A. Jacot

A Pseudo-Front Tracking Technique for the Modelling of Solidification Microstructures in Multi-Component Alloys

Abstract A two-dimensional model for the simulation of microstructure formation during solidification in multi-component systems has been developed. The model is based on a new pseudo-front tracking technique for the calculation of the evolution of interfaces that are governed by solute diffusion and the Gibbs–Thomson effect. The diffusion equations are solved in the primary solid phase and in the liquid using an explicit finite volume method formulated for a regular hexagonal grid. Volume elements located in the liquid phase undergo a transition to interfacial (or mushy) cells before being incorporated in the solid phase. This layer of interfacial elements, which always separates the solid from the liquid sub-domains, permits to handle the displacement of the interface in agreement with the flux condition at the interface. The interface curvature is obtained from the field of the signed distance to the interface, as reconstructed with a PLIC (piecewise linear interface calculation) technique. The concentrations at the solid–liquid interface are calculated using thermodynamic data provided by the phase diagram software Thermo-Calc [Sundman et al. CALPHAD 1987;9:153] . Different coupling strategies between the microstructure model and Thermo-Calc have been developed, in particular a computationally-efficient direct coupling using the TQ-interface of Thermo-Calc . After testing the accuracy of the model with respect to curvature calculation, comparisons are made with predictions obtained with the marginal stability theory, a one-dimensional front-tracking method and two-dimensional phase-field simulations of dendritic growth in binary alloys. The model is then used to describe the formation of several grains in an Al–1%Mg–1%Si alloy, as a function of the heat extraction rate and inoculation conditions. It is shown that the model is capable of reproducing the transition between globular and dendritic morphologies.

A two-dimensional diffusion model for the prediction of phase transformations: Application to austenitization and homogenization of hypoeutectoid Fe-C steels

A two-dimensional (2D) model has been developed for the prediction of diffusive phase transformations (e.g. α to γ). For that purpose, the diffusion equations are solved within each phase (α and γ) using an explicit finite volume technique formulated for a regular hexagonal grid. The discrete α/γ interface is represented by special volume elements α/γ. An α volume element undergoes a transition to an α/γ interface state before becoming γ. This procedure allows us to handle the displacement of the interface while respecting the flux condition at the interface. The model has been applied to the austenitization of a hypoeutectoid plain carbon steel during heating. Simulated microstructures showing the dissolution of ferrite particles in the austenite matrix are presented at different stages of the phase transformation. Specifically, the influence of the microstructure scale and of the heating rate on the transformation kinetics has been investigated. Reverse TTT-diagrams calculated with this 2D model are compared with experimental results from the literature and with the predictions of a simpler one-dimensional (1D) front-tracking calculation. Finally, it is shown that interface instabilities leading to the formation of dendrites can also be reproduced by such a model.

A combined model for the description of austenitization, homogenization and grain growth in hypoeutectoid Fe–C steels during heating

Reference LSMX-ARTICLE-1999-002View record in Web of Science Record created on 2005-11-22, modified on 2017-05-10

Modelling of reaustenitization from the pearlite structure in steel

Abstract A two-dimensional model has been developed for the description of the formation of austenite from lamellar pearlite in steel. The diffusion equation is solved in a small domain representative of a regular structure of lamellar pearlite. The solution is obtained using a finite element method with a deforming mesh and a remeshing procedure. The main assumption of the model is the condition of local equilibrium at the interfaces, including the curvature contribution and mechanical equilibrium of surface tensions at the triple junction where the ferrite, austenite and cementite phases meet. The velocity of the interface is deduced from a solute balance which involves the concentration given by the phase diagram modified by the Gibbs–Thomson effect. The model is used to predict the dissolution rate, the shape of the interface as well as the concentration field in austenite as a function of temperature. Both the transient and steady-state regimes are described. The model is first applied to a model alloy whose physical properties allow the problem to be solved for a wide range of lamellae spacings and temperature. Subsequently, the Fe–C system is examined and the numerical results are compared with experimental data from the literature. Finally, it is shown that the steady-state growth breaks down and the transformation occurs with a different regime at high superheating.

