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Publication# Two-Phase Modeling of Hot Tearing in Aluminum Alloys: Applications of a Semicoupled Method

Abstract

Hot tearing formation in both a classical tensile test and during direct chill (DC) casting of aluminum alloys has been modeled using a semicoupled, two-phase approach. Following a thermal calculation, the deformation of the mushy solid is computed using a compressive rheological model that neglects the pressure of the intergranular liquid. The nonzero expansion/compression of the solid and the solidification shrinkage are then introduced as source terms for the calculation of the pressure drop and pore formation in the liquid phase. A comparison between the simulation results and experimental data permits a detailed understanding of the specific conditions under which hot tears form under given conditions. It is shown that the failure modes can be quite different for these two experiments and that, as a consequence, the appropriate hot tearing criterion may differ. It is foreseen that a fully predictive theoretical tool could be obtained by coupling such a model with a granular approach. These two techniques do, indeed, permit coverage of the range of the length scales and the physical phenomena involved in hot tearing.

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Liquid

A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a nearly constant volume independent of pressure. It is one of the four fundamental states of matte

Aluminium alloy

An aluminium alloy (or aluminum alloy; see spelling differences) is an alloy in which aluminium (Al) is the predominant metal. The typical alloying elements are copper, magnesium, manganese, silicon

Pressure

Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled gage pressure) is the press

Hot tearing is one of the most severe defects observed in castings, e.g. in billets or sheet ingots of aluminum alloys produced by DC casting. It is due to both tensile strains and a lack of interdendritic feeding in the mushy zone. In order to predict this phenomenon at the scale of an entire casting, the two-phase averaged conservation equations for mass and momentum must be solved in the mushy (i.e. mixed solid and liquid) region of the material. In recent contributions, M'Hamdi et al [1] proposed a strongly coupled resolution scheme for this set of equations. The solution of the problem was obtained using a rheological model established by Ludwig et al [2] and that captures the partially cohesive nature of the mushy alloy. In the present contribution, the problem is addressed using a slightly different approach. The same rheological model (i.e. saturated porous media treatment) is used, but the influence of the liquid pressure is neglected at this stage. This assumption allows for a weakly coupled resolution scheme in which the mechanical problem is first solved alone using ABAQUS™ and user subroutines. Then the pressure in the liquid phase is calculated separately accounting for the viscoplastic deformation of the porous solid skeleton and solidification shrinkage. This is done with a code previously developed for porosity calculations, and that uses a refined mesh in the mushy zone [3]. This semi-coupled method was implemented and its numerical convergence studied from the point of view of both time step and mesh size. Guidelines for selecting these numerical parameters as well as the conditions under which the semi-coupled method may be applied are provided. The model was then applied to three cases, i.e. two tensile tests conducted on mushy alloys [4, 5] and the casting of an entire billet [6]. Experimental data was indeed available concerning these problems prior to the present work. This information was used for the validation of the thermal and mechanical models that were setup to describe these different cases. The results of the semi-coupled approach were also used to describe in more details these different castings. First of all, the numerical study of the mushy zone tearing test [5] proved helpful for distinguishing different fracture modes. The role of the high strain rate applied to the mushy alloy in this case was also outlined. Another tensile test, referred to as the rig test [4], was successfully modeled in the present framework. The numerical results could be used to quantify the redistribution of strain in the mushy sample. As a consequence, intrinsic properties of the material, such as its ductility, could be extracted from the results. This study also gave further insight about the conditions under which tearing occurs in the samples. Finally, the semi-coupled method was used to study the DC casting process. In this case, a real process performed under realistic conditions for the production of an industrial scale billet was modeled. As it is more complex and difficult to characterize experimentally, the conditions for hot tearing formation are less accessible. However, the isotherm velocity, the strain, the strain rate and the liquid pressure could be described reasonably accurately. It was thus possible to correlate experimental observations of the hot tear with various calculated indicators of hot tearing susceptibility. Even with this information, it remains difficult to formulate new hot tearing criteria because all the indicators follow a similar trend during the casting and their respective contributions can thus not be distinguished. The present work showed that the level of accuracy and detail that can be reached using two-phase models with appropriate materials properties and boundary conditions is satisfactory. It is indeed possible to model the relevant phenomena (heat flow, solid deformation and liquid feeding) at the scale of an entire casting. The variation of the different simulated fields can be described down to a scale of the order of a few millimeters. In that sense, this approach is one important aspect required to build a multiscale model for the problem of hot tearing. It is expected that coupling such a method with granular models (which cover length scales from a few microns to a few centimeters [7]) will allow for a more complete description of the phenomena at hand. In the future, the development of such a multiscale numerical tool may prove to be the most efficient way towards quantitative predictions of hot tearing formation in real solidification processes.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Hot tearing is a major defect of solidification processes and welding. It occurs in metallic alloys while they are in the semi-solid state and is caused by a lack of liquid feeding in the mushy zone to compensate solid network openings due to tensile and shear strains. These strains are localized in the thin liquid films that remain at grain boundaries. Therefore, it is important to know when the solid grains coalesce and percolate to form a fully coherent solid, i.e., when the material can transmit stresses and strains but also withstand thermal contraction without rupture. Coalescence is defined as the disappearance of a liquid film in between two distinct grains, forming a solid grain boundary, while percolation is defined as the gradual transition of isolated grains or clusters surrounded by a continuous liquid film to a continuous solid network across a domain, even if some liquid regions remain. If the simulation of hot tearing has to account for localization of strains and liquid feeding, it is mandatory to use a granular approach. Two previous theses done recently at EPFL have developed such granular models of hot tearing, first in 2D (PhD of S. Vernède) and then in 3D (PhD of M. Sistaninia). However, these models have used a Voronoi tessellation for the description of globular microstructures in which equiaxed grains were approximated by polygons and polyhedra, respectively. From the experimental point of view, an innovative method, based on the Bridgman furnace principle, has been developed. The standard Bridgman furnace has been substantially modified in order to obtain the desired globular-equiaxed grain structure and reduce macrosegregation in the quenched sample. This modified furnace has been characterized then in terms of thermal measurements and observed final grain size. The samples obtained with the modified Bridgman furnace were then observed post-mortem by ex situ X-ray tomography. In addition, X-ray tomography experiments under similar conditions have been performed in situ with the laser-heated furnace installed on the TOMCAT synchrotron beam line at PSI. A postprocessing analysis of the principal curvature distribution (ISD plots) was performed on both ex situ and in situ observations for various compositions of inoculated Al-Cu alloys. The ISD plots of the X-ray tomography observations indicated the formation of liquid cylinders along triple line and, at a more advanced solidification stage, the formation of liquid pocket at quadruple vertices. From the modeling point of view, a new mesoscopic model has been developed both for 2D and 3D simulations. This model was inspired by the granular model developed by Phillon et al. that considers polyhedral grains based on a Voronoi tessellation of space, but allow to obtain smoother grain shapes and more progressive coalescence. After its validation with multiphase-field simulations, the mesoscopic model has been used to predict the various percolation transitions of the solid phase. In addition, by estimating the solid moment at which the liquid starts to form isolated pockets, it was possible to predict the temperature and solid fractions for which the mushy zone is most vulnerable to hot tearing. The simulations have been performed for various conditions and copper compositions, indicating the nominal compositions that are more sensitive to hot tearing. Finally, the 3D mesoscopic model predictions were compared with the X-ray tomography observations.