Wave propagation through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces

Ultrasonics 82 (2018) 217–232

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Wave propagation through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces Fengyu Jiao a,b, Peijun Wei a,b,⇑, Yueqiu Li c a

Beijing Key Laboratory for Magneto-Photoelectrical Composite and Interface Science, University of Science and Technology Beijing, Beijing 100083, China Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China c Department of Mathematics, Qiqihar University, Qiqihar 161006, China b

a r t i c l e

i n f o

Article history: Received 12 April 2017 Received in revised form 1 July 2017 Accepted 21 August 2017 Available online 24 August 2017 Keywords: Piezoelectric solids Reflection/transmission coefficients Flexoelectric effect Transfer matrix Energy flux

a b s t r a c t Reflection and transmission of plane waves through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces are studied in this paper. The secular equations in the flexoelectric piezoelectric material are first derived from the general governing equation. Different from the classical piezoelectric medium, there are five kinds of coupled elastic waves in the piezoelectric material with the microstructure effects taken into consideration. The state vectors are obtained by the summation of contributions from all possible partial waves. The state transfer equation of flexoelectric piezoelectric slab is derived from the motion equation by the reduction of order, and the transfer matrix of flexoelectric piezoelectric slab is obtained by solving the state transfer equation. By using the continuous conditions at the interface and the approach of partition matrix, we get the resultant algebraic equations in term of the transfer matrix from which the reflection and transmission coefficients can be calculated. The amplitude ratios and further the energy flux ratios of various waves are evaluated numerically. The numerical results are shown graphically and are validated by the energy conservation law. Based on these numerical results, the influences of two characteristic lengths of microstructure and the flexoelectric coefficients on the wave propagation are discussed. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction It is known that the classical elastic theory does not suffice for an accurate and detailed description of corresponding mechanical behavior in the range of micro and nano scales. The main cause is the absence of internal characteristic length, characteristic of the underlying microstructure, from the constitutive equation in the classical elastic theory, and therefore the notable size effects observed experimentally could not be captured. In the problem of wave propagation, the classical elastic theory is also believed to be inadequate for a material possessing microstructure, in particular, when the wavelength of an incident wave is comparable to the length of the material microstructure. The same, in the classical piezoelectric elastic theory, no characteristic length is included in the constitutive relations. Therefore, the classical piezoelectric elastic theory cannot describe the mechanical and electrical behaviors of piezoelectric material in the micro or nano scale and size effects. Recently, Zubko et al. [1] studied the flexo⇑ Corresponding author at: Department of Applied Mechanics, University of Science and Technology Beijing, Beijing 100083, China. E-mail address: [email protected] (P. Wei). http://dx.doi.org/10.1016/j.ultras.2017.08.008 0041-624X/Ó 2017 Elsevier B.V. All rights reserved.

electric effect in piezoelectric solids. They discussed the presence of flexoelectric effect in many nanoscale systems and looked at its potential applications in the MEMS (micro electro mechanical system). Hu and Shen [2,3] studied the variational principles and governing equations in nano-dielectrics with the flexoelectric effect. By establishing the electric enthalpy variational principle for nano-sized dielectrics with the strain gradient and the polarization gradient effects, the governing equations and boundary conditions were given. Shu et al. [4] studied the symmetry of flexoelectric coefficients in crystalline medium. Their investigation indicated that the direct flexoelectric coefficients should be presented in 3
18 form and the converse flexoelectric coefficients in 6
9 form, rather than 6
6 form. Liang et al. [5] studied the Bernoulli–Euler dielectric beam model based on the straingradient effect. It was found that the beam deflection predicted by the strain gradient beam theory is smaller than that by the classical beam theory when the beam thickness is comparable to the internal length scale parameters. Zhang et al. [6] demonstrated an experiment on two designs of flexoelectric metamaterials. It was found that when a ferroelectric ceramic wafer is placed on a metal ring or has a domed shape, which is produced through the diffusion between two pieces of ferroelectric ceramic of different

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F. Jiao et al. / Ultrasonics 82 (2018) 217–232

compositions at high temperatures, an apparent piezoelectric response originating from the flexoelectric effect can be measured under a stress, and the apparent piezoelectric response of the materials based on the designs can be sustained well above the Curie temperature. Liu and Wang [7] investigated the sizedependent electromechanical properties of piezoelectric superlattices made of BaTiO3 and PbTiO3 layers, it was found that the strain gradient is giant at the interface between BaTiO3 and PbTiO3 layers, which will lead to the significant enhancement of polarization in the superlattice due to the flexoelectric effect, therefore, the influence of strain gradient at the interface becomes significant when the size of superlattice decreases. Although the flexoelectric effect has been studied in the above-mentioned literatures, the influences of the flexoelectric effect on the wave propagation in piezoelectric material have not been reported so far. The reflection and transmission of elastic wave through a slab with finite thickness was an everlasting interesting topic in the past decades. Caviglia and Morro [8,9] studied the wave propagation through an elastic slab and a viscoelastic slab, respectively. Tolokonnikov [10] studied the wave propagation through an inhomogeneous anisotropic slab. Larin and Tolokonnikov [11] further studied the wave propagation through a non-uniform thermoelastic slab. Hsia and Su [12] studied the wave propagation through a microporous slab characteristic of micropolar elasticity. Zhang et al. [13] also studied the wave propagation through a micropolar slab sandwiched by two elastic half-spaces. The influences of the micropolar elastic constants and the thickness of slab on the reflection and transmission waves were discussed in their paper. Li and Wei [14,15] studied the reflection and transmission of plane waves at the interface between two different dipolar gradient elastic halfspaces and the reflection and transmission through a microstructured slab sandwiched by two half-spaces. It was found that the microstructure effects make the propagating waves dispersive and create the evanescent waves that become the surface waves at the interface. The influences of three characteristic lengths, namely, the incident wavelength, the thickness of slab and the characteristic length of microstructure, on the reflection and transmission waves were analyzed. Because the sandwiched structure is widely met in the transducer, actuator, acoustic isolator, interface detection and so on, the researches on the wave propagation through a sandwiched slab are of importance theoretically and practically. In this paper, wave propagation through a piezoelectric slab sandwiched by two piezoelectric half-spaces with the flexoelectric effect taken into consideration is studied. The dispersive equation is derived from the general motion equation and the all possible partial waves in the piezoelectric material with the flexoelectric effect are discussed. The state vectors and the state transfer equation in the piezoelectric slab are also derived. Nontraditional interface conditions with the monopolar and dipolar tractions are considered, and the reflection and transmission coefficients are obtained in term of the transfer matrix. The amplitude ratios and further the energy flux ratios of the reflection and transmission waves are calculated numerically and the numerical results are validated by the check of energy conservation. Based on the numerical results, the influences of the flexoelectric coefficients and two characteristic lengths of microstructure on the reflection and transmission waves are discussed. 2. State vectors in the piezoelectric solid with the flexoelectric effect considered The general expression of the electric Gibbs free energy density function can be written as [5]

1 1 ekl Ek El þ cijkl Sij Skl  ekij Ek Sij  f ijkl Ei gjkl þ rijklm Sij gklm 2 2 1 þ g ijklmn gijk glmn ; 2

U¼

ð1Þ

where ekl and cijkl are the dielectric and elastic tensors, respectively. ekij is the piezoelectric tensor, f ijkl is flexoelectric coefficient tensor, rijklm denotes the coupling between the strain and strain gradient. The tensor g ijklmn represents the strain gradient effect. Sij is the strain tensor, gjkl is the strain gradient tensor and Ei is the electric field vector, which are defined, respectively, as Sij ¼ ðuj;i þ ui;j Þ=2, gijk ¼ Sij;k and Ei ¼ u;i , where u is the displacement, u is the electric potential, the comma indicates differentiation with respect to the spatial variables. It is noted that

Z

V

Z 1 r ijklm ðSij Skl Þ;m dV 2 V Z 1 ðSij Skl Þr ijklm nm dS: ðr ijklm ¼ rklijm Þ ¼ 2 S

r ijklm Sij gklm dV ¼

ð2Þ

This means that the fifth term in the right side of Eq. (1) is the contribution from the surface of material. Let

Us ¼

1 rijklm Sij gklm ; 2

ð3Þ

then, U s is the surface energy of unit surface area and it includes the contribution from the surface stresses. In the present work, the surface effects of material are neglected, namely, r ijklm is null. More2

over, g jkhmni is approximated by g jkhmni ¼ l1 dhi cjkmn [16], where l1 is internal characteristic length of microstructure. Then, the constitutive equations can be obtained from the electric Gibbs free energy as

rij ¼ cijkl Skl  ekij Ek ;

