## Abstract

The method of matched asymptotic expansions is applied to construct an asymptotic model for the Eshelby inhomogeneity problem in the case of a slender sufficiently rigid inhomogeneity. On the basis of the obtained asymptotic model, it is shown that the only infinitesimal perturbations of the elongated ellipsoid that preserve constancy of stresses inside the inhomogeneity are those into another elongated ellipsoid.

## 1. Introduction

### (a) Eshelby property

Eshelby's problem comprises two main formulations: the inclusion problem and the inhomogeneity problem (Kachanov *et al*. 2003). The inclusion problem can be formulated as follows. A given region *Q* (called inclusion) in an infinite linear elastic medium undergoes a prescribed eigenstrain (due to inelastic deformation, thermal expansion, etc.). Region *Q* is then elastically deformed into the initial shape and is inserted back. Of interest are the resulting elastic fields inside and outside *Q*.

Eshelby (1957) has proved that the strain field within a homogeneous ellipsoidal inclusion in an infinite isotropic elastic space is uniform, if the eigenstrain prescribed in the inclusion is uniform. Eshelby also stated that this property, which is now commonly referred to as the Eshelby property, does not hold for non-ellipsoidal inclusions.

Eshelby's inhomogeneity problem is formulated as follows. A given region *Q* (called inhomogeneity) in an infinite linear elastic space has elastic properties that are different from the surroundings. Of interest are the elastic fields inside and outside *Q* generated by remotely applied stresses.

The Eshelby property in the inclusion problem was investigated by many authors. Rodin (1996) showed that polyhedral inclusions cannot have the Eshelby property. Lubarda & Markenscoff (1998) proved that the Eshelby property does not hold for any inclusion bounded by a polynomial surface higher than the second degree as well as for any inclusion bounded by a non-convex surface. Using a geometric approach, Markenscoff (1998) showed that the class of shapes that may sustain constant eigenstresses form a nine-dimensional manifold embedded in the space of all possible shapes. She also proved that the only perturbations of the ellipsoid that maintain the Eshelby property are those that perturb the ellipsoid into another ellipsoid. However, the question of whether Eshelby's conjecture holds still remains unanswered.

Ru & Schiavone (1996) considered the anti-plane shear problem for an inhomogeneity embedded in an elastic medium, subjected at infinity to a uniform stress field. They proved that the state of deformation in the inclusion is a simple shear if and only if the curve enclosing the inclusion is an ellipse. In some recent works (Kawashita & Nozaki 2001; Wang & Xu 2004; Xu & Wang 2005), some special properties have been found for certain rotational symmetrical inclusions. Note that the non-ellipsoidal inclusion and inhomogeneity problems (Ru 2000; Pan 2004) have become increasingly important in recent years in view of industrial applications of non-ellipsoidal reinforcements in modern composite materials (Andrianov *et al*. 2002).

### (b) Statement of the inhomogeneity problem

In this paper, we discuss the issue of Eshelby's conjecture from the standpoint of asymptotic modelling. We consider an arbitrary slender inhomogeneity *Q*_{ϵ} embedded in an isotropic elastic space.

Let *ω*(*z*) be a family of domains on the plane ^{2} containing the origin and bounded by simple smooth closed contours ∂*ω*(*z*). Let *ϵ* be a small dimensionless parameter. Introducing a slender domain *Q*_{ϵ} with the varying cross section *ω*_{ϵ}(*z*), we put

We denote the Lamé constants of the matrix and the inhomogeneity *Q*_{ϵ} by *λ*, *μ* and *λ*_{j}, *μ*_{j}, respectively.

In space ^{3}, we consider the Lamé system(1.1)and(1.2)with the boundary conditions(1.3)and(1.4)and the following asymptotic condition at infinity:(1.5)Here, ** u**(

*ϵ*,

**) and**

*x*

*u*^{j}(

*ϵ*,

**) are the displacement vectors inside and outside**

*x**Q*

_{ϵ}, respectively, and

*v*^{0}(

**) is the displacement field generated by remotely applied stresses.**

*x*We consider the special case of stresses *σ* acting along the *Ox*_{3} axis, i.e.(1.6)where *E* and *ν* are Young's modulus and Poisson's ratio of the matrix, respectively.

