material. The latter is equivalent to the strain hardening rate that would be attained during deformation of the recrystallized material to a strain of (ε 1 + ε 2 ); that is, θ nr (ε 2 ) = θ r (ε 1 + ε 2 ). These hardening rates can be computed from continuous flow curves determined on materials of appropriate grain size. The value of θ f2 defines the temperaturecompensated time τ 5,2 for softening after the second pass and is also used to specify the strain hardening behavior in the third pass.

8 22 Microalloying for New Steel Processes and Applications Multiple Pass Rolling Likewise, assuming partial softening of X 2 after the second pass, the strain hardening rate at the end of the third pass can be calculated from: θ = θ ε X + X θ ε + ε + 1 X θ ε + ε + ε X, (12) ( ) [ ( ) ( ) ( )]( ) f, 3 r r r and so forth. This procedure can be repeated as many times as required as long as the calculated fractional softening after each successive pass falls between.5 and.95, or between other values specified at a given mill for a given product as the detectable onset and completion of softening (recrystallization). The required inputs are the strain hardening rates for recrystallized material at the different strains (i.e., at ε 1, ε 2, ε 3, ε 1 + ε 2, ε 2 + ε 3, ε 1 + ε 2 + ε 3, etc.). All of these can be obtained from a conventional continuous (uninterrupted) flow curve. In industrial rolling, the successive passes are generally performed at increasing strain rates and decreasing temperatures, that is, at increasing values of Z. To account for this, the input strain hardening rates must be taken from flow curves that correspond to the appropriate Z-values. Alternatively, they can be obtained from a single flow curve adjusted to the hot mill conditions using the unified flow curve approach described elsewhere [3, 4]. Evaluation of the retained strain and hence superposition of the flow curves or back extrapolation of the reloading curve to zero stress (which is sometimes employed to evaluate ε a ) are no longer necessary. 2 θf,1=θr(ε 1 ) θnr(ε 2 )=θr(ε2+ε1) θf,2 θr(ε2) True stress θ,2=θ,1x 1 +θf,1(1-x 1 ) θ,1 ε1 True strain ε2 Fig. 8. Strain hardening rate at the reloading offset strain ε 1 and at the end of reloading θ f,2 = θ r (ε 2 )X 1 + θ r (ε 2 +ε 1 )(1 X 1 ); X 1 is the fractional softening during unloading at ε 1. Note that θ r (ε 2 +ε 1 ) and θ r (ε 2 ) can be obtained from an uninterrupted flow curve. To quantify the kinetics of incomplete fractional softening in multipass deformation, the coefficients a and q are required. As already mentioned, they do not depend on the deformation conditions and do not need to be constantly adjusted during the calculation. They are simply input as constant model parameters. Since the approach described above provides better modeling flexibility, it can be especially useful for the dynamic control of rolling mills, as in the so-called feed-forward systems. These are intended to handle the variabilities of temperature, deformation resistance, gauge, etc. along the bar being rolled. If, for example, there is a colder portion in the bar coming into the mill due to a skid mark originating in the reheat furnace, the flow stress will be higher locally than in the rest of the bar. When the colder portion passes through the first stand, this causes a local increase in rolling force and in the exit thickness. Information about this increase is fed forward to the downstream

