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Publication# Numerical simulation of solid state phase transformations in gamma-TiAl alloys

Abstract

Gamma titanium aluminide (γ-TiAl) based alloys are very interesting materials for structural applications at elevated temperatures owing to the combination of high specific strength, good oxidation, creep and fatigue resistance. However, their relatively poor ductility and fracture toughness remain important obstacles for their industrial applications in engineering components. The mechanical properties of these alloys, namely the poor room temperature ductility, as well as the yield stress, creep and fracture resistance, can be significantly improved by controlling the microstructure. These alloys undergo solid state phase transformations upon cooling from the high temperature α phase, which results in formation of different types of microstructures depending on the cooling conditions and the alloy composition. Low and moderate cooling rates lead to the precipitation of the γ phase as parallel plates within the α matrix, followed by the α → α2 ordering reaction. As a result of this transformation, the characteristic lamellar structure of TiAl is produced. Rapid cooling leads to a massive transformation from α to γ. It has been shown that the massive microstructure can be used as a precursor to obtain a refined microstructure through a subsequent tempering in the α + γ phase field. However, obtaining a fully massive microstructure at all depths in a component can be difficult to achieve. Numerical simulation of microstructure formation can be used as a tool to anticipate the microstructure distribution in a component depending on the local chemistry and cooling conditions. The main objective of this work was to develop microstructure models describing the formation of the massive and lamellar microstructures. Two distinct modeling approaches have been used. The first approach is a deterministic model having for objective to predict the microstructure distribution in a cast and heat treated component. The model was designed to be integrated into a FEM or FDM heat flow solver in order to calculate microstructural quantities such as the proportion of phases and lamellar spacings at each nodal point of the mesh. The modeling approach is based on a combination of nucleation and growth laws which take into account the specific mechanisms of each phase transformation. Nucleation of massive and lamellar γ is described with classical nucleation theory, accounting for the fact that nuclei are formed predominantly at α/α grain boundaries. Growth of the massive γ grains is based on theory for interface-controlled reactions. A modified Zener model is used to calculate the thickening rate of the lamellar γ precipitates. The model incorporates the effect of particle impingement and coverage of the nucleation sites by the growing phases. The driving pressures of the phase transformations are obtained from Thermo-Calc based on the actual temperature and matrix composition. The deterministic approach was used to calculate CCT diagrams and lamellar spacings, which showed to be in good agreement with experimental data obtained from dedicated heat treatment experiments and from the literature. The model permitted investigating the influence of cooling rate, alloy chemistry and average α grain size upon the amount of massive γ and the average thickness and spacing of the lamellae. In particular, it indicated that the Al depletion of the α phase during lamellar precipitation seems to play an important role in the suppression of the massive transformation at moderate cooling rate and in the large lamellar spacings observed at low cooling rate. It was also found that Nb additions enhance the formation of massive γ by lowering the lamellar transformation temperature, increasing the T0 temperature and slowing down the lamellar growth due to the low diffusivity of Nb. The second approach was a cellular automaton (CA) model, which was used to describe the formation of the lamellar microstructure on the scale of a few microns. The objective of this approach was to make a direct description of the growth of a precipitate by an interfacial ledge mechanism, and then to evaluate whether the simplifications made in the deterministic model regarding this mechanism are appropriate. The modelling approach is based on the resolution of the diffusion equations in the α and γ phases. The ledge structure at the α/γ interface is described by introducing different types of cells in the CA grid. The CA model was used to describe the formation of the lamellar microstructure at isothermal conditions and at various cooling rates. The thickening kinetics of the lamellae, the lamellar spacings and the overall kinetics of the transformation were calculated with this approach and were compared with the results obtained with the deterministic model. It was found first that the thickening kinetics of the lamellae calculated with the deterministic model are slower than those calculated with the CA model. It was established that the method used in the deterministic model to account for the growth by a ledge mechanism always leads to a growth exponent of 0.5, which is characteristic of diffusion controlled reactions. In contrast, the CA approach yields growth exponents that are comprised between 0.5 and 1, as it is expected for growth in a mixed diffusion/interface controlled mode. The incorporation of a kinetic undercooling term into the deterministic model was found to be an appropriate correction. A good agreement was finally obtained in terms of lamellar spacings and overall kinetics. The underestimated thickening kinetics of the deterministic model was observed to be partly compensated by a higher nucleation rate at the beginning of the transformation. The overall kinetics of the lamellar transformation calculated with the deterministic model was found to be sufficiently accurate to correctly predict the competition between the lamellar and massive microstructures.

