D. Gurgul

Modelling of Dendritic Growth during Unidirectional Solidification by the Method of Cellular Automata

Modelling was carried out to investigate the internal dendrite grains structure formation from a liquid two-component solution in the area adjacent to a mould wall. For the simulation, our own model and computer program based on CAFD (Cellular Automata Finite Differences) were used. In modelling, the effect of process conditions and material-related parameters, e.g. nucleation temperature, heat exchange rate, interfacial energy, crystal orientation with respect to the casting wall, etc. on the nature of the dendritic grain growth was examined. It was demonstrated that the profile of concentration field in a near-mould-wall zone impedes the growth of the solid phase in the direct vicinity of the wall. A local melting down of the grains of a solid phase due to the segregation of admixtures reducing the alloy point of liquidus is also possible.

Simulation of the Ductile Iron Solidification Using a Cellular Automaton

The mathematical model of the globular eutectic solidification in 2D was designed. Pro¬posed model is based on the Cellular Automaton Finite Differences (CA-FD) calculation method. Model has been used for studies of the primary and of globular eutectic grains growth during the ductile iron (DI) solidification. A hyper-eutectic composition has been analyzed but this model can be used in the solidification modeling of hypo- and eutectic DI. The proposed model makes possible to trace the unrestricted growth of primary grains of two phases from the liq¬uid, transition from free to cooperative solidification, and cooperative growth of globular eutectic.

Cellular automaton modelling of ductile iron microstructure in the thin wall casting

The mathematical model of the globular eutectic solidification in 2D was designed. Proposed model is based on the Cellular Automaton Finite Differences (CA-FD) calculation method. Model has been used for studies of the primary austenite and of globular eutectic grains growth during the ductile iron solidification in the thin wall casting. Model takes into account, among other things, non-uniform temperature distribution in the casting wall cross-section, kinetics of the austenite and graphite grains nucleation, and non-equilibrium nature of the interphase boundary migration.

Stochastic Nature of the Casting Solidification Displayed by Micro-Modelling and Cellular Automata Method

Some aspects of stochastic nature of the solidification processes are described. Firstly, the influence of the random grains nucleation on the cooling curves repeatability in the thin wall casting is presented. Secondly, the foundations of an average shape prediction for geometry of ele¬mentary diffusion field (concept of the Averaged Voronoi Polyhedron, AVP) are shown for the mi¬cro-modelling of the diffusion limited growth. Stochastic nature of the grains nucleation and growth is taken into account in the solidification modelling based on the Cellular Automaton technique (CA).

Multiscale Modeling of Ductile Iron Solidification With Continuous Nucleation by a Cellular Automaton

The solidification of metals and alloys is a typical example of multiphysics and multiscale engineering systems. The phenomenon of different time and spatial scales should be taken into consideration in the modeling of a microstructure formation: heat diffusion, the components diffusion in the liquid and solid phases, the thermodynamics of phase transformation under a condition of inhomogeneous chemical composition of growing and vanishing phases, phase interface kinetics, and grains nucleation. The results of a two-dimensional modeling of the microstructure formation in a ductile cast iron are presented. The cellular automaton model (CA) was used for the simulation. The model takes into account the nucleation of two kinds of grains that appear inside of the liquid during solidification: austenite and graphite. The six states of CA cells correspond to the above-mentioned three phases (liquid, austenite and graphite) and to the three two-phase interfaces. A numerical solution was used for the modeling of concentration and temperature fields. The parabolic nonlinear differential equations with a source function were solved by using the finite element method and explicit scheme. In the mono-phase cells the source function is equal to zero. In the interface cells the value of the source function varies depending on the local undercooling. The undercooling value depends on the front curvature, the local temperature and the local chemical composition of the phases. Overlapping lattices with the same spatial step were used for concentration field modeling and for the CA. The time scale of the temperature field for this lattice is about 104 times shorter. Due to the above reasons, another lattice was used with a multiple spatial step and the same time step. The new grain nucleation of solid phases from a liquid is a phenomenon which must be taken into account for correct simulation of a polycrystalline structure formation. The cumulative distribution curve approach was used to calculate the number of substrates on which nucleation takes place as a function of under-cooling below the equilibrium temperature. An algorithm of continuous nucleation modeling during solidification is presented. The undercooling of solid phase grain nucleation was calculated on the basis of the inverse function of the above-mentioned cumulative distribution curve (fractile) with the argument equal to the random number generated in the interval 0[[ellipsis]]1 with uniform density. The domain of correct usage of this algorithm was analyzed.Copyright

Stereological Analysis of the Statistical Distribution of the Size of Graphite Nodules in DI

The estimate of a distribution law of the nodule diameters in a volume of cast iron provides information about the graphite nucleation kinetics, and also about the crystallization kinetics. This information is essential for building more accurate mathematical models of the alloy crystallization. The mapping of a Cumulative Distribution Function (CDF3) of radii for graphite nodules in ductile iron is presented on the base of a Probability Density Function (PDF1) of the chord length distribution for random sections of the sample at the planar cross-section.

Modelling of the Density Changes of Nodular Cast Iron During Solidification by CA-FD Method

Formation of the shrinkage defects in ductile iron castings is far more complicated phenomenon than in other casting alloys. In the presented paper changes the ductile iron density during solidification is analyzed. During the solidification path the influence of the temperature, phase fractions and phase composition is taking into account. Computer model, using cellular automata method, for estimation of changes in density of ductile iron during its solidification is applied. Results of the solidification modeling for Fe-C binary alloys with different composition in the castings with a different wall thickness are presented. As a result of calculations it was stated that after undercooling ductile iron below liquidus temperature volumetric changes proceed in three stages: pre-eutectic shrinkage (minimal in eutectic cast iron), eutectic expansion and the last shrinkage.

Using of the Averaged Voronoi Polyhedron for the Equiaxed Solidification Modeling

For the characterization of the equiaxed polycrystalline structure the Dirichlet tessellation is often used. The results of this space decomposition Voronoi polyhedrons are convex but not necessarily bounded. Size, volume and other characteristics of these bodies are the random variables. Parameters of the Averaged Voronoi Polyhedron are used in the presented paper for the modeling of the diffusion controlled peritectic transformation. Proposed model takes into account decreasing of the transformation interface surface in the remote regions of the diffusion field due to the probabilistic grains impingements. The results of the modeling are compared with the microstructure of the Pb-32 wt.% Bi alloy and thermal analysis results.