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Nickel-base superalloys display cuboidal precipitates aligned along the cubic directions, which are the elastic soft directions. At high precipitate volume fraction, the microstructure is often described as a regular array of precipitates organized on a simple cubic macro-lattice. In the present work, we use a stability analysis and 3D phase field simulations to show that such a regular array is in fact unstable whatever the volume fraction of precipitates. The two main instability modes are the longitudinal [100] mode and the transverse [110] mode along the [ 1 1 ¯ 0 ] eigenvector. We argue that these instabilities lead to formation of configurational defects closely related to experimentally observed branches and herringbone patterns. The roles of elastic anisotropy and elastic homogeneity are also discussed.

Grain growth during the solidification of an alloy is accompanied by the enrichment of the surrounding liquid phase in chemical species. The resulting diffusion fields induce interactions between the grains that are strongly dependent on their spatial arrangement. The influence of these diffusive interactions is not limited to the shapes and sizes of grains but also controls the overall kinetics of the transformation. In this study we investigate the role of the spatial arrangement of equiaxed grains in the average growth kinetics of an ensemble of grains. We perform full-field simulations of different grain ensembles using the mesoscopic grain envelope model (GEM). We show that the growth kinetics of randomly arranged ensembles is fundamentally different from that of periodic arrangements. We provide an explanation of this difference through analyses of the behaviour of individual grains in their local neighbourhood, defined by a Voronoi tessellation. Finally, we compare the full-field GEM simulations to state-of-the-art mean-field models and we point out the limitations of the latter in the prediction of dendritic growth.

The phase field method has emerged as the most powerful approach for modeling microstructure evolutions resulting from phase transformations. This approach provides a consistent thermodynamic framework in which many phenomena can be strongly coupled, such as diffusion, elasticity, and plasticity. The method is therefore particularly suited for studying the microstructure formation and evolution in nickel-base superalloys. In this chapter, we present the strength and limitations of the method and illustrate how the phase field methods have contributed to the understanding of the microstructure evolution in these alloys.

International audience; We investigate the growth of Widmanstätten structures in metallic alloys with phase field calculations. We show that elastic energy anisotropy associated with shear dominated transformations is alone sufficient to give rise to plate or needle like precipitates with constant lengthening rates at constant supersaturations. We show that Ivantsov analytical solutions still provide good estimates of the diffusion flux around tips even when elasticity is accounted for. A careful analysis demonstrates that, for shear dominated transformations, (i) the lengthening direction lies in the habit plane determined by one of the minima of the elastic kernel; (ii) the tip shape and size are equilibrium features, driven by elastic forces, rather than dynamically determined as usually assumed; (iii) the tip size of acicular precipitates can be rationalized by the value of the elastic kernel in the lengthening direction.

Formulating appropriate simulation models that capture the microstructure evolution at the mesoscale in metals undergoing thermomechanical treatments is a formidable task. In this work, an approach combining higher-order dislocation density based crystal plasticity with a phase-field model is used to predict microstructure evolution in deformed polycrystals. This approach allows to model the heterogeneous reorientation of the crystal lattice due to viscoplastic deformation and the reorientation due to migrating grain boundaries. The model is used to study the effect of strain localization in subgrain boundary formation and grain boundary migration due to stored dislocation densities. It is demonstrated that both these phenomena are inherently captured by the coupled approach.

The present contribution focuses on the nature of the β → α transformation in Ti-5333 alloy at intermediate temperatures. It is indeed still unclear whether this transformation is only controlled by bulk diffusion or whether interfacial kinetics may play a role. To address this issue, we have combined SEM and STEM-EDX to measure the concentrations of Al, Cr, Mo, V and Fe in α intragranular precipitates as well as in the abutting β matrix, paying a particular attention to the concentrations at flat interfaces corresponding to the precipitates habit planes. The comparison with Calphad calculations suggests that interfaces are not at equilibrium during the thickening of the plates.

