Containerless rapid solidification of undercooled Cu-Co peritectic alloys

Droplets of Co-16%Cu and Co-71.6%Cu peritectic alloys were solidified during containerless processing in a 3_m drop tube. The microstructures of Co-16%Cu alloy droplets were characterized by dendritic or equiaxed α-Co phase with a small amount of Cu-rich solid solution distributed on α-Co phase boundaries. Two thresholds of droplet diameter were observed for Co-16%Cu alloy at which "equiaxed-dendritic-equiaxed" morphological transitions occur to primary α-Co phase. This conspicuous refinement of primary α-Co grains results from the fragmentation of α-Co dendrites caused by recalescence effect. For Co-71.6%Cu alloy, the primary α-Co phase forms as coarse columnar dendrites in large droplets and equiaxed dendrites in small droplets. Theoretical calculations indicate that Marangoni migration contributes more to the growth of disperse Co-rich spheres by stimulating collision and coalescence than Stokes motion caused by the residual gravity in the falling Co-71.6%Cu alloy droplets.     

Modeling dendritic solidification of Al–3%Cu using cellular automaton and phase-field methods

We compared a cellular automaton (CA)–finite element (FE) model and a phase-field (PF)–FE model to simulate equiaxed dendritic growth during the solidification of cubic crystals. The equations of mass and heat transports were solved in the CA–FE model to calculate the temperature field, solute concentration, and the dendritic growth morphology. In the PF–FE model, a PF variable was used to identify solid and liquid phases and another PF variable was considered to determine the evolution of solute concentration. Application to Al–3.0 wt.% Cu alloy illustrates the capability of both CA–FE and PF–FE models in modeling multiple arbitrarily-oriented dendrites in growth of cubic crystals. Simulation results from both models showed quantitatively good agreement with the analytical model developed by Lipton–Glicksman–Kurz (LGK) in the tip growth velocity and the tip equilibrium liquid concentration at a given melt undercooling. The dendrite morphology and computational time obtained from the CA–FE model are compared to those of the PF–FE model and the distinct advantages of both methods are discussed.     

A ternary phase bi-scale FE-model for diffusion-driven dendritic alloy solidification processes

It is of high interest to describe alloy solidification processes with numerical simulations. In order to predict the material behavior as precisely as possible, a ternary phase, bi-scale numerical model will be presented. This paper is based on a coupled thermo-mechanical, two-phase, two-scale finite element model developed by Moj et al. [2], where the theory of porous media (TPM) [1] has been used. Finite plasticity extended by secondary power-law creep is utilized to describe the solid phase and linear visco-elasticity with Darcy's law of permeability for the liquid phase, respectively. Here, the microscopic, temperature-driven phase transition approach is replaced by the diffusion-driven 0D model according to Wang and Beckermann [3]. The decisive material properties during solidification are captured by phenomenological formulations for dendritic growth and solute diffusion processes. A columnar as well as an equiaxial solidification example will be shown to demonstrate the principal performance of the presented model. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

The morphological stability of dendritic growth from the binary alloy melt with an external flow

The morphological stability of dendritic growth from the binary alloy melt with an external flow is studied by means of the matched asymptotic expansion method and multiple variable expansion method. The uniformly valid asymptotic solution is obtained for the case of the large Schmidt number. The analytical result reveals that the stability of dendritic growth depends on a critical stability number above which dendritic growth is stable. The selection condition of dendritic growth determines the Peclet number, tip growth velocity, tip radius and oscillation frequency, which is significantly affected by the external flow. The stability mechanism of dendritic growth in the binary alloy melt with the external flow remains the same as that in pure melt. In the binary alloy melt with the external flow the solute concentration destabilizes the dendritic growth system. The numerical computation for various growth conditions demonstrates the variations of the critical stability number, tip growth velocity, tip radius, and oscillatory frequency with the undercooling, external flow and morphological number.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

The discrete nature of grain attachment models in simulations of equiaxed solidification

Grain refiner is often added to aluminum and magnesium alloys during solidification processing to encourage the development of a fine equiaxed grain structure. Numerical modeling of such processes face the challenge of considering the effect of free floating grains that nucleate on grain refiner particles, are advected in the bulk fluid flow, and eventually coalesce to form a permeable, rigid solid structure. While several models have been developed to consider the advection of solid grains, the attachment of these grains is uniformly treated on a discrete, cell-by-cell basis. In a previous study, channel segregates were observed in the predicted composition field of equiaxed solidification simulations and were found to exhibit an extreme grid dependence. These channels were examined in the present study in detail for two different grain attachment models, one that assumed coalescence occurs at a constant and uniform volume fraction solid, and one that considered the effects of the local solid velocity field. The mechanism of the initiation and propagation of these channels was explored, and their physical relevance considered. It was concluded that these defects were primarily numerical artifacts arising from the discrete nature of the grain attachment models, and therefore, necessarily occurred on the length scale of the grid spacing. Development of an alternative attachment model that avoid this numerical problem is the subject of ongoing research.     

