At the microscopic level, many materials are made of smaller and randomly oriented grains. These grains are separated by grain boundaries which tend to decrease the electrical and thermal conductivity of the material. The motion of grain boundaries is an important phenomenon controlling the grain growth in materials processing and synthesis. Mathematical modeling and simulation is a powerful tool for studying the motion of grain boundaries. The research reported in this thesis is focused on the numerical simulation and analysis of a coupled surface and grain boundary motion which models the evolution of grain boundary and the diffusion of the free surface during the process of grain growth. The “quarter loop” geometry provides a convenient model for the study of this coupled motion. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. They are coupled together at a triple junction. A front tracking method is used to simulate the migration. To describe the problem, different formulations are presented and discussed. A new formulation that comprises partial differential equations and algebraic equations is proposed. It preserves arc length parametrization up to scaling and exhibits good numerical performance. This formulation is shown to be well-posed in a reduced, linear setting. Numerical simulations are implemented and compared for all formulations. The new formulation is also applied to some other related problems. We investigate numerically the linear stability of the travelling wave solutions for the quarter loop problem and a simple grain boundary motion problem for both curves in two dimensions and surfaces in three dimensions. The numerical results give evidence that they are convectively stable. A class of high order three-phase boundary motion problems are also studied. We consider a region where three phase boundaries meet at a triple junction and evolve with specified normal velocities. A system of partial differential algebraic equations (PDAE) is proposed to describe this class of problems by extending the discussion for the coupled surface and grain boundary motion. The linear well-posedness of the system is analyzed and numerical simulations are performed.

It is well documented that the mechanical response of polycrystalline metals depends on the metal's microstructure, for example the dependence of yield strength on grain size (Hall-Petch effect). Local continuum approaches do not address the sensitivity of deformation to microstructural features, and are therefore unable to capture much of the experimentally observed behavior of polycrystal deformation. In this work, a crystal plasticity model is developed that predicts a dependence of yield strength on grain size without grain size explicitly entering into the constitutive equations. The grain size dependence in the model is the result of non-local effects of geometrically necessary dislocations (GNDs), i.e. GNDs harden both the material at a point and the surrounding material. The conventional FeFp kinematics for single crystals have been augmented based on a geometric argument that accounts for the grain orientations in a polycrystal. The augmented kinematics allows an initial GND state at grain boundaries and an evolving GND state due to sub-grain formation within the grain to be determined in a consistent manner. Numerically, these non-local affects are captured using a non-local integral approach rather than a conventional gradient approach. The non-local crystal plasticity model is used to simulate the tensile behavior in copper polycrystals with grain sizes ranging from 14 to 244 micron. The simulation results show a grain size dependence on the polycrystal's yield strength, which are qualitatively in good agreement with the experimental data. However, the Hall-Petch exponent predicted by the simulations is more like d-1 rather than d-0.5. The effects of different simulation parameters including grain shape and misorientation distribution did not greatly affect the Hall-Petch exponent. The simulation results indicate that the Hall-Petch exponent is sensitive to the grain boundary strength: the Hall-Petch exponent decreases as grain boundary strength decreases. The intragrain misorientations predicted by the non-local model were compared with experiments on polycrystalline nickel. Experimentally, the intragrain misorientations were tracked by electron back scatter diffraction (EBSD) at various strain levels from the same location. On average, the simulation results predicted enough misorientation throughout the sample. However, the model did not correctly predict the spatial details of the intragrain misorientation.

The microstructures and mechanical properties of the AZ31 Mg tubes fabricated by one-pass forward piercing tube extrusion operated at 250-400oC and 10-2-100 s-1 are examined. The grain size is refined from the initial ~75

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<p>Diffusion controlled phase transformations are studied bymeans of the phase-field method. Morphological evolution ofdendrites, grains and Widmanst\"atten plates is modeled andsimulated.</p><p>Growth of dendrites into highly supersaturated liquids ismodeled for binary alloy solidification. Phase-field equationsthat involve both temperature and solute redistribution areformulated. It is demonstrated that while at low undercoolingheat diffusion does not affect the growth of dendrites, i.e.solidification is nearly isothermal, at high cooling rates thesupersaturation is replaced by the thermal undercooling as thedriving force for growth.</p><p>In experiments many crystals with different orientationsnucleate. The growth of randomly oriented dendrites, theirsubsequent impingement ant formation of grain boundaries arestudied in two dimensions using the FEM on adaptive grids.</p><p>The structure of dendrites is determined by growthconditions and physical parameters of the solidifying material.Effects of the undercooling and anisotropic surface energy onthe crystal morphology are investigated. Transition betweenseaweeds, doublons and dendrites solidifying out of puresubstance is studied and compared to experimental data. Two-and three-dimensional simulations are performed in parallel onadaptive and uniform meshes.</p><p>A phase-field method based on the Gibbs energy functional isformulated for ferrite to austenite phase transformation inFe-C. In combination with the solute drag model, transitionbetween diffusion controlled and massive transformations as afunction of C concentration and temperature is established byperforming a large number of one dimensional calculations withreal physical parameters. In two dimensions, growth ofWidmanstaetten plates is governed by the highly anisotropicsurface energy. It is found that the plate tip can beapproximated as sharp, in agreement with experiments.</p><p><b>Keywords:</b>heat and solute diffusion, solidification,solid-solid phase transformation, microstructure, crystalgrowth, dendrite, grain boundary, Widmanstaetten plate,phase-field, adaptive mesh generation, FEM.</p>

