Eshelby's problem of non-elliptical inclusions

Zou, Wenming; He, Qi‐Chang; Huang, Mojia; Zheng, Quanan

doi:10.1016/j.jmps.2009.11.008

Cited by 125 publications

(76 citation statements)

References 31 publications

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“…Eshelby's conjecture on validity of his proposed method for only ellipsoidal inclusions has been addressed in Refs. [81][82][83][84][85][86][87][88]. However, neglecting the interaction of particles is an unrealistic assumption of Eshelby for materials with randomly dispersed particulate microstructure, even at a few percent volume fraction [89].…”

Section: Introductionmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

185

138

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The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill-Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE 2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two-and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.

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Section: Introductionmentioning

confidence: 99%

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Saeb

Steinmann

Javili

2016

Applied Mechanics Reviews

185

138

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“…The case of the so-called supersphere was considered in [20], which built on the work in [21][22][23]. For the case of the Newtonian potential problem and planar elastostatics, some analytical expressions have been derived for twodimensional problems where inhomogeneities are either polygonal or their shape can be described by finite Laurent expansions [24,25]. Further properties of the Eshelby tensor were deduced in [26,27], including the relationship of the averaged Eshelby tensor for non-ellipsoidal inhomogeneities to their ellipsoidal counterparts.…”

Section: Introductionmentioning

confidence: 99%

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Joyce

Parnell

2017

J Eng Math

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Understanding the fields that are set up in and around inhomogeneities is of great importance in order to predict the manner in which heterogeneous media behave when subjected to applied loads or other fields, e.g., magnetic, electric, thermal, etc. The classical inhomogeneity problem of an ellipsoid embedded in an unbounded host or matrix medium has long been studied but is perhaps most associated with the name of Eshelby due to his seminal work in 1957, where in the context of the linear elasticity problem, he showed that for imposed far fields that correspond to uniform strains, the strain field induced inside the ellipsoid is also uniform. In Eshelby's language, this corresponds to requiring a uniform eigenstrain in order to account for the presence of the ellipsoidal inhomogeneity, and the so-called Eshelby tensor arises, which is also uniform for ellipsoids. Since then, the Eshelby tensor has been determined by many authors for inhomogeneities of various shapes, but almost always for the case of uniform eigenstrains. In many application areas in fact, the case of non-uniform eigenstrains is of more physical significance, particularly when the inhomogeneity is non-ellipsoidal. In this article, a method is introduced, which approximates the Eshelby tensor for a variety of shaped inhomogeneities in the case of more complex eigenstrains by employing local polynomial expansions of both the eigenstrain and the resulting Eshelby tensor, in the case of the potential problem in two dimensions.

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“…Moreover, Taya (1997, 2000) highlight that the Eshelby's tensor at the centre and the averaged Eshelby's tensor over a polygonal inclusion are equal to that of a circular inclusion whatever the orientation of the inclusion. Using the irreducible decomposition of the Eshelby's tensor by Zheng et al (2006), Zou et al (2010) derive explicit expressions of the Eshelby's Tensor Field (ETF) and its average for a wide variety of non-elliptical inclusions. They formulate some remarks about the elliptical approximation to the average of ETF which is valid for a convex non-elliptical inclusion but becomes unacceptable for a non-convex non-elliptical inclusion.…”

Section: Introductionmentioning

confidence: 99%

“…They formulate some remarks about the elliptical approximation to the average of ETF which is valid for a convex non-elliptical inclusion but becomes unacceptable for a non-convex non-elliptical inclusion. Based on the results of Zou et al (2010) mainly the averaged Eshelby's tensor, Klusemann et al (2012) has investigated the effective responses of composites consisting of non-elliptical shape in the context of several homogenisation methods.…”

Section: Introductionmentioning

confidence: 99%

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Non-linear elastic moduli of Graphene sheet-reinforced polymer composites

Elmarakbi

Wang

Azoti

2016

International Journal of Solids and Structures

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The non-linear elastic moduli of the Graphene sheet-reinforced polymer composite are investigated using a combined molecular mechanics theory and continuum homogenisation tools. Under uniaxial loading, the linear and non-linear constitutive equations of the Graphene sheet are derived from a Taylor series expansion in powers of strains. Based on the modified Morse potential, the elastic moduli and Poisson's ratio are obtained for the Graphene sheet leading to the derivation of the non-linear stiffness tensor. For homogenisation purpose, the strain concentration tensor is computed by the means of the irreducible decomposition of the Eshelby's tensor for an arbitrary domain. Therefore, a mathematical expression of the averaged Eshelby's tensor for a rectangular shape is obtained for the Graphene sheet. Under the Mori-Tanaka micro-mechanics scheme, the effective non-linear behaviour is predicted for various micro-parameters such as the aspect ratio and mass fractions. Numerical results highlight the effect of such micro-parameters on the anisotropic degree of the composite.

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Eshelby's problem of non-elliptical inclusions

Cited by 125 publications

References 31 publications

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

Non-linear elastic moduli of Graphene sheet-reinforced polymer composites

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