AuthorG.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

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1. 1 Complex loading

In the context of process history in hot rolling, loading can be either mechanical or thermal. The mechanical loading usually refers to the roll separating force but is also associated with front/back tensions and loads accompanying imposed boundary conditions. Thermal loading is associated with reheating and cooling and should be included in this context.

Complex loading, by definition, occurs when there are rapid or abrupt changes in the loading path. During deformation, it can thus arise, for example, from rapid changes in strain rate and temperature, shear strain reversal, microstructural changes associated with dynamic recrystallisation etc and material inhomogeneities causing abrupt changes in stress levels as the stock progresses through the roll gap. Complex loading needs ideally to be considered in the context of the complete process route, but in the project is studied for at most 3 pass schedules including interpass cooling and reheating. In this case, complex loading is associated with changes in the deformation path such as interpass turns of the stock in H-V rolling and reversing of the stock or rolling direction in plate rolling. Load changes also arise as a result of static recrystallisation , cooling etc in the interpass.

It has been demonstrated in the final report of a previous ECSC project 1 (the forerunner to the current project) that different yield stresses may be obtained if different loading paths are used to obtain the same final dimensions. More generally, it has been shown that for certain metals, the thermomechanical history and microstructural evolution may significantly affect the properties and influence the magnitude of the loads and other operating parameters during processing 2. It is therefore important to establish, for which cases i.e. forming processes and steel qualities, the forming route is significant.

Significant effects are more likely to be observed when the path changes abruptly for instance if there is a substantial change in the strain angle, 4, defined by the equation


Where are the strain tensors at the point considered before and after the change in strain path and represented relative, to a coordinate frame, in the conventional manner, as a 3x3 array with normal strains along the diagonal and shear strains off the diagonal.

Application of the above formula assists in choosing deformation processes in which there is a significant change in strain angle. It also indicates that a complete reversal in strain, or even changes of 90° or more, may be difficult to realise in practice. Indeed, in phenomena such as shear reversal, although significant curvature in the strain path seems plausible, the change in strain angle is actually of the order of 60°. The changes likely to be observed in practice thus depend on the metal forming processes of commercial interest and on the experimental facilities available. The Department of Engineering Materials at SheffieldPage 12University has already carried out valuable work on the influence of strain path on metallurgical structure 3 using an ASP (arbitrary strain path) machine as well as a Gleeble machine and close liaison has been maintained through the IMMPETUS program 4.

1. 2 Constitutive modellling

In order to understand the relationship between microstructural evolution and process history for such complex loading paths, experimental testing alone is important but insufficient; it is essential also to predict, with sufficient accuracy, both the process parameters and microstructural characteristics for current and novel forming schedules. Consequently, an appropriate mathematical model needs to be selected and tested, if available. Therefore a review needs to be carried out of models by the partners, of models described in the literature and modelling facilities of third parties.

Many models, as to be expected, are available or potentially developable. These may be classified in several ways, each type having its potential uses and drawbacks 5. At one extreme are the empirical models obtained by fitting curves to experimental data without consideration of the reason, if it exists, for the relationships between the variables, and similarly, neural network models. At the other extreme are the phenomenological models based on 'fundamental' laws governing natural phenomena. The material properties in this case consist of 'fundamental' experimental data. Phenomenological models, in principle, can be applied to new problems with a minimum of new data i.e. sufficient to define the problem. In practice, the models may be very complex so that analytical solutions are rarely possible and so require numerical solutions obtained using finite elements etc. Generally, the models employed have been deterministic, but if it is difficult or impossible to observe a quantity, e.g. grain size then such a quantity can be replaced by a suitable statistical distribution; such models are termed probabilistic or stochastic if relating to temporal evolution. Generally, simplicity rather than complexity is required. One may wish to take on board the physical laws to predict the general behaviour without handling the difficult equations. This can sometimes be achieved by qualitative physical models.