ð4aÞ

rjkh ¼ f ijkh Ei þ l21 dhi cjkmn gmni ;

ð4bÞ

Dek ¼ ekl El þ ekij Sij þ f klmn glmn ;

ð4cÞ

where rij is the classical Cauchy stress tensor, rjkh is the higherorder stress tensor, and Dek is the electric displacement vector. It is noted that rij ¼ rji and rjkh ¼ rkjh . The kinetic energy density with consideration of the microinertial effect can be expressed as [14,15]

T¼

1 1 qu_ j u_ j þ ql22 u_ k;j u_ k;j ; 2 6

ð5Þ

where q is mass density, and l2 is the micro inertia characteristic length. Then, the variation of the electric Gibbs free energy density (electrostatic force is neglected) plus the kinetic energy density are [2,3,14], Z "

Z ðU þ TÞdV ¼

d

#

ðrij  rijm;m Þnj þ Dl ðnl Þrijm nj nm  Dj ðrijm nm Þ þ

V

a

Z  Z þ a

Z

ql22 € nj ui;j dui da 3

2 l2

q

€i  € i;jj Þdui dV ½ðrij  rijm;m Þ;j  ðqu u 3 Z rijm nj nm Ddui da þ ni Dei duda;

V

Dem;m dudV 

V

a

ð6Þ

where nj is the unit normal vector of the boundary of solid, Dj ð Þ ¼ ð Þ;j  nj nk ð Þ;k , Dð Þ ¼ nl ð Þ;l . The variation of work done by external forces can be expressed as

Z

dW ¼

Z

Z

F i dui dV þ V

Pi dui da þ a

Z

Ri Ddui da þ a

qduda; a

ð7Þ

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where F i is the body force (it is neglected in following), P i is the monopolar traction, Ri is the dipolar traction and q is the charge. According to the variation principle

Z

ðU þ TÞdV ¼ dW;

d

ð8Þ

V

the mechanical and electric governing equations, and the monopolar and dipolar tractions can be expressed as

€i  ðrij  rijm;m Þ;j ¼ qu

ql22 € 3

ui;jj ;

Dem;m ¼ 0:

ð9a; bÞ

where

8 r 9 > < xxx > = rxx ¼ rxxy ; > > : ;

3

q ¼ ni Dei :

ð10b; cÞ

Eqs. (4), (9) and (10) constitute the piezoelectric elastic theory taken the flexoelectric effect, the strain gradient effect and the micro-inertial effect into considered. Let the z-axis be the poling direction and the piezoelectric material is assumed to be transversely isotropic in the oxy coordinate plane. Then, Eqs. (4a)–(4c) reduce to [4]

8 rxx 9 > > > > > > > ryy > > > > > > =

2

c11 6 c12 6 6c zz 6 13 ¼6 > > 6 0 r yz > > > > 6 > > > > 4 0 r > > > : zx > ; rxy 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

9 3 8 Sxx > > > > > > 7 > Syy > > > > 7 > > = 7 < S > zz 7 7 > 7 > 2S > yz > > 7 > > > > 5 > 2S > > > : zx > ; 2Sxy 0:5ðc11  c12 Þ 3 2 0 0 e31 6 0 0 e31 7 7 8 Ex 9 6 < = 6 0 0 e33 7 7 6 6 7  Ey ; 6 0 e15 0 7 : ; 7 6 Ez 4 e15 0 0 5 0 0 0 0 0 0 0 0

2

8 3 gxx 9 M1 > > > > > 6 M2 7 8 9 > > gyy > > > > > 7 6 > < g > = 6 M 7 < Ex = zz 6 37 zz  þ G  E ¼ 6 ; 7 y 2gyz > > > 6 M4 7 : ; ryz > > > > > > > > > 7 6 E > > z > > > > 4 M5 5 2g > r > > > > > > > : zx > : zx > ; ; 2gxy rxy M6 2

8 e9 2 0 > < Dx > = Dey ¼ 6 4 0 > : e> ; e31 Dz 2

e11

0

6 þ4 0

e11

0

0

þ ½ M1

M2

0

0

0

e15

0

0

e15

0

e31

e33

0

0

3 8 9 0 > < Ex > = 7 0 5  Ey > > e33 : Ez ;

M3

M4

M5

ð11bÞ

0

l14

0

0

l14

0

6 M3 ¼ 4 0

l14

0 2

0

0

0

6 M4 ¼ 4 0

0

2

0 0 0

gxx ¼

3

7 0 5;

l14

2

l14

6 M2 ¼ 4 0

3

0

0

0

l14

0

0

l111 3

l11 3

0

7 0 5;

0

0

G32

3

7 0 5;

l11

2

6 l111 7 0 5; M5 ¼ 4 0 0 l111 0

gxxy ; > > : g ; 8 xxz 9 > < gyzx > = gyz ¼ gyzy ; > :g > ; yzz 2 G11 G12 26 G ¼ l1 4 G21 G22

0

0 7 0 5;

0

8 9 > < gxxx > =

G31

0

> :

l111 l111 0 3

6 M6 ¼ 4 l111

ð11aÞ 8 9 rxx > > > > > > ryy > > > > > > > =

l11

6 M1 ¼ 4 0

ð10aÞ Ri ¼ rijm nj nm ;

rzz ¼

8 r 9 > < zzx > =

rzzy ; > ; ryyz rxxz rzzz 8 8 8 r 9 r 9 r 9 > > > < yzx > < zxx > < xyx > = = = ryz ¼ ryzy ; rzx ¼ rzxy ; rxy ¼ rxyy ; > > > > > > : : : ; ; ; ryzz rxyz rzxz 2

q l2 € Pi ¼ rij nj ¼ ðrij  rijm;m Þnj þ Dl ðnl Þrijm nj nm  Dj ðrijm nm Þ þ 2 nj u i;j ;

8 r 9 > < yyx > = ryy ¼ ryyy ; > > : ;

8 9 > < gyyx > = gyy ¼ gyyy ; > :g > ; yyz 8 9 > < gzxx > = gzx ¼ gzxy ; > > : gzxz ; 3 G13 7 G23 5:

7 0 5; 0

8 9 > < gzzx > = gzz ¼ gzzy ; > > : g ; 8 zzz 9 > < gxyx > = gxy ¼ gxyy ; > :g > ; xyz

G33

The explicit expressions of elements in the matrix G are given in Appendix A. In the plane strain case, the displacement and the electric potential are only the function of x and z, namely,

u ¼ fuðx; z; tÞ; 0; wðx; z; tÞg;

u ¼ uðx; z; tÞ:

ð12a; bÞ

Then, Eq. (11) can be rewritten as

8 9 Sxx > > > > > > > > 3 > Syy > > > > > 0 > < S > = zz 7 05  > > 2S > > yz > > 0 > > > > > > > > 2Szx > > : ; 2Sxy

9 > > > > > > > > > = zz M6   ; > 2gyz > > > > > > > > > > 2gzx > > > > > > : 2g > ; xy

@u @w @u þ c13 þ e31 ; @x @z @z

ryy ¼ c12

rzz ¼ c13

@u @w @u þ c33 þ e33 ; @x @z @z

rzx ¼ c44

ryz ¼ rxy ¼ 0: ð11cÞ

8 gxx > > > > > gyy > > > > < g

rxx ¼ c11

@u 2 @2u 2 @2w ; þ l1 c11 2 þ l1 c13 @x @x@z @x 2 2 @u 2 @ u @ w 2 ¼ l14 þ l c13 2 ; þ l1 c11 @x@z 1 @z @z

@u @w @u þ c13 þ e31 ; @x @z @z ð13a; bÞ

  @u @w @u þ þ e15 ; @z @x @x ð13c; d; eÞ

rxxx ¼ l11 rxxz

@u 2 @2u 2 @2w ; þ l1 c12 2 þ l1 c13 @x @x@z @x 2 2 @u 2 @ u @ w 2 ¼ l14 þ l c13 2 ; þ l1 c12 @x@z 1 @z @z

ð14a; bÞ

ryyx ¼ l14 ryyz

ð14c; dÞ

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F. Jiao et al. / Ultrasonics 82 (2018) 217–232

