Ultrasonic Guided Waves in Piezoelectric Layered Composite with Different Interfacial Properties

Abstract

Combining the propagation model of guided waves in a multilayered piezoelectric composite with the interfacial model of rigid, slip, and weak interfaces, the generalized dispersion characteristic equations of guided waves propagating in a piezoelectric layered composite with different interfacial properties are derived. The effects of the slip, weak, and delamination interfaces in different depths on the dispersion properties of the lowest-order mode ultrasonic guided wave are analyzed. The theory would be used to characterize the interfacial properties of piezoelectric layered composite nondestructively.

1. Introduction

The acoustic wave propagation in piezoelectric materials has received considerable attention from engineering and scientific communities involved in nondestructive evaluation and transducer design. Numerous ultrasonic devices using piezoelectric materials are currently being developed for a variety of applications, such as SAW filters, ultrasonic transducers, and a variety of physical, chemical, and biological sensors [1, 2]. In aerospace and other structural applications, piezoelectric layered smart or intelligent composites are being used [3]. Many of these applications involve layered structures made up of piezoelectric materials.

Layered composite materials are usually made of different materials. Very thin adhesive interfacial layer serves as the bonding between adjacent layers. Some imperfections, such as cracks and small voids, often emerge at the interface and contribute to the fatigue of the composite. The interfacial property between layers is one of the key factors that determine the structural stability of composites. It is, therefore, very important to characterize the adhesive layer quality, which has been a main topic in nondestructive testing and evaluation of composite materials [4–8].

Ultrasonic wave is a mechanical wave, which reflects the elastic properties of the medium. The interfacial mechanical property is closely related to the coupling strength between layers in the structure and the level of acoustic impedance match between the bonding layer and the piezoelectric material. Because of its deep penetration ability, ultrasonic technique is among the most promising nondestructive evaluation methods for the interfacial layer characterization. There are some papers considering the wave propagation in this structure with perfect interface, that is, the stress and displacement across the interface are continuous [9–11].

This paper is concerned with the guided waves propagating in piezoelectric layered composite with different interfacial properties. First, the transfer matrix of the piezoelectric layer is derived. Then, based on the model of piezoelectric layered composite, the general dispersion equations of the guided waves propagating in it with rigid, slip, and weak interfaces are presented by using the transfer matrix techniques. Numerical analysis has been performed on typical Y-cut, Z-propagating lithium niobate multi-layered composite with slip, weak, and delamination interface in different depth. It is shown that the interface property and the interfacial defect depth have strong effect on the guided wave phase velocity. The theoretical work in this paper is intended to provide fundamental principles in nondestructive testing and evaluation of the interface of piezoelectric layered composite.

2. Transfer Matrix of Piezoelectric Layer

The transfer matrix method has long been used in modeling multi-layered structures [12]. The transfer matrix expresses the relationship of characterized parameters between the top and the bottom interface of a layer. The order of transfer matrix is determined by parameters of the layer material. To a piezoelectric layer, the transfer function consists of an 8∗8 matrix. Figure 1 shows the labeling system used in this paper. The waves propagate along the 𝑥₁ direction of a piezoelectric layer with its poling direction along 𝑥₃. We consider a two-dimensional problem assuming all field components are constant in the 𝑥₂ direction.

Figure 1 

Labeling system and coordinate system of a piezoelectric multi-layered composite.