Models can also be classified according to the scale or levels at which they operate, different but often relatable physical 'laws' being used at each level 6. The number of levels used and their definition in the literature varies somewhat, but broadly consist of (a) the macroscopic level, at which the operating parameters can be related to reduction and bulk temperature etc but not to the structural parameters (b) the mesoscopic level, at which the standard models for internal variables such as stress and strain, obtained by averaging over a length scale of c. 0.01–1.0 mm, are formulated (c) the 'microscopic' level, depending on whether optical or electron microscopy is used, dealing with evolution of grains and dynamics of dislocations. Between (b) and (c) a number of 'mezza' levels have sometimes usefully been introduced, allowing the gradual introduction of quantities such as hardness, drag stress, fraction recrystallised etc which reflect selected aspects of the microstructural behaviour. Also below (c), levels have been introduced, the 'submicroscopic' being relevant in this context and dealing with nucleation of grains from subgrains, or, in modelling damage, the migration of atomic vacancies to form voids. In reviewing and formulating constitutive models, attention has been focused on levels (b) and (c).

Constitutive equations describe the response of the body to applied loads resulting from the internal constitution of the material and so in this context include the various equations of elasticity, plasticity and heat transfer. A theory of constitutive behaviour, in which the constitutive equations are required to satisfy certain axioms and from which various types of constitutive behaviour/equations are classified, was formulated by Truesdell and Noll and Coleman 7,8. The application of this theory to continuum mechanics is described in detailPage 13in the book by Malvern 9 and the potential application to metal forming in particular described in a previous ECSC project 10. Of particular relevance to complex loading, hereditary behaviour is described by equations in which the stress is a functional of the previous history and, of possible relevance to rolling texture, simple materials are defined in which the stress is dependent on the history only of the deformation gradient. For some materials, viz. those with 'fading memory' only the recent history has a significant influence. In metal deformation, experiments have sometimes detected a slight lag in this response whereas in classical models of plastic deformation as used in rolling it is assumed that there is an immediate stress response to an applied displacement i.e. that the material satisfies an equation of state. These observations suggest that, to investigate the effect of deformation history, models along the above lines should be developed.

The equations referred to above are very difficult to solve efficiently and it is recommended that solutions be deferred. In the short term alternative ways can be found of incorporating process history the results of which can ultimately be compared with the those of the more complex equations and with experimental results, allowing the accuracy and efficiency of each solution approach to be assessed. One method is to update the programs with information on the deformation etc. gleaned from experimental results showing a history effect. It should also be noted that the equations do not directly predict the effects on metallurgical structure.

Not only is the metallurgical structure of a formed product a response of commercial interest in the project, it is a signature of the physical state of the latter and therefore a key factor in identifying, and understanding the effects of, process history. It is possible that the metallurgical structure of two products with the same external dimensions will be different, if they are formed by different process routes.

An alternative approach may therefore be adopted based on the assumption that there are state variables, related to characteristics of the microstructure, which encapsulate these history effects. Understanding the evolution of metallurgical structure may be achieved by means of constitutive equations describing the underlying microscopic behaviour of the material, in terms of the coupling between such structural variables and the thermomechanical variables, both during and between deformations. Several constitutive models are available, depending on the mesoscopic model adopted for the thermomechanical variables e.g. (elasto)(visco)plastic, small or finite strain 11,12, the accompanying structural variables 13 and the underlying assumptions based on thermodynamics 14,15. Comparative reviews of constitutive theories in cyclic plasticity have been given by Chan 16 and a unified treatment of the various viscoplastic theories given by Miller 17 and contained in the book by Krausz & Krausz 18. Such models can be implemented on the computer via a range of FE codes (see Refs. 19,20).

1. 3 Experimental investigation of constitutive behaviour and material characterisation

Development of such constitutive equations entails the carrying out of experimental trials to obtain data from which the coefficients can be extracted, to help validate them once formulated and to examine the influence of loading path on structural evolution, in particular recrystallisation. Various experimental techniques are used to carry out this examination and have been reviewed by Dzubinsky 21. Some are mechanical in nature using stress relaxation or double hit tests and others are direct observation methods using either optical or electron microscopy (e.g. EBSD/SEM, TEM).

