Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media

International Journal of Engineering Science 46 (2008) 1098–1110

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Reflection and refraction of plane waves at the interface between piezoelectric and piezomagnetic media Yu Pang a, Yue-Sheng Wang a,*, Jin-Xi Liu b, Dai-Ning Fang c a b c

Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China Department of Engineering Mechanics, Shijiazhuang Railway Institute, Shijiazhuang 050043, China Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e

i n f o

Article history: Received 29 November 2007 Received in revised form 17 April 2008 Accepted 26 April 2008 Available online 6 June 2008

Keywords: Elastic wave Reflection/refraction Interface Piezoelectric medium Piezomagnetic medium

a b s t r a c t The paper analyzes the reflection and refraction of a plane wave incidence obliquely at the interface between piezoelectric and piezomagnetic media. The materials are assumed to be transversely isotropic. Numerical calculations are performed for BaTiO3/CoFe2O4 material combination. Four cases, incidence of the coupled quasi-pressure (QP) and quasi-shear vertical (QSV) wave from BaTiO3 or CoFe2O4 media, are discussed. The reflection and transmission coefficients and energy coefficients varying with the incident angle are examined. Calculated results are verified by considering the energy conservation. Results show that the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP and QSV waves, evanescent electroacoustic (EA) and magnetic potential (MP) waves in the piezoelectric medium (BaTiO3), evanescent magnetoacoustic (MA) and electric potential (EP) waves in the piezomagnetic medium (CoFe2O4), among which the EA, MA, MP and EP waves propagate along the interface. The most amount of the incident energy goes with the waves that are the same type as the incident wave, while the energy arising from wave mode conversion occupies a less part of the incident energy. The electric energy in BaTiO3 is higher than the magnetic energy in CoFe2O4; they both attain their maximum values at/before the critical angle. Critical angles have little effect on evanescent waves except when the total reflection takes place. These results would provide useful complementary information for magnetoelectric composite materials. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Reflection and refraction of a plane wave at the interface between two dissimilar media is a fundamental topic in many fields such as seismology, geophysics, earthquake engineering, non-destructive evaluation, etc. This problem is intensively studied for both isotropic and anisotropic elastic solids; and well-known solutions can be found in many papers and monographs. It is impossible to embrace all the related references and textbooks on this matter. We just mention some of them, e.g. Ref. [1–7]. Compared with the isotropic cases, solution to the anisotropic problem is much more difficult because all kinds of the scattered waves including the incident wave are no longer pure pressure (P) or shear vertical (SV) waves [7]. In the past two decades, the reflection/refraction problems of piezoelectric materials received considerable attention. Piezoelectric solids are inherently anisotropic. Among others, Alshits et al. [8–10] contributed many efforts to this field. They solved various reflection problems of electroacoustic waves in a semi infinite piezoelectric medium based on eightdimensional Stroh formalism. Different from the purely elastic case, the acoustic wave propagating in piezoelectric solids

* Corresponding author. Tel.: +86 10 56188417; fax: +86 10 51682094. E-mail address: [email protected] (Y.-S. Wang). 0020-7225/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2008.04.006

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

1099

are accompanied by the waves of the quasistatic electric fields due to the electromechanical coupling effect. So these waves are also called electroacoustic waves. In recent years, the interest in device application of piezoelectric materials has attracted new attention to the reflection and refraction at the interface between a piezoelectric crystal and viscous conductive liquid for bulk waves liquid sensors [11], between piezoelectric ceramics and water for radiating acoustic energy transducers [12] and between 0 and 3 piezo-composite sensor and actuator for noise control devices [13], etc. More recently, Burkov et al. [14] reported the reflection and refraction of bulk acoustic waves at piezoelectric/vacuum and piezoelectric/elasticisotropic-medium interfaces under the action of an external electric field, and discussed the effects of the electric field on the reflection and refraction. From theoretical standpoints, Shuvalov and Lothe [15] demonstrated the reciprocity properties of reflection–transmission on a welded and electrically open interface between two piezoelectric media. Furthermore, they developed an approach to prove the reciprocity properties under various boundary conditions for the reflection–transmission problems. From 1970s, composites consisting of piezoelectric and piezomagnetic phases have attracted interest because they possess a novel magnetoelectric effect [16]. A lot of papers on experimental fabrication and theoretical prediction of the effective material constants for magnetoelectric composites have been published [17–20]. It is well-known that neither piezoelectric nor piezomagnetic phase has the magnetoelectric effect. But the composites have the remarkable magnetoelectric effect because of an elastic coupling between two different phases. In most published articles, the composites were modeled as homogeneous magnetoelectric materials, and fully coupling constitutive equations were constructed to take into account the coupling magnetic–electric–mechanical response [18]. Some researchers investigated wave propagation using the fully coupling constitutive equations with nonzero magnetoelectric coefficients, e.g. Wu et al. [21], Melkumyan [22], etc. Others, e.g. Liu et al. [23], Soh and Liu [24], studied wave propagation in magnetoelectric laminate composites consisting of piezoelectric and piezomagnetic materials as constituent phases using the constitutive equations with zero magnetoelectric coefficients for each constituent phase. To the best of authors’ knowledge, a few papers on wave propagation in such laminate composites have been published. Even the basic and important problem – the reflection and refraction at an interface between piezoelectric and piezomagnetic media has not been carefully investigated. This problem provides useful complementary information for wave propagation in layered or periodic magnetoelectric composite materials and structures. In this paper, the reflection and refraction of plane electroacoustic and magnetoacoustic waves incident obliquely at the interface between piezoelectric and piezomagnetic half-spaces are investigated. Both materials are assumed to be hexagonal (6 mm) crystals. The reflection and transmission coefficients (RTCs) are derived by solving a linear algebraic system [7]. Incident energy transmission and partition at the interface are also discussed. Numerical results for the variation of the RTCs and energy coefficients with the incident angle are presented for the BaTiO3/CoFe2O4 combination. This investigation is relevant to acoustic device application of magnetoelectric composite structures. 2. Basic equations and formal solutions For an anisotropic and linearly magneto-electric-elastic solid, the coupled constitutive relation can be written as [25] 32 3 2 3 2 s r C eT qT 76 7 6 7 6 ð1Þ e a 54 E 5; 4D5 ¼ 4e B

