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The hot working behaviour of two steels of opposite recrystallisation kinetics has been studied by monotonic compression and written as an isotropic hardening viscoplastic constitutive law with a state parameter that evolves with hardening, recovery and recrystallisation. The state parameter hypothesis implies an immediate response of the microstructure to any strain rate change and the simulation results obtained by applying this assumption are compared against corresponding experiments.

As post-deformation recrystallisation in varying strain rate experiments is sensitive to the path history, representative measures of the prior strain rate have been identified in stress relaxation experiments to be used in the process model.

A constitutive softening identified for both steels under shear strain reversal has been used to modify the constitutive law and predictions validated against torsion experiments.

Representative complex loading scenarios in rolling have been identified and analysed by simulations with the resulting Constitutive Law.

Results indicate that FEM hardening rates in compression and torsion testing are about the same, as opposed to classical torsion analysis that gives a larger DRX strain. The best fitting to stress relaxation after varying strain rate deformation is given by the time average prior strain rate. For bar rolling of the medium-C steel the process model predicts DRX at the bar core as opposed to the simplistic model based on area reduction that gives SRX across the whole section. For flat rolling of the non-recrystallising stainless steel and in spite of the differences in shear strain accumulation through passes, the effects on rolling forces between roughing and tandem configurations are minimal, this being due to the predominance of the deformation by rolling elongation. A comparison against corresponding rolling experiments shows in general good agreement in rolling force and power consumption, but the assumed full softening of the yield surface appears as a hypothesis too penalising.

CSM objectives within this project is to better understand the effect of complex loading on rolling variables, on the homogeneity of the rolled product and on tooling forces during hot rolling of:

a stainless duplex steel (2304) of extremely slow recrystallisation kinetics at the rolling temperatures explored.

a Carbon-Mn without micro alloying inhibitors (55Cr3) steel, having very fast recrystallisation kinetics

They meant to be representative of the extreme microstructural situations of large grain elongation and early refinement. The work is focused on representative passes of flat and long products rolling schedules.

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Classical hot working characterization procedures are based on testing at isothermal, nearly constant strain rate conditions and optimising the fitting to the resulting peak flow stress by using a function of temperature and strain rate. This classical approach delivers a viscoplastic-type of constitutive law that gives a meaningless flow stress with the vanishing of the strain rate, does not incorporate any history along the deformation path and is only suitable for steady state FEM simulation analysis [A3.1-A3.3].

Constitutive models incorporating some of the material history evolution [A3.4-A3.12] were next developed and interfaced to general purposes FEM codes on the assumption the invariants of strain, strain rate and flow stress computed from the general triaxial state in Metal Forming are equivalent to the corresponding measures of the mechanical testing. The early work was concerned with non-recrystallising steels [A3.6-A3.8]. It was later extended to austenite by incorporating post-deformation recrystallisation into interpass models that used different measures of the prior deformation critical strain defining the transition from static recrystallisation (SRX) to metadynamic recrystallisation (MDRX) [A3.1,A3.9-A3.11]. Integrated rolling-interpass models have been applied to determine the effects of rolling parameters on dimensional tolerances and forces and used to derivate simplified models for on-line control [A3.10-A3.11]. Through-process models of complete rolling lines that consider multiple structures [A3.1,A3.9] undergoing different degrees of hardening and volume fractions percolated by grain sizes have been also developed.

The basis of the constitutive formulation above is the assumption of immediate response of the material [A3.8, A3.14] to the changing temperature-strain rate conditions as does occur in the general Metal Forming case, where development of non-uniform distributions of deformation variables are unavoidable. Thus, become important to assess the departure of the FEM prediction based on this hypothesis against experimental testing at changing strain rate, where some delay in the material response might develop.

Previous ECSC funded research found a different mechanical behaviour from monotonic and strain reversing torsion testing [A3.8] of a duplex stainless steel. It was unclear whether the large difference was related to the very slow recrystallising steel then used, to the large percentage of ferrite present at the hot working temperature or to the straining mode. As rolling pass deformation involves also shear strain reversals, a constitutive description that contemplated the material hardening under shear being different than under normal loading was also required to investigate. This requires further investigation for the general three- dimensional case where not all the shear strain components undergo either reversal or suffer the same hardening than the normal components.

Due to the large amount of ferrite at rolling temperatures in the 2304 steel, this soft phase takes most of the strain thus preventing recrystallisation of austenite [A3.7]. For the general case of austenitic steels, recrystallisation during and past deformation needs to be incorporated into the constitutive law and related to the varying strain rate-temperature conditions found in rolling.

The experimental possibilities of following microstructural evolution in industrial mills at intermediate stages are very limited, because of the difficulties in both cutting samples and quenching them in times short enough to freeze the microstructure. The alternative of reproducing industrial rolling conditions in pilot mills is also restricted, because of their smaller geometric dimensions, their slower rolling speeds and the absence of interstand tensions. The presence of DRX and MDRX in the finishing rolling conditions has beenPage 210shown in torsion simulation tests and modelled using simple regression equations and spreadsheets [A3.15, A3.16]. Results from numerical simulation rolling models also support this view [A3.17-A3.20], while it is also suggested that the time to trigger DRX cannot be normally exceeded in the rolling pass [A3.21,A3.22].

The characteristics of SRX and MDRX have been discussed by Karjalainen and Perttula [A3.23] and those of MDRX by Hodgson [A3.24], among others. The onset and the kinetics of different softening mechanisms, DRX during deformation and static recrystallisation (SRX) or MDRX after the final pass or during interpass-times are temperature *T* and strain rate [VER SIMBOLO EN PDF ADJUNTO] dependent. Therefore, although the critical strain [VER SIMBOLO EN PDF ADJUNTO] to start DRX, its associated volume fraction *Xd* and the post-deformation softening fraction *Xs* can all be identified almost uniquely (within the experimental error limitations) in isothermal tests at constant strain rates, their extrapolation to the local conditions developed in metal forming is not straightforward.

A more detailed analysis of the problems above requires identification of the complex loading scenarios, definition of a constitutive framework general enough to be used in most of hot rolling situations and validation against experiments.

In this work, a full material characterisation of the two steels under study has been carried out and a constitutive model that considers work hardening, dynamic recovery, dynamic recrystallisation and static softening has been developed [A3.25]. The constitutive model has been experimentally validated for varying [VER SIMBOLO EN PDF ADJUNTO] conditions, coded in a coupled temperature-stress analysis procedure of the commercial FEM code Abaqus ® [A3.26] and applied to mechanical testing and typical passes of flat and bar rolling. Wherever possible, validation of the numerical predictions against the experiments has been carried out.

Classical Hot Working Theory (HWT) models only considers the two *primary* variables:

temperature *T *

strain rate [VER SIMBOLO EN PDF ADJUNTO]

Which although well defined in mechanical testing have only a *local meaning* when applied to the *localisations* that any Metal Forming process develops, as revealed by FEM simulations.

These two variables are normally recast in a single ‘Temperature-compensated strain rate’ or Zener-Hollomon variable:

[VER FORMULA EN PDF ADJUNTO] defined in terms of an activation energy *Qdef* for deformation and* R *= 8.31 J mol^{-1}K^{-1}.

In the context of Metal Forming FEM analysis, these two *primary variables do not have memory of their own*, their evolution is governed by the global Thermo-MechanicalPage 211incremental equilibrium of the deforming solid, this being driven by the kinematical boundary conditions defined by both the tooling and the symmetry planes and thermal boundary conditions.

These variables are related to the intrinsic process variables* history: *

strain [VER SIMBOLO EN PDF ADJUNTO]

elapsed time *t*

And to the

microstructural evolution of the particular material.

The range of the primary variables is only local to the time *t*. The range of the hereditary variables is larger, but they tend to fade away simultaneously with the emerging of new ones (i.e.: hardening of recrystallised grains, recrystallisation after subsequent deformation, etc.). As the constitutive model to be developed in this project aims at future applications in through-process modelling, the material description should keep the number of variables propagating throughout the rolling schedule to a minimum.

The simplest constitutive model enriches the classical Hot Working Theory by introducing a *‘strength’* or ‘*hardness’* factor *s *[A3.4-A3.9, A3.12] that keeps records of the hardening level achieved by the structure at a given stage of hot deformation as result of the competition of hardening and dynamic recovery (H+DRV). The strength and hardness names trace back to the pioneering work of Hart [A3.27], where *s* was a measure not depending on the primary variables *T* [VER SIMBOLO EN PDF ADJUNTO] as it happens in tensile or indentation cold testing.

In order to incorporate the material softening during and past deformation into the constitutive description, the *recrystallised volume fraction X* needs to be incorporated into the framework. To keep the number of variables confined, it is convenient to relate the recrystallised fractions to the single hardness parameter, which then becomes the only carrier of the microstructural evolution through the process [A3.1,A3.9]. Nevertheless, short- range secondary hereditary variables are needed, which are eliminated after their effects are transferred to the hardness carrier. They are:

the dynamic *Xd* and static *Xs* recrystallised volume fractions

the grain size

The laws defining the material evolution are obtained from experimental data at constant strain rate and temperatures. These equations need to be experimentally validated under varying strain rate conditions and then recast in incremental form to be integrated under the non-isothermal, through-strain rate localization paths that result in rolling.

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Representative complex loading scenarios of hot rolling are:

(a) loading is sensitive to strain rate and temperature (Hot Working Theory) and both these variables are not constant in the deformation zone as in the mechanical test.

(b) hardening rates for normal and shear deformation might be different.

(c) dynamic loading resulting from microstructural changes in the steel (DRX).

(d) dynamic loading as the result of shear strain reversal in the pass.

(e) uneven straining of the constituent phases [A3.37] in steels that are multiphase at the rolling temperatures.

In addition to the situations above, representative complex loading scenarios between passes arise from:

(f) reversal of the two normal strain components associated to the directions transversal to the rolling direction, in long product mills with alternating H-V stands.

(g) shear deformation *cycles *of different nature in reversing-roughing and tandem- finishing rolling lines.

To the purposes of linking subsequent passes throughout the rolling schedule, the microstructural state also needs to be updated accordingly to the static and post dynamic microstructural changes occurring at the interpass.

