Asbjørn Mo

Modeling of microsegregation in macrosegregation computations

A general framework for the calculation of micro-macrosegregation during solidification of metallic alloys is presented. In particular, the problems of back diffusion in the primary solid phase, of eutectic precipitation at the end of solidification, and of remelting are being addressed for an open system,i.e., for a small-volume element whose overall solute content is not necessarily constant. Assuming that the variations of enthalpy and of solute content are known from the solution of the macroscopic continuity equations, a model is derived which allows for the calculation of the local solidification path (i.e., cooling curve, volume fraction of solid, and concentrations in the liquid and solid phases). This general framework encompasses four microsegregation models for the diffusion in the solid phase: (1) an approximate solution based upon an internal variable approach; (2) a modification of this based upon a power-law approximation of the solute profile; (3) an approach which approximates the solute profile in the primary phase by a cubic function; and (4) a numerical solution of the diffusion equation based upon a coordinate transformation. These methods are described and compared for several situations, including solidification/remelting of a closed/open volume element whose enthalpy and solute content histories are known functions of time. It is shown that the solidification path calculated with method 2 is more accurate than using method 1, and that 2 is a very good approximation in macrosegregation calculations. Furthermore, it is shown that method 3 is almost identical to that obtained with a numerical solution of the diffusion equation (method 4). Although the presented results pertain to a simple binary alloy, the framework is general and can be extended to multicomponent systems.

Macrosegregation near a cast surface caused by exudation and solidification shrinkage

Abstract A one-dimensional mathematical model for the development of an exudated layer and macrosegregation near a cast surface is established. While forced convection leads to a highly segregated layer at the surface of the casting and a solute depleted zone near the surface, solidification shrinkage induced flow makes the negative segregation within the casting less pronounced. Parameter studies reveal that the heat transfer coefficient for chill contact, the metallostatic head, and the criterion for onset of exudation, all have a major influence on the resulting macrosegregation. The mathematical model is used to estimate macrosegregation development in the direct chill casting of an A1-4.5% Cu rolling slab. It is found that both forced convection and shrinkage-induced flow have a significant effect on the macrosegregation near the cast surface.

The influence of micro-scale solute diffusion and dendrite coarsening upon surface macrosegregation

A model problem and an associated mathematical model relevant for the macrosegregation formation close to and at the surface in aluminium direct chill casting are presented. The model is used to study how the solute diffusion within and coarsening of the dendrites influence the macrosegregation formation due to exudation and solidification shrinkage. It is found that the Scheil limit leads to the thickest and most concentrated exudated layer. For alloys having finite (but slow) solute diffusion, the model predicts a macrosegregation close to that obtained in the lever rule limit. If the secondary dendrite arm spacing is assumed to be equal to that found after complete solidification, the thickness of the exudated layer and the associated solute depletion within the casting are severely overestimated. For cases in which there is no exudation, the positive segregation due to solidification shrinkage is more pronounced the less the local solid solute diffusion is. Coarsening is shown to be of less importance in quantifying the macrosegregation when there is no exudation.

Mathematical modelling of surface segregation in aluminum DC casting caused by exudation

Abstract A mathematical model for the development of a segregated layer of exudated droplets during DC casting of aluminum ingots is established. The model accounts for the metallostatic pressure driven interdendritic melt flow through the mushy zone by a Darcy type equation, the surface segregation due to this melt flow, and the decrease of the total solute concentration in different positions of the mush as a result of the exudation. The solution domain for the governing differential equations is constituted by the mushy zone of the cast. The main physical phenomena included in the model have been studied in a simple one dimensional case study.

An internal variable description of solidification suitable for macrosegregation modeling

A mathematical description of solidification suitable for macrosegregation modeling is presented. The concept incorporates the effect of finite solid diffusion locally in the dendrites, and remelting can be modeled without any need to trace the solute concentrations in the dendritic structure during solidification. This is accomplished by interpreting the mean solute concentrations in the solid as internal variables on which the solid fraction depends and by accounting for the rate of change of these variables due to solidification, remelting, and solid diffusion by additional evolution equations. The material coefficients needed in the model are estimated for a ternary AlFeSi alloy of commercial purity, and the internal variable equations are incorporated in a simple model problem for interdendritic melt flow leading to macrosegregation. The results are compared to similar studies based on the lever rule and the Scheil approach, and it is shown that the choice of solidification description has a pronounced effect on the predicted amount of macrosegregation.