@u 2 @2u 2 @2w ; þ l1 c13 2 þ l1 c33 @x @x@z @x 2 2 @u 2 @ u @ w 2 þ l c33 2 ; ¼ l11 þ l1 c13 @x@z 1 @z @z

rzzx ¼ l14 rzzz

@u ; @z ! @u 2 @2u @2w ; þ ¼ l111 þ l1 c44 @x@z @x2 @z

ð14e; fÞ

ryzy ¼ rzyy ¼ l111 rzxx ¼ rxzx

rzxz ¼ rxzz ¼ l111 rxyy ¼ ryxy ¼ l111

ð14g; hÞ

! @u 2 @2w @2u þ 2 ; þ l1 c44 @x@z @z @x @u ; @x

ð14i; jÞ

rxxy ¼ ryyy ¼ rzzy ¼ ryzx ¼ rzyx ¼ ryzz ¼ rzyz ¼ rzxy ¼ rxzy ¼ rxyx ¼ ryxx ¼ rxyz ¼ ryxz ¼ 0: ð14kÞ   @u @w @u @2u @2w þ  e11 þ l11 2 þ l14 @z @x @x @x@z @x ! 2 2 @ u @ w ; þ l111 þ @z2 @x@z

Dex ¼ e15

Dey ¼ 0; Dez

ð15aÞ

are the bulk waves and six of them are the surface waves. Moreover, the two bulk waves are of the wave vector with positive projection along z axis and the other two bulk waves are of the wave vector with negative projection along z axis. Therefore, the four bulk waves are the incident and reflection waves, respectively. Let a1 ; a3 ; a5 ; a7 and a9 be the reflection quasi-longitudinal wave (QP), the reflection quasi-traverse wave (QSV), the reflection electric–acoustic wave (EA), the reflection P-type surface wave (SP) and reflection S-type surface wave (SS); while a2 ; a4 ; a6 ; a8 and a10 are the transmission QP wave, transmission QSV wave, transmission EA wave, transmission SP wave and transmission SS wave. The reflection and transmission angle hq can be determined by cot hq ¼ Realðaq Þ. Define the amplitude ratios of each coupled wave (q ¼ 1; 2;    ; 10)

8 13 ðaq ÞK 11 ðaq ÞK 23 ðaq Þ < Gq ¼ UU2q ¼ KK 21 ððaaq ÞK 1q 12 q ÞK 23 ðaq ÞK 13 ðaq ÞK 22 ðaq Þ

: Hq ¼ U3q ¼ K 11 ðaq ÞK 22 ðaq ÞK 12 ðaq ÞK 21 ðaq Þ U 1q K 12 ðaq ÞK 23 ðaq ÞK 13 ðaq ÞK 22 ðaq Þ

ð19Þ

Then, the displacement, the electric potential, the normal displacement gradient, the classical Cauchy stress, the electric displacement and the higher-order stress of the incident, reflection and transmission waves can be expressed as

fu; w; ug ¼

10 X   f1; Gq ; Hq gU 1q exp inðx þ aq z  ctÞ ;

ð20Þ

q¼1

ð15bÞ

@u @w @u @2w @2u þ e33  e33 ¼ e31 þ l11 2 þ l14 @x @z @z @x@z @z ! @2w @2u : þ þ l111 @x2 @x@z

:

fu;z ; w;z g ¼

10 X   infaq ; Gq aq gU 1q exp inðx þ aq z  ctÞ ;

ð21Þ

q¼1

ð15cÞ

frzx ; rzz ; rxx g ¼

10 X   infF 1q ; F 2q ; F 3q gU 1q exp inðx þ aq z  ctÞ ;

ð22Þ

q¼1

The plane waves propagating in the oxz coordinate plane are of the form

fu; w; ug ¼ fU 1 ; U 2 ; U 3 g exp½inðx þ az  ctÞ;

ð16Þ

where k ¼ kx ex þ kz ez is the wave vector and a ¼ kz =kx is the projection ratio. nð¼ kx Þ and c are the apparent wavenumber and the apparent speed, respectively. Inserting Eq. (16) into Eq. (9) leads to

2

K 11

6 4 K 21 K 31

K 12 K 22 K 32

3 8 9 8 9 K 13 > < U1 > = > = <0> 7 K 23 5  U 2 ¼ 0 : > : > ; > : > ; K 33 U3 0

ð17Þ

The explicit expressions of elements in the matrix K are given in Appendix B. The condition of existing non-trivial solution is

jK ij ðc; a; xÞj3
3 ¼ 0:

ð18Þ

It is the secular equation of wave motion. Different from the classical piezoelectric elastic theory, the apparent speed c is not only related to the wave propagation direction, but also related to the angular frequency x (n and x ¼ nc are taken as real quantities), so the wave propagating in piezoelectric material with the flexoelectric effect taken into consideration is dispersive. In general, for a given apparent wave speed c and angular frequency x, Eq. (18) is a polynomial of ten orders about a. The solutions of this polynomial stand for all possible wave modes. The real value of a stands for a bulk wave which propagates in oxz plane; The imaginary value of a stands for a surface wave which propagates along x axis and attenuates along z axis. The complex value of a stands for the bulk wave with decreasing or increasing amplitudes accompanied with propagation. This type of wave is physically unacceptable in the elastic solid without dissipation. For the transversely isotropic flexoelectric piezoelectric solid considered, the value of a indicates that there are ten possible partial waves. Four of them

fDez ; Dex g ¼

10 X   infF 4q ; F 5q gU 1q exp inðx þ aq z  ctÞ ;

ð23Þ

q¼1

frzxx ; rzxz ; rzzx ; rzzz ; rxxx ; rxxz g ¼

10 X fF 6q ; F 7q ; F 8q ; F 9q ; F 10q ; F 11q gU 1q q¼1

 
exp inðx þ aq z  ctÞ ; ð24Þ the explicit expressions of F iq ði ¼ 1—11Þ are given in Appendix C, and f1; Gq ; Hq g is the vibration form vector of each coupled wave. Let us consider the wave propagation through a piezoelectric slab with the thickness of h sandwiched between two piezoelectric elastic half-spaces, see Fig. 1. At the interface z ¼ 0 and z ¼ h, the monopolar traction, the dipolar traction and the surface charge are

€ i;z =3 ði ¼ x; zÞ; Pi ¼ riz  rizx;x  rizz;z  rixz;x þ ql2 u 2

Ri ¼ rizz ði ¼ x; zÞ; q ¼ Dez :

ð25aÞ ð25b; cÞ

Further, the explicit expressions of Eq. (25) can be expressed as

Px ¼

10 X

inM 1q U 1q exp½inðx þ aq z  ctÞ;

ð26aÞ

inM 2q U 1q exp½inðx þ aq z  ctÞ;

ð26bÞ

q¼1

Pz ¼

10 X q¼1

q¼

10 X inM 3q U 1q exp½inðx þ aq z  ctÞ; q¼1

ð26cÞ

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F. Jiao et al. / Ultrasonics 82 (2018) 217–232

Fig. 1. Reflection and transmission waves through a flexoelectric piezoelectric slab sandwiched by two piezoelectric half-spaces.

Rx ¼

10 X M 4q U 1q exp½inðx þ aq z  ctÞ;

ð26dÞ

q¼1

Rz ¼

10 X

where

"

AðzÞ ¼ M5q U 1q exp½inðx þ aq z  ctÞ:

ð26eÞ

q¼1

The explicit expressions of M iq ði ¼ 1—5Þ are given in Appendix D. Let U ¼ ½u; w; u; u;z ; w;z T and T ¼ ½P x ; Pz ; q; Rx ; Rz T . Define the state vector fðx; z; tÞ ¼ ðU; TÞT . Then, for the incident wave in the þ left half-space, the value of the state vector at z ¼ 0 and z ¼ h can be obtained from Eqs. (20), (21) and (26), namely,

fðz ¼ 0 Þ ¼ ðA0 U 01q þ AL U1 Þ exp½inðx  ctÞ; þ

R

fðz ¼ h Þ ¼ A

Uþ1

ð27aÞ

exp½inðx  ctÞ;

ð27bÞ

T T þ where U 1 ¼ ðU 11 ; U 13 ; U 15 ; U 17 ; U 19 Þ , U1 ¼ ðU 12 ; U 14 ; U 16 ; U 18 ; U 1ð10Þ Þ

and U 01q stands for the amplitude of the incident wave. The explicit expressions of A0 , AL and AR are given in Appendix E. When the flexoelectric effect, the strain gradient effect and the micro-inertial effect are ignored, the various quantities in Eq. (27) T T þ should be modified as U 1 ¼ ðU 11 ; U 13 ; U 15 Þ , U1 ¼ ðU 12 ; U 14 ; U 16 Þ ,

U ¼ ½u; w; uT and T ¼ ½P x ; P z ; qT . Correspondingly, A0 , AL and AR T

are also modified. The state vector fðx; z; tÞ ¼ ðU; TÞ reduces to 6 orders from 10 orders. All the expressions will reduce to the situation in the classical piezoelectric medium.