q

aT

l

H

where r, s, D, E, B and H, are the stress tensor, strain tensor, electric displacement, electrical field intensity, magnetic induction and magnetic field intensity, respectively; C, e and l are the stiffness tensor, dielectric, and magnetic permeability tensors, respectively; e, q and a are the piezoelectric, piezomagnetic and magnetoelectric coefficient tensors, respectively; the superscript ‘‘T” indicates transposition. For the specific case of a piezoelectric medium or piezomagnetic medium, the constitutive relations can be expressed in Eq. (1) by deleting coupled coefficient tensors e, q or a correspondingly. Specifically, the constitutive equations for the piezoelectric medium are given by 3e 2 3e 2 3e 2 s r C eT 0 6 7 6 7 6 7 ð2Þ e 05 4 E 5 4D5 ¼ 4e B

0

0

l

H

and those for the piezomagnetic medium are 3m 2 3m 2 3m 2 s r C 0 qT 6 7 6 7 6 7 0 5 4E5 ; 4D5 ¼ 40 e H B q 0 l

ð3Þ

where the superscript ‘‘e” and ‘‘m” denote quantities relative to piezoelectric and piezomagnetic materials, respectively. In this paper, we will consider the hexagonal (6 mm) crystals (transversely isotropic materials). We assume small displacement gradients and employ the linearized strain–displacement relation given by sJ ¼

1 ðruJ þ ðruJ ÞT Þ; 2

ð4Þ

1100

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

where uJ is the displacement vector with J = e or m. Based on the assumptions of electrostatics and magnetostatics, the electric field E and the magnetic field H are the gradient of an electric potential u and a magnetic potential /, respectively, i.e. EJ ¼ ruJ ;

HJ ¼ r/J :

ð5Þ

According to the quasistatic approximation and linearity assumption, the equations of motion with static electricity and magneticity, in the absence of body force and free charge, are r  rJ ¼ qJ

o2 uJ ; ot 2

r  DJ ¼ 0;

r  BJ ¼ 0:

ð6Þ

By substituting Eqs. (2) and (3) into Eq. (6), the governing equations can be obtained as r  ðCe rue þ eT rue Þ ¼ qe

o2 ue ot2

ð7Þ

r  ðerue  eeT rue Þ ¼ 0 r  ðle r/e Þ ¼ 0 for the piezoelectric medium, and r  ðCm rum þ qT r/m Þ ¼ qm

o2 um ot2

ð8Þ

r  ðemT rum Þ ¼ 0 m

m

m

r  ðqru  l r/ Þ ¼ 0 for the piezomagnetic medium. For the present hexagonal (6 mm) crystals (transversely isotropic materials), we will consider the motion in the sagittal plane (xz plane). According to Ref. [7], the formal solution of Eqs. (7) and (8) may be written as fuJ1 ; uJ3 ; uJ ; /J g ¼ U J exp½inðx þ aJ z  ctÞ;

ð9Þ J

where n is the x-component of the wave vector; c is the phase velocity along x; a is an unknown ratio of the wave vector components along the z- and x-directions; and UJ ¼ fU J1 ; U J2 ; U J3 ; U J4 gT are the amplitudes of the displacement, electric potential and magnetic potential. The generalized Snell’s law [7] has been taken into account in Eq. (9). Substitution from Eq. (9) into Eqs. (7) or (8) yields an eigen-value equation for the piezoelectric or piezomagnetic media which is written as KJ ðaJ ÞUJ ¼ 0;

ð10Þ

where the elements of the 4  4 matrix KJ are given in Appendix A. Setting the determinant of KJ in Eq. (10) equal to zero, we obtain an eighth-degree polynomial equation for aJ, as aJ8 þ AJ1 aJ6 þ AJ2 aJ4 þ AJ3 aJ2 þ AJ4 ¼ 0;

ð11Þ

where the coefficients AJ1 , AJ2 , AJ3 and AJ4 can be extracted from Eq. (10) and, for the sake of saving space, will not be given here. Eq. (11) admits four solutions for aJ2, or eight solutions that are restricted according to aJ1 ¼ aJ6 ;

aJ5 ¼ aJ2 ;

aJ4 ¼ aJ3 ;

aJ8 ¼ aJ7 :

aJq ,

For each we can use Eq. (10) to relate the wave amplitude ratios, 8 K 12 ðaeq ÞK 13 ðaeq Þ  K 11 ðaeq ÞK 23 ðaeq Þ > > W eq ¼ > > K 12 ðaeq ÞK 23 ðaeq Þ  K 13 ðaeq ÞK 22 ðaeq Þ > < K 11 ðaeq ÞK 22 ðaeq Þ  K 12 ðaeq ÞK 12 ðaeq Þ ueq ¼ > > K 12 ðaeq ÞK 23 ðaeq Þ  K 13 ðaeq ÞK 22 ðaeq Þ > > > : e /q ¼ 0 for the piezoelectric medium, and 8 m m m K 12 ðam q ÞK 14 ðaq Þ  K 11 ðaq ÞK 24 ðaq Þ > > Wm > q ¼ K ðam ÞK ðam Þ  K ðam ÞK ðam Þ > 12 q 24 q 14 q 22 q > < um q ¼ 0 > > m m m > K 11 ðam > q ÞK 22 ðaq Þ  K 12 ðaq ÞK 12 ðaq Þ > : /m q ¼ K ðam ÞK ðam Þ  K ðam ÞK ðam Þ 12 q 24 q 14 q 22 q

