Shear Waves in Functionally Graded Electro-Magneto-Elastic Media

Shear Waves in Functionally Graded Electro-Magneto-Elastic Media

David K. Gasparyan Institute of Mechanics National Academy of Sciences

Yerevan, Armenia

Karen B. Ghazaryan Institute of Mechanics National Academy of Sciences

Yerevan, Armenia

Abstract- In the framework of quasi-static approach the shear wave propagation is considered in functionally graded transversally isotropic hexagonal 6mm symmetry magnetoelectroelastic media (MEE). Assuming that in functionally graded MEE material elastic and electromagnetic properties vary in the same proportion in direction perpendicular to the MEE polling direction, special classes of inhomogeneity functions were found, admitting exact solutions for the coupled wave field and allowing to estimate the effects of inhomogeneity on the wave behavior. Exact solutions defining the coupled shear wave field in MEE can be used in many problems, e.g. shear surface waves propagation along the surface of a semi-infinite space, interfacial waves in a multilayered and periodic structure, Love type waves in a layer overlying a half-space, guided waves in an inhomogeneous waveguide, etc. Based on exact solutions, the localized wave propagation is studied for MEE layer with quadratic and inverse quadratic inhomogeneity profiles of material parameters varying continuously along the layer thickness direction. Dispersion equations are deduced analytically and for the BaTiO3CoFe2O4 MEE crystal the numerical results estimating effects of inhomogeneity are presented.

Keywords- piezomagnetic, piezoelectric, shear wave, waveguide, inhomogeneity

1. INTRODUCTION

   Recently, the propagation of coupled electromagnetic and elastic waves in magneto-electro-elastic (MEE) structures attracted much attention due to the wide range of application of these materials in smart structures. MEE materials are a class of new artificial composites that consist of simultaneous piezoelectric and piezomagnetic phases. The magnetoelectric effect of piezoelectricpiezomagnetic composites was first reported in [1]. Such materials rarely occur in nature and demonstrate weak magneto-electro-elastic effects. However in artificial composite materials the magneto-electro-elastic effect is notable, which makes such materials highly valuable in technological usage. Magneto-electro-elastic composites are built up by combining piezoelectric and piezomagnetic phases [2] to obtain a smart composite that presents not only the electro-mechanical and magneto-mechanical coupling, characteristic of constituent phases, but also a strong magnetoelectric coupling. In [2] the theoretical estimates are shown to be in agreement with available experimental results, and also show the interesting magnetoelectric behaviour of the composites. Review of the physics of such materials is given in [3], which begins with a brief summary of the historical perspective of the magnetoelectric composites since their

   appearance in 1972. The review concludes with an outlook of the exciting future possibilities and scientific challenges in the field of multiferroic magnetoelectric composites.

   A new method for the synthesis of artificial crystal with a strong electro-magneto-elastic interaction by combining electro-elastic BaTiO3 and magneto-elastic CoFe2O4 in a single crystal is presented in [4].

   Quasi-static approximation was used to research surface and bulk wave propagation in such materials [5-12]. The propagation of Bleustein-Gulyaev surface wave is investigated in [5] for transversely isotropic MEE materials. In [6] the existence of a new surface SH wave is stated for a cubic anisotropic MEE material. An analytical approach was used to investigate Love wave propagation in a layered MEE structure [7], where a solution of dispersion relations was obtained for magnetoelectrically open and short boundary conditions. In

   [8] Rayleigh waves are investigated in MEE half plane, the material of which is assumed to possess hexagonal 6 mm symmetry. In [9] it is shown that shear surface waves with twelve different velocities in cases of different magneto- electrical boundary conditions can be guided by the interface of two identical MEE half-spaces. The existence of shear surface wave travelling along the interface of two half spaces of different MEE materials is studied in [10]. The study of SH waves in a hetero-structure made from three different MEE materials with 6mm symmetry is given in [11]. In [12] the dispersion relation of the MEE three-dimensional, anisotropic and multi-layered thick plate vibration frequency has been derived. Applied problems of MEE beam and plate behaviour are studied in [13-17]. In [13-14] based on Timoshenko's beam theory an analytical model for MEE bimorph beam [13] is employed, to study its response to mechanical and electro- magnetic time varying loads. In [14] it was shown that the effect of shear deformation has a great influence on piezo- electro-elastic beams natural frequencies and mode shapes. The pyroelectric and pyromagnetic effects on magneto- electro-elastic plate with different boundary conditions under uniform temperature rise is studied in [15]. Based on Kirchhoff thin-plate theory, a closed form expressions for bending problem of MEE rectangular thin plates are derived in
   [16] and the exact solutions for the deformation behaviours of the fiber-reinforced the BaTiO3CoFe2O4 composites subjected to certain types of surface loads are analytically obtained. In [17] exact solutions are derived for anisotropic, simply-supported, multi-layered rectangular MEE plates under static loadings.

