Acta Mech Sin (010 6:41 46 DOI 10.1007/s10409-009-031-z RESEARCH PAPER An eien theory of waves in piezoelectric solids Shaohua Guo Received: 8 June 009 / Revised: 30 July 009 / Accepted: 3 September 009 / Published online: 1 December 009 he Chinese Society of heoretical and Applied Mechanics and Spriner-Verla GmbH 009 Abstract Based on the standard spaces of the physical presentation both the quasi-static mechanical approximation and the quasi-static electromanetic approximation of piezoelectric solids are studied here. he complete set of uncoupled elastic wave and electromanetic wave equations are deduced. he results show that the number and propaation speed of elastic waves and electromanetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromanetically anisotropic media and ones of mechanically anisotropic media. Based on these laws we discuss the propaation behaviour of elastic waves and electromanetic waves in the piezoelectric material of class 6 mm. Keywords Piezoelectric solid Standard spaces Elastic wave Electromanetic waves Modal equation 1 Introduction It is well known that the piezoelectric solids are both electromanetically anisotropic media and mechanically anisotropic ones. he theory of linear piezoelectricity is based on a quasi-static electromanetic approximation 1 3]. In this theory althouh the mechanical equations are dynamic the electromanetic equations are static and the electric field and the manetic field are not coupled. herefore it does not describe the wave behavior of electromanetic fields. S. Guo (B School of Civil Enineerin and Architecture Zhejian University of Science and echnoloy 31003 Hanzhou China e-mail: sh606@yahoo.com.cn S. Guo School of Civil Enineerin and Architecture Central South University 410075 Chansha China Electromanetic waves enerated by mechanical fields need to be studied in the calculation of radiated electromanetic power from a vibratin piezoelectric device 45] and are also relevant in acoustic delay lines 6] and wireless acoustic wave sensors 7] where acoustic fields produce electromanetic waves. When electromanetic waves are involved the complete set of Maxwell equation needs to be used coupled to the mechanical equations of equilibrium because the propaation speed of elastic waves is far less than one of electromanetic waves 8]. Althouh a lot of works about elastic waves in anisotropic solids have been done by in 9 11] the researches on elastic waves in anisotropic piezoelectric solids are still preliminary. Shear horizontal (SH waves in piezoelectric solids were studied by Li 1] usin scalar and vector potentials which results in a relatively complicated mathematical model of four equations. wo of these equations are coupled and the other two are one-way coupled. In addition a aue condition needs to be imposed. A different formulation was iven by Yan and Guo 13] which leads to two uncoupled equations. SH waves in piezoelectric solids over the surface of a circular cylinder of polarized ceramics were analyzed. Until now no explicit uncoupled equations of elastic waves or electromanetic waves in anisotropic piezoelectric media can be obtained because of the limitations of classical theory. In this paper the idea of standard spaces 14 18] is used to deal with both the Newton s equations of motion and the Maxwell s electromanetic equations. By this method the classical equations of motion and electromanetic equations under the eometric presentation can be transformed into the eien ones under the physical presentation. he former is in the form of vector and the latter is in the form of scalar. As a result a set of uncoupled modal equations of elastic waves and electromanetic waves are obtained each of which shows the existence of elastic sub-waves or and electromanetic sub-waves meanwhile the propaation velocity 13
