Wave propagation in a magneto-electroelastic

Science in China Series G: Physics Mechanics & Astronomy 008 SCIENCE IN CHINA PRESS Springer-Verlag www.scichina.com phys.scichina.com www.springerlink.com Wave propagation in a magneto-electroelastic plate WEI JianPing & SU XianYue State Key aboratory for urbulence and Complex System and Department of Mechanics and Aerospace Engineering Peking University Beijing 100871 China he wave propagation in a magneto-electro-elastic plate was studied. Some new characteristics were discovered: the guided waves are classified in the forms of the Quasi-P Quasi-SV and Quasi-SH waves and arranged by the standing wavenumber; there are many patterns for the physical property of the magneto-electro-elastic dielectric medium influencing the stress wave propagation. We proposed a self-adjoint method by which the guided-wave restriction condition was derived. After the corresponding orthogonal sets were found the analytic dispersion equation was obtained. In the end an example was presented. he dispersive spectrum the group velocity curved face and the steady-state response curve of a magneto-electro-elastic plate were plotted. hen the wave propagations affected by the induced electric and magnetic fields were analyzed. magneto-electro-elastic plate guided-wave restriction condition dispersive equation dispersive spectrum group velocity curved face steady-state response curve Magneto-electro-elastic dielectrics are widely used in the aerospace structures the equipments of the energy and chemical industries and they are the main part of the electron elements and the microelectronics mechanical systems. his kind of medium has piezoelectric piezomagnetic elastic dielectric and electromagnetic properties and there are no free electrons and electric flux in it so the electric and magnetic influences can only transmit in the forms of the induced electric and magnetic fields. But these media are very sensitive to the outside physical fields and they have many response patterns. In general there are coupling stress waves propagation and coupling mechanical and electromagnetic signs transmission which can affect the operation reliability and the control precision of these equipments and systems mentioned above. Because these media have coupled piezoelectric piezomagnetic and anisotropic elastic constitutive relations etc. the characteristics of the stress wave propagation are very complicated. If the effects of the boundary conditions are considered the coupling waves will reflect and refract at the boundaries thus forming complicated interferences and causing dispersive phenomena. Further- Received August 3 006; accepted May 15 007 doi: 10.1007/s11433-008-0059-9 Corresponding author (email: xysu@mech.pku.edu.cn) Supported by the National Natural Science Foundation of China (Grant Nos. 1057001 and 103040) Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

more the dynamic problem will become more difficult and complicated in magneto-electro-elastic structures. At present the Bessel-Fourier expansion method [1] and the Stroh method [3] are widely used to study the dynamic problems of the magneto-electro-elastic and piezoelectric structures but these two methods have some limitations. he Bessel-Fourier expansion method can be only applied in cylindrical structures and it requires that the physical property of the cross section is isotropic. he Stroh method is a complex method. heoretically it can solve all kinds of magneto-electro-elastic structures but this method is only applicable to two space arguments problems. he dispersive equations derived from these two methods contain transcendental functions so it is very difficult to plot the dispersive spectrum. Some researches have tackled the complicated magneto-electro-elastic guided-wave problems. Paul and Venkatesan [45] applied the Bessel-Fourier expansion method to studying the dispersive equation of a piezoelectric cylinder with a slightly changing cross section. It is shown that the dispersive equation changes with the slightly changing cross section. But it is very difficult to plot the dispersive curves so the difference between two slightly changing cross sections cannot be compared. Chen et al. [6] and Ding & Xu [7] studied the vibration of composite plates. In order to get the iteration matrix they selected two boundary conditions (elastic simply supported and rigid sliding supported) in the iterative process. It is very easy to find two group orthogonal sets for these two boundary conditions but it is very difficult to obtain the orthogonal sets for other boundary conditions to realize the iterative process. hese imply that there are internal relations between the boundary condition and the orthogonal sets. We have applied these two boundary conditions in studying the wave propagation in a piezoelectric cylinder [8]. We have also applied an approximate model in solving the guided-wave problems of the piezoelectric and magneto-electro-elastic cylinders [910] obtained the lower order modes and found that the electric and magnetic physical properties have different influences on the stress wave propagation in the piezoelectric and magneto-electro-elastic dielectric media. In this paper the self-adjoint method is presented in studying the wave propagation in the magneto-electro-elastic plate. he physical property of the magneto-electro-elastic dielectric medium the space structure and boundary condition of the guided-wave system are combined together in this method and a guided-wave restriction condition is derived to describe the inner relations of the guided-wave system in mathematics. After finding the orthogonal sets which satisfy the guided-wave restriction condition and mapping them from the time domain to the frequency domain the analytic dispersive equation the group velocity equation and the steady-state response are obtained completely. 1 Basic equation We consider the magneto-electro-elastic dynamic equations in a rectangular coordinate system. he relation between deformation and displacement is 1 su ( ) = ( u+ u ). (1) Relation between electric field and electric potential is E ( φ) = φ. () Relation between magnetic field and magnetic potential is 65 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

