## Abstract

This paper presents a closed-form solution for the arbitrary polygonal inclusion problem with polynomial eigenstrains of arbitrary order in an anisotropic magneto-electro-elastic full plane. The additional displacements or eigendisplacements, instead of the eigenstrains, are assumed to be a polynomial with general terms of order *M*+*N*. By virtue of the extended Stroh formulism, the induced fields are expressed in terms of a group of basic functions which involve boundary integrals of the inclusion domain. For the special case of polygonal inclusions, the boundary integrals are carried out explicitly, and their averages over the inclusion are also obtained. The induced fields under quadratic eigenstrains are mostly analysed in terms of figures and tables, as well as those under the linear and cubic eigenstrains. The connection between the present solution and the solution via the Green's function method is established and numerically verified. The singularity at the vertices of the arbitrary polygon is further analysed via the basic functions. The general solution and the numerical results for the constant, linear, quadratic and cubic eigenstrains presented in this paper enable us to investigate the features of the inclusion and inhomogeneity problem concerning polynomial eigenstrains in semiconductors and advanced composites, while the results can further serve as benchmarks for future analyses of Eshelby's inclusion problem.

## 1. Introduction

Eshelby's inclusion problem plays an important role in various engineering and physical fields [1,2], and continues to be an appealing subject in micromechanics and nanomechanics since the pioneering work of Eshelby [3–5]. Motivation for studying Eshelby's inclusion problem concerning anisotropic magneto-electro-elastic (MEE) materials arises from the wide applications of the composites involving MEE coupling (called multi-ferroics in materials and physics) as well as the many striking features found in these composites [6–8]. Furthermore, giant and universal magnetoelectric coupling effects can be designed and induced in many recent advanced soft materials [9]. Under two-dimensional deformation, Eshelby's inclusion problem in anisotropic elastic and piezoelectric (PE) planes was solved by Ru [10,11] using the conformal mapping method, by Pan [12,13] using Green's function method, and by Zou *et al.* [14] using a unified approach, among others. Also using Green's function method, Jiang & Pan [6] solved the polygonal inclusion problems in the corresponding MEE full, half- and bimaterial planes.

Most reported results on inclusion problems in anisotropic PE/MEE solids are limited to uniform eigenstrains. For an isotropic medium, Eshelby [5] showed that if the eigenstrain inside an ellipsoidal inclusion is in the form of a polynomial of an arbitrary order in Cartesian coordinates, then the induced strain field in the inclusion is also characterized by a polynomial of the same order. Rahman [15] derived the explicit polynomial form of the induced strain, and termed the results *Eshelby's polynomial conservation theorem*. Walpole [16], Asaro & Barnett [17], Kinoshita & Mura [18,19] and Mura & Kinoshita [20] obtained similar results for the inclusion problem in anisotropic solids. Besides the polynomial eigenstrains, other distributions of eigenstrains have also been considered. Sharma & Sharma [21] investigated the three-dimensional elastic state of inclusions in an infinite isotropic elastic solid under the eigenstrains with a Gaussian or an exponential distribution.

Although, in many practical applications, the eigenstrains may not be explicitly given in a polynomial form [21], they can usually be approximated by polynomial functions in the bounded domain. In particular, polynomial eigenstrains are found to be applicable in analysing the elastic interactions between the inhomogeneities [22–24] and induced fields in nanostructures by graded eigenstrains [25,26]. Thus, inclusion problems with polynomial eigenstrains are important both theoretically and practically. So far, however, only a very few works have been reported in this area. Nie *et al.* [27,28] studied the elliptical inhomogeneity problems in orthotropic materials under linear eigenstrains. Guo *et al.* [29] extended the polynomial order of the eigenstrains to quadratic and derived a closed-form solution for the elastic field of an elliptical inhomogeneity in orthotropic media with complex roots. More recently, based on Green's function method, Sun *et al.* [30] derived an explicit closed-form solution for a polygonal inclusion with a linear eigenstrain in the anisotropic piezoelectric full plane. The solution was later extended to the corresponding half-plane domain by Chen *et al.* [31] and to the corresponding quadratic eigenstrain case by Yue *et al*. [32]. When the eigenstrains are in the polynomial form of order 3 or higher, the explicit solution for this problem has not yet been reported in the literature.