CSM has developed over the years rolling models of different complexity for both flat and bar rolling [A3.2, A3.7-A3.11] using several FEM codes. The models used in this project are based on the commercial Abaqus® [A3.26] code.

To cut down the number of element in the mesh to a minimum, mirroring sections are substituted by boundary conditions on the symmetry planes wherever appropriate.

The stress analysis is performed considering the material is rate and temperature dependent and its history evolves under the constitutive options described.

The thermal solution considers internal conduction within the rollingstock, thermal radiation to the environment, heat transfer to the rolls and heat generation due to conversion of mechanical work both within the stock and at the friction interface.

In addition to the evolution of the unknowns of the system: displacement and temperatures and their derivate fields strain, stresses and thermal cooling/heating rates, a number of auxiliary variables are set up to monitor the onset of DRX and drive its integration,Page 213computation of mean strain rates for SRX/MDRX softening and introduction of softening due to strain reversals. All this features are explained in detail in the section §A3.3 and §A3.4.

One of the main advantages of the FEM analysis is that the material behaviour coded can be applied to a variety of deforming conditions. Thus, the constitutive law finally applied to the validation exercises against rolling results in §A3.5 were previously validated at different stages of its development on several testing conditions ranging from compression, torsion and strain reversals.

Table A3.1 gives the chemical composition of the steels studied in the project.

The 55Cr3 is a medium-C steel, rolled normally to bar and rods for fabrication of springs. It contains no recrystallisation inhibitors. The research on this steel is confined to the stands of the intermediate mill of ORI-MARTIN mill, near the cutting device where routine specimens are taken for inspection.

The 2304 duplex stainless steel is extremely slow in recrystallising, due to the fact that most of the strain during hot rolling is taken by the ferritic phase; it is finally rolled to sheet thickness for cold drawn domestic appliances. The interest in this research is in the THYSSEN-KRUPP ACCIAI SPECIALI TERNI hot rolling line.

The main thematic interconnections between the materials and process selected and the mechanical testing are shown in Table A3.3.

The two steels under study undergo opposite microstructural evolution during and after hot working. The medium-C steel recrystallises very fast [A3.10] both during and after deformation. Dynamic recovery during duplex stainless steel deformation counterbalances hardening giving a perfect steady state stress plateau [A3.7, A3.8] and no recrystallisation takes place after deformation at the standard hot working temperatures and interpass times.

Each of the two materials are associated to different final products and hence hot rolling routes. During duplex stainless steel flat rolling, grain elongation and accumulation of normal strain components throughout passes occur, without any reversal of them. The main shear strain component, however, undergoes a reversal under the roll gap and there is a residual shear strain accumulation through passes at about one quarter of the rolling stock [A3.7].

It is known that the medium-C 55Cr3 steel exhibits very fast recrystallisation kinetics and therefore a significant part of the rolling strain history is expected to be destroyed at interpass with grain refinement and growth overcoming grain elongation. The effective strain might even be reduced within the roll gap as result of dynamic recrystallisation in some passes of the schedule. This scenario is also bound to affect the shear strain reversal in the pass, although such components will not built up through-process accumulation as in the case of the 2304 steel. Those effects are expected to be maximum during the roughing pass, but at the finishing stands some strain accumulation and interaction with static and dynamic recrystallisation might take place. Also, as the result of alternate H-V roll positions,Page 214there is a reversing of normal strain components; depending on the rolling temperatures, such reversing might affect recrystallisation, as found by mechanical testing in a previous ECSC project [A3.8].

Hot working behaviour can be better studied by mechanical testing under controlled temperature strain rate conditions. Although the varying strain rate profiles that develop in rolling process cannot be reproduced experimentally, the mechanical test at constant rate conditions can be used to construct viscoplastic constitutive laws to be used in rolling FEM simulation. This transfer implies to validate first the immediate response of the material to sudden strain rate changes by mechanical testing. Relevant to the current complex loading project, differences in the hardening rate under normal and shear loading need to be investigated by compression and torsion.

The rolling strain rate is a sharply varying instantaneous measure, the history of which is not accounted even in the most sophisticated models. Consequently, prior rolling deformation strain rate measures have to be investigated to make use of the recrystallisation data obtained at constant strain rate mechanical testing.

Shear strain components suffer a reversal under the roll gap, both in flat and bar rolling. In addition, tandem H-V bar rolling configurations introduces a normal elongation-transverse spreading cycle that implies reversing of normal strain components. Mechanical testing represents a flexible tool to understand and quantify the effect of strain reversal on both hardening rate and recrystallisation kinetics.

A good interaction with the associated partners OULU and CEIT were maintained during the project. The early results were used for development of the constitutive model and the application of this to FEM simulations used for both defining conditions for further testing and validations.

For the 55Cr steel mechanically tested by both Centres, CSM took maximum care in cutting the samples from the same non central areas of the cross section, thus minimising the effects of segregation. CSM made a metallographic study on the speed of grain growth using different thermal cycles (Table A3.2). These results were then used to define a common preheating cycle by OULU and CEIT, thus avoiding excessive grain size scattering with a target grain size of 35 µm. It consisted of a heating up to 1050°C, holding 120 s, ramping at 2°C/s to the testing temperature (between 1050 and 950°C), which was next held for 15 s before testing. OULU also used the CSM results in Table A3.2 to generate different austenitic grain size at the deformation temperature in their study of the effect of such parameter on recrystallisation.

The University of Oulu performed monotonic compression, double hit compression, stress relaxation and tension/compression on the medium-C 55Cr3 steel. CEIT performed plane strain compression, and both monotonic and strain reversing torsion on both the 55Cr3 and 2304 steels.

CSM machined specimens of industrially as-cast and rolled material and distributed them to the partners the University of Oulu (55Cr3) and CEIT (55Cr3 and 2304).

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The raw material data were used by CSM to elaborate constitutive laws, accordingly to the general framework detailed in §A3.3. The constitutive laws were then implemented into a general purpose FEM program (§A3.3.3) and used for computing simulations and validation of the mechanical test results (§A3.3.3.1) and representative rolling scenarios (§A3.5).

Important interactions between partners were:

The University of Oulu validated the deformation response of the steel under *continuous and abruptly changed strain rate* against their results from isothermal *constant* strain rate. The results of this study justifies in a first approximation the use of a state parameter in the constitutive law developed by CSM for the medium-C steel as detailed in §A3.3.1. The FEM-implemented CL was then validated against OULU Gleeble compression and CEIT torsion testing. Of particular interest is the comparison of the results from numerical (immediate response) and experimental varying strain rate testing, which gives a measure of the delay in the response of the real material (§A3.3.3.1.2).

the recrystallisation kinetics results obtained from stress relaxation testing at constant and abruptly *changed* strain rate by OULU has allowed to calibrate/validate the best prior strain rate measure from general deformation (§A3.3.2.3) to be used in the interstand rolling model.

the results of the torsion experiments by CEIT have been used to compare torsion and compression hardening rates under monotonic loading conditions (§A3.3.3.1).

the amplitude of the shear strain reversals obtained from CSM flat and bar rolling simulations (§A3.4.2.1) was used in the definition of the torsion testing programmed by CEIT. Thus, shear strain reversals +0.06/-0.06 for flat rolling of duplex stainless steel and +0.15 / -0.15 for bar rolling of medium-C steel were selected.

CEIT torsion reversal data for both steels were used to modify the constitutive law for strain reversals and validate the FEM simulation results. (§A3.3.3.1.3).

The two hot rolling industrial lines of interest associated to the materials in §A3.2.1 are the ORI-MARTIN plant at Brescia and the THYSSEN-KRUPP ACCIAI SPECIALI TERNI.

The ORI-MARTIN mills produce bars of diameters ranging from 18 to 60 mm and wire rods from 42 to 5.5 mm. It uses as-cast feedstock of 140 mm x 140 mm preheated at typical exit temperatures of 1100°C, which are roughed in a single non-reversing rolling stand before entering two intermediate mills that have a total of 13 stands. The line then splits in three finishing mill branches accordingly to the final product: bar, rod or wire. The current research is focused on a stand of the intermediate mills placed before the splitting of the line and after the first loop, where the shear-cutter takes samples of rolled stock for routine checking. This choice allowed characterising the material entering the case study stand andPage 216therefore using a more realistic constitutive behaviour in the FEM simulation. Typical rolling temperatures for the case study stand for the medium-C steel studied are 1030°C- 1040°C with rolling speeds of the order or 1.7 m/s and pass reductions 0.3 -0.45.

The hot line of THYSSEN-KRUPP ACCIAI SPECIALI TERNI rolls slabs of 174 mm thickness to 1.5 to 12 mm. The 2304 steel slabs are preheated to around 1150°C and then rolled first in a reversible roughing stand in 5/7 passes and finally in the 7 stands of the tandem finishing mill. Typical rolling temperatures are 1050-1140°C in the roughing and 920 – 1000°C in the finishing mill. In some cases a 3-roughing passes pre-rolling is used [A3.7]. Typical rolling speeds are 1 to 2 m/s in the roughing and 1.4 to 12 m/s in the finishing, with pass reductions ranging from 0.15 to 0.45.

Representative rolling conditions were studied by rolling experiments at CSM two high stand Danieli, 0.475 m roll diameter pilot mill. Experimental evolution of temperature evolution from unloading, rolling forces and power consumption were recorded through the experiments.

The mechanical testing data produced by the University of Oulu and CEIT at constant[VER SIMBOLO EN PDF ADJUNTO] conditions were used to formulate a rate-dependent, isotropic hardening constitutive law for both materials.

The numerical fitting procedure follows the steps defined by Anand [A3.5], Brown et al [A3.6], modified by Bianchi et al [A3.7, A3.25].