H ( ψ ) = ψ (3) where s = [ ] u = [ ] E = [ ] H = [ ] φ and ψ are the strain tensor displacement vector s ij u i E i H i electric field vector magnetic field vector electric potential and magnetic potential respectively. he magneto-electro-elastic constitutive relations are σ = c s( u) E( φ) f H( ψ ) d (4) D= f s( u) + ε E( φ) + h H ( ψ ) (5) B = d s( u) + h E( φ) + μ H ( ψ ) (6) where σ = [ ] D = [ ] and B = [ ] are the stress tensor electric displacement vector and σ ij D i B i magnetic induction vector respectively. c = [ ] f = [ ] d = [ ] ε = [ ] h = [ ] c ijkl and μ = [ ] are the constants of the elasticity tensor piezoelectric tensor piezomagnetic tensor μ ij dielectric tensor electromagnetic tensor and permeability tensor respectively. he symmetric and transpose relations of the above tensors are c = c = c = c = f ijk ijkl jikl ijlk klij and ij = ji. he governing equations in the magneto-electro-elastic dielectric are σ = ρu (7) D = 0 (8) B = 0 (9) where ρ is the mass density. he superimposed dot indicates the differentiation with respect to the time parameter t. Here body force is neglected for simplicity. Substituting eqs. (1)-(6) into eqs. (7)-(9) the governing equation in the form of the displacements electric and magnetic potentials is U [ ( x t )] = 0 (10) where where U( x t) = U( x1 x x3 t) = [ u φ ψ]. he differential operator is ci11 k ik ρ tt ci1 k ik ci13 k ik fi1k ik di1k ik cik1 ik cik ik ρ tt cik3 ik fik ik dik ik = ci31 k ik ci3 k ik ci33 k ik ρ tt fi3k ik di3k ik fij1 ij fij ij fij3 ij εij ij hij ij dij1 ij dij ij dij3 ij hij ij μij ij ij = ( i j = 1 3) x x i j d ijk ijk ikj ε ij ijk = h ij jki (11) tt =. In derivation of the overall formulas we take t Einstein notation; while we do not take a special note will be given. Self-adjoint method If the steady-state response of the structure is considered namely U( x t) is the harmonic wave in formula (10) the coefficient matrix of U [ ( x t )] = 0 will be a Hermite s matrix. he characteristic of the Hermite s matrix is that all eigenvalues are real and the eigenvectors are orthogonal one WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 653

