## MULTISCALE MODELING OF DUCTILE IRON SOLIDIFICATION WITH CONTINUOUS NUCLEATION BY A CELLULAR AUTOMATON

### Citations

268 |
Cellular automata modeling of physical systems
- Chopard, Droz
- 1998
(Show Context)
Citation Context ...22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD The cellular automaton model is an idealization of a real system in which space and time are discrete [25]. Physical quantities also take a finite set of value. The presented model uses a set of six cell states for microstructure modeling: three mono-phase states – "liquid", "austenite", and "graphite" – and three two-phase states. At the beginning, all of the cells in the CA lattice are in the "liquid" state and have eutectic temperature. The liquid phase has eutectic composition (point E in Fig. 1). The analyzed domain is cooled at a constant cooling rate. When the temperature of the liquid drops below the liquidus, nucleation and growth of the solid grains are possible. Grain nucleation in indu... |

67 |
Morphological Stability of a Particle Growing by Diffusion or Heat Flow,
- Mullins, Sekerka
- 1963
(Show Context)
Citation Context ... the X axis and the normal direction of the grain interface. If the phase volume fraction in the interface cell increases up to 1, this cell varies they state from interface to appropriate one-phase. Additionally, this cell captures all of the adjacent ones: their states exchange to the appropriative interface. It is a well-known fact that heat and mass diffusion processes in a liquid near the solidification interface result in a growth of the disturbance with a low curvature. On the other hand, perturbations with a high curvature are dampened down by surface energy at the interphase boundary [31]. The results of this are the known morphological changes of the growing grains’ boundary shapes with the changes of growth velocity: plain, cellular, dendritic, seawead, and fractal. The computer modeling of the heat and mass diffusion processes together with the growing grain shape simulation by the cellular automata method make it possible to predict the structural evolution of the metallic alloy during solidification. The numerical solution of the nonlinear Fourier equation was used for heat flow in the analyzed domain: ( ) coolT qqTTc ++∇λ∇=τ∂ ∂ (4) where: T is the temperature, τ is the t... |

56 |
Quantitative PhaseField Modeling of Dendritic Growth in Two and Three Dimensions,
- Karma, Rappel
- 1998
(Show Context)
Citation Context ...he melt. Next, the austenite shell encompasses the graphite nodules. This envelope isolates the graphite grains from the liquid phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm sp... |

42 |
Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method.
- Warren, Boettinger
- 1995
(Show Context)
Citation Context ...he melt. Next, the austenite shell encompasses the graphite nodules. This envelope isolates the graphite grains from the liquid phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm sp... |

36 |
Iron - Binary Phase Diagrams
- Kubaschewski
- 1982
(Show Context)
Citation Context ...approach of the F-vector [32]. The angle θ between the growth direction (normal to the grain boundary) and the positive X-axis direction was calculated as follows: =θ ∑∑ ji jiji ji jiji fxfy , ,, , ,,arctan (9) where: fi,j is the volume fraction of the phase in the cell (i,j), and xi,j and yi,j are the relative coordinates of the adjacent cells. The summation in (9) concerning the 20 neighboring cells gives the best results of normal direction estimation [33]. The liquidus lines in the binary Fe-C thermodynamic diagram were approximated by a linear function using the data from [34]: − for the austenite liquidus: ( ) 2γ/ 0.1045898.40764.1789 CCCTEq,L −−= (10) − for the graphite liquidus: ( ) CCT grEq,L 7.367925.141/ +−= (11) Carbon solvus line in the austenite: ( ) CCT grEq, 3.297141.809γ/ += (12) The carbon concentrations in the above equations are in weight percents. The border curvature was estimated with the geometrical method proposed in [35]. The relative coordinates of the tracer points A, B, and C in the interface cell (see Fig. 2) were calculated on the basis of the normal direction and growing phase volume fraction. Front curvature in point A in Fig. 2 was esti... |

