Summary of progress
Author  G.J. Watts D.C.J. Farrugia B. Cheong Z. Husain M. Zhou I. Gutierrez D. J. Badiola J. H. Bianchi P. Vescovo O. Wiklund M. Karlberg M. Schmidtchen R Kawalla L. P. Karjalainen M. C. Somani 
Pages  1733 
Page 17
All the partners have been working in previous ECSC projects involving the modelling of metal forming and have brought with them a legacy of models of various types, as mentioned in Section 1.2, which have been reviewed for either immediate use in the project or further development in describing the influence of strain path on the evolution of microstructure and on the magnitudes of the operating parameters. The literature has also been reviewed for current developments in modelling and contact made with third party developers of software where relevant.
Page 18
The models can be divided into two groups viz. (a) empirical or analytical formulae for (i) the operating parameters in terms of reduction, bulk temperature, average strain rate (or speed), yield stress and friction and (ii) metallurgical parameters such as fraction recrystallised in terms of similar quantities to (i) but ostensibly at the mesoscopic scale and (b) finite element plasticity models for the prediction of the distribution of these quantities based on standard constitutive equations at this scale. Both these groups of models provide means for investigation of influence of thermomechanical history, which are straightforward but have limitations. In the first group, determination of any history effect generally requires taking some accompanying measurement(s) in a corresponding physical trial. For example

Load models require the mean forming resistance over the strain path, which is dependent on the microstructural behaviour but is estimated from experimental data using procedures such as those set up by TU Freiberg.

The probabilistic model used at the University of Oulu for the time to 50% recrystallisation in terms of mean stored energy expresses the latter in terms of the dislocation density. The dislocation density is not determined by recourse to an evolutionary model but estimated empirically in terms of the measured flow stress. Indeed this approach accompanied by appropriate experiments have readily demonstrated that the recrystallisation is not strongly dependent on the temperature history but determined essentially by the temperature prevailing in the interpass.

CEIT have viscoplastic and dislocation density based constitutive models, both largely analytical in nature. The first model takes into account the effect of strain reversal on the flow stress curve. However analytical modelling of the transient after reversal is difficult and the contribution of nonrecrystallised regions needs to be determined experimentally and is strongly dependent on the dynamically recrystallised fraction, which cannot be precisely determined in terms of a macroscopic external measure.
The intrinsic nature of the plasticity equations implies that the stress and strain distributions in the stock at any stage of the process are dependent in general on the deformation path taken to reach this stage where path is taken to mean a sequence of reductions and interpass turns. FE codes such as ABAQUS used by Corus and CSM can then determine the effect of this sequence on the mechanical state of the stock, and also on structural quantities such as dynamically recrystallised fraction and grain size (having checked that the critical strain has been attained) by means of user defined variables. Such codes do not however take into account the coupling between microstructural and thermomechanical quantities. Equations coupling these quantities can be introduced into ABAQUS using the *CREEP Option, provided that kinematic hardening is excluded, and without this exclusion by means of a user subroutine (i.e. UMAT for ABAQUS/Standard and VUMAT for ABAQUS/Explicit). Corus has developed an FE model, by means of the latter approach, incorporating kinematic and isotropic hardening; the numerical procedure for time integration of the constitutive equations may, however, converge very slowly or even diverge. An FE model taking into account the interaction between the mechanical variables, hardening, recovery, recrystallisation, and damage has also been developed by CSM in the previous project. It has been used to investigate strain reversal, in particular, the different material responses noticed by CEIT for shear and normal deformation. More work was necessary however to study the latter effects in rolling and to investigate the complex microstructural changes for both single phase and duplex structures. The interaction between deformation and dislocation behaviour has been studied in the previous project by MEFOS using the SandstromLagneborg, EngbergSiwecki and Bergstrom models relating flow stress,Page 19recrystallisation, dislocation and vacancy concentration including evolutionary equations for the latter two quantities and has been applied to the rolling of CMn and CMnNb steels in a single pass. The model has potential for further development and application to rolling.
From this review, the following recommendations were made for the FE models:

Augmentation of finite element models to include all processes of interest such as heat transfer, static softening, etc.

Utilisation, where possible, of fundamental models such as those based on dislocation density.

Incorporation in dislocation models of relevant processes such as nucleation and precipitation.

Use of efficient numerical techniques for time integration.

Determination of coefficients for all steel qualities allocated.