In this paper, we present a closed-form solution for an arbitrary polygonal inclusion in an anisotropic MEE full plane with eigenstrains characterized by a polynomial of arbitrary order in Cartesian coordinates. We also show that our solution reduces to the known results of uniform eigenstrains [6,8,13] and of linear eigenstrains [30]. Besides the induced fields listed in the tables and illustrated by the figures, we also analyse the singularity features at the vertex of the polygon and the averages of the induced fields over the inclusion.

This paper is organized as follows. In §2, for an inclusion with a general polynomial eigenstrain embedded in an anisotropic MEE full plane, both the induced fields and their averages over the inclusion are derived in terms of a set of basic functions involving boundary integrals. In §3, for an arbitrary *L*-side polygonal inclusion, we carry out the integrals explicitly so that the induced fields and their averages are expressed in terms of elementary functions. In §4, for the *L*-side regular polygonal (including circular) inclusions, we apply our solutions to the constant and linear eigenstrain cases and present also results for eigenstrains of higher orders. Conclusions are drawn in §5. For easy reference, we offer two appendixes in the electronic supplementary material (tables and appendices): the extended Stroh formulism for the MEE material is presented in appendix A, and the connection between the present solution and the solution via Green's function method is discussed in appendix B.

## 2. Solution of the magneto-electro-elastic inclusion problem due to eigenstrains of the general polynomial form

Let *Ω* be the (*x*_{1}, *x*_{2})-plane made up of a homogeneous MEE medium and *ω* a polygonal sub-domain of it. Let *ω* in the (*x*_{1}, *x*_{2})-plane and *Γ*=∂*ω* the boundary of *ω* (figure 1), with *ω* and *ω* and *ω* undergoes a group of extended additional displacements or eigendisplacements (consists of the elastic displacements, electric and magnetic potentials) characterized by a polynomial of arbitrary order. Then the corresponding eigenstrains to the eigendisplacements must also be in a polynomial form. In this paper, we symbolize the extended eigendisplacements by a general term of degree *M*+*N* as
*a* is the characteristic length of the inclusion, and *M* and *N* are two positive integers. The extended eigenstrains corresponding to **u***(*x*_{1},*x*_{2}) are determined by the relations

We denote by {*u*_{i},*ϕ*,*φ*} the elastic displacements and the electrical and magnetic potentials induced by the extended eigenstrain fields, and by **n** the unit outward normal of the boundary *Γ*. Then the continuity conditions for the elastic displacement and traction vectors across the boundary come to
*s* is to keep *ω* on the left-hand side as the Cartesian coordinate system is counter-clockwise orientated. This implies
*s* is an arc element of infinitesimal length at boundary point (*x*_{1},*x*_{2}). Substituting equations (A 8) and (A 11) of the electronic supplementary material, appendix A, and *E*_{1}=−*ϕ*_{,1},*E*_{2}=−*ϕ*_{,2},*H*_{1}=−*φ*_{,1},*H*_{2}=−*φ*_{,2} into (2.2)_{2}, (2.3) and (2.4) yields
_{1} and (2.7) gives the equivalent continuity conditions of the extended displacement **u** and stress function ** ψ** (refer to the electronic supplementary material, appendix A, for details) across the interface

*y*=

*x*

_{1}+

*ix*

_{2}∈

*Γ*is the complex coordinate of the boundary point.

Accounting for the general solution (A 8) in terms of the extended Stroh formalism as shown in the electronic supplementary material, appendix A, the continuity conditions (2.8) can be expressed by
**B**^{T} and **A**^{T}, respectively, and adding the resulting equations, we obtain
**1** being the 5×5 identity matrix.

As *f*_{I}(*z*_{I})(*I*=1,2,3,4,5) are five functions which are sectionally analytic with respect to *z*_{I}(=*x*_{1}+*p*_{I}*x*_{2}, with *p*_{I} being the eigenvalues with positive imaginary parts; see the electronic supplementary material, appendix A) in the entire complex plane except for *Γ*, it is more convenient to write **B**^{T}**u***(*y*) as functions of the complex variables *y*_{I} and *f*_{I}(*z*_{I}). Using
*Γ*, we can recast **B**^{T}**u***(*y*) as
*p*_{I}.

