## Abstract

Explicit expressions of Green’s function and its derivative for three-dimensional infinite solids are presented in this paper. The medium is allowed to exhibit a fully magneto-electro-elastic (MEE) coupling and general anisotropic behaviour. In particular, new explicit expressions for the first-order derivative of Green’s function are proposed. The derivation combines extended Stroh formalism, Radon transform and Cauchy’s residue theory. In order to cover mathematical degenerate and non-degenerate materials in the Stroh formalism context, a multiple residue scheme is performed. Expressions are explicit in terms of Stroh’s eigenvalues, this being a feature of special interest in numerical applications such as boundary element methods. As a particular case, simplifications for MEE materials with transversely isotropic symmetry are derived. Details on the implementation and numerical stability of the proposed solutions for degenerate cases are studied.

## 1. Introduction

Magneto-electro-elastic (MEE) materials are receiving increasing attention in smart structural applications owing to their ability to convert energy among the mechanical, electric and magnetic fields. Although single-phase MEE materials can be found in nature, practical applications are focusing on composites made by combining piezoelectric and piezomagnetic phases together, in which the coupling interaction between both the piezoelectric and piezomagnetic phases produces an electromagnetic coupling that can be several orders of magnitude higher than that observed in the single-phase MEE materials so far available. A review on the historical perspective, status and future directions of MEE composites can be found in Nan et al. (2008).

Green’s functions and their derivatives play a key role in the solution of many engineering problems. For instance, they are the basis to boundary element methods (BEM) (Brebbia & Dominguez 1992; Aliabadi 2002). Great effort has been devoted to deriving Green’s function and its derivative for general three-dimensional anisotropic elastic materials in a form suitable for their numerical implementation. The works by Fredholm (1900), Lifshitz & Rozentsveig (1947), Vogel & Rizzo (1973), Pan & Chou (1976), Wilson & Cruse (1978), Nakamura & Tanuma (1997), Ting & Lee (1997), Wang (1997), Sales & Gray (1998), Tonon *et al*. (2001), Phan *et al*. (2005) or Távara *et al*. (2008) among others should be cited here. Phan *et al*. (2005) and Lee (2003, 2009) derive explicit expressions for the derivative of three-dimensional Green’s function for fully anisotropic materials.

Many of the procedures developed for anisotropic elastic materials have been further applied to derive Green’s functions for three-dimensional piezoelectric materials. Relevant works by Deeg (1980), Benveniste (1992), Chen (1993), Dunn (1994), Wang & Zheng (1995), Dunn & Wienecke (1996), Akamatsu & Tanuma (1997), Pan & Yuan (2000) or Pan & Tonon (2000) may be cited. However, explicit expressions for the derivative of Green’s function are only available in this case for materials showing some class of elastic symmetry, but not for general anisotropic behaviour.

Furthermore, the above techniques have been recently extended to the analysis of MEE materials. Pan (2002) derived three-dimensional explicit Green’s functions in anisotropic MEE full space, half-space and bimaterials based on the extended Stroh formalism. In that work, a finite difference scheme is proposed to evaluate the derivatives of Green’s functions. Soh *et al*. (2003) presented an explicit three-dimensional Green’s function for infinite transversely isotropic MEE solid based on the potential theory. Wang & Shen (2002), Ding *et al*. (2005) and Hou *et al*. (2005) applied the potential function approach to derive Green’s functions for various MEE problems with transversely isotropic material symmetry. All these contributions are reviewed in the book by Qin (2007). However, explicit expressions for the derivative of Green’s function are not available in the literature, to the best of the authors’ knowledge, when it comes to three-dimensional fully anisotropic MEE materials.