Last Stage Solidification of Alloys: A Theoretical Study of Dendrite Arm and Grain Coalescence

Solidification of metallic alloys has been extensively studied (dendrite tip kinetics, microsegregation, coarsening of dendrite arms, etc.) but surprisingly, in the absence of eutectic reactions (i.e., low concentration alloys), little is known about the last stage of solidification when the primary phase regions impinge. Yet, the details of the final stages of solidification of multigrain and/or dendritic materials have significant impact on casting defects such as hot tearing. Indeed, models of hot tearing [2,3] require input of a value of fraction solid or temperature, sometimes called the coherency point, that defines the situation where the two-phase liquid plus solid material is sufficiently strong to withstand thermal contraction stresses without rupture.

Numerical Simulation of Solidification, Homogenization, and Precipitation in an Industrial Ni-Based Superalloy

AbstractA comprehensive simulation approach integrating solidification, homogenization, and precipitation during aging has been used to predict the formation of γ/γ′ microstructures in the AM1 nickel-based superalloy. The particle size distribution of intradendritic γ′ precipitates after aging was calculated with a multicomponent diffusion model coupled with CALPHAD thermodynamics for the equilibrium at the interface. The influence of residual microsegregation after homogenization and quenching was analyzed through different initial conditions obtained from calculations of the concentration profiles in the primary γ dendritic microstructure during solidification and the homogenization heat treatment. While the global sequence of precipitation remains qualitatively the same, substantial differences in the final volume fraction of γ′ precipitates were predicted between the core and the periphery of a former dendrite arm, for typical homogenization and aging conditions. To demonstrate the relevance of the developed simulation approach, the model was also used to investigate modified precipitation heat treatments. The simulations showed that relatively short heat treatments based on slow continuous cooling could potentially replace the extended isothermal heat treatments which are commonly used. Slow continuous cooling conditions can lead to similar γ′ precipitates radii and volume fractions, the main differences with isothermal heat treatments lying in a narrower particle size distribution.

A Numerical Model for the Description of Massive and Lamellar Microstructure Formation in Gamma-TiAl

A phenomenological modeling approach has been developed to describe the massive transformation and the formation of lamellar microstructures during cooling in gamma titanium aluminides. 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 both 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 calculated with a mathematical expression for interface-controlled reactions. A modified Zener model is used to calculate the thickening rate of the γ lamellar precipitates. The model incorporates the effect of particle impingement and rapid coverage of the nucleation sites due to growth. The driving pressures of the phase transformations are obtained form Thermo-Calc based on actual temperature and matrix composition. The model permitted investigating the influence of alloy chemistry, cooling rate and average α grain size upon the amount of massive γ and the average thickness and spacing of the lamellae. Calculated CCT diagrams were compared with experimental data collected from the literature and showed good agreement.

Multiphase-Field Modeling of Micropore Formation in Metallic Alloys

A multiphase-field model has been developed in order to study the evolution of micropores constrained to grow in a solid network, with a so-called pinching effect. The model accounts for the pressure difference due to capillarity between liquid and gas, the equilibrium condition at triple (solid-liquid-pore) lines, the partitioning and diffusion of dissolved gases such as hydrogen in aluminum alloys.The model is used to study the growth of a pore in a solid network, reconstructed from X-ray tomography observations. It was observed that such calculations can help to better understand the experimental observations, which have a limited temporal and spatial resolution. The predicted morphology is also compared with the experimental observation.Copyright

3D Phase-Field Simulation of Micropore Formation during Solidification: Morphological Analysis

A 3D multiphase-field (PhF) model has been developed in order to study the formation of a micropore constrained to grow in a solid network (i.e., pinching effect). The model accounts for the pressure difference due to capillarity between liquid and gas, the equilibrium condition at triple (solid-liquid-pore) lines, the partitioning and diffusion of dissolved gases such as hydrogen. From the predicted 3D morphology of the pore, entities such as the Interfacial Shape Distribution (ISD) are plotted and analyzed. It is shown that the mean curvature of the pore-liquid surface, and thus also the pressure inside the pore, is uniform. Despite the complex morphology of pores reconstructed using high-resolution X-ray tomography, the present PhF results suggest that a simple pinching model based on a spherical tip growing in between remaining liquid channels is a fairly good approximation.