Ch.-A. Gandin

A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes

Abstract A new algorithm based upon a 2-dimensional Cellular Automaton (CA) technique is proposed for the simulation of dendritic grain formation during solidification. The CA model takes into account the heterogeneous nucleation, the growth kinetics and the preferential growth directions of the dendrites. This new CA algorithm, which applies to non-uniform temperature situations, is fully coupled to an enthalpybased Finite Element (FE) heat flow calculation. At each time-step, the temperature at the cell locations is interpolated from those at the FE nodal points in order to calculate the nucleation-growth of grains. The latent heat released by the cells and calculated using a Scheil-type approximation is fed back into the FE nodal points. The coupled CA-FE model is applied to two solidification experiments, the Bridgman growth of an organic alloy and the one-dimensional solidification of an Al-7wt% Si alloy. In the first case, the predicted boundaries between grains are in good agreement with experiment, providing the CA cell size is of the order of the dendrite spacing. For the second experiment, the quality of the coupled CA-FE model is assessed based upon grain structures and cooling curves. The columnar-to-equiaxed transition and the occurrence of a recalescence are shown to be in good agreement with the model.

A 3D cellular automaton algorithm for the prediction of dendritic grain growth

Two- and three-dimensional (3D) Cellular Automaton (CA) algorithms are proposed for modelling the growth of dendritic grains from the liquid phase. These CA growth algorithms are validated for simple thermal situations by comparing the predicted grain shapes with those deduced from analytical models. The insight obtained by the 3D approach is demonstrated by studying the extension of a single dendritic grain in a squared platform (i.e. at a section change of a casting mould) under various conditions. In particular, the effects of crystallographic orientation, thermal gradient, velocity of the isotherms and growth kinetics are shown. This 3D CA growth algorithm, coupled with finite element heat flow calculations, will become a major tool for the prediction of dendritic grain structures in solidification processes.

Three-dimensional probabilistic simulation of solidification grain structures: Application to superalloy precision castings

A two-dimensional (2-D) probabilistic model, previously developed for the prediction of microstructure formation in solidification processes, is applied to thin section superalloy precision castings. Based upon an assumption of uniform temperature across the section of the plate, the model takes into account the heterogeneous nucleation which might occur at the mold wall and in the bulk of the liquid. The location and crystallographic orientation of newly nucleated grains are chosen randomly among a large number of sites and equiprobable orientation classes, respectively. The growth of the dendritic grains is modeled by using a cellular automaton technique and by considering the growth kinetics of the dendrite tips. The computed 2-D grain structures are compared with micrographie cross sections of specimens of various thicknesses. It is shown that the 2-D approach is able to predict the transition from columnar to equiaxed grains. However, in a transverse section, the grain morphology within the columnar zone differs from that of the experimental micrographs. For this reason, a three-dimensional (3-D) extension of this model is proposed, in which the modeling of the grain growth is simplified. It assumes that each dendritic grain is an octaedron whose half-diagonals, corresponding to the <100> crystallographic orientations of the grain, are simply given by the integral, from the time of nucleation to that of observation, of the velocity of the dendrite tips. All the liquid cells falling within a given octaedron solidify with the same crystallographic orientation of the parent nucleus. It is shown that the grain structures computed with this 3-D model are much closer to the experimental micrographie cross sections.

EBSD CHARACTERISATION AND MODELLING OF COLUMNAR DENDRITIC GRAINS GROWING IN THE PRESENCE OF FLUID FLOW

Abstract Columnar dendritic grains of steel grown in the presence of fluid flow (e.g. solidified on turning rolls) have been characterised by Electron Backscattered Diffraction (EBSD) technique. It is shown that grains have a random crystallographic orientation at the surfaces of the sheet in contact with the mould. In the middle of the sheet, the grains which have survived the growth selection mechanisms exhibit a 〈100〉 texture in which the average dendrite trunk direction is not exactly aligned with the thermal gradient (i.e. the normal to the surfaces of the sheet). It is tilted by about 15° toward the upstream direction. This deviation is examined by simulations of grain structure formation based on a three-dimensional Cellular Automaton (CA)–Finite Element (FE) (3D CAFE) model, which has been modified in order to account for fluid flow effects. The modified CA algorithm includes a growth kinetics of the dendrites which is a function of both the undercooling and fluid flow direction. It is validated by comparing the predicted shape of an individual grain growing under given thermal and fluid flow conditions with an analytical solution. The 3D CAFE predictions of the columnar grains grown in the presence of fluid flow are in good agreement with the experimental EBSD results.