Microstructure evolutions in Ni-based superalloys are investigated during [110] creep loading using 3D and 2D phase field simulations. A recently developed phase field model coupled to a crystal plasticity model based on dislocation densities is employed. The model uses a storage-recovery law for the dislocation density of each glide system and a hardening matrix to account for the short-range interactions between dislocations. We show that small misorientations of the tensile axis strongly modify the evolution: rafting is observed for small deviations, as opposed to a microstructure made of rod-like precipitates when loading is performed along a perfectly aligned [110] direction. Depending on the precise direction of the mechanical load, different evolutions are obtained accompanied by strong modification of the macroscopic creep behavior, explaining the variety of results observed experimentally. The relative role of inhomogeneous and anisotropic elastic and plastic driving forces is also investigated, plasticity being the main driving force for rafting in the considered case. In addition, our calculations show that the initial dislocation density slightly modifies the precipitates morphology whereas the creep curve is significantly impacted.

Fine equiaxed grain regions are frequently observed in additively manufactured Inconel 718 alloy. Based on a grain-to-grain orientation analysis using EBSD, many assemblies of neighboring grains in these regions have been found to display multiple-twins orientation relationships sharing common 〈 110 〉 directions and exhibiting 5-fold symmetry. This is an experimental proof that the grain refinement observed in inconel 718 is due to Icosahedral Short-Range Order (ISRO) mediated nucleation mechanism. This is the first report of ISRO-mediated nucleation of fcc nickel which widens the perspectives on the microstructures control in additive manufacturing of Ni-based superalloys.

International audience; This paper deals with the results of an investigation of low-temperature martensitic transformation in a metastable PuGa 1 at.% alloy. The kinetics of this process were studied experimentally via in situ XRD characterizations performed during isothermal holds. This revealed a double-C Time–Temperature–Transformation diagram similar to that reported in the literature for enriched alloys. The originality of our results lies in the demonstration of the existence of different martensite morphologies corresponding to the upper and lower parts of the TTT diagram. These microstructural variations seem to be the result of different accommodation mechanisms.The experimental nucleation rate study has suggested an autocatalytic character of the isothermal martensitic nucleation process as well as a large sensitivity to elastic and plastic strain. Indeed, the incubation period and subsequent increase in nucleation rate could be related to the appearance of new embryos rather than the growth of existing plates, while the partial character of the transformation may be attributed to interactions between plates that are being formed in the two-phase delta + alpha′ alloy. These hypotheses have been confirmed by the classical heterogeneous nucleation approach by Pati and Cohen, which takes into account the autocatalytic character of the transformation. This approach has also indirectly revealed the existence of restraining forces that reflect a high sensitivity of the energy barrier for nucleation to the alpha′ phase amount.

International audience; This paper discusses a coupled mechanics-phase field model that can predict microstructure evolution in metallic polycrystals and in particular evolution of lattice orientation due to either deformation or grain boundary migration. The modeling framework relies on the link between lattice curvature and geometrically necessary dislocations and connects a micropolar or Cosserat theory with an orientation phase field model. Some focus is placed on the underlying theory and in particular the theory of disloca-tions within a continuum single crystal plasticity setting. The model is finally applied to the triple junction problem for which there is an analytic solution if the grain boundary energies are known. The attention is drawn on the evolution of skew-symmetric stresses inside the grain boundary during migration.

International audience; Three continuum field theories are presented that account for the size-dependent behaviour of materials. The micromorphic medium is endowed with microdeformation degrees of freedom that describe the rotation and distortion of a triad of microstructural directions, like crystallographic lattice directions. It is a very general framework that can be specialized to strain gradient plasticity theory dedicated to the modelling of plastic events in metals and alloys. Both frameworks are developed here in the special case of crystal plasticity as a complete example of transition from micro-physical phenomena to continuum macro-modelling. Finally the phase field method is introduced in this landscape as a continuum modelling approach to the motion of phase boundaries and interfaces driven by thermodynamics and mechanics.