In this paper, new explicit expressions, in terms of Stroh’s eigenvalues, are obtained for the first-order derivative of Green’s function in three-dimensional fully anisotropic MEE materials. The derivation combines extended Stroh formalism, Radon transform and Cauchy’s residue theory. The procedure is an extension to the MEE case of that proposed for anisotropic elastic materials by Lee (2003). Furthermore, the procedure is generalized to cover mathematically degenerate materials, in the Stroh formalism context, via a multiple pole residue scheme. As a particular case, simplifications for MEE materials with transversely isotropic symmetry are also derived. Numerical implementation of the proposed solution is discussed and several numerical examples are presented to illustrate its validity and accuracy. In table 1 the nomenclature used in this article is summarized.

## 2. Preliminaries

### (a) Basic equations of linear magneto-electro-elasticity

Let {*x*_{i}} (*i*=1–3) be a Cartesian coordinate system in three dimensions. The equilibrium equations and the Maxwell equations under the assumption of static elastic fields are given by
2.1
2.2
2.3
where *σ*_{ij}, *D*_{i} and *B*_{i} are the components of Cauchy stress tensor, the electric displacements and the magnetic inductions, respectively, and *f*_{i}, *f*^{e} and *f*^{m} are the three components of body forces, the electric charge density and the electric current density, respectively. The infinitesimal strain tensor, the electric field *E*_{i} and the magnetic field *H*_{i} are defined as
2.4
2.5
where *u*_{i} are the components of the elastic displacement field, and *φ* and *ϑ* are the electric and magnetic potentials, respectively. Equations (2.1)–(2.5) are coupled through the following linear constitutive law (Soh & Liu 2005):
2.6
2.7
2.8
where *c*_{ijkl}, *ϵ*_{il} and *μ*_{il} denote the components of the elastic stiffness tensor, the dielectric permittivities tensor and the magnetic permeabilities tensor, respectively, and *e*_{ijk}, *q*_{ijk} and *λ*_{il} are the piezoelectric, piezomagnetic and magneto-electric coupling coefficients, respectively. The material constants tensors show the following symmetry conditions:
2.9
Moreover, the elastic constant, dielectric permittivities and magnetic permeabilities tensors are positive definite, i.e.
2.10
In this work, the extended notation introduced by Barnett & Lothe (1975) for piezoelectric materials is used. In this way, the linear MEE problem may be formulated in an elastic-like fashion by considering a generalized displacement vector extended with the electric potential and the magnetic potential as
2.11
a generalized stress tensor extended with the electric displacements and the magnetic inductions as
2.12
and an extended elasticity tensor with the following components:
2.13
By virtue of symmetries (2.9), *C*_{iJKm}=*C*_{mKJi} is satisfied. In the above definitions, the lowercase (elastic) and uppercase (extended) subscripts take values 1, 2, 3 and 1, 2, 3 (elastic), 4 (electric), 5 (magnetic), respectively so that the constitutive equations (2.6)–(2.8) can be rewritten together as
2.14
In the same way, equilibrium equations (2.1)–(2.3) can be expressed as
2.15
where *f*_{J} is the extended body force vector, defined as
2.16

### (b) Presentation of the problem

Consider a homogeneous and infinite medium in the three-dimensional space with linear anisotropic MEE behaviour. Let *δ*(**x**) be the Dirac delta function centred at the origin of a fixed Cartesian coordinate system {*x*_{i}} and *δ*_{JK} the five-dimension Kronecker delta. Green’s function is defined as a second order tensor in a five-dimension space with components *U*_{JK} such that it satisfies the partial differential equations (*extended equilibrium equations of Navier*) that result from combining equations (2.4), (2.5), (2.14) and (2.15) to yield
2.17
where the generalized body force vector corresponds to a point load at the origin *f*_{J}=*δ*_{JP}*δ*(**x**). Physically, Green’s function *U*_{KP}(**x**) represents the displacement in the *K*-direction (in the extended sense), in an infinite solid at point **x**≠0 owing to an unit force applied at the origin in the *P*-direction (also in the extended sense). Then, *U*_{KP}(**x**) is

the elastic displacement at point

**x**in the*x*_{K}-direction,*K*=1–3 due to (*a*) a mechanical force at the origin in*x*_{P}-direction,*P*=1–3; (*b*) a point electric charge*P*=4, or (*c*) a point electric current*P*=5;the electric potential at point

**x**,*K*=4 due to (*a*) a mechanical force at the origin in*x*_{P}-direction,*P*=1–3; (*b*) a point electric charge*P*=4; or (*c*) a point electric current*P*=5; andthe magnetic potential at point

**x**,*K*=5 due to (*a*) a mechanical force at the origin in*x*_{P}-direction,*P*=1–3; (*b*) a point electric charge*P*=4; or (*c*) a point electric current*P*=5.

