Solutions to the Pólya–Szegö Conjecture and the Weak Eshelby Conjecture

Eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform. He then conjectured that the converse is true, that is, that if the field inside an inclusion is uniform for all uniform loadings, then the inclusion is of elliptic or ellipsoidal shape. We call this the weak Eshelby conjecture. In this paper we prove this conjecture in three dimensions. In two dimensions, a stronger conjecture, which we call the strong Eshelby conjecture, has been proved: if the field inside an inclusion is uniform for a single uniform loading, then the inclusion is of elliptic shape. We give an alternative proof of Eshelby’s conjecture in two dimensions using a hodographic transformation. As a consequence of the weak Eshelby’s conjecture, we prove in two and three dimensions a conjecture of Pólya and Szegö on the isoperimetric inequalities for the polarization tensors (PTs). The Pólya–Szegö conjecture asserts that the inclusion whose electrical PT has the minimal trace takes the shape of a disk or a ball.     

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

Consider an infinite thermally conductive medium characterized by Fourier’s law, in which a subdomain, called an inclusion, is subjected to a prescribed uniform heat flux-free temperature gradient. The second-order tensor field relating the gradient of the resulting temperature field over the medium to the uniform heat flux-free temperature gradient is referred to as Eshelby’s tensor field for conduction. The present work aims at deriving the general properties of Eshelby’s tensor field for conduction. It is found that: (i) the trace of Eshelby’s tensor field is equal to the characteristic function of the inclusion, independently of the latter’s shape; (ii) the isotropic part of Eshelby’s tensor field over the inclusion of arbitrary shape is identical to Eshelby’s tensor field over a 2D circular or 3D spherical inclusion; (iii) when the medium is made of an isotropic material and when the inclusion has some specific rotational symmetries, the value of the Eshelby’s tensor field evaluated at the inclusion gravity center and the symmetric average of Eshelby’s tensor fields are both equal to Eshelby’s tensor field for a 2D circular or 3D spherical inclusion. These results are then extended, with the help of a linear transformation, to the general case where the medium consists of an anisotropic conductive material. The method elaborated and results obtained by the present work are directly transposable to the physically analogous transport phenomena of electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.     

The determination of the elastic field of a polygonal star shaped inclusion

Recently we found that the elastic field is uniform in a pentagonal star (five-pointed star inclusion) [1], and in a triangular inclusion [2], when an eigenstrain is distributed uniformly in these inclusions. This result is similar to the famous result of Eshelby (1957) that the elastic field is uniform in an ellipsoidal inclusion in an infinitely body when an eigenstrain is distributed uniformly in the ellipsoidal inclusion. We also found that for a Jewish star (Star of David or six points star) or a rectangular inclusion subjected to a uniform eigenstrain, the stress field is not uniform in these inclusions. These results also hold for two dimensional plane strain cases. Furthermore these analytical results are confirmed experimentally by photoelasticity method. In this paper, we investigate a more general inclusion of an m-pointed polygonal inclusion subjected to the uniform eigenstrain. We conclude that the stress field is uniform when m is odd number. This conclusion agrees with the speculation made by B. Boley after the author's talk at Shizuoka [2].     

Eshelby tensors for an ellipsoidal inclusion in a microstretch material

Eshelby tensors for an ellipsoidal inclusion in a microstretch material are derived in analytical form, involving only one-dimensional integral. As micropolar Eshelby tensor, the microstretch Eshelby tensors are not uniform inside of the ellipsoidal inclusion. However, different from micropolar Eshelby tensor, it is found that when the size of inclusion is large compared to the characteristic length of microstretch material, the microstretch Eshelby tensor cannot be reduced to the corresponding classical one. The reason for this is analyzed in details. It is found that under a pure hydrostatic loading, the bulk modulus of a microstretch material is not the same as the one in the corresponding classical material. A modified bulk modulus for the microstretch material is proposed, the microstretch Eshelby tensor is shown to be reduced to the modified classical Eshelby tensor at large size limit of inclusion. The fully analytical expressions of microstretch Eshelby tensors for a cylindrical inclusion are also derived.     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