In metallic polycrystals, an important descriptor of the underlying microstructure is the orientation of the crystal lattice of each grain. During thermomechanical processing, the microstructure can be significantly altered through deformation, nucleation of new subgrains and grain boundary migration. Cosserat crystal plasticity provides orientation as a degree of freedom and is therefore a natural choice for the development of a coupled framework to deal with concurrent viscoplasticity and grain growth. In order to take into account grain boundary motion, the Cosserat theory is adapted with inspiration from orientation phase-field models. This allows for the microstructure at a material point to evolve on the one hand due to deformation-induced lattice reorientation and on the other hand due to a sweeping grain boundary. With a proper separation of plastic evolution in the bulk of the grain and in the grain boundary, the model can successfully capture grain boundary migration due to lattice curvature and due to statistically stored dislocations.

International audience; We present the results of in-situ characterization of the phase transitions in metastable β-Ti alloy Ti5553 by contactless laser-based resonant ultrasound spectroscopy method and electrical resistance measurement in a four probe configuration. Phase transformations were studied during continuous heating from the room temperature to 700 °C with various heating rates. We showed that both methods provide complementary results and can be successfully used for observation of phase transitions in metastable β-Ti alloys.

Phase field modelling offers an extremely general framework to predict microstructural evolutions in complex systems. However, its computational implementation requires a discretisation scheme with a grid spacing small enough to preserve the continuous character of the theory. We present here a new formulation, which is intrinsically discrete, in which the interfaces are resolved with essentially one grid point with no pinning on the grid and an accurate rotational invariance, improving drastically the numerical capabilities of the method. We show that interfacial kinetic properties are reproduced with a high accuracy. Finally, we apply the model to a situation where conserved and non-conserved fields are coupled.

This paper studies a two-phase material with a microstructure composed of a hard brittle reinforcement phase embedded in a soft ductile matrix. It addresses the full three-dimensional nature of the microstructure and macroscopic deformation. A large ensemble of periodic microstructures is used, whereby the individual grains of the two phases are modeled using equi-sized cubes. A particular solution strategy relying on the Fast Fourier Transform is adopted, which has a high computational efficiency both in terms of speed and memory footprint, thus enabling a statistically meaningful analysis. This solution method naturally accompanies the regular microstructural model, as the Fast Fourier Transform relies on a regular grid.Using the many considered microstructures as an ensemble, the average arrangement of phases around fracture initiation sites is objectively identified by the correlation between microstructure and fracture initiation – in three dimensions. The results show that fracture initiates where regions of the hard phase are interrupted by bands of the soft phase that are aligned with the direction of maximum shear. In such regions, the hard phase is arranged such that the area of the phase boundary perpendicular to the principal strain direction is maximum, leading to high hydrostatic tensile stresses, while not interrupting the shear bands that form in the soft phase. The local incompatibility that is present around the shear bands is responsible for a high plastic strain. By comparing the response to a two-dimensional microstructure it is observed that the response is qualitatively similar (both macroscopically and microscopically). One important difference is that the local strain partitioning between the two phases is over-predicted by the two-dimensional microstructure, leading to an overestimation of damage.Using the many considered microstructures as an ensemble, the average arrangement of phases around fracture initiation sites is objectively identified by the correlation between microstructure and fracture initiation – in three dimensions. The results show that fracture initiates where regions of the hard phase are interrupted by bands of the soft phase that are aligned with the direction of maximum shear. In such regions, the hard phase is arranged such that the area of the phase boundary perpendicular to the principal strain direction is maximum, leading to high hydrostatic tensile stresses, while not interrupting the shear bands that form in the soft phase. The local incompatibility that is present around the shear bands is responsible for a high plastic strain. By comparing the response to a two-dimensional microstructure it is observed that the response is qualitatively similar (both macroscopically and microscopically). One important difference is that the local strain partitioning between the two phases is over-predicted by the two-dimensional microstructure, leading to an overestimation of damage.

A phase field model accounting for anisotropic elasticity as well as isotropic strain gradient viscoplasticity is employed to study the diffusion-controlled growth of acicular precipitates. Following some recent work (Cottura et al., 2014), it is shown that an anisotropic eigenstrain can generate microstructures with the morphological and kinetic feature of Widmanstatten plates. Then, we demonstrate that when isotropic viscoplasticity is considered, the plate lengthening remains stationary in isothermal conditions, but features slower growth rates and larger tip radii.