Classical Hot Working Theory (HWT) assumes the peak stress represents a balance between hardening and dynamic recovery and can be linked to a ‘Temperature- compensated strain rate’ (Equation (A3.1))*. *Figure A3.1 shows the medium-C steel experimental peak stress data in a range of temperature strain-rates and their fitting to a function of *Z*, which allows to write:

[VER FORMULA EN PDF ADJUNTO]

The fitting for the initial yield stress (Fig. A3.1) gives also a good correlation, but the slope (inverse of the strain rate sensitivity) and *s*-factor are different than for the peak fitting:

[VER FORMULA EN PDF ADJUNTO]

The *s*-variable has been named ‘hardness’, because of its independence of strain rate and temperature in the early models [A3.27]. This condition not necessarily occurs, as revealed by the different strain rate sensitivities *mo* and *ms* in Fig. A3.1. Consequently, it is convenientPage 217to take the steady state [VER SIMBOLO EN PDF ADJUNTO] as the reference and normalise the initial hardness accordingly [A3.7, A3.25]:

[VER FORMULA EN PDF ADJUNTO]

Also, the peak stress does not necessarily represent the hypothetical steady state if any DRX occurs. In such case, the [VER SIMBOLO EN PDF ADJUNTO] value has to be identified from extrapolation of early deformation stresses where DRX is absent [A3.8-A3.9]. The evolution of the material hardness *s* between the two extremes:

[VER FORMULA EN PDF ADJUNTO]

Evolves accordingly to a constitutive law that needs to be identified. Following previous work [A3.8-11,A3.25], the following equation was explored for the steels under study:

[VER FORMULA EN PDF ADJUNTO]

The flow stress for any intermediate state in Equation (A3.5) is then written with the same functional forms than Equations (A3.2-A3.3) as:

[VER FORMULA EN PDF ADJUNTO]

Tables A3.4-A3.5 give the constitutive parameters obtained for the two steels under study.

Curve 2 in Fig. A3.2 shows an example of this constitutive fitting to a 55Cr3 steel experimental data over a strain range up to 0.8. The light decay of the ‘steady state’ or ‘saturation’ hypothetical stress associated to [VER SIMBOLO EN PDF ADJUNTO] is due to heat generation.

By comparing the Level*I*constitutive predictions against the experimental data, the evolution of the flow stress softening [VER SIMBOLO EN PDF ADJUNTO] exclusively due to DRX, can be identified (e.g.: Curve 4 in Fig. A3.2) and written in terms of the DRX volume fraction *Xd* [A3.25] which in turn depends on strain rate, temperature and strain in excess of the DRX strain:

[VER FORMULA EN PDF ADJUNTO]

Consistently with the formulation of the first level (Equation (A3.6)), the evolution of the two history carriers hardness *sdef* and *Xd* past the [VER SIMBOLO EN PDF ADJUNTO] threshold are also written incrementally:

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[VER FORMULA EN PDF ADJUNTO]

Otherwise:

[VER FORMULA EN PDF ADJUNTO]

The total DRX fraction under varying [VER SIMBOLO EN PDF ADJUNTO] and *T *conditions are obtained as:

[VER FORMULA EN PDF ADJUNTO]

Curve 5 in Fig. A3.2 shows the resulting fitting to the experimental softening and Curve 3 the final fit to the flow stress.

The* integral* relationships for the history carriers *sdef* and *Xd* as obtained from the isothermal-constant- [VER SIMBOLO EN PDF ADJUNTO] mechanical testing are not suitable for FEM simulation of hot Metal Forming,because in any of these processes there is material flow through [VER SIMBOLO EN PDF ADJUNTO] and *T* localizations. For these conditions, it is preferred to use *incremental *forms of the history-carriers evolution as given by Equations (A3.6), (A3.9) and (A3.10), and integrate them (e.g.: Equation A3.13) along the material path at the particular quasi-static [VER SIMBOLO EN PDF ADJUNTO] and *T* conditions of each of the small time increments of a FEM analysis. The underlying assumption to this approach is the immediate reaction of the structure to the new local conditions, hypothesis that is discussed against experimental results in §A3.3.3.1.2.

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*Level III:*incorporation of static softening). Case study: 55 cr3 steel

An integral model for post-deformation volume fraction softening *Xs* was produced by the University of Oulu from double hit and stress relaxation [A3.28] axisymmetric compression:

[VER FORMULA EN PDF ADJUNTO]

These results require some re-casting in order to be used by the process models, because:

as opposed to the controlled isothermal conditions of the mechanical testing, the rolling-stock temperature varies in the interstand as consequence of loses by radiation and internal conduction from the hotter inner core to the surface.

there has not been a unique strain rate (in general also not a unique temperature) during prior rolling deformation.

Both issues being examined below. From the resulting *Xs *for the particular deformation- cooling conditions, the flow stress for subsequent deformation is computed with a *Level III* of the constitutive law using the law of mixtures:

[VER FORMULA EN PDF ADJUNTO] and the base equation (Equation (A3.7)). Available choices to compute the interstand softening *Xs* in the general Metal forming Case are examined below.

The FEM-thermal analysis subsequent to the rolling pass simulation proceeds at small time increments [VER SIMBOLO EN PDF ADJUNTO]. Taking advantage of this pacing, we can apply quasi-isothermally the Avrami’s integral relationship obtained from mechanical testing as described next.

If the volume fraction *Xi* at the *end* of time *ti *is made equal to the *starting *value *Xi+1* of the subsequent increment, an equivalent time *ti** for the new temperature *Ti+1 *results as:

[VER FORMULA EN PDF ADJUNTO] and the fraction recrystallised at *the end* of the interval *i+1*:

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[VER FORMULA EN PDF ADJUNTO] which allows defining the recursive relationship for the fraction *Xn* accumulated at time *tn:*

[VER FORMULA EN PDF ADJUNTO]

The [VER SIMBOLO EN PDF ADJUNTO] measure influences the reference time *t0.5 *(Equation (A3.15)) and consequently the speed of the recrystallisation kinetics (Equation (A3.14)). Moreover, the threshold for dynamic recrystallisation (Equation (A3.8)) that also delimits the SRX-DRX regimes,depends also on the prior [VER SIMBOLO EN PDF ADJUNTO]

Several [VER SIMBOLO EN PDF ADJUNTO] measures have been examined for varying strain rate histories available from FEM solutions:

a maximum strain rate along the considered deformation path:

[VER FORMULA EN PDF ADJUNTO]

a *strain-averaged* strain rate obtained by weighting the mid-increment *local *equivalent strain rate by its associated fractional equivalent strain increment:

[VER FORMULA EN PDF ADJUNTO] which might vary across the section; *pass* is the equivalent total strain of the pass computed along the considered material streamline.

A *time-averaged* strain rate, which also varies across the rolled section, obtained by weighting the mid-increment *local *equivalent strain rate by its associated fractional time increment:

[VER FORMULA EN PDF ADJUNTO] where # *tpass* is the total time taken by the deformation.

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In order to assess the most representative strain rate defining the *Xs* kinetics from prior deformations occurred at varying strain rates, the University of Oulu carried a series of stress relaxation experiments at the temperatures of 950°C and 1050°C on the 55Cr3 steel. CSM applied the prior deformation measures described in §A3.3.2.2 above, to the OULU mechanical testing conditions.

In accordance with typical rolling pass conditions, all OULU tests were carried up to total strains of about 0.5 to 0.6 and at abruptly changed [VER SIMBOLO EN PDF ADJUNTO] conditions. The recrystallised volume fraction *Xs* was compared against the Avrami fittings (Equation (A3.14)) corresponding to constant [VER SIMBOLO EN PDF ADJUNTO] (Fig. A3.3). Their results point to a shift of the recrystallisation Kinetics to the conditions defined by the second strain rate.

From the experiments in which the strain rate was changed from 10 s^{-1} to 1 s^{-1}, the prior strain rate measures computed by Equations (A3.21) and (A3.22) are:

[VER FORMULA EN PDF ADJUNTO]

For 1050°C, the *Xs* Avrami fittings (Equation (A3.14)) obtained using these rates in the *t*^{0.5 }computations (Equation A3.15) show that at the highest strain rate the best result is given by the strain-average strain rate initially, but for longer times the best performance is the time- averaged strain rate. For the testing at 950°C under the same conditions [A3.25], and at 1050°C for the slower strain rate levels (Fig. A3.3), the best results are obtained by using the time average. The time-average procedure, i.e. the mean strain rate, was also found reasonable by Karjalainen et al [A3.31] for a Ti-bearing steel, especially in the case of SRX and decreasing strain rate. They observed that the ‘effective strain rate’ was close to the final (second) strain rate.

The material law obtained from the deformation conditions under Gleeble axisymmetric compression was next implemented as rate-dependent plasticity into the commercial FEM code Abaqus®.

The FEM simulations make the assumption that the second invariant of the stress and strain components (the equivalent stress and strain) do not depend on the complexity of the deformation state. As this varies in the several situations considered: compression, torsion, bar and flat rolling, it is important to validate the simulations against the experimental results and assess whether corrections to the basic constitutive behaviour obtained in §A3.3.1 and §A3.3.2 are required.

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Figure A3.4 shows the application of the FEM model to simulate a Gleeble compression test at 1050°C and 1 s^{-1}, revealing the excellent agreement between the raw experimental axial load data obtained by OULU and the simulation by CSM. It is interesting to note also the uniformity of the plastic strain distribution up to plastic strains around 0.6.

Next, the model was applied to a torsion experiment from CEIT under the same nominal hot working conditions. Figure A3.5(a) shows a detail of the simulating conditions, in which the symmetry plane at mid-gauge length has been exploited to cut down computing.

The resulting plastic strain distribution reproduces well its linear dependencies across the specimen radius (Fig. A3.5(b)) and amount of twist 4(proportional to the time at constant twist rate conditions in Fig. A3.5(c)) assumed by both the Fields-Backoffen [A3.29] and effective radius [A3.30] classical analyses. The linearity of the strain with the twist(time) breaks down at the specimen shoulder from strains about 0.33 (Fig. A3.5(c), strain rate= 1 s- ^{1}). In general, this local breakdown develops only on a very narrow band that is about 4% of the specimen half-gauge length (Fig. A3.5(d)).