M1 11 ðzÞN11 ðzÞ

#

M1 11 ðzÞN12 ðzÞ

1 N21 ðzÞM21 ðzÞM1 11 ðzÞN11 ðzÞ N22 ðzÞM21 ðzÞM11 ðzÞN12 ðzÞ

:

The explicit expressions of elements in matrix Mi1 ði ¼ 1; 2Þ and matrix Nij ði ¼ 1; 2; j ¼ 1; 2Þ are given in Appendix F. Eq. (28) is called a state transfer equation. Theoretically, the relation between the state vectors at two sides of the slab can be obtained from the state transfer equation. Define the transfer matrix Bðz; z0 Þ, which relates the state vectors at z and z0 by

f0 ðzÞ ¼ Bðz; z0 Þf0 ðz0 Þ:

ð29Þ

Inserting Eq. (29) into Eq. (28) leads to

dBðz; z0 Þ ¼ AðzÞBðz; z0 Þ; Bðz0 ; z0 Þ ¼ I: dz

ð30Þ

Eqs. (28) and (30) are applied to the general heterogeneous slab with the material parameters changing with the location. The solution of Eq. (30) is usually obtained by the complex Magnus series [17]. In this paper, the flexoelectric piezoelectric slab is assumed to be homogenous. Then, M and N are constant matrixes, and thus A is also constant matrix. The solution of Eq. (30) can be expressed as

Bðz; z0 Þ ¼ eAðzz0 Þ ;

ð31Þ

namely

BðhÞ ¼ Bðz0 þ h; z0 Þ ¼ eAh :

ð32Þ

3. Reflection/transmission coefficients and energy conservation

After the transfer matrix of the flexoelectric piezoelectric slab is obtained, the state vectors at the left and right sides of the slab can be related by

Let U0 ¼ ½u; w; u; u;z ; w;z T and T0 ¼ ½P x ; P z ; q; Rx ; Rz T in the flexoelectric piezoelectric slab. Define the state vector

f0 ðz ¼ h Þ ¼ BðhÞf0 ðz ¼ 0þ Þ:

T

f0 ðx; z; tÞ ¼ ðU0 ; T0 Þ , f0 ðx; z; tÞ is a function of position. The specific values of the state vector at the left and right sides of the slab are different. By the introduction of the state vector, the motion equations of second order of derivative, i.e. Eq. (9), can be recast into the first-order matrix differential equation

df0 ¼ AðzÞf0 ; dz

ð28Þ



ð33Þ

The perfect interface conditions require that the displacement components, the electric potential, the normal displacement gradient components, the monopolar traction components, the normal components of the electric displacement, and the dipolar traction components, are continuous across the interface. In other words, the state vectors are continuous across the perfect interface, namely,

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F. Jiao et al. / Ultrasonics 82 (2018) 217–232 

þ

fðz ¼ 0 Þ ¼ f0 ðz ¼ 0þ Þ; f0 ðz ¼ h Þ ¼ fðz ¼ h Þ:

ð34Þ

Recall Eq. (27), Eq. (33) can be rewritten as

0 3 31 2 U 12 U 11 B 7C 6 U 7 6 B 6 14 7 6 U 13 7C B 7 7C 6 6 L6 B 0 0 7 7C AR 6 6 U 16 7 ¼ BðhÞ  BA U 1q þ A 6 U 15 7C: B 7 7C 6 6 @ 4 U 18 5 4 U 17 5A U 1ð10Þ U 19 2

Q i ðtÞ ¼ rij u_ j  rijm u_ j;l nm nl þ uD_ i ; ð35Þ

Eq. (35) relates the reflection and transmission waves with the incident wave. In order to obtain the reflection and transmission coefficients from Eq. (35), let



1

R

W ¼ B ðhÞ  A ¼

W1 W2

"

 ;

0

A ¼

A01 A02

# ;

 AL ¼

A1



A2

:

Then, Eq. (35) can be recast into



A1

W1

A2

W2



R

"



T

¼

# A01 ; A02

ð36Þ

R ¼ ðR1 ; R2 ; R3 ; R4 ; R5 ÞT ;

T2 ¼

U 13 U 01q

;

U 14 U 01q

;

R3 ¼ T3 ¼

U 15 U 01q U 16 U 01q

; ;

T ¼ ðT 1 ; T 2 ; T 3 ; T 4 ; T 5 ÞT ; R4 ¼ T4 ¼

U 17 U 01q U 18 U 01q

; ;

R5 ¼ T5 ¼

U 19 U 01q

;

U 1ð10Þ U 01q

R1 ¼ T1 ¼

U 11 U 01q

U 12 U 01q

Q z ðtÞ ¼ Pi u_ i  Ri nj u_ i;j þ uni D_ i ;

Q i ðtÞ ¼ rij u_ j þ uD_ i ;

ð42Þ

Q z ðtÞ ¼ rzj u_ j þ uD_ z :

ð43Þ

The average energy fluxes carried by the reflection and transmission waves in a period are





þ

ð38Þ

Correspondingly, Eq. (35) is modified as

8 8 9 91 0 > > < U 12 > < U 11 > = = B C AR U 14 ¼ B0 ðhÞ  @A0 U 01q þ AL U 13 A: > > > > : : ; ; U 16 U 15

T

Q i ðtÞdt:

ð44Þ

0

ð45Þ

ð37Þ

f00 ðz ¼ h Þ ¼ fðz ¼ h Þ:

Z

1 Q zq ¼ n2 cðM 1q þ M 2q Gq þ M3q Hq  M 4q aq  M5q Gq aq ÞðU 1q  U 1q Þ; 2

;

where f00 ðx; z; tÞ ¼ ðu; w; u; P x ; P z ; qÞT ; z 2 ð0þ ; h Þ. B0 ðhÞ is of 6
6 orders while BðhÞ in Eq. (32) is of 10
10 orders. The relation between B0 ðhÞ and BðhÞ is given in Appendix G. After such a manipulation, the continuous conditions of the state vector, i.e. Eq. (34), can be modified as

fðz ¼ 0 Þ ¼ f00 ðz ¼ 0þ Þ;

1 T

Inserting Eqs. (20)–(21) and (26) into Eqs. (41) and (44) leads to

are the reflection coefficients of QP, QSV, EA, SP and SS waves in the left half-space, and the transmission coefficients of QP, QSV, EA, SP and SS waves in the right half-space, respectively. And q ¼ 2 indicates the incident QP wave, q ¼ 4 indicates the incident QSV wave. When the flexoelectric effect, the strain gradient effect and the micro-inertial effect are ignored in the half-spaces at the two sides of the flexoelectric piezoelectric slab, the state vectors fðz ¼ 0 Þ þ and fðz ¼ h Þ at the two half-spaces are of 6 orders while the state  vectors in the slab, namely, f0 ðz ¼ 0þ Þ and f0 ðz ¼ h Þ, are of 10 orders. Therefore, Eq. (34) needs modification. Recall that the dipolar tractions disappear at the two half-spaces, the dipolar tractions in the slab should tend to zero at the two boundaries, namely,  Rx ¼ Rz ¼ 0 at both z ¼ 0þ and z ¼ h . By using these prerequisite conditions, Eq. (29) can be modified as 

ð41Þ

where the first term on the right hand side is related to the work rate done by the monopolar tractions, the second term to the work rate done by the dipolar tractions and the third term to the work rate done by the electric filed. In the classical piezoelectric solid (without consideration of the flexoelectric effect, the strain gradient effect and the micro-inertial effect), Eqs. (40) and (41) reduce to

;

;

f00 ðz ¼ h Þ ¼ B0 ðhÞf00 ðz ¼ 0þ Þ;

ð40Þ

and the normal energy flux is

Qi ¼

where

R2 ¼

no contribution to the energy fluxes into and out of the flexoelelctric piezoelectric slab. The energy fluxes carried by the reflection and transmission bulk waves can be evaluated by