In the piezoelectric plate, the coupled wave equations using the quasistatic approximation are𝜌𝜕²𝑢_𝑗𝜕𝑡²−𝑐_{𝑖𝑗𝑘𝑙}𝜕²𝑢_𝑘𝜕𝑥_𝑖𝑥_𝑙−𝑒_𝑘𝑖𝑗𝜕²𝜙𝜕𝑥_𝑖𝑥_𝑘𝑒=0,_𝑖𝑘𝑙𝜕²𝑢_𝑘𝜕𝑥_𝑖𝑥_𝑙−𝜀_𝑖𝑘𝜕²𝜙𝜕𝑥_𝑖𝑥_𝑘=0,(1) in which 𝜌 is the density of material, 𝑢_𝑘 is the particle displacement vector, and 𝜙 is the electric potential. 𝑐_{𝑖𝑗𝑘𝑙}, 𝑒_𝑖𝑗𝑘, and 𝜀_𝑖𝑘 are, respectively, the tensor components of the elastic stiffness measured at constant electric field, the piezoelectric constant, and the dielectric permittivity measured at constant strain. The subscripts 𝑖,𝑗,𝑘, and 𝑙 range over the values 1, 2, and 3, and the summation convention on repeated subscripts is employed. The solution of the form is assumed as𝑢𝜙_𝑗==𝛼𝛼_𝑗4exp𝑖𝑘𝑏𝑥₃exp𝑖𝑘𝑥₁−𝑣𝑡,exp𝑖𝑘𝑏𝑥₃exp𝑖𝑘𝑥₁−𝑣𝑡,(2) where 𝑏 is the attenuation coefficient. Substituting (2) into (1), we obtainΓ_𝑖𝑗−𝛿^𝑖𝑗𝜌𝑣²𝛼_𝑗=0,(3) where 𝑣 is the phase velocity of the wave. The symmetric matrix Γ is given byΓΓ_{𝑖𝑗𝑖4}=𝑐_3𝑖3𝑗𝑏²+𝑐_1𝑖3𝑗+𝑐_3𝑖1𝑗𝑏+𝑐_1𝑖1𝑗,Γ₄₄=𝑒_3𝑖3𝑏²+𝑒_1𝑖3+𝑒_3𝑖1𝑏+𝑒_1𝑖1,=−𝜀₃₃𝑏²+𝜀₁₃+𝜀₃₁𝑏+𝜀₁₁.(4) Equation (3) has a nontrivial solution only if ||Γ_𝑖𝑗−𝛿^𝑖𝑗𝜌𝑣²||=0.(5) For every value of 𝑣, (5) has eight eigenvalues of 𝑏. There are also eight four-component eigenvectors corresponding to the eight eigenvalues. Thus, the general solution can be written as𝑢_𝑗=×ex_𝑛p𝐶𝑖_𝑛𝑘𝛼𝑥_𝑗(𝑛)1exp𝑖𝑘𝑏^(𝑛)𝑥₃𝜙=_𝑛−𝑣𝑡,×exp𝐶𝑖_𝑛𝑘𝛼𝑥_4(𝑛)1exp𝑖𝑘𝑏^(𝑛)𝑥₃−𝑣𝑡,𝑛=1∶8,(6) where 𝐶_𝑛 is the weighting coefficients.

The normal components of stress and electric displacement of piezoelectric materials are𝜎_3𝑗=𝑐_{3𝑗𝑘𝑙}𝐷₃𝜕𝜕𝑢𝑥^𝑘_𝑙+𝑒^𝑘3𝑗𝜕𝜕𝑥𝜙_𝑘,=𝑒_3𝑘𝑙𝜕𝜕𝑢𝑥^𝑘_𝑙−𝜀^3𝑘𝜕𝜕𝑥𝜙_𝑘.(7)

Assume that the displacement, stress, electric displacement, and potential are known at the top interface of the plate. According to (6) and (7), the weighting coefficients can be written as⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐶₁𝐶₂𝐶₃𝐶₄𝐶₅𝐶₆𝐶₇𝐶₈⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦=[𝑀]_−1top⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_top.(8)

At the bottom interface, the displacement, stress, electric displacement, and potential are⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_bottom=[𝑀]_bottom⋅[𝑀]⁻¹_top⋅⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢¹𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_top.(9) The matrix product in this equation now relates the characterized parameters between the top and the bottom interface of a piezoelectric layer and is defined as the transfer matrix [𝑃],[𝑀]_bottom[𝑀]⁻¹_top=[𝑃].(10)