## 3. Three-dimensional Green’s function for anisotropic magneto-electro-elastic solids

### (a) Integral expression of Green’s function

In order to obtain Green’s function, a possible approach is via the Radon transform technique (e.g. Wang & Achenbach 1995; Wang 1997; Pan 2002; among others).

For MEE materials, it can be shown that *U*_{JK}(**x**) admits the following integral representation (Pan 2002):
3.1
where **d** is a vector variable in the space-fixed coordinates {*x*_{i}}, *Ω*(**d**) is any closed surface enclosing the origin, and the Christoffel tensor is introduced the components of which are
3.2
with well-defined inverse. In particular, the integration can be carried out over the surface of a unit sphere |**d**|=1. Expression (3.1) can be further reduced to (e.g. Wang & Achenbach 1995)
3.3
where *r*=|**x**| and ; **n*** being an unit vector such that , so that the integral is done along the unit circumference |**n***|=1 on the plane normal to (figure 1).

The unit vector **n*** in equation (3.3) on the oblique plane can be represented in terms of the arbitrary parameter *ψ* as (Ting & Lee 1997)
3.4
where *n*_{i} and *m*_{i} are the components of any two mutually orthogonal unit vectors such that is a right-handed triad. Figure 1 shows a sketch of the unit sphere centred at the origin and the geometrical relation between the **x**, , **m**, **n** and **n*** vectors. Then, the Christoffel tensor *Γ*_{JK}(**n***) can be rewritten as (Ting & Lee 1997)
3.5
being
3.6
Then, the integral in equation (3.3) can be expressed in terms of the parameter *ψ* to yield
3.7
where the fact that ** Γ**(

*ψ*) is a periodic function in

*ψ*with periodicity

*π*has been used. Introducing the change of variable and noting that 3.8 and 3.9 it follows that (Ting & Lee 1997) 3.10 with 3.11

Thus, the solution (3.10) may be written as a singular term multiplied by a modulation function *H*_{JK} (Wilson & Cruse 1978)
3.12
where the modulation function *H*_{JK}(**x**) depends on the direction of **x** but not on its modulus *r*
3.13
The matrix **H** is one of the three extended Barnett–Lothe tensors which is symmetric and . Hence, **U**(**x**) is also symmetric and **U**(**x**)=**U**(−**x**). Moreover, **H** and **U** are independent of the choice of the unit vectors **m** and **n** on the oblique plane, as physically expected.

Let be the adjoint of *Γ*_{JK}, defined as , *H*_{JK} can then be further expressed as
3.14

### (b) Explicit expression of Green’s function

Therefore, finding an explicit expression of Green’s function is equivalent to find it for the extended Barnett–Lothe tensor *H*_{JK}. It is well known that the kernel of integral (3.14) has five complex poles with positive imaginary part that corresponds to the roots of the 10th-order polynomial equation (Ting 1996)
3.15
In Stroh’s formalism context, these roots are Stroh’s eigenvalues *p*_{α}. The other five roots are the conjugate of the remainder. The determinant in equation (3.15) can be factorized as
3.16
with the bar denoting complex conjugate and with . Since is an analytic function, the integration in equation (3.14) can be done by Cauchy’s residue theory to yield
3.17

In the above equation, it has been assumed that the five Stroh’s eigenvalues are different. Therefore, this result is only valid for mathematical non-degenerate materials. This expression is in full agreement with the expression presented in Pan (2002) by taking into account equation (3.12), that and *a*_{11} in Pan’s work is |**T**| in the present work.