20 |
Modelling of microstructure formation in solidification processes,
- Rappaz
- 1989
(Show Context)
Citation Context ...and aren’t superimposed beforehand. The following physical phenomenon are taken into account in the modeling: the releasing of the latent heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD T... |

7 |
A Coupled Finite Element-Cellular Automaton Model for The Prediction Of Dendritic Grain Structures in Solidification Processes",
- Gandin, Rappaz
- 1994
(Show Context)
Citation Context .... The analyzed domain is cooled at a constant cooling rate. When the temperature of the liquid drops below the liquidus, nucleation and growth of the solid grains are possible. Grain nucleation in industrial alloys has a heterogeneous nature. The substrates for the nucleus are randomly distributed in the bulk. Bulk distribution of differently-sized substrates also has a random nature. The undercooling value of substrate activation is a function of its size. Functional relationship between the active substrate fraction and undercooling ∆T should be a feature of the probability distribution law [26]. The undercooling value of each phase should be calculated relative to the liquidus lines that are represented in Fig. 1 by solid lines. The nucleation laws based on Gaus [27], Weibull [28] and lognormal [29] distributions are known. The method of continuous nucleation modeling used will be present in the next part of this paper. The state of the CA cell with the active nucleus varies from "liquid" to "austenite" or "graphite". The states of adjacent liquid cells are changed to the appropriate interface. The new phase growth and volume fraction changes are only possible in the interface cells... |

6 |
Atomistic computation of liquid diffusivity, solid-liquid interfacial free energy, and kinetic coefficient in Au and Ag.",
- Hoyt, Asta
- 2002
(Show Context)
Citation Context ... real temperature Tr is equal to the sum of capillary undercooling ∆Tκ and kinetic undercooling ∆Tµ: 2 Copyright © 2010 by ASME Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use 3 Copyright © 20xx by ASME Figure 1. SCHEME OF THE IRON-CARBON BINARY PHASE DIAGRAM. µκ ∆+∆=− TTTT rEq (1) where ∆Tκ = Γκ, Γ is the Gibbs-Thomson coefficient, and κ is a front curvature. The scheme of liquidus lines positions after accounting for the capillary effect for convex grains is shown in Fig. 1 by dashed lines. Basing on [30], it has been assumed in the computations that the interface migration rate is a linear function of the local kinetic undercooling ∆Tµ: µ ∆µ= Tu (2) where µ is the kinetic growth coefficient. The increment of the new phase volume fraction in the interface cells ∆f over one time step ∆τ in the square CA cells of size a was calculated using the equation proposed in [35]: ( )θ+θ τ∆=∆ sincosa u f (3) where θ is the angle between the X axis and the normal direction of the grain interface. If the phase volume fraction in the interface cell increases up to 1, this cell varies they state from interfac... |

5 |
Multi-scale computational modelling of solidification phenomena,
- Rafii-Tabar, Chirazi
- 2002
(Show Context)
Citation Context ...he graphite nodules. This envelope isolates the graphite grains from the liquid phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm space, solute distribution in the solids, etc.... |

3 |
A quantitative dendrite growth model and analysis of stability concepts,
- Beltran-Sanchez, Stefanescu
- 2004
(Show Context)
Citation Context ...and aren’t superimposed beforehand. The following physical phenomenon are taken into account in the modeling: the releasing of the latent heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD T... |

3 |
A Three Dimensional Modified Cellular Automaton Model for the Prediction of Solidification Microstructures",
- Zhu, Hong
- 2002
(Show Context)
Citation Context ...and aren’t superimposed beforehand. The following physical phenomenon are taken into account in the modeling: the releasing of the latent heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD T... |

3 |
Modelling of Dendritic Growth During Unidirectional Solidification by the Method of Cellular Automata",
- Burbelko, Fraś, et al.
- 2010
(Show Context)
Citation Context |

2 |
A three-dimensional sharp interface model for the quantitative simulation of solutal dendritic growth",
- Pan, Zhu
- 2010
(Show Context)
Citation Context |