It is known that the stress–strain state of a fibre-reinforced composite structure and the asymptotic ansatz primarily depend on the relation between the elastic moduli of matrix, *E*, and inhomogeneity, *E*_{j}. For example, if the matrix is relatively pliable, the inhomogeneity can be assumed undeformable in the first-order approximation, and the stress–strain state of the entire composite structure can be analysed stage by stage. At the first and second stages, deformation of the matrix and the inhomogeneity are determined, respectively. It was shown by Argatov & Nazarov (1993) that the so-called resultant problem (asymptotic model) does not allow any asymptotic decomposition if the elastic inhomogeneity is sufficiently rigid.

Thus, we consider the special case when(1.7)where does not depend on the parameter *ϵ*.

We note that the relation (1.7) is very restrictive; however, all results of the subsequent analysis will be true in a more general case of a rigid inhomogeneity when . In the case of a pliable inhomogeneity when as *ϵ*→0, the asymptotic solution should be constructed in another way.

Let us construct an approximate (but asymptotically exact as *ϵ*→0) solution of problem (1.1)–(1.5) in the case (1.6) under the condition (1.7). Using the obtained asymptotic model, we will show that the only infinitesimal perturbations of the elongated ellipsoid that preserve constancy of stresses inside the inhomogeneity are those into another elongated elliptical ellipsoid.

## 2. Asymptotic modelling

### (a) Outer asymptotic representation of the displacement field in the matrix

Let *T*^{(3)}(** x**) be a solution of the Thomson problem of an elastic space subjected to a unit point force applied at the origin of coordinates and directed along the

*Ox*

_{3}axis. We have (e.g. Kachanov

*et al*. 2003)(2.1)where

*δ*

_{i,k}is Kronecker's delta.

The displacement vector field of an elastic space subjected to the remotely applied stresses and the concentrated loads distributed along the axis of the inhomogeneity *Q*_{ϵ} with a specified line density *p*(*s*) is given by the following formula:(2.2)

It is easy to see that, due to singularities in the kernel (2.1), we may not simply put ** y**=0 in the integral (2.2).

We assume that the function *p*(*z*) is differentiable on the interval [−*l*,*l*]. Hence, it is not hard to show that, as , we will have(2.3)and(2.4)where(2.5)

Note that detailed investigations of the remainders in asymptotic relations (2.3) and (2.4) can be found in the works of Handelsman & Keller (1967*a*,*b*), Kalker (1977) and Fedoryuk (1987).

### (b) Inner asymptotic representation of the displacement field in the matrix

In the neighbourhood of the inhomogeneity *Q*_{ϵ}, we introduce the ‘stretched’ variables(2.6)

The inner asymptotic representation of the solution ** u**(

*ϵ*,

**) to the original problem has the form(2.7)**

*x*Substituting the stretched variables (2.6) into the vector equation (1.1) and retaining the leading asymptotic terms, we obtain the following scalar equation:(2.8)

In the case of the relatively rigid inhomogeneity *Q*_{ϵ}, we have the boundary condition(2.9)where *w*(*z*) is the principal term of the asymptotic expansion of in *Q*_{ϵ}.

Within the framework of the method of matched asymptotic expansions, formulae (2.8) and (2.9) should be supplemented by the conditions characterizing the behaviour of the function *V*_{3}(** η**;

*z*) at infinity. Thus, according to (2.2), (2.3) and (2.4), we obtain the following asymptotic condition:(2.10)where

*α*=1/2(1−

*ν*).

