Microstructural Evolution in Inhomogeneous Elastic Media

Jou, Herng-Jeng; Leo, Perry H.; Lowengrub, John

doi:10.1006/jcph.1996.5581

Cited by 147 publications

(163 citation statements)

References 49 publications

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“…In order to determine the derivative of non-homogeneous variables, we utilize Eq. [19]. Differentiating Eq.…”

Section: ½20mentioning

confidence: 99%

“…They have also estimated symmetry breaking as a function of particle size for different combinations of elastic anisotropies and misfits. Jou et al [19] use a boundary integral method to study single precipitate growth as well as two-precipitate interactions. Kolling et al [20] use a finite element and optimization-based method to predict the equilibrium precipitate shape for misfitting particles with dilatational eigenstrain.…”

Section: Introductionmentioning

confidence: 99%

“…The authors also compare with the results from their corresponding boundary-integral-based modeling methods. [19] The authors also highlight the advantage of using the diffuse-interface schemes where merging and splitting of particles can be handled easily in contrast to the sharp-interface methods which require re-meshing. A similar dynamic evolution model has also been used to predict equilibrium shapes in 3Ds in Reference 28.…”

Section: Introductionmentioning

confidence: 99%

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Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Bhadak

Sankarasubramanian

Choudhury

2018

Metall Mater Trans A

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Coherency misfit stresses and their related anisotropies are known to influence the equilibrium shapes of precipitates. Additionally, mechanical properties of the alloys are also dependent on the shapes of the precipitates. Therefore, in order to investigate the mechanical response of a material which undergoes precipitation during heat treatment, it is important to derive the range of precipitate shapes that evolve. In this regard, several studies have been conducted in the past using sharp-interface approaches where the influence of elastic energy anisotropy on the precipitate shapes has been investigated. In this paper, we propose a diffuse interface approach which allows us to minimize grid-anisotropy-related issues applicable to sharp-interface methods. In this context, we introduce a novel phase-field method where we minimize the functional consisting of the elastic and surface energy contributions while preserving the precipitate volume. Using this technique, we reproduce the shape bifurcation diagrams for the cases of pure dilatational misfit that have been studied previously using sharp-interface methods and then extend them to include interfacial energy anisotropy with different anisotropy strengths which has not been a part of previous sharp-interface models. While we restrict ourselves to cubic anisotropies in both elastic and interfacial energies in this study, the model is generic enough to handle any combination of anisotropies in both the bulk and interfacial terms. Further, we have examined the influence of asymmetry in dilatational misfit strains along with interfacial energy anisotropy on precipitate morphologies.

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“…In order to determine the derivative of non-homogeneous variables, we utilize Eq. [19]. Differentiating Eq.…”

Section: ½20mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Bhadak

Sankarasubramanian

Choudhury

2018

Metall Mater Trans A

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“…However, the exact treatment of the differential operator requires evaluating exponentials of the approximation matrix for the linear differential operator. For periodic systems, this calculation is cheap both in CPU and storage because the approximation matrix can be diagonalized in the Fourier space [1][2][3][4][5][6][7]. For non-periodic systems, in which the approximation matrices are not diagonal, storage and calculation of exponentials of the matrices are significantly more expensive.…”

Section: Introductionmentioning

confidence: 99%

“…Different approximation of the integral involving nonlinear term ℱ(u) results in either the integration factor (IF) method or the exponential time differencing (ETD) method [8]. For example, the second order integration factor Adams-Bashforth method (IFAB2 [9]) has the form (3) and the second order ETD method [2,9] has a form: (4) In (3) and (4), is a matrix of size n × n for a system with only one diffusive species, and u k is the approximate solution at the kth time step. The computational cost for updating u k+1 at one time step is of order of n 2 due to the three vector-matrix multiplications associated with and in (3).…”

Section: Introductionmentioning

confidence: 99%

Compact integration factor methods in high spatial dimensions

Nie

Wan

Zhang

et al. 2008

Journal of Computational Physics

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The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N × N spatial points, the exponential matrix is of a size of N 2 × N 2 based on direct representations. The vector-matrix multiplication is of O(N 4 ), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two-and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two-or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N 2 ) and O(N 3 ), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.

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Simulation of Microstructural Evolution Using the Field Method

Wang

Chen

2002

Characterization of Materials

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Morphological pattern formation and its spatio‐temporal evolution are among the most intriguing natural phenomena governed by nonlinear complex dynamics. These phenomena form the major study subjects of many fields including biology, hydrodynamics, chemistry, optics, materials science, ecology, geology, geomorphology, and cosmology, to name a few. In materials science, the morphological pattern is referred to as microstructure, which is characterized by the size, shape, and spatial arrangement of phases, grains, orientation variants, ferroelastic/ferroelectric/ferromagnetic domains, and/or other structural defects. These structural features usually have an intermediate mesoscopic length scale in the range of nanometers to microns. Microstructure plays a critical role in determining the physical properties and mechanical behavior of materials. Today the primary task of material design and manufacture is to optimize microstructures— by adjusting alloy composition and varying processing sequences—to obtain the desired properties. Microstructural evolution is a kinetic process that involves the appearance and disappearance of various transient and metastable phases and morphologies. The optimal properties of materials are almost always associated with one of these nonequilibrium states, which is “frozen” at the application temperatures. Theoretical characterization of the transient and metastable states represents a typical nonequilibrium nonlinear and nonlocal multiparticle problem. This problem generally resists any analytical solution except in a few extremely simple cases. With the advent of and easy access to high‐speed digital computers, computer modeling is playing an increasingly important role in the fundamental understanding of the mechanisms underlying microstructural evolution. In this article, methods recently developed for simulating the formation and dynamic evolution of complex microstructural patterns are reviewed. First, a brief account of the basic features of each method, including the conventional front‐tracking method (also referred to as the sharp‐interface method) and the new techniques without front‐tracking are given. Detailed description of the simulation technique and its practical application focuses on the field method since it is the most familiar and it is a more flexible method with broader applications. The need to link the mesoscopic microstructural simulation with atomistic calculation as well as with macroscopic process modeling and property calculation is also briefly discussed.

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Microstructural Evolution in Inhomogeneous Elastic Media

Cited by 147 publications

References 49 publications

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Phase-Field Modeling of Equilibrium Precipitate Shapes Under the Influence of Coherency Stresses

Compact integration factor methods in high spatial dimensions

Simulation of Microstructural Evolution Using the Field Method

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