another for different eigenvalues. Eringen and Suhubi [11] got that the squares of the wave frequencies are positive real numbers in studying the differential operator of isotropic material and the eigenvectors construct a complete orthogonal set. Gurtin [1] proposed a method to construct the orthogonal sets for the isotropic guided-wave systems and he applied this method to getting some group orthogonal sets in one- and two-dimensional endless guided-wave systems. In this paper we still apply the characteristic of the Hermite s matrix and construct the complete orthogonal sets to solve the magneto-electro-elastic guided-wave problem. he differential operator is a self-adjoint operator so it satisfies the following integral: { V ( x) t U [ ( x)] t U ( x) t V [ ( x )]}d t V = 0 (1) Ω where U( x t) = [ u φ1 ψ1] and V( x t) = [ v φ ψ] are two arbitrary orthogonal modes of structure Ω. Substituting the two orthogonal modes into the differential eq. (11) and calculating the following formula yield V ( x t) U [ ( x t)] U ( x t) V [ ( x t)] = { v [ c s( u) + φ1 f + ψ1 d]} + { φ[ f s( u) φ1 ε ψ1 h]} + { ψ[ d s( u) φ1 h ψ1 μ]} { u [ c s( v) + φ f + ψ d]} { φ1[ f s( v) φ ε ψ h]} { ψ1[ d s( v) φ h ψ μ]} ρv u + ρu v. (13) If we consider the harmonic wave propagation in this guided-wave system the displacements electric and magnetic potentials will be written as iωt U( x t) = B U( x )e (14) where ω is the circular frequency; i = 1; B U( x ) is the mode; B is the wave amplitude matrix; U( x )= [ U ] 1 U U3 U4 U 5 is the orthogonal sets. Substituting eq. (13) into (1) and considering the harmonic wave eq. (14) we obtain { V ( x) t U [ ( x)] t U ( x) t V [ ( x)]}d t V Ω = { V ( x) [ U( x)] n U ( x) [ V( x)] n }ds Ω = 0 where n is the external normal vector of the structure surface Ω c s( u) + φ1 f + ψ1 d σ[ U( x)] U [ ( x)] = f su ( ) φ1 ε ψ1 h = DU [ ( x)] ( ) φ [ ( )] 1 ψ d s u h 1 μ BU x c s( v) + φ f + ψ d σ[ V( x)] V [ ( x)] = f sv ( ) φ ε ψ h = DV [ ( x)]. ( ) φ [ ( )] ψ d s v h μ BV x On the structure surface in the form of the harmonic wave eq. (15) equals (15) (16) 654 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

V ( x) [ U( x)] n U ( x) [ V( x)] n = 0 on Ω (17) and eq. (17) can be written in the following determinant form: Um m( U) = det 0 Vm m( V) where blocking matrix m = diag( U 1 U U 3 U 4 U 5 ) 1j nj jnj 3j j 4j j 5j j on Ω (18) V = diag( V V V V V ) and m = U m 1 3 4 5 diag( n n n ) are all diagonal matrixes. Furthermore we obtain au + b ( U ) n = 0 i i i ij j av + b ( V ) n = 0 (i from 1 to 5; not to do summation over i on Ω ) (19) i i i ij j where a i and b i ( i = 1 3 4 5) are dimensional constants and will not equal zero at the same time. Because the orthogonal sets U( x ) and V( x ) have the same form only one of the two formulas in eq. (19) needs to be taken au + b ( U ) n = 0 (i from 1 to 5; not to do summation over i on Ω ). (0) i i i ij j Now eq. (0) contains a lot of information. Firstly it holds on the structure surface involving every boundary condition; secondly it satisfies arbitrary structures because of the external normal vector; thirdly it fully reflects the constitutive relation of the magneto-electro-elastic dielectric medium; fourthly it gives the restriction condition between the displacements electric and magnetic potentials and the stress electric displacement and magnetic induction. So we name eq. (0) the guided-wave restriction condition in the magneto-electro-elastic guided-wave system. We can also get that there exists a group of different orthogonal set matching every boundary condition. his proves that different boundary conditions adopt different orthogonal sets in refs. [6-8]. When a i and b i ( i = 1 3 4 5 ) only equal 1 or 0 there are 3 (= 5 ) groups of boundary conditions in the same normal direction. But most of them only have meaning in mathematics and will not appear in a real situation. We name these boundary conditions self-adjoint boundary conditions to make a distinction with the other boundary conditions. Eq. (0) implicates the constitutive relation. If the boundary condition only has the displacements electric and magnetic potentials namely a i = 1 and b i = 0 ( i = 1 3 4 5 ) which means that the constraint of the constitutive relation is removed we need to add the constitutive relation to constrain the solution space in this case. So we know that the self-adjoint method describes the inner relations among the physical property of the magneto-electro-elastic dielectric the space structure and the boundary condition of the guided-wave system. hen we need only find the corresponding orthogonal sets to thoroughly solve the guided-wave problem. In detail for a real guided-wave system if the corresponding orthogonal sets are found by which the displacements electric and magnetic potentials that obey the operator eq. (10) are derived then the differential eq. (11) is applied to mapping wave propagation from the time domain to the frequency domain. he relation between the wavenumber and the frequency is decided namely the dispersive equation is obtained. WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 655

