The equilibrium shape of an elastically inhomogeneous inclusion

Schmidt, Ingo; Groß, Dietmar

doi:10.1016/s0022-5096(97)00011-2

Cited by 67 publications

(52 citation statements)

References 18 publications

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“…Our work can be seen as a diffuse-interface counterpart of previous work by Voorhees et al [10] and other sharp-interface models by Schmidt and Gross [14] and Jog et al [17] as well as the level-set-based FEM method proposed by Duddu et al [22] and Zhao et al [23,24] Herein, we use a modification of the volume-preserved Allen-Cahn evolution equation that was proposed previously by Nestler and colleagues, [44] wherein the Allen-Cahn equation prescribing the evolution of a given order parameter is modified such that the integrated change in the volume that is computed over the entire domain of integration returns zero. The motivation for proposing a diffuse-interface model is threefold: firstly given that the Allen-Cahn dynamics for the order parameter ensures energy minimization, there is no requirement for an additional optimization routine that is used in corresponding works by Schmidt and Gross [14] and Jog et al [17] Secondly, complicated discretization and solution routines adopted in Reference 10 can be avoided allowing for an easy extension from two dimensions (2Ds) to 3Ds. Thirdly, a diffuse-interface approach allows for an easy coupling of elastic and surface energy anisotropies.…”

Section: Introductionsupporting

confidence: 62%

“…This is also remarked in Schmidt and Gross, [14] although the algorithms of arriving at the interface solutions are different. Also, for the more generic case in the presence of interfacial energy anisotropy, cj should be replaced with r Á n; where n is the Cahn-Hoffmann zeta vector [54] as in Reference 10, and which can be also shown in the present phase-field model in the sharp-interface limit.…”

Section: ½55mentioning

confidence: 67%

“…In terms of computational efficiency, this is more costly than our model because our evolution equation concerns the update of the order parameter / whose variation is only over a finite length defined by the diffuse-interface region that is much smaller in extent compared to the corresponding solution of the composition field over the entire integration domain. Apart from this, there is also a physical difference which stems from the fact that our misfits and stiffness tensors are enslaved to the value of the order parameter /: This implies that the condition of diffusional and mechanical equilibrium may be decoupled as is the situation in the classical Johnson and Cahn model [9] or the numerical FEM models, [10,14,17,23,27] where the composition in the matrix and precipitate satisfies just the Laplace equation with the constraint of the same diffusion potential. However, for the diffuse-interface models that use the composition to parameterize the variation of the misfits and stiffness across the interface, the diffusional and mechanical equilibrium problems become coupled.…”

Section: ½55mentioning

confidence: 99%

“…The solution set for the coordinates thereby constitutes the equilibrium shape of the particle. The method has also been extended into three dimensions (3Ds) in the work by Thompson and Voorhees [12] and Li et al [13] Schmidt and Gross [14] use a boundary integral technique to perform the energy minimization in order to determine the equilibrium precipitate shapes in an elastically inhomogeneous and anisotropic medium, which is also used for studying multi-particle interactions in Reference 15. As an extension, Schmidt et al [16] have determined the 3D equilibrium precipitate morphologies using a boundary integral method.…”

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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Section: Introductionsupporting

confidence: 62%

Section: ½55mentioning

confidence: 67%

Section: ½55mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Recently, Schmidt and Gross [37,38,39], Orlikowski et al [36,31], Li and Chen [23,24], and Lee [18,19,17] have reported results for elastically anisotropic and inhomogeneous systems. Schmidt and Gross investigated the effect of inhomogeneity on the equilibrium shape and stability of a single precipitate (either in all of space or in a periodic box) in cubic anisotropic media.…”