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The basis of the stress–relaxation (SR) test 22,23 is that the decrease in stress following deformation can be divided into 3 discrete stages consisting of creep, recrystallisation and stress relaxation of soft material. An equation has been proposed 24 enabling the fraction recrystallised to be ascertained from the characteristics of the curve, however stress remaining during the test can sometimes promote climbing of unstable dislocations and thereby accelerate recovery. Stress relaxation tests can thus prove insensitive so double hit tests are used instead in such cases. In the double hit test (DH), the fraction recrystallised is calculated by finding the changes induced in the second deformation; this is achieved either by back extrapolation, comparing areas or by comparing the difference between the mean yield and an offset yield stress for both hits. It is generally less reliable than stress relaxation when the latter can be employed.

In optical metallography (OM), the fraction recrystallised is found in section(s) of a suitably prepared sample either by finding the ratio of the total area of regions recrystallised to the area observed, by similarly comparing intercepts on a given line with the total length or by finding the fractional number of a points in a random set that fall within the recrystallised region. The method is relatively simple to apply, although sometimes time consuming, but significantly subjective. It is difficult to apply for low carbon steels and when observing complex microstructures that have been austenite quenched. Austenite structures can be difficult to detect when transformation has occurred and bainite or pearlite have been formed; they are easier to observe when martensite has been formed but this generally requires a rapid cooling rate and thin sections. It can also be difficult to distinguish between recrystallised and deformed grains particularly if the sample has been subjected to a compression that was axisymmetric. OM is also used to measure grain sizes and to describe the local distribution (usually assumed to be lognormal) by calculating the average grain size and standard deviation. If there is not a wide variation in grain size, and the original grain sizes are equiaxed then calculation of the major and minor axes of the deformed grain can be employed to calculate the strain distribution using the Hensler-Gifkins method 25; the strains can then be used in model validation.

Of the techniques considered, EBSD mapping is potentially the most accurate. The electron backscatter patterns of Kikuchi bands produced by diffraction of a beam from a Scanning Electron Microscope, on striking the crystalline sample, are used to obtain Orientation Imaging Micrographs which provide a visual characterisation of the orientation characteristics of the microstructure. By combining these micrographs with a band contrast map, a measure of the sharpness of the EBSD pattern above the background is obtained, which, in principle, enables recrystallised and non-recrystallised regions to be identified. A number of micrographs are employed e.g. image quality maps, orientation spread maps and inverse pole figure maps. The former describes the quality of the pattern and provides a qualitative indication of the microstrain distribution, a high IQ value suggesting recrystallisation. The orientation spread for each grain is obtained by averaging, over points of the grain, the absolute deviation from the mean orientation of the grain; a small value indicates recrystallisation. Inverse Pole Figure Maps use colour to represent the orientation, a predominance of white suggesting little likelihood of recrystallisation. For maximum accuracy, the maps should be used in combination. The EBSD/SEM method, however, does have some limitations (a) as for OM, it can only determine the fraction recrystallised if no phase transformation has occurred after recrystallisation and does not easily differentiate between deformed and recrystallised grains, particularly if the sample has undergone axisymmetric compression, (b) the accuracy depends on the equipment used, a tungsten filament being less accurate than a field emission system.


[1] Ingham, P M: 'The Effect of Strain Reversal and Strain-Time Path on Constitutive Relationships for Metal Rolling/Forming Processes', Final Report ECSC Project, 7210.EC/811, 2000.

[2] Davenport, S B, Higginson, R L and Sellars, C M: 'The effect of strain path on material behaviour during hot rolling of FCC metals', Phil. Trans. R-Soc, London A (1999), 357, pp1645-1661.

