Modélisation stochastique de la solidification eutectique équiaxe
The solidification of eutectic alloys has been described by means of stochastic methods. These methods allow some of the hypotheses which are usually made in deterministic models to be relaxed, in particular those related to the description of grain impingement. Deterministic solidification models assume that grains are spherical, are stationary and have a unique growth rate. Stochastic methods consider a volume of liquid metal within which a given number of grains can nucleate (according to a theoretical or empirical relationship). The growth rate being given by a deterministic relation, the evolution of each grain can be individually traced during solidification. This is the main difference with respect to deterministic models, which generally consider a mean grain radius. The stochastic aspect arises mainly from the random location of the nuclei in the volume of the melt. The stochastic methods developed in the present work couple the heat balance of a small specimen with a calculation of the nucleation and growth of the grains. No model is required for grain impingement since it is directly accounted for by these methods. Although the most spectacular result induced by such models is an image of the evolution of the microstructure during solidification, it is not the only interest of these methods. These stochastic methods are developed and applied to three particular cases: grains which move during solidification, nodular cast iron solidification, for which the growth rate is different for each grain, solidification in a thermal gradient, a situation for which the grains are no longer spherical in shape. In these three cases, deterministic models are not able to describe the solidification correctly. In the first case, it is shown that the movement of the grains has a strong influence on the resulting microstructure and on the cooling curve. The effect of sedimentation is more pronounced than that of a random movement, which is homogeneous at the scale of the sample. Many original stereological results are presented for different types of grain movement and interaction. In the case of nodular cast iron, it is shown that the grain radius-dependent growth rate modifies both size distributions and microstructures, but has little effect on the cooling curve and grain density. It is shown that the method of Saltykov used to obtain volumetric grain size distributions from metallographic sections is very imprecise and thus should be avoided. In most cases, a simple stereological relationship based on a unique grain size is sufficient. It is also shown that a 2-dimensional cut through the simulated 3-dimensional microstructure strongly smoothes any differences seen in the volumetric grain size distributions. In the case of laser remelting, the grains are elongated in the direction of the thermal gradient. The asymmetry of the grains is studied as a function of the process parameters (temperature gradient and scanning velocity) and of the nucleation parameters (nucleation undercooling and grain density), using 2D and 3D stochastic models and an analytical 1D model. The grain asymmetry is not specific to high temperature gradients and growth rates: it can also be observed under conventional casting conditions. These three stochastic models provide a very useful tool for computer metallography and stereology. They give a realistic picture of the grain structure during the entire solidification process. If the models developed in this work are focused on small samples, their extension to more complex geometries and thermal conditions is possible, the only limitation of such tools arising from the memory space and power of present day computers.