Development/application of models for multipass rolling including effects of cooling especially in the interpass.
Concerning the application of analytical and empirical models, it was recommended that supporting measurements be replaced by a constitutive model where possible. It is also important that the skills and expertise gained in measurement techniques be used to full advantage (a) to provide constitutive data for use in FE models both locally and by the other partners and (b) to investigate experimentally the effect of a range of complex loading paths on structural evolution.
Experimental work has been carried out at the University of Oulu to investigate the influence of deformation history, in particular, changes in temperature and strain rate, during uniaxial compression/tension on the flow stress and subsequent softening behaviour in both the hot deformed austenite and warmdeformed ferrite phase (see Section 2.2 (v)).
It was found that when the strain rate decreases abruptly, in the dynamic recovery regime, the flow stress lags behind the behaviour of a mechanical equation of state, so that it is finally higher than that at the corresponding constant final strain rate, close to the stress at the constant mean strain rate of the whole deformation history. The static recrystallisation rate lags even more than the flow stress from the behaviour following the mechanical equation of state, so that following the abrupt decrease in the softening rate remains close to that at the high strain rate before the drop. Since the effect of strain rate on recrystallisation kinetics is generally weak, the mean strain rate can be used for modelling with reasonable accuracy. The recrystallisation behaviour, however, is affected quite significantly in the instances that the strain rate drop takes place after a strain resulting in dynamic recrystallisation. The above effects have been shown to be independent of the steel composition.
Similar work has also been carried out by Corus for compression and by CEIT in torsion tests.
Page 20
The influence of temperature increase was investigated only for the austenite phase. The results generally indicated that the temperature history seems to have only an insignificant influence on the flow stress. Similarly its effect on subsequent softening behaviour is minimal.
The influence on flow stress and static recrystallisation kinetics of strain reversal in axisymmetric deformation has been investigated by the University of Oulu and Corus for CMnNb steels supplied by Corus and various other steel qualities. Tests by all partners have shown that strain reversal gives rise to a noticeable Bauschinger effect, a transient characterised by a plateau. The recrystallisation kinetics is strongly retarded by the reversal, especially at small prestrains. This effect seems to be quite independent of the chemical composition of the steel. A strain reversal coupled with a drop in strainrate is found to decrease the SRX rate even more. Contrary to this, partial recrystallisation after the first deformation before reversing the strain enhances slightly the recrystallisation. CEIT have shown that strain reversal in torsion coupled with an increase in strain rate tends to increase the stress after the change and to shorten or remove the plateau. The effects from the above two coupled tests can be related to an excess or shortage of dislocations in relation to those due to the dissolution of the substructure.
Torsion tests have been carried out on CMnNb, medium C and duplex stainless steels in order to investigate the effect of strain reversal on the flow stress under a broad range of conditions and on the recrystallisation kinetics.
For CMnNb steels, full – or even relatively large partial –reversal gives recrystallisation times shorter than those for the prestrain only but longer than if the total absolute strain is applied monotonically. For small reverse strains, the recrystallisation time is approximately the same, or even longer if the prestrain is low i.e. c. 0.15, the longest times occurring for reverse strains close to the end of the plateau.
For medium C steel, tests have been carried out to simulate the sequence of shear reversals taking place during a rolling schedule assuming no softening between passes and to obtain constitutive experimental data for the CSM FE programs (pure shear). It has been notably observed that the flow stress is lower in a 2nd pass with a prior reversal than without. This effect has also been observed in the nonrecrystallising duplex stainless steel. Multipass tests have also been carried out to determine the softening taking place during given holding times applied after different strain paths. For forward and reverse strains of 0.1, interpass times greater than 10 s are required to initiate static softening, but the softening is significantly increased if a higher strain of 0.2 is used. A series of cyclic tests of small strain produce stresses in successive passes significantly lower than those in a monotonic test having the same equivalent strain.
The objective of this subtask was to investigate the effects of successive hits of various magnitudes applied isothermally, but varying the loading direction on the flow stress and recrystallisation kinetics of Type 316 stainless steel. In particular it was aimed at comparing the effects of a single hit and a series of hits of the same total strain whether applied on the same or on different pairs of faces (i.e. indifferent directions relative to axes in the specimen.
Page 21
The order of the hits and the operating temperatures viz. 900°C, 1000°C and 1100°C were also examined. To maintain isothermal conditions, specimens were quenched and reheated between hits.
The changes of orientation of the specimen relative to the platens in the main reflect the configurational changes occurring in industrial rolling processes such as turning of a bar during edge rolling and cross rolling of plates. More general changes e.g. turning of the whole stock in more than 1 plane, although not representative of bulk processes in conventional rolling, may nevertheless be applicable locally to regions of the stock undergoing complex deformation e.g. near the web flange interface in beam rolling.
Stress relaxation tests carried out to measure the recrystallisation indicate that

The time for recrystallisation in a single hit decreases notably with strain and marginally with increasing temperature.

Recrystallisation behaviour is marginally dependent on the order of the hits.

Multiple hits of the same total magnitude as a single hit give a lower recrystallisation and, triple hits each of magnitude (~20%) applied in various directions can inhibit recrystallisation entirely.