[3] Angella, G: 'Strain path, flow stress and microstructure evolution of a austenitic stainless steel at high temperature', Ph.D. Thesis (University of Sheffield), 2002.

[4] Wynne, B: Dept. of Engineering Materials, Univeristy of Sheffield (Private communication).

[5] "Process Modelling", Ch. 8 in 'Advanced Physical Chemistry for Process Metallurgy', Ed. Maeda, M, Academic Press, 1997.

[6] Weinan, E and Engquist, B: 'Multiscale Modelling and Computation', Vol. 50, No. 9.

[7] Truesdell, C and Noll, W: 'The non-linear field theories of mechanics in 'Encyclopedia of Physics', ed. Flugge, S, 1965, Vol. 3/3, Berlin, Springer-Verlag.

[8] Coleman, B D: 'Thrmodynamics of materials with memory' Arch. Rational, Mech. Anal, 1964, Vol. 17, pp1-46.

[9] Malvern, L E: 'Introduction to the mechanics of a continuous medium', publ. Prentice Hall, Inc. 1969.

[10] ECSC Research Contract 7210.EA/805: 'Three-Dimensional Model for Metal Flow under Triaxial Strain Conditions', British Steel Corporation, 1983.

[11] Hohenemser, R and von Prager, W: Z Angew. Math. Mech., 1932, Vol. 12, pp216-226.

[12] Oldroyd, J G: 'A rational formulation of the equations of plastic flow for a Bingham solid' Proc. Cambridge Phil Soc., 1947, Vol. 43, pp100-105.

[13] Perzyna, P: 'On the constitutive equations for work hardening and rate sensitive plastic materials', Bull. Acad. Pol. Sci. Ser. Sci. Technol, 1964, Vol 12, (4), pp199-206.

[14] Rice, J R: 'Inelastic constitutive relations for solids: An internal-variable theory and its application for metal plasticity', J. Mech. Phys. Solids, 1971, Vol. 19, pp433-455.

[15] Lematre, J and Chaboche, J L: 'Mecanique des materiaux solides', Dunod, Bordes, Paris, English edition, CUP, Cambridge, UK, 1985.

[16] Chan, D S, et al: 'A survey of unified constitutive theories', 2nd symp. NASA LEWIS, Non-Linear Constitutive Relat. High Temp. Appl., Cleveland Ohio, NASA Conf. Publ. 2365, 1969.

[17] Miller, A H, ed.: 'Unified constitutive equation for plastic deformation and creep of engineering alloys', Elsevier Applied Science, NY, 1987.

[18] Krausz, A S and Krausz, K (eds.): 'Unified Constitutive Laws of Plastic Deformation', Academic Press, 1996.

[19] Bodner, S R and Partom, Y: 'Constitutive equations for elastic viscoplastic strain- hardening materials', J. Appl. Mech., 42, pp385-389.

[20] Chaboche, J L: 'Viscoplastic constitutive equations for the description of cyclic and anisotropic behaviour of metals', Bull. Acad. Pol. Sci., Sem., Sci, Ser.

[21] Dzubinsky, M: 'Comparison of recrystallisaton kinetics established by stress relaxation, double hit, optical metallography methods and using orientation maps obtained by SEM/EBD', Report Code, STC/SMT MMM/KR/08616/2002/A, (Reference Source no. 107498), Corus Research, Development & Technology, Swinden Technology Centre, 2002.

[22] Sellars, C M In: Sellars, C M, Davies, C J eds. Proc. Conf. on Hot Working and Forming Processes, London, Metal Society, 1980, pp3-15.

[23] Karjalainen, L P and Perttula, J: ISIJ, 1996, Vol. 36 (6), pp729-736.

[24] Karjalainen, L P: Mater. Sci. Technol. 1995, Vol. 11, pp557-565.

[25] Hensler, J W and Gifkins, R C: 'The estimation of slip strain during creep', J-Inst. Metals, 1963-64, Vol. 92, p340.

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