   In [18] functionally graded materials, whose properties vary continuously in space were used to improve the efficiency of BleusteinGulyaev waves for a hexagonal 6 mm piezoelectric crystal. Assuming that the elastic stiffness, the piezoelectric constant, the dielectric constant, the mass density of MEE material vary in the same proportion with a single space variable special classes of the inhomogeneity functions were founded allowing the exact solution of surface wave propagation problem.

   Functionally graded materials (FGM) are inhomogeneous elastic bodies whose properties vary continuously with space. The FGM structure has attracted wide and increased attentions of scientists and engineers. FGM plays an essential role in the most advanced integrated systems for vibration control and health monitoring. The progress in the characterization, modelling, analysis and principal developments of FGM was reviewed in [19, 20].

   x Dx y Dy 0

   x Bx y By 0

   x xz y yz t,tW

   yz G yW e y y Dy e yW y y By yW y y

   xz GxW ex x Dx exW x x Bx xW x x

   In pure elastic FGM materials surface wave propagation was discussed in [21-26].

   Here the W x, y,t is the elastic displacement directed

   In [27, 28] the propagation of shear electroelastic

   along z -direction, , xz x, y,t , yz x, y,t

   are mechanical

   monochromatic localized waves in functionally graded piezoelectric layer is studied, where the influence of

   stresses, the Dx x, y,t , Dy x, y,t , Bx x, y,t , By x, y,t ,

   inhomogeneity function on dispersion of shear wave is

   (x, y,t) , (x, y,t)

   are electric displacements, the

   analysed and numerical comparison between wave speeds of magnetic inductions, the electric potential and the magnetic

   homogeneous and inhomogeneous layers is carried out. potential, correspondingly, ,G c44 , e e15 , d15 , re

   Surface Love waves are considered in [29] for a layered structure with inhomogeneous piezoelectric layer. The behaviour of Lamb waves in the functionally graded piezoelectricpiezomagnetic plate with material parameters varying continuously along the thickness direction is investigated in [30], where the power series technique is employed to find dispersion curves numerically. In [31] the BleusteinGulyaev waves are studied by analytical technique in a functionally graded transversely isotropic MEE half- space, in which all parameters change exponentially along the depth direction. All the above mentioned solutions were based on quasi-static approximation of a problem [32], where the derivatives of time in Maxwells electro-dynamic equations

   were ignored. Such approximation allows high precision

   the bulk density, elastic, piezoelectric, piezomagnetic and magneto-elastic modulus respectively, and are the

   x y tt tt

   dielectric permittivity and magnetic permeability coefficients, while x; y ; 2 2 .

   To exemplify the problem and provide insights of shear waves propagation in functionally graded piezo-electro- magneto-elastic media the following model is considered.

   The material parameters in the MEE medium gradually change along y-direction having the same function variation properties

   solutions regarding the influence of the electro-magnetic field on the properties of elastic fields, however is limited in finding out the coupled wave processes in magneto-electro-

   G( y) G0 f ( y); 0 f ( y); e e0 f ( y)

   0 f ( y); 0 f ( y); 0 f ( y); o f ( y)

   elastic materials. More specifically, quasi-static definition cannot be used to describe the reflection and refraction of electro-magnetic waves [33], coupling effects of electro- magnetic and elastic fields which causes polariton interaction in MEE periodic structure [34]. Based on the full complete set of electrodynamics dynamics equations and the elasticity theory equations in [35] the two-dimensional equations are

   Here f ( y) is the inhomogeneity function which will be specified later.

   Equations (1) and (2) can be considered as a set of first order six differential equations with six sought functions yz , Dy , By ,W,, .

   derived describing coupled wave process in MEE medium of hexagonal symmetry, where it was stated that contrary to the quasi-static approximation under dynamic approach the plane

   The functions

   sought functions.

   xz , Dx , Bx

   can be defined from (3) via the

   and anti-plane deformations are coupled.