We now state the following theorem [34,35]: let *Γ* be a simple, closed, regular, positively oriented curve enclosing the origin, and let *b*(*t*) (*t*∈*Γ*) be a Hölder continuous function, namely for *t*,*τ*∈*Γ* satisfying |*b*(*t*)−*b*(*τ*)|≤*C*|*t*−*τ*|^{α}, *C*>0, *α*∈(0,1] on *Γ*, then the degenerated Privalov (or Riemann–Hilbert) problem *f*^{out}(*t*)=*f*^{in}(*t*)−*b*(*t*) has the general solution
*Γ*_{I} directly yield
*f*_{I}(*z*_{I}), namely

Based on the solved analytical functions *f*′_{I}(*z*_{I}), the derivatives of the extended displacements and also the stresses are then given by
*ε*_{IJ}=0.5(*u*_{I,J}+*u*_{J,I}) and *B*_{1M}=−*p*_{M}*B*_{2M} with *u*_{I,3}=*u*_{I,4}=*u*_{I,5}≡0, and the notation
*total*, but the induced strain only. This is different from the solution based on Green's function method. The connection between the solutions by the two methods is discussed in detail in appendix B (included in the electronic supplementary material). We also point out that our solution is in a unified form (*M* and *N* are arbitrary integers), so it is convenient in real calculations. However, Green's function method becomes more and more tedious when dealing with problems involving eigenstrains of higher order.

Therefore, we have solved the induced strain and stress fields by an inclusion with polynomial eigenstrains of arbitrary order. These fields can all be expressed by the basic functions introduced by equation (2.20). Furthermore, to find the averages of the induced fields, one needs only to find the corresponding average of these basic functions, which is discussed below.

Using the formula [36]
*ω* as
*ω*| denotes the area of the inclusion originally defined as the interior of the curve *Γ*. Thus, the average of

## 3. Explicit solution for a polygonal inclusion

As a general description for the shape of an inclusion is to approximate it by an *L*-side polygon, uniform eigenstrain problems of the polygonal inclusion in various material domains were reported, including elastic, piezoelectric and MEE materials [37–40]. In this section, we derive the explicit expressions of the basic function *L*-side polygonal inclusion with polynomial eigenstrains of arbitrary order.

In the following discussion, symbols *p*,*y*,*z*,… can be freely replaced by *p*_{I},*y*_{I},*z*_{I},…. We now define the points on the *i*-th side of the polygonal inclusion in the complex plane, in terms of parameter *t*(0≤*t*≤1), as
*y*_{i} and *y*_{i+1} are the two end points of the side. We further denote *w*_{i} as the relative position vector *w*_{i}=*y*_{i}−*z* and *s*_{i} as the *i*-th side vector *s*_{i}=*y*_{i+1}−*y*_{i}=*w*_{i+1}−*w*_{i}.

For a point *y* on the *k*-th side, *y*=*y*_{k}+*s*_{k}*t*, we can construct the following expressions:
*L* is the side number of the polygon and

In order to avoid the logarithmic singularity in the calculation, we further decompose the average of the basic function into two parts, i.e.

We now parametrize point *z* from the *j*-th side and point *y* from the *k*-th side by
*w*=*y*−*z*=*y*_{k}+*s*_{k}*t*−*y*_{j}−*s*_{j}*τ*=*s*_{j,k}+*s*_{k}*t*−*s*_{j}*τ*, where use has been made of the notations *s*_{j,k}=*y*_{k}−*y*_{j} and *s*_{k}=*y*_{k+1}−*y*_{k}=*s*_{k,k+1}. After some straightforward but long derivations, the two averages in equations (3.7) and (3.8) can be explicitly expressed as
*k*=*j*;
*k*=*j*+1;
*j*=*k*+1; and
*j*≠*k*)∩(*j*+1≠*k*)∩(*j*≠*k*+1). As for *Ξ*_{jk} in the expression of *Θ*_{jk} and *Ξ*_{jk}, use has been made of the notations

## 4. Numerical examples

The solutions presented above are in analytical form and are for the general polygonal inclusion with eigenstrains characterized by a single polynomial term of arbitrary order. Thus, they can be used to deal with polygonal inclusions under any polynomial eigenstrains by a simple process of superposition. Furthermore, the solutions also contain those in the corresponding anisotropic piezoelectric plane as special cases (by just neglecting the magnetic potential *φ* and the magnetic induction **B** with also *q*_{ijk}=0 and *d*_{ij}=0). By further letting the electric quantities and coupling coefficients be zero, our solutions will be reduced to those of the corresponding anisotropic elastic full-plane cases. In this section, we first verify our solutions against some existing and simple ones, including the uniform eigenstrain case in MEE media [8] and the linear eigenstrain case in piezoelectric media [30]. After that, we will present some numerical results associated with polynomial eigenstrains of higher orders in *L*-side polygons, including the distributions and the singularity features of the induced fields.

