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Three-dimensional Green’s functions for infinite anisotropic piezoelectric media
International Journal of Solids and Structures 44 (2007) 1680–1684 www.elsevier.com/locate/ijsolstr
Short Communication
Three-dimensional Green’s functions for infinite anisotropic piezoelectric media Xiangyong Li *, Minzhong Wang State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, 100871 Beijing, People’s Republic of China Received 11 November 2005; received in revised form 9 June 2006 Available online 20 June 2006
Abstract A direct, effective and concise method is adopted in this paper to find out the Green’s functions for infinite anisotropic piezoelectric media. The partial differential equations satisfied by the Green’s functions turn into a set of inhomogeneous algebraic equations after by using Fourier transform. Then inverse transform the solutions of the algebraic equations, the Green’s functions can be expressed by contour integral. Finally, the explicit expression can be obtained for the Green’s functions by using residual theory. The method demonstrated in this paper is easier to follow by people without knowledge of Radon transform, which has been used to obtain the Green’s functions by others. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Green’s function; Fourier transform; Contour integral; Residual theory
1. Introduction Because of the importance of Green’s functions for solving three dimensional eigenstrain problems such as inclusion problems in anisotropic piezoelasticity, Elastostatic Green’s functions in 3D anisotropic media have been studied by many researcher, for example, Freedholm (1900), Lifshitz and Rozenzweig (1947), Synge (1957), Willis (1965), Mura and Kinoshita (1971), Pan and Chou (1976), Wang (1997). For anisotropic piezoelastic media, Masayuki and Kazumi (1997) used Stroth theory to find out the 3D Green’s functions, while Pan and Tonon (2000) extended the results obtained by Wang (1997) to piezoelectricity, and the method of Radon transform was used in their paper. Radon transform, which has relations to other integral transforms such as Fourier transform, Gegenbauer transform and Hough transform, has been used widely in many fields (Deans, 1983). It was also used in elasticity: Willis (1971) made extensive use of the Radon transform of the traction vector in developing a method for determining the stresses in a composite body consisting of dissimilar isotropic elastic halfspaces bonded over a circular region, and Willis (1972) extended the method to the dual problem of the stress analysis of *
Corresponding author. Tel.: +86 01052763578. E-mail address: [email protected] (X. Li).
0020-7683/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.06.021
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
1681
two dissimilar elastic halfspaces that are perfectly bonded together at all points of their interface except over a circular region which delimits a crack; Wu (1998) generalized the Stroh Formalism to 3-Dimensional Anisotropic Elasticity. However, the method of Radon transform is not very easy to implement in deriving Green’s functions. The reason is that one must use the plane integral representation for Dirac delta function firstly. And in papers (Wang, 1997 and Pan and Tonon, 2000), the processing of an integral over a rectangular parallelepiped is not very elegant. Aimed to find out the Green’s functions more easily, a simple and convenient method i.e. Fourier transform is adopted in this paper. Firstly, the Green’s functions can be expressed by a contour integral after by using Fourier transform, secondly, the contour integral is changed into an infinite integral by changing the variables, finally them can be expressed by algebraic expression after by using residual theory. The results given by this paper are the same as that given by Ernian Pan and Fulvio Tonon, but the method in this paper seems to be more easier to people without knowledge of Radon transform. 