2. STATEMENT OF THE PROBLEM

   For a transversely isotropic magneto-electro-elastic hexagonal symmetry 6 mm medium with z-axis normal to the plane of isotropy, polling direction of which coincides with z- axis direction, the anti-plane equations in Cartesian coordinate system x, y, z can be written as

   Considering a wave with the circular frequency and wave number k we present all functions in the form of plane harmonic wave travelling along the x – direction

   yz y y

   , D , B ,W ,,x, y,t

   2

   0 0 0 0

   y , D y , B y ,W y , y , y

   0 00e00

   oyz oy oy

   0 0 0

   e

   Introducing vectors

   expi kx t

   0 0 0 0 0

   and substituting (6) into (5) it is straightforward to derive the following matrix equations

   oyz oy oy

   y , D , B T

   0 0 0

   U W , , T

   we can rewrite the anti-plane equations as the set of first order differential equations in a matrix form

   y y MU y

   U M L

   0 0 p 0

   0 0 q 0

   N L U

   with respect to the new unknown vector functions

   y y , D y B y T

   yU y N y

   Here M , S are the following matrixes

   Here

   0 oyz oy oy

   0 0

   U y W y , S y F y T

   G e

   M

   1 1 1

   M

   1

   0 0 0 0

   0 1 0

   e

   0 E p2 k 2

   0 0 0 1

   N T 1

   2

   G k 2 e

   E

   1 1

   T k 2

   0 0 0 0 0

   e

   N0 0 1 0

   0 0 1

   Defining new functions

   oyz y, D oy y, B oy y

   and new

   auxiliary potentials

   S y, F y

   oyz y oyz

   as

   y

   f y

   Lp

   Lq

   d P y dy

   d P y dy

   Doy y D oy

   y

   f y

   P y 1 df

   2 f dy

   Boy y B oy y f y W0 y W0 y f y

   0 0 0 0 0 0 0

   p 1 2

   y

   S y0 W0 y0 F y 0

   E G e 1

   0

   0

   0

   0 f y

   2

   E k 2

   F y0 S y 0 W0 y0

   y

   0 f y

   Now by substituting the vector U0 from (7) into (8) and the

   0

   0

   where

   vector from (8) into (7) we come to the two decoupled sets of equations

   p2

   1 p2 1

   1 p2 1 L L

   Case (ii)

   0 0 0 0 q p oyz

   0 1 0 Lq Lp D oy 2

   0 0 1 L L B

   f ( y) Acosh b B sinh b y

   q p oy

   oyz

   b

   k 2 D

   B

   oy

   oy

   Depending on whether b is positive, negative, or equal to zero several kind of functions can be found. Assumptions (14,15) are obviously somewhat artificial and narrow the class of functions characterizing the inhomogeneity. Nevertheless it allows solving the problem for functionally graded MEE

   p2 0 0 Lp LqW

   0 W

   0

   materials and estimating the effect of inhomogeneity on the wave dispersion relations of waves. Let us note that these

   p q

   0 1 0 L L S k 2 S

   inhomogeneity functions profiles do not depend on material properties and can be used for both piezomagnetic,

   0 0 1 L L F F

   Noting that

   p q

   piezoelectric and pure elastic materials. For piezoelectric

   materials these inhomogeneity functions (14,15) were derived also in [18]. Some types of these functions have been used also in [17, 25, 26, 28] where shear wave prorogation in elastic and piezoelectric media was studied.

   d 2

   dP 2

   Lp Lq dy2 dy P

3. GOVERNING EQUATIONS AND SOLUTIONS

   d y 2

   2

   Lq Lp dy2 P

   dP

   dy

   The equations for the first type of inhomogeneity functions can be cast as:

   we come to the following two ways of finding the exact solutions if we assume that

   Case (i):

   d 2

   oyz r 2

   dP

   dy

   P2 b

   dy2

   d 2 D

   oyz

   d 2 B

   1 p2 0 oy bD

   0 oy b B

   0

   dy2

   oy dy2 oy

   or

   0

   0

   d 2 D

   dy2

   oy q2 D

   0;

   d 2 B

   dy2

   oy q2 B 0

   oy

   oy

   dP P2 b

   r b p2 k 2 ; q

   b k 2

   where b is a constant. Since

   dP

   dy

   P2

   1 d 2 f 1 2

   Now the solutions of this system of second order differential equations with constants coefficient can be easily found:

   Doy y C1 expqy A1 exp qy

   dy

   f 1 2

   dx2

   Boy

   y C2 expqy A2 exp qy

   P

   2 dP

   dy

   1 d 2 f dx2

   0 y

   0 C1 exp qy A1 exp qy

   f 0

   we get the two types of inhomogeneity functions admitting the exact solutions of (10) and (11)

   Case (i)

   2

   0 C2 exp qy A2 exp qy

   0

   C3 expry C4 exp ry

   Solutions for W0 y,0 y,0 y follow from (6, 7)

   f ( y) Acosh

   b

   B sinh

   b

   b y

   or

   ery C r P y e ry A r P y

   Here c1,c2 ,c3 , a1, a2 , a3 are arbitrary constants.

