On the generalized Barnett Lothe tensors for monoclinic piezoelectric materials

Transcription

1 International Journal of Solids and Structures 44 (00) On the generalized Barnett Lothe tensors for monoclinic piezoelectric materials J.Y. Liou a, *, J.C. Sung b a Department of Civil Engineering, Kao Yuan University, Kaohsiung 8151, Taiwan, ROC b Department of Civil Engineering, National Cheng Kung University, Tainan 0101, Taiwan, ROC Received 8 September 00; received in revised form 1 November 00 Available online 8 December 00 Abstract The three generalized Barnett Lothe tensors L, S and H, appearing frequently in the investigations of the two-dimensional deformations of anisotropic piezoelectric materials, may be expressed in terms of the material constants. In this paper, the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x = 0 are constructed based on the extended Stroh formalism. Then the three generalized Barnett Lothe tensors are calculated from these eigenvectors and are expressed explicitly in terms of the elastic stiffness instead of the reduced elastic compliance. The special case of transversely isotropic piezoelectric materials is also presented. Ó 00 Elsevier Ltd. All rights reserved. Keywords: Stroh formalism; Generalized Barnett Lothe tensors; Monoclinic piezoelectricity; Transversely isotropic piezoelectricity; Anisotropic solid 1. Introduction It is known that Stroh formalism is mathematically elegant and technically powerful in determining the two-dimensional deformations of anisotropic elastic solids (Stroh, 1958; Ting, 199). The three real matrices L, S and H, called Barnett Lothe tensors appear often in the solutions of many anisotropic linear elastic boundary value problems. Due to its importance, the explicit expressions of Barnett Lothe tensors have been investigated by many researchers for purely elasticity problem. The most general anisotropic materials without any symmetry plane assumed were considered by Wei and Ting (1994) and Ting (199). Other related works for anisotropic elastic materials with special symmetry plane considered are cited in the paper, e.g., by Soh et al. (001). However, all their representation of the three real matrices L, S and H are essentially in terms of the reduced elastic compliances. * Corresponding author. Present address: No. 181, Jhongshan Rd., Lujhu Township, Kaohsiung County 8151, Taiwan, ROC. Tel.: ; fax: address: ljy@cc.kyu.edu.tw (J.Y. Liou) /$ - see front matter Ó 00 Elsevier Ltd. All rights reserved. doi:10.101/j.ijsolstr

2 The extension of the Stroh formalism to the investigation of anisotropic piezoelectricity was made very earlier by Barnett and Lothe (195). In recent years, to better understanding the intrinsic coupling effect between mechanical and electrical field, this extended Stroh formalism is widely used in the piezoelectric research field. In applying this extended formalism to anisotropic piezoelectricity problem, similar to the purely elasticity problem, solutions may often be expressed in terms of three real 4 4 matrices L, S and H, or now called generalized Barnett Lothe tensors (Ting, 199). As noted by Soh et al. (001), based on the extended Stroh formalism, the eigenvalues and the associated eigenvectors are not able to be expressed explicitly for piezoelectric materials which make the explicit expressions of the generalized Barnett Lothe tensors unavailable. To circumvent this, the modified Lekhinitskii formalism was adopted by them (Soh et al., 001) to construct the eigenvalues and the corresponding eigenvectors for the general anisotropic piezoelectric materials and the explicit expressions of the generalized Barnett Lothe tensors for transversely isotropic piezoelectric materials are presented. Their results are all in terms of the reduced elastic compliances. The approach of constructing the eigenvalues and the corresponding eigenvectors from the Lekhinitskii formalism was earlier used by many researchers e.g. Suo (1990), Ting (199) for purely elastic materials. In this paper, we construct the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x = 0 based on the extended Stroh formalism in a straightforward manner. The eigenvalues and eigenvectors are directly related to the elastic stiffness instead of the reduced elastic compliance. With algebraic calculations (Wei and Ting, 1994), the explicit expressions of the three generalized Barnett Lothe tensors are presented in a concise form. Results for the special case of transversely isotropic piezoelectric materials are also given. Below is the plan of our work. In Section, the extended Stroh formalism is outlined. Then in Section the eigenvalues and eigenvectors are constructed for monoclinic piezoelectric materials with the symmetry plane at x = 0. In Section 4 the generalized Barnett Lothe tensors are computed. In Section 5, the special case of transversely isotropic piezoelectric materials is presented, and finally in Section concludes the work.. The extended Stroh formalism J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) In this section, the extended Stroh formalism is introduced in the following. The convention that all Latin indices range from 1 to (except indicated) and repeated indices imply summation are all followed. Bold-faced symbol stands for either column vectors or matrices depending on whether lower-case or uppercase is used. In a rectangular coordinate system x i (i = 1,,), it is known that for two-dimensional piezoelectric materials the generalized displacement vector u =[u 1,u,u,u 4 ] T (u i, i = 1,,: the elastic displacements; u 4 : electric potential) and the generalized stress function vector, u =[u 1,u,u,u 4 ] T are expressed as u ¼ RefAfðzÞg; u ¼ RefBfðzÞg; ð:1þ ð:þ A ¼½a 1 ; a ; a ; a 4 Š; B ¼½b 1 ; b ; b ; b 4 Š; f ðzþ ¼½f 1 ðz 1 Þ; f ðz Þ; f ðz Þ; f 4 ðz 4 ÞŠ T ; ð:þ ð:4þ ð:5þ the superscript T appearing above stands for the transpose and z k = x 1 + p k x. Unknown complex number p k and constant vector a k are determined by the eigenrelation Ua k ¼ 0 ðk ¼ 1; ; ; 4Þ; ð:þ U ¼½Qþp k ðr þ R T Þþp k TŠ; Q ¼ c i1k1 e 1i1 ; R ¼ c i1k e 1i e T 1k1 a 11 e T k1 a 1 ; T ¼ c ik e i e T k a ð:þ : ð:8þ

