Pure shear elastic surface waves in magneto-electro-elastic

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by stress free plane cuts with different magneto-electrical properties in ... layers on their traction free plane surfaces with different magneto-electric properties.
New pure shear acoustic surface waves guided by cuts in magneto-electro-elastic materials

Arman Melkumyan Department of Mechanics, Yerevan State University, Alex Manoogyan Str. 1, Yerevan 375025, Armenia E-mail: [email protected]

ABSTRACT It is shown that new pure shear acoustic surface waves with five different velocities can be guided by stress free plane cuts with different magneto-electrical properties in magneto-electro-elastic materials. The possibility for the surface waves to be guided by a cut in pairs, which is reported in this paper, is new in magnetoelectro-elastic materials and has no counterpart in piezoelectric materials. The five velocities of propagation of the surface waves are obtained in explicit forms. It is shown that the possibility for the surface waves to be guided in pairs disappears and the number of surface waves decreases from 5 to 1 if the magneto-electro-elastic material is changed to a piezoelectric material.

PACS 46.25.Hf; 46.25.Cc; 77.65.Dq

Bleustein [1] and Gulyaev [2] have shown that an elastic shear surface wave can be guided by the free surface of piezoelectric materials in class 6 mm, and later Danicki [3] has described the propagation of a shear surface wave guided by an embedded conducting plane. In this paper the existence of pure shear acoustic surface waves guided by infinite plane cuts in transversely isotropic magneto-electro-elastic [4-8] materials in class 6 mm is investigated. The medium can also be considered as a system of two magneto-electro-elastic half-spaces with infinitesimally thin layers on their traction free plane surfaces with different magneto-electric properties. Discussing different magneto-electrical boundary conditions on the free surfaces, pure shear surface waves with 5 different velocities of propagation are obtained. In this paper it is also reported that in magneto-electro-elastic materials the surface waves can be guided in pairs, which has no counterpart in piezoelectric as well as in elastic materials. It is expected that these waves will have numerous applications in surface acoustic wave devices. 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. By 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

1

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. In the present paper the equations of interest 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 = ∂ 2 /∂x12 + ∂ 2 /∂x22 is the two-dimensional Laplacian operator, 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 − ε11ϕ,1 − d11φ,1 , D2 = e15 u3, 2 − ε11ϕ,2 − d11φ,2 ,

(2)

B1 = q15 u3,1 − d11ϕ,1 − μ11φ,1 , B2 = q15 u3,2 − d11ϕ,2 − μ11φ,2 . Solving Eqs. (1) for ∇ 2 u3 , ∇2ϕ and ∇2φ it has been found that after defining functions ψ and χ by

ψ = ϕ − mu3 , χ = φ − nu3 ,

(3)

the solution of Eqs. (1) is reduced to the solution of

∇ 2 u3 = ρ c44−1u3 , ∇2ψ = 0 , ∇2 χ = 0 ,

(4)

where m=

e15 μ11 − q15 d11

ε11 μ11 − d

2 11

, n=

q15 ε11 − e15 d11

ε11 μ11 − d112

,

(5)

and c44 = c44 + ( e152 μ11 − 2e15 q15 d11 + q152 ε11 ) ( ε11 μ11 − d112 ) = c44e + ε 11−1 ( d11 e15 − q15 ε 11 )

2

ε μ11 − d112 ⎞⎟⎠

⎛ ⎜ 11 ⎝

2

= c44m + μ11−1 ( d11 q15 − e15 μ11 )

ε μ11 − d112 ⎞⎟⎠ .

⎛ ⎜ 11 ⎝

2

(6)

In Eqs. (6) c44 is magneto-electro-elastically stiffened elastic constant, c44e = c44 + e152 ε11 is electroelastically stiffened elastic constant and c44m = c44 + q152 μ11 is magneto-elastically stiffened elastic constant. With the analogy to the electro-mechanical coupling coefficient ke2 = e152 coupling coefficient km2 = q152



m 11 44

c



e 11 44

c

) and the magneto-mechanical

) introduce the magneto-electro-mechanical coupling coefficient

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

= e152

2 ( c44 ε11 ) + c44−1ε11−1 ( q15 ε11 − e15 d11 ) ⎛⎜⎝ ε11 μ11 − d112 ⎞⎟⎠

= q152

2 ( c44 μ11 ) + c44−1 μ11−1 ( e15 μ11 − q15 d11 ) ⎛⎜⎝ ε11 μ11 − d112 ⎞⎟⎠ .

(7)

From Eqs. (5)-(7) it follows that 2 e15 m + q15 n = c44 kem , ε11 m + d11 n = e15 , d11 m + μ11 n = q15 ,

( e15 μ11 − q15 d11 ) m = c44 μ11 kem2 − q152 ( q15ε11 − e15 d11 ) n = c44 ε11 kem2 − e152

,

(8)

.

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 ,

(9)

B1 = −d11ψ ,1 − μ11 χ,1 , B2 = −d11ψ ,2 − μ11 χ,2 . From the condition of the positiveness of energy using Eqs. (2) one has that c44 > 0 , ε11 > 0 , μ11 > 0 , ε11 μ11 − d112 > 0 .