A phase field model accounting for inhomogeneous and anisotropic elasticity has been used to investigate the consequences of inhomogeneous elasticity on the formation and coarsening of cuboidal microstructures in Ni-based superalloys. For that purpose, two- and three-dimensional calculations have been carried out considering high volume fractions of γ ′ precipitates, and accounting for their ordered crystal structure. The analysis of a substantial number of cases with different contrasts of elastic constants highlights the primary role played by the inhomogeneity Δ C ′ of the shear modulus C ′ on the shapes and spatial arrangements of the precipitates. This can be attributed to the sensitivity of the mechanisms involved in coarsening to Δ C ′ . Finally, scanning electron micrographs of the AM1 superalloy are statistically analyzed and discussed in light of the calculations.

The microstructure evolution due to thermomechanical treatment of metals can largely be described by viscoplastic deformation, nucleation and grain growth. These processes take place over different length and time scales which present significant challenges when formulating simulation models. In particular, no overall unified field framework exists to model concurrent viscoplastic deformation and recrystallization and grain growth in metal polycrystals. In this work a thermodynamically consistent diffuse interface framework incorporating crystal viscoplasticity and grain boundary migration is elaborated. The Kobayashi--Warren--Carter (KWC) phase field model is extended to incorporate the full mechanical coupling with material and lattice rotations and evolution of dislocation densities. The Cosserat crystal plasticity theory is shown to be the appropriate framework to formulate the coupling between phase field and mechanics with proper distinction between bulk and grain boundary behaviour., Accepted manuscript. Funded under the European Union (EU) Horizon 2020 Marie Sklodowska-Curie action; MIGRATE, grant number 707392

A step-by-step modeling approach, from the Phenomenological Theory of Martensite (PTM) to full field elasto-viscoplastic calculations, has been used to investigate the isothermal martensitic transformation in Pu-1at.%Ga. Based on the different calculations, an explanation is proposed for the peculiar morphological arrangements observed in this alloy at transformation temperatures near -20 °C.

A multi-phase field model describing the anisotropic growth of the {332} twinning system in β-metastable titanium alloys is hereby presented considering the geometrically linear and non-linear frameworks. Phase fields are associated with each twin variant of the {332} system. The lattice transformation describing the twinning systems is incorporated in the model via relevant eigenstrain tensors. We illustrate the capability of our model by investigating the formation of twins networks, within both geometrically linear and non linear frameworks. In particular, it is shown that the predicted microstructures exhibit strong differences, resulting from a difference in the selection of the variant habit planes.

Metastable β-Ti alloys exhibit various solid-solid phase transitions. Our study is focused on the characterization of the diffusion controlled β→ωiso phase transition. The particles of ω phase play an important part in thermomechanical treatment since they serve as heterogeneous nucleation sites for precipitation of finely dispersed particles of hexagonal α phase. The in-situ observation of the growth of particles of ω phase could be difficult by conventional techniques. However, it was shown recently that the ω phase significantly influences the elastic constants of the material, and the different forms of ω phase have different effects on the elastic anisotropy, as well as on the internal friction coefficients. Therefore, the β→ω phase transformation could be in-situ observed by the precise measurement of the tensor of elastic constants. In this contribution, we present the study of the kinetics of the β→ωiso phase transformation by resonant ultrasound spectroscopy. The polycrystalline samples of TIMETAL LCB alloy were in-situ examined by this technique during isothermal and non-isothermal ageing at temperatures up to 300 °C.

International audience; Recent advances in phase field modelling include the description of elastoviscoplastic material behaviour of the phases combined with diffusion and phase transformation. The corresponding models can be classified into two main groups of theories, referred to as interpolation and homogenisation models in the present work. It is shown that both approaches strongly differ concerning the question of inheritance of plastic deformation after the passing of a phase transformation front. Inheritance of plastic deformation is to be distinguished from the inheritance of the microstructure hardening and the corresponding dislocation structures. That is why the analysis is performed in the absence of hardening in the constitutive model. Finite element simulations of the growth of elastic misfitting precipitates embedded in a rate-independent elastoplastic matrix material reveal that the interpolation model allows for total inheritance of plastic deformation in contrast to the homogenisation model. The residual stress field and the growth kinetics are shown to be impacted by this essential property of the models. The results suggest that new models should be designed that allow for partial and controlled inheritance.