3 510 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) The material constants appearing in the matrices Q, R and T which are taking some special values for some indices are arisen from the constitutive equations for a linear piezoelectric material (Suo et al., 199) which are given by r ij ¼ c ijkl e kl e kij E k ; D i ¼ e ikl e kl þ a ik E k ; ð:9þ e ij and E i are the strains and electric fields, r ij and D i are the stresses and electrical displacements, and c ijkl, e kij and a ij are the elastic stiffnesses, piezoelectric-stress and dielectric constants, respectively. The generalized stress function vector expressed in Eq. (.) may be employed to evaluate the stress and electric displacement components directly by the formula as t 1 ¼½r 11 ; r 1 ; r 1 ; D 1 Š T ¼ ou ; t ¼½r 1 ; r ; r ; D Š T ¼ ou : ð:10þ ox ox 1 The column vector of matrix B =[b 1,b,b,b 4 ] appearing in Eq. (.4) is related to the column vector of matrix A =[a 1,a,a,a 4 ] in the following form b k ¼ðR T þ p k TÞa k ; k ¼ 1; ; ; 4: ð:11þ Matrices A and B defined above possess the orthogonality relations (Ting, 199) from which three real matrices L, S and H, which are called the generalized Barnett Lothe tensors (Ting, 199), may be defined as L ¼ ibb T ; S ¼ iðab T IÞ; H ¼ iaa T ; ð:1þ i = 1 and I is a 4 4 unit real matrix. Matrices L and H are symmetric and non-singular. The defined three generalized Barnett Lothe tensors are not independent. They are related by HL SS ¼ I; SH þ HS T ¼ 0; LS þ S T L ¼ 0: ð:1a; b; cþ As has been pointed out by Soh et al. (001), the explicit expressions of A and B can not be easily obtained for anisotropic piezoelectric material based on the extended Stroh formalism. It would be even more difficult to obtain explicit expressions for real matrices L, S and H. In the following sections, we will construct matrices A and B directly from the extended Stroh formalism and will present the three generalized Barnett Lothe tensors of L, S and H in explicit forms for monoclinic piezoelectric materials.. The explicit expressions of A and B for monoclinic piezoelectric material In this section the explicit expressions of A and B will be constructed for monoclinic piezoelectric materials based on the extended Stroh formalism. For monoclinic piezoelectric materials of class m, with the symmetry plane at x =0(Nye, 1985), the constitutive equation (Eq. (.9)) is r 11 c 11 c 1 c c 1 e 11 e 1 0 e 11 r c 1 c c 0 0 c e 1 e 0 e r c 1 c c c 4 c 5 c e 1 e 0 e r 0 0 c 4 c 44 c e 4 e r 1 ¼ 0 0 c 5 c 45 c e 5 e 1 ; ð:1þ r 1 c 1 c c 0 0 c e 1 e 0 e 1 D 1 e 11 e 1 e e 1 a 11 a 1 0 E 1 4 D 5 4 e 1 e e 0 0 e a 1 a 0 54 E 5 D e 4 e a E contracted notations e ia and c ab (a, b =1,,...,) have been used here for e ikl and c ijkl, respectively. The matrix U =[Q + p(r + R T )+p T] defined in Eq. (.) for this kind of material becomes

4 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) ðc p þ c 1 p þ c 11 Þ ðc p þðc 1 þ c Þp þ c 1 Þ 0 ðe p þðe 1 þ e 1 Þp þ e 11 Þ ðc p þðc 1 þ c Þp þ c 1 Þ ðc p þ c p þ c Þ 0 ðe p þðe 1 þ e Þp þ e 1 Þ U ¼ ðc 44 p þ c 45 p þ c 55 Þ 0 5 : ðe p þðe 1 þ e 1 Þp þ e 11 Þ ðe p þðe 1 þ e Þp þ e 1 Þ 0 ða p þ a 1 p þ a 11 Þ Letting the determinant of U equal to zero, an equation for the determination of the eigenvalues is produced as juj ¼jc 44 p þ c 45 p þ c 55 jd ¼ 0; which can be separated into the following two equations, i.e., jc 44 p þ c 45 p þ c 55 j¼0; ðc p þ c 1 p þ c 11 Þ ðc p þðc 1 þ c Þp þ c 1 Þ ðe p þðe 1 þ e 1 Þp þ e 11 Þ d ¼ ðc p þðc 1 þ c Þp þ c 1 Þ ðc p þ c p þ c Þ ðe p þðe 1 þ e Þp þ e 1 Þ ¼ 0: ðe p þðe 1 þ e 1 Þp þ e 11 Þ ðe p þðe 1 þ e Þp þ e 1 Þ ða p þ a 1 p þ a 11 Þ The root corresponding to Eq. (.4) with positive imaginary part is easily obtained as p ¼ c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 45 þ i c 44 c 55 c 45 ; ð:þ c 44 while the rest of the roots corresponding to Eq. (.5) with positive imaginary part may be expressed as p 1 ¼ m 1 þ in 1 ; p ¼ m þ in ; p 4 ¼ m 4 þ in 4 ; ð:þ in which m k (k = 1,,4) are real, n k (k = 1,,4) are real and positive. Once all eigenvalues are determined, we are then in a position to determine the corresponding eigenvectors a and b. To proceed this, we first note that the eigenvectors a may be written as a k ¼½a 1k ; a k ; 0; a 4k Š T ðk ¼ 1; ; 4Þ; a ¼½0; 0; a ; 0Š T ; ð:8þ a 1k, a k, a 4k (k = 1,,4) and a are unknown constants to be determined. The zero elements of a in Eq. (.8) are easily understood from the structure of the matrix U stated in Eq. (.). As to the eigenvector b, the relationship between a and b in Eq. (.11) implies that b is of the form b k ¼½b 1k ; b k ; 0; b 4k Š T ðk ¼ 1; ; 4Þ; b ¼½0; 0; 1; 0Š T ; ð:9þ b 1k, b k and b 4k (k = 1,,4) are also unknown constants. Without loss of generality the unit length of eigenvector b has been taken in Eq. (.9). The condition of r 1 = r 1 (from Eqs. (.) and (.10)) implies that b 1k and b k are actually related to each other by b 1k ¼ p k b k ðk ¼ 1; ; 4Þ: ð:10þ With this relation the eigenvectors b k =[b 1k,b k,0,b 4k ] T (k = 1,,4) may be further expressed as b k ¼½ p k ; 1; 0; h k Š T ðk ¼ 1; ; 4Þ; ð:11þ we have chosen b k =1(k = 1,,4) and a new variable is taken for b 4k, i.e., h k = b 4k (or h k = b 4k /b k if b k 5 1) for the simplicity. Note that among the four elements of the eigenvectors b k (k = 1,,4) shown in Eq. (.11), only one element h k is left to be determined. Therefore it is desirable to express the unknown elements of a k all in terms of h k. This may be achieved by using the relation between a k and b k (k = 1,,4) as shown in Eq. (.11), i.e., ð:þ ð:þ ð:4þ ð:5þ a k ¼ðR T þ p k TÞ 1 b k ¼ Vb k ; ð:1þ