An interpass turn of 90° enhances the early recrystallisation when one of the reductions is large (~40%) but significantly retards it when both hits are moderate (~20%).
The influence of strain reversal and strainrate change on flow stress behaviour and evolution of recrystallisation has been investigated by the University of Oulu also for the ferrite phase, using compression testing and optical and electron microscopes. This information may be especially useful in the development of future models for the ferritic rolling. As the softening of ferrite is affected strongly by the recovery, stress relaxation techniques could not be used to reveal the rerystallisation rate, but the doublehit compression technique had to be applied. The influence of strain rate change has been studied for the various IF steels and hi TiNb steels. The results obtained are qualitatively similar to those in the austenite phase. When the strainrate decreases abruptly, the flow stress and, even more noticeably, the static recrystallisation rate lag behind those of a mechanical equation of state. After the change, the cells appear somewhat larger than when the high strain rate is maintained indicating a rapid decrease in the driving force for static recrystallisation. The misorientation distribution of grain boundaries is, however, unaffected. In particular for small reversed strains, the strain reversal retards the recrystallisation rate of ferrite – by as much as 8 times – and significantly coarsens the ferrite grain size and the cell structure. It also tends to decrease the average misorientation across the grain boundaries. Strain rate tests have also been carried out on 55Cr3 steel to obtain constitutive data in cooperation with CSM for use in its rolling model. Tests have also been modeled using the model developed at MEFOS to understand the relationship between flow stress changes and dislocation dynamics (see Section WP4).
Page 22
Two series of trials using Type 316 stainless steel bars were carried out on the local Cavendish mill, the first to compare the effects on structure when rolling with and without an interpass turn of 90° between 2 successive 20% passes at an initial temperature of ~1100°C, and the second to investigate the effects of the above when combined with various reheating strategies in the interpass.

In the first set of trials, lower recrystallisation was observed at the surface but the loads in the second pass are lower suggesting that retardation of the recrystallisation is localised.

In the second set of trials, four bars were rolled in the first pass at a lower temperature than in the first set of trials, viz. 1040°C. The bars were then quenched and two reheated to 1040°C to promote recrystallisation and two to 950°C to inhibit it. One bar in each of these sets was rolled in the second pass without a prior turn. The other bars were turned respectively in transverse and horizontal planes.
Conclusive evidence cannot be given on the effect on recrystallisation since determination of the latter by optical metallography has been considered unreliable and also the grain sizes observed varied widely, especially for the bar reheated to 950°C and not turned. The fact that the loads in the second pass were similar for both reheating temperatures suggests that little recrystallisation has been promoted. This is also suggested, albeit weakly, from pole figure maps obtained by EBSD.
The set of grain size results give an overall view of the maximum rather than the average grain sizes attained and suggest that the maximum is greater in the specimens that have been turned, the substantial differences are however more likely to reflect material heterogeneity than a retardation in grain refinement.
The hypothesis made following these tests is that any effects of the stock rotations and reheating strategies on recrystallisation in a rolling context are secondary.
The objectives of this work were (a) to investigate, for various rolling temperatures, the effect of an interpass turn through 180° on the recrystallisation kinetics and evolution of austenitic grain structure in CMnNb steel plates (b) to perform a benchmark test showing the effect of stressstrain history on the rolling force and microstructural evolution. Rolling trials were carried out both on the plate mill at Swinden Technology Centre and the reversing stand mill at TU Freiberg.
Plate rolling schedules, with and without reversal, were first designed and implemented at STC, in conjunction with FE and metallurgical modelling, to guide the plate rolling trials at Freiberg. The rolling temperatures employed were 1020°C and 1050°C and the plates were reduced in thickness from 30 mm in 3 passes of draughts 5 mm, 3 mm and 3 mm with and without reversal. Measurements of loads and spread were in good agreement with values predicted by the models. However, although large grain sizes were predicted by the metallurgical model, the austenitic grain structure could not always be differentiated becausePage 23of the presence of phases such as bainite, and decarburisation at the surface and where the presence of martensite did allow this structure to be detected, the grain sizes varied widely.
In the schedules at Freiberg, plates were employed of thickness 6.7 mm, this having been calculated as the maximum value for which the cooling rate would be sufficiently high to produce martensite. However, trials were also carried out using the 30 mm thick plates, as supplied by Corus, the temperature being more stable and controllable than in the plates of reduced thickness. The stock was reduced in thickness, firstly by 40% and then by a further 50%, with and without reversal between the passes, the plates being moved manually to the roll bite in the first case, and turned through 180° in the second. Rolling temperatures of 1000°C, 900°C and 850°C were employed. Small differences were observed between the dimensionless forming resistance (and hence the loads) with and without turns for the 6.7 mm thick plate, it being higher in the former case at 900°C but lower at 1000°C (undecidable at 850°C because of the different L/h values actually used).
As in the trials at STC, decarburisation occurred and no martensite was produced in the thicker plate but was produced in the initially 6.7 mm thick plate. Inhomogeneous grain size distributions were obtained, the grain size tending to be smaller in the plates that had undergone reversal rolling, supporting the hypothesis that the different strain pass histories give rise to different average grain sizes. A combination of visioplastic techniques, analytical and FE models (as described in Sections 2.4 and 2.5 for tasks WP4 and WP5) have been used to obtain predictions which can be compared with experimental results. The FE results obtained by Corus and MEFOS for this benchmark application are discussed in Section 3.6 for task WP6.
The objective of this task is to compare the development of austenitic grain size in CMnNb steel at low speeds with that at high speeds such as occur in rod rolling. TU Freiberg has collaborated with Corus in the use of its high speed rod mill (top speed ~ 32 m/s) and also in the use of its plate mill (as described above) in providing results at the lower end of the speed range (~1 m/s). Moderately low speeds of the order of 5 m/s, attainable on the rod mill were also employed. A constant roll pass design (round – oval; 10 mm >6 mm) was employed for all trials except those at low speed. In high speed rolling, reheating temperatures of 1200°C and 1150°C were employed, for times of 10 s and 60 s approximately). All stands F1 –F4 were used. Air cooling was used, At low speed speeds, trials were carried out using all stands F1 –F4 and water cooling, F1F2 and water bath and F1 only with the water bath.
Information from Corus at STC on the dependence of recrystallisation time on temperature, initial grain size (in this case 50 microns) and strain was employed in assessing these trials.
The experimental results, however, show only slight differences in grain structure for the various cases considered viz. higher rolling speeds produce smaller grain sizes when used in conjunction with higher input of thermal energy (i.e. higher reheating temperatures or reheating times) and so are comparable with those produced by reversal at the lower speeds in plate rolling. Notable differences in load are found viz. higher values at higher temperatures (at constant speed), probably arising from changes in material behaviour associated with phase transformation.
Page 24
Information on the materials examined by the various partners and the corresponding metallurgical processes studied has been input into a template which has been involved in the testing of most of the metals. The information is reproduced in Appendix 7. Details of the materials and associated tests and models for the individual partners are provided in Table 4(af).
The results from the various testing facilities of the partners for the materials listed in these tables are outlined below.