International audience; Coherency loss of misfitting precipitates by nucleation of dislocations at the interface is investigated using a nonlinear elastic model. This is made possible because the model inherently takes into account, in addition to long-range elastic interactions, phenomena involving dislocation cores such as dislocation nucleation and cross-slip. It is also shown that the model naturally delivers viscous motion laws above a threshold stress. The mechanism of prismatic punching proposed by Ashby and Johnson is confirmed by the present work for small precipitates. For larger precipitates, more complex situations are predicted which involve several dislocations, beyond the scope of the available analyses, but often observed experimentally.

A phase field model accounting for inhomogeneous and anisotropic elasticity has been developed to study the growth of acicular precipitates. The diffusion-controlled growth of an isolated precipitate is investigated in the presence of elastic stresses generated by anisotropic transformation eigenstrains in isothermal conditions. It is shown that elastic anisotropy qualitatively changes the growth mechanisms and that a stationary regime may be reached. In light of these conclusions, the elastic anisotropies of several metallic alloys featuring Widmanstatten structures are analyzed. The growth regimes predicted by our analysis are consistent with experimental observations; this strongly suggests the primary importance of elastic anisotropy in the formation of these microstructures.

Careful experiments are reported about the influence of applied stress (both compressive and tensile) on the formation of α’ martensite in PuGa 1 at.% at low temperatures. For that purpose, original sample-holders have been designed to apply controlled loadings at low temperature during in situ X-ray diffraction (XRD) analysis and dilatometry experiments. Based on the Patel and Cohen model together with an original transformation pathway involving particular combinations of partial dislocations to approach an Invariant Plane Strain condition, the mechanical driving force and the resulting change in Ms temperature have been computed for the different experimental conditions. First, very good agreement is achieved between experiments and calculations concerning the variation of Ms with applied stress. Although strong asymmetry is observed between compression and tension, it is shown that α’ can still appear under tension when accounting for the proper orientation of the plates. Second, in situ XRD during several isothermal treatments at various applied stresses provide additional elements to complete our understanding of the transformation process. In particular, the kinetics can be explained by two features specific to autocatalytic processes that are emphasised by the applied stresses: (i) stress induced nucleation of new plates that gives rise to cobweb spatial patterns starting from the specimen surface; (ii) strain induced nucleation associated with the necessary plastic relaxation of the matrix, demonstrated by the broadening of X-ray diffraction peaks. The occurrence of this plastic relaxation with the increase in internal stresses can also explain the saturation of the kinetics below complete transformation.

A thick interface model is proposed aiming at predicting the whole spectrum of conditions at an austenite/ferrite interface during ferrite growth in ternary Fe–C–X alloys, from full equilibrium to paraequilibrium, with intermediate cases as the most interesting feature. It is shown that the different kinetics can be obtained by tuning only the ratio between the diffusivities inside the thick interface and in bulk austenite. The predictions of the model for the kinetic transitions in different Fe–C–Ni alloys with a single value of this ratio are in good agreement with available measurements. In particular, sudden growth stops observed in alloys with high substitutional species contents are explained by the transition from paraequilibirium to local equilibrium non partitioning growth modes.

International audience; This paper presents a fully coupled glide-climb crystal plasticity model, whereby climb is controlled by the diffusion of vacancies. An extended strain gradient crystal plasticity model is therefore proposed, which incorporates the climbing of dislocations in the governing transport equations. A global-local approach is adopted to separate the scales and assess the influence of local diffusion on the global plasticity problem. The kinematics of the crystal plasticity model is enriched by incorporating the climb kinematics in the crystallographic split of the plastic strain rate tensor. The potential of the fully coupled theory is illustrated by means of two single slip examples that illustrate the interaction between glide and climb in either bypassing a precipitate or destroying a dislocation pile-up.