Numerical modelling of stress formation in solidifying and cooling castings

This paper is concerned with calculating stresses in two–component alloy castings solidifying in metal moulds. The main reasons for stress formation are mechanical resistance of the mould and non–homogeneous temperature gradients. Stresses appear in castings when a coherent solid phase skeleton has formed, which may occur in wide range of solid phase fraction. The need for taking the thermal resistance of air gaps into account is exposed.     

Numerical modelling of solidification of castings made of two‐component alloys

The paper deals with a numerical modelling of equiaxed microstructure formation during the solidification of two‐component alloys. The basic enthalpy formulation was applied to model the solidification. The equiaxed grain size was dependent on the average cooling velocity at the moment when the liquid metal reaches the liquidus temperature. The experimentally determined dependence between grain radius and cooling velocity was used in the calculation of average grain radii distribution.     

一种守恒型间断跟踪法用于处理一维Euler方程组中的若干问题

1引言间断跟踪法(front-tracking)是数值求解双曲型守恒型方程(组)的一种重要的数值方法,其主要特点是把间断作为移动的内边界来处理,光滑区域中的数值解用计算光滑解有效的数值方法来求解,而间断的移动和间断两侧的数值解的修正要满足Rankine-Hugoniot条件．我们称这样的跟踪法为传统的间断跟踪法(见[3],[14])．本文的第二作者多年来研究设计了一种基于解的守恒性质的间断跟踪法(见[11],[12]),其最主要的特点是以解的守恒性作为跟踪的机制,而不是象传统的间断跟踪法那样利用     

The development of a cellular automaton-finite volume model for dendritic growth

A cellular automaton to track the solid–liquid interface movement is linked to finite volume computations of solute diffusion to simulate the behavior of dendritic structures in binary alloys during solidification. A significant problem encountered in the CA formulation has been the presence of artificial anisotropy in growth kinetics introduced by a Cartesian CA grid. A new technique to track the interface movement is proposed to model dendritic growth in different crystallographic orientations while reducing the anisotropy due to grid orientation. The model stability with respect to the numerical parameters (cell size and time step) for various operating conditions is examined. A method for generating an operating window in Δt and Δx has been identified, in which the model gives a grid-independent set of results for calculated dendrite tip radius and tip undercooling. Finally, the model is compared to published experimental and analytical results for both directional and equiaxed growth conditions.     

Liquid structure of undercooled Cu70Ni30 alloy

Experiments of X-ray diffraction for liquid Cu70Ni30 alloy above and below its liquidus (1 230℃) have been carried out. By the analysis of experimental results, it is discovered that difference between structures of liquid and undercooled Cu70Ni30 alloy is their cluster sizes. The correlation radius of cluster is 1.125 nm and the atom number of cluster is 403 at 1 250\_1 400℃, and they are 1.3 nm and 704 respectively at the undercooled liquid state (1 200℃). The structure of liquid Cu70Ni30 alloy is fcc short order and its solid structure, fcc, is kept from liquid fcc short order.     

A cellular automaton technique for modelling of a binary dendritic growth with convection

A two-dimensional model for the simulation of a binary dendritic growth with convection has been developed in order to investigate the effects of convection on dendritic morphologies. The model is based on a cellular automaton (CA) technique for the calculation of the evolution of solid/liquid (s/l) interface. The dynamics of the interface controlled by temperature, solute diffusion and Gibbs–Thomson effects, is coupled with the continuum model for energy, solute and momentum transfer with liquid convection. The solid fraction is calculated by a governing equation, instead of some approximate methods such as lever rule method [A. Jacot, M. Rappaz, Acta Mater. 50 (2002) 1909–1926.] or interface velocity method [L. Nastac, Acta Mater. 47 (1999) 4253; L. Beltran-Sanchez, D.M. Stefanescu, Mat. and Mat. Trans. A 26 (2003) 367.]. For the dendritic growth without convection, mesh independency of simulation results is achieved. The simulated steady-state tip velocity are compared with the predicted values of LGK theory [Lipton, M.E. Glicksmanm, W. Kurz, Metall. Trans. 18(A) (1987) 341.] as a function of melt undercooling, which shows good agreement. The growth of dendrite arms in a forced convection has been investigated. It was found that the dendritic growth in the upstream direction was amplified, due to larger solute gradient in the liquid ahead of the s/l interface caused by melt convection. In the isothermal environment, the calculated results under very fine mesh are in good agreement with the Oseen–Ivanstov solution for the concentration-driven growth in a forced flow.     

