Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

Zhao, Xujun; Bordas, Stéphane Pierre Alain; Qu, Jianmin

doi:10.1016/j.jmps.2015.04.008

Cited by 24 publications

(14 citation statements)

References 62 publications

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“…In particular, the γ ′ shape attracts extensive interest and has been widely studied for optimization of the mechanical properties of alloys. The γ ′ precipitates often evolve from an initial spherical shape to a cuboidal one, consistent with theoretical approaches indicating that precipitation shape strongly depends on particle size [11][12][13][14][15] . Many alloys with fcc crystal structure have an elastically soft <100> direction and a hard <111> direction.…”

Section: Introductionsupporting

confidence: 81%

Morphology Evolution of γ′ Precipitates during Isothermal Exposure in Wrought Ni-Based Superalloy Inconel X-750

2017

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The morphological evolution of γ ′ precipitates and lattice mis t with isothermal aging were closely investigated in wrought Ni-based superalloy Inconel X-750. The γ ′ morphology dramatically changes in terms of shape, distribution, coalescence and coherency at the γ/γ ′ interface. These processes and their dependence on temperature are summarized as a γ ′ morphology map together with a time-temperature-precipitation (TTP) diagram through quantifying relevant morphological parameters. The lattice mis t was measured by X-ray diffraction and is positive; it decreases from 0.6% at room temperature to 0.1% at the aging temperature. These results suggest that the morphological changes of the γ ′ precipitates are attributable to very low lattice mis t, the interaction of the elastic eld, the volume fraction of the precipitates and incoherence in γ/γ ′ interface.

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Section: Introductionsupporting

confidence: 81%

Morphology Evolution of γ′ Precipitates during Isothermal Exposure in Wrought Ni-Based Superalloy Inconel X-750

2017

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“…This is similar to the condition that is derived by Zhao et al, [23,24] wherein ðx b À x a Þ is nothing but the term n Á ðR b À R a Þ Á n; with R being the Eshelby momentum tensor ðf el I À r Á ruÞ [53] (u is the displacement field) that is used in the paper. Similarly, for the condition of equilibrium, v n is identically zero at all the interface points for which we have…”

Section: ½55supporting

confidence: 63%

“…The shape of the precipitate will be described by a fixed contour line of this function. 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.…”

Section: Introductionmentioning

confidence: 56%

“…Consequently, the mechanical equilibrium and the equilibration of the composition field may need to be performed in a coupled fashion along with the solution to the Allen-Cahn equation for the order parameter update under the constraint of constant volume. This is similar to the chemo-mechanical model level-set-based FEM approach proposed by Zhao et al, [24] which also incorporates interfacial stresses. Extension of our model along these lines particularly for addressing the situations of multi-component alloys is possible.…”

Section: ½55mentioning

confidence: 84%

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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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“…Mesoscale modeling, which examines materials at the nanometer to micron length scale, can improve understanding of superalloy microstructural evolution. Mesoscale studies have investigated both the equilibrium shapes of individual γ precipitates [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the coarsening behavior of multi-precipitate systems [25][26][27][28][29][30][31][32][33][34][35][36], with the former generally studied via sharp-interface models and the latter primarily via phase field models, though both approaches have been taken to study both problems. Studies on the equilibrium shapes of precipitates have been primarily 2D in nature [10-13, 15, 18-20, 23, 24], while fewer 3D studies have been performed [14,16,17,21,22].…”

Section: Introductionmentioning

confidence: 99%

Predicting the morphologies of γʹ precipitates in cobalt-based superalloys

et al. 2017

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Cobalt-based alloys with γ/γ microstructures have the potential to become the next generation of superalloys, but alloy compositions and processing steps must be optimized to improve coarsening, creep, and rafting behavior. While these behaviors are different than in nickel-based superalloys, alloy development can be accelerated by understanding the thermodynamic factors influencing microstructure evolution. In this work, we develop a phase field model informed by first-principles density functional theory and experimental data to predict the equilibrium shapes of Co-Al-W γ precipitates. Three-dimensional simulations of single and multiple precipitates are performed to understand the effect of elastic and interfacial energy on coarsened and rafted microstructures; the elastic energy is dependent on the elastic stiffnesses, misfit strain, precipitate size, applied stress, and precipitate spatial distribution. We observe characteristic microstructures dependent on the type of applied stress that have the same γ morphology and orientation seen in experiments, indicating that the elastic stresses arising from coherent γ/γ interfaces are important for morphological evolution during creep. The results also indicate that the narrow γ channels between γ precipitates are energetically favored, and provide an explanation for the experimentally observed directional coarsening that occurs without any applied stress.

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Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

Cited by 24 publications

References 62 publications

Morphology Evolution of γ′ Precipitates during Isothermal Exposure in Wrought Ni-Based Superalloy Inconel X-750

Morphology Evolution of γ′ Precipitates during Isothermal Exposure in Wrought Ni-Based Superalloy Inconel X-750

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

Predicting the morphologies of γʹ precipitates in cobalt-based superalloys

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