Figure A3.6(a) shows the evolution of the thermal field at different surface locations along the specimen length. The results have been obtained on the assumptions of a constant initial temperature 1050°C, internal conduction, adiabatic heating with a 0.9 conversion factor of mechanical work into heat, zero flux towards the shoulder, no radiation loses but also no electric heating. The higher temperature localization towards the middle of the gauge length (Fig. A3.6(c)) justifies the specimen break up there, result that is not available in the conventional Fields-Backoffen model. The thermal evolution in time is shown in Figure A3.6(b).

A comparison of the raw torque-time experimental and simulation results is shown in Fig. A3.7. The agreement is very good in the hardening-dynamic recovery region and up to early dynamic recrystallisation but the FEM predictions start to deteriorate at strains beyond 0.7. This is adequate for the bar rolling schedule of the steel under consideration, where the maximum pass strain is no larger than 0.8 and is completely erased in the following interpass due to the fast MDRX recrystallisation kinetics. Two possible reasons can explain the mismatch at larger strains in Fig. A3.7. The first is the breakdown of the thermal conditions imposed in the simulation to follow the real conditions of the experiment. The second is that the numerical predictions were computed with a constant grain size that in the current simulation version does not evolve as DRX progresses. Should the mean grain size be updated by incorporating the DRX grain refinement, this is bound to lower the critical strain to DRX and hence to accelerate the softening, thus bringing the simulation torque closer to the observed experimental values.

Figure A3.8 shows a comparison of the resulting stresses obtained by the different methods. The experimental stress level from the Gleeble compression data is about the same that the one obtained from the raw torque curve using the conventional classical model [A3.29-A3.30] based on the product of a power of strain times power of strain rate. However, the strain to peak stress-hence the threshold for DRX- predicted by the torsion classical model is larger than both the current model and the compression experiments.

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It can also be seen in Fig. A3.8 how the Gleeble experimental strain-stress data are very well matched by the FEM local stresses in torsion at the middle of its specimen gauge length. Even the stress at the torsion specimen shoulder does not exhibit a large departure from the compression-testing curve.

The University of Oulu next carried out experiments in which the constant strain rate was abruptly changed at a certain strain [VER SIMBOLO EN PDF ADJUNTO] under both increasing and decreasing conditions simulating rolling. The experiments were done with [VER SIMBOLO EN PDF ADJUNTO] occurring at both the hardening- dynamic recovery region and past the onset of DRX (Figs. A3.9 and A3.10). A comparison of the results against the constant strain rate testing showed that within the experimental scatter the flow stress of the second [VER SIMBOLO EN PDF ADJUNTO] regime tends to approach asymptotically the single- [VER SIMBOLO EN PDF ADJUNTO] curve [A3.25]. There is always a short transient that is the composite effect of the strain at which the applied [VER SIMBOLO EN PDF ADJUNTO] stabilises at its new value, dependent on the response of the Gleeble simulator, and a purely material response lagging behind the mechanical state [A3.31,A3.32].

CSM constitutive model in §A3.3.1 is based on the instantaneous response of the structure to the new strain rate conditions. Thus, a comparison of the FEM (instantaneous) and experimental (real material) results under the same kinematics condition gives an estimate of the delay in the material response.

The application of the FEM simulation §A3.3.3 to the experiments, enforcing the same kinematical conditions by using exactly their ramping displacements, avoids any Gleeble machine dynamic effect in the simulation. The results in Figs. A3.9 and A3.10 show in general terms a reasonable agreement. Some scattering is apparent at the highest strain rate due to the experimental difficulty in maintaining this constant. The difference is bigger near the yield point, where even the improvement given by Equation (A3.4) fails to fit properly. The agreement, however, in general tends to improve on reaching stress levels near saturation. The clear effect of recrystallisation on the stress levels after strain rate changes can also be seen.

For all increasing and decreasing strain rates the FEM (immediate response) predictions comes earlier than in the real material. The delay is of the order of 0.02 in strain.

Both in flat and bar working stocks, the rolling process involves a sharp and fast shear strain reversal under the roll gap (Figs. A3.14 and A3.15) and a subsequent inversion at the next stand if interstand microstructural changes do not erase the history (§A3.4.3 and §A3.4.4). The set of torsion experiments agreed with CEIT reflected that amount of shear strain and inversion times, with a sequence composed of immediate reversal followed by reversal after a holding time of about 1 s.

CEIT torsion experimental results clearly showed a lower flow stress level on the loading subsequent to the first reversal [CEIT Appendix in this Report], respect to the monotonic case. It is important to note that such behaviour not only appeared in the extremely fast recrystallisation non-micro alloyed medium-C steel in the austenite region currently studied, but was also previously detected in a non-recrystallising duplex stainless steel [A3.8, A3.35].

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The current constitutive model so far described is based on a flow stress that relies on an ever-increasing equivalent plastic strain (Equation (A3.26)) that does not react to reversals. Consequently, in order to simulate the experimental results within this constitutive framework, the resulting hardness from Equations (A3.6 ) and (A3.9) has to be softened after the first reversal.

Two hypothesis were examined and implemented in the FEM code:

(a) anisotropy creep *yield surface *[A3.26,A3.36]. defined as:

[VER FORMULA EN PDF ADJUNTO] where the coefficients *F*,….*N* are function of ‘*potential ratios*’ defined in terms of stresses of single deformation modes (i.e.: compression, torsion,etc.) as:

[VER FORMULA EN PDF ADJUNTO]

The effects of shear strain [VER SIMBOLO EN PDF ADJUNTO] reversals were then simulated by modifying the *Rij *as follows: *Rij = *1 for any monotonic loading

[VER FORMULA EN PDF ADJUNTO]

In this hypothesis *only* the shear stress component that underwent strain reversal was intended to be reduced. The early FEM results produced with this hypothesis for the torsion reversal experiments were not conclusive and showed little difference respect to the conventional isotropic hardening constitutive law that evolves in terms of the ever increasing plastic strain.

(b) rate dependent isotropic plasticity with the whole yield surface softened after the first reversal as:

[VER FORMULA EN PDF ADJUNTO]

The simulation attempts by using Equations (A3.24)-(A3.25) to introduce a Shear Strain Reversal Softening (SSRS) of the flow stress in the constitutive law should be regarded as an approximation to the real complex loading situation and considers a flat stress level for any strain beyond the first reversal, with elimination of the hardening rate.

A comparison of the experimental results against FEM predictions using the monotonic loading constitutive model (Equations (A3.6) and (A3.9) for the history carrier and the basic Equation (A3.7) for the stress) and the new level accounting for the softening due to reversals (Equation (A3.25)) is shown in Fig. A3.11. Note that torque has been plotted as its absolute value to compare changes of level after reversal. The agreement is good enough to consider the constitutive model validated for the torsion conditions.

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**(WP5) **

Practically all current FEM work on process modelling uses constitutive laws obtained from the stress/strain data resulting under the nearly uniform strain state achieved in mechanical testing and this is carried at constant strain rate [VER SIMBOLO EN PDF ADJUNTO] and temperature *T*. These conditions contrast with rolling deformation, where several deformation modes might become activated at varying [VER SIMBOLO EN PDF ADJUNTO] and *T*.

Two plastic strain measures are found in FEM codes. They are the equivalent strain:

[VER FORMULA EN PDF ADJUNTO] where [VER SIMBOLO EN PDF ADJUNTO] is the always-positive strain rate defined from its components as:

[VER FORMULA EN PDF ADJUNTO] and the effective strain:

[VER FORMULA EN PDF ADJUNTO] updated from the [VER SIMBOLO EN PDF ADJUNTO] strain components, these not necessarily increasing monotonically. The last condition is particularly true for the shear strain components developed towards the rolling stock border, which change sign as revealed by the very well known ‘Neutral Point’ of flat rolling.

The measures given by Equation (A3.26) and Equation (A3.28) only coincide under proportional loading conditions, which is not always the case of the complex loading situation that develops in metal forming.

The constitutive law-FEM model developed and validated in §A3.3 was next applied to rolling case studies. The first stage §A3.4.2.1 - §A3.4.2.2 of this application implies the underlying hypotheses:

in presence of localizations, the material reacts instantaneously to any new [VER SIMBOLO EN PDF ADJUNTO] conditions, therefore following an equation of state.

the constitutive law is invariant in terms of the equivalent total strain in spite of the differing deformation state found in the tests and in the rolling pass.

While a second stage §A3.4.3-§A3.4.4 considers:

incorporation of Shear Strain Reversal Softening (*SSRS*) effects.

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Tables A3.6 and A3.7 show typical conditions used in the FEM simulations of both bar and flat rolling.

Temperature is the process variable of more impact to the constitutive behaviour. Figure A3.12 shows the temperature distribution predicted by the FEM analysis for the medium-C steel at the 8th stand of the bar mill studied and detail of its evolution in time during the subsequent interpass. In spite of the fast cooling of the rolling stock surface when contacting the rolls, its temperature recovers within the interpass due to conduction from the internal core. For the conditions studied in Table A3.6, the temperature difference between core and surface at the entry of the subsequent stand is about 90°C, with a 25°C maximum temperature rise due to conversion of mechanical work.

The process variable of second constitutive importance is the strain rate. Figure A3.13 shows examples of strain rate distributions predicted by the model for the stand under study. The pattern of the effective strain rate normally used in the constitutive description is shown in Fig. A3.13(a) and the contribution of its main normal and shear components in Fig. 13(b and c) respectively; the meaningful values of these last justify a constitutive characterisation by both compression and torsion. Even though the shear strain in the pass is relatively low (Figs. A3.14-A3.15), its reversal entails a high shear strain rate change (-11 to +12 s-1 in Fig. A3.13(c)). As the strain rate vanishes outside the roll gap, a meaningful value is required to compute recrystallisation kinetics; Fig. A3.13(d) shows the average measure computed accordingly to §A3.3.2.2.

