... This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.

Abstract not found

Elastic interactions arising from a di#erence of lattice spacing between two coherent phases can have a strong influence on the phase separation (coarsening) behaviour of alloys. If the elastic moduli are di#erent in the two phases, the elastic interactions may accelerate, slow down or even stop the phase separation process. If the material is elastically anisotropic, the precipitates can be shaped like plates or needles instead of spheres and can arrange themselves into highly correlated patterns . Tensions or compressions applied externally to the specimen may have a strong e#ect on the shapes and arrangement of the precipitates. In this paper, we review the main theoretical approaches that have been used to model these e#ects and we relate them to experimental observations. The theoretical approaches considered are (i) `macroscopic' models treating the two phases as elastic media separated by a sharp interface (ii) `mesoscopic' models in which the concentration varies continuously a...

We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation whose velocity is a non-local quantity depending on the whole shape of the dislocation line. We study the special cases where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton-Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model. AMS Classification: 35F25, 35D05. Keywords: Dislocation dynamics, Peach-Koehler force, eikonal equation, Hamilton-Jacobi equations,

Contents 1. Introduction 2 2. The diffuse interface model 7 3. Existence for the diffuse interface system 12 3.1. The gradient flow structure 12 3.2. Assumptions 15 3.3. Weak solutions 16 3.4. The implicit time discretisation 17 3.5. Uniform estimates 21 3.6. Proof of the existence theorem 25 3.7. Uniqueness for homogeneous linear elasticity 26 4. Logarithmic free energy 29 4.1. A regularised problem 32 4.2. Higher integrability for the strain tensor 36 4.3. Higher integrability for the logarithmic free energy 42 4.4. Proof of the existence theorem 45 5. The sharp interface limit 46 5.1. The \Gamma--limit of the elastic Ginzburg--Landau energies 52 5.2. Euler--Lagrange equation for the sharp interface functional 60 6. The Gibbs--Thomson equation as a singular limit in the scalar case 70 7. Discussion 79 8. Appendix 81 9. Notation 86 References 90 1 1. Introduction We study a mathematical model describing phase separation in multi-- component alloy

Properties of the solidification front in a hypercooled liquid, socalled because the temperature of the resulting solid is below the melting temperature, are derived using a phase field (diffuse interface) model. Certain known properties for hypercooled fronts in specific materials are reflected within our theories, such as the presence of thin thermal layers and the trend towards smoother fronts (with less pronounced dendrites) when the undercooling is increased within the hypercooled regime. Both an asymptotic analysis, to derive the relevant free boundary problems, and a rigorous determination of the inner profile of the diffusive interface are given. Of particular interest is the incorporation of anisotropy and general microscale interactions leading to higher order differential operators. These features necessitate a much richer mathematical analysis than previous theories. Anisotropic free boundary problems are derived from our models, the simplest of which involves determining t...

We propose a model to study, on an atomistic scale, the effects of elastic misfit strains on the growth and coarsening of domains in phase-separating alloys. The model considered is a two-dimensional square crystal lattice with periodic boundary conditions and with an A or B atom near each site. To model the elastic interaction, nearest and next-nearest neighbors are connected by springs with longitudinal and transverse stiffnesses. In addition, there is a chemical interaction between nearest neighbours, favouring like against unlike pairs. A mean field analysis predicts phase separation into coherent phases, below a temperature which (for a typical set of elastic parameters) increases with the strength of the elastic forces. This critical temperature is less than that for separation into incoherent phases. The analysis also predicts how the anisotropy determines the shapes of domains at early times. A Monte Carlo procedure for implementing Kawasaki dynamics to represent diffusion in t...

A multiple-order-parameter theory of ordering on a binary face-centered-cubic (fcc) crystal lattice is used to model diffuse interphase boundaries and provide expressions for the anisotropy of the kinetic coefficient that characterizes the speed of the order-disorder boundary. The anisotropy is varied parametrically with the ratio of two gradient energy coefficients. In contrast to the results from single-order parameter theories, the orientation dependence of the kinetic coefficient differs significantly from that of the surface energy. Although the interfacial free energy anisotropy from this model is not strong enough to eliminate any orientations in the (three-dimensional) equilibrium shapes, the kinetic coefficient is sufficiently anisotropic to eliminate some orientations during growth. The long-time kinetic growth shapes show the development of edges and corners in a definite sequence. Permanent address: Department of Mathematical Sciences, University of Delaware, Newark, DE, ...

Abstract The peculiar behavior of active crystals is due to the presence of evolving phase mixtures the variety of which depends on the number of coexisting phases and the multiplicity of symmetryrelated variants. According to Gibbs ’ phase rule, the number of phases in a single-component crystal is maximal at a triple point in the p-T phase diagram. In the vicinity of this special point the number of metastable twinned microstructures will also be the highest — a desired effect for improving performance of smart materials. To illustrate the complexity of the energy landscape in the neighborhood of a triple point, and to produce a workable example for numerical simulations, in this paper we construct a generic Landau strain-energy function for a crystal with the coexisting tetragonal (t), orthorhombic (o), and monoclinic (m) phases. As a guideline, we utilize the experimental observations and crystallographic data on the t-o-m transformations of zirconia (ZrO 2), a major toughening agent for ceramics. After studying the kinematics of the t-o-m phase transformations, we re-evaluate the available experimental data on zirconia polymorphs, and propose a new mechanism for the technologically important t-m transition. In particular, our proposal entails the softening of a different tetragonal modulus from the one previously considered in the literature. We derive the simplest expression for the energy function for a t-o-m crystal with a triple point as the lowest-order polynomial in the relevant

In the case of Lipschitz extremals of vectorial variational problems an important class of strong variations originates from smooth deformations of the corresponding non-smooth graphs. These seemingly singular variations, which can be viewed as combinations of weak inner and outer variations, produce directions of differentiability of the functional and lead to singularity-centered necessary conditions on strong local minima: an equality, arising from stationarity, and an inequality, implying configurational stability of the singularity set. To illustrate the underlying coupling between inner and outer variations we study in detail the case of smooth surfaces of gradient discontinuity