CMnNb steel is produced by Corus and it has been decided that this should be one of the major steels used by several partners for comparison purposes. Accordingly, it has been supplied to TU Freiberg for benchmark plate rolling trials and to CEIT for torsion testing. Steels of a similar composition and CMnNbTi steels, have also been used by the University of Oulu in its compression/tension testing programme. It has been used by Corus in this project in strain reversal tests and plate rolling trials.
The torsion testing has found that the flow stress stagnates following reversal, probably due to strain induced precipitation in the niobiumcarbonitrides. Most significantly, it transforms in the range 750°C to 800°C, in a manner dictated by the cooling rate. If the latter is sufficiently high, corresponding to a time t8/5 of less than 1.3 s martensite is produced which is beneficial to the detection of austenite, otherwise phases such as bainite and pearlite are produced which mask the austenitic structure.

CMn steels (i.e. without the addition of Niobium) have been used to a limited extent by Corus, having been extensively used in the previous project. In particular, the dependence of recrystallisation on parameters such as grain size and interpass time was sought to help determine an appropriate schedule for the rolling trials on the rod mill at TU Freiberg. In particular, it was found that coarsening the initial grain size significantly increases the recrystallisation time making partial recrystallisation possible before entry to the next pass.

CMn steels with the addition of sulphur have also been tested by the University of Oulu and characteristically have an insignificant grain size evolution below 1200°C.

The medium C steel 55Cr3 has also been investigated by several of the partners, being used in the rolling trials on the ORIMARTIN bar mill, modelled by CSM and supplied to CEIT and the University of Oulu for obtaining torsion and tensile/compressive constitutive data and for validation purposes. It is characterised by rapid recrystallisation and grain growth, thereby wiping out strain history effects in the bar rolling process.
The evolution of grain size in a medium C steel has also been studied by Corus to assist the heat treatment process at TU Freiberg in achieving an initial grain size of 200 microns for the high speed rod rolling trials.

Low C steels have been investigated by the University of Oulu and are interesting because (a) they are insensitive to stress relaxation; (b) dynamic and static recovery are difficult to distinguish; (c) they exhibit considerable retardationPage 25of softening when the strain rate is changed abruptly; (d) thermal history can exert a significant influence on the structural evolution e.g. the formation of homogenous polygonal ferrite on cooling at 5°C/s to 500°C followed by heating to 700°C.

The duplex stainless steel 2304 has been included in the multilevel model developed by CSM, having sharply contrasting properties to the medium C steel mentioned above, viz. zero recrystallisation and hence an accumulation of strain throughout rolling.
It has been supplied to CEIT for torsion testing and has been investigated in both ascast and wrought conditions. These have significantly different structural properties. In the wrought state, the austenite and ferrite occur in incoherent parallel bands; during deformation the austenite tends to act as rigid fibres in the soft ferrite, thereby accommodating the torsional shear strain and producing linear strain hardening. In the ascast condition, the austenite is mixed with, and coherently attached to the delta ferrite grains. During deformation, the austenite blocks tend to align with the direction of the deformation and to become separated from the ferrite giving rise to damage and induced softening.
Differences between the stressstrain curves for reversal are small except at high strains. The stress –strain curves are however significantly different from those observed in single phase materials. Moreover, the dissolution/rebuilding of the dislocation substructure during the transient is unobservable because of the different mechanical behaviour of the two phases.