The process variable of third constitutive importance is the strain, which is related to the rolling reduction. Figure A3.14 shows the evolution of the strain components as revealed by a simulation of a typical flat rolling pass of duplex stainless steel 2304, at three through- thickness locations: (a) midplane, (b) a quarter thickness and (c) underneath stock skin. The results have been produced for a rolling reduction r=25.8%, corresponding to the engineering strain 0.264 for plane strain conditions, using the base constitutive law *Level I *described in §A3.3.1.1. It can be clearly seen that the rolling-thickness directions (*rd,td*) shear strain component near the strip symmetry mid-plane tends to zero but its absolute amount increases towards the rollingstock surface with a very well defined reversal. For this case study the amplitude of the reversal is from –0.052 to +0.044 at quarter thickness and from –0.07 to +0.064 at a point 1.05 mm under the contact skin**.**

Bar-rolling simulation results for a typical round-oval pass, for an engineering area reduction AR=38 % are shown in Fig. A3.15. The simulation has been produced with the *Level II *constitutive law for 55Cr3 medium-C steel introduced in §A3.3.1.2 in which the hardness is modified to incorporate DRX. As in the flat rolling case, the shear strain components towards the midplane of the bar parallel to the roll axis (points (a) and (d)) are very small. Their value increase towards the border in contact with the rolls (points (b) and (c), located about 2 mm under the skin of the rolled bar). The shear component defined by the pair (*rd, td*) always undergoes some reversal in the range +0.25 to –0.08 (point (b)) and +0.15/-0.14 (point (c)). The component (*rd, wd*) undergoes a non-monotonic shear straining but without significant reversion. Near the inflection point of the grooved roll (c) there is also a significant shear strain component (*td, wd*).

The simulation results for both flat and bar rolling in Fig. A3.14 and Fig. A3.15 clearly show that as a consequence of the shear strain reversal at the outer layers of the stock, the morePage 227realistic effective strain computed with Equation (A3.28) is smaller than the equivalent value obtained from Equation A3.26.

Another important result of the FEM simulation is their difference respect to the conventional area reduction AR industrial parameter for bar rolling. While in the case of flat rolling the AR=25.8% overall parameter is close to the 0.27-0.29 equivalent strains (Fig. A3.14), in the case of bar rolling an AR=38% implies equivalent strains around 0.7 (Fig. A3.15).

Figure A3.16 shows the evolution of the DRX volume fraction *Xd* predicted by Equation A3.13, which is obtained by adding up Equation A3.10 contributions each increment the strain, exceeds the instantaneous [VER SIMBOLO EN PDF ADJUNTO] given by Equation (A3.8). The maximum *Xd* occurs around the centre streamline *(I)* and decreases towards the contacting surface *(II) *(Fig. A3.16). The last result is due to the increase of [VER SIMBOLO EN PDF ADJUNTO] (Equation (A3.8)) mainly as consequence of the *T*–drop produced by the chilling effects of the rolls (Fig. A3.12(c)), combined with the lower H resulting from the reversal of its shear strain components towards the bar border (Fig. A13.15). Figure A3.16 also indicates that the bar’s outer layers will undergo SRX as consequence of *Xd* =0 while the core will suffer MDRX.

Interstand softening influences the material strength and therefore the demand to the tooling at the subsequent stand. Two main issues needs to be considered:

The strain rate influences both the critical strain for dynamic recrystallisation [VER SIMBOLO EN PDF ADJUNTO] (Equation (A3.8)) and determines whether the subsequent softening is static or metadynamic (Equation (A3.15)). As rolling deformation involves material flow with varying strain rate, a representative strain rate needs to be defined.

The simplest Homogeneous Deformation Model (*HDM*) considers that the whole volume of the bar under the rolling gap was subjected to a mean, constant strain rate obtained as a ratio of the Area Reduction divided by the roll-gap residence time. A more realistic model uses a strain rate obtained from Equivalent Rectangular areas, Constant Volume Elongation and residence time *HDM(EQR+CVEL)* [A3.33 – A3.34]. As the strain rates computed by FEM models vary locally and are much higher than those predicted by simplified models, the alternative measures introduced in §A3.3.2 are now applied to rolling.

A comparison of the recrystallisation predictions for different prior strain rate definitions for the stand under study is shown in Table A3.8 and discussed next:

The over simplistic *HDM(AR)* predicts:

[VER FORMULA EN PDF ADJUNTO]

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[VER FORMULA EN PDF ADJUNTO] results those are constant across the product section.

On the contrary, the measures introduced in §A3.3.2 vary across the section. Their computation along streamlines of the FEM solution, for the two typical exit locations in Fig. (A3.16), gives for the maximum strain rate along the path:

[VER FORMULA EN PDF ADJUNTO] and for the average prior strain rate measures:

[VER FORMULA EN PDF ADJUNTO]

Although the FEM average strain rate measures gives larger strain and critical recrystallisation strain than the HDM-EQRCVEL model, both of them predict MDRX along the considered streamlines. The FEM averages along path, however, are preferred because of the validation exercise described in §A3.3.2.3 (Fig. A3.3).

The recrystallised volume fraction *X* at the interpass is traditionally computed in terms of the elapsed time and the local temperature, assumption that overestimates the recrystallisation Kinetics. Consistently with the treatment of all the other history variables in the constitutive framework of this research, the Avrami equation is also written *incrementally* and integrated under the non-isothermal conditions determined by the interpass cooling as detailed for the general case in §A3.3.2.1.

Figure A3.17(a-c) shows the amount of FEM pass-strain in excess of the instantaneous dynamic strain determined with some of the [VER SIMBOLO EN PDF ADJUNTO] definitions above, for the 55Cr3 medium C steel and rolling pass subsequent to the microstructural changes in (Fig. A3.16), with an initial insterstand temperature 1040°C. No substantial differences are seen, result which is due to the predominance of the temperature factor in Equation (A3.15) and the low exponent affecting the strain rate in [VER SIMBOLO EN PDF ADJUNTO]. Equation (A3.8), Table A3.4).

Recrystallisation is very fast at the end of the following interstand (Fig. A3.17(d)) and no significant differences in volume fraction *Xs* results by applying the various models [A3.11], although the scatter in grain sizes from SRX and MRDX can be significant. Two regions develop slower recrystallisation kinetics: a fraction of the contacting layers cooled by the rolls and a zone at the free surface; although this last remains at higher temperature than the former, has underwent lower strain in the previous rolling pass.

The effects of the different strain rate hypotheses on the rolling forces are almost negligible because of the extension in volume of the fully recrystallised area and the small product cross section. The torque, however, which depends on the material strength at the border, shows some minimal sensitivity to the choice of the strain rate (Fig. A3.17(e)).

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The different constitutive response of the material to single straining modes examined in §A3.3.3.1 needs next to be mapped upon the hot rolling case, where many of these modes coexist (Fig. A3.14 and Fig. A3.15). The constitutive model validation against mechanical testing has shown that for *monotonic* loading the constitutive law from the axial *compression *Gleeble experiments gives also good approximation to *shear strains* up to about 0.8 (Figs. A3.7 and A3.8). The next issue to consider is how to incorporate the effect of *shear strain reversals* into the rolling model.

Of the two hypothesis examined in connection with the torsion twist reversal, and in view of the results validated in Fig. A3.11, the second of them given by Equation (A3.25) was selected with

[VER FORMULA EN PDF ADJUNTO] where *rd* and *td* are the main rolling and thickness direction respectively. The switching during the FEM rolling simulation from the conventional hardness resulting from Equations (A3.6), (A3.9) and (A3.16) to the softened hardness in Equation (A3.25) is driven by an auxiliary variable that monitors the change in sign of the shear strain component given by Equation (A3.33).

Figure A3.19 compares the strain distribution across a rolled bar section for the conventional and modified constitutive law obtained for the 55Cr3 medium-C steel. As the result of the material softening due to reversal (Fig. A3.18) the second model predicts a bigger plastic strain on the same area (Fig. A3.19).

The modified *SSRS* constitutive law predicts a stress softening at the bar border not found in the conventional isotropic model (Fig. A3.20), which in turn affects the demand to the tooling. The results in Fig. A3.21 for an intermediate rollingstock exiting at 52 mm diameter, show that the differences in rolling forces are almost negligible, but there is 12% difference in the predicted torque. For the wider rolling stocks resulting in flat rolling, both forces and power consumption show difference to the incorporation of the strain reversal softening effect in the constitutive law (§A3.5.2).

The differences between the equivalent (Equation (A3.26)) and the strain-reversal- influenced effective plastic strain (Equation (A3.28)), shown in Fig. A3.14 and Fig. A3.15, are also reflected in the pace of the dynamic recrystallisation kinetics. For the bar rolling pass of the medium C-steel examined, DRX is maximum towards the centre of the stock but the difference resulting from both strain definitions is minimal due to the small shear strains (Fig. A3.22, point (a)).

Although DRX at the outer layers of the rolling stock (e.g. point (b)) is of lower magnitude because of the chilling effects of the rolls raising the threshold [VER SIMBOLO EN PDF ADJUNTO], the influence of the shear strain reversal on both effective strain and recrystallised volume fraction is significant.

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There is a marked difference on the effect of the strain reversal on the two materials under study. The duplex stainless steel does not suffer DRX and therefore the outer layers of the strip undergo in each rolling pass a full shear strain reversal (Fig. A3.14(b and c)). On the other hand, the very fast recrystallisation non-micro alloyed medium-C steel develops DRX within the deformation zone reducing the magnitude of the strain that propagates. On the assumption that all strain components undergo the same softening than the effective strain:

[VER FORMULA EN PDF ADJUNTO] *unrecovered* strains smaller than the kinematics predictions of *Level I* will result.

Figure A3.23 shows the reducing effect of DRX on the strains predicted by the FEM model. The presence of substantial DRX at the bar core has a significant effect on the effective strain (point a), but its effect on the small *shear *components there is minimal.

Because DRX diminishes towards the bar border ( (*II*) in Fig. A3.16 and point (b) in Fig. A3.22), its effects on the *unrecovered strains* also diminish (Fig. A3.23, point (b)) respect to the core. In particular, there is almost no effect on the shear strain. The *unrecovered shear strains* at the bar outer layers are therefore significant and might preset reversals at the entry of the next rolling stand if the SRX/MDRX in the following interpass is not complete.