5 51 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) V 11 V 1 0 V 14 V ðr T þ p k TÞ 1 V 1 V 0 V 4 ¼ V 0 5 ; ð:1þ V 41 V 4 0 V 44 explicit expressions of the non-zero elements of matrix V are listed in Appendix A. Substituting Eqs. (.8), (.11) and (.1) into Eq. (.1), we end up with the following a 1k ðp k Þ¼ðj 11 h k ðp k Þþj 10 Þ=D; ð:14aþ a k ðp k Þ¼ðj 1 h k ðp k Þþj 0 Þ=D ðk ¼ 1; ; 4Þ; ð:14bþ a 4k ðp k Þ¼ðj 41 h k ðp k Þþj 40 Þ=D; ð:14cþ a ðp Þ¼ðc 45 þ p c 44 Þ 1 i ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; c 44 c 55 c 45 ð:14dþ D is defined in Eq. (A), and j 11 ¼ðc e c e Þp k þðc e 1 þ c e c e 1 c e Þp k þ c e 1 c e 1 ; j 10 ¼ðe þ c a Þp k þðc a þ c a 1 þ e e þ e e 1 Þp k þðc a 1 þ c a þ e e 1 þ e þ e e 1 Þp k þ e e 1 þ c a 1 ; j 1 ¼ðc e c e Þp k þðc e 1 þ c 1 e c 1 e c e 1 Þp k þ c 1 e 1 c 1 e 1 ; j 0 ¼ ðc a þ e e Þp k ðc a þ c a 1 þ c 1 a þ e e 1 þ e e 1 þ e Þp k ðc a 1 þ c 1 a 1 þ c 1 a þ e e 1 þ e 1 e þ e 1 e 1 Þp k ðe 1 e 1 þ c 1 a 1 Þ; j 41 ¼ðc c c Þp k þðc 1c c 1 c Þp k þ c 1 c c 1 c ; j 40 ¼ðc e c e Þp k þðc e 1 þ c e c 1 e c e Þp k þðc e 1 c 1 e c 1 e Þp k þ c e 1 c 1 e ; ð:15aþ ð:15bþ ð:15cþ ð:15dþ ð:15eþ ð:15fþ It is observed from Eq. (.14) that the dependence of all the elements of eigenvectors a k (k = 1,,4) on the element h k are in a simple form. The only unknown element h k can be determined by substituting Eq. (.14) into Eq. (.) by enforcing the eigenvectors a k to satisfy the eigenrelation. By doing so, a vector equation is arrived for h k q Ua k ¼½q 1 ðh k Þ; 0; 0; q 4 ðh k ÞŠ T ¼ 0: ð:1þ Choosing for example the condition of q 4 (h k )=0,h k can be determined as h k ðp k Þ¼ b 4p 4 k þ b p k þ b p k þ b 1p k þ b 0 n 4 p 4 k þ n p k þ n p k þ n 1p k þ n 0 ; ð:1þ