3. Wave Equations in Piezoelectric Layered Composite

The model and the coordinate system of the piezoelectric layered composite are shown in Figure 1. Waves propagate along the 𝑥₁ direction. In the transfer matrix method, the intermediate interfaces are eliminated so that the fields in all layers are described solely in terms of external boundary conditions. The displacement, stress, electric displace, and potential must be continuous across a rigid interface between two layers; ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗+1,top=[𝑃]_𝑗⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗,top.(11) This process can be continued layer by layer, resulting in the follwing equation;⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑛,bottom=[𝑄]⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢¹𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_1,top,(12) where 𝑛 is the last layer and [𝑄] is the system matrix consisting of the matrix product of layer matrices;[𝑄]=[𝑃]₁⋅[𝑃]₂⋯[𝑃]_𝑛−1⋅[𝑃]_𝑛.(13) The wave must satisfy the appropriate electrical and mechanical boundary conditions on both surfaces of the composite. The top and bottom of composite are free. So the normal stresses at the extreme interfaces (top and bottom) must be zero. Then the mechanical boundary conditions are𝜎𝜎₃₁_𝑛,bottom=𝜎₃₁_1,top=0,(14)₃₂𝜎_𝑛,bottom=𝜎₃₂_1,top=0,(15)₃₃_𝑛,bottom=𝜎₃₃_1,top=0.(16) The metallization on plate surfaces is assumed to consist of a perfectly conducting film of negligible thickness. Thus, the presence of the metal film does not alter the mechanical boundary conditions. However, the electrical boundary conditions vary according to whether it is open (unmetallized) or shorted (metallized). We consider the case where both surfaces are open circuit condition. The electrical boundary conditions are𝐷𝐷₃_𝑛,bottom=𝑘𝜀₀[𝜙]_𝑛,bottom,(17)₃_1,top=−𝑘𝜀₀[𝜙]_1,top,(18) where 𝜀₀ is the dielectric constant of the vacuum. According to (14)–(17), (12) can be written as[𝐴]⎡⎢⎢⎢⎢⎢⎢⎣𝑢¹𝑢₂𝑢₃𝜙⎤⎥⎥⎥⎥⎥⎥⎦_1,top[𝐴]=⎡⎢⎢⎢⎢⎢⎢⎣𝑄=0,₄₁𝑄₅₁𝑄₄₂𝑄₄₃𝑄₄₈1+𝑘𝜀₀𝑄₆₁𝑄₅₂𝑄₅₃𝑄₅₈1+𝑘𝜀₀𝑄₇₁𝑄₆₂𝑄₆₃𝑄₆₈1+𝑘𝜀₀−𝑘𝜀₀𝑄₈₁𝑄₇₂−𝑘𝜀₀𝑄₈₂𝑄₇₃−𝑘𝜀₀𝑄₈₃𝑄₇₈−𝑘𝜀₀𝑄₈₈1+𝑘𝜀₀⎤⎥⎥⎥⎥⎥⎥⎦.(19)For this equation to be satisfied, the submatrix must be singular, ||𝐴||=0.(20) Equation (20) is the dispersion equation of the piezoelectric layered composite with rigid interfacial condition.

If there is a slip interface between the 𝑗th and the (𝑗+1)th layers, the interface can only transfer normal displacement and stress. The normal displacement and stress of two layers are continuous, and the shear stresses are zero. So the interfacial boundary conditions are𝑢₃𝜎_{𝑗,bottom31}=𝑢₃_𝑗+1,top,𝜎_,𝑗,bottom=0,₃₁𝜎_𝑗+1,top=0,₃₂𝜎_𝑗,bottom=0,₃₂𝜎_j+1,top=0,₃₃𝐷₃_j,bottom=𝜎₃₃_j+1,top,[𝜙]_{j,bottomj,bottom}=𝐷₃_j+1,top,=[𝜙]_𝑗+1,top.(21) The other interfaces of the composite are all rigid, therefore⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢₁𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑛,bottom=[𝐷]⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢¹𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗+1,top,(22) where[𝐷]=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢[𝑃₁]_𝑗+1⋅[𝑃]_𝑗+2…[𝑃]_𝑁−1⋅[𝑃]_𝑛,𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_𝑗,bottom=[𝐸]⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑢¹𝑢₂𝑢₃𝜎₃₁𝜎₃₂𝜎₃₃𝐷₃𝜙⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦_1,top,(23) where[𝐸]=[𝑃]₁⋅[𝑃]₂…[𝑃]_𝑗−1⋅[𝑃]_𝑗.(24) By combining (22)–(24) with (21), the dispersion equation of the piezoelectric layered composite with a slip interfacial condition becomes||||||||||||𝐷₄₁𝐷₄₂𝐹₁₁𝐹₁₂𝐷₅₁𝐷₅₂𝐹₂₁𝐹₂₂𝐷₆₁𝐷₆₂𝐹₃₁𝐹₃₂𝐷^∗71𝐷^∗72𝐹₄₁𝐹₄₂||||||||||||=0,(25) where⎡⎢⎢⎢⎢⎢⎢⎣𝐹₁₁𝐹₁₂𝐹₂₁𝐹₂₂𝐹₃₁𝐹₃₂𝐹₄₁𝐹₄₂⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣𝐷₄₃𝐷₄₆𝐷₄₇𝐷₄₈𝐷₅₃𝐷₅₆𝐷₅₇𝐷₅₈𝐷₆₃𝐷₆₆𝐷₆₇𝐷₆₈𝐷^∗73𝐷^∗76𝐷^∗77𝐷^∗78⋅⎡⎢⎢⎢⎢⎢⎢⎣𝐸₃₁⎤⎥⎥⎥⎥⎥⎥⎦𝐸₃₂𝐸₃₃𝐸^∗38𝐸₆₁𝐸₆₂𝐸₆₃𝐸^∗68𝐸₇₁𝐸₇₂𝐸₇₃𝐸^∗78𝐸₈₁𝐸₈₂𝐸₈₃𝐸^∗88⎤⎥⎥⎥⎥⎥⎥⎦⋅⎡⎢⎢⎢⎢⎢⎢⎣𝐺1001₁₁𝐺₁₂𝐺₂₁𝐺₂₂⎡⎢⎢⎣𝐺₁₁⎤⎥⎥⎥⎥⎥⎥⎦,𝐺₁₂𝐺₂₁𝐺₂₂⎤⎥⎥⎦=⎡⎢⎢⎣−𝐸₄₃−𝐸^∗48−𝐸₅₃−𝐸^∗58⎤⎥⎥⎦⁻¹⋅⎡⎢⎢⎣𝐸⁴¹𝐸₄₂𝐸₅₁𝐸₅₂⎤⎥⎥⎦.(26) The rows 1 to 6 and 8 of the matrix [𝐷]^∗ are the same as those of matrix [𝐷]. The row 7 of matrix [𝐷]^∗ satisfies𝐷^∗7𝑗=𝐷_7𝑗−𝑘𝜀₀𝐷_8𝑗.(27) The columns 1 to 6 and 8 of matrix [𝐸] are the same as those of matrix [𝐸]^∗. The column 7 of the matrix [𝐸] satisfies𝐸^∗𝑗7=𝐸_𝑗7+𝑘𝜀₀𝐸_𝑗8.(28) If there is a weak interface between the 𝑗th and the (𝑗+1)th layers, the “spring” model can be used to characterize the interface property [13, 14]. So the interfacial boundary conditions are𝜎𝜎₃₃₃₁_𝑗,bottom=𝜎₃₃_𝑗+1,top=𝐾_𝑛𝑢³_𝑗,bottom−𝑢₃_𝑗+1,top,𝜎₃₂_𝑗,bottom=𝜎₃₁_𝑗+1,top=𝐾_𝑡1𝑢¹_𝑗,bottom−𝑢₁_𝑗+1,top,_𝑗,bottom=𝜎₃₂_𝑗+1,top=𝐾_𝑡2𝑢²_𝑗,bottom−𝑢₂_𝑗+1,top,(29) where 𝐾_𝑛, 𝐾_𝑡1, and 𝐾_𝑡2 are the normal stiffness coefficient, the shear stiffness coefficient of the 𝑥₁-dimension and 𝑥₂-dimension, respectively.