   W y 3 3

   0 E k 2 p2

   E k 2 p2

   All these solutions can be useful in tackling the problems

   0 0

   0 y

   eqy E p2 C C q P y ery C

   r P y

   of shear surface wave propagation over the semi-infinite space surface, interfacial wave in multilayer and periodic structures, Love type waves in a layer overlying a half-space, guided

   0 0 1 0 2 0 3

   0 0

   E k 2 p2

   0 0 1 0 2 0 3

   e qy E p2 A A q P y e ry A r P y

   E k 2 p2

   waves in waveguides , etc.

4. SHEAR LOCALIZED WAVES IN WAVEGUIDE

   Here we shall limit ourselves by considering the localized

   0 0

   0 y

   wave propagation in waveguide 0 y h, x , when 1 p 0 .

   eqy E p2 C C q P y ery C r P y

   0 0 1 0 2 0 3

   0 0

   E k 2 p2

   The quadratic and the inverse quadratic inhomogeneity profiles will be considered according to case b 0 (14, 15), a 0

   0 0 1 0 2 0 3

   e qy E p2 A A q P y e ry A r P y

   • E k 2 p2

   Case (i):

   f y 1 ay p

   0 0

   Case (ii):

   f y 1 ay p

   Here C1 ,C2 ,C3 , A1 , A2 , A3 are arbitrary constants.

   The equations and solutions for the second type of inhomogeneity functions: Case (ii):

   The following magneto-electro-elastic contact conditions are within our interest:

   Symmetric conditions

   d 2W dy2

   0

   0 r 2W

   0;

   2

   d S 2

   q S 0

   dy2

   yz 0; 0; 0; y 0; y h

   d 2 F 2

   0; 0; D

   0; y 0; y h

   dy2

   q F 0

   yz y

   yz 0; 0; By 0; y 0; y h

   W0 y c1 expry a1 expry

   S y c2 expqy a2 expqy

   F y c3 expqy a3 expqy

   Asymmetric conditions

   W 0; 0; 0; y 0

   Solutions for

   y, D y, B y, y, y follow

   0; 0; 0; y h

   from (6, 8)

   oyz oy oy 0 0

   yz

   Doy

   Boy

   y c2eqy q P y a2e qy q P y

   3 3

   y c eqy q P y a e qy q P y

   Substituting the solutions (17, 18, 20, 21) into the boundary conditions (22-25), the homogeneous systems of equations with respect to the constants will be obtained. Equating the determinants of simultaneous sets of equations to zero we can obtain dispersion equations.

   Introducing dimensionless parameters

   y eqy c c 1 q P y

   0 1 0 2 0 0

   3 0

   c ery E r P y

   K K

   • 2 K K

   +e qy a

   a 1 q P y

   K

   e e

   1 2

   1 0 2 0 0

   a e ry E r P y

   2 2

   3 0

   0 0 0 3 0 1 0 2

   y 1 ery c eqy c c

   e0 0

   G

   G

   Ke ; K

   0 0 0 0

   1 e ry a e qy a a

   0

   ; d kh

   0 0 3 0 1 0 2

   y 1 ery c eqy c c 0 0

   0 0 0 3 0 2 0 1

   0 0 3 0 2 0 1

   1 e ry a e qy a a

   the dispersion equations according to a symmetric boundary condition (22, 23) can be written as

   Case (i):

   2d 2 p3 1 a K 1 K 1 sech d sech dp

   a2 dp p2 K 1 p2 K tanh d 1 K p tanh dp

5. NUMERICAL ANALYSIS OF DISPERSION EQUATIONS AND DISCUSSION OF RESULTS

   1. Symmetric boundary conditions

      Dispersion equations (26-30) impose a relationship between dimensionless phase speed of localized wave 1

      d 2 p2 1 aK 2 1 K 2 p2 a2 p2 K 1 p2 2

      p 1 2 0 and wave number k (d kh) .

      Case (ii):

      tanh dp tanh d 0

      All numerical calculations will be carried out for the MEE crystal BaTiO3CoFe2O4 with electro-magneto-elastic coupling coefficient K 0.612 [16].