6 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) b 4 ¼ c e 1 a c e 1 a þ c e a 1 c e a 1 þ e 1 e e e e 1 ; ð:18aþ b ¼ c e 11 a c 1 e 1 a c e 1 a þ c e 1 a þ c e 1 a 1 c e 1 a 1 þ c 1 e a 1 c e a 1 þ c e a 1 c e 1 a 1 c e a 11 þ c e a 11 þ e 1 e e þ e 1 e e 1 e 1 e e 1 e 1 e e e 1 þ e e 11; ð:18bþ b ¼ c e a 11 þ c 1 e a 11 c e a 11 c e 1 a 11 þ c 1 e a 1 c 1 e 1 a 1 c e 1 a 1 c e 1 a 1 þ c 1 e a 1 þ c e 1 a 1 þ c e 11 a 1 þ c e 11 a c 1 e 1 a c 1 e 1 a þ e 11 e e þ e 11 e e 1 e 1 e e 1 e 1 e e 1 e 1 e e 1 e 1 e 1 þ e 1 e ; ð:18cþ b 1 ¼ c e 11 a c 1 e 1 a c 1 e 1 a 1 c e 1 a 1 c 1 e 1 a 1 þ c 1 e a 1 þ c e 11 a 1 c e 1 a 11 þ c 1 e a 11 þ c 1 e a 11 þ e 11 e e 1 þ e 11 e e 1 þ e e 11 e 1 e e 1 e 1 e 1 e 1 ; b 0 ¼ c 1 e a 11 c e 1 a 11 þ c e 11 a 1 c 1 e 1 a 1 þ e 11 e e 1 e 1 e 1 ; n 4 ¼ c c a c e e þ c e þ c e c a ; n ¼ c e 1 e þ c e 1 e c e e 1 c e 1 e þ c e 1 e þ c 1 e c 1 e e c e e 1 c 1 c a þ c 1 c a þ c c a 1 c a 1; ð:18dþ ð:18eþ ð:18fþ ð:18gþ n ¼ c c a 11 þ c 1 c a 1 c 1 c a 1 c 1 c a þ c 1 c a c a 11 þ c e 11 e c e e 11 c 1 e e 1 c e 1 e 1 þ c 1 e 1 e þ c e 1 e þ c e 1 e 1 þ c 1 e e þ c e 1 c e e 1 c 1 e 1 e c e 1 e 1 þ c e 1 c 1e ; ð:18hþ n 1 ¼ c 1 e 1 c 1e 1 e 1 þ c e 11 e 1 c e 1 e 11 þ c e e 11 c 1 e 1 e þ c e 1 e 1 c e 1 e 1 þ c 1 e e 1 þ c 1 e 1 e c e 11 e þ c 1 c a 11 c 1 c a 11 c 1 c a 1 þ c 1 c a 1 ; n 0 ¼ c 1 c a 11 c 1 c a 11 þ c 1 e 1 e 1 c e 1 e 11 þ c e 11 e 1 c 1 e 1 ; ð:18iþ ð:18jþ It can be verified that the values of h k determined above by the condition of q 4 (h k ) = 0 automatically satisfy the other condition, i.e., q 1 (h k ) = 0. So far we have constructed the elements of matrices A and B in terms of the elastic stiffness. In summary, matrices A and B for monoclinic piezoelectric materials with the symmetry plane at x = 0 are expressed in the following form: a 11 ðp 1 Þ a 1 ðp Þ 0 a 14 ðp 4 Þ p 1 p 0 p 4 a 1 ðp 1 Þ a ðp Þ 0 a 4 ðp 4 Þ A ¼ 0 0 ðc 45 þ p c 44 Þ ; B ¼ ; ð:19þ a 41 ðp 1 Þ a 4 ðp Þ 0 a 44 ðp 4 Þ h 1 h 0 h 4 a ik (i, k = 1,,4) are given by Eq. (.14) and h k (k = 1,,4) is given by Eq. (.1). The structure of the eigenvectors a and b has been studied by many researchers, e.g., Soh et al. (001), all are from the point of view of Lekhinitskii formalism. We emphasize here that the same structure can also be constructed from the point of view of Stroh formalism. One feature of the present approach is that the eigenvectors obtained above are directly expressed in terms of the elastic stiffness, piezoelectric-stress and dielectric constants. For later computations of L, S and H, we need inverse of B which is given by h h 4 p 4 h p h 4 0 p 4 p B 1 ¼ 1 h 4 h 1 p 1 h 4 p 4 h 1 0 p 1 p 4 jbj jbj 0 5 ; ð:0þ h 1 h p h 1 p 1 h 0 p p 1

7 514 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) jbj ¼h 1 ðp p 4 Þþh ðp 4 p 1 Þþh 4 ðp 1 p Þ: ð:1þ 4. Explicit expressions of L, S and H for monoclinic piezoelectric material With the expressions of A and B constructed in Eq. (.19), the three real matrices L, S and H defined in Eq. (.1) can now be determined. As noted by Ting (199), Wei and Ting (1994) and Soh et al. (001), a direct substitution of A and B into Eq. (.1) would lead to an undesired algebraic calculation for satisfying the requirement of orthogonality relations (Ting, 199). An alternative approach adopted by Ting (199) and Soh et al. (001) is followed which employs the inverse of the generalized surface impedance matrix M (Lothe and Barnett, 19), which is written as Y ¼ M 1 iab 1 ¼ L 1 isl 1 : ð4:1þ Note that M is a non-singular Hermitian matrix (Suo et al., 199), as is the matrix M 1. Taking the real and imaginary part of Y defined in Eq. (4.1), we may obtain L 1 and SL 1, respectively. The results are Y 11 Y 1 0 Y 14 0 ^Y 1 0 ^Y 14 Y L 1 1 Y 0 Y 4 ¼ Y 0 5 ; ^Y ^Y 4 SL 1 ¼ ; ð4:þ Y 14 Y 4 0 Y 44 ^Y 14 ^Y Y 11 ¼ i½a 11 ðh h 4 Þþa 1 ðh 4 h 1 Þþa 14 ðh 1 h ÞŠ=jBj; ð4:aþ Y ¼ i½a 1 ðh p 4 h 4 p Þþa ðh 4 p 1 h 1 p 4 Þþa 4 ðh 1 p h p 1 ÞŠ=jBj; ð4:bþ Y 44 ¼ i½a 41 ðp 4 p Þþa 4 ðp 1 p 4 Þþa 44 ðp p 1 ÞŠ=jBj; ð4:cþ Y 1 ¼ Imf½a 11 ðh p 4 h 4 p Þþa 1 ðh 4 p 1 h 1 p 4 Þþa 14 ðh 1 p h p 1 ÞŠ=jBjg; ð4:dþ ^Y 1 ¼ Ref½a 11 ðh p 4 h 4 p Þþa 1 ðh 4 p 1 h 1 p 4 Þþa 14 ðh 1 p h p 1 ÞŠ=jBjg; ð4:eþ Y 14 ¼ Imf½a 11 ðp 4 p Þþa 1 ðp 1 p 4 Þþa 14 ðp p 1 ÞŠ=jBjg; ð4:fþ ^Y 14 ¼ Ref½a 11 ðp 4 p Þþa 1 ðp 1 p 4 Þþa 14 ðp p 1 ÞŠ=jBjg; ð4:gþ Y 4 ¼ Imf½a 1 ðp 4 p Þþa ðp 1 p 4 Þþa 4 ðp p 1 ÞŠ=jBjg; ð4:hþ ^Y 4 ¼ Ref½a 1 ðp 4 p Þþa ðp 1 p 4 Þþa 4 ðp p 1 ÞŠ=jBjg; ð4:iþ 1 Y ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð4:jþ c 44 c 55 c 45 Taking the inverse of L 1, we obtain the matrix L as ðy Y 44 Y 4 Þ=D L ðy 14 Y 4 Y 1 Y 44 Þ=D L 0 ðy 1 Y 4 Y 14 Y Þ=D L ðy 14 Y 4 Y 1 Y 44 Þ=D L ðy 11 Y 44 Y 14 L ¼ L 0 ðy 1 Y 14 Y 4 Y 11 Þ=D L Y ; ð4:4þ ðy 1 Y 4 Y 14 Y Þ=D L ðy 1 Y 14 Y 4 Y 11 Þ=D L 0 ðy 11 Y Y 1 Þ=D L D L ¼ Y 11 Y Y 44 þ Y 1 Y 14 Y 4 Y 11 Y 4 Y Y 14 Y 44Y 1 : ð4:5þ Using the identity S =(SL 1 )L, the matrix S is found to be