Precipitation decomposition in β metastable phase in TI 5553 alloy were characterized by electrical resistivity and high energy X-ray diffraction for two thermal paths, isothermal transformation at 325°C after direct quenching, or cooling to room temperature and further ageing at 325°C. Both treatments led to different transformation sequences with formation of ω and/or α''. Mean cell parameters of β and α'' were characterized all along the transformation. Results suggest a transformation with poor solute partitioning and building of large stresses in each phase.

International audience; Two topological parameters derived from pattern formation theory are introduced, and their relevance for the description of the elasticity-induced coarsening in Ni-based single-crystal superalloys is investigated. For that purpose, and given the crucial role of the spatial arrangement of the precipitates on microstructure evolution, two-dimensional phase field calculations of ideal microstructures undergoing slight perturbations have been carried out. The response of the system against Eckhaus and zigzag perturbations is discussed and quantitatively described by means of the developedtopological parameters.; Deux paramètres topologiques dérivés de la théorie des structures dissipatives (pattern formation) sont introduits, et leur potentiel à décrire l'influence de l'élasticité sur la coalescence dans les superalliages monocristallins à base de nickel est investigué. Pour cela, étant donné le rôle crucial de l'arrangement spatial des précipités sur les évolutions microstructurales, nous avons réalisé des simulations de type champ de phase avec des microstructures 2D périodiques, soumises à de légères perturbations topologiques. La réponse du système vis-à-vis de perturbations de type Eckhaus et zigzag est discutée, et analysée quantitativement au moyen des paramètres topologiques développés.

International audience; Because of the enormous range of time and space scales involved in dislocation dynamics, plastic modeling at macroscale requires a continuous formulation. In this paper, we present a rigorous formulation of the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. First, we focus on the underlying coarse-graining procedure. Our work reveals that both a spatiotemporal convolution and an ensemble average are required and that the emerging correlation-induced stresses are scale dependent. Each of these stresses can be expanded into the sum of two components. The first one depends on the local values of the dislocation densities and always opposes the sum of the applied stress and long-range mean field stress generated by the geometrically necessary dislocation (GND) density; this stress acts as a friction stress. The second component depends on the local gradients of the dislocation densities and is inherently associated to a translation of the elastic domain; therefore, it acts as a back stress. Finally, we show that that these friction and back stresses contain symmetry-breaking components that were missing in previous continuous formulations and that make, at mesoscale, the local stress experienced by dislocations depend on the sign of their Burgers vectors.

A phase field model is coupled to strain gradient crystal plasticity based on dislocation densities. The resulting model includes anisotropic plasticity and the size-dependence of plastic activity, required when plasticity is confined in region below few microns in size. These two features are important for handling microstructure evolutions during diffusive phase transformations that involve plastic deformation occurring in confined areas such as Ni-based superalloys undergoing rafting. The model also uses a storage-recovery law for the evolution of the dislocation density of each glide system and a hardening matrix to account for the short-range interactions between dislocations. First, it is shown that the unstable modes during the morphological destabilization of a growing misfitting circular precipitate are selected by the anisotropy of plasticity. Then, the rafting of γ ′ precipitates in a Ni-based superalloy is investigated during [100] creep loadings. Our model includes most of the important physical phenomena accounted for during the microstructure evolution, such as the presence of different crystallographic γ ′ variants, their misfit with the γ matrix, the elastic inhomogeneity and anisotropy, the hardening, anisotropy and viscosity of plasticity. In agreement with experiments, the model predicts that rafting proceeds perpendicularly to the tensile loading axis and it is shown that plasticity slows down significantly the evolution of the rafts.