Eshelby’s problem in an anisotropic multiferroic bimaterial plane

We solve analytically the Eshelby’s problem in an anisotropic multiferroic bimaterial plane. The solution is based on the extended Stroh formalism of complex variables, and is valid for the inclusion of arbitrary shapes, described by a Laurent polynomial, a polygon, or the one bounded by a Jordan curve. Furthermore, the results in the corresponding half plane and full plane can be reduced directly from the bimaterial-plane solution. As such, the solution unifies the complex variable method and the Green’s function method, extending further to the multiferroic bimaterial plane of general anisotropy. The essential eigenfunctions are also identified by which the induced fields can be simply determined. Numerical results are presented to investigate the features of these eigenfunctions as well as the strain, electric and magnetic fields (components of the extended Eshelby tensor). Particularly, we present the values of these fields at the center of the N-side regular polygonal inclusion and also the average values of these fields over the inclusion area. The effect of the half-plane traction-free surface condition as well as the effect of various couplings on the induced fields is discussed in detail. For the N-side regular polygonal inclusion, it is found that, when the inclusion is in the full plane, both the center and average values of the Eshelby tensor are independent of the side number N, except for N = 4. We further show that the piezoelectric and piezomagnetic coupling coefficients could significantly affect the Eshelby tensor. These features should be useful in controlling the Eshelby tensor for the design of better multiferroic composites. Typical contours of the field quantities in and around the inclusion bounded by both straight and curved line segments in a multiferroic bimaterial plane are also presented.     

On the solution of the dynamic Eshelby problem for inclusions of various shapes

In many dynamic applications of theoretical physics, for instance in electrodynamics, elastodynamics, and materials sciences (dynamic variant of Eshelby’s inclusion and inhomogeneity problems) the solution of the inhomogeneous Helmholtz equation (‘dynamic’ or Helmholtz potential) plays a crucial role. In materials sciences from such a solution the dynamical fields due to harmonically transforming eigenfields can be constructed. In contrast to the static Eshelby’s inclusion problem (Eshelby, 1957), due to its mathematical complexity, the dynamic variant of the problem is comparably little touched. Only for a restricted set of cases, namely for ellipsoidal, spheroidal and continuous fiber-inclusions, analytical approaches exist. For ellipsoidal shells we derive a 1D integral representation of the Helmholtz potential which is useful to be extended to inhomogeneous ellipsoidal source regions. We determine the dynamic potential and dynamic variant of the Eshelby tensor for arbitrary source densities and distributions by employing a numerical technique based on Gauss quadrature. We study a series of examples of Eshelby problems which are of interest for applications in materials sciences, such as for instance cubic and prismatic inclusions. The method is especially useful to be applied in self-consistent methods (e.g. the effective field method) if one looks for the effective dynamic characteristics of the material containing a random set of inclusions.     

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation 2

The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

The Eshelby property of sliding inclusions

The present paper solves the problem of sliding ellipsoidal/elliptical inclusion with the Eshelby property. Results show that the sliding ellipsoidal/elliptical inclusions can have uniform eigenstresses if the prescribed uniform eigenstrains fulfill certain prerequisites. Solutions and prerequisites are obtained for ellipsoidal and elliptical inclusions, respectively. It is shown that the eligible uniform eigenstrains inducing uniform eigenstresses can be shear or non-shear eigenstrains, depending on the geometric shape and the material constants of the inclusion. The study indicates that inclusions of degenerated form, like spheroids, spheres and circles, may also maintain uniform eigenstresses. At last, the corresponding discussion for the inhomogeneous sliding inclusion problem with both eigenstrain and remote loading is also given.     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

