A mathematical model for the recovery and recrystallization of austenite during a multi-pass hot rolling was developed. Contrasting to the conventional model, in which the condition of austenite before each deformation is expressed only by the average grain structure with homogeneous dislocation density, in the present model, the austenite is expressed taking into account the microstructure heterogeneity. Thereby, the distribution of grain size and dislocation density can be predicted. In modelling the microstructural change in the multi-pass hot deformation process, austenite was divided into a large number of small elements having, in each, different grain sizes as well as different dislocation densities. The distribution of grain size and dislocation density can be expressed by summing up the conditions of all the elements. The optimum number of elements were determined to be 100 considering the accuracy of prediction and the calculation time. The comparison of recrystallized fraction between the present and conventional models indicated that the present model gave much-improved results, especially when partially recrystallized austenite was hot rolled as often experienced in the latter stage of hot strip mill rolling and in the entire process of plate mill rolling.

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The temporal evolution and morphology of two-dimensional grain growth are simulated by Monte Carlo simulation techniques. In the simulation, the anisotropy of the grain boundary energy is incorporated into the model. Compared with the case in which no anisotropy of boundary energy is assumed, the suppression of grain growth is observed and the grain size and the edge number distributions become broad. The occurrence of the wetting phenomena is considered to be responsible for the broadness of the grain size distribution in the structure simulated by a model in which anisotropy of boundary energy is incorporated. The pinning effect of precipitates on growth kinetics is also studied by the model. The size of pinned grain is found to varied with inverse square root of the particle volume fraction. The size distribution of pinned grain structure is narrower than that of the pinning-free grain structure with the same anisotropy of grain boundary energy. The average size of n-sided grain is proportional to the grain edge number, n. The effect of the anisotropy of grain boundary energy on the n-dependence of the average grain size is not evident. The nearest neighbor sides correlation like Aboav-Weaire relation is observed in both grain structures computed by Potts model and modified Potts model. The conversion of Monte Carlo step to real time is attempted based on diffusion controlled mechanisms. The estimated grain size with use of the conversion formula in pure iron at 1273 K is in good agreement with that evaluated by the empirical formula derived on the basis of experimental result.

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The isothermal precipitation behavior of Nb(C, N) in austenite has been investigated using steels of different carbon contents. The observed progress of the precipitation in extra low carbon steel is much faster and size of the precipitates is apparently larger than those in steels with higher carbon content even though their supersaturations are the same. To explain this phenomenon, the local equilibrium at the austenite/Nb(C, N) interface has been introduced into the classical nucleation theory and the spherical growth theory, and a generalized precipitation model has been proposed which can predict the precipitation behavior of extra low carbon steels as well as that of HSLA steels. As a conclusion, this study clearly shows that the kinetics of MC-type precipitation is influenced not only by the precipitate-forming atomic product, M×C, but also by its ratio, M/C.

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A computer model which predicts complex precipitation behavior quantitatively in Nb-Ti bearing steel has been developed on a theoretical basis. The solubility and composition of the complex precipitates, and the chemical driving force of the precipitates from supersaturated austenite are estimated by means of thermodynamic analysis of regular solution composed of four-binary compounds. The change in dislocation density which acts as a nucleation site during hot working is calculated by using dislocation theory. And the time dependence of volume fraction and the particle radius of strain induced precipitation are also predicted on the basis of classical nulceation theory. In order to estimate the effect of deformation on nucleation, the change in elastic energy of dislocation with nucleation is calculated.
Experimental results showed that combination of Nb and Ti addition, decreased the solubility of carbonitrides and accelerated the precipitation rate from supersaturated austenite because of the formation of complex precipitates. Such experimental results are in good agreement with the prediction by the present model. And both the acceleration of precipitation rate and the refinement of precipitates particles due to hot deformation are also quantitatively explained.

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The development of a software package DICTRA for simulation of diffusional reactions in multicomponent alloys and a databank for multicomponent diffusivities is described. Applications concerning heat treatment of low-alloy steels and martensitic stainless steels are discussed.

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A computer model is developed to simulate the time-temperature-transformation (TTT-)diagram for the proeutectoid ferrite transformation in Fe-C and Fe-C-X alloys, where X is a substitutional alloying element, from the diffusion growth theory which assumes para- and/or local equilibrium of solute atoms at advancing phase interfaces. The incubation time of ferrite allotriomorph nucleus calculated from the nucleation theory is assumed to be the initiation time of transformation. The influence of the enrichment (or depletion) of solute atoms in the untransformed austenite matrix on subsequent transformation is incorporated by using the quasi-stationary approximation to calculate the diffusion field in the matrix. Both the planar growth and the growth of spherical interfaces (toward the center of sphere) are considered. The model reproduces the basic features of proeutectoid ferrite transformation in iron alloys during isothermal holding and can be used to predict approximately and quickly the TTT-diagram for the transformation when the alloy composition and the austenite grain size are given.