2. Basic equations of linear piezoelectricity With the notation introduced by Barnett and Lothe (1975), the elastic displacement and electric potential, the elastic strain and electric field, the stress and electric displacement, and the elastic and electric moduli can be united together as ui I ¼ 1; 2; 3 uI ¼ ð1Þ / I ¼4 I ¼ 1; 2; 3 cij cIj ¼ ð2Þ Ej I ¼ 4 rij J ¼ 1; 2; 3 ð3Þ riJ ¼ Di J ¼ 4 8 C ijkl J ; K ¼ 1; 2; 3 > > > < elij J ¼ 1; 2; 3; K ¼ 4 ð4Þ C iJKl ¼ > eikl J ¼ 4; K ¼ 1; 2; 3 > > : eil J ; K ¼ 4 In this and the following sections, the elastic displacement and the electric potential, defined by Eq. (1), will be called generalized displacements, and the elastic strain and the electric field, defined by Eq. (2), will be called generalized strains, and the elastic stress and electric displacement, defined by Eq. (3), will be called generalized stresses. Additionally, it should be noted that the lowercase and uppercase subscripts in Eqs. (1)–(4) take on the range 1–3 and 1–4, respectively. Therefore, with the concise notations above, the constitutive relations and the equilibrium equations can be expressed by riJ ¼ C iJKl cKl
ð5Þ
riJ ;i þ F J ¼ 0;
ð6Þ
and respectively, where the generalized body forces FJ are defined by FJ J ¼ 1; 2; 3 FJ ¼ Q J ¼ 4
ð7Þ
3. Integral expression for Green’s functions Let d(x) = d(x1, x2, x3) be the Dirac delta function centered at the origin of a space-fixed Cartesian coordinates (O, x1, x2, x3) and dJP be the fourth-rank Kronecker delta. The Green’s functions are the fundamental
1682
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
solutions of Eq. (6) caused by a generalized point force. Mathematically, the Green’s functions GKP(x)(K,P = 1, 2, 3, 4) is defined by the following partial differential equations: C iJKl GKP ;li ðxÞ ¼ dJP dðxÞ:
ð8Þ
We now take Fourier transforms in the x1-, x2- and x3-directions for GKP(x), thus Z 1 e KP ðnÞ ¼ G GKP ðxÞeixn dx
ð9Þ
1
where n = (n1, n2, n3), and Z 1 e KP ðnÞeixn dn: GKP ðxÞ ¼ G
ð10Þ
1
Then Eq. (8) turns into a set of algebraic equations, e KP ðnÞnl ni ¼ dJP : C iJKl G
ð11Þ
Solving the algebraic Eq. (11), we can obtain e KP ðnÞ ¼ C1 dJP ¼ C1 ; G JK KP
ð12Þ
where CJK = CiJKlnlni. Taking Fourier inverse transforms for Eq. (12), we arrive at Z 1 Z 1 AKP ðnÞ ixn 3 3 1 ixn e GKP ðxÞ ¼ ð2pÞ CKP ðnÞe dn ¼ ð2pÞ dn: 1 1 DðnÞ By the transformation of n ! n, GKP(x) can also be expressed as Z 1 AKP ðnÞ ixn 3 e dn: GKP ðxÞ ¼ ð2pÞ 1 DðnÞ
ð13Þ
ð14Þ
In Eqs. (13) and (14), D(n) and AKP(n) are the determinant and adjoint of matrix CJK(n), respectively. From the definition of matrix CJK(n), D(n) and AKP(n) are homogeneous of degree of 8 and 6 in n, respectively. Define the volume element dn as dn = n2 dn dS, where n = (nini)1/2 and S is the surface element on the unit sphere S2 in the n-space, centered at the origin of the coordinates ni (see Fig. 1). And denote n = nn0, x = xx0, where n0 and x0 are the unit vector in the directs of n and x, respectively, and x = (xixi)1/2. By adding Eqs. (13) and (14), in the sphere coordinates, we have 3
GKP ðxÞ ¼
ð2pÞ 2
Z
Z
1
dn 1
einxn
0
x0
S2
AKP ðn0 Þ dSðn0 Þ: Dðn0 Þ
ð15Þ
ξ3 y
O
ξ
x dS
θ
ξ0
ϕ
ξ1
S1
ξ2
S2 Fig. 1. The unit sphere S2 in the n-space. Green’s functions at point x is expressed by a line integral along S1 which lies on the plane perpendicular to x.