Except for the verification of the linear eigenstrain case, we choose the same material as used by Zou & Pan [8]. The detailed material properties can be easily found in the electronic supplementary material. In calculations, the dimensions of stress, electric displacement and magnetic induction are, respectively, in 10^{11} Pa, 10 C m^{−2} and 10^{3} Wb m^{−2}, and the electric and magnetic fields are in 10^{10} V m^{−1} and 10^{8} A m^{−1}.

### (a) Validation against existing results

Inclusion problems with uniform eigenstrains have been most widely studied and many analytical and numerical results can be found in literature. In this paper, we choose the newly reported results by Zou & Pan [8] to compare with. We consider a regular *L*-side polygon inscribed inside a unit circle, centred at the origin point and with one of the vertices on the positive *x*_{1}-axis. When the polygonal domain undergoes a non-zero eigenstrain *ε*_{h}=*ε*_{11}+*ε*_{22}, the deviatoric strain *E*-/*H*-fields *L*. Our results are in full agreement with those in [8].

For the linear eigenstrain case, we assume that a circular inclusion with radius 10 nm undergoes a hydrostatic eigenstrain *L*-side polygon with *L*=100. It is noted that the difference between the present results and those in [30] based on Green's function methods is the eigenstrain. More specifically, our solutions give only the induced fields while those based on Green's function method are the total fields.

### (b) Fields induced by quadratic eigenstrains

In this subsection, we apply our solution to study the MEE polygonal inclusion problem with quadratic eigenstrains. In general, the characteristic length can be assigned to be the radius of its minimal circumcircle of the polygon. In what follows, we always assume that the polygonal inclusion can be inscribed in a unit circle centred at the origin point of the Cartesian coordinate system, i.e. *a*=1, and undergoes two kinds of quadratic eigenstrain

First we consider *L*-side regular polygons (*L*=3, 4, 5, 10, 20, 50 and 100), and calculate the induced hydrostatic strain *ε*_{h}, the deviatoric strain *ε*_{d} and the *E*-/*H*-fields *E*_{h} and *H*_{h} under the eigenstrains *x*_{1}=*x*_{2}) within the inclusion and their averages over the inclusion, from which we can conclude that all the induced fields are convergent to those in the circular inclusion with increasing number *L*.

In what follows, we present the variation of the extended strains along different lines induced by the eigenstrains *L*-side polygon with *L*=100. Figure 2*a*–*d* shows the variation of the strain components (*ε*_{11},*ε*_{22},*E*_{2},*H*_{2}) along the *x*_{1}-axis, *x*_{2}-axis and the diagonal line (*x*_{1}=*x*_{2}) induced by the eigenstrain *a*–*c* shows the variation of strain components (*ε*_{12},*E*_{1},*H*_{1}) under *x*_{1}=*x*_{2}) experience jumps across the boundary of the inclusion (at *x*_{1}=*x*_{2}=±0.707). This is because of the fact that, while the extended displacements are continuous across the boundary here, their derivatives in the *x*_{1}- and *x*_{2}-directions are not. It should also be mentioned that the strain components (*ε*_{12},*E*_{1},*H*_{1}) under eigenstrain *ε*_{11},*ε*_{22},*E*_{2},*H*_{2}) under *x*_{1}- and *x*_{2}-axes.

Shown in figure 4*a*–*d* are the contours of the extended strains, both inside and outside the inclusion, induced by the quadratic eigenstrain *a* that the hydrostatic strain *ε*_{h} is symmetric about the *x*_{1}- and *x*_{2}-axes, and that inside the inclusion it is negative. The concentrations of *ε*_{h} can be found near the interface between the inclusion and the matrix. Figure 4*b* shows that there are four concentrations of *ε*_{d} near the interface and two minimum values inside the inclusion. Figure 4*c* shows that the induced *E*_{h} field reaches maximum at the intersection of the *x*_{1}-axis and the interface. Figure 4*d* shows that the concentrations of *H*_{h} are also located on the interface.

Figure 5*a*–*d* shows the contours of the induced strains under the eigenstrain *a* shows that *ε*_{h} is antisymmetric about the *x*_{1}- and *x*_{2}- axes, both inside and outside the inclusion, and that it is negative in the first and third quadrants and tensile in the second and fourth quadrants within the inclusion. Figure 5*b* shows that the concentrations of *ε*_{d} are located at the top and the bottom of the inclusion, and two points with local minima can be found in the upper and lower parts of the inclusion. Figure 5*c*,*d* shows that the contours of *E*_{h} and *H*_{h} are similar within the inclusion. They are discontinuous across the whole interface between the inclusion and the matrix with their concentrations being located near the boundary of the inclusion.