ð12Þ W Jq

¼

U J2q =U J1q ,

uJq

¼

U J3q =U J1q

and /Jq ¼

U J4q =U J1q ,

as

ð13Þ

ð14Þ

for the piezomagnetic medium. The third equation of Eq. (13) implies that the magnetic potential is uncoupled with the mechanical displacements in the piezoelectric material. Similarly, the second equation of Eq. (14) means that electric potential and mechanical displacements are uncoupled in the piezomagnetic material. Combining Eq. (13) with (2), we can use superposition to rewrite the formal solutions for displacements, stresses, electric potential, electric displacement, magnetic potential and magnetic induction as

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

fue1 ; ue3 ; ue ; /e g ¼

6 8 X X f1; W eq ; ueq ; 0gU e1q exp½inðx þ aeq z  ctÞ þ f0; 0; 0; 1gU e1q exp½inðx þ aeq z  ctÞ;

ð15aÞ

q¼7

q¼1

frexz ; rezz ; Dez ; Bez g ¼

1101

6 X

infF e1q ; F e2q ; F e3q ; 0gU e1q exp½inðx þ aeq z  ctÞ þ

8 X

inf0; 0; 0; F e4q gU e1q exp½inðx þ aeq z  ctÞ;

ð15bÞ

q¼7

q¼1

where 8 e F 1q ¼ ce44 ðW eq þ aeq Þ þ e15 ueq > > > e > < F ¼ ce þ ce W e ae þ e33 ue ae 2q q q 13 33 q q > F e3q ¼ e31 þ e33 W eq aeq  eeq ueq aeq > > > : e F 4q ¼ le33 aeq

ð16Þ

:

With reference to the relation (12) and by inspection of Eqs. (13), (15a), (15b) and (16) we recognize the following properties, 8 e F 16 ¼ F e11 ; F e15 ¼ F e12 ; F e14 ¼ F e13 > > > e < F 26 ¼ F e21 ; F e25 ¼ F e22 ; F e24 ¼ F e23 ð17Þ > F e ¼ F e31 ; F e35 ¼ F e32 ; F e34 ¼ F e33 > > : 36 e e F 48 ¼ F 47 which are direct consequences of the hexagonal (6 mm) crystals symmetry. The similar analysis for the piezomagnetic material gives m m m fum 1 ; u3 ; u ; / g ¼

6 8 X X m m m m f1; W m f0; 0; 1; 0gU m q ; 0; /q gU 1q exp½inðx þ aq z  ctÞ þ 1q exp½inðx þ aq z  ctÞ;

m m m frm xz ; rzz ; Dz ; Bz g ¼

6 X

ð18aÞ

q¼7

q¼1

m m m m infF m 1q ; F 2q ; 0; F 4q gU 1q exp½inðx þ aq z  ctÞ þ

i¼1

8 X

m m inf0; 0; F m 3q ; 0gU 1q exp½inðx þ aq z  ctÞ;

ð18bÞ

i¼7

where 8 m m m m F ¼ cm > 44 ðW q þ aq Þ þ h15 /q > > 1q > > m m m m > m m < Fm 2q ¼ c 13 þ c 33 W q aq þ h33 /q aq m m > > Fm > 3q ¼ e33 aq > > > : m m m m m F 4q ¼ h31 þ h33 W m q aq  l33 /q aq

The similar properties as in Eq. (17) exist 8 m m m Fm Fm F 16 ¼ F m > 11 ; 15 ¼ F 12 ; 14 ¼ F 13 > > m m m m m m < F 26 ¼ F 21 ; F 25 ¼ F 22 ; F 24 ¼ F 23 > Fm ¼ F m > 37 > : 38 m m m F 46 ¼ F m Fm Fm 41 ; 45 ¼ F 42 ; 44 ¼ F 43

ð19Þ

ð20Þ

3. Solution to reflection and refraction Consider the problem shown in Fig. 1. The originating medium and continuing medium occupy the spaces z P 0 and z 6 0, respectively. The x-axis is taken along the interface and the z-axis is directed vertically downwards. For the oblique incidence of the coupled plane quasi-pressure (QP) wave from the piezoelectric medium at the interface z = 0, all kinds of scattered waves are depicted in Fig. 1. The transmitted wave fields consist of the transmitted QP, quasi-shear vertical (QSV), evanescent magnetoacoustic (MA) and electric potential (EP) waves. And the reflected wave fields make up of the reflected QP, QSV, evanescent electroacoustic (EA) and magnetic potential (MP) waves. For the case in which the originating medium is the piezoelectric material and continuing medium is the piezomagnetic one, the displacement and stress fields of the incident wave are shown as X I inf1; W eq ; ueq gU e1q exp½inðx þ aeq z  ctÞ; ð21aÞ fue1 ; ue3 ; ue g ¼ q¼1;2;3

frexz ; rezz ; Dez gI ¼

X

infF e1q ; F e2q ; F e3q gU e1q exp½inðx þ aeq z  ctÞ;

q¼1;2;3

where q = 1, 2, 3 represent the incidence of the coupled QP, QSV and evanescent EA waves, respectively. After reflection at the interface z = 0, the reflected wave fields in originating medium may be written as

ð21bÞ

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Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