4 S. Guo propaation direction polarization direction and space pattern of these sub-waves can be completely determined by the modal equations. Modal constitutive equation of piezoelectric body For a piezoelectric but non-manetizable dielectric body the constitutive equations are the followin σ = c S e E (1 D = e S + ε E ( B = μ H (3 where the dielectric permittivity matrix ε manetic permeability matrix μ and elastic matrix c can be spectrally decomposed as follows 14 18] ε = (4 μ = Ɣ (5 c = (6 where = diaλ 1 λ λ 3 λ 4 λ 5 λ 6 ] = diaω 1 ω ω 3 ]Ɣ = diaγ 1 γ γ 3 ] are the matrixes of eien dielectric permittivity eien manetic permeability and eien elasticity respectively; ={φ 1 φ φ 3 φ 4 φ 5 φ 6 } = {ϕ 1 ϕ ϕ 3 } = {ϑ 1 ϑ ϑ 3 } are the modal matrixes of electric media manetic media and elastic media respectively which are both orthoonal and positive definite matrixes and satisfy = I = I = I. Projectin the electromanetic physical qualities of the eometric presentation such as the electric field intensity vector E manetic field intensity vector H manetic flux density vector B electric displacement vector D stress vector σ and strain vector S into the standard spaces of the physical presentation we et σi Si = φi σ i = 1...m (7 = φi S i = 1...m (8 D I = ϕ I D I = 1...n (9 E I = ϕ I E I = 1...n (10 BI = ϑ I B I = 1...n (11 HI = ϑ I H I = 1...n (1 where the stars express the modal variables m( 6 and n( 3 are number of the mechanical and electromanetic independent subspaces respectively. Equations (7 (1 show the electromanetic and mechanical physical qualities under the physical presentation they can also be written in the form of matrix as follows σ = σ S = S (13 D = D E = E (14 B = B H = H. (15 Substitutin Eqs. (13 (15 into Eqs. (1 (3 respectively and multiplyin them with the transpose of modal matrix in the left we have D = e S + ε E (16 B = μ H (17 σ = c σ e E. (18 Let G = e G = e that is a coupled piezoelectric matrix and usin Eqs. (4 (6 and (13 (15 we et D = GS + E (19 B = ƔH (0 σ = S G E. (1 Rewritin the above equations in the form of scalar we have D I = ω I E I + IjS j I = 1...n j = 1...m (sum to j ( BI = γ I HI I = 1...n (3 σi = λ i Si ij E J i = 1...m J = 1...n(sum to J. (4 Equations ( (4 are just the modal constitutive equations for anisotropic piezoelectric body in which Ij = {ϕ I } e]{φ j } ij ={φ i} e] {ϕ J } ij = Ji are the coupled piezoelectric coefficients. 3 Eien expression of the quasi-static electromanetic approximation and the quasi-static mechanical approximation he classical Maxwell s equations and Newton s equation in passive reion of solids are the followin e IJK H KJ = Ḋ I (5 e IJK E KJ = Ḃ I (6 σ ji j = ü i. (7 hese equations can be written as eien ones under physical Representation 14 17] { I }E I = t{ϑi }B I I = 1...n (8 { I }H I = t {ϕ I }D I I = 1...n (9 i σ i = tt si i = 1...m (30 13
An eien theory of waves in piezoelectric solids 43 where i ={φi } ]{φi } is the stress operator 1415] } is the electromanetic operator 1617] in which { I ] 11 0 0 0 31 1 0 0 3 0 1 = 0 0 33 3 31 0 0 3 3 ( + 33 1 31. 