Figures 6*a*–*d* and 7*a*–*d* show the strain fields inside and outside the square inclusion (inscribed in a unit circle and with one vertex on the positive *x*_{1}-axis) induced by the eigenstrains *a*–*d* that all the induced strains are singular at the right and left vertices but non-singular at the top and bottom vertices. Furthermore, all the fields are discontinuous across the boundary of the square inclusion. Except for the small domains near the top and the bottom vertices, the *ε*_{h} field is negative in the inclusion, as shown in figure 6*a*. Figure 6*b*,*c* shows that the contours of *ε*_{h} and *E*_{h} are similar and have two local minima in the lower and the upper part of the inclusion. The contours of the magnetic field *H*_{h} are particularly interesting at the right and left vertices of the inclusion, with its singularity being in different features along different directions, as shown in figure 6*d*. Figure 7*a*–*d* shows that, under the eigenstrain *a* further shows that the hydrostatic strain *ε*_{h} is antisymmetric about both the *x*_{1}- and *x*_{2}-axes, but non-singular at all four vertices. The induced strains *ε*_{d},*E*_{h} and *H*_{h}, however, are singular at the right and left vertices, as shown in figure 7*b*–*d*.

Figures 8*a*–*d* and 9*a*–*d* show the contours of the induced fields inside and outside a regular isosceles triangular inclusion (inscribed in a unit circle and with one vertex on the positive *x*_{1}-axis) under the eigenstrains *a*–*d* that, at the vertex on the positive *x*_{1}-axis, all the induced fields are singular; however, at the other two vertices of the triangle, only slight concentrations are observed. Furthermore, across the vertical line of the triangle, all the contours are continuous whilst they are not when crossing the two inclined sides of the triangle. It is noted from figure 8*a* that the hydrostatic strain *ε*_{h} is symmetric with respect to the *x*_{1}-axis, and it is negative within the inclusion. It should also be pointed out that the magnetic field *H*_{h} has a considerable concentration with magnitude of 104.4×10^{4} A m^{−1} at the right vertex (figure 8*d*). Under the eigenstrain *a*–*d* shows that field concentrations occur at all three corners of the triangular inclusion. Figure 9*a* shows that the hydrostatic strain *ε*_{h} is antisymmetric with respect to the *x*_{1}-axis. The field concentrations at the left two vertices shown in figure 9*b*–*d* are almost the same as those at the right vertex located on the positive *x*_{1}-axis.

### (c) Eigenstrains with different orders

To study the effect of different orders of polynomial eigenstrains on the induced extended strain fields, we present in this subsection the strains inside and outside a unit circular inclusion induced by the eigenstrains *a*=1.

Shown in figure 10*a*–*d* are the contours of *ε*_{h} (×10^{−1}), *ε*_{d} (×10^{−1}), *E*_{h} (×10^{8} V m^{−1}) and *H*_{h} (×10^{4} A m^{−1}), respectively, induced by the linear eigenstrain *a* shows that the hydrostatic strain *ε*_{h} within the inclusion is a linear function of *x*_{1}, as expected, and that it is further antisymmetric about the *x*_{2}-axis both inside and outside the inclusion. Concentrations occur at six locations near the interface between the inclusion and matrix. Figure 10*b* shows that the deviatoric strain field *ε*_{d} varies linearly in *x*_{2} within the inclusion with its maximum being at the top and bottom of the inclusion. Figure 10*c* shows that the electric field *E*_{h} has concentrations at six locations near the interface between the inclusion and matrix, and has one minimum in the centre of the inclusion. It is interesting to note from figure 10*d* that the magnetic field *H*_{h} in the inclusion varies linearly in the radial direction and shows a couple of concentrations when approaching the interface between the inclusion and matrix.

Figure 11*a*–*d* shows the contours of the induced fields by the cubic eigenstrain *a* shows that the hydrostatic strain *ε*_{h} varies nonlinearly along *x*_{1}-axis, as compared with that induced by the linear eigenstrain in figure 10*a*. However, the contour shapes in the matrix are very similar to those shown in figure 10*a*, except for different contour values. The deviatoric strain *ε*_{d}, on the other hand, varies nonlinearly along the *x*_{2}-axis, as compared with the corresponding linear result in figure 10*b*. The electric and magnetic fields *E*_{h} and *H*_{h} in the inclusion vary along the radial direction with concentrations being mostly located on the interface between the inclusion and matrix (figure 11*c*,*d*).