3 Wave propagation in a magneto-electro-elastic plate Plate is an easily manufactured simple structure with a widely applying range. It is very easy to install the detecting devices on its surfaces. If the solution of the wave propagation and the mechanical and electromagnetic signs transmission (energy transportation) characteristic is obtained it will help us in applying the magneto-electro-elastic dielectric medium and structure. he oyz plane is in the middle of the infinite plate; the x-axis is normal to the middle plane. he plate is high. Figure 1 is the map of the exhibition of the magneto-electro-elastic plate. Figure 1 he map of the exhibition of magneto-electro-elastic plate. 3.1 Wave propagation governing equation in a magneto-electro-elastic plate he magneto-electro-elastic constitutive relation considered in this paper is σ xx c11 c1 c13 f31 d31 xx σ yy c1 c11 c13 f31 d s yy 31 σ zz c13 c13 c33 f33 d s 33 σ yz zz c44 f15 d s 15 yz σ zx c44 f15 d s 15 zx σ xy c s 55 xy = Dx. f15 ε11 h 11 E x D f y 15 ε11 h11 E y D f z 31 f31 f33 ε33 h33 E z B d x 15 h11 μ11 H x B d y 15 h11 μ11 H y B d31 d31 d33 h33 μ 33 z H z Substituting constitutive eq. (1) into differential eq. (11) yields s (1) 656 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

c11 + c 55 + c 44 ρ ( c 1 + c55) x y z t xy ( c + c ) c + c + c ρ xy x y z t = ( c13 + c44) ( c13 + c44) xz yz ( f15 + f31) ( f15 + f31) xz yz ( d15 + d31) ( d15 + d31) xz yz 1 55 55 11 44 ( c13 + c44) ( f15 + f31) ( d15 + d31) xz xz xz ( c13 + c44) ( f15 + f31) ( d15 + d31) y z y z y z c + + c ρ f + + f d + + d x y z t x y z x y z So 44 33 15 33 15 33 f15 + + f33 ε 11 + ε33 h 11 + h33 x y z x y z x y z d15 + d + 33 h 11 + h 33 μ 11 + μ 33 x y z x y z x y z () U [ ( x t )] = 0 (3) is the wave propagation governing equation about the constitutive eq. (1) where the displacements electric and magnetic potentials are U( x t) = U( x y z t) = [ u φ ψ] = [ u v w φ ψ]. (4) On the surface of the plate the guided-wave restriction condition can be simplified as au 1 ± b1σ xx = 0 av ± bσ yx = 0 aw 3 ± b3σ zx = 0 (5) a4φ ± b4dx = 0 a5ψ ± b5bx = 0. Now after giving the boundary condition the guided-wave system will be decided. If the orthogonal sets satisfying the restriction eq. (5) can be found the dynamic problem will be thoroughly solved. 3. Dispersive equation Here we select a group self-adjoint boundary condition B1: WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 657

