EFFECT OF UNIAXIAL INITIAL STRESSES, PIEZOELECTRICITY AND THIRD ORDER ELASTIC CONSTANTS ON THE NEAR-SURFACE WAVES IN A STRATIFIED HALF-PLANE

Dispersion curves of a system consisting of a piezoelectric covering layer and metal half-plane under the uniaxial initial stresses perpendicular to the wave propagation direction are obtained within the scope of the Threedimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies. The elasticity relations of the metal half-plane material are described by the Murnaghan potential. The numerical results are discussed for PZT-2 covering layer and aluminum half-plane material. The influence of the initial stresses, the piezoelectricity of the covering layer, and the third order elastic constants on the near-surface wave propagation velocity is illustrated.


INTRODUCTION
Initial stresses can arise from the material thermal expansion, in the absence of applied loads (residual stress), and intentionally to improve the material performance or unintentionally from the manufacturing process. The effect of initial stresses on wave propagation in a piezoelectric stratified half-plane system is widely studied.
Propagation behavior of Love waves in a layered piezoelectric half-plane with initial stresses has been studied for electrically open and short cases. Results have shown the important effect of initial stress on the Love wave propagation and stress distribution [1]. Also same system model under the slowly varying inhomogeneous initial stresses has been studied to show the effect of initial stresses on dispersion curves of Love waves [2]. In a system consisting of an isotropic layer, a fiber-reinforced layer and isotropic half-plane under the influence gravity dispersion equation of Love waves have been investigated [3]. The combined effect of initial stress and finite deformation on the wave propagation velocity of Love waves at the boundary between a layer and a half-space is discussed [4]. According to the results, wave propagation velocity increases under the stretching stress and decreases under the compression stress with increasing dimensionless wavenumber kh .
Quasi-longitudinal waves under initial stresses in a system consisting of two facing half-planes have been investigated, and the results have shown that initial stresses cause a significant change in the velocity of surface waves [5]. The effect of initial stress on the reflection coefficients has been studied for an initially stressed piezoelectric half plane [6].
Torsional waves in a system consisting of a magneto-viscoelastic layer and an inhomogeneous half-plane with linear variation in rigidity and density have been investigated [7]. Furthermore, in a system consisting of an initially stressed inhomogeneous layer and an a linearly varying inhomogeneous half-space the propagation of torsional surface waves. According to this study the inhomogeneity of the constituents of the system and the initial stress affects the torsional wave propagation velocity considerably [8]. Torsional wave propagation velocity decreases with the increasing compressional initial stress and increasing with increasing dimensionless wavenumber kh .
However, these studies have not taken into consideration the third order elastic constants. Longitudinal wave propagation velocity in the two-layered circular hollow cylinder with the aluminum, steel and tungsten material layers has been studied under the uniaxial initial stretching and compression stress [9]. According to this study, the third order elastic constants of the materials affects considerably the axisymmetric wave propagation velocity.
The influence of the third order elastic constants on near-surface wave propagation velocity has been shown in a metal elastic stratified half-plane [10]. Near-surface waves in metal half-plane covered with piezoelectric layer have been investigated, and the dispersion curves of the system have been obtained [11]. The effect of initial stresses along the wave propagation direction on the near surface wave dispersion curves considering the third order elastic constants for this system is studied [12]. For this system, results have been developed for different pairs of materials: PZT-2, PZT-4, PZT-6B for the covering layer; steel, aluminum for the half-plane in the electrically open and short cases [13]. In a PZT/Metal/PZT sandwich plate, Lamb waves have been investigated for the system both in the presence and absence of the initial stresses [14,15].
In this study, a system consisting of a piezoelectric covering layer and a metal elastic half-plane under the uniaxial initial stresses perpendicular to the wave propagation direction is investigated. The effect of initial stresses, the piezoelectricity of the covering layer, and the third order elastic constants on near-surface wave propagation velocity for this system is evaluated.