ð39Þ

Now, let us examine the energy fluxes carried by the reflection and transmission waves. The incident wave can input the energy flux into the piezoelectric slab while the reflection and transmission bulk waves will extract energy from the piezoelectric slab. The surface waves propagate along the interface, and thus have

where the superscript ‘‘” indicates the conjugated complex. The averaged energy flux carried by a coupled wave in the classical piezoelectric medium through an unit area vertical to the x and z axis can be estimated by

(

Q xq ¼ 12 n2 c½F 3q þ Gq F 1q þ Hq F 5q ðU 1q  U 1q Þ Q zq ¼ 12 n2 c½F 1q þ Gq F 2q þ Hq F 4q ðU 1q  U 1q Þ

:

ð46Þ

The averaged energy flux along the propagation direction nq can be obtained by

Q q ¼ Q nq ¼ Q xq cosðnq ; xÞ þ Q zq cosðnq ; zÞ:

ð47Þ

Due to the exponential attenuation (the attenuation term is enbq z ) of the mechanical and electrical quantities at the equiphase plane for the surface wave (aq ¼ ibq ) , the energy flux along the x axis is dependent upon the selection of the unit area. For convenience of comparison, the unit area is selected as ly
lz ¼ n
1=n. Thus, the energy flux of the surface wave should R 1=n 2bq be multiplied by a discount factor, i.e. n 0 e2nbq z dz ¼ 1e2bq for R0 2bq for the right piezoelectric half-space and n 1=n e2nbq z dz ¼ e 2b1 q the left piezoelectric half-space. Defined the energy reflection and transmission coefficients as the ratios of the energy flux (along the respective propagation direction) of the reflection and transmission waves to the incident wave, namely,

ERs ¼

Q Rs I Q QPðQSVÞ

;

ETs ¼

Q Ts I Q QPðQSVÞ

;

ðs ¼ QP; QSV; EAÞ:

ð48Þ

Because there is no energy dissipation taken place in the piezoelectric material and the flexoelectric piezoelectric slab due to the purely elastic nature, the normal energy flux balance requires

 . E ¼ Q RzQP þ Q RzQSV þ Q TzQP þ Q TzQSV Q IzQPðQSVÞ ¼ 1:

ð49Þ

Eq. (49) implies that the total energy flux which flows into the slab is equal to the total energy flux which flows out from the slab in a period. The energy conservation law can be used to validate the numerical results in the next section.

223

F. Jiao et al. / Ultrasonics 82 (2018) 217–232 Table 1 The physical constants of ALN and BaTiO3 [18,19].

ALN BaTiO3 C ij : GPa, eij : C=m2 ,

a

C 11

C 33

C 13

C 44

e33

e15

e31

e11

e33

q

410 166

390 162

100 78

120 43

1:55 18:6

0:48 11:6

0:58 4:4

0:071 11:2

0:084 12:6

3230 5800

eij : 109 C2 =N m2 , q: kg=m3 .

1.0

b l1 l1 l1 l1

0.8

0.02 0.03 0.04 0.05

0.04

l1 l1 l1 l1

0.03

0.02 0.03 0.04 0.05

R

R

E QP

E QSV

0.6 0.02

0.4

0.01 0.2

0.0

0.00 0

10

20

30

40

50

60

70

80

90

0

10

20

30

40 50 I / degree QP

60

70

80

90

30

40 50 I QP/degree

60

70

80

90

l1 l1 l1 l1

0.02 0.03 0.04 0.05

I degree QP/

c 0.012

d l1 l1 l1 l1

0.008

0.02 0.03 0.04 0.05

1.0

0.8

l1 l1 l1 l1

E

T

R EA

E QP

0.6

0.4

0.02 0.03 0.04 0.05

0.004 0.2

0.000

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

I degree QP/

e

0.12

f0.005 l1 l1 l1 l1

0.10

0.08

0.02 0.03 0.04 0.05

0.004

T

E EA

T

E QSV

0.003 0.06

0.002 0.04 0.001

0.02

0.00

0.000 0

10

20

30

40

50

I QP/degree

60

70

80

90

0

10

20

30

40 50 I / degree QP

Fig. 2. The influences of the characteristic length l1 on the reflection and transmission coefficients in the case of incident QP wave. (l11 ¼ 9, h ¼ 0:2).

60

70

80

90

l14 ¼ 10, l111 ¼ 11, l2 ¼ 0:01,

224

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

a

b

1.0

l2 l2 l2 l2

0.8

0.02 0.03 0.04 0.05

0.06

l2 l2 l2 l2

0.05

0.04

0.02 0.03 0.04 0.05

R

R

E QP

E QSV

0.6 0.03

0.4 0.02 0.2

0.01

0.0

0.00 0

10

20

30

40

50

60

70

80

90

0

10

20

30

40 50 I QP/degree

60

70

80

90

30

40 50 I / degree QP

60

70

80

90

I QP/degree

c

d

0.012

l2 l2 l2 l2

0.008

0.02 0.03 0.04 0.05

1.0

0.8

l2 l2 l2 l2

T

R

E QP

E EA

0.6

0.4 0.004

0.02 0.03 0.04 0.05

0.2

0.000

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

I QP/degree

e

f

0.07

l2 l2 l2 l2

0.06

l2 l2 l2 l2

0.004

0.02 0.03 0.04 0.05

0.003 T

E EA

0.04

T

E QSV

0.05

0.02 0.03 0.04 0.05

0.005

0.03

0.002

0.02 0.001 0.01 0.00

0.000 0

10

20

30

40

50

60

70

80

90

I QP/degree

0

10

20

30

40 50 I QP/degree

Fig. 3. The influences of the characteristic length l2 on the reflection and transmission coefficients in the case of incident QP wave. (l11 ¼ 9, h ¼ 0:2).

4. The numerical results and discussion In this numerical example, the flexoelectric piezoelectric sandwiched slab is assumed to be BaTiO3 and the two half-spaces are assumed to be ALN. The material constants of two piezoelectric solids, BaTiO3 and ALN, are listed in Table 1. The incident wave is assumed to be in the left half-space and propagates through the

60

70

80

90

l14 ¼ 10, l111 ¼ 11, l1 ¼ 0:1,

sandwiched slab. The transmission waves and the reflection waves appear in the right and left half-spaces, respectively, see Fig. 1. Two cases, the incident QP wave and the incident QSV wave, are both considered. In order to demonstrate expediently the influences of the flexoelectric effect, the strain gradient effect and the microinertial effect in the sandwiched slab, the flexoelectric effect, the strain gradient effect and the micro-inertial effect in the two

225

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

half-spaces are ignored and the following non-dimensional forms

increase while the transmission coefficients decrease (except QSV

of material and geometric parameters, h ¼ h=k, l1 ¼ l1 =k, l2 ¼ l2 =k, l11 ¼ l11 =ke33 , l14 ¼ l14 =ke33 , l111 ¼ l111 =ke33 (where k is the wavelength of the incident wave), are introduced.

wave) with the increasing of the characteristic lengths l1 and l2 . Compared with the other waves, the QSV wave is more sensitive

Figs. 2and 3 show the influences of the characteristic lengths l1

the characteristic lengths l1 and l2 upon the reflection and transmission coefficients of the QSV wave may have opposite trend with the increasing of the incident angle (see Figs. 2b, e and 3b, e). Recall

and l2 upon the reflection and transmission coefficients in the case of incident QP wave. It is noted that the reflection coefficients

a

to the characteristic lengths l1 and l2 . Moreover, the influences of

b

1.0 11 11

0.8

11 11

0.15

10 15 20 25

10 15 20 25

11 11 11 11

0.10

R

R

E QP

E QSV

0.6

0.4 0.05 0.2

0.0

0.00 0

c

10

20

30

40 50 I QP/degree

60

70

80

90

0

d

0.012 11 11 11 11

10 15 20 25

10

20

30

40 50 I QP/degree

60

70

80

90

30

40 50 I / degree QP

60

70

80

90

1.0

0.8

0.008 0.6

E QP

11

R

T

E EA

10 15 20 25

11

11

0.4

11

0.004 0.2

0.000

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

I QP/degree

e

0.08 11 11 11

0.06

11

f

10 15 20 25

0.005 11 11

0.004

11 11

10 15 20 25

E EA

0.04

T

T

E QSV

0.003

0.002

0.02 0.001

0.00

0.000 0

10

20

30

40

50

I QP/degree

60

70

80

90

0

10

20

30

40 50 I / degree QP

60

70

80

90

Fig. 4. The influences of the flexoelectric coefficient l11 on the reflection and transmission coefficients in the case of incident QP wave. (l14 ¼ 9, l111 ¼ 11, l1 ¼ 0:1, l2 ¼ 0:01, h ¼ 0:2).