u = 0 σ = 0 σ = 0 D = 0 B = 0 x = ±. (6) xy xz It means that the plate surface cannot move in the x-axis direction but it can slide in oyz plane and the electric and magnetic fields are in an open circuit state at the boundary. Because the plate is limited in the x-axis direction we apply the standing wave superposition method to forming the wave propagation in the x-axis direction and thus we assume the displacements electric and magnetic potentials in the following forms: x iωt i( ny+ kz ωt) m m m m= 0 m= 0 U( x t) = [ u v w φ ψ] = U ( x)e = B U ( x)e = Au m m Bv m m Cmwm Dmφm Emψm m= 0 m= 0 m= 0 m= 0 m= 0 x i( ny+ kz ωt) e where BU m m( x ) is the mode B m is the wave amplitude matrix U m ( x ) is the orthogonal sets A B m m C m m D and potentials respectively. (7) E m are the coefficients of the displacements electric and magnetic b = [ A B C D E ] is the wave amplitude vector and B m = m m m m m diag( Am Bm Cm Dm E m). m is the standing wavenumber and mπ/ will be the nonnegative integer. n and k are the propagating wavenumbers along y- and z-axis directions. In our model the wave which keeps the standing waveform propagates along the propagating wave direction. For the same standing waveform the propagating waves are independent for different propagating wavenumbers. From the above assumption we find the orthogonal sets satisfying the self-adjoint boundary condition B1 which are mx π mx π mx π sin cos cos um = vm = wm = mx π cos mx π sin mx π sin mx π mx π cos cos φm = ψ m =. (8) mx π sin mx π sin When m is even the first lines are adopted; when m is odd the second lines are adopted. When m s we obtain [ Um( x) U s( x)]dx= 0. (9) Whether the different modes are orthogonal one another is decided by the standing wavenumber. So we get the complete orthogonal sets. Substituting eqs. (7) and (8) into (3) for a fixed standing wavenumber m we obtain 1 ( ω m ; n k ) U m ( x y z ) = 0 (30) where the matrix 1 ( ω m; n k) is a Hermite s matrix. Its elements are in Appendix. he differential eq. () is applied to mapping the wave propagation from the time domain to the frequency domain so matrix 1 ( ω m; n k) gives the relation between the wavenumber and the frequency of the guided-wave system. When the wave arrives the displacements electric and 658 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

magnetic fields will not equal zero simultaneously. So the dispersive equation of the plate about the boundary condition B1 is F( ω m; n k) = det[ 1( ω m; n k)] = 0. (31) he elements of the matrix 1 ( ω m ; n k ) are all polynomials. F( ω m; n k) = 0 is a cubic ij polynomial equation about the circular frequency ω so ω can be solved and analytically expressed by the parameters m n and k in other words we get the analytical dispersive equation. here are three ω for every group wavenumber m n and k. We denote the three wavenumber-frequency groups as ( ω i m; n k) i = 1 3. here is the relation the phase velocity c so is the phase velocity equation. ω = sc among the frequency the wavenumber s = m + n + k and Fsc ( m; n k ) = 0 (3) 3.3 Group velocity equation he x-axis direction is the standing wave direction. When the wavenumber m is given the energy exchange along this direction is zero in one period time. he group velocity can be defined as dω cg = where wavenumber s1 = n + k. he relation between the group velocity and the ds1 wavenumber-frequency group is that the propagating wave keeps the standing waveform and propagates along the direction decided by n and k with the speed of this group velocity. And the group velocity is applied to describing the speed of the energy flux so it is less than the wave propagation velocity in the infinite body with the same medium in the same direction. hus the group velocity curved face is decided by a uniform continuity function. Obtaining the value of the group velocity of n and k in the group velocity curved face we can arrive at this pair (n k) through an arbitrary route. Assume n= αk k 0 where α is the propagating wavenumber ratio. So 1 k α 1 s = + and s 1 ds 1 = ndn+ kd k. Differentiating eq. (31) produces F F F F s1 F s1 F df = dn+ dk + dω = ds1 + ds1 + dω n k ω n n k k ω = 1 F F ds1 + dω = 0 α + 1 k ω and 1 F F + c g = 0 α + 1 k ω then eliminating ω with eq. (31) the group velocity equation is Gc ( g m; n k ) = 0. (33) F F When k = 0 there is + c g = 0. n ω 3.4 Steady-state response he relation between the wavenumber-frequency groups ( ω i m; n k) i = 1 3 and the matrix 1 ( ω m; n k) will be discussed again. First the matrix 1 ( ω m; n k) is blocked and then it becomes WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 659