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
Integrate Eq. (15) with respect to n, we have 2 Z ð2pÞ AKP ðn0 Þ dSðn0 Þ: dðxn0 x0 Þ GKP ðxÞ ¼ 0 2 2 Þ Dðn S
1683
ð16Þ
Denoting the angle between n0 and x0 by h, we have n0 x0 ¼ cos h; dðn0 x0 Þ ¼ sin h dh; dSðn0 Þ ¼ sin h dh du; where u is defined on the plane perpendicular to x. Therefore, we can obtain Z 1 Z 2p 1 AKP ðn0 Þ GKP ðxÞ ¼ 2 du dðn0 x0 Þdðxn0 x0 Þ 8p 1 Dðn0 Þ 0 I 1 AKP ðn0 Þ ¼ 2 du; 8p x S 1 Dðn0 Þ
ð17Þ
where S1 is the unit circle on S2 intersected by the plane perpendicular to x. 4. Algebraic expression for Green’s functions Noting that n0 = eiu, then Eq. (17) can be evaluated by Z p=2 1 AKP ðeiu Þ du: GKP ðxÞ ¼ 2 4p x p=2 Dðeiu Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi Define u = arctanf and R ¼ 1 þ f2 , and substitute them to Eq. (18), then we have Z 1 1 AKP ðRei arctan f Þ GKP ðxÞ ¼ 2 df 4p x 1 DðRei arctan f Þ Z 1 1 AKP ðp þ fqÞ df; ¼ 2 4p x 1 Dðp þ fqÞ
ð18Þ
ð19Þ
where p and q are two fixed unit vectors(see Fig. 2), p ? q and Rei arctanf = Reiu = p + fq. Note that D(n) has non-real roots only and that these occur in complex conjugate pairs, since the differential Eq. (8) are elliptic and the coefficients in D(n) are all real. Therefore the eight-order polynomial equation of f Dðp þ fqÞ ¼ 0;
ð20Þ
has eight roots, four of them being the conjugate of the remainder. With these roots, D(p + qf) can be expressed as Dðp þ fqÞ ¼
8 X k¼0
akþ1 fk ¼ a9
4 Y
ðf fm Þðf fm Þ;
ð21Þ
m¼1
q
p O
ϕ R
ζ
Fig. 2. The unit vectors p and q on the plane where the unit circle S1 located.
1684
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
where a9 is the coefficient of f8, Im fm > 0, (m = 1, 2, 3, 4), and fm is the conjugate of fm. For simplicity, we just give the expression for the case that the roots of D(p + fq) = 0 are all distinct. Then in terms of residual theory, the Green’s functions can be finally expressed explicitly as GKP ðxÞ ¼
4 X 1 AKP ðp þ fqÞ Im : Q4 2px m¼1 a9 ðfm fm Þ k¼1;k6¼m ðfm fk Þðfm fk Þ
ð22Þ
5. Conclusion From the expression equation (22) for the Green’ functions, we can see that our result is the same as that Pan and Tonon (2000) gave. But the method used in this paper is more concise and direct than the Radon transform to obtain the Green’ functions. Acknowledgment The authors acknowledge the support of the National Natural Science Foundation of China Grants No. 10372003. References Barnett, D.M., Lothe, J., 1975. Dislocation and line charges in anisotropic piezoelectric insulators. Phys. Status Solidi B 67, 105–111. Deans, S.R., 1983. The Radon Transform and Some of its Applications. John Wiley & Sons, New York. Freedholm, I., 1900. Sur les equation de l’equilibre d’un corps solide elastique. Acta Math. 23, 1–42. Lifshitz, I.M., Rozenzweig, L.N., 1947. On the construction of the Green’s tensor for the basic equation of the theory of elasticity of an anisotropic infinite medium. Zh. Eksp. Teor. Fiz. 17, 783–791. Masayuki, Akamatsu, Kazumi, Tanuma, 1997. Green’s function of anisotropic piezoelectricity. Proc. Roy. Soc. London A 453, 473–487. Mura, T., Kinoshita, N., 1971. Green’s functions for anisotropic elasticity. Phys. Status Solidi B 47, 607–618. Pan, Y.C., Chou, T.W., 1976. Point force solution for an infinite transversely isotropic solid. J. Appl. Mech. 43, 608–612. Pan, E., Tonon, F., 2000. Three-dimensional Green’s functions in anisotropic piezoelectric solids. Int. J. Solids Struct. 37, 943–958. Synge, J.L., 1957. The Hypercircle in Mathematical Physics. Cambridge University Press. Wang, C.Y., 1997. Elastostatic fields produced by a point source in solids of general anisotropy. J. Eng. Math. 32, 41–52. Willis, J.R., 1965. The elastic interaction energy of dislocation loops in anisotropic media. Q. J. Mech. Appl. Math. 18, 419–433. Willis, J.R., 1971. Interfacial stresses induced by arbitrary loading of dissimilar elastic half-spaces joined over a circular region. J. Inst. Maths Appl. 7, 179–197. Willis, J.R., 1972. The penny-shaped crack on an interface. Q. J. Mech. Appl. 25, 367–385. Wu, K.C., 1998. Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity. J. Elasticity 51, 213–225.