8 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) S 11 S 1 0 S 14 S 1 S 0 S 4 S ¼ ; ð4:þ S 41 S 4 0 S 44 S 11 ¼½^Y 1 ðy 1 Y 44 Y 14 Y 4 Þþ^Y 14 ðy 14 Y Y 1 Y 4 ÞŠ=D L ; S 1 ¼½^Y 1 ðy 14 Y 11Y 44 Þþ^Y 14 ðy 11 Y 4 Y 1 Y 14 ÞŠ=D L ; S 14 ¼½^Y 1 ðy 11 Y 4 Y 1 Y 14 Þþ^Y 14 ðy 1 Y 11Y ÞŠ=D L ; S 1 ¼½^Y 1 ðy Y 44 Y 4 Þþ^Y 4 ðy 14 Y Y 1 Y 4 ÞŠ=D L ; S ¼½^Y 1 ðy 14 Y 4 Y 1 Y 44 Þþ^Y 4 ðy 11 Y 4 Y 1 Y 14 ÞŠ=D L ; S 4 ¼½^Y 1 ðy 1 Y 4 Y 14 Y Þþ^Y 4 ðy 1 Y 11Y ÞŠ=D L ; S 41 ¼½^Y 14 ðy Y 44 Y 4 Þþ^Y 4 ðy 14 Y 4 Y 1 Y 44 ÞŠ=D L ; S 4 ¼½^Y 14 ðy 14 Y 4 Y 1 Y 44 Þþ^Y 4 ðy 11 Y 44 Y 14 ÞŠ=D L; S 44 ¼½^Y 14 ðy 1 Y 4 Y 14 Y Þþ^Y 4 ðy 1 Y 14 Y 11 Y 4 ÞŠ=D L : The matrix H may be determined by the expression given by Eq. (.1a), i.e., H ¼ L 1 þ SðSL 1 Þ: ð4:aþ ð4:bþ ð4:cþ ð4:dþ ð4:eþ ð4:fþ ð4:gþ ð4:hþ ð4:iþ ð4:8þ The result for H is H 11 H 1 0 H 14 H 1 H 0 H 4 H ¼ Y 0 5 ; ð4:9þ H 14 H 4 0 H 44 H 11 ¼ Y 11 ½^Y 1 ðy 11Y 44 Y 14 Þþ^Y 1 ^Y 14 ðy 1 Y 14 Y 11 Y 4 Þþ^Y 14 ðy 11Y Y 1 ÞŠ=D L; H 1 ¼ Y 1 ½^Y 1 ðy 1Y 44 Y 14 Y 4 Þþ^Y 1 ^Y 4 ðy 1 Y 14 Y 11 Y 4 Þ þ ^Y 1 ^Y 14 ðy 14 Y Y 1 Y 4 Þþ^Y 14 ^Y 4 ðy 11 Y Y 1 ÞŠ=D L; H 14 ¼ Y 14 ½^Y 14 ðy 14Y Y 1 Y 4 Þþ^Y 1 ^Y 4 ðy 14 Y 11Y 44 Þ þ ^Y 1 ^Y 14 ðy 1 Y 44 Y 14 Y 4 Þþ^Y 14 ^Y 4 ðy 11 Y 4 Y 1 Y 14 ÞŠ=D L ; H ¼ Y ½^Y 1 ðy Y 44 Y 4 Þþ^Y 1 ^Y 4 ðy 14 Y Y 1 Y 4 Þþ^Y 4 ðy 11Y Y 1 ÞŠ=D L; H 4 ¼ Y 4 ½^Y 4 ðy 11Y 4 Y 1 Y 14 Þþ^Y 1 ^Y 4 ðy 14 Y 4 Y 1 Y 44 Þ þ ^Y 1 ^Y 14 ðy Y 44 Y 4 Þþ^Y 14 ^Y 4 ðy 14 Y Y 1 Y 4 ÞŠ=D L ; H 44 ¼ Y 44 ½^Y 14 ðy Y 44 Y 4 Þþ^Y 14 ^Y 4 ðy 14 Y 4 Y 1 Y 44 Þþ^Y 4 ðy 11Y 44 Y 14 ÞŠ=D L: ð4:10aþ ð4:10bþ ð4:10cþ ð4:10dþ ð4:10eþ ð4:10fþ It should be noted that the structures of the generalized Barnett Lothe tensors have been given by Soh et al. (001) for transversely isotropic piezoelectric materials. Our results presented above are valid for monoclinic piezoelectric materials. Both structures appear the same, however, more elements are not identically zero for monoclinic piezoelectric materials. For example there are two extra elements of matrix L which are not identically zero, i.e., L 1 5 0, L Besides, our results are expressed in terms of elastic stiffness instead of reduced elastic compliances as did in the paper by Soh et al. (001).