where A ρω I R 1 ( ω m; n k) = R S A I R S. 111 11 113 114 115 144 145 ρω = 11 1 13 = 14 15 = 154 155 131 13 133 134 135 Because det[ 1( ω m; n k)] = 0 the equation 1 ( ω m; n k) b = 0 (35) has a nonzero solution. he vector is denoted as * * * 1 3 4 5 1 4 5 1. [ ] b= b b b b b = b b b = b b Eq. (35) changes to so then we obtain In eq. (38a) the matrix eigenvector must exist denoted by A I R = 0 * R S ( b ) * ρω ( b1 ) * * 1 (34) (36) ( A ρω I)( b ) + R( b ) = 0 (37a) * * 1 R ( b ) + S( b ) = 0 (37b) 1 * ρω 1 ( A RS R I)( b ) = 0 (38a) * 1 * = 1 ( b ) S R ( b ). (38b) 1 A RS R is a Hermite s matrix and * * * 1 ω1 1 ω 1 ω3 b ( ) b ( ) b ( ). And we have * * 1 ω 1 ρω is the eigenvalue so the ( ) b i b ( ωj) = 0 i j. (39) In eq. (39) the wave amplitude vectors which are formed by the three wave amplitudes about the displacement are orthogonal at the same group wavenumber with the different frequencies. his wave amplitude vector decides the orthogonal modes when the three group wavenumbers are 1 the same. If the matrix RS R = 0 this modal will degenerate to the elastic guided-wave system. 1 he influences of the induced electromagnetic field come from matrix RS R. In formula (38b) * * * the wave amplitude vectors b ( ω ) b ( ω ) b ( ω ) are derived so the wave amplitude vectors b( ω i m; n k) 1 3 * * ω 1 3 i = are obtained. But we need point out that [ ] b( ωi) b ( ωj) 0 i j and ( ) b i b ( ωj) 0 i j at the same group wavenumber. his phenomenon comes from the assumption of this modal; in detail it comes from eqs. (8) and (9). But it surpasses the discussion range in this paper so we will not debate it any more. When the wave amplitude vector is obtained for the arbitrary standing wavenumber m the steady-state response is derived: B ( ω ) U ( ) (not to do summation over m) (40) m m x 660 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

b ( ω) diag[ b ( ω) b ( ω) b ( ω) b ( ω) b ( ω)] and where m = 1m m 3m 4m 5m 1m m 3m b ( ω) + b ( ω) + b ( ω) = 1. 4 Dispersive spectrum group velocity curved face and steady-state response curve In this part the spectrum the group velocity curved face and the steady-state response curve will be plotted. In order to analyze the wave propagation affected by the physical properties of the magneto-electro-elastic dielectric medium we will plot the curves not only about the magneto-electroelastic plate but also about the piezoelectric piezomagnetic and elastic plates which adopt the corresponding parameters from the data of the magneto-electro-elastic plate. For example when we consider the piezoelectric plate we take the parameters c 11 c 1 c 13 c 33 c 44 c 55 f 15 f 31 f 33 ε 11 ε 33 and ρ. In computation the following parameters are taken: c 11 = 166 GPa c 1 = 77 GPa c 13 = 78 GPa c 33 = 16 GPa c 44 = 43 GPa c 55 = (c 11 c )/ f 15 = 11.6 C/m f 31 = 4.4 C/m f 33 = 18.6 C/m ε 11 = 11. nf/m ε 33 = 1.6 nf/m d 15 = 550 N/Am d 31 = 580.3 N/Am d 33 = 669.7 N/Am h 11 = 5.0 10 1 Ns/VC h 33 = 3.0 10 1 Ns/VC μ 11 = 5.0 10 5 Ns /C μ 33 = 1.0 10 5 Ns /C and ρ = 7500 kg/m 3. Before plotting the figures we introduce the non-dimensional frequency Ω = ω/ c phase wave velocity C = c/ c S and group velocity Cg = cg / cs where c S = c 44 / ρ. he symbols of MEEP PEP PMP and EAP represent the magneto-electro-elastic piezoelectric piezomagnetic and elastic plates respectively. 4.1 Dispersive spectrum Figure shows the dispersive spectrum of MEEP about the Quasi-P Quasi-SV and Quasi-SH waves with m = 0 and the Quasi-P wave with m = 1. For every standing wavenumber there are three curved faces and they are corresponding to the Quasi-P Quasi-SV and Quasi-SH waves. he S Figure Dispersive spectrum of MEEP. WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 661