Short Communication
Three-dimensional Green’s functions for infinite anisotropic piezoelectric media Xiangyong Li *, Minzhong Wang State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, Peking University, 100871 Beijing, People’s Republic of China Received 11 November 2005; received in revised form 9 June 2006 Available online 20 June 2006
Abstract A direct, effective and concise method is adopted in this paper to find out the Green’s functions for infinite anisotropic piezoelectric media. The partial differential equations satisfied by the Green’s functions turn into a set of inhomogeneous algebraic equations after by using Fourier transform. Then inverse transform the solutions of the algebraic equations, the Green’s functions can be expressed by contour integral. Finally, the explicit expression can be obtained for the Green’s functions by using residual theory. The method demonstrated in this paper is easier to follow by people without knowledge of Radon transform, which has been used to obtain the Green’s functions by others. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Green’s function; Fourier transform; Contour integral; Residual theory
1. Introduction Because of the importance of Green’s functions for solving three dimensional eigenstrain problems such as inclusion problems in anisotropic piezoelasticity, Elastostatic Green’s functions in 3D anisotropic media have been studied by many researcher, for example, Freedholm (1900), Lifshitz and Rozenzweig (1947), Synge (1957), Willis (1965), Mura and Kinoshita (1971), Pan and Chou (1976), Wang (1997). For anisotropic piezoelastic media, Masayuki and Kazumi (1997) used Stroth theory to find out the 3D Green’s functions, while Pan and Tonon (2000) extended the results obtained by Wang (1997) to piezoelectricity, and the method of Radon transform was used in their paper. Radon transform, which has relations to other integral transforms such as Fourier transform, Gegenbauer transform and Hough transform, has been used widely in many fields (Deans, 1983). It was also used in elasticity: Willis (1971) made extensive use of the Radon transform of the traction vector in developing a method for determining the stresses in a composite body consisting of dissimilar isotropic elastic halfspaces bonded over a circular region, and Willis (1972) extended the method to the dual problem of the stress analysis of *
Corresponding author. Tel.: +86 01052763578. E-mail address: [email protected] (X. Li).
0020-7683/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2006.06.021
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
1681
two dissimilar elastic halfspaces that are perfectly bonded together at all points of their interface except over a circular region which delimits a crack; Wu (1998) generalized the Stroh Formalism to 3-Dimensional Anisotropic Elasticity. However, the method of Radon transform is not very easy to implement in deriving Green’s functions. The reason is that one must use the plane integral representation for Dirac delta function firstly. And in papers (Wang, 1997 and Pan and Tonon, 2000), the processing of an integral over a rectangular parallelepiped is not very elegant. Aimed to find out the Green’s functions more easily, a simple and convenient method i.e. Fourier transform is adopted in this paper. Firstly, the Green’s functions can be expressed by a contour integral after by using Fourier transform, secondly, the contour integral is changed into an infinite integral by changing the variables, finally them can be expressed by algebraic expression after by using residual theory. The results given by this paper are the same as that given by Ernian Pan and Fulvio Tonon, but the method in this paper seems to be more easier to people without knowledge of Radon transform. 