The second Eshelby problem and its solvability

It is still a challenge to clarify the dependence of overall elastic properties of heterogeneous materials on the microstructures of non-elliposodal inhomogeneities (cracks, pores, foreign particles). From the theory of elasticity, the formulation of the perturbance elastic fields, coming from a non-ellipsoidal inhomogeneity embedded in an infinitely extended material with remote constant loading, inevitably involve one or more integral equations. Up to now, due to the mathematical difficulty, there is almost no explicit analytical solution obtained except for the ellipsoidal inhomogeneity. In this paper, we point out the impossibility to transform this inhomogeneity problem into a conventional Eshelby problem by the equivalent inclusion method even if the eigenstrain is chosen to be non-uniform. We also build up an equivalent model, called the second Eshelby problem, to investigate the perturbance stress. It is probably a better template to make use of the profound methods and results of conventional Eshelby problems of non-ellipsoidal inclusions.     

The Arithmetic Mean Theorem of Eshelby Tensor for a Rotational Symmetrical Inclusion

In 1997, H. Nozaki and M. Taya found numerically that for any regular polygonal inclusion except for a square, both the Eshelby tensor at the center and the average Eshelby tensor over the inclusion domain are equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Then in 2001, these remarkable properties were mathematically justified by Kawashita and Nozaki. In this paper, a more radical property is presented for a rotational symmetrical inclusion: For any N-fold (N is an integer greater than 2 and unequal to 4) rotational symmetrical inclusion, the arithmetic mean of the Eshelby tensors at N rotational symmetrical points in the inclusion is the same as the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. It follows that the Eshelby tensor at the center and the average Eshelby tensor over the rotational symmetrical inclusion domain are identical to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion as well. This paper shows that although the Eshelby property does not hold for non-ellipsoidal inclusions, the Eshelby tensor for a rotational symmetrical inclusion satisfies the arithmetic mean property. Mathematics Subject Classifications (2000) 73C02.     

Solutions of inhomogeneity problems with graded shells and application to core-shell nanoparticles and composites

This paper first presents the Eshelby tensors and stress concentration tensors for a spherical inhomogeneity with a graded shell embedded in an alien infinite matrix. The solution is then specialized to inhomogeneous inclusions in finite spherical domains with fixed displacement or traction-free boundary conditions. The Eshelby tensors in the infinite and finite domains and the stress concentration tensors are especially useful for solving many problems in mechanics and materials science. This is demonstrated on two examples. In the first example, the strain distributions in core-shell nanoparticles with eigenstrains induced by lattice mismatches are calculated using the Eshelby tensors in the finite domains. In the second example, the Eshelby and stress concentration tensors in the three-phase configuration are used to formulate the generalized self-consistent prediction of the effective moduli of composites containing spherical particles within the framework of the equivalent inclusion method. The advantage of this micromechanical scheme is that, whilst its predictions are almost identical to the classical generalized self-consistent method and the third-order approximation, the expressions for the effective moduli have simple closed forms.     

Circular inclusions in anti-plane strain couple stress elasticity

The solution for a circular inclusion with a prescribed anti-plane eigenstrain is derived. It is shown that the components of the Eshelby tensor within the inclusion, corresponding to a uniform eigenstrain, can be either uniform or non-uniform, depending on the imposed interface conditions. The stress amplification factors due to circular void or rigid inclusion in an infinite medium under remote anti-plane shear stress are calculated. The failure of the couple stress elasticity to reproduce the classical elasticity solution in the limit of vanishingly small characteristic length is indicated for a particular type of boundary conditions. The solution for a circular inhomogeneity in an infinitely extended matrix subjected to remote shear stress is then derived. The effects of the imposed interface conditions, the shear stress and couple stress discontinuities, and the relationship between the inhomogeneity and its equivalent eigenstrain inclusion problem are discussed.     

Eshelby tensor construction for a weakly anisotropic elastic medium

A method of constructing the interior Eshelby tensor for a weakly anisotropic elastic medium is proposed.     

On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).