9 Materials Science Forum Vols stands so that the roll gaps and rolling forces can be adjusted accordingly in anticipation of the approaching colder portion to correct for the gauge variation. When corrections to the downstream stands are made in this way, the pass reductions (and consequently the pass speeds) are changed locally causing further variations in the softening kinetics. This in turn causes further variations in the microstructure already altered locally by the lower temperature. Using traditional modeling methods and employing the retained strain, it is problematic to account for these variations because too many parameters need to be simultaneously and locally adjusted and the information relating to these adjustments propagated downstream. By contrast, with the θ-based approach, the value of the strain hardening rate in successive stands can be readily computed, the changes in the softening kinetics assessed and fed forward, and then the pass reductions corrected accordingly to minimize the variations in gauge and microstructure. Strain-Induced Precipitation The interaction of recrystallization and precipitation at hot rolling temperatures is a distinctive feature of microalloyed steels. To estimate the possible effect of precipitates, the equilibrium precipitation/solution temperature T sol must first be evaluated. The general industrial practice is that, during slab reheating, all the alloying elements are brought into solid solution. Even in the case of direct hot rolling, the same principle generally applies and rolling is initiated at temperatures at which all the elements are still in solution. When the bar is cooled during rolling, precipitation can occur statically or dynamically (that is, simultaneously with deformation). Another possibility is that the precipitation in austenite can occur after deformation but prior to recrystallization. In this case, as well as in the dynamic case, the precipitation is usually referred to as strain-induced. The onset of precipitation depends strongly on the mill configuration and the related temperature profile [9, 1 and references therein]. While in conventional plate and hot strip mills, the bar temperature decreases progressively, in mills with coilboxes, it can increase in the course of rolling and, depending on steel chemistry, be brought above T sol. Thus, modeling the deformation behavior in the course of precipitation is rather complicated and requires a model for the kinetics of precipitation and dissolution. In the industrial modeling of the hot rolling of microalloyed steels, especially in dynamic control, it is important to be able to predict the moment at which precipitation can begin to interfere with deformation and softening. In the model proposed by Dutta and Sellars [11], the time t ps required for the isothermal precipitation of Nb carbonitrides from supersaturated austenite to start (i.e., the time to attain 5 % volume fraction of precipitation, which is assumed to be a detectable amount) is given by 1 27 B [ ] ( ) t ps = A Nb exp exp, (13) 3 2 RT T ln Ks where K s is the supersaturation ratio, and A and B are constants with the former also depending on the Mn and Si contents. Further details can be found elsewhere [12]. The time t ps is counted from the moment the steel was last at T sol during cooling from temperatures high enough to dissolve all the precipitates. Deformation significantly affects the kinetics of precipitation. This is because, in practice, the nucleation of Nb(C, N) precipitates takes place heterogeneously on preferential sites, the densities of which increase with deformation. Based on this, the strain-induced precipitation start time t def ps for a given rolling pass can be represented as [11] [ ] ( ) def B t ps = Cε Z Nb exp exp (14) 3 2 RT T ln Ks

10 24 Microalloying for New Steel Processes and Applications It is evident that the strain-induced precipitation start time t def ps can be expressed in terms of the static precipitation start time t ps as def 1. 5 ( C A) t ε t ps = Z. (15) The above expression is problematic as it conflicts with the expectation that, in the absence of deformation (i.e., at ε = Z = ), t def ps should be equal to t ps ; that is, with ε, t def ps t ps. This does not follow from Eq. 14, which predicts that t def ps in the absence of deformation. Thus the time t def ps in Eqs. 14 and 15 should be understood as the change in the precipitation start time induced by deformation. The failure to account for this discrepancy introduces an additional difficulty in the modeling of strain-induced precipitation during hot rolling. There is an important similarity between the present two deformation-induced phenomena recrystallization and strain-induced precipitation; this is reflected in the similarity between Eq. 2 (recrystallization) and Eq. 14 (precipitation). The driving force for recrystallization is the stored energy, which requires the prior application of strain. The driving force for precipitation is the decrease in the Gibbs free energy (chemical potential) of the precipitating elements. The overall kinetics of these two processes are sharply different [13]; the latter displays the characteristic C- shape associated with driving force control at high temperatures and diffusive growth control at low temperatures. By contrast, in the former process, the start time decreases continuously with temperature, indicating that recrystallization is controlled by diffusive growth over the whole range. In the case of strain-induced precipitation, there is an additional, non-equilibrium driving force that can be related to dissipation of the stored strain energy. For strain-induced precipitation, the total driving force includes both the equilibrium and non-equilibrium contributions. Alternatively, the presence of dislocations and excess vacancies in the deformed material can be considered to increase the rates of both pipe and bulk diffusion, leading to an acceleration of the growth kinetics def in this way. This reasoning supports the view advanced above that t ps in Eq. 14 represents the difference in precipitation start time induced by the additional dislocations. The model can then be restated as def ps.5 ( C A)( 1+ ε ) 1 t ps = t ps Z, (16) Such a description requires revised constants, as well as strain and Z exponents. Eq. 15 and its revised form, Eq. 16, show that the start time for strain-induced precipitation, being inversely proportional to the strain, should decrease with increasing strain somewhat in the manner of the strain hardening rate. Based on the similarities between the kinetics of recrystallization and strain-induced precipitation referred to above, it can be of interest to substitute the deformation term (1 + εz.5 ) with a function of the strain hardening rate. Indeed, when a steel is deformed into the steady state at a constant temperature and strain rate, the densities of dislocations and other preferential sites for precipitation become independent of strain, especially in the case of DRX. Accordingly, the start time for strain-induced precipitation should no longer depend on the strain. Since the extent and rate of approach of steady state deformation can be described by the strain hardening rate, it is reasonable to assume that the precipitation rate, as well as t def ps, can also be expressed as functions of θ, as was in the case for t 5 for recrystallization. In fact, both processes are growth or diffusion controlled, so that their kinetics should be describable in terms of somewhat similar combinations of the stored energy (dislocation density) and its rate of accumulation. Considerable work is needed to explore these possibilities. If they prove to be of at least descriptive validity and the kinetics of strain-induced precipitation can be expressed in terms of the strain hardening rate as was shown above for the recrystallization kinetics, then the entire metallurgical model of the rolling mill can be significantly simplified, making it more adaptive and more applicable to dynamic process control.