MATHEMATICAL FORMULATION OF THE PROBLEM
The considered system consisting of a metal elastic half-plane covered by a piezoelectric layer with thickness h is subjected uniaxial initial stresses perpendicular to the interface between the constituents as shown in Figure 1. We associate the interface plane between the covering layer and half-plane with the Lagrangian coordinate system 1 2 3 Ox x x which in the natural state coincide with Cartesian coordinates. The piezoelectric covering layer and elastic half-plane occupy the domains  The initial stresses in the covering layer and half-plane are defined within the framework of the classical linear theory of the electro-elasticity as follows: (1),0 (1),0 (1),0 22 1 21 11 According to [16,17], the equations of motion within the scope of classical linear theory of electroelasticity are as follows: where () Ox axis. Constitutive equations of piezoelectric covering layer can be written as follows: (1) (1) (1 where the ( (1) f  are elasticity constants, piezoelectric constants, dielectric constants, and electric potential, respectively.
The material of the half-plane is non-linear pure elastic. Therefore, the motion of the half-plane can be described with the first and second equation in Eq. (2). The constitutive equations of the half-plane material are given by Murnaghan potential ( Φ ).
where (2) λ and (2) μ are Lamé constants; (2) a , (2) b , and (2) c are the third order elasticity constants determined by ultrasonic methods; ( (2) 3 A are the first three invariants of the strain tensor and they are as follows: In Eq. (5), the components of the strain tensor are as follows: The components of the stress tensor can be defined as follows: Linearizing the Eq. (7), the constitutive equations of the elastic half-plane can be written as follows: (2) 12 1 The complete contact conditions exist between the covering layer and half-plane ( The boundary condition for the electric potential is as follows: The boundary condition for the electric displacement is as follows: The boundary conditions for the mechanical stresses on the top surface of the covering layer ( 2 xh  ) are as follows: The boundary condition for the electric potential is as follows: The boundary condition for the electric displacement is as follows: The boundary conditions Eqs. (11) and (14) for the electrically shorted circuit, and the boundary conditions Eqs. (12) and (15) for the electrically open circuit should be satisfied.
The following decay conditions are satisfied for the near-surface waves in which there is no reflection of waves travelling along the negative 2 x direction in the half-plane. (2) 0 ij σ  , (2)