2 |
Interaction Between Single Grain Solidification and Macro Segregation: Application of a Cellular Automaton -Finite Element Model",
- Guillemot, Gandin, et al.
- 2007
(Show Context)
Citation Context |

2 |
Simulation of Weld Solidification Microstructure and its Coupling to the Macroscopic Heat And Fluid Flow Modelling", Modelling and Simulation
- Pavlyk, Dilthey
- 2004
(Show Context)
Citation Context |

2 |
Modelling of Non-Equilibrium Solidification in Ternary Alloys:
- Jarvis, Brown, et al.
- 2000
(Show Context)
Citation Context |

2 |
Nonequilibrium Kinetics of Phase Boundary Movement in Cellular Automaton Modelling",
- Burbelko, Fraś, et al.
- 2006
(Show Context)
Citation Context |

2 |
Modeling of microstructure evolution in regular eutectic growth",
- Zhu, Hong
- 2002
(Show Context)
Citation Context ...t heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD The cellular automaton model is an idealization of a real system in which space and time are discrete [25]. Physi... |

2 |
Computational modeling of microstructure evolution in solidification of aluminum alloys",
- Zhu, Hong, et al.
- 2007
(Show Context)
Citation Context ...t heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD The cellular automaton model is an idealization of a real system in which space and time are discrete [25]. Physi... |

2 |
Nucleation and grains density - A theoretical model and experimental verification.
- Fraś, Wiencek, et al.
- 2001
(Show Context)
Citation Context ...ion in industrial alloys has a heterogeneous nature. The substrates for the nucleus are randomly distributed in the bulk. Bulk distribution of differently-sized substrates also has a random nature. The undercooling value of substrate activation is a function of its size. Functional relationship between the active substrate fraction and undercooling ∆T should be a feature of the probability distribution law [26]. The undercooling value of each phase should be calculated relative to the liquidus lines that are represented in Fig. 1 by solid lines. The nucleation laws based on Gaus [27], Weibull [28] and lognormal [29] distributions are known. The method of continuous nucleation modeling used will be present in the next part of this paper. The state of the CA cell with the active nucleus varies from "liquid" to "austenite" or "graphite". The states of adjacent liquid cells are changed to the appropriate interface. The new phase growth and volume fraction changes are only possible in the interface cells. The kinetic undercooling of the mother liquid phase is a thermodynamic driving force of the new grains' growth. Total undercooling at the solid-liquid interface, hence the difference betwe... |

1 |
The Invention of Ductile Cast Iron... In Millis' Own Words", Modern Casting,
- Millis
- 1998
(Show Context)
Citation Context ...tectic microstructure is a result of the simultaneous growth of the grains of two (or more than two) solid phases from the melt. This structure is a base for numerous technical alloys. The physical and mechanical properties of such materials depend not only on the properties and volume fraction of each phase but also on the grains’ morphology. Nodular graphite cast iron – so-called Ductile Iron (DI) – has high mechanical properties in the as-cast state. The properties of such materials can be upgraded as a result of further heat treatment. Ductile Cast Iron has been known only since the 1940s [1], but it has grown in relative importance and currently represents about 30% (23,84 million metric tons) of cast iron production in the word. Global casting production in 2008 was 93,449 million metric tons (e.g. cast iron - 68,298 million metric tons) [2]. The microstructure peculiarity of DI is the nodular shape of the grains of one phase – graphite. Another phase that grows directly from the melt during solidification is austenite, a solid solution of carbon in a crystal lattice of γ-Fe. The graphite and austenite grains are nucleated in the liquid phase. At first these grains grow directly... |

1 |
Polycrystalline patterns in far-from-equilibrium freezing: a phase field study",
- Granasy, Pusztai, et al.
- 2006
(Show Context)
Citation Context ...he melt. Next, the austenite shell encompasses the graphite nodules. This envelope isolates the graphite grains from the liquid phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm sp... |

1 |
Multiscale modelling of solidification microstructures, including microsegregation and microporosity in an Al-SiCu alloy",
- unknown authors
- 2004
(Show Context)
Citation Context ...he graphite nodules. This envelope isolates the graphite grains from the liquid phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm space, solute distribution in the solids, etc.... |