As the result of its negligible interstand recrystallisation [A3.7-A3.8,A3.35], the 2304 steel is ideal to study the effects of strain accumulation and residual shear strain after reversal. The torsion reversal tests performed by CEIT for the 2304 steel [A3.7-A3.8, A3.35 and current report] showed a Shear Strain Reversal Softening (*SSRS)* that has been incorporated into the constitutive model as a full contraction of the yield surface (Equation (A3.25)) once a the shear strain (Equation (A3.33)) first reverses.

The constitutive model defined by the CL-*Levels I *(§A3.3.1.1) and *IV* (§A3.3.3.1.3) was next applied to FEM simulation of both roughing and tandem configurations. Figure A3.24 shows the areas affected by shear strain reversal during two consecutive passes. The extension of the *SSRS* area through thickness does not present significant differences between the two configurations, with the only exception of a band at the midplane.

The evolution of the residual shear strain at different thickness is shown in Fig. A3.25. It is clear that for the same reductions, the reversing stands deliver a stock with larger shear strain accumulation and this builds up further through passes.

Figure A3.26 shows FEM predictions of specific rolling forces and power consumption per unit width for the roughing and tandem configurations, under the classical (*NO- SSRS)* and *SSRS *constitutive assumptions at equal initial temperature and pass reduction. As in the matching experiments (§A3.5), the simulation assumes that the microstructure in the first pass has been homogenized during the reheating, eliminating any cumulative effects. It can be seen that the difference between the *NO-SSRS* and *SSRS* predictions in the second pass is significant. At equal temperature, rolling speed and reductions, both the load andPage 231power demanded by the second pass appears to be slightly lower in the case of roughing sequences than in tandem sequences (Fig. A3.26).

A case study to assess the amount of normal strain reversal taking place between the H-8th and V-9th stands of ORI-MARTIN bar mill was carried out. At the H-8th stand exit, the effective strain H distribution across the oval section ranges from 0.4 to 0.73 (Fig. A3.15), while the normal strain component [VER SIMBOLO EN PDF ADJUNTO] in the rolling direction exhibits a reduced scatter from +0.35 to +0.38. The normal strain components in the direction *transversal to rolling*, however, show more variation. Thus, the [VER SIMBOLO EN PDF ADJUNTO] component varies from -0.15 at the free surface to -0.73 at the core (Fig. A3.27 (a)), the negative sign indicating fibber compression across the whole bar section. Full recrystallisation (*X=*1) at the 8th –9th interpass destroys the previous strain history, with the net result of an [VER SIMBOLO EN PDF ADJUNTO] at the 9 th stand varying from a –0.20 fibber compression at the tool/free surface edge to a +0.71 fibber elongation at the core (Fig. A3.27(b)). If recrystallisation is incomplete, however, some reversal will take place under the roll gap at the 9th stand. Assuming that all strain components undergo the same softening than the equivalent strain, the accumulated strain component[VER SIMBOLO EN PDF ADJUNTO] at the (*I+1*)th stand can be written as:

[VER FORMULA EN PDF ADJUNTO]

In addition to this strain reduction due to recrystallisation, the normal components [VER SIMBOLO EN PDF ADJUNTO] will undergo additional reduction due to their reversal. In the extreme case of recrystallisation absence (*X*=0), the resulting [VER SIMBOLO EN PDF ADJUNTO] varies from +0.11 elongation near the symmetry midplane to –0.5 shortening at the tool/free surface border (Fig. A3.24(c)). Although this variation is dependent on the groove geometries, the general trend is a reduction of the spread of the transversal-to-rolling-direction normal strains due to the alternating H-V configuration.

The general incomplete recrystallisation case (0 X ) will fall between the two extreme situations shown in Fig. A3.24(b and c). Thus, Fig. A3.24(d) shows the [VER SIMBOLO EN PDF ADJUNTO] computed for the *X *fraction in Fig. A3.17.

For the particular 55Cr3 steel and rolling conditions examined, recrystallisation is complete almost elsewhere and therefore the [VER SIMBOLO EN PDF ADJUNTO] strains will not either propagate or undergo significant reversal during deformation in the next rolling stand. This scenario is not bound to change in the following mill stands because of the temperature raise due to mechanical working and the effect of such variable on the *t0.5* (Equation A3.15).

An important result from Fig. A3.24 is that there is more straining *under *the second roll gap in the directions transversal to rolling when previous interpass recrystallisation has been full. Strain reversals due to incomplete recrystallisation in presence of H-V stands tends to cancel out strain accumulation on those directions. The rolling-thickness directions *shear*Page 232strain also undergoes reversal under the roll pass (Fig. A3.15), and this is also sensitive to the magnitude of interstand tensions.

Plates of both 2304 duplex stainless steel and 55Cr3 medium-c steel were rolled in the 470 mm diameter, two-high Danieli mill at CSM. The 35-40 mm thick specimens were reheated at the temperatures detailed in Table A3.9 for 1 hour. Accordingly to the experimental testing carried by CEIT and OULU, the selected rolling temperature and reductions define non-recrystallising conditions for the 2304 steel and full interpass recrystallisation in the 55Cr3 steel. thus, the experiments on the 2304 could be used to investigate cumulative effects between passes, whilst the rolling on the 55Cr3 defines almost independent passes.

Continuous data were acquired from the mill load cells at intervals of 0.05 s and from a thermocouple embedded at midthickness of the plate at intervals of 0.5 s. Figure A3.28 shows an example of the temperature records for the case study A30 in Table A3.9.

Both the tension and current motor consumption were recorded at the much higher frequency of 8192 acquisitions per second (Fig. A3.29(a and b). These data were later decoded using a Fourier Transform algorithm which eliminates the contributions of frequencies greater than the first significant noise frequency and returns the data thus filtered. Figure A3.29(c and d) shows typical experimental rolling force and power consumption data obtained for the two passes of case study A30.

The experimental results were next used to validate the predictions of the fem model. For the first pass of the non-recrystallising 2304 steel and for all independent passes of the fast recrystallisation 55Cr3 steel, the predicted rolling forces were within a -4% to + 14% error for the 2304 steel and +8% to +18% for the 55Cr3 steel (Table A3.9). The associated power consumption is in an error band respect to experiments of –7% to +9% (2304) and -11% to +6% (55Cr3).

For the 2304 steel, the absence of recrystallisation makes the effect of the strain reversal in the *first *pass to influence the *second* pass. Figs. A3.27, A3.30 and A3.31, show that both the separating force and the power consumption for the second pass differ depending on whether the SSRS or NO-SSRS constitutive hypothesis is used. The resulting error of the force FEM predictions, was between –24% and +7% (one exception at +11%) for the SSRS hypothesis and –9% and 12% (one exception at +19%) for the NO-SSRS. The associated error in power was –10% to +16% (with one exception at +27%) for the SSRS and 0 % to 19% (exception at +33%) for the NO-SSRS. The two main causes of errors are: the slowdown of the roll speed when the mill is loaded and the definition of a steady state with the short rollingstock length used (300 mm).

The numerical predictions for force and power under both the SSRS and classical NO-SSRS hypothesis are compared against the experimental data for one case study in Figs. A3.30 and A3.31. The analysis is restricted to the second pass, where any softening due to strain reversal could be detected and both roughing and tandem configuration are examined. Being practically impossible to attain the same temperature in all the experiments, allPage 233computations have been converted to the temperature of the roughing R2 pass, which is then taken as a reference, by using

[VER FORMULA EN PDF ADJUNTO] and the Sims’ mean strain rate [A3.38] was used to compute the Zener-Hollomon parameter Z (Equation (A3.1)).

Both the experimental and FEM predictions (Figs. A3.30-A3.31) indicate that once this conversion has been made, the power/force demanded by a reversing pass is only marginally lower than for the equivalent conditions tandem pass.

Figure A3.32 compares the measured power at the 8th stand of ORI-MARTIN mill against the FEM torque prediction, this converted to power using the nominal speed and a transmission efficiency overall factor 0.85.

The research in this project aimed at identifying a general framework for development of constitutive equations for representative complex loading scenarios. constitutive laws incorporating not only the reaction to the instantaneous variables strain rate and Temperature -as normally assumed in classical hot working theories- but also including the material history have been developed for two steels of extremely different recrystallisation kinetics. The resulting constitutive law have been implemented into a fem code for thermal- stress analysis and extensively applied and to mechanical testing and rolling cases and the results validated against corresponding experiments.

The main outcomes of the project for the bar rolling of the 55Cr3 steel are:

Hardening rates from compression and torsion are comparable. Monotonic torsion results have been reproduced by a FEM simulation using a constitutive law from monotonic compression up to strains of 0.7.

The stress levels, hardening/softening behaviour and location of peak strains of both compression and torsion modes predicted by the FEM model are about the same up to strains 0.7. This result contrasts with the classical torsion analysis, which predicts about the same stress level but larger DRX strains.

The steel nearly follows an equation of state in presence of strain rate changes, with a strain delay about 0.02.

Shear strain reversal in torsion introduces a material softening that has been implemented within the isotropic hardening constitutive framework as a weakening of the whole yield surface.

The effects of different prior strain rate measures on the interpass recrystallisation have been examined. In spite of the large scattering in thesePage 234[VER SIMBOLO EN PDF ADJUNTO] values for the rolling pass studied, the effect on rolling forces is minimal because of the massive recrystallised volume. There is however, a minimum effect on the torque because its dependency on the steel strength at the nonrecrystallised outer layer.

The medium-C spring steel exhibits very fast recrystallisation kinetics. For the round-oval, 8th pass of the intermediate mill considered, the hypothesis of immediate reaction of the microstructure to the instantaneous [VER SIMBOLO EN PDF ADJUNTO], *T* conditions leads to DRX in a wide central area of the bar. Recrystallisation at the end of the following interpass is nearly complete apart from two small zones at the rolled and free surfaces. Both the FEM model and the second HDM model (EQRCVEL) with compatible constant volume elongation predict DRX and MDRX, in contrast with the over simplistic first HDM model based only on area reduction that predicts SRX.

The reversal of the normal strain components in the direction perpendicular to rolling during bar rolling is significant, but its effect for the considered H-V stands is minimised by interstand recrystallisation. There is more normal straining in the directions transversal to rolling when recrystallisation in the previous interpass has been complete.