propagating waves are arranged by the standing wavenumber. But for a better view the curved faces about the Quasi-SV and Quasi-SH waves with m = 1 are not plotted. In this figure the whole property of the dispersive spectrum is displayed. he dispersive curves affected by the induced electric and magnetic fields are compared in detail in Figures 3 and 4. In every figure the three upper maps and the three lower maps are respectively the dispersive frequency curves and the phase velocity curves. From left to right the maps are corresponding to the Quasi-P Quasi-SV and Quasi-SH waves. In these figures the standing wave Figure 3 Influence of the induced electric and magnetic fields on the dispersive curves. Figure 4 Influence of the induced electric and magnetic fields on the dispersive curves. 66 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

number is m = 1 and the propagating wavenumbers are n = 5 and k = 5 respectively. It means that the wave propagation directions asymptotically tend to z- and y-axis directions respectively. Because the physical properties in oyz plane are anisotropic we need to observe the changes from these two directions. he stress wave propagations affected by the induced electric and magnetic fields along these two directions are obviously different. In Figure 3 the dispersive curve of the Quasi-P wave is affected by the induced electric and magnetic fields in all bands. he dispersive curve of the Quasi-SV wave is only affected by the induced electric and magnetic fields in the low frequency band. But in some part the influence of the induced magnetic field vanishes. he dispersive curve of the Quasi-SH wave is barely affected by the induced electric and magnetic fields. In Figure 4 the dispersive curve of the Quasi-P wave is only affected by the induced electric and magnetic fields in the low frequency band. he dispersive curve of the Quasi-SV wave is affected by the induced electric and magnetic fields in all bands. But in some part the influence of the induced magnetic field vanishes. he dispersive curve of the Quasi-SH wave is hardly affected by the induced electric and magnetic fields. 4. Group velocity curved face In Figure 5 the group velocity curved faces are displayed. Every curved face is uniformly continuous. he group velocity curves affected by the induced electric and magnetic fields are compared in detail in Figure 6. From up to down the maps are corresponding to the Quasi-P Quasi-SV and Quasi-SH waves. he group velocity curves affected by the induced electric and magnetic fields along y- and z-axis directions are obviously different. here are the same regularities as discussed in Figures 3 and 4. In the low frequency band the group velocity is small which implies that there is more influence from the structure. When the propagating wavenumber increases the influence from the structure is weaker and the group velocity tends to the wave velocity in the infinite body in the same direction. ines Pz SVz SHz Py SVy and SHy are the three waves velocities along the z- and y-axis directions in the infinite body. Figure 5 Group velocity curved face of MEEP. WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 663

Figure 6 Influence of the induced electric and magnetic fields on the group velocity curves. 4.3 Steady-state response curve Figure 7 shows the steady-state response curves of MEEP where the normalized amplitudes of the displacement are plotted. From up to down the maps are corresponding to the Quasi-P Quasi-SV and Quasi-SH waves. he three steady-state response modes are corresponding to the extensional thickness-twist and thickness-shear modes of the plate. he displacement curves of the Quasi-P wave affected by the induced electric and magnetic fields are compared in detail in Figure 8. From up to down the maps are corresponding to w v and u. he influences of the induced electric and magnetic fields on the displacement are very small. Figure 7 Steady-state response curves of MEEP. 664 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666

Figure 8 Influence of the induced electric and magnetic fields on the steady-state response curves. 5 Conclusions he stress wave propagations affected by the induced electric and magnetic fields in the magneto-electro-elastic plate were studied. Some new characteristics about the guided waves in the magneto-electro-elastic plate were discovered. hey are (1) he guided stress waves are classified in the forms of the Quasi-P Quasi-SV and Quasi-SH waves corresponding to the extensional thickness-twist and thickness-shear modes of the plate and are arranged by the standing wavenumber. () he propagating waves affected by the induced electric and magnetic fields along y- and z- axis are obviously different each other. (3) he Quasi-P waves are affected by the induced electric and magnetic fields only in the lower frequency band when the wave propagation direction tends to y-axis; they are thoroughly affected by the induced electric and magnetic fields when the wave propagation direction tends to z-axis. (4) he Quasi-SV waves are affected by the induced electric and magnetic fields only in the lower frequency band when the wave propagation direction tends to z-axis; they are thoroughly affected by the induced electric and magnetic fields when the wave propagation direction tends to y-axis. But in some part of the low frequency band the influence of the induced magnetic field vanishes. (5) he Quasi-SH wave is hardly affected by the induced electric and magnetic fields. (6) When the propagating wavenumber is low the influence from the structure is large. When the propagating wavenumber increases the influence from the structure is weak. (7) he displacements are slightly affected by the induced electric and magnetic fields. All of these solutions will help us to study the magneto-electro-elastic dielectrics and structures in the future. Appendix he elements of 1 ( ω mnk ; ) : ij WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666 665