## 4. Derivative of the three-dimensional Green’s function for anisotropic magneto-electro-elastic solids

### (a) Integral expression of the derivative of Green’s function

From the integral expression (3.1), the first-order derivatives of Green’s function with respect to *x*_{m} are given by
4.1
with being the derivative of the Dirac delta function. Following an analogous procedure to that by Wang & Achenbach (1995) for the anisotropic elastic case, the vector **d** is split into (figure 1), where and , so that equation (4.1) can be rewritten as
4.2
where the first integration is defined along the unit circumference |**n***|=1. Carrying out the second integration with respect to *b* yields
4.3
where
4.4

Taking into account that , it follows after some manipulations that 4.5 so that 4.6

Provided that the kernel of the integral in equation (4.3) has a periodicity of *π*, the derivative of Green’s function can be expressed as a function of the parameter *ψ* introduced in equation (3.4) to yield
4.7
An analogous expression for the elastic problem was derived for the first time by Barnett (1972) using the Fourier transform. Deeg (1980) and Chen (1993) derived equivalent expressions for piezoelectric solids and in this work is now presented for MEE materials.

Equation (4.7) has been used to numerically evaluate the derivative of Green’s function with a Romberg integration scheme in Barnett (1972) for anisotropic elastic materials. For piezoelectric materials, it has been implemented by Chen & Lin (1993) by using a standard Gauss quadrature.

### (b) Explicit expression of the derivative of Green’s function

Following the work by Lee (2003) for anisotropic elastic materials, it is defined
4.8
Considering again the change of variable , introducing the definition
4.9
and taking into account equations (3.8) and (3.9), the *M*_{ijPKMN} integral can be written as
4.10
where **T** has been previously defined in equation (3.6), *p*_{α} are Stroh’s eigenvalues and the function
4.11
has been introduced. This function is analytic everywhere in the upper half-plane (Im(*p*)>0) and the kernel in the integral (4.10) has five complex double poles with positive imaginary part corresponding to the roots of |** Γ**(

*p*)|

^{2}=0.

Applying Cauchy’s residue theory, an explicit expression for *M*_{ijPKMN} may be obtained as the sum of residues of the kernel at the poles *p*_{α} to yield
4.12
Each residue can be explicitly evaluated from
4.13
where
4.14
and
4.15
and *p*_{6}=*p*_{1}, *p*_{7}=*p*_{2}, *p*_{8}=*p*_{3} and *p*_{9}=*p*_{4}.

Therefore, the derivative of Green’s function may be written as
4.16
in terms of the following modulation function:
4.17
that only depends on the orientation of but not on its modulus *r*.

Because of the symmetry of the adjoint matrix , the components *M*_{qsPKMJ} satisfy the following symmetry conditions:
4.18
The matrix *B*_{ij} is also symmetric, resulting in an additional symmetry:
4.19
These symmetries allow to reduce considerably the number of components *M*_{qsPKMJ} to be calculated, and must be considered in the numerical implementation. In the fully coupled problem, *M*_{qsPKMJ} represents 5625 components, but only 720 of them are different.

In BEM applications, it is necessary to evaluate the traction fundamental solution *T*_{JK} which represents the *J*-component of the generalized traction vector produced by a generalized *x*_{K}-direction point force. Their expression follows easily from the derivative of displacement Green’s function as
4.20
where *η*_{i} are the components of the external unit normal vector to the boundary at the observation point where traction fundamental solution is being evaluated. In some BEM approaches, the second-order derivatives of Green’s function are required as well, like in dual or mixed formulations for fracture analysis. They may be obtained following a similar procedure to the one described in this paper for the first-order derivative. Their integral expressions are included in appendix A for completeness.

At this point, it is important to point out that explicit expressions for the derivative of Green’s function in MEE materials had been previously obtained only for transversely isotropic materials (Soh *et al*. 2003; Hou *et al*. 2005). For general anisotropy only numerical evaluation by a finite difference scheme had been reported in bibliography (Pan 2002). An explicit expression for anisotropic MEE materials is presented for the first time in the present work, as defined by equation (4.16).