At the microscopic level, many materials are made of smaller and randomly oriented grains. These grains are separated by grain boundaries which tend to decrease the electrical and thermal conductivity of the material. The motion of grain boundaries is an important phenomenon controlling the grain growth in materials processing and synthesis. Mathematical modeling and simulation is a powerful tool for studying the motion of grain boundaries. The research reported in this thesis is focused on the numerical simulation and analysis of a coupled surface and grain boundary motion which models the evolution of grain boundary and the diffusion of the free surface during the process of grain growth. The “quarter loop” geometry provides a convenient model for the study of this coupled motion. Two types of normal curve velocities are involved in this model: motion by mean curvature and motion by surface diffusion. They are coupled together at a triple junction. A front tracking method is used to simulate the migration. To describe the problem, different formulations are presented and discussed. A new formulation that comprises partial differential equations and algebraic equations is proposed. It preserves arc length parametrization up to scaling and exhibits good numerical performance. This formulation is shown to be well-posed in a reduced, linear setting. Numerical simulations are implemented and compared for all formulations. The new formulation is also applied to some other related problems. We investigate numerically the linear stability of the travelling wave solutions for the quarter loop problem and a simple grain boundary motion problem for both curves in two dimensions and surfaces in three dimensions. The numerical results give evidence that they are convectively stable. A class of high order three-phase boundary motion problems are also studied. We consider a region where three phase boundaries meet at a triple junction and evolve with specified normal velocities. A system of partial differential algebraic equations (PDAE) is proposed to describe this class of problems by extending the discussion for the coupled surface and grain boundary motion. The linear well-posedness of the system is analyzed and numerical simulations are performed.

No description available.

The effect of thermo-mechanical processing by deformation and annealing on the grain boundary configuration of commercially pure Ni-200 is reported in this thesis. Ni-200 is unalloyed, thus avoiding the complex effects associated with alloying elements on the formation and development of different types of grain boundaries. One step strain-recovery with strain levels in the range of 3% to 7.5% (with 1.5% intervals) and annealing temperatures in the range of 800ºC to 1000ºC (with 100ºC intervals) were used in processing. The effects of parameters such as strain level, annealing temperature, annealing time and grain growth on grain boundary configurations were studied. Using Orientation Image Microscopy (OIM) it was found that the Fsp (fraction of special grain boundaries) value of strained samples annealed in the range of 800ºC to 1000ºC began to increase after a critical length of time, after which the Fsp value increased quickly and becoming a maximum in 2~4 minutes. The length of the critical annealing time for the increase of Fsp was shorter in the material with the higher levels of strain at a constant annealing temperature. Also the critical annealing time was shorter when annealed at higher temperatures under a fixed level of strain. The Fsp value increased to 80% from an as received value of about 30% in the samples with varying strain levels. However, the Fsp values only increased from 30% to 45% in the material without strain. Due to grain boundary migration, the Fsp values increased with grain size and became a maximum during the heat treatment of the strained material. In the material without strain however even when grain growth occurred, limited improvement in Fsp values occurred showing that contribution of strain is very important to the formation of special boundaries. By varying the strain levels, annealing temperatures and times, material with high Fsp values in a wide range of grain size can be obtained. Under the present processing conditions used however, multi-cycle was not helpful to the improvement of Fsp. TEM observations indicated dislocation tangles occurred near the grain boundary of the 1x6% strained samples. These dislocation tangles decreased with time at 800˚C and were reduced considerably after 20 minutes. Thermodynamic and kinetic models were used in the calculations of twin density-grain size relationships. The results indicated that the contribution of strain is equivalent to the increase of grain boundary energy, which provided an extra driving force and improved probability of twin embryo formation.

Nanocrystalline materials are polycrystalline solids with grain size in the nanometer range (< 100nm), which have been found to exhibit superior properties such as high magnetic permeability and corrosion resistance, as well as a considerably increase of strength when compared with their coarse grain counterparts. All those improved properties are attributed to the high volume fraction of grain boundaries (GBs). However, the high density of GBs brings a large amount of excess enthalpy to the whole system, making the nanostructures unstable and suffer from severe thermal or mechanical grain growth. In order to maintain the advantageous properties of nanocrystalline materials, it is necessary to stabilize GB and inhibit grain growth. While alloying has been found to be an effective way of achieving stabilized nanocrystalline metal alloys experimentally, the direct quantification of solute effects on GB motion still poses great challenge for investigating thermal stability of general nanocrystalline materials. In this research, impurity segregation and solute drag effects on GB motion were investigated by extending the interface random-walk method in direct molecular dynamics simulations. It was found that the GB motion was controlled by the solute diffusion perpendicular to the boundary plane. Based on the simulation results at different temperatures and impurity concentrations, the solute drag effects can be well modeled by the theory proposed by Cahn, Lücke and Stüwe (CLS model) more than fifty years ago. However, a correction to the original CLS model needs to be made in order to quantitatively predict the solute drag effects on a moving GB.