Eshelby’s circular cylindrical inclusion with polynomial eigenstrains in transverse direction by residue theorem

Yu, X.-W.; Wang, Z.-W.; Wang, H.; Ningyi, Leng,

doi:10.1007/s00419-020-01831-y

Cited by 6 publications

(5 citation statements)

References 26 publications

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“…The elastostatic stress tensor component 𝜎 xx inside of the cylindrical region with eigenstrains 𝜖 00 xx and 𝜖 00 yy , surrounded by elastic medium, is given by Equation ( 26). [42] 𝜎…”

Section: Numerical Experiments 2: Elastic Deformation Of Materials Su...mentioning

confidence: 99%

“…The elastostatic stress tensor component

\sigma _{xx}

inside of the cylindrical region with eigenstrains

\epsilon _{xx}^{00}

and

\epsilon _{yy}^{00}

, surrounded by elastic medium, is given by Equation (26). [ 42 ]

\begin{equation} \sigma _{xx}=-\frac{E}{2(1+v)}{\left(\frac{3}{4(1-v)}\epsilon _{xx}^{00}+\frac{1}{4(1-v)}\epsilon _{yy}^{00} \right)} \end{equation}

…”

Section: Numerical Experimentsmentioning

confidence: 99%

See 1 more Smart Citation

Full Field Model Describing Phase Front Propagation, Transformation Strains, Chemical Partitioning, and Diffusion in Solid–Solid Phase Transformations

Pohjonen

2023

Advcd Theory and Sims

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A novel mathematical formulation is presented for describing growth of phase in solid-to-solid phase transformations and it is applied for describing austenite to ferrite transformation. The formulation includes the effects of transformation eigenstrains, the local strains, as well as partitioning and diffusion. In the current approach the phase front is modeled as diffuse field, and its propagation is shown to be described by the advection equation, which reduces to the level-set equation when the transformation proceeds only to the interface normal direction. The propagation is considered as thermally activated process in the same way as in chemical reaction kinetics. In addition, connection to the Allen-Cahn equation is made. Numerical tests are conducted to check the mathematical model validity and to compare the current approach to sharp interface partitioning and diffusion model. The model operation is tested in isotropic 2D plane strain condition for austenite to ferrite transformation, where the transformation produces isotropic expansion, and also for austenite to bainite transformation, where the transformation causes invariant plane strain condition. Growth into surrounding isotropic austenite, as well as growth of the phase which has nucleated on a grain boundary are tested for both ferrite and bainite formation.

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Section: Numerical Experiments 2: Elastic Deformation Of Materials Su...mentioning

confidence: 99%

“…The elastostatic stress tensor component

\sigma _{xx}

inside of the cylindrical region with eigenstrains

\epsilon _{xx}^{00}

and

\epsilon _{yy}^{00}

, surrounded by elastic medium, is given by Equation (26). [ 42 ]

\begin{equation} \sigma _{xx}=-\frac{E}{2(1+v)}{\left(\frac{3}{4(1-v)}\epsilon _{xx}^{00}+\frac{1}{4(1-v)}\epsilon _{yy}^{00} \right)} \end{equation}

…”

Section: Numerical Experimentsmentioning

confidence: 99%

Full Field Model Describing Phase Front Propagation, Transformation Strains, Chemical Partitioning, and Diffusion in Solid–Solid Phase Transformations

Pohjonen

2023

Advcd Theory and Sims

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show abstract

“…A plane strain condition is assumed since the geometrical model is thick in the longitudinal direction of fibers [31]. For two-dimensional problems, the stress and strain fields of a circular inclusion with polynomial eigenstrains can be evaluated by residue theorem [32].…”

Section: Circular Inhomogeneitiesmentioning

confidence: 99%

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method

Wang

2021

Mathematics and Mechanics of Solids

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Traditional equivalent inclusion method provides unreliable predictions of the stress concentrations of two spherical inhomogeneities with small separation distance. This paper determines the stress and strain fields of multiple ellipsoidal/elliptical inhomogeneities by equivalent inhomogeneous inclusion method. Equivalent inhomogeneous inclusion method is an inverse of equivalent inclusion method and substitutes the subdomains of matrix with known strains by equivalent inhomogeneous inclusions. The stress and strain fields of multiple inhomogeneities are decomposed into the superposition of matrix under applied load and each solitary inhomogeneous inclusion with polynomial eigenstrains by the iteration of equivalent inhomogeneous inclusion method. Multiple circular and spherical inhomogeneities are respectively used as examples and examined by the finite element method. The stress concentrations of multiple inhomogeneities with small separation distances are well predicted by equivalent inhomogeneous inclusion method and the accuracies improve with the increase of eigenstrain orders. Equivalent inhomogeneous inclusion method gives more accurate stress predictions than equivalent inclusion method in the problem of two spherical inhomogeneities.

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“…Generally, one can observe that research on the behavior of transforming inclusions is still in progress; see, e.g., the recent work on circular (cylindrical) inclusions with a non-uniform eigenstrain [8,9] in this journal. In the present study, particle shapes are represented by ellipsoids.…”

Section: Introductionmentioning

confidence: 99%

Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

et al. 2022

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Thermodynamic modeling of the development of non-spherical inclusions as precipitates in alloys is an important topic in computational materials science. The precipitates may have markedly different properties compared to the matrix. Both the elastic contrast and the misfit eigenstrain may yield a remarkable generation of elastic strain energy which immediately influences the kinetics of the developing precipitates. The relevant thermodynamic framework has been mostly based on spherical precipitates. However, the shapes of actual particles are often not spherical. The energetics of such precipitates can be met by adapting the spherical energy terms with shape factors. The well-established Eshelby framework is used to evaluate the elastic strain energy of inclusions with ellipsoidal shapes (described by the axes a, b, and c) that are subjected to a volumetric transformation strain. The outcome of the study is two shape factors, one for the elastic strain energy and the other for the interface energy. Both quantities are provided in the form of easy-to-use diagrams. Furthermore, threshold elastic contrasts yielding strain energy shape factors with the value 1.0 for any ellipsoidal shape are studied.

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Eshelby’s circular cylindrical inclusion with polynomial eigenstrains in transverse direction by residue theorem

Cited by 6 publications

References 26 publications

Full Field Model Describing Phase Front Propagation, Transformation Strains, Chemical Partitioning, and Diffusion in Solid–Solid Phase Transformations

Full Field Model Describing Phase Front Propagation, Transformation Strains, Chemical Partitioning, and Diffusion in Solid–Solid Phase Transformations

Multiple ellipsoidal/elliptical inhomogeneities embedded in infinite matrix by equivalent inhomogeneous inclusion method

Strain and interface energy of ellipsoidal inclusions subjected to volumetric eigenstrains: shape factors

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