Type 316 L stainless steel has been used by Corus in the experimental testing and modelling of multidirectional compression and HV rolling. Optical metallography of specimens before thermomechanical processing has indicated the presence of delta ferrite with some twinning of the grains with evidence of inclusions and, in the interdendritic spacing, of the original ascast microstructure. This causes difficulties in the initial definition of the grain size, ostensibly low at around 9 microns, and in its measurement at stages in its evolution. This was borne out in some of the optical metallographs in the HV rolling trials, with a wide variation in grain size in the material being manifest even after a homogenising heat treatment prior to the trials.
Unlike the duplex stainless steel 2304, Type 316 stainless steel recrystallises, albeit moderately slowly. There is some evidence from the trials that the recrystallisation is retarded locally in the stock that has undergone an interpass turn but overall it was considered that, if occurring, this is a secondary effect. Indeed, for some of the tests, it was not possible, using optical metallography, to establish whether or not recrystallisation was substantial or nonexistent in tests involving either REX promoting or REX inhibiting reheats. These findings were also reflected, because of the poor image quality, in IQ maps obtained by EBSD. Pole figure maps however give some indication of very slight recrystallisation.

A number of interstitial free steels have been examined by the University of Oulu, inc. TiIF, Nb –Ti IF. 'Strainrate drop' tests on the Gleeble have shown that the final flow stress is higher than that at the lower strain rate but below that at the initial high strain rate. For Ti –IF the relaxation kinetics occurs over a longer time than for the austenite. Similar relaxation curves are found for the Nb–TiIF but the recrystallisation in the ferrite is not apparent. Examination of the dislocationPage 26structure using electron microscopy, following the strainrate drop tests, have shown that larger cells are formed than when the strainrate is maintained at the constant higher rate, but that the misorientation of the grain boundaries is not affected. Analogous examination for strain reversal tests demonstrates also larger cells but a change viz. a decrease in the average misorientation of the grain boundaries.

Strain reversal on Armco iron specimens has also been studied by the University of Oulu and the tests showed it gives rise to considerably decreased rate of SRX and grain sizes much coarser than those for austenite.

CCrV steel is a tool steel, primarily of interest to the NORDIC community and the rolling of such steel has been modelled by MEFOS. For use in the model, flow stress data has been sampled by SIMR at strain rates between 0.8 and 30 s 1 and at temperatures between 800 and 1200°C.
In the evaluation of constitutive equations (Task WP1) described in Section 3.1, it was established that the models generally employed were intrinsically limited in their analysis of the effects of complex loading paths, specifically not including the coupling between the local structural variables and the thermomechanical variables. It was also emphasised that reliable and accurate experimentation is vital in demonstrating the path dependence and for providing constitutive data for the models.
Many of the experimental results show a dependence of the flow stress behaviour on the strain path, in particular that the flow stress lags behind that predicted by an equation of state. There may also be changes induced by the deformation history in the grain size related to the recrystallisation event because the extent and speed of recrystallisation can also be affected by the deformation path. At a deeper level, other structural changes occur e.g. in dislocation density which influences recrystallisation rate or changes the hardness. To gain a more complete understanding of the forming process such structural variables therefore need to be included when formulating mathematical models for the process. Various types of models can be constructed, and many possible combinations of variables can be included as mentioned earlier. This variety is reflected in the models of the various partners.

It is possible, by a number of means, to elicit some of the effects of deformation history, bypassing the coupling between the thermomechanical and structural variables, one of these being the capture of experimental data on such effects and the input to a database and incorporation of relevant information into updated numerical models. An example of this approach is the use of visioplasticity in conjunction with correlation based image processing techniques to investigate the effects of reversal in plate rolling. These techniques enable, for various FF & FR schedules, the strain and stress fields to be calculated from examination of the deformation of a grid marked in the stock. This information has then been incorporated in FE models applied to such processes.

An empirical model of the complete stress strain curve (including the transient) for strain reversal has been constructed from experimental assessment of several parameters. Some of the equations are sensitive to small changes in some of the coefficients but good agreement with experimental results is obtained provided that filtering techniques are applied to the data.
Page 27

In order to better understand the underlying factors controlling the evolution of the flow stress, in particular, its lagging behind the mechanical equation of state following abrupt changes in strain rate, the University of Oulu has derived stress strain curves by adaptation of the analytical model developed by MEFOS relating the flow stress, under constant deformation conditions, as well as under decreasing strain rate to the total dislocation density. In adapting the model, several assumptions had to be made in order to establish the duration of the transient, the remobilisation curve for immobile dislocations and the evolution of the friction stress in this period etc. The stress strain curves could be fitted well with those measured. A model somewhat similar has been applied by the University of Oulu and CSM to investigate DRX and MDRX in rod rolling.
The models described above are useful for describing the stress behaviour even though they do not provide any direct information on the microstructural evolution associated with the complex strain path.

A sophisticated analytical model for the fraction recrystallised, including the effects of strain reversal, has been developed by CEIT using an Avrami equation in which the dependence of t0.5 on strain path is taken into account. This is achieved by expressing the stored energy driving the recrystallisation (to which the above time is related) as a sum of contributions from the internal dislocation density and the misorientation across the cell boundaries; the balance between these contributions depends on the reversal and is determined from the pre stress.
Numerical models predicting the distribution and evolution of dislocation density and other structural variables in conjunction with those of stress and strain etc have also been developed and are described below.

Corus has developed, in conjunction with Birmingham University, a unified viscoplastic model which takes into account the interaction between dislocation dynamics, grain size evolution, recovery, recrystallisation and isotropic but not kinematic hardening. The primary variables are thus the relative dislocation density, instantaneous grain size, the volume fraction recrystallised and the strain and stress tensors.