## 5. Mathematical degenerate materials

Expressions (3.17) and (4.12)–(4.15) are not defined when there is any repeated Stroh’s eigenvalue. Such situation may happen depending both on the material properties *C*_{iJKm} and the direction of the vector **x**. Degenerate cases are not so sporadic and it is actually necessary to provide a scheme for Green’s function evaluation of general validity. On the other hand, numerical instabilities are observed in quasi-degenerate cases when Stroh’s eigenvalues are sufficiently close. In this work, a multiple pole residue approach is proposed in order to overcome such degeneracies and obtain accurate results.

Let the point *p*_{o} be a pole of order *m* of a function *f*(*p*). A formula for evaluating the residue of *f*(*p*) at this pole is given as (e.g. Sveshnikov & Tikhonov 1978; Phan *et al*. 2005)
5.1
Equation (5.1) allows to write explicit solutions for degenerated cases. At most, there are *N* () distinct Stroh’s eigenvalues *p*_{α} of *m*_{α} multiplicity. Hence, a general expression, valid for degenerate and non-degenerate materials, of the extended Barnett–Lothe tensor *H*_{JK} may be obtained as
5.2
In particular, when *N*=5 (*m*_{α}=1, ∀*α*), expression (5.2) is reduced to equation (3.17).

Similarly, general expressions both for degenerate and non-degenerate materials may be derived for the *M*_{ijPKMN} components to yield
5.3
As expected, when all Stroh’s eigenvalues are distinct (*N*=5), expression (5.3) is reduced to equations (4.12) and (4.13).

## 6. Implementation and numerical validation

### (a) Implementation details

Once Stroh’s eigenvalues are obtained, expressions for Green’s function (equations (3.12) and (5.2)) and the derivatives of Green’s function (equations (4.16), (4.17), (5.2) and (5.3)) are fully explicit. Stroh’s eigenvalues follow either from solving 10th-order characteristic equation (3.15), or alternatively, they can be obtained by solving the linear eigenproblem defined by (Ting 1996)
6.1
where
6.2
with **Q**, **R** and **T** defined in equation (3.6) and the superscript T denoting transpose.

It has been shown that it is not possible to obtain a general solution for Stroh’s eigenvalues in terms of the material constants and the position (Head 1979). This difficulty can be overcome by using any of the numerical routines available in commercial packages to obtain eigenvalues. In this implementation, the DEVLRG subroutine of IMSL (2009) library has been used. An alternative approach would be to extend to the MEE case the procedure developed by Sales & Gray (1998) and Phan *et al*. (2005) for anisotropic elastic materials, who directly computed the roots of the characteristic equation using Newton’s method in conjunction with Horner’s algorithm for efficient polynomial evaluation.

One of the crucial aspects for the numerical implementation is to establish a criterion for deciding when two Stroh’s eigenvalues, say *p*_{α} and *p*_{β}, are actually equal. An extensive numerical investigation has led us to the conclusion that the expressions of *H*_{JK} for degenerate materials with coincedent poles (equation (5.2)) should be used when
6.3
where stands for modulus and is Stroh’s eigenvalues with the largest modulus. Expression (5.3) has a different behaviour. Thus, two Stroh’s eigenvalues are considered equal for the evaluation of *M*_{ijPKMN} when
6.4

During the evaluation of the solutions there are functions that can be evaluated once and subsequently be treated as constant throughout the computation process of the 45 components of and 15 components of *U*_{JK}. It is the case of functions (equation (4.14)) and (equation (4.15)), the denominator in *Φ*_{ijKLMN} (equation (4.11)) and |**T**| (equation (3.6)). The same happens with the functions *B*_{ij} (only six components need to be computed), the adjoint matrix (only 15 components need to be computed) and the derivatives with respect to *p*. Then, the computing of the different *Φ*_{ijKLMN} components and its derivatives results in the adequate combination of *B*_{ij} and matrix components and its derivatives. In appendix B some simplifications of these functions for materials with transversely isotropic symmetry are presented.