Unconditionally Stable Splitting Methods for the Shallow Water Equations

The front-tracking method for hyperbolic conservation laws is combined with operator splitting to study the shallow water equations. Furthermore, the method includes adaptive grid refinement in multidimensions and shock tracking in one dimension. The front-tracking method is unconditionally stable, but for practical computations feasible CFL numbers are moderately above unity (typically between 1 and 5). The method resolves shocks sharply and is highly efficient. The numerical technique is applied to four test cases, the first being an expanding bore with rotational symmetry. The second problem addresses the question of describing the time development of two constant water levels separated by a dam that breaks instantaneously. The third problem compares the front-tracking method with an explicit analytic solution of water waves rotating over a parabolic bottom profile. Finally, we study flow over an obstacle in one dimension.     

Liquid structure of undercooled Cu70Ni30 alloy

Experiments of X-ray diffraction for liquid Cu₇₀Ni₃₀ alloy above and below its liquidus (l 230°C) have been carried out. By the analysis of experimental results, it is discovered that difference between structures of liquid and undercooled Cu₇₀Ni₃₀ alloy is their cluster sizes. The correlation radius of cluster is 1.125 nm and the atom number of cluster is 403 at l 250—l 400°C, and they are 1.3 nm and 704 respectively at the undercooled liquid state (1 200°C). The structure of liquid Cu₇₀Ni₃₀ alloy is fcc short order and its solid structure, fcc, is kept from liquid fcc short order.     

Rapid Freezing of Dilute Alloys

The solidification process of a dilute binary alloy with completemixing of the liquid and no diffusion in the solid is modelled,when a low temperature is imposed either directly or convectivelyat one end. The resulting moving-boundary problem has time-varyingcritical temperature which depends on the location of the interface.Approximations to the solution are constructed as perturbationexpansions in powers of the Stefan number. As an example thesolidification of a copper-nickel alloy is considered.     

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.     

Remote stationary wave field generated by local perturbing sources in a flow of stratified fluid

A linear formulation is used to study the problem of stationary waves formed in a uniform flow of an inviscid incompressible vertically stratified fluid past a point source or a mass dipole. Formulas are derived representing the characteristics of the wave field in the form of the sum of single integrals. A method is developed for constructing complete asymptotic expansions of the integrals obtained for large distances from the wave generator, including uniform expansions near the leading fronts of the separate modes. Approximate solutions of the problem in question exist (/1–4/ et al.). The behaviour of the characteristics of the wave field near the leading fronts of internal waves was studied in /5, 6/. In the case of a deep liquid the asymptotic form uniform in the neighbourhood of the leading fronts is expressed in terms of Fresnel integrals /5/, and in the case of a liquid of finite depth by Airy functions /6/. Examples of the exact solution of the problem are given in /7/.     

Microstructure analysis on IN 718 alloy round rod by FEM in the hot continuous rolling process

Based on physical metallurgy rules and experiential equations, models for microstructure analysis on IN 718 alloy in the round rod hot continuous rolling process has been developed using the finite element method (FEM) on the software ANSYS/LS-DYNA. The dynamic and metadynamic recrystallization models in and after deformation, the grain growth models in the compensated reheating process for IN 718 alloy are regressed, and corresponding processes are involved in these models. For a real rolling practice, the calculated central grain sizes were examined and are in good agreement with the measured ones. The element in the center of the workpiece is a typical one possessing the maximum of the effective strain, the temperature and the grain size in the rolling process. In the hot continuous rolling process, the relationship between the final grain size of the typical element and the inlet velocity of the first stand has been regressed by FE analysis, and the lower rolling speed is beneficial to the grain refinement.     

Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case

The paper is concerned with the existence and qualitative features of entire solutions for delayed reaction diffusion monostable systems. Here the entire solutions mean solutions defined on the $ (x,t)\in\mathbb{R}^{N+1} $. We first establish the comparison principles, construct appropriate upper and lower solutions and some upper estimates for the systems with quasimonotone nonlinearities. Then, some new types of entire solutions are constructed by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_* $ and propagation directions and a spatially independent solution, where $c_*>0$ is the critical wave speed. Furthermore, various qualitative properties of entire solutions are investigated. In particularly, the relationship between the entire solution, the traveling wave fronts and a spatially independent solution are considered, respectively. At last, for the nonquasimonotone nonlinearity case, some new types of entire solutions are also investigated by introducing two auxiliary quasimonotone controlled systems and establishing some comparison theorems for Cauchy problems of the three systems.