2. Basic equations of linear piezoelectricity With the notation introduced by Barnett and Lothe (1975), the elastic displacement and electric potential, the elastic strain and electric field, the stress and electric displacement, and the elastic and electric moduli can be united together as ui I ¼ 1; 2; 3 uI ¼ ð1Þ / I ¼4 I ¼ 1; 2; 3 cij cIj ¼ ð2Þ Ej I ¼ 4 rij J ¼ 1; 2; 3 ð3Þ riJ ¼ Di J ¼ 4 8 C ijkl J ; K ¼ 1; 2; 3 > > > < elij J ¼ 1; 2; 3; K ¼ 4 ð4Þ C iJKl ¼ > eikl J ¼ 4; K ¼ 1; 2; 3 > > : eil J ; K ¼ 4 In this and the following sections, the elastic displacement and the electric potential, defined by Eq. (1), will be called generalized displacements, and the elastic strain and the electric field, defined by Eq. (2), will be called generalized strains, and the elastic stress and electric displacement, defined by Eq. (3), will be called generalized stresses. Additionally, it should be noted that the lowercase and uppercase subscripts in Eqs. (1)–(4) take on the range 1–3 and 1–4, respectively. Therefore, with the concise notations above, the constitutive relations and the equilibrium equations can be expressed by riJ ¼ C iJKl cKl
ð5Þ
riJ ;i þ F J ¼ 0;
ð6Þ
and respectively, where the generalized body forces FJ are defined by FJ J ¼ 1; 2; 3 FJ ¼ Q J ¼ 4
ð7Þ
3. Integral expression for Green’s functions Let d(x) = d(x1, x2, x3) be the Dirac delta function centered at the origin of a space-fixed Cartesian coordinates (O, x1, x2, x3) and dJP be the fourth-rank Kronecker delta. The Green’s functions are the fundamental
1682
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
solutions of Eq. (6) caused by a generalized point force. Mathematically, the Green’s functions GKP(x)(K,P = 1, 2, 3, 4) is defined by the following partial differential equations: C iJKl GKP ;li ðxÞ ¼ dJP dðxÞ:
ð8Þ
We now take Fourier transforms in the x1-, x2- and x3-directions for GKP(x), thus Z 1 e KP ðnÞ ¼ G GKP ðxÞeixn dx
ð9Þ
1
where n = (n1, n2, n3), and Z 1 e KP ðnÞeixn dn: GKP ðxÞ ¼ G
ð10Þ
1
Then Eq. (8) turns into a set of algebraic equations, e KP ðnÞnl ni ¼ dJP : C iJKl G
ð11Þ
Solving the algebraic Eq. (11), we can obtain e KP ðnÞ ¼ C1 dJP ¼ C1 ; G JK KP
ð12Þ
where CJK = CiJKlnlni. Taking Fourier inverse transforms for Eq. (12), we arrive at Z 1 Z 1 AKP ðnÞ ixn 3 3 1 ixn e GKP ðxÞ ¼ ð2pÞ CKP ðnÞe dn ¼ ð2pÞ dn: 1 1 DðnÞ By the transformation of n ! n, GKP(x) can also be expressed as Z 1 AKP ðnÞ ixn 3 e dn: GKP ðxÞ ¼ ð2pÞ 1 DðnÞ
ð13Þ
ð14Þ
In Eqs. (13) and (14), D(n) and AKP(n) are the determinant and adjoint of matrix CJK(n), respectively. From the definition of matrix CJK(n), D(n) and AKP(n) are homogeneous of degree of 8 and 6 in n, respectively. Define the volume element dn as dn = n2 dn dS, where n = (nini)1/2 and S is the surface element on the unit sphere S2 in the n-space, centered at the origin of the coordinates ni (see Fig. 1). And denote n = nn0, x = xx0, where n0 and x0 are the unit vector in the directs of n and x, respectively, and x = (xixi)1/2. By adding Eqs. (13) and (14), in the sphere coordinates, we have 3
GKP ðxÞ ¼
ð2pÞ 2
Z
Z
1
dn 1
einxn
0
x0
S2
AKP ðn0 Þ dSðn0 Þ: Dðn0 Þ
ð15Þ
ξ3 y
O
ξ
x dS
θ
ξ0
ϕ
ξ1
S1
ξ2
S2 Fig. 1. The unit sphere S2 in the n-space. Green’s functions at point x is expressed by a line integral along S1 which lies on the plane perpendicular to x.