9 51 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) The explicit expressions of A, B and L, S, H for transversely isotropic piezoelectric material Since transversely isotropic piezoelectric material has wide applications, it is of interest to present in this section the explicit expressions for matrices A and B and the corresponding generalized Barnett Lothe tensors. For transversely isotropic piezoelectric materials with thex -axis parallel to the poling direction, the constitutive equation can be obtained from monoclinic piezoelectric materials by letting c 45 = c 1 = c = c = c 4 = c 5 =0, c = c 1, c = c 11, c = c 44, e 11 = e 1 = e 1 = e = e 5 = a 1 =0, e = e 1 and e 4 = e 1. With these vanishing elements, the matrix U defined in Eq. (.) simplifies to ðc 44 p þ c 11 Þ ðc 1 þ c 44 Þp 0 ðe 1 þ e 1 Þp ðc 1 þ c 44 Þp ðc p þ c 44 Þ 0 ðe p þ e 1 Þ U ¼ ðc 44 p þ c 55 Þ 0 5 ; ð5:1þ ðe 1 þ e 1 Þp ðe p þ e 1 Þ 0 ða p þ a 11 Þ c 55 =(c 11 c 1 )/. The roots corresponding to juj = 0 can be classified into two types. For the type I the roots are pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 ¼ in 1 ; p ¼ in ; p ¼ i c 55 =c 44 ; p 4 ¼ in 4 ; ð5:þ which show that all four roots are purely imaginary. As to type II the roots are pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 1 ¼ in 1 ; p ¼ m þ in ; p ¼ i c 55 =c 44 ; p 4 ¼ m þ in ; ð5:þ which show that two of them (p 1 and p ) are purely imaginary and the rest two (p and p 4 ) have non-zero real parts but with equal imaginary parts (Suo et al., 199). The corresponding eigenvectors a and b expressed in the forms of Eqs. (.8) and (.9) are related to each other. By adopting the relation a k =(R T + p k T) 1 b k = Vb k ( non-zero elements of matrix V for transversely isotropic piezoelectric materials are listed in Appendix A) the elements of a k may be related to h k again by Eq. (.14) but now the parameters shown in Eq. (.15) should be replaced by j 11 ¼ðc e 1 c 44 e Þp k ; j 10 ¼ c a þ e p k þðc 44a þ e e 1 Þp k ; j 1 ¼ c 44 e p k c 1e 1 ; j 0 ¼ ðc 1 a þ e e 1 þ c 44 a Þp k e 1e 1 ; j 41 ¼ c c 44 p k þ c 44c 1 ; j 40 ¼ðc e 1 c 1 e c 44 e Þp k þ c 44e 1 ; ð5:4aþ ð5:4bþ ð5:4cþ ð5:4dþ ð5:4eþ ð5:4fþ and a ¼ðp c 44 Þ 1 ¼ p ffiffiffiffiffiffiffiffiffiffiffi i : ð5:5þ c 44 c 55 The substitution of the eigenvector a k into Eq. (.) leads to an equation for the unknown h k which is determined as h k ðp k Þ¼ b 4p 4 k þ b p k þ b 0 n 4 p 4 k þ n p k þ n ; ð5:þ 0

10 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) b 4 ¼ c e 1 a þ e 1 e ; b ¼ c 1 e a 11 þ c 44 e a 11 c e 1 a 11 c 1 e 1 a e 1 e e 1 þ e e 1 ; b 0 ¼ c 44 e 1 a 11 e 1 e 1 ; n 4 ¼ c c 44 a þ c 44 e ; n ¼ c 44 e e 1 þ c e 1 e 1 c 1 e e 1 þ c e 1 þ c c 44 a 11 c 1 c 44 a ; n 0 ¼ c 44 c 1 a 11 c 1 e 1 : ð5:aþ ð5:bþ ð5:cþ ð5:dþ ð5:eþ ð5:fþ Although matrices A and B for transversely isotropic piezoelectric materials are still expressed in the form as p those in Eq. (.19) (except the element a is replaced by a ¼ i= ffiffiffiffiffiffiffiffiffiffiffi c 44 c 55 Þ, we may further express the matrices A and B in a more explicit form depending on what types of the p s are. For type I, using the fact that elements of h k (p k ) (Eq. (5.)) and a 1k (p k )(k = 1,,4) (Eq. (.14a)) are all real, and the fact that elements of a k (p k ), a 4k (p k )(k = 1,,4) (Eqs. (.14b) and (.14c)) are all purely imaginary, matrices A and B 1 may be further expressed as a 11 ðp 1 Þ a 1 ðp Þ 0 a 14 ðp 4 Þ iimfa 1 ðp 1 Þg iimfa ðp Þg 0 iimfa 4 ðp 4 Þg A ¼ i pffiffiffiffiffiffiffiffi c 44 c ; ð5:8þ iimfa 41 ðp 1 Þg iimfa 4 ðp Þg 0 iimfa 44 ðp 4 Þg h h 4 iðimfp 4 gh Imfp gh 4 Þ 0 iimfp 4 p g B 1 ¼ 1 h 4 h 1 iðimfp 1 gh 4 Imfp 4 gh 1 Þ 0 iimfp 1 p 4 g jbj jbj 0 5 ; ð5:9þ h 1 h iðimfp gh 1 Imfp 1 gh Þ 0 iimfp p 1 g jbj ¼i½h 1 Imfp p 4 gþh Imfp 4 p 1 gþh 4 Imfp 1 p gš; ð5:10þ which is purely imaginary. For type II, noting that p 4 ¼ p (Eq. (5.)), and h 4 ¼ h (Eq. (5.)), the matrices A and B 1 are further written as a 11 ðp 1 Þ a 1 ðp Þ 0 a 1 ðp Þ a 1 ðp 1 Þ a ðp Þ 0 a ðp Þ A ¼ i pffiffiffiffiffiffiffiffi c 44 c ; ð5:11þ a 41 ðp 1 Þ a 4 ðp Þ 0 a 4 ðp Þ iimfh g Refp h g 0 Refp g B 1 ¼ 1 h h 1 p 1 h þ p h 1 0 p 1 þ p jbj jbj 0 5 ; ð5:1þ h 1 h p h 1 p 1 h 0 p p 1 jbj ¼h 1 Refp gþimfp 1 gimfh g Refh p g: ð5:1þ Here jbj is real since p 1 is purely imaginary, and h 1 is real for type II. For both type I and type II, matrices L 1, SL 1 and the generalized Barnett Lothe tensors L, S and H are all expressed in the same following form Y ^Y 1 0 ^Y 14 0 Y L 1 0 Y 4 ¼ Y 0 5 ; ^Y SL 1 ¼ ; ð5:14þ 0 Y 4 0 Y 44 ^Y