13 0 13 1 ( 11 + 33 3 1 1 0 13 3 ( + 11 (31 For quasi-static electromanetic approximation we have Accordin to the principle of operator Eq. (39 becomes ij ( Jk ω J δ ik + λ i ] i s i = tt s i i = 1...m J = 1...n (sum toj. (40 hey are the equations of elastic waves in quasi-static electromanetic approximation of piezoelectricity in which the propaation speed of elastic waves is the followin λ i + J Jk ij ω J δ ik v i =. (41 I D I = 0 I = 1...n (3 4. Electromanetic waves where I ={ϕ I } ]{ϕ I } is the electric displacement operator 1617] in which 11 1 31 ] = 1 3. (33 13 3 33 Substitutin Eqs. ( (4 into Eqs. (8 (9 (34 we have { I }E I = t{ϑ I }γ I H I I = 1...n (4 { I }H I = t {ϕ I }(ω I E I + IjS j I = 1...n (43 For quasi-static mechanical approximation we have i σ i = 0 i = 1...m. (34 4 Modal equation of waves in piezoelectric solids 4.1 Elastic waves Substitutin Eq. ( into Eq. (3 we have ω I I E I + I Ijs j = 0. (35 he above can also be rewritten as follows Ji J E J + J si = 0 ω J J = 1...n i = 1...m (sum to i. (36 Accordin to the principle of operator Eq. (36 becomes E J = Jk sk ω J J = 1...n k = 1...m (sum to k. (37 In the same way substitutin Eq. (4 into Eq. (30 we have i λ is i i ij E J = tts i. (38 Usin Eq. (37 Eq. (38 becomes i λ isi + i Jk ij δ ik si = tt si ω J i = 1...m J = 1...n. (39 i (λ i S i ij E J = 0 i = 1...m. (44 ransposin Eq. (4 and multiplyin it with { I } and also usin Eq. (43 we have { I }{ I } E I = tt{ϑ I } {ϕ I }γ I (ω I E I + IjS j. (45 Let L I ={ I } { I } and ξ I ={ϑ I } {ϕ I }. Equation (45 can be written as L I E I + ttξ I γ I ω I E I = ttξ I γ I Ij S j. (46 From Eq. (44 we have ij E J. (47 i S i = i λ i Usin the principle of operator and chanin the index we have S jk j = EK j = 1...6 (sum tok. (48 λ j Substitutin Eq. (48 into Eq. (46 we et the equations of electric fields ] L I E I + jk ttξ I γ I ω I + Ij δ IK E I λ = 0 j (sum to j K. (49 13
44 S. Guo In the same way we can et the equations of manetic fields ] L I H I + jk ttξ I γ I ω I + Ij δ IK HI = 0 λ j (sum to j K. (50 Equations (49 and (50 are just the eien equations of electromanetic waves in piezoelectric solids for quasi-static electromanetic approximation the speed of electromanetic waves is the followin c I = 1 ]. (51 ξ I γ I ω I + jk Ij λ j δ IK 5 Application In this section we discuss the propaation laws of elastic waves and electromanetic waves in a polarized ceramics poled in the x 3 -direction. he material tensors in Eqs. (1 (3 are represented by the followin matrices under the compact notation 1] c 11 c 1 0 0 0 c 1 c 11 0 0 0 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 44 0 0 0 0 0 0 c 66 0 0 e 31 0 0 e 31 0 0 e 33 0 e 15 0 e 15 0 0 0 0 0 ε 11 0 0 0 ε 11 0 0 0 ε 33 μ 11 0 0 0 μ 11 0 0 0 μ 33 (5 where c 66 = 1 (c 11 c 1. here are four independent mechanical eien spaces 14 15] as follows W mech = W (1 1 φ 1 ] W (1 φ ] W ( 3 φ 3 φ 6 ] W ( 4 φ 4 φ 5 ] (53 where φ 1 = (λ 1 c 11 c 1 + c13 1 1 λ ] 1 c 11 c 1 0 0 0 φ 3 = 1 1 0 0 0 0] φ i = ξ i i = 4 5 6 (54 λ 1 = c (c11 11 + c 1 + c 33 + c 1 + c 33 ± + c13 λ 3 = c 11 c 1 λ 4 = c 44. (55 hen we have φ 1 = φ 1 3 φ 3 = 1 1 0 0 0 1] 3 φ 4 = 0 0 0 1 1 0]. here are two independent electric or manetic eien spaces 1617] as follows W ele = W ( 1 ϕ 1 ϕ ] W (1 ϕ 3 ] (56 W ma = W ( 1 ϑ 1 ϑ ] W (1 ϑ 3 ] (57 where = diaε 11 ε 11 ε 33 ] Ɣ = diaμ 11 μ 11 μ 33 ] (58 1 0 0 1 0 0 = 0 1 0 = 0 1 0. (59 0 0 1 0 0 1 hen we have ϕ 1 = ϑ 1 = 1 1 0] ϕ = ϑ =0 0 1]. It is seen that the electric subspaces are the same as manetic ones for polarized ceramics. hus the physical quantities the coupled coefficients and the correspondin operators of polarized ceramics are calculated as follows S1 = a 1S 11 + S + b 1 S 33 ] 3 S3 = 3 (S 11 S + S 1 S4 = (S 3 + S 31 ( 1 = a 1 x + y + b 1 z 3 = x 3 + y 4 = 1 ( x + y + z + xy 13