Matrix *Γ*_{JK}(*p*) and its adjoint depend on the choice of the two mutually orthogonal unit vectors **n** and **m** on the oblique plane with normal . However, the and *U*_{JK}(**x**) tensors and, of course, and *U*_{JK,q}(**x**) are invariants with the choice of **n** and **m** (Ting 1996) as it has been previously mentioned. So, they can be adequately selected in order to obtain the most simplified expressions for , *B*_{ij}(*p*) and |**T**|. This fact is exploited in the simplifications of appendix B for materials with transversely isotropic symmetry. In this implementation, the two vectors are taken as and , where . For evaluations on *x*_{3}-axis these vectors are not defined and **n**={1,0,0} and **m**={0,1,0} are considered.

### (b) Validation examples

Following the works of Pan (2002) for MEE materials and Pan & Tonon (2000) for piezoelectric materials, two coupled materials are considered for comparison purposes. The first one (material A) is a transversely isotropic material with the elastic and piezoelectric properties of BaTiO_{3} and the piezomagnetic coefficients of CoFe_{2}O_{4}, but no electromagnetic coupling (Pan 2002). Material B is a transversely isotropic piezoelectric material (Pan & Tonon 2000). A summary of material constants for these materials is listed in tables 2 and 3. Moreover, for transversely isotropic materials it is satisfied
6.5

In the works of Pan & Tonon (2000) and Pan (2002), a finite difference scheme is used in order to compute the derivatives of Green’s function, so that they are approximated by 6.6 6.7 6.8

Table 4 presents results for the *σ*_{11} component of the generalized stress tensor at point **x**={1,1,−1} due to a point mechanical force in the *x*_{i}-direction (Ξ_{11i}) for material A. According to the constitutive law, these values are calculated from
6.9
6.10
6.11

Very good agreement is observed between results computed with the present explicit solution and those evaluated by a finite difference scheme with *h*=10^{−6}, as in Pan (2002). Relative errors below 10^{−9} are obtained between both sets of results.

Table 5 shows the *U*_{JK,2} values for material B at point **x**={1,1,1}. Once more, very good agreement is observed between results computed with the present solution and those obtained by a finite difference scheme, as in Pan & Tonon (2000), with relative errors below 10^{−9}.

### (c) Exploration examples

Two more cases are next considered. First, results for a fully anisotropic MEE are computed both by the present approach and the finite differences scheme described in the previous section. The material is a transversely isotropic BaTiO_{3}–CoFe_{2}O_{4} composite material, with volume fraction *V*_{f}=0.5 and an arbitrary orientation of the principal axis of the material with respect to the global coordinate system, defined by the transformation matrix
6.12
Such a rotation leads to fully populated constitutive tensors. Material properties are taken from Song & Sih (2003) and are summarized in table 6 after rotation (material C).

Table 7 presents the results obtained for the derivatives *U*_{JK,3} at point **x**={1,1,1}. Once more, very good agreement is observed between the present explicit solution and the finite differences values (*h*=10^{−6}).

The last example focuses on a degenerate material case, namely a transversely isotropic material the properties of which coincide with those of material A neglecting the piezoelectric (*e*_{ikm}=0) and piezomagnetic (*q*_{ikm}=0) couplings. For observation points along the *x*_{3}-axis the five Stroh’s eigenvalues are coincident when the source point is located at the origin, as figure 2*a* shows. The analytic elastic solution by Pan & Chou (1976) is considered for comparison. In such case, expressions (3.17) and (4.12)–(4.15) are not defined on the *x*_{3}-axis and are unstable close to it (say for angles *ϕ* between the *x*_{3}-axis and the position vector , ). This can be clearly seen in figure 2*b*, where the *T*_{11} component of the generalized traction fundamental solution defined in equation (4.20) is plotted against the *ϕ* angle for *r*=1 and an outward unit normal defined by . Figure 2*b* plots together the exact solution by Pan & Chou (1976) and the solution evaluated with expressions (5.2) and (5.3), assuming that the five Stroh’s eigenvalues are either coincident (*N*=1) or distinct (*N*=5). For angles , Stroh’s eigenvalues are almost coincident, which results in a quasi-degeneracy and a numerically unstable solution when the expressions for *N*=5 are used. Within this range the expressions for *N*=1 provide a stable and accurate solution. For angles , the expressions for *N*=5 fully coincide with the analytical solution proposed by Pan & Chou (1976).

