Pure shear elastic surface waves in magneto-electro-elastic materials

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Department of Mechanics, Yerevan State University, Alex Manoogyan Str. 1, .... e d u d ε χ φ ε μ. −. = −. −. ,. (3) the solution of Eqs. (1) is reduced to the solution of. 2. 1 ..... in Eqs. (13) together with Eqs. (9), Eqs. (10)-(11) leads to a system of six.
Eleven shear surface waves guided by inclusions in magneto-electro-elastic materials

Arman Melkumyan

a)

Department of Mechanics, Yerevan State University, Alex Manoogyan Str. 1, Yerevan 375025, Armenia

Eleven pure shear surface waves are shown to exist in magneto-electro-elastic materials, which contain different plane inclusions inside them. The plane inclusions affect the electric and magnetic fields only, so that the medium remains mechanically continuous across it. Expressions for velocities of propagation of these 11 surface waves are obtained in explicit forms. It is expected that these waves will have several applications in surface acoustic wave devices.

a)

Electronic mail: [email protected]

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Bleustein1 and Gulyaev2 have shown that an elastic shear surface wave can be guided by the free surface of piezoelectric materials in class 6 mm, and then Danicki3 has described the propagation of a shear surface wave guided by an embedded conducting plane. In this paper we consider an infinite magneto-electro-elastic4-8 medium which contains embedded infinitesimally thin plane with different electric and magnetic properties and study the possibilities for surface waves to be guided by it. The medium can also be considered as a system of two perfectly bonded magneto-electro-elastic half-spaces with infinitesimally thin layers in the plane of bonding. Discussing different magneto-electrical boundary conditions in the plane of bonding we obtain surface waves with eleven different velocities. Let x1 , x2 , x3 denote rectangular Cartesian coordinates with x3 oriented in the direction of the sixfold axis of a magneto-electro-elastic material in class 6 mm. Introducing electric potential ϕ and magnetic potential φ , so that E1 = −ϕ,1 , E2 = −ϕ, 2 , H1 = −φ,1 , H 2 = −φ,2 , the five partial differential equations which govern the mechanical displacements u1 , u2 , u3 , and the potentials ϕ , φ , reduce to two sets of equations when motions independent of the x3 coordinate are considered. The equations of interest in the present paper are those governing the u3 component of the displacement and the potentials ϕ , φ , and can be written in the form c44 ∇ 2 u3 + e15 ∇ 2ϕ + q15 ∇ 2φ = ρ u3 , e15 ∇ 2 u3 − ε11∇ 2ϕ − d11∇ 2φ = 0 ,

(1)

q15 ∇ 2 u3 − d11∇ 2ϕ − μ11∇ 2φ = 0 ,

where ∇ 2 is the two-dimensional Laplacian operator, ∇ 2 = ∂ 2 /∂x12 + ∂ 2 /∂x22 , ρ is the mass density, c44 , e15 , ε11 , q15 , d11 and μ11 are elastic, piezoelectric, dielectric, piezomagnetic, electromagnetic and magnetic constants, and

the superposed dot indicates differentiation with respect to time. The constitutive equations which relate the stresses Tij ( i, j = 1, 2,3 ), the electric displacements Di ( i = 1, 2, 3 ) and the magnetic induction Bi ( i = 1, 2, 3 ) to u3 , ϕ and φ are T1 = T2 = T3 = T12 = 0 , D3 = 0 , B3 = 0 , T23 = c44 u3, 2 + e15ϕ, 2 + q15φ, 2 , T13 = c44 u3,1 + e15ϕ,1 + q15φ,1 , D1 = e15 u3,1 − ε 11ϕ,1 − d11φ,1 , D2 = e15 u3, 2 − ε11ϕ, 2 − d11φ, 2 , B1 = q15 u3,1 − d11ϕ,1 − μ11φ,1 , B2 = q15 u3, 2 − d11ϕ, 2 − μ11φ, 2 .

2

(2)

Solving Eqs. (1) for ∇ 2 u3 , ∇ 2ϕ and ∇ 2φ we find that after defining functions ψ and χ by

ψ =ϕ −

e15 μ11 − q15 d11 q ε −e d u3 , χ = φ − 15 11 15 211 u3 , 2 ε11 μ11 − d11 ε11 μ11 − d11

(3)

the solution of Eqs. (1) is reduced to the solution of ∇ 2 u3 = ρ c44−1u3 , ∇ 2ψ = 0 , ∇ 2 χ = 0 ,

(4)

where c44 = c44 + ( e152 μ11 − 2e15 q15 d11 + q152 ε11 ) ( ε11 μ11 − d112 )

= c44e + ε11−1 ( d11 e15 − q15 ε11 )