m π m 111 = c11 + c 55n + c44k ρω 11 = c1 + c55 4 imkπ m imkπ = ( 1) ( d + d ) imnπ ( 1) ( ) m imkπ = ( 1) ( f + f ) m+ 1 imnπ = ( 1) ( c + c ) m 113 = ( 1) ( c13 + c44) 114 15 31 115 15 31 11 1 55 m π 1 = c55 + c 11n + c44k ρω 13 = 13 + 44 4 = ( f + f ) nk ( d d ) nk 14 15 31 15 = 15 + 31 imkπ m+ 1 131 = ( 1) ( c13 + c44 ) 13 = 13 + 44 m π 133 = c44 + c 44n + c33k ρω 4 m π 135 = d15 + d 15n + d33k 4 = ( f + f ) nk 14 15 31 m π 144 = ε11 ε 11n ε33k 4 ( c c ) nk ( c c ) nk m π 134 = f15 + f 15n + f33k 4 imkπ ( 1) ( ) m+ 1 141 = f15 + f31 m π 143 = f15 + f 15n + f33k imkπ 4 m+ 1 151 = ( 1) ( d15 + d31) 15 = 15 + 31 m π 153 = d15 + d 15n + d33k 4 m π 155 = μ11 μ 11n μ33k 4. m π 145 = h11 h 11n h33k 4 ( d d ) nk m π 154 = h11 h 11n h33k 1 Karl F G. Wave Motion in Elastic Solids. New York: Dover Publications Inc. 1991 Paul H. A note on the static behavior of simply-supported laminated piezoelectric cylinders. Int J Solids Struct 1997 34(9): 3781-3794 3 ing C. A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials. Int J Solids Struct 00 39(1-): 547-5445[DOI] 4 Paul H S Venkatesan M. Wave propagation in a piezoelectric solid cylinder of arbitrary cross section. J Acoust Soc Am 1987 8(6): 013-00[DOI] 5 Paul H S Venkatesan M. Wave propagation in a piezoelectric solid cylinder of arbitrary cross section of class 6 (bone). Int J Engin Sci 1989 7(7): 847-853[DOI] 6 Chen W Q Xu R Q Ding H J. On free vibration of a piezoelectric composite rectangular plate. J Sound Vibr 1998 18(4): 741-748[DOI] 7 Ding H J Xu R Q. Free axisymmetric vibration of laminated transversely isotropic annular plates. J Sound Vibr 000 30(5): 1031-1044[DOI] 8 Wei J P Su X Y. Wave propagation in a piezoelectric rod of 6 mm symmetry. Int J Solids Struct 005 4(13): 3644-3654 [DOI] 9 Wei J P Su X Y. Solving wave propagation along axis direction in piezoelectric hollow cylinder by integral average method. Acta Sci Nat Univ Pekinensis (in Chinese) 005 41(1): 76-81 10 Wei J P Su X Y. Wave propagation and energy transportation along cylindrical piezoelectric-piezomagnetic material. Acta Sci Nat Univ Pekinensis (in Chinese) 006 4(3): 310-314 11 Eringen A C Suhubi E S. Elastodynamics vol.. inear heory. New York: Academic Press 1975 1 Gurtin M E. he inear heory of Elasticity. Berlin Heidelberg New York: Springer-Verlag 197 4 666 WEI JianPing et al. Sci China Ser G-Phys Mech Astron Jun. 008 vol. 51 no. 6 651-666