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To make quantitative prediction of microstructure in HSLA steels produced by isothermal or continuous cooling transformation, transformation kinetics of various phases were modeled thermodynamically. Five kinds of phases, i.e. polygonal ferrite, Widmanstätten ferrite, pearlite, bainite and martensite, and 10 alloying elements were taken into account. A program was developed on the basis of this model. Out line of the involved transformation kinetics are presented in this paper. The effects of work-hardening of austenite on the nucleation and growth rates of various phases are discussed and a mathematical model of transformation kinetics from work-hardened austenite is also presented.

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A mathematical model for calculating phase transformations in steels during rapid heating and cooling is presented. It is based on a rule of additivity. The isothermal kinetics are modelled by Johnson-Mehl-Avrami law. The model describes the kinetics of austenitization during heating, the state of austenite at the end of heating (carbon content, grain size), the kinetics of transformations during cooling, the final microstructure and hardness. The model is worked out firstly on dilatometric specimens without thermal gradients in order to validate the modelling and the input data. Then the application of the model to massive cylinders heated up and cooled down with high thermal gradients is presented.

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Thermodynamic and kinetic modelling was performed to describe the decomposition of austenite and the precipitation of alloy carbides in Nb microalloyed steels. Phase analysis of the Fe-C-Mn and the Fe-C-Nb systems was carried out considering local equilibrium and paraequilibrium, their nucleation behavior (matrix/grain boundary) and the shear energy during transformation. From the analysis, equilibrium transformation temperatures of each phase, the Gibbs energy changes for nucleation and compositions of elements at the phase boundary were derived. The transformation kinetics were formulated using classical nucleation and growth theories. Both local equilibrium and para-equilibrium conditions were considered to evaluate the partitioning of Mn during growth. The calculated TTT, CCT diagrams and the volume fraction of each phase were compared with the experimental results, and the most probable behaviors of γ/α transformation and NbC precipitation in Nb steels were discussed.

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The heavy-gauge high strength steel plates with low yield ratio (YR) have been applied to the steel-frames of high-rise buildings to utilize the benefits of their high strength and uniformability. The yielding behavior of steels is influenced by the microstructural factors such as microstructure, grain size, volume fraction and shape of each phase, dislocation density and so on. In the present paper, the microstructural morphologies to control the yielding behavior of the steel were investigated by FEM analysis. The influences of shape, volume fraction and mechanical properties of each phase on yielding behavior were analized by using dual-phase and tri-phase models. From a series of FEM analysis, the appropriate microstructural morphologies to decrease the YR were clarified. Results obtained are summarized as follows; (1) Uniform distribution of circle high-hardness phase in low-hardness phase, (2) Around 50% volume fraction of low-hardness phase, (3) An increase in the ratio of yield strength of high-hardness phase to that of low-hardness phase.

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An approach to predict mechanical properties of hot-rolled multi-phase steels referring to the stress-strain curves is proposed. Different from a conventional approach of regression analysis about the relationships between properties and chemical compositions and processing factors, a proposed one is based on the analysis and application of stress-strain curve: several commonly used mechanical properties such as yield strength, tensile strength, uniform elongation, total elongation, work-hardening exponent (n) and Vickers hardness, are derived systematically from the stress-strain curve of a multi-phase steel, which is calculated by using concentration factor, i.e., strain partition ratio and stress-strain curves of constituent phases. Stress-strain curves of individual component structures such as ferrite, pearlite, bainite, and martensite are expressed by Swift's equation. Physical background of the concentration factor is discussed by examining theoretical models of deformation for two-phase materials. Evaluation of plastic relaxation related to microstructural topology might be the most difficult point of this approach and some trials are presented.

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Computer modelling of microstructural changes and the relationships between microstructure and mechanical properties of hot rolled steel products has been under active development in the research community. A new technology emerges from the modelling studies, attempting to predict the microstructural changes occurring during hot rolling and cooling of steels and to control their mechanical properties so that production is carried out under the optimum processing condition. This computer aided prediction and control technology is practiced using the mathematical models based on physical metallurgy. The model describes quantitatively the transformation behaviors during hot working and cooling, such as recrystallization, grain growth, precipitation and phase transition from austenite to ferrite. It also clarifies the relations existing among the processing condition, microstructure and the final mechanical properties. Various models of alloy steels as well as plain carbon steels have been developed for the last 10 years. It has been expected that the models would widely be applied to practice in the steel industry in the near future and make a great contribution to quality control and process optimization. Some prospective areas where the models are applied are guarantee of mechanical properties throughout the coil length, elimination of tensile tests, decrease in property variation, automatic resetting of processing conditions, save in alloys, and developing new process control models. However, in order to meet this expectation, there are many obstacles to be cleared with regard to model refining, understanding on physical metallurgy, sensor development, process control and quality design systems. It was also suggested that a close cooperation among researchers and engineers from different disciplines would be indispensable to accomplish the goal.