MEFOS has also formulated a model in conjunction with SIMR and MIKRAB based on dislocation density and vacancy concentration and including the effect of both these quantities on recovery and recrystallisation. A major softening process is recrystallisation; so that both metadynamic and dynamic recrystallisation have been incorporated by means of a sophisticated model developed in conjunction with SIMR and MIKRAB which also includes nucleation, growth and coarsening of the new grains. Notably, nucleation is not described using classical theory (which is inappropriate for recrystallisation) but is based on the abnormal growth of subgrains, embryos growing normally being unlikely to nucleate into mature grains. Grain growth at the metal surface and in the interior is considered separately, the driving forces being the reduction of stored energy due to reduction of grain boundary area, the latter parameter acting as a restraint at the surface. Coarsening is the mechanism for the evolution of the interior grains. After recrystallisation, the recrystallised and nonrecrystallised parts behave differently concerning DRX. The model is also applicable in principle to multistep deformation.
Page 28
The flow stress is given in terms of the dislocation density by means of a modified Hirsch equation. The associated constitutive equations have been incorporated into an elastoplastic finite element model.
Although dislocation based models such as those above are potentially very useful in providing an understanding of material behaviour at microscopic and possibly sub microscopic levels, they require many experimental data to realise this potential and provide an impetus for obtaining equipment and developing techniques for the acquisition of these data. Even without all these data they can nevertheless be used to carry out sensitivity and 'whatif' analyses.
Alternatively, a 'topdown' approach can be followed by extending a standard mesoscopic model to progressively deeper levels by augmenting the number of variables to include quantities related to the microstructural evolution such as fraction recrystallised, back stress and hardness.
CSM has formulated a constitutive model constructed at several levels incorporating respectively hardening and dynamic recovery, dynamic recrystallisation, and static softening, using experimental data obtained from torsion tests at CEIT and compression tests at the University of Oulu. At the first level, the model is a modification of the classical inverse sinh law in which the flow stress is expressed not only as a function of the primary variables  strain rate and temperature combined in the ZenerHollomon parameter, but also of a strength or hardness factor, denoted by s where s is a hereditary variable, dependent on the strain history and evolving according to a prescribed differential equation. This factor corresponds to the hot strength reached without dynamic recrystallisation. The effect of the latter is incorporated at the next level via a secondary hereditary variable, viz. the DRX volume fraction evolving, when activated, according to an Avrami type equation dependent on strain, strainrate and temperature. It is determined assuming that the structure responds instantaneously with the changing local conditions. The interpass (i.e. post deformation) softening is calculated, taking into account both static and metadynamic recrystallisation, using a model provided by the University of Oulu for axisymmetric compression. To apply to rolling schedules, procedures have been devised for identifying and computing prior strain rate measures to apply to the process. The effect of softening on flow stress is then calculated using the method of mixtures. The effect of strain path, in this case shear strain reversal, is introduced at a further level by locally changing the yield surface following such a reversal. Two hypotheses have been considered for defining this change viz. the flow stresses reduce uniformly (SSRS) or that only the shear stress component that underwent reversal is reduced. The model thus maintains the assumption of instantaneous response but takes into account the metallurgical evolution.
Incorporation of a constitutive law into a finite element model of rolling requires the following steps:

Choice of constitutive equation including combination of variables appropriate for the process considered, if several options are available.

Experimental processing of a sufficient number of specimens of the materials allocated over a range of conditions, related to the independent variables of the constitutive equations and representative of the rolling conditions considered, toPage 29accurately determine the required constitutive response, i.e. in this case the flow stress.

Suitable software e.g. based on nonlinear regression or genetic algorithms to filter these data and to determine the coefficients of the constitutive equations.

Choice of suitable numerical techniques for the solution of these equations, in practice, efficient numerical techniques for the accurate integration of the rate equations.

Incorporation of these equations and the techniques for their solution into a form that is recognisable by the FE package employed i.e. corresponding to one of the listed types of deforming material or, more commonly, into a user subroutine in the format stipulated and written in an acceptable language.
The constitutive data have been incorporated in different ways by the respective partners reflecting the constitutive models adopted and the experimental facilities accessed.

In the load model used by TU Freiberg, the method of dimensions (i.e. the PI theorem) is used to relate the ratio of load to mean yield stress multiplied by the contact area to a minimal number of dimensionless variables aspect ratios of the rolling geometry and the coefficient of friction. The mean yield stress is determined from the stress strain curve (for the reduction, strain rate and temperature operative) by an appropriate integration procedure. The method of representing the stressstrain is left open. FE codes are also used as mentioned above into which constitutive data for a range of materials may be input together with information obtained by the visioplasticity technique.

A vast amount of constitutive data for a range of materials deformed under several types of complex loading conditions has been generated using the Gleeble at the University of Oulu. These have generally been presented graphically but software based on regression analysis is available to fit equations if required. Constitutive models using the data for rolling applications in this project have been developed by other partners primarily CSM. However, models for ferritic rolling can be developed in future using the data obtained for materials such as Armco iron and low carbon steel.