1 |
Three-dimensional phase-field simulations of directional solidification",
- Plapp
- 2007
(Show Context)
Citation Context ... phase. From this moment the graphite grains grow due to carbon diffusion from the liquid solution to the nodules’ surface through the solid solution layer. The austenite grains still solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm space, solute distribution in the solids, etc.) are the results of the modeling and aren’t superimposed beforehand. The fol... |

1 |
Mathematical Modeling of the Dendrite Growth in the Undercooled Melt",
- Umantsev, Vinogradov, et al.
- 1985
(Show Context)
Citation Context ...l solidify directly from the melt, but on the "graphite nodule – austenite envelope" border, this phase vanishes and allows spheroid growth. For direct modeling of the microstructure formation during solidification, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm space, solute distribution in the solids, etc.) are the results of the modeling and aren’t superimposed beforehand. The following physical phenomenon are taken into account in the modeling: the releasing of the latent heat of phase transformations in the phase interface and heat flow, the redistribution ... |

1 |
Modeling of the Dendrite Structure Evolution",
- Umantsev, Vinogradov, et al.
- 1986
(Show Context)
Citation Context ...ication, two multiscale computational methods are used: the phase-field method (PF) [3, 4, 5] and the cellular automata method (CA) [6,7]. Models based on the first one remain to date computationally too costly [8]. The first mathematical model of dendritic solidification of metals and alloys based on the CAFD method (Cellular Automata – Finite Differences) was presented by Umantsev et al. [9]. The authors’ next publication disclosed the effect of initial undercooling, obtained by modeling, on the morphology of thermal dendrite and the structural evolution of its secondary arms in micro-scale [10]. In CA modeling the outer grain shape and inner structure (eg. secondary dendrite arm space, solute distribution in the solids, etc.) are the results of the modeling and aren’t superimposed beforehand. The following physical phenomenon are taken into account in the modeling: the releasing of the latent heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into co... |

1 |
Simulation of Solidification Grain Structures with a Multiple Diffusion Length Scales Model",
- Mosbah, Bellet, et al.
- 2009
(Show Context)
Citation Context |

1 |
A 3D Cellular Automaton Model of Coupled Growth in Two Component Systems",
- Spittle, Brown
- 1994
(Show Context)
Citation Context ...t heat of phase transformations in the phase interface and heat flow, the redistribution of the solutes between the different phases and diffusional mass transport as well as the equilibrium temperature changes near the curved grain boundaries (GibbsThomson effect.) It is possible to take into consideration the non-equilibrium character of the phase transformations. The model’s development for one-phase microstructure evolution is the subject of much research [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. The question of eutectic solidification in a superimposed temperature condition was solved in [21, 22, 23, 24]. The purpose of the present work is to develop a twodimensional model to simulate the formation of the DI structure during solidification in a non steady-state temperature condition. The model takes into account continuous nucleation, the separate growth of graphite nodules and austenite dendrites at the first solidification stage as well as the following cooperative growth of the graphite-austenite eutectic in the binary Fe-C system. A DESCRIPTION OF THE MODEL AND SOLUTION METHOD The cellular automaton model is an idealization of a real system in which space and time are discrete [25]. Physi... |

1 |
Modeling of mrcrostructure formation in slolidification processes",
- Rappaz
- 1989
(Show Context)
Citation Context ... Grain nucleation in industrial alloys has a heterogeneous nature. The substrates for the nucleus are randomly distributed in the bulk. Bulk distribution of differently-sized substrates also has a random nature. The undercooling value of substrate activation is a function of its size. Functional relationship between the active substrate fraction and undercooling ∆T should be a feature of the probability distribution law [26]. The undercooling value of each phase should be calculated relative to the liquidus lines that are represented in Fig. 1 by solid lines. The nucleation laws based on Gaus [27], Weibull [28] and lognormal [29] distributions are known. The method of continuous nucleation modeling used will be present in the next part of this paper. The state of the CA cell with the active nucleus varies from "liquid" to "austenite" or "graphite". The states of adjacent liquid cells are changed to the appropriate interface. The new phase growth and volume fraction changes are only possible in the interface cells. The kinetic undercooling of the mother liquid phase is a thermodynamic driving force of the new grains' growth. Total undercooling at the solid-liquid interface, hence the di... |