Tran-QSV Tran-QP

{F }

m T

continuing medium

MA EP

δ →0 π

-π

X

MP EA

originating medium

Incident wave QP or QSV

{F } {F }

e R

e I

Ref-QSV

Ref-QP

Z

Fig. 1. Reflection and refraction of waves at an interface between piezoelectric and piezomagnetic solids and energy flow into and out of the interface.

fue1 ; ue3 ; ue gR ¼

X

inf1; W eq ; ueq gU e1q exp½inðx þ aeq z  ctÞ;

ð22aÞ

q¼6;5;4

where q = 6, 5 and 4 correspond to the reflected QP, QSV and evanescent EA waves, respectively. Because of the coupled magnetic–electric effect at the interface, the reflected wave fields should include the reflected evanescent MP wave which is uncoupled from the mechanical displacement so that it satisfies the boundary conditions. We identify this reflected evanescent MP wave with q = 8 and write f/e gR ¼ inU e18 exp½inðx þ ae8 z  ctÞ:

ð22bÞ

According to the constitutive relation (2), the corresponding reflected stress and electric displacement fields have the form X infF e1q ; F e2q ; F e3q gU e1q exp½inðx þ aeq z  ctÞ ð23aÞ frexz ; rezz ; Dez gR ¼ q¼6;5;4

and the reflected magnetic inductions originated from the reflected evanescent MP wave are fBex gR ¼ inle11 U e18 exp½inðx þ ae8 z  ctÞ; fBez gR ¼ inle33 ae8 U e18 exp½inðx þ ae8 z  ctÞ: The transmitted wave fields in the continuing medium are shown as X m m T m m m inf1; W m fum 1 ; u3 ; / g ¼ q ; /q gU 1q exp½inðx þ aq z  ctÞ:

ð23bÞ

ð24aÞ

q¼1;2;3

Similarly, we should consider the transmitted evanescent EP wave which is also uncoupled from the mechanical displacements in the piezomagnetic medium m fum gT ¼ inU m 17 exp½inðx þ a7 z  ctÞ:

The corresponding transmitted stress and magnetic induction fields are X m T m m m m m infF m frm 1q ; F 2q ; F 4q gU 1q exp½inðx þ aq z  ctÞ xz ; rzz ; Bz g ¼

ð24bÞ

ð25aÞ

q¼1;2;3

and the transmitted electric displacements originated from the transmitted evanescent EP wave are T m m m fDm x g ¼ ine11 U 17 exp½inðx þ a7 z  ctÞ; T m m m m fDm z g ¼ ina7 e33 U 17 exp½inðx þ a7 z  ctÞ;

ð25bÞ

where we identify the transmitted QP, QSV, evanescent MA and evanescent EP waves with q = 1, 2, 3, 7, respectively. The continuity conditions at the interface z = 0 are fue1 gI þ fue1 gR ¼ fue1 gT ; e I

e R

m T

fu g þ fu g ¼ fu g ; I

R

T frexz g þ frexz g ¼ frm xz g ; T fDez gI þ fDez gR ¼ fDm z g ;

fue3 gI þ fue3 gR ¼ fue3 gT ; f/e gI þ f/e gR ¼ f/m gT ; T frezz gI þ frezz gR ¼ frm zz g ; T fBez gI þ fBez gR ¼ fBm z g :

ð26Þ

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

1103

Now substituting Eqs. (21)–(25) into the boundary condition Eq. (26) and invoking Eqs. (12), (17) and (20), we obtain eight linear simultaneous equations for the amplitudes of the reflected and transmitted waves. The matrix form of these equations can be expressed as 32 e 3 2 3 2 1 1 1 0 1 1 1 0 1 1 0 1 U 16 76 e 7 6 W e 6 We We We 0 Wm Wm 0 Wm W e5 W e4 0 7 7 6 6 6 5 4 6 5 4 76 U 15 7 6 76 e 7 6 e 72 6 e 3 e e 7 7 6 u6 ue5 ue4 6 6 0 0 0 1 0 u5 u4 0 7 U e11 u6 U 14 7 7 7 6 6 6 76 e 7 6 0 6 0 6 e 7 /m 0 /m 0 0 1 /m 0 0 1 7 76 U 12 7 6 6 6 5 4 76 U 18 7 ð27Þ 76 e 7: 6 e m m m 76 m 7 ¼ 6 e e e 7 7 7 6 F 16 F e15 F e14 6 6 0 F 16 F 15 0 F 14 F 15 F 14 0 4 U 13 5 F 76 U 11 7 6 16 7 6 e e e 76 m 7 6 F e 6 Fe 0 F m F m 0 F m F e25 F e24 0 7 7 U 17 6 26 F 25 F 24 6 26 25 24 76 U 12 7 26 76 m 7 6 7 6 e m e e e 5 5 5 4 F 36 F e35 F e34 4 4 0 0 0 F 38 0 F 36 F 35 F 34 0 U 17 0

0

0

F e48

F m 46

F m 45

0

F m 44

Um 13

0

0

0

F e48

The reflected/transmitted coefficient (RTC) is defined by the ratio of the reflected/transmitted amplitudes to the incident amplitude. The RTCs may be obtained by solving these equations. It should be mentioned that the wave amplitudes of U e11 , U e12 , U e13 and U e17 on the right-hand side of Eq. (27) stand for the incidence of QP, QSV, evanescent EA and MP waves in the piezoelectric originating medium, respectively. If the originating medium is the piezomagnetic material, the RTCs can be obtained by following the similar procedure. For the sake of brevity we shall not pursue this any further. 4. Energy transmission and partition For practical engineering applications in areas such as signal processing, transduction and frequency control, it may be important to examine the transmission and partition of the incident wave energy at the interface [26]. In this section, we will examine this problem. The time averaged energy flux carried by a wave is defined as [2] F¼

x 2p

Z

p x

PðtÞ dt;