Section: Introductionmentioning

confidence: 99%

Microstructural Evolution in Orthotropic Elastic Media

Leo

Lowengrub

Nie

2000

Journal of Computational Physics

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We consider the problem of microstructural evolution in binary alloys in two dimensions. The microstructure consists of arbitrarily shaped precipitates embedded in a matrix. Both the precipitates and the matrix are taken to be elastically anisotropic, with different elastic constants. The interfacial energy at the precipitatematrix interfaces is also taken to be anisotropic. This is an extension of the inhomogeneous isotrpic problem considered by H.-J. Jou et al. (1997, J. Comput. Phys. 131, 109). Evolution occurs via diffusion among the precipitates such that the total (elastic plus interfacial) energy decreases; this is accounted for by a modified Gibbs-Thomson boundary condition at the interfaces. The coupled diffusion and elasticity equations are reformulated using boundary integrals. An efficient preconditioner for the elasticity problem is developed based on a small scale analysis of the equations. The solution to the coupled elasticity-diffusion problem is implemented in parallel. Precipitate evolution is tracked by special non-stiff time stepping algorithms that guarantee agreement between physical and numerical equilibria. Results show that small elastic inhomogeneities in cubic systems can have a strong effect on precipitate evolution. For example, in systems where the elastic constants of the precipitates are smaller than those of the matrix, the particles move toward each other, where the rate of approach depends on the degree of inhomogeneity. Anisotropic surface energy can either enhance or reduce this effect, depending on the relative orientations of the anisotropies. Simulations of the evolution of multiple precipitates indicate that the elastic constants and surface energy control precipitate morphology and strongly influence nearest neighbor interactions. However, for the parameter ranges considered, the overall evolution of systems with large numbers of precipitates is primarily driven by the overall reduction in surface energy. Finally,

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Homogeneous Second‐Phase Precipitation

Wagner

Kampmann²,

Voorhees

2013

Materials Science and Technology

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The sections in this article are Introduction General Considerations General Course of an Isothermal Precipitation Reaction Thermodynamic Considerations–Metastability and Instability Decomposition Mechanisms: Nucleation and Growth versus Spinodal Decomposition Thermodynamic Driving Forces for Phase Separation Experimental Techniques for Studying Decomposition Kinetics Microanalytical Tools Direct Imaging Techniques Scattering Techniques Experimental Problems Influence of Quenching Rate on Kinetics Distinction of the Mode of Decomposition Precipitate Morphologies Experimental Results Factors Controlling the Shapes and Morphologies of Precipitates Early Stage Decomposition Kinetics Cluster‐Kinetics Approach Classical Nucleation–Sharp Interface Model Time‐Dependent Nucleation Rate Experimental Assessment of Classical Nucleation Theory Non‐Classical Nucleation–Diffuse Interface Model Distinction Between Classical and Non‐Classical Nucleation Diffusion‐Controlled Growth of Nuclei from the Supersaturated Matrix The Cluster‐Dynamics Approach to Generalized Nucleation Theory Spinodal Theories The Philosophy of Defining a ‘Spinodal Alloy' —Morphologies of ‘Spinodal Alloys' Monte Carlo Studies Coarsening of Precipitates General Remarks The LSW Theory of Coarsening Extensions of the Coarsening Theory to Finite Precipitate Volume Fractions Other Approaches Towards Coarsening Influence of Coherency Strains on the Mechanism and Kinetics of Coarsening–Particle Splitting Numerical Approaches Treating Nucleation, Growth and Coarsening as Concomitant Processes General Remarks on the Interpretation of Experimental Kinetic Data of Early Decomposition Stages The Langer and S chwartz Theory ( LS Model) and its Modification by K ampmann and W agner ( MLS Model) The Numerical Model ( N Model) of K ampmann and W agner ( KW ) Decomposition of a Homogeneous Solid Solution General Course of Decomposition Comparison Between the MLS Model and the N Model The Appearance and Experimental Identification of the Growth and Coarsening Stages Extraction of the Interfacial Energy and the Diffusion Constant from Experimental Data Decomposition Kinetics in Alloys Pre‐Decomposed During Quenching Influence of the Loss of Particle Coherency on the Precipitation Kinetics Combined Cluster‐Dynamic and Deterministic Description of Decomposition Kinetics Self‐Similarity, Dynamical Scaling and Power‐Law Approximations Dynamical Scaling Power‐Law Approximations Non‐Isothermal Precipitation Reactions Acknowledgements

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The equilibrium shape of an elastically inhomogeneous inclusion

Cited by 67 publications

References 18 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

Microstructural Evolution in Orthotropic Elastic Media

Homogeneous Second‐Phase Precipitation

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