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
Integrate Eq. (15) with respect to n, we have 2 Z ð2pÞ AKP ðn0 Þ dSðn0 Þ: dðxn0 x0 Þ GKP ðxÞ ¼ 0 2 2 Þ Dðn S
1683
ð16Þ
Denoting the angle between n0 and x0 by h, we have n0 x0 ¼ cos h; dðn0 x0 Þ ¼ sin h dh; dSðn0 Þ ¼ sin h dh du; where u is defined on the plane perpendicular to x. Therefore, we can obtain Z 1 Z 2p 1 AKP ðn0 Þ GKP ðxÞ ¼ 2 du dðn0 x0 Þdðxn0 x0 Þ 8p 1 Dðn0 Þ 0 I 1 AKP ðn0 Þ ¼ 2 du; 8p x S 1 Dðn0 Þ
ð17Þ
where S1 is the unit circle on S2 intersected by the plane perpendicular to x. 4. Algebraic expression for Green’s functions Noting that n0 = eiu, then Eq. (17) can be evaluated by Z p=2 1 AKP ðeiu Þ du: GKP ðxÞ ¼ 2 4p x p=2 Dðeiu Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi Define u = arctanf and R ¼ 1 þ f2 , and substitute them to Eq. (18), then we have Z 1 1 AKP ðRei arctan f Þ GKP ðxÞ ¼ 2 df 4p x 1 DðRei arctan f Þ Z 1 1 AKP ðp þ fqÞ df; ¼ 2 4p x 1 Dðp þ fqÞ
ð18Þ
ð19Þ
where p and q are two fixed unit vectors(see Fig. 2), p ? q and Rei arctanf = Reiu = p + fq. Note that D(n) has non-real roots only and that these occur in complex conjugate pairs, since the differential Eq. (8) are elliptic and the coefficients in D(n) are all real. Therefore the eight-order polynomial equation of f Dðp þ fqÞ ¼ 0;
ð20Þ
has eight roots, four of them being the conjugate of the remainder. With these roots, D(p + qf) can be expressed as Dðp þ fqÞ ¼
8 X k¼0
akþ1 fk ¼ a9
4 Y
ðf fm Þðf fm Þ;
ð21Þ
m¼1
q
p O
ϕ R
ζ
Fig. 2. The unit vectors p and q on the plane where the unit circle S1 located.
1684
X. Li, M. Wang / International Journal of Solids and Structures 44 (2007) 1680–1684
where a9 is the coefficient of f8, Im fm > 0, (m = 1, 2, 3, 4), and fm is the conjugate of fm. For simplicity, we just give the expression for the case that the roots of D(p + fq) = 0 are all distinct. Then in terms of residual theory, the Green’s functions can be finally expressed explicitly as GKP ðxÞ ¼
4 X 1 AKP ðp þ fqÞ Im : Q4 2px m¼1 a9 ðfm fm Þ k¼1;k6¼m ðfm fk Þðfm fk Þ
ð22Þ
5. Conclusion From the expression equation (22) for the Green’ functions, we can see that our result is the same as that Pan and Tonon (2000) gave. But the method used in this paper is more concise and direct than the Radon transform to obtain the Green’ functions. Acknowledgment The authors acknowledge the support of the National Natural Science Foundation of China Grants No. 10372003. References Barnett, D.M., Lothe, J., 1975. Dislocation and line charges in anisotropic piezoelectric insulators. Phys. Status Solidi B 67, 105–111. Deans, S.R., 1983. The Radon Transform and Some of its Applications. John Wiley & Sons, New York. Freedholm, I., 1900. Sur les equation de l’equilibre d’un corps solide elastique. Acta Math. 23, 1–42. Lifshitz, I.M., Rozenzweig, L.N., 1947. On the construction of the Green’s tensor for the basic equation of the theory of elasticity of an anisotropic infinite medium. Zh. Eksp. Teor. Fiz. 17, 783–791. Masayuki, Akamatsu, Kazumi, Tanuma, 1997. Green’s function of anisotropic piezoelectricity. Proc. Roy. Soc. London A 453, 473–487. Mura, T., Kinoshita, N., 1971. Green’s functions for anisotropic elasticity. Phys. Status Solidi B 47, 607–618. Pan, Y.C., Chou, T.W., 1976. Point force solution for an infinite transversely isotropic solid. J. Appl. Mech. 43, 608–612. Pan, E., Tonon, F., 2000. Three-dimensional Green’s functions in anisotropic piezoelectric solids. Int. J. Solids Struct. 37, 943–958. Synge, J.L., 1957. The Hypercircle in Mathematical Physics. Cambridge University Press. Wang, C.Y., 1997. Elastostatic fields produced by a point source in solids of general anisotropy. J. Eng. Math. 32, 41–52. Willis, J.R., 1965. The elastic interaction energy of dislocation loops in anisotropic media. Q. J. Mech. Appl. Math. 18, 419–433. Willis, J.R., 1971. Interfacial stresses induced by arbitrary loading of dissimilar elastic half-spaces joined over a circular region. J. Inst. Maths Appl. 7, 179–197. Willis, J.R., 1972. The penny-shaped crack on an interface. Q. J. Mech. Appl. 25, 367–385. Wu, K.C., 1998. Generalization of the Stroh Formalism to 3-Dimensional Anisotropic Elasticity. J. Elasticity 51, 213–225.