Similarly many data have been obtained by CEIT for a range of steel qualities and used in the model developed by CSM. Moreover, the approach used by CEIT to construct equations of the flow stress curve can be usefully employed by partners in FE rolling applications.

Corus, for input into ABAQUS, have used, in the main, constitutive data in tabular format obtained from a variety of sources – results obtained on the local Gleeble, the literature, using the commercial software package ZMAT and more recently the local materials database. Alternatively, equations available in ABAQUS such as the overstress power law have sometimes been used. The dislocation based constitutive model formulated in this project, described in the previous section (and the viscoplasticity model formulated in the previous project and also used in this project) are incorporated into ABAQUS by either a UMAT or VUMAT routine. The constitutive equations in the previous model presented problems for some materials and applications. Firstly a good fit was obtained only for part of the range of strains etc and secondly the constitutive equationsPage 30are 'stiff', causing the standard backward difference methods employed in the numerical integration to converge extremely slowly or even diverge for some applications. In the recent model, flow stress data were obtained for Type 316 stainless steel from delayed double hit tests on the Gleeble. GA software has been used to fit the coefficients in the model equations for application to the HV bar rolling trials and predictions are readily obtained using the computer hardware currently installed.

In the MEFOS modelling activity, CMnNb and CCrV steels have been considered. Flow stress data for the latter quality have been obtained from SIMR as described in Section 2.3. Flow stress data for CMnNb steel have been collected from the pilot plant at TU Freiberg at temperatures between 800°C and 1200°C and at strain rates of 0.1, 1 and 10 s^{1}. These values are compared with corresponding values obtained from solutions of the constitutive equations obtained for a comprehensive set of combinations of values of the coefficients appearing in the constitutive equations. The coefficients chosen for the model are those providing the closest fit of theoretical and experimental data, in the topology defined by the root mean square error. Various optimisation algorithms for minimising this error using direct search, gradient methods, NewtonRaphson and quasiNewton techniques have been explored as the ease with which particular functions neighbouring the data are found can be significantly dependent on the search technique employed (the latter method is generally efficient as it does not require updating the Hessian matrix). Difficulties have been experienced in getting the best fit, especially for the CMnNb steel but also for the CCrV steel, related to fitting polynomials to the frictional stress at very low strains for which data were not available. Some difficulties can be avoided by the use of screening to eliminate spurious data, interpolation to include 'missing' data regions, filtering to smooth oscillations (as observed for the strain rate) and normalising to handle wide variation in magnitudes of the respective gradients.
Problems were encountered in compiling and linking the SIMR and MIKRAB recrystallisation software with LS Dyna on the current pc platform, which were solved only through contact with the proprietor of the latter code, Livermore Software Technology Corporation.
The time integration of the constitutive equations is achieved using the Euler method since small time steps are used in the explicit LSDyna FE package installed on the computer platform recently installed; this gives results up to 100 times faster than the RungeKutta method used previously, without loss of accuracy.
In the plasticity model, maintaining points in stress space on (or below, if elastic) the stress surface has caused problems, as the radial return method previously employed is not implemented in LSDyna which, to be solved, have required direct introduction by the user of procedures such as penalty methods.
The constitutive model was incorporated into an FE code intended to model the forwardreverse rolling of plates at Freiberg. Consequently, it had to include features such as heat transfer viz. radiation, convection and conduction and interpass behaviour. Conduction and thermal contact have proved to be inefficient in searching for elements in contact and to give only small improvements to the solution and therefore are not currently incorporated. Modelling of the interpass behaviour has proved time consuming because of thePage 31small time steps used in the explicit LSDyna code but, as remarked above, is very efficient in modelling the deformation.

