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.     

Proof of the Strong Eshelby Conjecture for Plane and Anti-plane Anisotropic Inclusion Problems

Based on the Stroh formalism for anisotropic elasticity and the complex variable function method, we prove in this paper that the strong Eshelby conjecture holds for simply-connected anisotropic inclusion problems under plane or anti-plane deformation. The interfaces can be either perfect or dislocation-like. For these inclusion problems, if the induced stress field inside the inclusion is uniform for a single uniform eigenstrain, the inclusion is of the elliptic shape. Thanks to the generality of the proof method, we obtain also alternative proofs of the strong Eshelby conjecture for isotropic inclusion problems, which are given in the Appendix.     

An irreducible polynomial functional basis of two-dimensional Eshelby tensors

The two-dimensional(2D) Eshelby tensors are discussed. Based upon the complex variable method, an integrity basis of ten isotropic invariants of the 2D Eshelby tensors is obtained. Since an integrity basis is always a polynomial functional basis, these ten isotropic invariants are further proven to form an irreducible polynomial functional basis of the 2D Eshelby tensors.     

含强约束界面片状夹杂复合陶瓷热膨胀系数预报

以含强约束界面相片状夹杂复合陶瓷的细观结构为基础,建立含片状夹杂、强约束界面相、基体氛围和有效介质组成的四相模型,将片状夹杂、强约束界面相和基体氛围构成的三相胞元看作复合夹杂,根据Eshelby理论,确定了含同向片状夹杂复合陶瓷的有效热膨胀系数的解析表达式,复合陶瓷为横观各向同性,有2个独立的热膨胀系数.定量分析表明含同向片状夹杂的强约束界面复合陶瓷的有效热膨胀系数具有明显的尺度效应.     

复合材料动态粘弹性能的细观研究

利用细观力学的Eshelby等效夹杂方法研究了颗粒增强复合材料的动态粘弹性力学性能,分析了材料复模量随夹杂体积分数、载荷频率之间的变化规律,给出了许多有意义的结论,为复合材料结构的优化设计及应用提供了理论基础。     

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.     

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.     

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.     

含圆形夹杂的无穷大平面电致伸缩材料的应力分析

利用电致伸缩基本方程,采用伪总应力和复变函数解法,并利用级数展开方法得出了含圆形夹杂的无限大电致伸缩材料应力场,在一般情况下,与Eshelby夹杂理论不同,在电致伸缩材料圆形夹杂内部应力场是非均匀的．     

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.     

On the Nonlinear Continuum Theory of Dislocations: A Gauge Field Theoretical Approach

A nonlinear continuum theory of material bodies with continuously distributed dislocations is presented, based on a gauge theoretical approach. Firstly, we derive the canonical conservation laws that correspond to the group of translations and rotations in the material space using Noether’s theorem. These equations give us the canonical Eshelby stress tensor as well as the total canonical angular momentum tensor. The canonical Eshelby stress tensor is neither symmetric nor gauge-invariant. Based on the Belinfante-Rosenfeld procedure, we obtain the gauge-invariant Eshelby stress tensor which can be symmetric relative to the reference configuration only for isotropic materials. The gauge-invariant angular momentum tensor is obtained as well. The decomposition of the gauge-invariant Eshelby stress tensor in an elastic and in a dislocation part gives rise to the derivation of the famous Peach-Koehler force.     

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.     

Stress fields and Eshelby forces on half-plane inhomogeneities with eigenstrains and strip inclusions meeting a free surface

The stress field due to a half-plane inhomogeneity with plane eigenstrain is obtained by a limiting procedure from the one of a circular Eshelby inhomogeneity/inclusion. This field, which requires tractions to be applied at infinity to be sustained, has minimum strain energy versus any other superposed homogeneous one, and is the Eshelby solution inside plus the Hill jump conditions. By superposition, the stresses due to an infinite strip (Eshelby property domain) inhomogeneity with eigenstrain are obtained, and, by superposition periodic strips or laminates can be obtained. By cancelling the stresses on a free-surface, strips of inclusions meeting a free surface are solved. They exhibit tensile stresses under the free surface, and logarithmic singularities in the tensile stress at the vertex, which may initiate cracking. The Eshelby self-forces on the boundary of circular and half-plane inhomogeneities are computed.     

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.     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

Closed form solution and numerical analysis for Eshelby’s elliptic inclusion in plane elasticity

This paper presents a closed form solution and numerical analysis for Es- helby＇s elliptic inclusion in an infinite plate. The complex variable method and the confor- real mapping technique are used. The continuity conditions for the traction and displace- ment along the interface in the physical plane are reduced to the similar conditions along the unit circle of the mapping plane. The properties of the complex potentials defined in the finite elliptic region are analyzed. From the continuity conditions, one can separate and obtain the relevant complex potentials defined in the inclusion and the matrix. From the obtained complex potentials, the dependence of the real strains and stresses in the inclusion from the assumed eigenstrains is evaluated. In addition, the stress distribution on the interface along the matrix side is evaluated. The results are obtained in the paper for the first time.     

SMA短纤维增强复合材料在热载下的相交特性

在形状记忆合金（ＳＭＡ）复合材料研究中,相变特性的研究是一个主要的工作．基于Ｅｓｈｅｌｂｙ的等效夹杂模型和Ｍｏｒｉ和Ｔａｎａｋａ的场平均法,考虑到ＳＭＡ材料的强物理非线性,发展了增量型的等效夹杂模型（ＩｎｃｒｅｍｅｎｔａｌＥｑｕｉｖａｌｅｎｔＩｎｃｌｕｓｉｏｎＭｏｄｅｌ）．考虑在某一温度循环条件下讨论形状记忆合金短纤维增强的铝基复合材料在热载下的相变行为．特别研究了ＳＭＡ短纤维复合材料在变温过程中纤维几何尺寸、体积分数等参数对ＳＭＡ复合材料的相变行为和ＳＭＡ内残余应力等的影响．这些工作对于指导材料设计和了解ＳＭＡ复合材料热机械特性是颇有意义的．     

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.     

EFFECTIVE SPECIFIC HEATS OF MULTI-PHASE THERMOELASTIC COMPOSITES

This paper studies the effective properties of multi-phase thermoelastic composites. Based on the Helmholtz free energy and the Gibbs free energy of individual phases, the effective elastic tensor, thermal-expansion tensor, and specific heats of the multi-phase composites are derived by means of the volume average of free-energies of these phases. Particular emphasis is placed on the derivation of new analytical expressions of effective specific heats at constant-strain and constant-stress situations, in which a modified Eshelby’s micromechanics theory is developed and the interaction between inclusions is considered. As an illustrative example, the analytical expression of the effective specific heat for a three-phase thermoelastic composite is presented.