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F. Jiao et al. / Ultrasonics 82 (2018) 217–232

that the strain gradient effect and the micro-inertial effect are ignored in the piezoelectric half-space. Therefore, the increasing

micro-inertial effect contribute to the mode transformation. The

of the characteristic lengths l1 and l2 in the sandwiched slab implies that the impedance mismatch increases. This can explain why the reflection waves increase while the transmission waves

and l2 upon the reflection and transmission QSV waves with the increasing of the incident angle can be explained by the anisotropic properties of the piezoelectric slab at the xoz coordinate plane.

decrease when the characteristic lengths l1 and l2 increase. The phenomenon that the QSV wave is more sensitive to the character-

Although the influences of the characteristic lengths l1 and l2 upon the reflection and transmission waves are of similarity, the differences still exist and can be explained by considering the fact that

istic lengths l1 and l2 implies that the strain gradient effect and the

a

opposite trend of the influences of the characteristic lengths l1

b

1.0 14 14

0.8

14 14

10 15 20 25

0.25

10 15 20 25

14 14

0.20

14 14

0.15

R

R

E QP

E QSV

0.6

0.4

0.10

0.2

0.05

0.0

0.00 0

10

20

30

40

50

60

70

80

90

0

10

20

30

40 50 I degree QP/

60

70

80

90

30

40 50 I degree / QP

60

70

80

90

I degree QP/

c 0.012

d 14 14 14 14

10 15 20 25

1.0

0.8

0.008 0.6

E QP

R

T

E EA

10 15 20 25

14 14 14

0.4

14

0.004 0.2

0.000

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

I degree QP/

e

f

0.2 14 14 14 14

10 15 20 25

0.005

10 15 20 25

14

0.004

14 14 14

T

E EA

T

E QSV

0.003 0.1

0.002

0.001

0.0

0.000 0

10

20

30

40

50

I degree QP/

60

70

80

90

0

10

20

30

40 50 I degree / QP

60

70

80

90

Fig. 5. The influences of the flexoelectric coefficient l14 on the reflection and transmission coefficients in the case of incident QP wave. (l11 ¼ 9, l111 ¼ 11, l1 ¼ 0:1, l2 ¼ 0:01, h ¼ 0:2).

227

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

the increasing of the characteristic length l1 enhances the effective elasticity of the material while the increasing of the inertial characteristic length l2 enhances the effective inertance of the material. The influences of the flexoelectric coefficients upon the reflection and transmission coefficients in the case of incident QP wave are shown in Fig. 4, Figs. 5 and 6. It is observed that the flexoelectric effect has evident influences upon all type of reflection and transmission waves. But the influences of the flexoelectric coeffi-

a

cients l11 , l14 and l111 are different from each other. The flexoelectric coefficients l14 and l111 have nearly same influences upon the reflection and transmission waves and have completely opposite influences compared with the flexoelectric coefficient l11 except the reflection QP wave. With regard of the influences upon the reflection QP wave, the flexoelectric coefficients l14 and l11 have nearly same influences while have completely opposite influences compared with the flexoelectric coefficient l111 . Moreover, com-

1.0

b 111 111

0.8

111 111

10 15 20 25

0.08

10 15 20 25

111 111 111

0.06

111

R

R

E QP

E QSV

0.6 0.04

0.4

0.02 0.2

0.0

0.00 0

10

20

30

40

50

60

70

80

90

0

10

20

30

40 50 I / degree QP

60

70

80

90

30

40 50 I / degree QP

60

70

80

90

I QP/degree

c 0.012

d 111 111 111 111

10 15 20 25

1.0

0.8

0.008 0.6

E QP

R

T

E EA

10 15 20 25

111 111 111

0.4

111

0.004 0.2

0.000

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

I QP/degree

e

f

0.18

10 15 20 25

111 111 111 111

0.12

0.005 111 111

0.004

111 111

10 15 20 25

T

T

E EA

E QSV

0.003

0.002 0.06 0.001

0.00

0.000 0

10

20

30

40

50

I QP/degree

60

70

80

90

0

10

20

30

40 50 I / degree QP

60

70

80

90

Fig. 6. The influences of the flexoelectric coefficient l111 on the reflection and transmission coefficients in the case of incident QP wave. (l11 ¼ 9, l14 ¼ 11, l1 ¼ 0:1, l2 ¼ 0:01, h ¼ 0:2).

228

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

pared with the other reflection and transmission waves, the reflection and transmission QSV waves are more sensitive to the flexoelectric effect. This phenomenon can be explained by that the QSV wave is of shortest wave length among all type of waves. Fig. 7 shows the influences of the characteristic length l1 upon the reflection and transmission coefficients in the case of incident QSV wave. In the case of incident QSV wave, there is a critical angle

(about 38.40) which is corresponding with that the reflection and transmission QP waves become the surface waves. At the vicinity of the critical angle, the reflection and transmission coefficients exhibit dramatic change. It is observed that the influences of the strain gradient effect upon the reflection and transmission waves are similar with that in the case of incident QP wave. The influences of the micro-inertial effect upon the reflection and transmis-

b 1.0

4

a

l1 0.02 l1 0.04 l1 0.06 l1 0.08

3

l1 0.02 l1 0.04 l1 0.06 l1 0.08

0.8

E QSV

2

R

R

E QP

0.6

0.4

1 0.2

0

0.0 0

10

20

30

40

50

60

70

80

90

0

10

20

30

40 50 I / degree QSV

60

70

80

90

30

40 50 I / degree QSV

60

70

80

90

I QSV/degree

c 0.4

d4 l1 0.02 l1 0.04 l1 0.06 l1 0.08

3

T

E QP

R

E EA

0.3

l1 0.02 l1 0.04 l1 0.06 l1 0.08

0.2

0.1

2

1

0.0

0 0

10

20

30

40

50

60

70

80

90

0

10

20

I QSV/degree

e

1.0

f

l1 0.02 l1 0.04 l1 0.06 l1 0.08

0.8

0.25

l1 0.02 l1 0.04 l1 0.06 l1 0.08

0.20

0.15

T

T

E EA

E QSV

0.6

0.4

0.10

0.2

0.05

0.0

0.00 0

10

20

30

40

50

I QSV/degree

60

70

80

90

0

10

20

30

40 50 I / degree QSV

Fig. 7. The influences of the characteristic length l1 on the reflection and transmission coefficients in the case of incident QSV wave. (l11 ¼ 9, h ¼ 0:2).

60

70

80

90

l14 ¼ 10, l111 ¼ 11, l2 ¼ 0:01,

229

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

sion waves are also similar to the case of incident QP wave (the corresponding figures are not provided for brevity). Fig. 8 shows the influences of the flexoelectric coefficient l11 upon the reflection and transmission coefficients in the case of incident QSV wave. It is observed that the flexoelectric coefficient l11 has not nearly influences upon the reflection and transmission waves before the critical angle. This can be explained by that the QSV wave is uncoupled with the electric potential when it propa-

gates along the normal direction, and the critical angle is a little bit small, so the influences of the flexoelectric coefficients are nearly void before the critical angle. The influences of the flexoelectric coefficients l14 and l111 are similar with l11 (the corresponding figures are not provided for brevity). In order to validate the numerical results, the energy conservation into and out of the slab is checked for both of incident QP wave and QSV wave and are shown graphically in Fig. 9. The exact

5

a

11 11

4

11 11

b 1.0

10 15 20 25

11

0.8

11 11

0.6

R

R

E QP

E QSV

3

2

0.4

1

0.2

0

0.0 0

c

10 15 20 25

11

10

20

30

40 50 I degree QSV/

60

70

80

90

0

d

0.4 11 11 11

0.3

11

10 15 20 25

10

20

30

40 50 I QSV/degree

60

70

80

90

30

40 50 I degree QSV/

60

70

80

90

6

10 15 20 25

11

5

11 11 11

T QP

0.2

E

R

E EA

4

3

2 0.1 1

0.0

0 0

10

20

30

40

50

60

70

80

90

0

10

20

I QSV/degree

e

f

1.0 11 11

0.8

11 11

10 15 20 25

0.25 11 11

0.20

11 11

0.15

T

T

E EA

E QSV

0.6

10 15 20 25

0.4

0.10

0.2

0.05

0.0

0.00 0

10

20

30

40

50

I QSV/degree Fig. 8. The influences of the flexoelectric coefficient l2 ¼ 0:01, h ¼ 0:2).