ð28Þ

p x

where P(t) is the energy flux density which is a function of the time; and x is the angular frequency. For the case of the electrostatics and magnetostatics approximation, the energy flux density in the piezoelectric material is determined as [27] _ P ¼ r  u_ þ uD

ð29Þ

and in magnetoelastic media the power flow of a magneto-elastic wave without considering the dispersion induced by magnetization can be expressed as [28] _ P ¼ r  u_  /B;

ð30Þ

where the symbol ‘‘” denotes the differentiation with respect to time t. Inserting Eqs. (29) and (30) into Eq. (28) yields the time averaged energy fluxes of the scattered waves in piezoelectric and piezomagnetic media, respectively. Considering the periodicity of the problem along the interface (x-axis), we examine the energy fluxes into and out of a thin slice of the material with infinitesimal thickness (d ? 0) and 2p in length containing the interface, (see Fig. 1). The scattered bulk waves can input to or extract energy from the thin slice. The evanescent waves propagate carrying energy along the interface. They have no contribution to the energy fluxes into and out of the thin slice because of its infinitesimal thickness. (Even the slice is of finite thickness, the energy fluxes carried by the evanescent waves into and out of the slice are equal because of periodicity.) So here we only evaluate the time averaged energy fluxes over a period of all the scattered bulk waves. The energy flux flowing into the slice, i.e. the input energy flux of the incident wave, is given by 1 I 2 fF e g ¼  nxðF e16 þ W e6 F e26 þ ue6 F e36 ÞðU e11 Þ 2

ð31Þ

and the energy flux outflowing from the slice, carried by the reflected and transmitted waves, is given by 1 1 fF e gR ¼  nxðF e16 þ W e6 F e26 þ ue6 F e36 ÞðU e16 Þ2  nxðF e15 þ W e5 F e25 þ ue5 F e35 ÞðU e15 Þ2 2 2

ð32Þ

1 1 m m m 2 m m m 2 m m m m fF m gT ¼  nxðF m nxðF m 11 þ W 1 F 21 þ /1 F 41 ÞðU 11 Þ  12 þ W 2 F 22 þ /2 F 42 ÞðU 12 Þ : 2 2

ð33Þ

and

We define the energy coefficients as the time averaged energy fluxes of the reflected or refracted waves compared with that of the incident wave, i.e. {Fe}R/{Fe}I and {Fm}T/{Fe}I. From the energy balance of the slice, we should have

1104

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

fF e gR e I

fF g

þ

fF m gT fF e gI

¼ 1;

ð34Þ

which can be used as a check on the numerical computations. 5. Numerical results and discussion The formulation and discussion in the previous sections are quite general. However, in order to carry out computations and to give explicit exploration to the problem, we will consider a particular example. We take the BaTiO3/CoFe2O4 bi-material system for detailed computation. Both piezoelectric material, BaTiO3, and piezomagnetic material, CoFe2O4, are hexagonal crystals with the material properties listed in Tables 1 and 2, respectively. Liu and Chue [29] have indicated that the negative permeability l11 of CoFe2O4 is unreasonable. Here we take l11 = l22 = l33 = 157  106 Ns2/C2. We will consider four cases: QP wave incidence from BaTiO3 and CoFe2O4 (Case I) and QSV wave incidence from BaTiO3 and CoFe2O4 (Case II). The RTCs and energy coefficients for different incident angles varying from 1° to 90° are computed. For convenience of discussion, the slowness curves on the x–z plane for BaTiO3/CoFe2O4 are plotted in Fig. 2. The reflected and refracted waves for the QP wave incidence from BaTiO3 are also shown. As we know, all critical angles can be calculated from the slowness curves.

Table 1 Material constants for the piezoelectric solid BaTiO3 (Cij in 109N/m2, eij in C/m2, eij in 109C2/(Nm2), lij in 106Ns2/C2 and q in kg/m3) [30] C11 = C22

C12

C13 = C23

C33

C44 = C55

C66 = 0.5(C11–C12)

166

77

78

162

43

44.5

e31 = e32 4.4

e33 18.6

e24 = e15 11.6

q 5300

e11 = e22 11.2

e33 12.6

l11 = l22 5

l33 10

Table 2 Material constants for the piezomagnetic solid CoFe2O4 (Cij in 109N/m2, hij in N/Am, eij in 109C2/(Nm2), lij in 106 Ns2/C2 and q in kg/m3) [30] C11 = C22

C12

C13 = C23

C33

C44 = C55

C66 = 0.5(C11–C12)

286

173

170.5

269.5

45.3

56.5

h31 = h32 580.3

h33 699.7

h24 = h15 550

q 5800

e11 = e22 0.08

e33 0.093

l11 = l22 590

l33 157

CoFe2O4 QSV QP

QP QSV

BaTiO3 Fig. 2. Slowness curves of BaTiO3/CoFe2O4 on x–z plane and reflected and refracted waves for QP wave incidence from BaTiO3.