(10)

From Eqs. (6), (7) and (10) one has that c44 ≥ c44 , c44 ≥ c44e , c44 ≥ c44m ; c44 = c44 if and only if e15 = 0 , q15 = 0 ;

c44 = c44e if and only if d11 e15 = ε11 q15 ;

(11)

c44 = c44m if and only if μ11 e15 = d11 q15 ;

3

and 2 kem ≥ e152

( c44 ε11 ) ,

2 kem = e152

( c44 ε11 )

if and only if d11 e15 = ε11 q15 ;

2 kem = q152

( c44 μ11 )

if and only if μ11 e15 = d11 q15 ;

2 kem ≥ q152

( c44 μ11 ) ,

0 ≤ kem < 1 ;

(12)

kem = 0 if and only if e15 = 0 , q15 = 0 ; 2 if d11 = 0 then c44 = c44 + e152 ε11−1 + q152 μ11−1 , kem = e152 / ( c44 ε11 ) + q152 / ( c44 μ11 ) .

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

μ A = μ B = μ , d A = d B = d , ε A = ε B = ε , q A = qB = q , c A = cB = c , cAe = cBe = c e , cAm = 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 → −∞ ,

(13)

and the mechanical boundary conditions on the plane boundaries of the half-spaces require that TA = TB = 0 on x2 = 0 .

(14)

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 ) ⎤⎦ ,

(15)

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 ) ⎤⎦ ,

(16)

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

4

in the half-space x2 < 0 . These expressions satisfy the conditions (13) 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 .

(17)

Now the mechanical boundary conditions (14) together with different magneto-electrical contact conditions on x2 = 0 must be satisfied. In the present paper the following cases of the magneto-electrical contact conditions on x2 = 0 are of our interest: 1a) DA = DB = 0 , BA = BB , φ A = φB ; 1b) DA = DB = 0 , φ A = φB = 0 ; 1c) DA = DB = 0 , BA = 0 , φB = 0 ; 2a) ϕ A = ϕ B = 0 , BA = BB = 0 ; 2b) DA = DB , ϕ A = ϕ B , BA = BB = 0 ; 2c) ϕ A = 0 , DB = 0 , BA = BB = 0 ; 3a) DA = DB , ϕ A = ϕ B , BA = BB , φ A = φB ; 3b) ϕ A = ϕ B = 0 , φ A = φB = 0 ; 3c) DA = 0 , ϕ B = 0 , BA = 0 , φB = 0 ;

(18)

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

ϕ A = 0 , DB = 0 , BA = 0 , φB = 0 ;

7a) ϕ A = 0 , DB = 0 , BA = BB , φ A = φB ; 7b) DA = DB , ϕ A = ϕ B , BA = 0 , φB = 0 ; 8)

DA = DB = 0 , BA = BB = 0 .

Each of the 17 groups of conditions in Eqs. (18) together with Eqs. (14), (15)-(16) leads to a system of six linear homogeneous algebraic equations for w0 A , ψ 0 A , χ 0 A , w0 B , ψ 0 B , χ 0 B , the existence of nonzero solution of which requires that the determinant of that system be equal to zero. This condition for the determinant together

5

with Eq. (17) determines the surface wave velocities Vs = ω ξ1 . In the case of 1a) of Eqs. (18) this procedure leads to a surface wave with the following velocity:

(

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

2

).

(19)

The same velocity is obtained in the cases 1b) and 1c). Each of the cases 2a), 2b) and 2c) leads to a surface wave with velocity

(

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

2

),

(20)

and each of the cases 3a), 3b) and 3c) leads to a surface wave with velocity Vs23 = ( c ρ ) (1 − ke4m ) .

(21)

The case 8) does not lead to any surface wave. Each of the cases from 4a) to 7b) leads to its own pair of surface waves with velocities Vs 2 , Vs 3 in the cases 4a) and 4b); Vs1 , Vs 3 in the cases 5a) and 5b);

Vs1 , Vs 2 in the case 6); Vs 4 , Vs 5 in cases 7a) and 7b);

where Vs24 = ( c ρ ) (1 − α 2 ) , Vs25 = ( c ρ ) (1 − β 2 ) ,

α=

⎞ 1 εμ ⎛ 3εμ − d 2 2 e2 q 2 kem − − − Q⎟, ⎜ 2 2εμ − d 2 ⎝ εμ cε c μ ⎠

β=

⎞ 1 εμ ⎛ 3εμ − d 2 2 e2 q 2 kem − − + Q⎟, 2 ⎜   2 2εμ − d ⎝ εμ cε c μ ⎠ 2

⎛ 3εμ − d 2 2 2εμ − d 2 e2 q 2 ⎞ − Q=⎜ kem − ⎟ −4 cε c μ ⎠ εμ ⎝ εμ

( e μ − 2eqd + q ε ) = 2

2

2

(22)

⎛ 2 q2 ⎞ ⎛ 2 e2 ⎞ ⎜ kem − ⎟ ⎜ kem − ⎟ c μ ⎠ ⎝ cε ⎠ ⎝

+ 2( qε − ed ) q 2 + 2( e μ − qd ) e 2 + ( μ e 2 − ε q 2 ) 2

(εμ c )

2

2

2

.

From Eqs. (22) it follows that

(1 − β )(1 − α ) =

εμ 2εμ − d 2

⎡⎛ εμ − d 2 e2 q 2 ⎤ e2 q 2 ⎞ 2 2 + (1 − kem + + ) ⎢⎜ ⎟ (1 − kem ) + 2 ⎥ . cε c μ ⎠ c εμ ⎦ ⎣⎝ εμ

6

(23)

Using Eqs. (10)-(12), (22) and (23) one has that

Q ≥ 0 ; 0 ≤ α ≤ β