60

70

80

90

0

10

20

30

40 50 I / degree QSV

60

70

80

90

l11 on the reflection and transmission coefficients in the case of incident QSV wave. (l14 ¼ 9, l111 ¼ 11, l1 ¼ 0:1,

230

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

a

b

1.01

1.01

l1 0.02 l1 0.04 l1 0.06 l1 0.08

1.00

E

E

l1 0.02 l1 0.03 l1 0.04 l1 0.05

0.99

1.00

0.99 0

10

20

30

40 50 I QP/degree

60

70

80

90

0

10

20

30

40 50 I QSV/degree

60

70

80

90

Fig. 9. Check on the energy flux conservation among the incident wave, reflection waves and transmission waves in the case of incident QP wave (a) and QSV wave (b). (l11 ¼ 9, l14 ¼ 10, l111 ¼ 11, l2 ¼ 0:01, h ¼ 0:2).

energy conservation is reflected by index E ¼ 1. It is observed that the deviation from E ¼ 1 is less than one percent at the total incident angle range. This validates the present numerical results to some extent. 5. Conclusion Although, the wave propagation in the piezoelectric solid is studied extensively, the influences of the microstructure effects upon the wave propagation are rarely reported. In the present work, the reflection and transmission behaviors of wave propagation through a sandwiched slab are considered and the microstructure effects, including the flexoelectric effect, the strain gradient effect and the micro-inertial effect are investigated. Based on the theoretic analysis and the numerical results, some conclusions can be drawn as following. (1) The microstructure effects create the new modes of wave propagation, namely, the P-type surface wave and the Stype surface wave. These new modes are dispersive due to the microstructure effects and will disappear in the classical piezoelectric solids. (2) Although the strain gradient effect and the micro-inertial effect have similar influences upon the reflection and transmission waves, the deviations still exist. This can be explained by the fact that the strain gradient effect enhances the effective elasticity of the material while the inertial effect enhances the effective inertance of the material. (3) The flexoelectric effect has evident influences upon all type of reflection and transmission waves. But the flexoelectric coefficients l14 and l111 have same influences upon the reflection and transmission waves except the reflection QP wave, while the flexoelectric coefficient l11 has opposite influences compared with the flexoelectric coefficients l14 and l111 . With regard of the reflection QP wave, the flexoelectric coefficients l11 and l14 have same influences while have opposite influences when compared with the flexoelectric coefficient l111 . (4) The QSV wave and the EA wave are sensitive to the microstructure effects. This implies that the microstructure effects, including the flexoelectric effect, the strain gradient effect and the micro-inertial effect, can enhance not only the mode transformation but also the electromechanical coupling.

Acknowledgments The work is supported by Fundamental Research Funds for the Central Universities (FRF-BR-15-026A) and National Natural Science Foundation of China (No. 10972029). Appendix A

2

G11

c11 6 0 6 6 6 0 ¼6 6c 6 12 6 4 0 0

G12 ¼ GT21

0

0

c12

0

c11

0

0

c12

0

c11

0

0

0

0

c11

0

c12

0

0

c11

6 6 6 6 G22 ¼ 6 6 6 6 4

c13

0 0 0 0

0

0

0

0 c33

0

0

0

0 0

0 c33 0 0 0 c44

0 0

0

0

0

c11 3 0 0 0 07 7 7 0 07 7; 0 07 7 7 0 05

0

0

c33

3

0 7 7 7 c12 7 7; 0 7 7 7 0 5

0 c12 0 0 2 0 0 c13 0 6 0 c 0 0 13 6 6 6 0 0 c13 0 ¼6 6c 0 0 6 13 0 6 4 0 c13 0 0 0

2

0

0 c44

3

0 7 7 7 0 7 7; 0 7 7 7 0 5

0 0 0 0 0 c44 2 c44 0 0 0 0 6 0 c 0 0 0 44 6 6 6 0 0 c44 0 0 G33 ¼ 6 6 0 0 0 0:5ðc  c Þ 0 11 12 6 6 4 0 0 0 0 0:5ðc11  c12 Þ 0

0

0

0

G13 ¼ G23 ¼ G31 ¼ G32 ¼ ½06
6 :

0

0 0 0 0 0 0:5ðc11  c12 Þ

3 7 7 7 7 7; 7 7 7 5

231

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

Appendix B

Appendix E

 T A0 ¼ 1 Gq Hq inaq inGq aq inM 1q inM2q inM3q M 4q M 5q ;

2

K 11 ¼ c11 þ c44 a2 þ l1 n2 ½c11 þ ðc11 þ c44 Þa2 þ c44 a4   qc2 2

 ql2 n2 c2 ð1 þ a2 Þ=3;

2 2

K 12 ¼ K 21 ¼ ðc13 þ c44 Þa þ ðc13 þ c44 Þl1 n2 að1 þ a2 Þ; K 13 ¼ ðe15 þ e31 Þa  il11 n  iðl14 þ 2l111 Þna2 ; K 22 ¼ c44 þ c33 a þ 2

q

2 l2 n2 c2 ð1

2 l1 n2 ½c44

þ ðc33 þ c44 Þa þ c33 a   qc 2

4

2

þ a Þ=3; 2

K 23 ¼ e15 þ e33 a2  il11 na3  iðl14 þ 2l111 Þna; K 33 ¼ e11  e33 a2 ;

1

1

6 G1 6 6 6 H1 6 6 ina 1 6 6 6 inG1 a1 L A ¼6 6 inM 11 6 6 6 inM21 6 6 inM 31 6 6 4 M41 M51 1

K 31 ¼ ðe15 þ e31 Þa þ il11 n þ iðl14 þ 2l111 Þna2 ;

2

K 32 ¼ e15 þ e33 a2 þ il11 na3 þ iðl14 þ 2l111 Þna:

6 G2 6 6 6 H2 6 6 ina2 6 6 6 inG2 a2 R A ¼6 6 inM 12 6 6 6 inM 22 6 6 inM 32 6 6 4 M 42

Appendix C

F 1q ¼ c44 ðaq þ Gq Þ þ e15 Hq ; F 2q ¼ c13 þ c33 aq Gq þ e33 aq Hq ; F 3q ¼ c11 þ c13 aq Gq þ e31 aq Hq ; F 4q ¼ e31 þ e33 aq Gq  e33 aq Hq þ in½l11 Gq a2q þ l14 aq þ l111 ðaq þ Gq Þ;

F 6q ¼ inl111 Hq aq 

2 l1 n2 c44 ðGq

1

G3

G5

G7

H3

H5

H7

ina3

ina5

ina7

inG5 a5 inM 15

inG7 a7 inM17

inM 23

inM 25

inM27

inM 33

inM 35

inM37

M 43

M 45

M47

M 53 1

M 55 1

M58 1

G4

G6

G8

M59 1

þ aq Þ;

H4

H6

H8

ina4

ina6

ina8

inG4 a4 inM 14

inG6 a6 inM16

inG8 a8 inM18

inM 24

inM26

inM28

inM 34

inM36

inM38

M 44

M46

M48

M 54

M56

M58

 A0 ¼ 1 Gq

Hq

inF 1q

2

1

1

2

F 8q ¼ inl14 Hq  l1 n2 ðc13 þ c33 Gq aq Þ; 2

F 9q ¼ inl11 Hq aq  l1 n2 ðc13 þ c33 Gq aq Þaq ;

F 11q ¼ inl

a

c13 Gq aq Þ; þ c13 Gq aq Þaq :

Appendix D

M5ð10Þ

inF 2q

inF 4q

T

1

3

6 G 6 1 6 6 H1 L A ¼6 6 inF 11 6 6 4 inF 21

H3 inF 13 inF 23

G5 7 7 7 H5 7 7; inF 15 7 7 7 inF 25 5

inF 41

inF 43

inF 45

G3

2

;

1

1

1

3

6 G 6 2 6 6 H2 R A ¼6 6 inF 12 6 6 4 inF 22

H4 inF 14 inF 24

G6 7 7 7 H6 7 7: inF 16 7 7 7 inF 26 5

inF 42

inF 44

inF 46

G4

3 1 0 0 0 0 6 0 1 0 0 0 7 7 6 7 6 6 þ inðf þ f Þ e þ inðf þ f Þ  e f f e 333 33 M11 ¼ 6 313 3113 3131 3313 3331 3133 3333 7 7; 2 2 2 2 7 6 inl1 c1311 inl1 c1331 f 3133 l1 c1313 l1 c1333 5 4 2