11 518 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) and 1 Y Y 0 44 Y 0 4 Y Y 44 Y L ¼ Y 4 4 Y Y Y 0 ; ð5:15aþ 4 5 Y 0 4 Y 0 Y 4 Y Y 44 Y Y 44 Y 4 ^Y 0 14 Y 4 ^Y 1 Y 44 ^Y 0 1 Y 4 ^Y 14 Y Y Y 44 Y Y 4 Y 44 Y 4 ^Y 1 S ¼ Y ; ð5:15bþ 4 5 ^Y 14 Y Y 11 ^Y 1 Y 44 ^Y 1 ^Y 14 Y 4 þ^y 14 Y Y Y 44 Y 4 H ¼ 0 Y ^Y 1 Y 11 0 Y 4 ^Y 1 ^Y 14 Y 11 ; ð5:15cþ 0 0 Y Y 4 ^Y 1 ^Y 14 Y 11 0 Y 44 ^Y 14 Y 11 with the elements shown above taken different values for different type roots. For type I the elements appearing in Eqs. (5.14) and (5.15) are Y 11 ¼ i½ðh h 4 Þa 11 þðh 4 h 1 Þa 1 þðh 1 h Þa 14 Š=jBj; ð5:1aþ Y ¼ i½h 4 ða p 1 a 1 p Þþh ða 1 p 4 a 4 p 1 Þþh 1 ða 4 p a p 4 ÞŠ=jBj; ð5:1bþ Y 44 ¼ i½p 1 ða 4 a 44 Þþp ða 44 a 41 Þþp 4 ða 41 a 4 ÞŠ=jBj; ð5:1cþ Y 4 ¼ i½p 1 ða a 4 Þþp ða 4 a 1 Þþp 4 ða 1 a ÞŠ=jBj; ð5:1dþ ^Y 1 ¼ i½imfp 1 gðh 4 a 1 h a 14 ÞþImfp gðh 1 a 14 h 4 a 11 ÞþImfp 4 gðh a 11 h 1 a 1 ÞŠ=jBj; ð5:1eþ ^Y 14 ¼ i½imfp 1 gða 1 a 14 ÞþImfp gða 14 a 11 ÞþImfp 4 gða 11 a 1 ÞŠ=jBj; ð5:1fþ 1 Y ¼ p ffiffiffiffiffiffiffiffiffiffiffi ; c 44 c 55 ð5:1gþ which are all real and for type II the elements of Eqs. (5.14) and (5.15) are Y 11 ¼ ½h 1 Imfa 1 ðp Þg a 11 ðp 1 ÞImfh g Imfa 1 ðp Þh gš=jbj; ð5:1aþ Y ¼ ½Imfa 1 ðp 1 ÞgRefp h g Imfp 1 grefa ðp Þh g h 1 Imfa ðp Þp gš=jbj; ð5:1bþ Y 44 ¼ ½Imfa 41 ðp 1 ÞgRefp g Imfp 1 grefa 4 ðp Þg Imfa 4 ðp Þp gš=jbj; ð5:1cþ Y 4 ¼ ½Imfa 1 ðp 1 ÞgRefp g Imfp 1 grefa ðp Þg Imfa ðp Þp gš=jbj; ð5:1dþ ^Y 1 ¼ ½h 1 Refa 1 ðp Þp g a 11 ðp 1 ÞRefp h g Imfp 1 gimfa 1 ðp Þh gš=jbj; ð5:1eþ ^Y 14 ¼ ½Refa 1 ðp Þp g a 11 ðp 1 ÞRefp g Imfp 1 gimfa 1 ðp ÞgŠ=jBj; ð5:1fþ 1 Y ¼ p ffiffiffiffiffiffiffiffiffiffiffi ; ð5:1gþ c 44 c 55 which are all real again. Note again that the structures of the generalized Barnett Lothe tensors given above have the same form as those given by Soh et al. (001). Their results are, however, in terms of the reduced elastic compliances. Numerical results for some practical piezoelectric materials calculated by the present formulae are given in Appendix B. The material constants (Ou and Chen, 00) corresponding to these practical piezoelectric materials are listed in Table 1 of Appendix B. The non-zero elements of matrices Y and the three generalized Barnett Lothe tensors corresponding to these practical piezoelectric materials are given in Table of Appendix B.