ε μ11 − d112 ⎞⎟⎠

2 ⎛ ⎜ 11 ⎝

= c44m + μ11−1 ( d11 q15 − e15 μ11 )

ε μ11 − d112 ⎞⎟⎠

2 ⎛ ⎜ 11 ⎝

(5)

is magneto-electro-elastically stiffened elastic constant, c44e = c44 + e152 ε11 is piezoelectrically stiffened elastic constant and c44m = c44 + q152 μ11 is piezomagnetically stiffened elastic constant. With the analogy to the piezoelectric coupling factor ke2 = e152



e 11 44

c

) and the piezomagnetic coupling factor k

2 m

= q152



m 11 44

c

) introduce

piezoelectromagnetic coupling factor 2 −1 kem = c44 ( e152 μ11 − 2e15 q15 d11 + q152 ε11 ) (ε11μ11 − d112 )

= e152 ( ε11c44 ) + c44−1ε11−1 ( d11e15 − q15ε11 ) = q152

ε μ − d112 ⎞⎟⎠

2 ⎛ ⎜ 11 11 ⎝

2 ( μ11c44 ) + c44−1 μ11−1 ( d11q15 − e15 μ11 ) ⎛⎜⎝ ε11 μ11 − d112 ⎞⎟⎠ .

(6)

Using the introduced functions ψ and χ and the magneto-electro-elastically stiffened elastic constant, the constitutive Eqs. (2) can be written in the following form: T23 = c44 u3,2 + e15ψ ,2 + q15 χ , 2 , T13 = c44 u3,1 + e15ψ ,1 + q15 χ ,1 , D1 = −ε11ψ ,1 − d11 χ ,1 , D2 = −ε11ψ , 2 − d11 χ , 2 ,

(7)

B1 = − d11ψ ,1 − μ11 χ ,1 , B2 = − d11ψ ,2 − μ11 χ , 2 .

Introduce short notations e = e15 , μ = μ11 , d = d11 , ε = ε11 , q = q15 , c = c44 , c e = c44e , c m = c44m , c = c44 , w = u3 , T = T23 , D = D2 , B = B2 and use subscripts A and B to refer to the half-spaces x2 > 0 and x2 < 0 ,

respectively. As the materials in the half spaces x2 > 0 and x2 < 0 are identical, one has that eA = eB = e ,

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μ A = μ B = μ , d A = d B = d , ε A = ε B = ε , q A = qB = q , cA = cB = c , c Ae = cBe = c e , c Am = cBm = c m , cA = cB = c .

The conditions at infinity require that wA , ϕ A , φ A → 0 as x2 → ∞ , wB , ϕ B , φB → 0 as x2 → −∞ ,

(8)

and the mechanical bonding conditions require that wA = wB , TA = TB on x2 = 0 .

(9)

Consider the possibility of a solution of Eqs. (3)-(4) of the form wA = w0 A exp ( −ξ 2 x2 ) exp ⎡⎣i (ξ1 x1 − ω t ) ⎤⎦ ,

ψ A = ψ 0 A exp ( −ξ1 x2 ) exp ⎡⎣i (ξ1 x1 − ω t ) ⎤⎦ ,

(10)

χ A = χ 0 A exp ( −ξ1 x2 ) exp ⎡⎣i (ξ1 x1 − ω t ) ⎤⎦ , in the half space x2 > 0 and of the form wB = w0 B exp (ξ 2 x2 ) exp ⎣⎡i (ξ1 x1 − ω t ) ⎦⎤ ,

ψ B = ψ 0 B exp (ξ1 x2 ) exp ⎡⎣i (ξ1 x1 − ω t ) ⎤⎦ ,

(11)

χ B = χ 0 B exp (ξ1 x2 ) exp ⎡⎣i (ξ1 x1 − ω t ) ⎤⎦ , in the half space x2 < 0 . These expressions satisfy the conditions (8) if ξ1 > 0 and ξ 2 > 0 ; the second and the third of Eqs. (4) are identically satisfied and the first of Eqs. (4) requires c (ξ12 − ξ 22 ) = ρω 2 .