The influence of the shear strain reversal on DRX is higher towards the rolling stock surface than at its centre, lowering there the DRX volume fraction predicted by the equivalent strain.

Because of its distribution on the bar section, DRX has a minimal effect on reducing the value of the shear strain towards the workpiece surface contacting the rolls. Thus, in general most of the shear strain that underwent reversal in a pass will reverse again at the next stand. However, in the case of the fast recrystallisation medium-C studied, SRX/MDRX will virtually erase previous strain during the interpass, thus preventing any further reversal.

The main outcomes of the project for the flat rolling of the 2304 steel are:

In spite of the differences in the through-passes accumulation of reversing shear strain under reversing and tandem rolling configurations, the extension of the area affected by shear strain reversal through thickness is almost the same.

In contrast with the bar mill, the rolling stock width in flat rolling enhances differences in both forces and power consumption in the 2nd pass. When all conditions are converted to the same temperature, the differences between tandem and reversing configurations are however negligible.

A comparison of experimental and simulation rolling results shows that the hypothesis of full softening of the yield surface after the shear strain reversal is too penalising. The experiments values are bounded between these SSRS predictions and the one predicted by the *Level I* that does not consider strain reversal effects.

A common result to both steels and process are:

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CEIT reported a development of material softening due to shear strain reversal.

This feature has been consequently incorporated into both steel constitutive descriptions with the results for the particular conditions reported above.

The delay in the microstructural response resulting from comparison of the instantaneous numerical solution and the real material response might have a strong impact on DRX during bar rolling, where the residence time is very short. For a typical 5 ms residence time, the results in §A3.3.3.1.2 suggest that the delay in strain rate adjustment might take place even within the gap [A3.25].

[A3.1] Bianchi, J H: ‘From Computational Models of Hot Deformation to Process Design and Control’, *4th World Congress on Computational Mechanics*, Buenos Aires, S. Idelsohn, E.Oñate y E. Dvorkin eds., CIMNE publ, Barcelona (1998).

[A3.2] Mirabile, M, Bianchi, J, Buenten, R, Buessler, P, Ingham, P, Mamalis, M,Monfort, G, Requejo, F and Turpel, P: ‘Application of FEM to Hot Rolling and Deep Drawing’, *Technical Steel Research Final Report N° EUR 15803*, Office for Official Publ. of the E.U., Luxembourg (1997).

[A3.3] Montmitonet, P and Buessler, P: ‘A review of the theoretical analysis of rolling in Europe’, *ISIJ Int.*, 31 (1991), pp525-538.

[A3.4] Perdrix, C: ‘Caracteristiques d’ecoulement plastique du metal’, *Technical Steel **Research Final Report EUR 8697FR*, Official Publications of the EU, Luxembourg (1987).

[A3.5] Anand, L: ‘Constitutive equations for rate dependent deformation of metals’.*ASME J. of Eng. Mater. and Technol.,* 104 (1982), pp12-17.

[A3.6] Brown, S B, Kim, K and Anand, L: ‘An internal variable constitutive model for hot working of metals’. *International Journal of Plasticity*, 5 (2), (1989), pp95-130.

[A3.7] Bianchi, J H and Urcola, J J: ‘Development of a model for hot strip rolling of stainless duplex and non oriented magnetic steels’, *Technical Steel Research Final Report EUR 18378*, Official Publications of the EU, Luxembourg (1998).

[A3.8] Zhou, M, Wiklund, O, Karjalainen, L P, Bianchi, J H, Gutierrez, I and Peura, P:‘The effect of strain reversal and strain-time path on constitutive relationships for metal rolling/forming processes’, *Technical Steel Research Final Report EUR 19891, *Official Publications of the EU, Luxembourg (2001).

[A3.9] Bianchi, J H: ‘Coupling of microstructural rate effects with the Mechanics of rolling deformation’, in: *Proc. 5*^{th}* Int. Conf. on Computational Plasticity*, Barcelona, Cimme press, D.R.J. Owen et al. ed. (1997), pp1276-1283.

[A3.10] Thorp, J H, Bianchi, J H, Dahm, B, Mehren, D, Johnson, C P and Dionisi Vici, F:‘Development of systems to aid flexible rolling of long products to closer tolerances by on-line calculation of loaded roll gap, control of stand speed and reheat’. *Technical Steel Research Final Report EUR 20473*, Official Publications of the EU, Luxembourg (2002).

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[A3.11] Bianchi, J H and Dionisi, F: ‘Effects of temperature, rolling speed and preset variations on the loaded roll gap response of a bar mill stand’, *Proc. 14 Conference, *The IAS, Buenos Aires, C. Genzano ed. (2002), pp103-112.

[A3.16] Maccagno, T M, Jonas, J J and Hodgson, P D: ‘Spreadsheet modelling of grain size evolution during rod rolling’, *ISIJ Int.,* (1996), 36(6), pp720-728.*th Rolling *

[A3.12] Estrin, Y and Mecking, H: ‘A unified phenomenological description of work hardening and creep based on one-parameter models’, *Act. Met.,* (1984), 32(1), pp57-70.

[A3.13] Bianchi, J H: ‘A viscoplastic constitutive law with internal variables for Multipass Hot Rolling’, *Proc. 4*^{th}* Int. Conf. on Computational Plasticity*, Barcelona, Pineridge Press, D.R.J. Owen et al. eds. (1995), pp2283-2292.

[A3.14] Urcola, J J and Sellars, C M: 'Effect of Changing Strain Rate on Stress-Strain Behaviour during High Temperature Deformation', *Acta Metal,* 1987, 35, pp2637- 2647.

[A3.15] Maccagno, T M and Jonas, J J: ‘Correcting the effects of static and metadynamic recrystallisation during the laboratory simulation of rod rolling’, *ISIJ Int*., (1994), 34 (7), pp607-614. [A3.17] Choi, S, Lee, Y, Hodgson, P and Woo, J S: ‘Feasibility study of partial recrystallisation in multi-pass hot deformation and application to calculation of mean flow stress’, *J. Mater. Process. Technol*., (2002), 125-126, pp63-71.

[A3.18] Lee, Y, Choi, S and Hodgson, P D: ‘Integrated model for thermo-mechanical controlled process in rod (or bar) rolling’, J*. Mater. Process. Technol*., (2002), 125-126, pp678-688.

[A3.19] Serajzadeh, S, Mirbagheri, H and Karimi Taheri, A: ‘Modelling the temperature distribution and microstructural changes during hot rolling of low carbon steel’, *J. Mater. Process. Technol.,* (2002), 125-126, pp89-96.

[A3.20] Anelli, E: ‘Application of mathematical modelling to hot rolling and controlled cooling of wire rods and bars’, *ISIJ Int*., (1992), 32(3), pp440-449.

[A3.21] De Ardo, A J: ‘Fundamental aspects of the Physical Metallurgy of Thermomechanical Processing of Steels’, in: Proc. Int. Conf. on Physical Metallurgy of Thermomechanical Processing of Steels and Other Metals, *THERMEC-88*, Tokyo, the ISIJ, I. Tamura ed. (1988), pp20-29.

[A3.22] DeArdo, A J: ‘Modelling of deformation processing: wonderful tool or wishful thinking’, in: *Proc. Int. Symp. Mathematical Modelling of Hot Rolling of Steel*, Metallurgical Society of The Canadian Institute of Mining and Metallurgy, Montreal, S. Yue ed. (1990), pp220-238.

[A3.23] Karjalainen, L P and Perttula, J: ‘Characteristics of static and metadynamic recrystallization and strain accumulation in hot-deformed austenite as revealed by the stress relaxation method’, *ISIJ Int*., (1996),36 (6), pp729 - 736.

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[A3.24] Hodgson, P D: ‘The Metadynamic Recrystallisation of Steels’*, *Int. Conf. Thermomechanical Processing of Steels and Other Materials, *THERMEC-97, *The Minerals, Metals and Materials Society, T. Chandra et al. eds (1997), pp121- 131.

[A3.25] Bianchi, J H and Karjalainen, L P: ‘Modelling of dynamic and metadynamic recrystallisation during bar rolling of a medium carbon spring steel’, *Journal Materials Processing Technologies, *Elsevier, (2005), Vol. 160, No. 3, pp267-277.

[A3.26] HKS, ABAQUS *User’s manual* (2001-2004). [A3.30] Fields, D and Backoffen, W: ‘Determination of the strain hardening characteristics by torsion testing’, *Proc. ASTM,* 1957, 57, pp1259-1272.

[A3.31] Karjalainen, L P, Porter, D and Peura, P: ‘Recrystallization of Ti-microalloyed steels under constant and varying deformation conditions’. Proc. 4 on Recrystallization and Related Phenomena, *Proceedings of ReX’99 (JIMIS- 10),* (1999), pp721-726.

[A3.27] Hart, E W: ‘Constitutive relations for the non-elastic deformation of metals’,*ASME Journal of Eng. Mat. and Tech*., (1976), 98, pp193-202.

[A3.28] Karjalainen, L P: ‘Stress relaxation method for investigation of softening kinetics in hot deformed steels’, *Mater. Sci. Technol*., (1995), 11,(6), pp557-565.

[A3.29] Pöhlandt, K and Tekkaya, A: ‘Torsion testing plastic deformation to high strain rates’, *Mat. Sc. and Tech*., Nov. 1985, 1, pp973-977.th Intern. Conf.

[A3.32] Karjalainen, L P, Somani, M C, Peura, P and Porter, D: ‘Effects of strain rate changes and strain path on flow stress and recrystallization kinetics in Nb- bearing Microalloyed Steels’, *Proc. Int. Conf. Thermomechanical Processing of Steels, London, IOM,* (2000), pp130-139.

[A3.33] Lee, Y: ‘Calculating model of mean strain in rod rolling process’, *ISIJ Int.,* (1999),39 (9), pp961-964.

[A3.34] Lee, Y, Kim, H J and Hwang, S M: ‘Analytic model for the prediction of mean effective strain in rod rolling process’, *J. Mater. Process. Technol*., (2001), 114, pp129-138.