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2022A phenomenological modelling approach has been developed to describe the massive transformation and the formation of lamellar microstructures during cooling in binary gamma titanium aluminides. The modelling approach is based on a combination of nucleation and growth laws which take into account the specific mechanisms of each phase transformation. Nucleation of massive and lamellar gamma is described with classical nucleation theory, accounting for the fact that nuclei are formed predominantly at alpha/alpha grain boundaries. Growth of the massive gamma grains is based on theory for interface-controlled reactions. A modified Zener model is used to calculate the thickening rate of the gamma lamellar precipitates. The model incorporates the effect of particle impingement and coverage of the nucleation sites by the growing phase. The driving pressures of the phase transformations are obtained from Thermo-Calc based on the actual temperature and matrix composition. CCT diagrams and lamellar spacings calculated with the model are in good agreement with experimental data obtained from dedicated heat treatment experiments and from the literature. The model permitted investigating the influence of cooling rate, alloy chemistry and average alpha grain size upon the amount of massive gamma and the average thickness and spacing of the lamellae. In particular it indicates that the Al depletion of the alpha phase during lamellar precipitation seems to play an important role in the suppression of the massive transformation at moderate cooling rate and in the large lamellar spacings observed at low cooling rate. (C) 2008 Elsevier Ltd. All rights reserved.

In the first part of this work a comprehensive model of the continuous hardening of 3D-axisymmetric steel components by induction heating has been developed. In the model, the Maxwell and heat flow equations are solved using a mixed numerical formulation : the inductor and the workpiece are enmeshed with finite elements (FE) but boundary elements (BE) are used for the solution of the electromagnetic equations in the ambient air. This method allows the inductor to be moved with respect to the workpiece without any remeshing procedure. The heat flow equation is solved for the workpiece using the same FE mesh. For the thermal boundary conditions, a net radiation method has been implemented to account for grey diffuse bodies and the viewing factors of the element facets are calculated using a "shooting" technique. These calculations have been coupled to a metallurgical model describing the solid state transformations that occur during both heating and cooling. 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The transformation of ferrite into austenite has been described using a pseudo-front tracking finite volume approach for solving the diffusion equation in a 1D, 2D or 3D domain. At the start of the computation, the volume is made of ferritic particles and austenitic zones resulting from the pearlite dissolution. The model allows to calculate the kinetics of the phase transformation as a function of the temperature and the initial microstructure. Although the comparison of the transformation kinetics with experimental results was quite satisfactory, it appeared that the calculated kinetics were slightly slower. This effect has been attributed to the other alloying elements which are contained in the Ck45 steel used for the experiments. This discrepancy has also been explained by a stereological effect which is due to the calculation in a 2D section instead of a 3D domain as demonstrated by a comparison of 2D and 3D simulations. The austenitic grain growth has been described with two models based respectively on a Monte Carlo technique and a mechanical approach. A methodology for obtaining a correspondence between the simulation time scale and real time has been presented and applied to the Ck45 steel. The value of the exponent n of the grain growth law d = Κ t1/n (where d is the mean diameter and t the time) has been determined for both models. The mechanical model (n=2) turned out to describe perfectly the case of normal grain growth as it occurs in liquid-gas systems. However the results of the Monte Carlo simulation (n=2.3) are in better agreement with the non-ideal behaviour of the Ck45 steel. The influence of the impurities and particles which are present in real materials should be taken into account in both models if a more quantitative agreement is to be obtained. Finally, a combined model coupling the various steps of the austenitisation process has been proposed. It allowed to show that the assumption consisting in dividing the process in three separate steps is valid in most cases. This combined model is a first attempt for a comprehensive modelling of the austenitisation process.