We remark that the solutions presented above are for one of the polynomial terms. In real applications, contributions from such individual terms need to be simply added together. In so doing, however, one should keep in mind that the compatibility relations should be satisfied when constructing the eigenstrains of different orders, especially for polynomials of order 2 or higher. That is to say, one must construct some eigenstrains in a group, and cannot give a non-zero eigenstrain component alone, such as

### (d) Singularity at the vertices of a polygonal inclusion

Our presentation above shows that the field concentration and/or singularity may exist at the vertices of the polygonal inclusion. In this subsection, we will carry out the essential analysis in terms of the fundamental formulae. It is well known that the solutions for the elastic polygonal inclusion problems exhibit logarithmic singularity around the vertices, in both isotropic and anisotropic materials [41]. Many factors may contribute to the logarithmic singularity of the induced field, including material properties, geometry of the inclusion and eigenstrains. However, the singularity features associated with a polygonal inclusion can be clarified by analysing the basic functions

From equation (3.4) for the basic functions and one of their factors *Λ*(*z*) in equation (3.5), we observe that the logarithmic term may appear only under the condition *y*_{k}, can be extracted as

To illustrate this important conclusion, we consider again the triangular and square inclusion cases as discussed in §4.2, but investigate only the singularity features of the basic functions *p*=1.901*i* and further let *M*=3, *N*=0 (corresponding to one group of quadratic eigenstrains). Again, the regular triangle and square are oriented in such a way that one of their vertices is on the positive *x*_{1}-axis located at *x*_{1}=1. Figure 12*a*,*b* shows the variation of the real parts of the functions *x*_{1}-axis for different pairs of (*q*,*r*) and for both the triangular and square inclusions. It is observed clearly from figure 12*a*,*b* that, except for the case (*q*,*r*)=(3,0), results corresponding to all other pairs of (*q*,*r*) show singularity at the vertex. This is because equation (3.4) does not contain the term with *q*,*r*)=(3,0); however, for all other pairs of (*q*,*r*), the term with *x*_{1}-axis.

## 5. Concluding remarks

In this paper, we have derived closed-form fields induced by polynomial eigenstrains of arbitrary order within a polygonal inclusion in the anisotropic MEE full plane. The solutions are complete and general, and contain many existing solutions in the literature as special cases, including those of anisotropic or piezoelectric polygonal inclusion problems under constant or linear eigenstrains. The methodology presented in this paper is further comprehensive for dealing with inclusion problems under polynomial eigenstrains in anisotropic MEE materials regardless of the shape of the inclusion.

In deriving the solutions, we first formulate the prescribed eigendisplacements in a general polynomial form, rather than the eigenstrains as usually adopted. Based on the extended Stroh formulism in MEE materials, we then attribute the solution of the problem to a set of basic functions involving boundary integrals, which are mainly characterized by the geometry of the inclusion and the order of polynomial in the eigenstrains. Finally, for the case of polygonal inclusions, the integrals are carried out explicitly. Numerical results have proved that *Eshelby's polynomial conservation theorem* is valid for the MEE inclusion problem with eigenstrains of quadratic polynomials. For a circular inclusion, concentrations of induced fields are mainly distributed near the interface, but the number of concentrations, the value and the symmetry depend on the order of the polynomial in terms of variables *x*_{1} and *x*_{2} in the prescribed eigendisplacements. For polygonal inclusions, concentrations are located at the vertices with possible singularity there. As benchmarks, we have also listed in tables in the electronic supplementary material the fields induced by quadratic eigenstrains in an *L*-side regular polygon with *L*=3, 4, 5, 10, 20, 50 and 100 for future reference.

## Authors' contributions

E.P. suggested the problem and main examples. W.N.Z. conceived the mathematical model and the analytic method, mainly performed the integral solution, derived and wrote the electronic supplementary material, appendix B, and supervised the whole process of writing the paper. Y.G.L., as a doctoral student under the supervision of W.N.Z., wrote the paper, derived the explicit solutions for polygonal inclusions with polynomial eigenstrains and performed the calculation for illustrations and tables. E.P. and W.N.Z. further helped Y.G.L. to check the solutions, interpret the computational results and polish the paper. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

This work was supported by the National Science Foundation of China (NSFC), grant no. 11372124, which is greatly appreciated by the first two authors.

- Received October 29, 2014.
- Accepted June 1, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.