Grain Texture Evolution during the Columnar Growth of Dendritic Alloys

The grain selection that operates in the columnar zone of a directionally solidified (DS) INCONEL X750 superalloy has been investigated using standard metallography and an automatic indexing technique of electron backscattered diffraction patterns (EBSPs). From the crystallographic orientations measured at 90,000 points in a longitudinal section, the grain structure was reconstructed. The grain density as measured by the inverse of the mean linear intercept was found to be a decreasing function of the distance from the chill. The evolution of the 〈100〉 pole figures along the columnar zone of the casting and the distribution of the angle θ characterizing the 〈100〉 direction of the grains that is closest to the temperature gradient were then deduced from the EBSPs measurement. It was found that, near the surface of the chill, the θ distribution was close to the theoretical curve calculated for randomly oriented grains. As the distance from the chill increased, the measured θ distribution became narrower and was displaced toward smaller θ values. At 2 mm from the chill, the most probable orientation of the grains was found to be about 0.21 rad (12 deg). The information obtained with the EBSPs was then compared with the results of a three-dimensional stochastic model (3D SM) describing the formation of grain structure during solidification. This model accounts for the random location and orientation of the nuclei, for the growth kinetics and preferential 〈100〉 growth directions of the dendrites. Although this model assumes a uniform temperature within the specimen, the simulation results were found to be in good agreement with the EBSPs measurement.

Prediction of a process window for the investment casting of dendritic single crystals

Large dendritic single crystal blades operating in jet engines or land-based turbines are commonly produced in industrial plants. However, the production cost of some of these parts remains high due to the several types of defect which may appear during the different stages of the investment casting process. The prevention of stray crystal formation during solidification is the main problem which has to be faced by engineers. Analytical developments are first presented in order to define conditions under which single crystal growth is favored. This process window is described in a space where the thermal gradient and the withdrawal speed of a Bridgman furnace apparatus are the controlling parameters. The limits of this process window are then refined in order to determine the influence of other parameters such as the crystallographic orientation of the single dendritic grain, the size of the platform in which the single grain has to extend, the growth kinetics of the dendrite tips, or the existence of a lateral thermal gradient.

Orientation Dependence of Primary Dendrite Spacing

The orientation dependence of the primary dendrite spacing is examined through the solidification of a succinonitrile-3.61 wt pct acetone alloy with a Bridgman-type device. Primary dendrite spacing has been studied as a function of the primary dendrite trunk orientation with respect to the thermal gradient direction under different growth rate conditions. The observations show that the orientation dependence of the primary dendrite spacing can be related to the formation of new primary dendrites at divergent grain boundariesvia the branching mechanism. A branching-based model is developed to study the interaction between secondary and tertiary dendrite arms. Based on this model, a simple analytical relationship is proposed to account for the orientation dependence of the primary dendrite spacing.

Analytical and Numerical Predictions of Dendritic Grain Envelopes

An analytical model is developed for the prediction of the shape of dendritic grain envelopes during solidification of a metallic alloy in a Bridgman configuration (i.e. constant thermal gradient and cooling rate). The assumptions built into the model allow a direct comparison of the results with those obtained from a previously developed cellular automaton-finite element (CAFE) model. After this comparison, the CAFE model is applied to the study of the extension of a single grain into an open region of liquid after passing a re-entrant corner. The simulation results are compared with experimental observations made on a directionally solidified succinonitrile-acetone alloy. Good agreement is found for the shape of the grain envelopes when varying the orientation of the primary dendrites with respect to the thermal gradient direction, the velocity of the isotherms or the thermal gradient.

Process Modelling and Microstructure [and Discussion]

Among the many routes which are used for the processing of high-temperature materials, solidification plays a key role. Several modelling tools are now available for the simulation of the interconnected macroscopic phenomena associated with any casting process (heat exchange, mould filling, convection, stress development, etc.). Based upon finite-difference (FD) or finite-element (FE) techniques, these models solve the continuity equations of mass, energy, momentum, solute species, averaged over the liquid and solid phases. As such, macroscopic models do not account for the detailed phenomena occurring at the scale of the microstructure. For that reason, a stochastic cellular automaton (CA) model has been developed recently for the prediction of the grain structure formation in solidification processes, in particular during the investment casting of superalloys. Such a microscopic model considers the heterogeneous nucleation of grains at the surface of the mould and in the bulk of the liquid, the growth kinetics and preferential growth directions of the dendrites and the microsegregation. The microscopic CA model has been coupled to FE heat flow computations in order to predict the grain structure at the scale of a casting. It is shown that microstructural features and crystallographic textures can be simulated as a function of the casting conditions and alloy composition.