METHOD OF SOLUTION
Substituting Eq. (3) into Eq. (2), the equations of electro-elastic motion of piezoelectric covering layer is obtained as follows: (1) (1) Considering the harmonic wave propagation is in the 1 Ox axis direction, the displacements and electric potential of the covering layer can be represented as follows: where A , B , and C are unknown constants, k is wave number, ω is angular frequency, and b is a parameter to be determined.
Substituting Eq. (18) into Eq. (17), we obtain the following equations to find the unknown constants A , B , and C in Eq. (18).
(1), 2 In order to find a non-trivial solution of A , B , and C , following equation must be satisfied: , and some mathematical manipulations should be made for the complex roots of (1) n b [11]. (1) n A are six unknown constants used instead of the unknown constants A , B , and C in Eq. (18).
By taking the piezoelectric and dielectric constants of the piezoelectric material as zero, the covering layer turns into a transversal isotropic material without piezoelectricity. In Figure 2, first four lowest modes of the dispersion curves are obtained and plotted with dashed lines for the system which has a transversal isotropic covering layer. In this figure, solid lines represent the dispersion curves of the system which has a PZT-2 covering layer. Dispersion curves in Figure 2 are obtained for the case where piezoelectric layer is electrically shorted, and there are not initial stresses in the system. In Figure 2 (1) () c c e ερ  . In Figure 2, it is seen that the near-surface wave propagation velocity in the system with piezoelectric character is higher than in the system without piezoelectricity. Also in the higher values of kh , effect of piezoelectricity on wave propagation velocity is higher than in the lower values of kh . Same results in this figure is also available in [13], therefore it shows the accuracy of the PC algorithm.
Second, third and fourth dispersion curves for the system consisting of a PZT-4 covering layer and an aluminum metal elastic half-plane for the case where there are not initial stresses in the system are obtained in the paper [11]. These dispersion curves for the considered system are also obtained in [12] and [13] with the algorithm used in present investigation for the case where there are not initial stresses in the system. However, the first lowest mode dispersion curves for the system consisting of a PZT-4 covering layer and an aluminum metal elastic halfplane for the case where there are not initial stresses in the system are not given in the paper [11]. Present investigation and the study in [18] show that for the system consisting of a PZT-4 covering layer first lowest mode dispersion curves exist only before a certain value of the dimensionless wave number kh . Therefore, these results can be taken as validation of the testing of the algorithm used in the present investigation.
As the initial stresses on the covering layer and half-plane are equal ( (1),0 (2),0 22 22 0 σ σ P  ), uniaxial initial stresses on the system can be given as follows: (1) In order to show the influence of the initial stresses on the near-surface wave propagation velocity, following notation will be used: .In Eq. (31), c is the wave propagation velocity for the system with uniaxial initial stresses, and c is the wave propagation velocity for the system without initial stresses.
Figs. 3, 4, and 5 show the dependencies between the parameter η and kh for three different magnitude of the initial stretching and compression stresses ( (1) ψ  0.001, 0.002 , 0.003 ). In these figures, (a), (b), (c), and (d) show η and kh relation for the first, second, third, and fourth modes of the dispersion curves, respectively. This relation is shown with dashed lines for the system which has transversal isotropic covering layer and with solid lines for the system which has PZT-2 covering layer.    Figure 3 shows the influence of the uniaxial initial stretching ( (1) 0 ψ  ) on the near-surface wave propagation velocity. In this figure, it is seen that the near-surface wave propagation velocity in the system under uniaxial initial stretching is higher than in the system without initial stresses. However, the piezoelectricity of the covering layer reduces the increment of wave propagation velocity. Figure 4 shows the influence of the uniaxial initial compression ( (1) 0 ψ  ) on the near-surface wave propagation velocity. In this figure, it is seen that the near-surface wave propagation velocity in the system under uniaxial initial compression is lower than in the system without initial stresses. However, piezoelectricity of the covering layer reduces the decrease of wave propagation velocity. Figs. 3 and 4 show that in the lower values of kh , effect of uniaxial initial stretching on wave propagation velocity is higher than in the higher values of kh . Also, this effect is significant in the first mode of the dispersion curves.
In Figure 3 and 4, third order elastic constants are not taken into account ( (2) (2) (2) 0 abc    ). Figure 5 shows the influence of the uniaxial initial compression on the near-surface wave propagation velocity by taking third order elastic constant into consideration ( (1) 0 ψ  , (2) (2) , , 0 abc ). In Figure 5, it is seen that third order elastic constants effect results significantly, especially in the lower values of kh .

CONCLUSION
Effects of the piezoelectricity of covering layer, third order elastic constants of the half-plane, and uniaxial initial compression and stretching, which is perpendicular to the wave propagation direction, on near-surface wave propagation velocity are investigated.
For concrete numerical investigations, the half-plane material is selected as aluminum and the covering layer material is selected as PZT-2. First four lowest modes of the dispersion curves are obtained for different cases in the system; without initial stresses (i), under uniaxial initial stretching (ii), under uniaxial initial compression without considering the third order elastic constant (iii), under uniaxial initial compression considering the third order elastic constant (iv). It is assumed that the positive polarization direction of the piezoelectric covering material coincide with the 2 Ox axis. Furthermore, the top and the bottom surfaces of the covering layer are electrically shorted.
According to the results, without the piezoelectric character of the covering layer, near-surface wave velocity is lower than in the system with piezoelectric character. Uniaxial initial stretching or compression significantly increases or reduces the near-surface wave propagation velocity. Also, the piezoelectricity of the covering layer affects the wave propagation velocity, considerably. Moreover, third order elastic constants have a remarkable effect on wave propagation velocity which increases or decreases depending on the values of kh .