Let *ζ*=*f*(*Z*), *Z*=*η*_{1}+i*η*_{2}, be a conformal mapping that transfers the region into the exterior of the unit circle, i.e. |*ζ*|>1. Then, the function *G*_{∞}(*z*, ** η**), called Green's function with a pole at infinity, vanishes on ∂

*ω*(

*z*) and admits the asymptotic representation(2.11)where

*R*(

*z*) is the outer conformal radius of the set (Landkof 1972).

In addition to (2.11), we havewhere *ν* is the unit outer normal to ∂*ω*(*z*).

We put(2.12)

Observe that the matrix shear stresses on the boundary ∂*Q*_{ϵ} are determined by the formulae

Finally, taking into account formulae (2.10)–(2.12), as a result of the matching procedure, we derive the following integral equation:(2.13)where the integral operator ** J** is defined by formula (2.5).

### (c) Asymptotic representation of the displacement field in the inhomogeneity

Following the algorithm developed by Gol'denveizer (1962) for the asymptotic analysis of elliptic problems in thin domains, we take the asymptotic representation of the displacement vector *u*^{j}(*ϵ*, ** x**) in

*Q*

_{ϵ}in the following form:

The function *w*(*z*) must satisfy the differential equation(2.14)with the boundary conditions(2.15)Here, *S*_{ϵ}(*z*) is the area of the cross section *ω*_{ϵ}(*z*). Note that *S*_{ϵ}(*z*)=*ϵ*^{2}|*ω*(*z*)|, where |*ω*(*z*)| is the area of the domain *ω*(*z*).

Thus, in the case when geometry of the inhomogeneity *Q*_{ϵ} is prescribed, relations (2.13)–(2.15) constitute a coupled system for the unknown functions *p*(*z*) and *w*(*z*).

### (d) Asymptotic model for the interaction of a slender inhomogeneity with matrix

Equation (2.13) considers with equations looked at by Tuck (1964), Fedoryuk (1981) and Maz'ya *et al*. (1981). Owing to their analysis, we have the equalities(2.16)where *P*_{n}(*ζ*) is the Legendre polynomial, *μ*_{0}=0, and(2.17)

Thus, on the basis of formula (2.16), the following representation holds:(2.18)

Observe also that, from (2.17), it follows that:where *C* is the Euler constant.

Since *μ*_{n}→+∞ as *n*→+∞, we find that equation (2.13), which depends on ln *ϵ*, turns out not to be solvable for all right-hand sides for a sequence of values of *ϵ* that condenses to zero. Thus, we shall use the regularizing procedure presented by Maz'ya *et al*. (1981) and consider the following equation:(2.19)Here, the operator *J*_{ϵ} is defined by the formula (see formulae (2.5) and (2.18))(2.20)where the integer part is denoted by the square brackets.

Observe that the regularization (2.20) of the operator ** J** enables one to construct the asymptotic solution of equation (2.13) that leaves on the r.h.s. a discrepancy with superpower decrease as

*ϵ*→0 and, consequently, to obtain the asymptotic solution of the original problem.

Thus, equation (2.19) together with equation (2.14) and the boundary conditions (2.15) constitute the regularized asymptotic model for the determination of the functions *p*(*z*) and *w*(*z*) in the case of the prescribed geometry of the inhomogeneity *Q*_{ϵ}.

## 3. Eshelby property

### (a) The Eshelby property for an elongated ellipsoidal inhomogeneity

Suppose the slender inhomogeneity *Q*_{ϵ} has the elliptic cross section *ω*_{ϵ}(*z*) with semi-axes *ϵR*(*z*)(1+*m*) and *ϵR*(*z*)(1−*m*). Since the conformal mapping *Z*=*R*(*z*)(*ζ*+*mζ*^{−1}) transfers the exterior of the unit circle into the region , the quantity *R*(*z*) coincides with the outer conformal radius of the set .

In the case of the ellipsoidal inhomogeneity *Q*_{ϵ}, we have(3.1)where *r*(*z*)=*ϵR*(*z*) is the outer conformal radius of the set .