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Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

5.1. Case I – QP wave incidence Figs. 3 and 4 show the RTCs and energy coefficients varying with the incident angle for the QP wave incidence from BaTiO3 and CoFe2O4, respectively. For the QP wave incidence from BaTiO3, a critical angle appears at 48.9°, where the transmitted QP wave becomes evanescent propagating along the interface. Figs. 3a and 4a depict the RTCs for the reflected and transmitted QP and QSV waves. In both cases, the RTC for the reflected QP waves increases with the incident angle increasing until the incident angle exceeds the critical angle (if exists); then it almost keeps unchanged (Fig. 3a). However, the values for the other three waves generally decrease with the increase of the incident angle, except that the transmitted QP wave increases slightly in the middle values of the incident angle. If a critical angle exists, such as in Fig. 3a, the RTC for these three waves first increase, attain their peaks at the critical angle, then decrease. Obviously, the critical angle has remarkable influence not only on the transmitted QP wave but also on all other scattered bulk waves. Figs. 3b and 4b show the energy coefficients of the energy flux normal to the interface for the reflected and transmitted QP and QSV waves. With the increase of the incident angle, the values for both reflected and transmitted QSV waves in two cases first increase from zero to peaks then reduce to a rather low lever at 90°; the values for the transmitted QP waves decrease; and those for the reflected QP waves increase. After the incident angle exceeds the critical angle (if exists), the values for the reflected QP wave keep unchanged (Fig. 3b). The critical angle has little effect on the values for QSV waves. It is easily verified that Eq. (35) is satisfied in both cases. From the two figures we observe that the transmitted/reflected QP waves carry most part of the incident energy flux at smaller/bigger incident angles. The energy flux of the QSV waves, which comes into being from the wave mode conversion, is much less than that of the QP waves. Figs. 3c and 4c show the RTCs for the EA and MA waves. With the increase of the incident angle, the values for these two waves first increase from zero near the normal incidence, attain their maximum values, then decrease and finally vanish at the grazing incidence. Because the EA and MA waves are evanescent, the critical angle has little effect on these waves. Figs. 3d and 4d depict the RTCs for the EP and MP waves. The values for the EP and MP waves all start at their maximum values near the normal incidence and decrease abruptly to a rather low level at a small incident angle, and then gradually to zero at the grazing incidence. We argue that the possible reason for this phenomenon is the limitation of the quasistatic approximation of the electro- and magneto-static equations [26]. But till now, it is not clear why it fails to describe a tiny vicinity of the normal incidence. The latest research [31] for piezoelectric media also noticed this and pointed out that the quasi-electrostatic approximation is absolutely adequate for solving the reflection–transmission problems in piezoelec-

0.25

0.20

1.6

1.2

0.15

0.8

0.10

0.4

0.05

0.0

0.00 0

10

20

30

40

50

60

70

80

0.030 1.0

RTC for QSV

RTC for QP

2.0

b

0.30

0.015 0.4

0.010

0.2

0.005

0.0

0.000

90

0

10

20

d

0.18

40

50

60

70

80

90

0.12 0.10 0.08

0.015

0.06 0.010 0.04 0.005

0.02

RTC for Reflected MP( ×108A/m)

0.14

0.020

10

Ref-MP Tran-EP

2.5

0.16

0.025

3.0

8 2.0 6

1.5 1.0

4 0.5 2

0.0 -0.5

0

0.00

0.000

-0.02 0

10

20

30

40

50

60

Incident Angle (in Degree)

70

80

90

RTC for Transmitted EP ( ×1010V/m)

Ref-EA Tran-MA

RTC for Transmitted MA

RTC for Reflected EA

30

Incident Angle (in Degree)

0.035

0.030

0.020

0.6

Incident Angle (in Degree)

c

0.025

Ref-QP Tran-QP Ref-QSV Tran-QSV

0.8

Energy coefficients for QSV

Ref-QP Tran-QP Ref-QSV Tran-QSV

2.4

Energy coefficients for QP

a

-1.0 0

10

20

30

40

50

60

70

80

90

Incident Angle (in Degree)

Fig. 3. Variations of reflection and transmission coefficients and energy coefficients with the incident angle for the QP wave incidence from BaTiO3 medium.

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

a

b

0.12

Ref-QP Tran-QP Ref-QSV Tran-QSV

1.0

0.10

0.6 0.06 0.4

0.04

0.2

0.02

0.0

0.00

RTC for QSV

RTC for QP

0.08

Ref-QP Tran-QP Ref-QSV Tran-QSV

0.8

4

3

0.6 2 0.4 1 0.2 0 0.0

Energy coefficient for QSV( ×10-3)

0.8

5 1.0

Energy coefficient for QP

1106

-1

0

10

20

30

40

50

60

70

80

90

0

10

20

Incident Angle (in Degree)

d

0.008 0.004 0.006 0.003 0.004

0.002

0.001

0.000

0.000 20

30

40

50

60

Incident Angle (in Degree)

70

80

90

70

80

90

3.5

10

3.0 2.5

8

2.0 6

1.5 1.0

4

0.5 2

0.0 -0.5

0

8

0.002

10

60

Ref-EP Tran-MP

10

RTC for transmitted EA

RTC for reflected MA

0.006

0.005

0

50

RTC for Transmitted MP( ×10 A/m)

Ref-MA Tran-EA

0.010

40

Incident Angle (in Degree)

0.012

RTC for Reflected EP ×10 V/m)

c

30

-1.0 0

10

20

30

40

50

60

70

80

90

Incident Angle (in Degree)

Fig. 4. Variations of reflection and transmission coefficients and energy coefficients with the incident angle for the QP wave incidence from CoFe2O4 medium.

