Global well-posedness of a nonlinear Fokker-Planck type model of grain growth

Analytical modelling of grain growth in metals and alloys in the presence of growing and dissolving precipitates—I. Normal grain growth

In this study, four different finite element level-set (FE-LS) formulations are compared for the modeling of grain growth in the context of polycrystalline structures and, moreover, two of them are presented for the first time using anisotropic grain boundary (GB) energy and mobility. Mean values and distributions are compared using the four formulations. First, we present the strong and weak formulations for the different models and the crystallographic parameters used at the mesoscopic scale. Second, some Grim Reaper analytical cases are presented and compared with the simulation results, and the evolutions of individual multiple junctions are followed. Additionally, large-scale simulations are presented. Anisotropic GB energy and mobility are respectively defined as functions of the mis-orientation/inclination and disorientation. The evolution of the disorientation distribution function (DDF) is computed, and its evolution is in accordance with prior works. We found that the formulation called “Anisotropic” is the more physical one, but it could be replaced at the mesoscopic scale by an isotropic formulation for simple microstructures presenting an initial Mackenzie-type DDF.

We have considered normal grain growth in thin films attached to a substrate and shown that shear-coupled grain boundary migration results in the build-up of elastic stresses in the film. We proposed a semi-quantitative model of grain growth, combining the elements of disclinations theory with the Burke-Turnbull model of normal grain growth. We have shown that shear-coupled grain boundary migration may lead to stagnation of normal grain growth, whereas for high coupling factor values, the grains in the film do not grow at all. Finally, we discussed the possible mechanisms of stress relaxation and resulting activation energies of grain growth.

Making use of Anelli's idea and an own model for isothermal grain growth, a new model for the grain growth during a continuous reheating process was deduced. This model makes use of the real three dimensional grain diameter instead of the mean linear intercept distance as a measure for the grain size. Two series of experiments have been undertaken: in the first series, the validity of the model for several intermediate temperatures has been investigated, while in the second series, the influence of the heating rate on the validity of the model has been verified. In both cases a quite reasonable matching between the prediction of the model and the experimentally determined three dimensional grain size has been found.

Linear bubble model of abnormal grain growth

We prove the existence of self-similar solutions to the Fradkov model for two-dimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbours (topological class). For the proof we introduce a finite maximal topological class and study an appropriate upwind-discretization of the time dependent problem in self-similar variables. We first show that the resulting finite dimensional differential system has nontrivial steady states. Afterwards we let the discretization parameter tend to zero and prove that the steady states converge to a compactly supported self-similar solution for a Fradkov model with finitely many equations. In a third step we let the maximal topology class tend to infinity and obtain self-similar solutions to the original system that decay exponentially. Finally, we use the upwind discretization to compute self-similar solutions numerically.

Grain growth stagnation in thin films due to shear-coupled grain boundary migration

A variational formulation and a double-grid method for meso-scale modeling of stressed grain growth in polycrystalline materials

Three-dimensional modeling of grain structure growth within ceramic tool material

Previous models for grain growth are usually based on Beck's formula, which are inadequate for quantitative prediction of austenite grain growth during reheating of as cast microstructures in microalloyed steels. The applications of these empirical grain growth models are limited to some particular categories of steels, such as Nb, Nb–Ti and Ti–V microalloyed steels, etc. In this study, a metallurgically based model has been developed to predict the austenite grain growth kinetics in microalloyed steels. This model accounts for the pinning force of second phase particles on grain boundary migration, in which the mean particle size with time and temperature is calculated on the basis of the Lifshitz–Slyozov–Wagner (LSW) particle coarsening theory. The volume fraction of precipitates is obtained according to the thermodynamic model. The reliability of the model is validated by the agreement between theoretical predictions and experimental measurements in the literature.

Accelerated Potts model for grain growth – Application to an IF steel

Recent developments in the modeling of grain growth and coarsening are briefly considered. Common characteristics of both these phenomena are briefly pointed out. A formulation based on stochastic consideration is proposed. This formulation assumes that the rate of growth of an individual grain or a precipitate is not entirely determined by its size but has a random component to it. This leads to a Fokker-Planck Equation for the size distribution. It can then be shown that there is indeed a unique state (the self-similar state) which is in general reached from an arbitrary initial state. The case of two dimensional grain growth is treated in detail as an example. Grain size distribution is obtained from these considerations, which is in good agreement with experiments.

Two finite element level-set (FE-LS) formulations are compared for the modeling of grain growth of 316L stainless steel in terms of grain size, mean values, and histograms. Two kinds of microstructures are considered: some are generated statistically from EBSD maps, and the others are generated by the immersion of EBSD data in the FE formulation. Grain boundary (GB) mobility is heterogeneously defined as a function of the GB disorientation. On the other hand, GB energy is considered as heterogeneous or anisotropic, which are, respectively, defined as a function of the disorientation and both the GB misorientation and the GB inclination. In terms of mean grain size value and grain size distribution (GSD), both formulations provide similar responses. However, the anisotropic formulation better respects the experimental disorientation distribution function (DDF) and predicts more realistic grain morphologies. It was also found that the heterogeneous GB mobility described with a sigmoidal function only affects the DDF and the morphology of grains. Thus, a slower evolution of twin boundaries (TBs) is perceived.

Modelling of the grain size probability distribution in polycrystalline nanomaterials

Modeling of grain growth in two dimensions