The constitutive model developed at CSM is capable of operating at several levels, in particular with or without recrystallisation and shear strain reversal, it is capable of handling many of the structural changes occurring in a hot rolling schedule and can vary markedly depending on the type of rolling process and the steel quality rolled. In particular, it can model; (a) the rolling of bars of mediumC steel for which recrystallisation may be very fast resulting in annihilation of strain history and associated with grain refinement and there is also a reversal of shear strains which modifies the DRX (b) the rolling of stainless steel plates, at relative low temperatures at which the recrystallisation is nonexistent so that normal strains accumulate throughout the passes and grains elongate. The shear strains may reverse but there will be some accumulation of residual shear strain.
Accordingly, it has been incorporated into ABAQUS FE coupled temperature displacement models of bar and flat rolling.
Constitutive data for the medium C steel have been obtained from results of torsion tests at CEIT and compression tests at the University of Oulu with comparable hardening rates.
(a) Shear strain reversal in torsion produces material softening; this has been implemented as an isotropic contraction of the yield surface.
(b) Following a change in strain rate, the steel almost follows an equation of state, with a strain transient of about 0.02. Analogous delay in the microstructural response may have a significant effect on dynamic recrystallisation especially in bar rolling where the interpass times are extremely short.
The data for the duplex stainless steel have been obtained from torsion tests at CEIT. Shear strain reversal has been modelled using both hypotheses.
For both steels shear strain reversal produces material softening and accordingly has been incorporated into both constitutive equations.
(i) The model was used to assess the amount of normal strain reversal occurring between the H8th and V9th passes of the OriMartin bar mill in the rolling of the fast recrystallising 55Cr3 medium C spring steel. Reversal of the shear strain is predicted in the rolling plane within the 8th pass. Predictions using the selective softening hypothesis for reversal differ very little from the those obtained using the classical isotropic hardening law; alternatively, use of the full softening hypothesis has no effect on the rolling forces but a significant effect on torque. The shear strain reversal influences the process of dynamic recrystallisation especially towards the rolling stock surface where the DRX volume is reduced, which occurs in a central but wide area of the bar, therefore having minimal effect on the shear strain near the roll stock surface. The presence of suchPage 32an alternating HV stand reverses the strain components in directions transverse to rolling, so cancelling accumulation effects. The only cumulative strain component is the one in the rolling direction; since recrystallisation is completed almost everywhere in the interpass period, the effect on the 9th pass is minimal and prevents any further reversal.
(ii) Levels I and IV of the model have been applied to the flat rolling of the essentially nonrecrystallising duplex stainless steel 2304 in roughing and tandem configurations. The shear strain therefore accumulates from pass to pass, especially for the reversing stands; the extent of the area affected is however almost the same. Use of the full strain hypothesis is found to be too excessive on softening and consequently has a significant effect on torque and power, predictions of roll force and power in each second pass being higher when the NOSSRS hypothesis is employed. Predictions for the tandem mill are noticeably higher than those for the reversing stand but, when compared under the same thermal conditions, the difference, in contrast, becomes marginal. In contrast with results for the bar mill, these predictions differ significantly from those for the first pass. (Comparison with values measured in the mill is covered in Section WP6).

Finite element models developed by Corus have been applied to selected experimental tests and rolling trials for all the types of complex loading path considered. Where measurements have been possible, predictions have generally been in good agreement e.g. for loads and torques and spread in the forwardreverse plate rolling trials. Quantitative comparison of strains has not been possible because of unsuitability of the HenslerGifkins technique for the metallurgical structures employed but normal strains e.g. for the multidirectional compression tests are qualitatively as expected. Application of the dislocation based model to the bar rolling trials predicts: (a) for pass 1, an essentially uniform distribution of grain size and dislocation density, but not of Mises stress, this being maximum at the quarter depth in the midplane position; (b) little recrystallisation after the interpass as suggested by the experimental results.

The TU Freiberg model, including subroutines for the stressstrain behaviour and implemented using the FE code MARC has been applied to the plate rolling schedules considered for CMnNb steel. Differences in the shear deformation during the second pass have been indicated between forward and reverse rolling. The flow lines during the passes have been computed from the predicted fields and compared with results obtained using visioplasticity measurements. The deformation of the flow lines and the wire grid are very similar apart from small deviations near the exit of the rolling gap in the middle of the rolling sample as well as in an area located at a quarter of the sample thickness below the surface. The deviations are probably due to the out  of  plane displacement of the specimen surface, frictional effects, the influence of thermal boundary conditions and the use of a 2D rather than a 3D FE model.
Page 33
From computational studies the origin of the inhomogeneity and differences in recrystallisation behaviour have been identified.

The MEFOS model has been validated by applying it to compression and to rolling. The first application was set up to validate the flow stress behaviour i.e. to check that the data had been sampled correctly. FE predictions, at several temperatures and strain rates, of the development of the yield stress during the compression were compared with curves obtained from the constitutive model. Good agreement was obtained for CCrV steel, and for CMnNb steel except near the plateau and where the second derivatives are the largest i.e. the model does not explain the softening phenomena at strains around 0.4. The second application was to assess the model for the benchmark application i.e. the plate rolling trial at TU Freiberg. Predictions indicate; (a) a maximum effective stress within the deformation zone; (b) relaxation in the rolling direction; (c) an increase in dislocation density during rolling; (d) higher effective plastic strain at the middle than at the front section; (e) almost constant vacancy concentration.
High values of dislocation density are predicted by the model; it has not been possible to confirm these values experimentally.

The constitutive model developed by CSM has been set up using data for 55Cr3 steel from the Gleeble compression machine at the University of Oulu and from the torsion machine at CEIT and has also been validated experimentally. In the first case, for a test temperature of 1050°C and a strain rate of 1 s^{1} excellent agreement was obtained between predicted and experimental loads and stress strain curves. The torsion tests used the same primary variables as the Gleeble and comparison was made between corresponding values of torque. Very good agreement was obtained for strains relevant for rolling i.e. up to 0.7. Predictions of strain obtained from the FE model also agreed well with those calculated using the analytical Fields – Backoffen model. Stressstrain curves obtained using the FE model fit well with those obtained using the analytical model. Results were subsequently compared with test results on the Gleeble with abruptly changing strain rate conditions. Differences in stress were found, indicative of the errors made by assumption of instantaneous response in the model; these in general were small, results for the material occurring at a strain of c. 0.02 later than predicted. The model was then validated for rolling both on experimental and industrial mills. Reasonable agreement was obtained for the loads. For industrial validation, the ORIMARTIN mill was chosen. Predictions of load evolution are in excellent agreement with the trace from the mill.