tric materials with the exception of the incidence under a small angle of the order of va /vel to the normal to the interface (where va and vel are the velocities of the sound and electromagnetic waves). The critical angle has little effect on the EP and MP waves because they are evanescent. Fig. 5 shows the energy coefficients of the electric and magnetic energies carried by the QP waves in two cases. The electric and magnetic energies are calculated according to Eqs. (29) and (30), respectively, in which mechanical energy flux densities are excluded. As a reference, electric or magnetic energy carried by the incident QP wave is also depicted in Fig. 5. The energies carried by QSV waves, which arising from the wave mode conversion, are not depicted in this figure because they are about 102 order of those for QP waves. With the increase of the incident angle, the energy coefficients of electric energy in BaTiO3 increase from zero near the normal incidence to a maximum value at the critical angle in Fig. 5a and at 40° in Fig. 5b, and then reduce to zero at the grazing incidence. Magnetic energy in CoFe2O4 attains a lower peak at a smaller incident angle and a highest peak at a bigger incident angle. When the incident angle is beyond the critical angle (if exists), the magnetic energy in CoFe2O4 vanishes because the wave becomes evanescent with the magnetic energy concentrated near the surface. The magnetic energy carried by the bulk waves in CoFe2O4 is smaller than the electric energy carried by the bulk waves in BaTiO3. This is due to the fact that BaTiO3 has a greater capacity to transform mechanical energy into electric energy than CoFe2O4 to transform mechanical energy into magnetic energy. 5.2. Case II – QSV wave incidence In this section, we discuss the QSV wave incidence from BaTiO3 and CoFe2O4 media. Figs. 6 and 7 show the RTCs and energy coefficients varying with the incident angle in two cases, respectively. For the wave incidence from BaTiO3, two critical angles exist. The first one is 22.31°, where the transmitted QP wave becomes evanescent; and the second one is 32.7°, at which the reflected QP wave becomes evanescent. But for the wave incidence from CoFe2O4, there exists three critical angles at 24.2°, 37.36° and 78.7°, at which the reflected QP, transmitted QP and transmitted QSV waves take turns to become evanescent waves. Figs. 6a and 7a show the RTCs for the reflected and transmitted QP and QSV waves. With the increase of the incident angle, the values for the transmitted QSV wave first decrease then increase in both cases; and to a peak near 71°, then reduce to the lowest at the last critical angle (see Fig. 7a). However, the values for other three waves generally increase with the incident angle increasing with fluctuating at the first two critical angles. After the incident angle is beyond the last critical angle,

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Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

b

0.025

0.015 0.006 0.010 0.004

0.005

0.002

0.000

0.000

0

10

20

30

40

50

60

70

80

0.006 0.016

RQP IQP TQP

0.005

0.014 0.012

0.004 0.010 0.003

0.008 0.006

0.002 0.004 0.001

0.002 0.000

0.000

Energy coefficient for electric in BaTiO3

0.008

Magnetic energy in CoFe2O4

0.020

Electric energy in BaTiO3

0.010

RQP IQP TQP

Energy coefficient for magnetic in CoFe 2O4

a

-0.002

90

0

10

20

Incident Angle (in Degree)

30

40

50

60

70

80

90

Incident Angle (in Degree)

Fig. 5. Variations of energy coefficients of the electric and magnetic energy with the incident angle for the QP wave incidence from BaTiO3 and CoFe2O4 media.

0.9

1.1

0.8

1.0

1.0

0.9 0.8 0.7

0.5

0.6

0.4

0.5

0.3

0.4 0.3

0.2

0.2 0.1

Ref-QP Tran-QP Ref-QSV Tran-QSV

0.025

RTC for QSV

RTC for QP

0.6

1.1

0.020

0.7 0.6

0.015 0.5 0.4

0.010

0.3 0.2

0.005

0.0

0.0

-0.1

-0.1 10

20

30

40

50

60

70

80

0.1 0.0

0.000

-0.1

90

0

10

20

30

40

50

60

70

80

90

Incident Angle (in Degree)

Incident Angle (in Degree)

0.15

0.3

0.10

0.2

0.05

0.1

0.00

0.0

6

2.0

Ref-MP Tran-EP 1.5

5

1.0

4

0.5

3

0.0

2

-0.5

1

0

-1.0 0

10

20

30

40

50

60

Incident Angle (in Degree)

70

80

90

RTC for transmitted EP( ×109V/m)

0.4

RTC for transmitted MA

0.20

RTC for Reflected MP( ×107A/m)

d

0.5

0.25

Ref-EA Tran-MA

RTC for Reflected EA

0.8

0.1

0

c

0.9

Energy coefficient for QSV

Ref-QP Tran-QP Ref-QSV Tran-QSV

0.7

b 0.030 Energy coefficient for QP

a

0

10

20

30

40

50

60

70

80

90

Incident Angle (in Degree)

Fig. 6. Variations of reflection and transmission coefficients and energy coefficients with the incident angle for the QSV wave incidence from BaTiO3 medium.

the total reflection phenomenon takes place and the RTCs for the transmitted waves increase whereas those for the reflected waves keep unchanged (see Fig. 7a). It is obvious that the first two critical angles have little effect on the QSV waves, while that the last critical angle has significant effect on all scattered bulk waves. Figs. 6b and 7b show the energy coefficients for the reflected and transmitted bulk waves in two cases. With the increase of the incident angle, the values for the reflected and transmitted QP waves in two cases first increase, attain their peaks, and then reduce to zero at their corresponding critical angles. In both cases, the energy coefficients for the reflected QSV waves increase and those for the transmitted QSV waves decrease with the incident angle increasing. After the incident angle is

1108

4.0 3.5

0.030 1.0

5

0.025

4

2.5

3

1.5 2 1.0

RTC for QSV

2.0

1

0.5 0.0

0

Ref-QP Tran-QP Ref-QSV Tran-QSV

0.020

0.8

0.6 0.015 0.4

0.010

0.005

0.2

0.000

0.0

Energy Coefficient for QSV

3.0

RTC for QP

b

6

Ref-QP Tran-QP Ref-QSV Tran-QSV

Energy Coefficient for QP

a

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

-0.5 0

10

20

30

40

50

60

70

80

90

0

10

20

Incident Angle (in Degree)

c 0.7

0.2 0.2 0.1

0.1 0.0

0.0

-0.1 30

40

50

60

Incident Angle (In Degree)