We put(3.2)(3.3)Thus, equations (2.13)–(2.15) will be satisfied exactly, provided the following equations hold:(3.4)and(3.5)

Note that under the condition (1.7), the resultant problem of (3.4) and (3.5) does not allow any asymptotic decomposition (in powers of the parameter *ϵ*).

### (b) Absence of the Eshelby property for slender non-ellipsoidal inhomogeneities

Assuming that the strain within the inhomogeneity *Q*_{ϵ} is constant (see formula (3.3)), we shall have(3.6)and, consequently,(3.7)

Note that in equation (3.7), the constant of integration, which describes the rigid body translative displacement of the inhomogeneity, has been omitted due to the condition *w*(0)=0. This condition holds for any inhomogeneity that is symmetrical with respect to the *Ox*_{1}*x*_{2} plane. In the general case, this constant of integration should be determined from the following equation:However, owing to the symmetry of the remotely applied stresses (1.6), we do not consider the case of a non-symmetrical inhomogeneity *Q*_{ϵ}.

Taking into account the conjecture (3.6) and equation (2.14), we derive the following equation:(3.8)

In view of formulae (3.7), (1.6) and (2.19), we obtain(3.9)

It is clear that the substitution of expressions (3.1) and (3.2) into equations (3.8) and (3.9) turns them into equations (3.4) and (3.5), provided(3.10)

Now, we put(3.11)and(3.12)where *ρ*(*z*/*l*) and *δ*(*z*/*l*) are infinitesimal variations.

Substituting expressions (3.11) and (3.12) into equations (3.8) and (3.9) and linearizing them with respect to the variations *ρ*(*z*/*l*) and *δ*(*z*/*l*), we derive the following equations:(3.13)(3.14)

Taking into account relations (3.5) and (3.10), we find(3.15)and, consequently, equation (3.13) takes the form(3.16)

Finally, introducing the dimensionless coordinate *ζ*=*z*/*l* in equations (3.14) and (3.16), we obtain the system(3.17)

(3.18)

Let us show that the system of equations (3.17) and (3.18) has only a trivial solution.

We put(3.19)

Substituting expansions (3.19) into equation (3.17) and taking into consideration the equationwe obtain the following equality:(3.20)

Now, using the formula (Gradshtein & Ryzhik 1965)we readily find(3.21)

Now, substituting expansions (3.19) into equation (3.18) and taking into account formulae (3.20) and (3.21), we derive the following infinite system of linear algebraic equations:(3.22)(3.23)Here,(3.24)and(3.25)

It is evident that *b*_{n}+1≠0, *n*=1, 2, …. Hence, from equations (3.22) and (3.23), *n*=2, 4, …, it immediately follows that *A*_{n}=0, *n*=2, 4, ….

Further, it is easy to see that the system of equation (3.23), *n*=1, 3, …, has a trivial solution *A*_{n}=0, *n*=1, 3, …, and the following formal non-trivial solution:(3.26)

Since as *n*→+∞, the series with terms (3.24) and (3.25) is divergent and therefore the infinite product in (3.26) is also divergent. Thus, the system of (3.17) and (3.18) has only a trivial solution.

Finally, observe that the quantities *σ*_{0} and *r*_{0} were assumed to be fixed by relations (3.4), (3.5) and (3.15). Therefore, we proved that there is not any infinitesimal perturbation of the fixed elongated ellipsoid (3.1) that preserves the constancy of the fixed stress (3.6) inside the inhomogeneity. Obviously, changing the value of *σ*_{0}, we find another value of the parameter *r*_{0}. Thus, we have that the only infinitesimal perturbations of the elongated ellipsoid that preserve constancy of stresses inside the inhomogeneity are those into another elongated ellipsoid.

## Acknowledgments

The authors thank both reviewers for their constructive comments.

## Footnotes

- Received October 26, 2007.
- Accepted January 4, 2008.

- © 2008 The Royal Society