[A3.35] Iza-Mendia, A, Piñol-Juez, A, Urcola, J J and Gutierrez, I: Microstructural and mechanical Behaviour of a Duplex stainless Steel under Hot working Conditions, *Metallurgical and Materials Transactions*, 1998, Vol. 29A, pp2975-2986.

[A3.36] Hill, R: *Proc. Roy. Soc.,* A 193 (1948), pp281-310.

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[A3.37] Sellger, R, Agren, J, Hillert, M, Hoglund, L and Bianchi, J: ‘Control of phase transformation during processing of partially bainitic multi-component strip steels for controlling the work-hardening characteristics,*Technical Steel Research, Final Report N° EUR 20582,* Office for Official Publ. of the E.U., Luxembourg (2003).

[A3.38] Sims, R B: ”The calculation of roll force and roll torque in hot rolling mills”, *Proc. **Inst. Mech. Eng*., (1954), 168, pp191-200.

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Table A3.1: Chemical composition of the steels under study

2304 duplex stainless steel | ||||||||
---|---|---|---|---|---|---|---|---|

Cr | Ni | Mo | Mn | Si | P | Sppm | C | N |

23.1 | 4.83 | 0.22 | 1.3 | 0.51 | 0.024 | 10 | 0.025 | 0.095 |

Medium-C 55Cr3 steel | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

C | S | P | Mn | Si | Cu | Sn | Cr | Ni | Mo | Al |

0.55 | 0.002 | 0.007 | 0.78 | 0.31 | 0.10 | 0.005 | 0.76 | 0.07 | 0.02 | 0.004 |

Table A3.2 Thermal treatment trials used to characterise the grain growth kinetics. R.T.= room temperature; W.Q= water quenching.

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Table A3.3 Process-Mechanical Testing links of CSM work.

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Table A3.4: Parameters of the isotropic hardening viscoplastic constitutive law for the medium-C 55Cr steel

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Table A3.5: Parameters of the isotropic hardening viscoplastic constitutive law for the 2304 duplex stainless steel

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Table A3.6: Typical parameters of a simulation for a round-oval rolling pass, intermediate mill, medium-C steel

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Table A3.7: Typical parameters of a simulation for a flat rolling pass, 2304 steel; IM = finishing stand of industrial mill, PM = pilot plant mill

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Table A3.8: Comparison of dynamic recrystallisation strain for different prior strain rate definitions for the streamlines exiting the roll gap at positions (I) and (II) in Fig. A3.16

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Table A3.9: Comparison of experimental and FEM simulation results. Flat rolling in the 470 mm diameter two high Danieli mill a at CSM

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Fig. A3.2 History effects during monotonic loading considered by the proposed constitute model constructed from Oulu’s Gleebe compression data. H=hardening, DRV=dynamic recovery,DRX=dynamic recrystallisation. Medium-C 55Cr3 steel tested at 1050°C, 1 s-1.

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Fig. A3.1: Yield and peak stresses in term of the temperature compensated strain rate. Comparison of the constitutive fitting against monotonic axisymmetric compression Gleeble data for medium C steel 55Cr3 from the University of Oulu.

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Fig. A3.3: Effect of prior strain rate on static softening.Comparison of Oulu experimental results from constant and changing strain rate with CSM FEM model mean strain rate measures. Stress relaxation experiments at 1050°C on the medium-C 55Cr3 steel.

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Fig. A3.4: FEM simulation of a Gleeble monotonic axisymmetric compression test, 55Cr3 medium-C steel at 1050 °C and 1 s-1: (a)comparison of experimental and simulation raw force results (b) detail of the octave section used in the FEM simulation at (constant) plastic strain 0.587.

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Fig. A3.5: FEM simulation of a monotonic torsi on test, 55Cr3 medium-C steel at 1050 °C and 1 s-1; (a) details of the half specimen and boundary conditions; (b) resulting local strain distribution; (c) comparison of classical model and local FEM strains; (d) progress of strain localisation in time.

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Fig. A3.6: FEM simulation of a monotonic torsi on test, 55Cr3 medium-C steel at 1050 °C and 1 s-1; (a) temperature distribution; (b) and (c):progress of thermal localization.

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Fig. A3.7: FEM simulation of a monotonic torsion test, 55Cr3 medium-C steel at 1050 °C and 1 s-1. Comparison of experimental and simulation raw torque results.

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Fig. A3.8: Comparison of stresses from different techniques, monotonic torsion test, 55Cr3 medium-C steel at 1050 °C and 1 s-1. Note the agreement between compression experiment and FEM simulation at the torsion specimen mid-gauge length and the difference in strain to peak stress between compression and classical torsion analysis.

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FIG. A3.9: Stepped increasing strain rate Gleeble compression, 55Cr3 medium-C steel at 1050°C. Comparison of the University of Oulu experiments and CSM-CL predictions. Note the effect of DRX

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Fig. A3.10: Stepped decreasing strain rate Gleeble compression, 55Cr3 medium-C steel at 1050°C. Comparison of the University of Oulu experiments and CSM-CL predictions. Note the effect of DRX

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Fig. A3.11: FEM simulation of a torsion test, 55Cr3 medium-C steel at 1050°C and 1 s-1. Comparison of monotonic and strain reversal results from CEIT experiments and CSM FEM simulations under the base (NO-SSRS) and modified (SSRS) constitutive laws:(a) absolute value of raw torque; (b) equivalent stress.

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Fig. A3.12: Typical thermal distribution predicted by the FEM simulation, round-oval 8th pass of ORI-MARTIN bar mill.

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Fig. A3.13: Typical strain rate distribution predicted by the FEM simulation, round-oval 8th pass of ORI-MARTIN bar mill: (a) t otal effective strain rate; (b) strain rate normal component due to the elongation in the rolling direction; (c) shear strain rate component in the rolling-thickness direction; (d) average strain rate;*rd *=rolling direction,*td*=thickness direction,*wd*=width direction.

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Fig. A3.14: Evolution of strain components and their equivalent measures within the roll gap. Flat rolling of duplex stainless 2340. Entry thickness= 20 mm, rolling reduction=25.8%, friction coefficient 0.35; *rd *=rolling direction,*td*=thickness direction,*wd*=width direction.

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Fig. A3.15: Evolution of strain components and their equivalent measures within the roll gap. R ound-oval 8th pass, medium-C 55C r3 steel.Entry diameter 55 mm, rolling area reduction=38%, friction coefficient 0.35; *rd *=rolling direction,*td*=thickness direction,*wd*=width direction.

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Fig. A3.16: Recrystallised volume fraction predicted by the model at the exit of the 8th round-oval pass, T=1040°C, rolling speed =1.73 m/s; (I) and (II) are streamlines at the roll gap exit; medium-C 55Cr3 steel.

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Fig. A3.17: FEM softening predictions for the 8th-9th interstand of ORI-MARTIN bar mill: (a)-(c) effect of different prior stra in rate measures on the difference between strain and DRX strain; (d) evolution of recrystallisation in the interpass; (e) effect of the prior strain rate choice on the resulting torque; medium-C 55Cr3 steel.

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Fig. A3.18: Area affected by shear strain reversal during 7thstand oval-round pass, area reduction 27%, initial section 39 x 90 mm.

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Fig. A3.19: FEM prediction of cross-section local strains for the conventional (NO-SSRS) and modified (SSRS) Constitutive Law; 7th oval-round pass; medium-C 55Cr3 steel.

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Fig. A3.20: FEM prediction of local stress during rolling for the conventional (NO-SSRS) and modified (SSRS) Constitutive Law;7th oval-round pass; medium-C 55Cr3 steel.

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Fig. A3.21: Comparison of the Rolling Force and Rolling Torque predicted by the conventional (NO-SSRS) and modified (SSRS) Constitutive Law for the 7th oval-round pass; medium-C 55Cr3 steel.

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Fig. A3.23: Effect of DRX on shear strain reversal; 7th oval-round pass, medium-C 55Cr3 steel.

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Fig. A3.22: Effect of shear strain reversal on DRX; 7th oval-round pass, medium-C 55Cr3 steel.

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Fig. A3.24: Evolution and through-passes propagation of the area affected by the strain reversal in a non- recrystallising 2304 steel during flat rolling: (a) first pass; (b) second pass in reversing mode; (c) second pass in tandem mode.

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Fig. A3.25: Evolution of the shear strain component during two consecutive passes at several through thickness:(a) pass 1– reversing pass 2; (b) pass 1– tandem pass 2. Flat rolling, 1000°C, two 40% reduction passes from Ho= 35 mm.

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Fig. A3.26: Comparisons of the predicted power consumption and specific roll force for two consecutive roughing and tandem passes under the NO-SSRS and SSRS constitutive hypotheses in the second pass. Non-recrystallising 2304 steel, 1000°C, two 40 % reduct passes from Ho= 35 mm.

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Fig. A3.27: Accumulation of strain components between c onsecutive H-V stands. Case study: normal strain component [VER SIMBOLO EN PDF ADJUNTO] in the thickness direction; (a) round-oval 8th pass; (b) oval-oval 9th pass after full interpass recrystallisation; (c) oval-ov al 9th pass without recrystallisation (hypothetical case not applicable to the medium-C steel under study); (d) oval-oval 9th pass after recrystallisation in the interstand computed with the medium prior strain rate.

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Fig. A3.28: Experimental temperature evolution during the two rolling passes in case study A30 (Table A3.9)

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Fig. A3.29: Typical experimental rolling forces and power consum ption data: (a) and (b) raw electrical data; (c) and (d) load cells data for forces and FT-processed power data for two consecutive passes (case study A30).

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Fig. A3.30: Comparison of experimental and FEM predicted rolling force for the second pass. Flat rolling, 1000°C, pass reduction 40% H1=21mm

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Fig. A3.31: Comparison of experimental and FEM predicted power consumption for the second pass. Flat rolling, 1000°C, pass reduction 40% from H1=21 mm.

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Fig. A3.32: Comparison of FEM prediction and experimental measurement, rod rolling of medium-C steel 55Cr3, Temperature = 1020 roll diameter = 0.433 m, Area Reduction = 23%. Conversion efficiency factor = 0.85

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