70

80

90

RTC for Reflected EP( ×1010V/m)

RTC for Reflected MA

0.3

RTC for Transmitted EA

0.3

0.4

20

60

70

80

90

5

4

1.0 3 0.8 2 0.6 1

0.4

0

0.2

RTC for Transmitted MP( ×107A/m)

0.4

10

50

Ref-EP Tran-MP

1.2

0.5

0

40

d

0.5

Ref-MA Tran-EA

0.6

30

Incident Angle (In Degree)

-1

0.0 0

10

20

30

40

50

60

70

80

90

Incident Angle (in Degree)

Fig. 7. Variations of reflection and transmission coefficients and energy coefficients with the incident angle for the QSV wave incidence from CoFe2O4 medium.

beyond the third critical angle, the value for the reflected QSV wave increase to unit whereas that for the transmitted QSV wave reduces to zero (see Fig. 7b). Eq. (35) is satisfied in both cases. From the two figures we observe that the transmitted/ reflected QSV waves carry most part of the incident energy at smaller/bigger incident angles. The energy of the QP waves, which comes into being from the wave mode conversion, is much less than that of the QSV waves. Figs. 6c and 7c depict the RTCs for the EA and MA waves in two cases. With the increase of the incident angle, the values increase until the third critical angle (if exists as in Fig. 7c) where the transmitted EA wave decreases to a minimum and the reflected MA wave increases to a maximum. The first two critical angles have little effect on the evanescent EA and MA waves in both cases. Figs. 6d and 7d depict the RTCs for the EP and MP waves. The values increase with the increase of the incident angle in both cases. After the incident angle exceeds the third critical angle (if exists as in Fig. 7d), the RTCs for the two waves keep almost unchanged. 6. Concluding remarks By solving a set of linear algebraic equations, we have obtained the reflection and transmission coefficients (RTCs) and energy coefficients for the QP and QSV waves incident obliquely at the interface between piezoelectric and piezomagnetic media. Numerical calculations are performed for BaTiO3/CoFe2O4 material combination. Numerical results of the RTCs and energy coefficients are presented for the QP/QSV wave incidence from BaTiO3 and CoFe2O4 media, respectively. The results show: (1) For the QP or QSV wave incidence from BaTiO3 or CoFe2O4 medium, the reflected and transmitted wave fields in the sagittal plane consist of six kinds of waves, i.e. the coupled QP waves, QSV waves, evanescent EA wave in the piezoelectric medium (BaTiO3), evanescent MA wave in the piezomagnetic medium (CoFe2O4), the uncoupled evanescent MP wave in the piezoelectric medium (BaTiO3) and evanescent EP wave in the piezomagnetic medium (CoFe2O4). (2) With the increase of the incident angle, the RTCs for the transmitted and reflected QP and reflected QSV wave increase before they become evanescent in turns; but those for the transmitted QP and reflected QSV waves decrease if no critical angle exists (see Fig. 4a). The behavior of the transmitted QSV wave is much more complex.

Y. Pang et al. / International Journal of Engineering Science 46 (2008) 1098–1110

1109

(3) With the increase of the incident angle, the RTCs for the MA, EA, MP and EP waves increase for the QSV wave incidence; the values for the MA and EA waves first increase then decrease while those for the MP and EP waves decrease for the QP wave incidence. Critical angle has little effect on these evanescent waves except when the total reflection takes place (see Figs. 7c and d). (4) The most amount of the incident energy goes with the wave which is the same type as the incident wave, while the energy which is from wave mode conversion occupies a less part of the incident energy. The energy fluxes normal to the interface carried by the waves that come from wave mode conversion attain their maximum values at/near the corresponding critical angles. (5) The electric energy in BaTiO3 and the magnetic energy in CoFe2O4 attain their maximum values at/before the critical angle. The electric energy in BaTiO3 is higher than the magnetic energy in CoFe2O4 because BaTiO3 has more capacity to transform mechanical energy into electric energy than CoFe2O4 to transform mechanical energy into magnetic energy.

Acknowledgement The authors are grateful to the support by the National Natural Science Foundation of China under Grant Nos. 10672108 and 10632020 and the key project of the Ministry of Education of China under Grant No. 206014. Appendix A The elements of the symmetric matrix Ke in Eq. (10) for the piezoelectric medium are K e11 ¼ C e11 þ C e44 ae2  qe c2 ; K e22 K e44

¼ ¼

C e44 le11

þ C e33 ae2  þ le33 ae2 ;

e 2

q c ; K e14 ¼

K e12 ¼ ðC e13 þ C e44 Þae ; K e23 ¼ e15 þ e33 ae2 ; K e24 ¼ K e34 ¼ 0:

K e13 ¼ ðe15 e K 33 ¼ ðee11

þ e31 Þae ; þ ee33 ae2 Þ;

ðA:1Þ

The elements of the symmetric matrix Km in Eq. (10) for the piezomagnetic medium are m m m2 Km  qm c2 ; 11 ¼ C 11 þ C 44 a

Km 22 Km 44

¼ ¼

Cm 44

m2 þ Cm  qm c2 ; 33 a m m m2 ðl11 þ l33 a Þ; K m 13

m m m Km 12 ¼ ðC 13 þ C 44 Þa ;

Km 24 ¼

m2

¼ h15 þ h33 a

Km 23

¼

Km 43

;

m Km 14 ¼ ðh15 þ h31 Þa ; m m m2 Km ; 33 ¼ e11 þ e33 a

ðA:2Þ

¼ 0:

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