An efficient method for computing two-phase equilibria in multicomponent alloys is proposed. It relies on a hybrid scheme using as a first step the Quick Hull algorithm with the general dimension Beneath-Beyond algorithm of Barber et al. (1996) [1] adapted for computing the convex hull of the Gibbs energy hypersurface of multicomponent two-phase alloys. First, the salient features of our method are illustrated with calculations of isothermal ferrite–austenite equilibria in Fe–C–Cr, described by the Compound Energy Formalism with two sublattices. Then, computational efficiency is investigated to suggest strategies to achieve the best compromise between efficiency and robustness. Finally, successive equilibrium calculations in a Fe–C–Cr-Mo steel over a large temperature range show the benefit of computing the convex hull before performing the conventional Newton–Raphson search.

High energy X-ray diffraction is a powerful tool, able to follow phase transformations during complex thermal or thermo-mechanical treatments. High energy allows one to study volumic specimens of a few mm3 and get successive data within a few seconds or less. The technique is described with different experimental setups (heating devices, detectors for diverse acquisition times) allowing diverse ranges for heating and cooling rates. Three examples are considered to illustrate the results obtained by using high energy X-ray diffraction. The first one corresponds to a simple diffusive phase transformation during an isothermal thermal path for the α–β transformation in a titanium alloy, highlighting the diffusive character considering the cell parameter evolutions of the parent phase. The second one illustrates the precipitation sequences observed during ageing of a β-metastable phase in a titanium alloy that was not obtained by TEM. The last example illustrates the phase evolutions during ageing of a martensitic steel showing the complexity of cell parameters evolution and some evolutions of the stress state.

A cyclic phase transformation concept has been proposed to investigate the growthkinetics of the austenite (γ) to ferrite (α ) and vice versa in Fe-Mn-C and Fe-C alloys. In the caseof cyclic partial transformations in Fe-Mn-C alloys, two new and special stages are observed:a stagnant stage in which the degree of transformation does not vary while the temperaturechanges and an inverse phase transformation stage, during which the phase transformationproceeds in a direction contradictory to the temperature change. The local equilibrium (LE)and paraequilibrium (PE) are both applied to analyzing the new observations. The stagnantstage was found to be caused by the Mn partitioning, while the inverse phase transformationstage was due to equilibrium conditions not being reached at the transition temperatures.A mixed-mode model is applied to simulating the cyclic phase transformation in Fe-C alloy,and it is found that the cyclic phase transformation concept is a very promising method forinvestigating the interface mobility.

A series of cyclic partial phase transformation experiments has been performed to investigate the growth kinetics of the austenite to ferrite phase transformation, and vice versa, in Fe–Mn–C alloys. Unlike the usual phase transformation experiments (100% parent phase → 100% new phase), in the case of cyclic partial transformations two special stages are observed: a stagnant stage in which the degree of transformation does not vary while the temperature changes, and an inverse phase transformation stage, during which the phase transformation proceeds in a direction contradictory to the temperature change. The experimental results have been analyzed using paraequilibrium and local equilibrium diffusional growth models. Only the local equilibrium model was shown to predict the new features of the cyclic phase transformation kinetics. The stagnant stage was found to be due to Mn partitioning, while the inverse phase transformation is caused by non-equilibrium conditions when switching from cooling to heating and vice versa.

In the present study we focus on the precipitation processes during heating and ageing of β-metastable phase in the near β Ti-5553 alloy. Transformation processes have been studied using continuous high energy X-Ray Diffraction (XRD) and electrical resistivity for two different states of the β-metastable phase. Microstructures have been observed by electron microscopy. Different transformation sequences are highlighted depending on both heating rate and chemical composition of the β-metastable phase. At low temperatures and low heating rates, the hexagonal ωisophase is first formed as generally mentioned in the literature. Increasing the temperature, XRD evidences the formation of an orthorhombic phase (α’’), which evolves toward the hexagonal pseudo compact α phase. For higher heating rates or for richer composition in β-stabilizing elements of the β-metastable phase, ω phase may not form and α’’ forms directly and again transforms into α phase. A direct transformation from β-metastable to a phase is observed for the highest heating rate. The formation of the metastable ωisoand α’’ phases clearly influences the final morphology of α.