The dispersion equation of the piezoelectric layered composite with a weak interfacial condition is given by||||||||||||𝑍₄₁𝑍₄₂𝑍₄₃𝑍₄₈𝑍₅₁𝑍₅₂𝑍₅₃𝑍₅₈𝑍₆₁𝑍₆₂𝑍₆₃𝑍₆₈𝑍₇₁𝑍₇₂𝑍₇₃𝑍₇₈||||||||||||=0,(30) where[𝐻]=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣100−1𝐾_𝑡10100−01000𝐾_𝑡200100−0100𝐾_𝑛00000[𝑍00000]=00000[𝐷10000]010000010000010000001_∗0⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,⋅[𝐻]⋅[𝐸]^∗.(31)

4. Numerical Results and Discussion

The analysis presented in the previous section has been applied to calculate the characteristics of the 𝐴₀ and 𝑆₀ modes propagating in a Y-cut, Z-propagating lithium niobate layered composite. The material constants of the LiNiO₃ come from [15]. Since this paper deals with the effect of the interfacial properties of composite on the dispersion characteristics of the guided waves propagating, it is assumed that each layer has the same material, crystal type, cut direction, and thickness. The piezoelectric plate has 40 layers, and the thickness of each layer is 25 μm. In this paper, the structure with a defective interface is represented as m/n. The left and right number of the symbol “/” represent the number of layers on and under the defective interface.