2

2 l2 n2 c2 Gq

M 2q ¼ F 2q  F 8q  F 9q aq  F 7q  q

aq =3; M3q ¼ F 4q ;

M 4q ¼ F 7q ; M5q ¼ F 9q :

2

2

inl1 c3311

M21

7 7 7 H10 7 7 ina10 7 7 7 inG10 a10 7 7: inM1ð10Þ 7 7 7 inM2ð10Þ 7 7 inM3ð10Þ 7 7 7 M4ð10Þ 5

Appendix F

M 1q ¼ F 1q  F 6q  F 7q aq  F 11q  ql2 n2 c2 aq =3;

2

3

G10

2

F 10q ¼ inl

7 7 7 7 7 ina9 7 7 7 inG9 a9 7 7; inM 19 7 7 7 inM 29 7 7 inM 39 7 7 7 M49 5 G9

H9

When the microstructure effects are ignored, these matrixes reduce to

F 7q ¼ inl111 Hq  l1 n2 c44 ðGq þ aq Þaq ;

2 2 11 Hq  l1 n ðc11 þ 2 2 14 Hq q  l1 n ðc 11

3

1

inG3 a3 inM 13

M 52

F 5q ¼ e15 ðaq þ Gq Þ  e11 Hq þ in½l11 þ l14 Gq aq þ l111 ðaq þ Gq Þaq ;

1

2

inð1 þ l1 n2 Þc1113

6 2 6 inð1 þ l1 n2 Þc3113 6 6 ¼6 ine113  n2 f 1113  n2 f 1131 6 2 2 2 6 c 4 1313  l1 n ðc1313 þ c1111 Þ þ ql2 x2 =3 2

c3313  l1 n2 ðc3313 þ c3111 Þ

2

ine311 þ n2 f 3111

0

2

ine331 þ n2 f 3311

0

ine13

inf 1133

ine133  n2 f 1313  n2 f 1331 2

c1333  l1 n2 ðc1333 þ c1131 Þ 2

2

f 3333 l1 c3313 l1 c3333

inð1 þ l1 n2 Þc1133 inð1 þ l1 n2 Þc3133

2

2

inl1 c3331

2

e313 þ inðf 3131 þ f 3113 Þ il1 nc1113 2

c3333  l1 n2 ðc3333 þ c3131 Þ þ ql2 x2 =3 e333 þ inðf 3331 þ f 3313 Þ il1 nc3113

0

3

7 7 7 7 inf 1333 7; 7 2 il1 nc1133 7 5 0

2

il1 nc3133

232

F. Jiao et al. / Ultrasonics 82 (2018) 217–232

2

N11

0 0

0 0

6 6 6 2 2 ¼6 ine31 6 ðine311  n f 3111 Þ ðine331  n f 3311 Þ 6 0 0 inf 1133 4 0

2

N21

6 6 6 6 ¼6 6 6 4

2

inf 1333

0

2

2

"

0

B ðhÞ ¼

1 B11 ðhÞ  B12 ðhÞB1 42 ðhÞB41 ðhÞ B13 ðhÞ  B12 ðhÞB42 ðhÞB43 ðhÞ 1 B31 ðhÞ  B32 ðhÞB1 42 ðhÞB41 ðhÞ B33 ðhÞ  B32 ðhÞB42 ðhÞB43 ðhÞ

2

n2 ð1 þ l1 n2 Þc1131

n2 ð1 þ l1 n2 Þc3111

2

n2 ð1 þ l1 n2 Þc3131  qx2  ql2 n2 x2 =3

n2 ðe111 þ inf 1111 Þ

n2 ðe131 þ inf 1311 Þ

inð1 þ

0 0 0

0

0

2

2 l1 n2 Þc1311 2 l1 n2 Þc3311

3

2

2

2 2

inð1 þ l1 n2 Þc3331

0

0

0 0 0

3

b11 ðhÞ 6 B11 ðhÞ ¼ 4 b21 ðhÞ b31 ðhÞ 2 b16 ðhÞ 6 B13 ðhÞ ¼ 4 b26 ðhÞ

b22 ðhÞ b32 ðhÞ b17 ðhÞ b27 ðhÞ

b36 ðhÞ b37 ðhÞ 

2 b14 ðhÞ 7 6 b23 ðhÞ 5; B12 ðhÞ ¼ 4 b24 ðhÞ b34 ðhÞ b33 ðhÞ 3 2 b19 ðhÞ b18 ðhÞ 7 6 b28 ðhÞ 5; B14 ðhÞ ¼ 4 b29 ðhÞ b39 ðhÞ b38 ðhÞ

b12 ðhÞ b13 ðhÞ

3

b15 ðhÞ

3

7 b25 ðhÞ 5; b35 ðhÞ 3 b1ð10Þ ðhÞ 7 b2ð10Þ ðhÞ 5; b3ð10Þ ðhÞ

   b44 ðhÞ b45 ðhÞ ; B22 ðhÞ ¼ ; b51 ðhÞ b52 ðhÞ b53 ðhÞ b54 ðhÞ b55 ðhÞ " #   b49 ðhÞ b4ð10Þ ðhÞ b46 ðhÞ b47 ðhÞ b48 ðhÞ B23 ðhÞ ¼ ; B24 ðhÞ ¼ ; b59 ðhÞ b5ð10Þ ðhÞ b56 ðhÞ b57 ðhÞ b58 ðhÞ B21 ðhÞ ¼

2

b41 ðhÞ b42 ðhÞ b43 ðhÞ

b61 ðhÞ b62 ðhÞ b63 ðhÞ

3

2

b64 ðhÞ b65 ðhÞ

3

6 7 6 7 B31 ðhÞ ¼ 4 b71 ðhÞ b72 ðhÞ b73 ðhÞ 5; B32 ðhÞ ¼ 4 b74 ðhÞ b75 ðhÞ 5; b81 ðhÞ b82 ðhÞ b83 ðhÞ b84 ðhÞ 2 3 b69 ðhÞ b66 ðhÞ b67 ðhÞ b68 ðhÞ 6 6 7 B33 ðhÞ ¼ 4 b76 ðhÞ b77 ðhÞ b78 ðhÞ 5; B34 ðhÞ ¼ 4 b79 ðhÞ b89 ðhÞ b86 ðhÞ b87 ðhÞ b88 ðhÞ 2

"

0 0

#

:

3

7 0 07 7 7 n2 e11 0 0 7; 7 7 ine113 þ n2 ðf 1131 þ f 1113 Þ 0 0 5 n2 ðe131  inf 1311 Þ

ine133 þ n2 ðf 1331 þ f 1313 Þ 0 0

References

7 0 0 07 7 0 0 07 7: 7 0 0 05 0 0 0

Appendix G

2

n2 ðe111  inf 1111 Þ

inð1 þ l1 n2 Þc1331

7 6 6 0 60 0 0 0 07 6 0 7 6 6 7 6 ¼ 6 0 0 1 0 0 7; N22 ¼ 6 0 0 6 7 6 6 40 0 0 1 05 4 1 0 0 0 0 0 1 0 1

b85 ðhÞ 3 b6ð10Þ ðhÞ 7 b7ð10Þ ðhÞ 5; b8ð10Þ ðhÞ

# " # b91 ðhÞ b92 ðhÞ b93 ðhÞ b94 ðhÞ b95 ðhÞ ; B42 ðhÞ ¼ ; bð10Þ1 ðhÞ bð10Þ2 ðhÞ bð10Þ3 ðhÞ bð10Þ4 ðhÞ bð10Þ5 ðhÞ " # b96 ðhÞ b97 ðhÞ b98 ðhÞ B43 ðhÞ ¼ ; bð10Þ6 ðhÞ bð10Þ7 ðhÞ bð10Þ8 ðhÞ " # b99 ðhÞ b9ð10Þ ðhÞ B44 ðhÞ ¼ ; bð10Þ9 ðhÞ bð10Þð10Þ ðhÞ B41 ðhÞ ¼

where bij is the element of BðhÞ at i-th line and j-th column. B0 ðhÞ can be obtained by

n2 ð1 þ l1 n2 Þc1111  qx2  ql2 n2 x2 =3

inð1 þ

N12

3 1 0 7 0 17 7 0 07 7; 7 0 05 0 0

0 0

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