## 7. Conclusion

In this paper, explicit three-dimensional first-order derivatives of full space Green’s functions in fully anisotropic MEE materials have been derived for the first time. The theoretical deduction combines the Radon transform, the generalized Stroh formalism and Cauchy’s residue theory. Expression results in terms of Stroh’s eigenvalues and no integration need to be performed in the proposed approach.

An expression for Green’s function has also been presented and is in agreement with that presented by Pan (2002). However, this solution does not remain valid for mathematical degenerate cases when Stroh’s eigenvalues are coincident, meanwhile numerical instabilities are observed in quasi-degenerate cases when Stroh’s eigenvalues are nearly equal. In order to amend this situation, a multiple pole residue approach has been proposed in the present paper. Thus, both the obtained Green’s function and its derivative cover all the possible mathematically degenerate and non-degenerate materials. Several numerical examples have been presented and discussed in order to validate the derived expressions.

The obtained solution is valid for pure elastic, piezoelectric, piezomagnetic or magneto-electric materials as particular cases of the fully coupled problem by setting the appropriate coefficients to zero.

## Acknowledgements

This work was supported by the Ministerio de Ciencia e Innovacin of Spain and the Consejera de Innovacin, Ciencia y Empresa of Andaluca (Spain) under projects DPI2007-66792-C02-02 and P06-TEP-02355. The author F. C. B. is gratefully acknowledged to the Junta de Andaluca of Spain for financial support throughout the Excellence Scholarship Program.

## Appendix A. Second-order derivative of Green’s function

Following the same procedure presented here, an integral expression for the second-order derivative may be obtained as A1 where the singularity is one order higher than the first-order derivative (equation (4.7)). Following again the work by Lee (2003), the new integral is defined as A2

With the change of variable , the *N*_{ijklMNPQRS} integral can be reduced to a form suitable to be computed through Cauchy’s residue theory, as it has been shown for *M*_{ijPKMN}. Thus, an explicit expression for the second-order derivative may be obtained as
A3

## Appendix B. A simplification for transversely isotropic symmetry

Following Ting & Lee (1997) and Akamatsu & Tanuma (1997) among others, some simplifications for transversely isotropic materials are next described. Any plane that contains the *x*_{3}-axis is a plane of material symmetry so that, without loss of generality, the plane *x*_{2}=0 is next taken. A point on this plane with coordinates such that is considered. Let **n**={0,1,0} and , where *ϕ* (figure 3) is given by , 0≤*ϕ*≤*π*. The Barnett–Lothe tensor for these points has the following structure:
B1
Regarding the rotational material symmetry, and therefore of the solution of the problem, a general expression for the extended Barnett–Lothe tensor **H** for any point can be obtained as
B2
where ** Ω** is a proper orthogonal transformation matrix in five dimensions defined by
B3
where with 0<

*θ*<2

*π*(figure 3

*a*). The present choice for the

**n**and

**m**vectors simplifies the structure of

**Q**,

**R**and

**T**matrix leading to a more compact expression for transversely isotropic Green’s function.

Furthermore, components , , and (*j*=1 and 3) and , , , , , , , , , and vanish for points on the plane *x*_{2}=0. Then, a general expression for the modulation function for transversely isotropic material is obtained as
B4
where *Ω*^{(3)} is the proper orthogonal transformation matrix in three dimensions defined by
B5

## Footnotes

- Received July 23, 2009.
- Accepted September 25, 2009.

- © 2009 The Royal Society