Figure 2 shows the Lamb wave phase velocities of the 𝐴₀ and 𝑆₀ modes as a function of frequency with a delamination interface in different depth. The dispersion characteristic of 𝐴₀ and 𝑆₀ modes in an YZ LiNiO₃ plate looks very similar to that of the lowest order Lamb waves propagating in a plate of isotropic plate [16]. However, It should be noted that 𝐴₀ and 𝑆₀ modes in piezoelectric composite have all three components of particle displacement, that is, the waves are more general Lamb waves. As the frequency increases, the 𝐴₀ and 𝑆₀ mode velocities tend asymptotically to the SAW velocity. Further, it can be shown that the phase velocity of the 𝐴₀ mode increases when the depth of the delamination interface increases while the phase velocity of the 𝑆₀ mode decreases when the depth of the delamination interface increases. Figure 3 shows the Lamb wave phase velocity of the 𝐴₀ mode as a function of frequency with a slip interface in different depth. Except for the 20/20 structure, the phase velocity does not tend asymptotically to a constant velocity when the frequency increases. If the frequency less than 2.5 MHz is considered, the phase velocity of the Lamb wave decreases with the depth when it is less than one-half of the plate thickness, but increases with the depth when it is more than one-half of the plate thickness.

Figure 2 

The 𝐴0 mode dispersion curve of YZ LiNiO3 multi-layered plate with a delamination interface in different depth.

Figure 3 

The 𝐴0 mode dispersion curve of YZ LiNiO3 multi-layered plate with a slip interface in different depth.

When the interface is weak, the stiffness 𝐾_𝑛 and 𝐾_𝑡𝑖 are finite values. Figure 4 shows the Lamb wave phase velocity of the 𝐴₀ mode as a function of frequency with a weak interface in different depth. It is assumed that 𝐾_𝑛 and 𝐾_𝑡2 are infinite, and that 𝐾_𝑡1 is 5 × 10¹⁵ N/m³. Like the condition of the slip interface, the phase velocity does not show saturation when the frequency increases, except for the 20/20 structure. From Figures 3 and 4, one can find that the dispersion characteristic with a weak interface is between that of a slip interface in the same depth and that of the ideal rigid interface. Furthermore, the dispersion characteristic with a weak interface in the central plate is the same as that of a slip interface.

Figure 4 

The 𝐴0 mode dispersion curve of YZ LiNiO3 multi-layered plate with a weak interface in different depth.

Figure 5 shows the 𝐴₀ mode phase velocity versus the defect depth at 1 MHz. When the central interface of the composite has a delamination interface, the phase velocity is the same as that when it has a slip interface. When the depth of the delamination interface is less than half of the thickness of the composite, the phase velocity is less than that of a slip interface in the same depth, and vice versa. The phase velocity with a delamination and slip interface is less than that with a weak interface in the same depth.

Figure 5 

The phase velocity of 𝐴0 mode versus the default depth with the 1 MHz frequency.

Figure 6 presents the effect of the interface stiffness constants 𝐾_𝑡1on the phase velocity of the 𝐴₀ mode in the 10/30 composite structure. In the calculation 𝑓 is equal to 1 MHz, and 𝐾_𝑛 and 𝐾_𝑡2 are infinite. It reveals that the phase velocity of the 𝐴₀ mode increases with the increasing 𝐾_𝑡1. When 𝐾_𝑡1 approaches zero, the phase velocity approaches that of a composite with a slip interface. When 𝐾_𝑡1 approaches infinity, the phase velocity approaches to that of a composite with a rigid interface. Very large variation of the phase velocity of the 𝐴₀ mode can be seen when 𝐾_𝑡1 varies in the range of 10¹²–10¹⁵ N/m³.

Figure 6 

The effect of the interface stiffness constants 𝐾𝑡1 (N/m3) on phase velocity of 𝐴0 mode.

Figure 7 shows the Lamb wave phase velocity of the 𝑆₀ mode as a function of frequency with a rigid, slip, and weak interface in the 10/30 composite structure. Unlike 𝐴₀ mode, the interface properties have very little effect on the phase velocity of 𝑆₀ mode.

Figure 7 

The 𝑆0 mode dispersion curve of YZ LiNio3 multi-layered plate with a default interface in 10/30 structure.

5. Conclusions

We have conducted a theoretical investigation of the velocity variation of guided waves propagating in piezoelectric multi-layered composite with different interfacial properties. General dispersion equations for a composite that contains rigid, slip, and weak interfaces are derived by using the transfer matrix technique. The phase velocities of the lowest antisymmetry and symmetry modes have been numerically computed on a YZ lithium niobate layered composite with slip, weak, and delamination interface in different depth. As the depth of the defective interface changes, the velocity of the lowest anti-symmetry mode changes drastically. But the velocity of the lowest symmetry mode is almost not effected by the interface characteristics. When the interface stiffness coefficient varies in the range of 10¹²–10¹⁵ N/m³, the velocity of the lowest anti-symmetry mode has the largest change.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (10904073), the Natural Science Research Project of Jiangsu Higher Education Institutions (10 KJB510011), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.