11 Materials Science Forum Vols Conclusions The kinetics of softening after hot deformation have been shown to depend on the strain hardening rate at the end of preloading. The effects of the preloading conditions (strain, strain rate and temperature) on the post-deformation softening kinetics can be described in terms of their effects on the strain hardening rate prior to unloading. This allows for: 1) A radical reduction in the number of experiments required to generate sufficient data for modeling the post-deformation softening behavior; 2) A description of strain accumulation due to incomplete interpass softening in multipass hot rolling in terms of the strain hardening rate. 3) Significant simplification of the extrapolation of laboratory results to industrial high strain rate multipass processing. References [1] P.D. Hodgson and R.K. Gibbs: ISIJ Int., 32 (1992), 1329 [2] E.I. Poliak and J.J. Jonas: - this Proceedings [3] E.I. Poliak and J.J. Jonas: ISIJ Int., 43 (23), 684 [4] E.I. Poliak and J.J. Jonas: ISIJ Int., 44 (24), 1874 [5] S.H. Cho, K.B. Kang and J.J. Jonas: ISIJ Int., 41 (21), 766. [6] P.D. Hodgson, S.H. Zahiri and J.J. Whale, ISIJ Int., 44 (24), 1224 [7] C.M. Sellars, and J.A. Whiteman: Metal. Sci., 13 (1979), 187 [8] E.I. Poliak and J.J. Jonas: Acta Mater., 44 (1996), 127. [9] J.J. Jonas: ISIJ Int. 4, (2), No. 8, 731 [1] F. Siciliano, Jr. and J.J. Jonas: Metall. Trans.A, v. 31A (2), 511 [11] B. Dutta and C.M. Sellars: Mat.Sci. Technol., v. 3 (1987), 197 [12] D.Q. Bai, S. Yue, W.P. Sun and J.J. Jonas: Metall. Trans. A, 24A (1993), 2151 [13] J.J. Jonas: 2 nd Int. Conf. Thermomech. Processing, (Verlag Stahleisen, Germany, 24), 35

12 Microalloying for New Steel Processes and Applications 1.428/ A New Model for Interpass Softening Based on the Strain Hardening Rate Prior to Unloading 1.428/ DOI References [1] P.D. Hodgson and R.K. Gibbs: ISIJ Int., 32 (1992), /isijinternational [3] E.I. Poliak and J.J. Jonas: ISIJ Int., 43 (23), /isijinternational [4] E.I. Poliak and J.J. Jonas: ISIJ Int., 44 (24), /isijinternational [5] S.H. Cho, K.B. Kang and J.J. Jonas: ISIJ Int., 41 (21), /isijinternational [6] P.D. Hodgson, S.H. Zahiri and J.J. Whale, ISIJ Int., 44 (24), /isijinternational [8] E.I. Poliak and J.J. Jonas: Acta Mater., 44 (1996), / (95)146-7 [9] J.J. Jonas: ISIJ Int. 4, (2), No. 8, /isijinternational [12] D.Q. Bai, S. Yue, W.P. Sun and J.J. Jonas: Metall. Trans. A, 24A (1993), /BF