      For fixed h in the long wave approximation d 1 from

      2d 2 p 1 a K 1 K 1 sech d sech dp

      1. and (27) it follows that 1 1 K 0.790 , in the

         1 2

         a2 d 1 K p tanh d K tanh dp

         short wave approximation

         d 1 we

         d 2 1 a 1 K 2 p2 K 2 a2 tanh dp tanh d 0

         have 2 1 2K 1 K 0.927 .

         The dispersion equations according to the boundary condition (24) for Case (ii) have the form

         2 1 a pd 2 1 K K – K sech d sech dp -1

         a2 d 1 K 1 K p tanh d – K – K tanh dp –

         In short wave approximation the localized wave speed coincides with the speed of the BleusteinGulyaev surface wave for electrically shorted and traction free interface of MEE half-space [5]. Let us note that the inhomogeneity does not affect the phase speeds of both short and long waves and that for the pure elastic layer (K Ke 0) all dispersion

         equations (26-30) have no solutions corresponding to

         – a2 1 K 2 – d 2

         1 a 1 K

         2 p2 K – K 2

         localized wave.

         For any value of the inhomogeneity parameter a , equation

         • tanh dp tanh d 0

           (26) has one root in the interval d d0

           and two roots in the

           The dispersion equation according to the boundary condition (24) can be obtained by replacing K Ke in (28).

           The dispersion equations according to an asymmetric boundary condition (25) can be written as

           Case (i):

           interval d d0 (see Tab.1). The corresponding dispersion curves d diagrammatically shown on Fig.1.

           a2 d 2 p K p2 1 K

           dp a2 d 2 ad 2 p2 1 K a3 K tanh d

           d a3 p2 a2 a3 1 ad 2 p2 K tanh dp

           a a2 1 ad 2 p2 a2 1 p2 K

           tanh dp tanh d 0

           a	Case(i)	Case(ii)
           	d 0	d 0
           0	5.27	5.27
           0.5	5.35	5.93
           1	5.47	7.22
           2	568	9.62
           5	6.12	14.22

           Fig.1 Structure of dispersion curves Tabl.1

           0

           In Tabl.1 the data for the function d a are presented for

           Case (ii):

           1 ad p 1 K tanh d K tanh dp

           a tanh dptanh d 0

           several values of inhomogeneity parameter. As it follows from the data of Tabl.1 the inhomogeneity in the case of inverse qadratic inhomogeneity profile sufficiently affects at the location structure of the roots and extends the interval where the one localised wave exist. In the case of quadratic inhomogeneity profile the effect of inhomogeneity factor is very weak.

   2. Asymmetrical boundary conditions

   Contrary to the results of symmetrical boundary conditions, the results corresponding to asymmetrical boundary conditions are qualitatively different in Case (i) and Case (ii).

   For short waves, equations (29, 30) have no solutions corresponding to localized waves and may have only one solution corresponding to localized waves for certain values of parameters d, a .

   In Fig.2 and Fig.3 in the phase plane of parameters d, a the curves of functions d0 a are presented defining the regions where equations (29, 30) have no solutions corresponding to localized waves. In Fig. 2, 3 shaded regions correspond to the regions of parameters d , a where the equations have no solutions (a localized wave does not exist). Outside of shaded regions the solutions corresponding to localized waves do exist for any values of d, a including the points of curves d0 a .

   Fig.2 Localized wave existence region for Case (i )

   Fig.3 Localized wave existence region for Case (ii)

   As it follows from the analysis of Fig.2 and Fig.3 the inhomogeneity factor plays an important role in the localized wave propagation behavior. Increasing the parameter a in Case(i) leads to the elimination of localized shear wave for any value of d 2.64,6.31 , while in Case(ii) results in the

   appearance of localized wave for any value of

   d 0.79, 2.64.

6. CONCLUSIONS

Two classes of inhomogeneity functions are defined providing the exact solution of shear wave propagation in 6mm symmetry magneto-electro elastic functionally graded media. Solutions of the wave field are derived, which can be used in the problems of shear surface wave propagation over the semi-infinite space surface, interfacial wave in multilayer and periodic structures, Love type waves in a layer overlying a half-space, guided waves in waveguides, etc. The quadratic and inverse quadratic inhomogeneity profiles were considered in the several boundary problems of shear guided localized wave propagation in MEE waveguide. The dispersion equations are deduced analytically and by means of the

numerical analysis for the BaTiO3CoFe2O4 MEE crystal the effects of the inhomogeneity are discussed in detail.

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