1 |
The Application of Some Probability Density Function of Heterogeneous Nucleation",
- Fraś, Wiencek, et al.
- 2006
(Show Context)
Citation Context ...lloys has a heterogeneous nature. The substrates for the nucleus are randomly distributed in the bulk. Bulk distribution of differently-sized substrates also has a random nature. The undercooling value of substrate activation is a function of its size. Functional relationship between the active substrate fraction and undercooling ∆T should be a feature of the probability distribution law [26]. The undercooling value of each phase should be calculated relative to the liquidus lines that are represented in Fig. 1 by solid lines. The nucleation laws based on Gaus [27], Weibull [28] and lognormal [29] distributions are known. The method of continuous nucleation modeling used will be present in the next part of this paper. The state of the CA cell with the active nucleus varies from "liquid" to "austenite" or "graphite". The states of adjacent liquid cells are changed to the appropriate interface. The new phase growth and volume fraction changes are only possible in the interface cells. The kinetic undercooling of the mother liquid phase is a thermodynamic driving force of the new grains' growth. Total undercooling at the solid-liquid interface, hence the difference between the equilibrium ... |

1 |
Numerical simulation of dendrite morphology and grain growth with modified cellular automata,
- Dilthley, Pavlik
- 1998
(Show Context)
Citation Context ...ce cells the value of the heat and mass sources for the finite-difference scheme are: τ∆ ∆ = ββ→α f LqT (7) ( ) τ∆ ∆ −= ββα f CCqC (8) where Lα→β is the volumetric latent heat of α→β transformation, Cα and Cβ are the carbon concentrations in the vanishing and growing phases, and ∆fβ is the growth of the new phase volume fraction during the time step. The source function (7) calculated for the elements of the dense mesh was integrated over the area of the elements of the sparse one. The normal direction of the grain boundary in the interface cells was determined by the approach of the F-vector [32]. The angle θ between the growth direction (normal to the grain boundary) and the positive X-axis direction was calculated as follows: =θ ∑∑ ji jiji ji jiji fxfy , ,, , ,,arctan (9) where: fi,j is the volume fraction of the phase in the cell (i,j), and xi,j and yi,j are the relative coordinates of the adjacent cells. The summation in (9) concerning the 20 neighboring cells gives the best results of normal direction estimation [33]. The liquidus lines in the binary Fe-C thermodynamic diagram were approximated by a linear function using the data from [34]: − for the austenite liq... |

1 |
Analysis of causes and means to reduce artificial anisotropy in modelling of the solidification process on cellular automaton",
- Burbelko, Kapturkiewicz, et al.
- 2007
(Show Context)
Citation Context ...a of the elements of the sparse one. The normal direction of the grain boundary in the interface cells was determined by the approach of the F-vector [32]. The angle θ between the growth direction (normal to the grain boundary) and the positive X-axis direction was calculated as follows: =θ ∑∑ ji jiji ji jiji fxfy , ,, , ,,arctan (9) where: fi,j is the volume fraction of the phase in the cell (i,j), and xi,j and yi,j are the relative coordinates of the adjacent cells. The summation in (9) concerning the 20 neighboring cells gives the best results of normal direction estimation [33]. The liquidus lines in the binary Fe-C thermodynamic diagram were approximated by a linear function using the data from [34]: − for the austenite liquidus: ( ) 2γ/ 0.1045898.40764.1789 CCCTEq,L −−= (10) − for the graphite liquidus: ( ) CCT grEq,L 7.367925.141/ +−= (11) Carbon solvus line in the austenite: ( ) CCT grEq, 3.297141.809γ/ += (12) The carbon concentrations in the above equations are in weight percents. The border curvature was estimated with the geometrical method proposed in [35]. The relative coordinates of the tracer points A, B, and C in the interface cell (see Fig. 2) were cal... |