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By collaborative work the Sheffield Leicester Integrated Model for Microstructural Evolution in Rolling (SLIMMER) has been developed for hot rolling of flat products. The background physical metallurgy is presented together with the expressions used to describe microstructure evolution for a range of ferrous and non-ferrous metals. The finite difference thermal model at the heart of SLIMMER computes heat loss to air, descalers, rolls and water cooling while allowing for oxidation and deformation heating. The use of temperature compensated time enables isothermally determined equations for microstructure evolution to be applied to practical non-isothermal conditions. Rolling loads and torques are calculated using Sims theory with an accurate prediction of mean flow stress. Examples of rolling niobium microalloyed steel plate and the effect of initial grain size illustrate the capabilities of SLIMMER and show some of the validation of the predictions.

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The static recrystallization characteristics (grain size, kinetics) have been established for Ti-V-(Nb) austenites and used as the basis for a theoretical evaluation of microstructural evolution during hot rolling of plate. In this context, particular attention has been focussed on so-called recrystallization controlled rolling, whereby a fine as-rolled ferrite garin size is obtained via transformation from an austenite which has been substantially grain refined via static recrystallization. The model is shown to forecast a behaviour which is in acceptable accord with practical plate-rolling experience. Furthermore, the rolling model has been used in a systematic theoretical investigation of the effect of principal rolling variables on the degree of microstructural refinement during processing of Ti-V-(Nb) steels via recrystallization controlled rolling.

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The combination of a mathematical model for recrystallization of austenite and a computer simulation for strain and temperature analyses were performed in order to predict the austenite grain size distribution in the through-thickness direction. The experimental grain size distribution was measured using laboratory one-pass and two-pass rolling tests in low carbon steels.
Mathematical models for the critical strain for dynamic recrystallization, the grain size of dynamic and static recrystallizations, the fraction of dynamic and static recrystallizations and the grain growth of dynamically- and statically-recrystallized austenites are made to predict the austenite structure, based on the results of compression tests using hot deformation simulators. The temperature calculated with the finite element method (FEM) and the finite difference method (FDM) corresponded well with the temperature measured at the center and the quarter point of thickness of plates. Grain size distribution in the through-thickness direction increased with coarsening the initial austenite grain size and at rolling reduction ranging from 20 to 40%. The calculated grain size distributions obtained by computer simulation corresponded well with the experimental ones obtained with rolling tests at rolling reductions of 30 and 40%. The calculated values were, however, smaller than the experimental ones at a large rolling reduction of more than 50%. Since the difference between both values increased with increasing fraction of dynamic recrystallization, it is necessary for the mathematical model for it to improve.

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Computer models for the simulation of grain size development during thermomechanical rolling and the resulting strength properties have been developed for the prediction of the material properties of microalloyed HSLA steel plates. The main input variables for calculations are steel composition, deformation-time-temperature schedule during rolling and cooling rate after rolling. A comparison of calculated and experimentally determined grain sizes as well as strength properties shows a good consistency. The models can therefore efficiently support the development of new steels and the planning of altered production conditions.

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For manufacturing the steel plates with good mechanical properties by Thermo-Mechanical Control Process (TMCP), the manufacturing conditions are necessary to be controlled in an integrated manner through the reheating step to the accelerated cooling step. The present mathematical model for the prediction of the microstructural evolution and the mechanical properties of the steel plates produced by TMCP enables this integrated control. The present report describes the flow and the each equation of the model and the comparison between the calculated results with this model and the data obtained by the rolling experiments.
The model consists of four modules such as Reheating, Rolling, Cooling and Mechanical Properties in which the metallurgical phenomena occurring at each step are calculated. The characteristic of the model is shown as follows:
(1) The effect of work hardening of austenite by rolling in non recrystallization temperature region is represented as the effect of average dislocation density (ρ) accumulated in austenite grains. The nucleation rate of intragranular ferrite and that of grain boundary ferrite are distinctively formulated as functions of ρ.
(2) The effects of microalloying elements are considered in the formulation of metallurgical phenomena and mechanical properties such as pinning and solute drag effects on the grain growth of austenite at the reheating step, the restraining effect on recovery and recrystallization of work-hardened austenite at the rolling step, the restraining effect of solute elements on the progress of transformation at the cooling step, and the solution hardening and the precipitation hardening for the mechanical properties of the plates.