GLEEBLE MACHINE DETERMINATION OF CREEP LAW PARAMETERS FOR THERMALLY INDUCED DEFORMATIONS IN ALUMINIUM DC CASTING

By means of a Gleeble machine, the flow stress at steady-state creep in an AA3103 aluminium alloy has been measured for temperatures and strain rates relevant for thermally induced deformations in DC casting. The strain rate has been determined by measuring the global radial strain rate at the specimen center by an extensometer, and the stress has been set equal to the force in the axial direction divided by the cross-section area. The parameters of Garofalos equation have been fitted to the resulting steady-state stress and strain rate. Such a method is based upon the assumption of homogeneous stress and strain rate fields. In the Gleeble machine, the specimens are heated by the Joule effect leading to axial temperature gradients, and the specimen geometry is noncylindrical. The resulting inhomogeneities in the stress and strain rate fields are studied by finite element modeling, and it is shown that although they can be significant, the global radial strain rate and the axial force divided by the cross...By means of a Gleeble machine, the flow stress at steady-state creep in an AA3103 aluminium alloy has been measured for temperatures and strain rates relevant for thermally induced deformations in DC casting. The strain rate has been determined by measuring the global radial strain rate at the specimen center by an extensometer, and the stress has been set equal to the force in the axial direction divided by the cross-section area. The parameters of Garofalos equation have been fitted to the resulting steady-state stress and strain rate. Such a method is based upon the assumption of homogeneous stress and strain rate fields. In the Gleeble machine, the specimens are heated by the Joule effect leading to axial temperature gradients, and the specimen geometry is noncylindrical. The resulting inhomogeneities in the stress and strain rate fields are studied by finite element modeling, and it is shown that although they can be significant, the global radial strain rate and the axial force divided by the cross-section area at the specimen center can be relatively close to what the respective strain rate and stress values would have been if the conditions actually were homogeneous.

THE EFFECT OF WORK HARDENING ON THERMALLY INDUCED DEFORMATIONS IN ALUMINIUM DC CASTING

This article documents a series of physical direct chill casting simulations performed on specimens of an AA3103 alloy by means of a Gleeble machine. During the experiments the specimens are subjected to thermal and straining histories similar to those experienced by material points in an ingot during the casting process due to thermal stresses. The measured stress is compared to the stress given by a steady-state creep law for the measured temperature and strain rate versus time. The creep law gives a good fit for temperatures above 400oC but increasingly overestimates the stress level as the temperature decreases below this level because of the increasing importance of work hardening. Since thermally induced straining occurs in the entire temperature interval in the casting process, it is concluded that more sophisticated constitutive modeling than the creep law is needed.

Macrosegregation Caused by Deformation of the Mushy Zone

An experimental set-up for investigating macrosegregation formation due to deformation of an isotropic metallic mushy zone is presented. In the experiment, a semisolid Al-5.9wt%Cu sample is isothermally and non-uniformly compressed. Concentration and eutectic fraction are measured along selected lines, after quenching the sample. Results show that interdendritic liquid is pressed out of the central part of the sample to its outer part, increasing the concentration in this region. The experimental test is then addressed by a two-phase continuum model recently presented elsewhere. The modelling results show the same tendencies observed experimentally, although local variations in composition are not well correlated. Suggestions for future work are made.

Constitutive Equation for Thermal Strain in the Mushy Zone during Solidification of Aluminium Alloys

A constitutive equation for thermal strain in the mushy zone has recently been established [1]. The parameters in this constitutive relation are in the present study determined for the commercial alloys A356, AA2024, AA6061 and AA7075 in addition to an Al-4 wt% Cu alloy by combining experimentally measured contraction of a cast sample with thermomechanical simulations. The constitutive equation for thermal strain in the mushy zone reflects that there is no thermal strain in the solid part of the mushy zone at low solid fractions and that the thermal strain in the mushy zone approaches thermal contraction in fully solid as the solid fraction increases towards one. Experiments were performed at cooling rates in the range from 2 to 5.5 °C/s. The solid fractions when the tested alloys start to contract, gsth , are in the range from 0.63 to 0.94. Grain refinement increases gsth for all the tested alloys.

Computation of Macrosegregation due to Solidification Shrinkage

A system of partial differential equations and a set of constitutive relation modelling macro segregation development during solidification of a binary alloy is presented. The development of a finite element simulator for the numerical solution of these equations using Diffpack is discussed. In particular we consider the solution of systems of PDEs using object oriented programming techniques.