12 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) In the papers by Ou and Chen (00) and Liu (005), numerical results of matrices A and B are presented for transversely isotropic piezoelectric materials. To validate our procedure, we take the results of PZT-5H for comparison. With the material constants shown in Appendix B for PZT-5H, the results of Y =iab 1 calculated from the paper by Liu (005) are 0:0155 0:009i 0 0:008i 0:00i 0: :01 Y ¼ iab 1 ¼ : ; ð5:18þ 0:00815i 0: :0459 while the results from our formulae are 0:0155 0:009i 0 0:0084i 0:009i 0: :018 Y ¼ iab 1 ¼ : ; ð5:19þ 0:0084i 0: :0458 which show that both are in good agreement.. Conclusions We construct the eigenvalues and eigenvectors for monoclinic piezoelectric materials of class m, with the symmetry plane at x = 0 based on the extended Stroh formalism. The approach is effective and straightforward similar to those researchers who employ the modified Lekhinitskii formalism. Our eigenvalues and eigenvectors are directly related to the elastic stiffness instead of the reduced elastic compliance. The explicit expressions of the three generalized Barnett Lothe tensors are presented, all are in terms of the elastic stiffness. Results for the special case of transversely isotropic piezoelectric materials are also given. It is noted that the present approach may be applied to the determination of Barnett Lothe tensors for the general anisotropic elastic materials which is under investigation. Appendix A The non-zero elements of matrix V in Eq. (.1) for monoclinic piezoelectric materials are as follows: V 11 ¼ e þ c a Þp k þðc a þ e e þ c a 1 þ e 1 e Þp k þðc a 1 þ e e 1 =D; V 1 ¼ ðc a þ e e Þp k þ c a þ e þ c a 1 þ e e 1 pk þðc a 1 þ e e 1 Þ =D; V 14 ¼ ðc e c e Þp k þðc e þ c e 1 c e c e 1 Þp k þðc e 1 c e 1 Þ =D; V 1 ¼ ðc a þ e e Þp k þðc 1a þ e e 1 þ c a 1 þ e e 1 Þp k þðc 1 a 1 þ e 1 e 1 Þ =D; V ¼ e þ c a p k þðc 1a þ e 1 e þ c a 1 þ e 1 e Þp k þðc 1 a 1 þ e 1 e 1 Þ =D; V 4 ¼ ðc e c e Þp k þðc ða1þ 1e þ c e 1 c 1 e c e 1 Þp k þðc 1 e 1 c 1 e 1 Þ =D; V 41 ¼ ðc e c e Þp k þðc 1e c e 1 Þp k þðc 1 e c e 1 Þ =D; V 4 ¼ ðc e c e Þp k þðc e 1 c 1 e Þp k þðc e 1 c 1 e Þ =D; V 44 ¼ ðc c c Þp k þðc 1c c 1 c Þp k þðc 1 c c 1 c Þ =D; V ¼ðc 44 p k þ c 45 Þ 1 ;

13 50 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) Table 1 Material constants for practical piezoelectric ceramics Unit PZT-4 PZT-5H PZT-B PZT-A PZT- BaTiO c Nm c Nm c Nm c 10 9 Nm c Nm e 1 Cm e Cm e 1 Cm a C (Vm) a 10 9 C (Vm) Table The non-zero elements of matrices Y, L, S and H for practical piezoelectric ceramics PZT-4 PZT-5H PZT-B PZT-A PZT- BaTiO Type II Type II Type I Type II Type II Type II p i 1.011i 0.51i 1.i 0.99i 0.991i p i i.1015i i i i p 1.09i i 1.411i i 0.995i 0.90i p i i 1.014i i i i Y Y Y Y Y Ŷ Ŷ L L L L L S S S S H H H H H D ¼ D p k þ D p k þ D 1p k þ D 0 ; D ¼ c a c c a þ c e e c e c c ; D ¼ c 1 c a c 1 c a c c a 1 þ c a 1 þ c e e 1 c e 1 e þ c e 1 e c e 1 e þ c 1 e e þ c e e 1 c e 1 e c 1 e ; ðaþ D 1 ¼ c 1 c a c 1 c a þ c 1 c a 1 c 1 c a 1 c 1 e e c 1 e 1 e þ c 1 e þ c e e 1 þ c 1 e e 1 þ c e 1 e 1 c e e 1 c e 1 e 1 ; D 0 ¼ c 1 c a 1 c 1 c a 1 c 1 e 1 e þ c 1 e 1 e þ c e 1 e 1 c e 1 e 1 : The non-zero elements of matrix V for transversely isotropic piezoelectric materials are as follows:

14 J.Y. Liou, J.C. Sung / International Journal of Solids and Structures 44 (00) V 11 ¼ c a þ e p k =D; V 1 ¼ ðc 44 a þ e e 1 Þp k =D; V 14 ¼ ðc 44 e c e 1 Þp k =D; V 1 ¼ðc 1 a þ e e 1 Þp k =D; V ¼ c 44 a p k þ e 1e 1 =D; V 4 ¼ c 44 e p k c 1e 1 =D; V 41 ¼ðc 1 e c e 1 Þp k =D; V 4 ¼ c 44 e p k e 1 =D; V 44 ¼ c 44 c p k c 1 =D; V ¼ðc 44 p k Þ 1 ; D ¼ p k c c 44 a þ c 44 e p k þðc e 1 e 1 c 44 e 1 e c 1 e e 1 c 44 c 1 a Þ : ða4þ ðaþ Appendix B See Tables 1 and. References Barnett, D.M., Lothe, J., 195. Dislocations and line charges in anisotropic piezoelectric insulators. Phys. Status Solidi (b), Liu, J.X., 005. Comments and author s reply on Explicit expressions of eigenvalues and eigenvectors for transversely isotropic piezoelectric materials by Z.-C. Ou and Y.-H. Chen (Acta Mech. 1, 1-19, 00). Acta Mech. 1, Lothe, J., Barnett, D.M., 19. Integral formalism for surface waves in piezoelectric crystals. J. Appl. Phys. 4, Nye, J.F., Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford. Ou, Z.C., Chen, Y.H., 00. Explicit expressions of eigenvalues and eigenvectors for transversely isotropic piezoelectric materials. Acta Mech. 1, Soh, A.K., Liu, J.X., Fang, D.N., 001. Explicit expressions of the generalized Barnett Lothe tensors for anisotropic piezoelectric materials. Int. J. Eng. Sci. 9, Stroh, A.N., Dislocations and cracks in anisotropic elasticity. Philos. Mag., 5 4. Suo, Z., Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. Lond. A 4, Suo, Z., Kuo, C.M., Barnett, D.M., Willis, J.R., 199. Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Solids 40, 9 5. Ting, T.C.T., 199. Barnett Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x =0.J. Elasticity, Ting, T.C.T., 199. Anisotropic Elasticity: Theory and Application. Oxford University Press, Oxford. Ting, T.C.T., 199. New explicit expression of Barnett Lothe tensors for anisotropic linear elastic materials. J. Elasticity 4, 50. Wei, L., Ting, T.C.T., Explicit expressions of the Barnett Lothe tensors for anisotropic materials. J. Elasticity, 8.