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Mezomodeling of Solidificatnion Usin a Cellular Automaton", UWND AGH, Krakow (in Polish).
- Burbelko
- 2004
(Show Context)
Citation Context ... (1) where ∆Tκ = Γκ, Γ is the Gibbs-Thomson coefficient, and κ is a front curvature. The scheme of liquidus lines positions after accounting for the capillary effect for convex grains is shown in Fig. 1 by dashed lines. Basing on [30], it has been assumed in the computations that the interface migration rate is a linear function of the local kinetic undercooling ∆Tµ: µ ∆µ= Tu (2) where µ is the kinetic growth coefficient. The increment of the new phase volume fraction in the interface cells ∆f over one time step ∆τ in the square CA cells of size a was calculated using the equation proposed in [35]: ( )θ+θ τ∆=∆ sincosa u f (3) where θ is the angle between the X axis and the normal direction of the grain interface. If the phase volume fraction in the interface cell increases up to 1, this cell varies they state from interface to appropriate one-phase. Additionally, this cell captures all of the adjacent ones: their states exchange to the appropriative interface. It is a well-known fact that heat and mass diffusion processes in a liquid near the solidification interface result in a growth of the disturbance with a low curvature. On the other hand, perturbations with a high curvature are d... |

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Revealing and characterising solidification structure of ductile cast iron",
- Rivera, Boeri, et al.
- 2002
(Show Context)
Citation Context ...undercooling increases at both solidification fronts. The thermodynamic driving force of the crystallization of both phases increases, resulting in accelerated migration of the grain boundaries towards one another. The scheme of carbon distribution in the liquid between the austenite and graphite grains (with a non-equilibrium border concentration on both interfaces) is shown at the top part of Fig. 1. As follows from Fig. 4f, each austenite grain can cover several graphite nodules. These results are in the good correlation with the experimental investigation of solidification structure of DI [36,37]. Examples of the real microstructure of industrial ductile iron are presented in Fig. 5. The history of mean temperature changes in the analyzed domain obtained in the simulation is presented in Fig. 6. a) b) c) 7 Copyright © 2010 by ASME Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use 8 Copyright © 20xx by ASME d) e) f) Figure 4. THE STAGES OF DI MICROSTRUCTURE FORMATION (MODELING); TIME, s: a) 2; b) 10; c) 17; d) 24; e) 33; f) 46. Figure 5. REAL MICROSTRUCTURE OF DI, NONETCHED. Figure 6. COOLING CURV... |

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Examination of the solidification macrostructure of spheroidal and flake graphite cast irons using DAAS and ESBD",
- Rivera, Calvillo, et al.
- 2008
(Show Context)
Citation Context ...undercooling increases at both solidification fronts. The thermodynamic driving force of the crystallization of both phases increases, resulting in accelerated migration of the grain boundaries towards one another. The scheme of carbon distribution in the liquid between the austenite and graphite grains (with a non-equilibrium border concentration on both interfaces) is shown at the top part of Fig. 1. As follows from Fig. 4f, each austenite grain can cover several graphite nodules. These results are in the good correlation with the experimental investigation of solidification structure of DI [36,37]. Examples of the real microstructure of industrial ductile iron are presented in Fig. 5. The history of mean temperature changes in the analyzed domain obtained in the simulation is presented in Fig. 6. a) b) c) 7 Copyright © 2010 by ASME Downloaded From: https://proceedings.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use 8 Copyright © 20xx by ASME d) e) f) Figure 4. THE STAGES OF DI MICROSTRUCTURE FORMATION (MODELING); TIME, s: a) 2; b) 10; c) 17; d) 24; e) 33; f) 46. Figure 5. REAL MICROSTRUCTURE OF DI, NONETCHED. Figure 6. COOLING CURV... |