A model has been developed which is able to predict the kinetics of β → α transformation in industrial multicomponent titanium alloys during complex heat treatments. It isbased on (i) analytical nucleation and growth laws based on simple geometric representationsof the di erent morphologies commonly observed in these alloys; (ii) the assumption of localequilibrium at interfaces, handled within the CalPhaD framework; (iii) averaged solute balancesin each morphology. The potentialities of the model will be demonstrated on the Ti17 industrialalloy upon isothermal holdings and cooling from the β phase field.

The partitioning of alloying elements between a and b phases was measured at different steps of the isothermal transformation at 710 and 610°C in Ti 17 alloy using EDX analyzer in the TEM. In addition, the transformation crystallography was determined. No differences in substitution elements were observed for a same transformation temperature in the a phase, however the composition varied with the transformation temperature. For the partial transformed specimens, gradient in composition were obtained. Results are compared to calculated compositions using ThermoCalc software and Saunders database.

Tliquidus was proposed as the interface temperature (Seiler and Froment, 2000), for various severe accident situations for thermalhydraulic steady state. This proposal was made on the basis of the analysis of solidification front stability in thermalhydraulic steady state for volumetrically heated corium pools and was extended to reactor transients with slow solidification rates that are controlled by the longterm decrease in residual power. The conclusions were corroborated by prototypic corium and variable solidification rates obtained by experimental approaches (Dauvois et al., 2000; Journeau et al., 2003) for corium containing small amounts of silica or none at all. When the concentration in silica increases (approximately above 10 wt%), it was concluded from the experiments that a plane-front situation could not be obtained. The present work offers a theoretical approach to the maximum time delay that is necessary for mass transfer and full phase-segregation in volumetrically heated liquid pools bounded by a crust. It is concluded that full segregation is obtained for in-vessel situations within time delays that are shorter or of the same order of magnitude as the characteristic time for the corium pool to form and evolve to a quasi-steady-state situation. The characteristic time delay for mass transfer associated with simulant material experiments is also determined. Phase segregation can also be obtained for corium–concrete interaction, provided that the silica content is less than approximately 10 wt%. However in the latter case, more complex phenomena occur at the interface due to the interaction with sparging gas (such as porous medium formation) which requires a different model approach.

We propose a description of incoherent and semi-coherent interfaces within the framework of diffuse interface modelling: accounting for the discontinuity of tangential displacements is an issue addressed by introducing a new field, called the incoherency field, located in the diffuse interface and able to release the shear stresses tangential to the interface. After having demonstrated the ability of the model to recover trends obtained by sharp interface models of sliding interfaces, the process of loss of coherency is investigated by assuming a local process which would result from the competition between the shear stress release at the interface and the interfacial energy increase.

International audience; A general constitutive framework is proposed to incorporate linear and nonlinear mechanical behaviour laws into a standard phase field model. In the diffuse interface region where both phases coexist, two mixture rules for strain and stress are introduced, which are based on the Voigt/Taylor and Reuss/Sachs well-known homogenization schemes and compared to the commonly used mixture rules in phase field models. Finite element calculations have been performed considering an elastoplastic precipitate growing in an elastic matrix in order to investigate the plastic accommodation processes.

A multiphase model for Ti–6Al–4V is proposed. This material is widely used in industrial applications andsoneeds accurate behaviour modeling. Tests have been performed inthe temperature range from 25 Cto1020 1C and at strain rates between 10 3 s and 1 s 1 . This allowed the identification of a multiphasemechanical model coupled with a metallurgical model. The behaviour of each phase is calibrated by solv-ing an inverse problem including a phase transformation model and a mechanical model to simulate testsunder thermomechanical loadings. A scale transition rule (b-rule) is proposed in order to represent theredistribution of local stresses linked to the heterogeneity of plastic strain. Finally this model is appliedto two laser assisted processes: direct laser fabrication and laser welding. 2009 Elsevier B.V. All rights reserved. 1. IntroductionTitanium alloys are extensively used in leading industries, dueto their low density and high specific strength (Rm/ q ) at elevatedtemperature (aeronautics, nuclear energy) and their compatibilitywith human tissues (biomedical applications). The most populartitanium alloy is Ti–6Al–4V (wt.%). This is a