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The mathematical model for the prediction of microstructural evolution and mechanical properties shown in the previous report is integrated with the process models those calculate the strain and temperature distribution through thickness at multiple points in the plate during the rolling and the cooling steps for the adoption to a commercial production line. The experiments that cover the wide range of the chemical compositions and the manufacturing conditions of steel plates were conducted and the obtained results such as microstructures and mechanical properties of the plates were compared with the calculated results of the present integrated model. The main points of the present report are summarized as follows:
(1) For the observation of austenite (γ) grain size during the interpass time of the rolling, the Thermal-Sampler that could cut off and quench a small specimen within 10 sec after the previous rolling pass was equipped in the commercial rolling line. The γ grain size calculated by using the present model almost agreed with the observed ones.
(2) In the case of the air-cooled plates after the rolling whose microstructures were mainly ferrite (α) and pearlite, the calculated microstructures such as α grain size, the volume fraction and the microhardness of each phase showed good agreement with the observed ones. As a result, the calculated mechanical properties agreed well with the observed ones.
(3) In the case of the acceleratedly cooled plates after the rolling whose tensile strength was above 600 MPa and major microstructure was bainite, the accuracy of the calculated results of microstructures and mechanical properties were inferior a little to the cases of the air-cooled plates. Even in the plates whose tensile strength was above 600 MPa, the calculated results were confirmed to be practically adoptable for the plates whose major microstructure consists of ferrite.

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The computer simulation model of microstructural evolution on the basis of chemical thermodynamics and classical nucleation and growth theory has been developed. The metallurgical phenomena in thermomechanical treatment of steel, such as austenite grain growth, recrystallization and growth, carbonitride precipitation and austenite to ferrite phase transformation can be predicted by the model. The influences of steel chemistry and thermomechanical condition on the transformed microstructure of 0.10C-1.5Mn-0.35Nb steel (high C-high Mn steel) and 0.06C-1.25Mn-0.035Nb steel (low C-low Mn steel) are evaluated by computer simulation. With the increase of C and/or Mn concentrations, the volume fraction of second phase increases and the ferrite grain size is refined C-high Mn steel, the transformed microstructure consists of ferrite and pearlite phases at lower cooling rates and/or larger effective austenite interfacial area per unit volume, S_V. The volume fraction of second phase increases with the increase of cooling rate and/or the decrease of S_V value. The second phase of the steel at higher cooling rates is bainite. The transformed microstructure of low C-low Mn steel consists of ferrite and bainite phases. The influences of the rolling condition and the cooling rate on the transformed microstructure are smaller in low C-low Mn steel. However, the transformed microstructures is apparently influenced by the start temperature of accelerated cooling.

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Mathematical models for predicting microstructural evolution and mechanical properties of hot strips have been reviewed. The metallurgical features of the hot strip rolling are discussed and the fundamental idea of their modelling is introduced.
As applications of the mathematical models, the on-line prediction of the microstructure and strength, the resistance to hot deformation and the cooling curves affected by the heat evolution due to transformation are given.
Finally, the future work and prospective of the mathematical model in hot strip rolling is also presented.

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A mathematical model has been developed for predicting phase transformation of hypoeutectoid steels during continuous cooling, with a special reference to the application to the temperature simulation in hot strip rolling. The model contains only a small number of experimental parameters so as to be used in wide ranges of temperature and chemical composition. Coupled with a FEM heat-transfer model, this model predicts the temperature of steel strips on the run-out table of a hot strip mill with great accuracy. This model is applied to the investigation of the effect of processing conditions on hardness of 0.5 mass%C steel sheets produced in the hot strip mill. The results show that the calculation by the model prior to the production brings useful information to attain steel sheets with high quality.

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A set of integrated mathematical models for simulating hot rolling and controlled cooling of wire rods and bars has been developed through extensive laboratory research work and validation against carefully monitored results from industrial mills.
Experimental tests have been carried out on C-Mn and eutectoid steels selected as representative of the various applications of wire rods and bars.
Static and dynamic recrystallization of austenite, fraction of transformed austenite, final microstructures and mechanical properties are all calculated by modelling physical phenomena and using quantitative relationships between the microstructural and kinetic parameters and the process variables, i.e. strain, strain rate, temperature and time.
The models have been applied to predict the microstructure evolution during hot rolling and to investigate the effect of working conditions and recrystallization mechanisms on the formation of heterogeneous austenitic microstructures.
The effects of the cooling pattern on the temperature profile and the austenite phase transformation have also been studied to prevent: coarse pearlite and martensite formation at the centre of wire rods which have cores enriched in C and Mn; surface hardening of bars when water tube cooling systems are used to control the temperature at the cooling beds.
The models provide an important insight into the process that is beneficial to enhance the quality of long products.

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