An eien theory of waves in piezoelectric solids 45 E 1 = (E 1 + E H1 = (H 1 + H E = E 3 H = H 3 ( L 1 = 1 x y + z xy L ( = x + y ξ I = 1 I = 1 11 = 0 1 = 0 31 = 0 41 = e 15 1 = a 1 (e 31 + b 1 e 33 = a (e 31 + b e 33 3 = 0 4 = 0 11 = 0 1 = 0 13 = 0 14 = e 15 1 = a 1(e 31 + b 1 e 33 = a (e 31 + b e 33 3 = 0 4 = 0 where a 1 = (λ 1 c 11 c 1 + c13 b 1 = λ 1 c 11 c 1. hus we have v 1 = λ 1 (60 v = λ + a 1 (e 31+b 1 e 33 ε 11 + a (e 31+b e 33 ε 33 (61 v3 = c 11 c 1 (6 v4 = c 44 + e 15 ε 11 (63 c1 = 1 ] μ 11 ε 11 + a 1 (e 31+b 1 e 33 λ + e 15 c 44 (64 c = 1 μ ε + a (e 31+b e 33 λ ]. (65 It is seen that for quasi-static electromanetic approximation there exist four elastic waves in polarized ceramics solids. wo of them are the quasi-dilatation wave such as v 1 and v and the other two are the quasi-shear waves such as v 3 and v 4. Only two waves (v and v 4 are affected by the piezoelectric coefficients which speed up the propaation of the second and fourth waves. For quasi-static mechanical approximation there exist two electromanetic waves in polarized ceramics solids both are affected by the piezoelectric coefficients which slow down the propaation of electromanetic waves. 6 Conclusions he Maxwell s equations coupled to the mechanical equations of equilibrium and the mechanical equations of motion coupled to the equation of static electric field are studied here in the standard spaces of the physical presentation. he complete set of uncoupled electromanetic waves equations and elastic waves for anisotropic piezoelectric solids are deduced. he results show that the number of electromanetic waves and elastic waves in piezoelectric solids is determined by both the subspaces of electromanetically anisotropic media and ones of mechanically anisotropic media. the propaation speed of elastic waves is affected by eien elasticity and electromanetically anisotropic media. For the piezoelectric material of class 6 mm it is seen that there exist four elastic waves but only two waves are affected by the piezoelectric coefficients two electromanetic waves are affected by the piezoelectric coefficients. References 1. iersten H.F.: Linear Piezoelectric Plate Vibrations. pp. 0 50. Plenum New York (1969. Erinen A.C. Mauin G.A.: Electrodynamics of Continua. pp. 30 60. Spriner New York (1990 3. Mindlin R.D.: Electromanetic radiation from a vibratin quartz plate. Int. J. Solids Struct. 9(3 697 70 (197 4. Lee P.C.Y.: Electromanetic radiation from an A-cut quartz plate under lateral-field excitation. J. Appl. Phys. 65(5 1395 1399 (1989 5. Lee P.C.Y. Kim Y.G. Prevost J.H.: Electromanetic radiation from doubly rotated piezoelectric crystal plates vibratin at thickness frequencies. J. Appl. Phys. 67(9 6633 664 (1990 6. Oliner A.A.: Acoustic Surface Waves. pp. 45 60. Spriner New York (1978 7. Sedov A. Schmerr J.R. et al.: Some exact solutions for the propaation of transient electroacoustic waves. I. Piezoelectric halfspace. Int. J. En. Sci. 4(3 557 568 (1986 8. Dieulesaint E. Royer D.: Elastic Wave in Solids Applications to Sinal Processin. pp. 68 90. John & Sons New York (1980 9. in.c..: Lonitudinal and transverse waves in anisotropic elastic materials. Acta Mech. 185( 147 164 (006 10. in.c..: ransverse waves in anisotropic elastic materials. Wave Motion 44(1 107 119 (006 11. in.c..: An explicit secular equation for surface waves in an elastic material of eneral anisotropy. Q. J. Mech. Math. 55( 97 311 (00 1. Li S.: he electromaneto-acoustic surface wave in a piezoelectric medium: the Bleustein Gulyaev mode. J. Appl. Phys. 80(10 564 569 (1996 13. Yan J.S. Guo S.H.: Piezoelectromanetic waves uided by the surface of a ceramic cylinder. Acta Mech. 181( 199 05 (006 14. Guo S.H.: An eien theory of rheoloy for complex media. Acta Mech. 198( 53 60 (008 13
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