(12)

Now the elastic contact conditions (9) together with different magneto-electrical contact conditions on x2 = 0 must be satisfied. The following cases of magneto-electrical contact conditions on x2 = 0 are of our interest in the present paper: 1a) ϕ A = ϕ B , DA = DB , φ A = φB = 0 ; 1b) DA = DB = 0 , φ A = φB = 0 ; 2a) ϕ A = ϕ B = 0 , φ A = φB , BA = BB ; 2b) ϕ A = ϕ B = 0 , BA = BB = 0 ;

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3) ϕ A = ϕ B = 0 , φ A = φB = 0 ; 4) DA = DB = 0 , BA = 0 , φB = 0 ; 5) ϕ A = 0 , DB = 0 , BA = BB = 0 ; 6) DA = 0 , ϕ B = 0 , BA = 0 , φB = 0 ; 7) ϕ A = 0 , DB = 0 , φ A = φB = 0 ;

(13)

8) ϕ A = ϕ B = 0 , BA = 0 , φB = 0 ; 9) ϕ A = 0 , DB = 0 , BA = 0 , φB = 0 ; 10) ϕ A = ϕ B , DA = DB , BA = 0 , φB = 0 ; 11) ϕ A = 0 , DB = 0 , φ A = φB , BA = BB ; 12a) ϕ A = ϕ B , DA = DB , φ A = φB , BA = BB ; 12b) DA = DB = 0 , φ A = φB , BA = BB ; 12c) ϕ A = ϕ B , DA = DB , BA = BB = 0 ; 12d) DA = DB = 0 , BA = BB = 0 . Each of the 17 groups of conditions in Eqs. (13) together with Eqs. (9), Eqs. (10)-(11) leads to a system of six homogeneous algebraic equations for w0 A , ψ 0 A , χ 0 A , w0 B , ψ 0B , χ 0 B , the existence of nonzero solution of which requires that the determinant of that system must be equal to zero. This condition for the determinant together with Eq. (12) determines the surface wave velocities Vs = ω ξ1 . In the case of 1a) of Eqs. (13) this procedure leads to a surface wave with velocity

(

2 − e 2 / ( cε ) ⎤⎦ Vs21 = ( c ρ ) 1 − ⎡⎣ kem

2

).

(14)

The same velocity is obtained in the case 1b). The cases 2a) and 2b) both lead to a surface wave with velocity

(

2 Vs22 = ( c ρ ) 1 − ⎡⎣ kem − q 2 / ( c μ ) ⎤⎦

2

).

(15)

Each of the cases 3) to 11) leads to its own one surface wave and the corresponding surface wave velocities are Vs23 = ( c ρ ) (1 − ke4m ) ;

(

2 Vs24 = ( c ρ ) 1 − 14 ⎡⎣ kem − e 2 / ( cε ) ⎤⎦

2

); 5

(

2 − q 2 / ( c μ ) ⎤⎦ Vs25 = ( c ρ ) 1 − 14 ⎡⎣ kem

2

);

4 Vs26 = ( c ρ ) (1 − 14 kem );

(

2

);

(

2

);

2 − 12 e 2 / ( cε ) ⎤⎦ Vs27 = ( c ρ ) 1 − ⎡⎣ kem

2 Vs28 = ( c ρ ) 1 − ⎡⎣ kem − 12 q 2 / ( c μ ) ⎤⎦

(16)

2 ⎛ ⎡ ⎛ e2 q 2 ⎞⎤ ⎞ 1 2 V = ( c ρ ) ⎜ 1 − ⎢ kem − ⎜ + ⎟⎥ ⎟ ; ⎜ ⎢⎣ 2 ⎝ cε c μ ⎠ ⎥⎦ ⎟ ⎝ ⎠ 2 s9

2 ⎛ ⎛ εμ 2 ⎞ ⎞ ⎡ k 2 − e 2 / ( cε ) ⎦⎤ ⎟ ; Vs210 = ( c ρ ) ⎜ 1 − ⎜ 2 ⎟ ⎣ em ⎜ ⎝ 2εμ − d ⎠ ⎟ ⎝ ⎠ 2 ⎛ ⎛ εμ 2 ⎞ ⎞ ⎡ k 2 − q 2 / ( c μ ) ⎤⎦ ⎟ . Vs211 = ( c ρ ) ⎜ 1 − ⎜ 2 ⎟ ⎣ em ⎜ ⎝ 2εμ − d ⎠ ⎟ ⎝ ⎠

The cases 12a) to 12d) do not lead to any surface wave. If the magneto-electro-elastic material degenerates to a piezoelectric material, so that q → 0 , d → 0 , the surface waves that have velocities Vs1 , Vs 4 , Vs10 disappear and Vs 2 , Vs 3 , Vs 8 → Vbg =

(c

Vs 5 , Vs 6 , Vs 7 , Vs 9 , Vs11 →

e

ρ )(1 − ke4 ) ;

(c

e

ρ )(1 − 14 ke4 ) ,

(17)

where Vbg is the Bleustein-Gulyaev surface wave speed, so that the number of surface waves decreases from 11 to 2 when the magneto-electro-elastic material is changed to a piezoelectric material.

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2

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3

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4

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5

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6

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7

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8

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