ABSTRACT

It is theoretically considered the propagation (first evidence) of new dispersive shear-horizontal (SH) acoustic waves in the piezoelectromagnetic (magnetoelectroelastic) composite plates. The studied two-phase composites (BaTiO₃-CoFe₂O₄ and PZT-5H-Terfenol-D) possess the piezoelectric phase (BaTiO₃, PZT-5H) and the piezomagnetic phase (CoFe₂O₄, Terfenol-D). The mechanical, electrical, and magnetic boundary conditions applied to both the upper and lower free surfaces of the plate are as follows: the mechanically free, electrically closed, and magnetically closed surfaces. As a result, the fundamental modes of two new dispersive SH-waves recently discovered in book [Zakharenko, A.A. (2012) ISBN: 978-3-659-30943-4] were numerically calculated. It was found that for large values of normalized plate thickness kd (k and d are the wavenumber and plate half-thickness, respectively) the velocities of both the new dispersive SH-waves can approach the nondispersive SH-SAW velocity of the piezoelectric exchange surface Melkumyan (PEESM) wave. It was also discussed that for small values of kd, the experimental study of the new dispersive SH-waves can be preferable in comparison with the nondispersive PEESM wave. The obtained results can be constructive for creation of various technical devices based on (non)dispersive SH-waves and two-phase smart materials. The new dispersive SH-waves propagating in the plates can be also employed for nondestructive testing and evaluation. Also, it is obvious that the plates can be used in technical devices instead of the corresponding bulk samples for further miniaturization.

Keywords:Piezoelectromagnetics, Magnetoelectric Effect, Acoustic SH-Waves in Plates, Wave Dispersion

References

2. Theory and Results

First of all, it is necessary to mention the high symmetry propagation directions for the transversely isotropic piezoelectromagnetic materials of class 6 mm. In these propagation directions, the propagation of the SH-waves possessing the anti-plane polarization must be coupled with both the electrical and magnetic potentials. This means that such propagation directions can also support the propagation of the purely mechanical Lamb waves possessing the in-plane polarization. As a result, this report has no interest in the propagation of purely mechanical Lamb waves. So, the wave propagation direction is along the plate surface and perpendicular to both the surface normal and the sixfold symmetry axis of the studied material of class 6 mm. The surface normal and the sixfold symmetry axis must be also perpendicular to each other [53] -[55] . It is possible to state that many such propagation directions can be found. However, the SH-wave speed in such propagation directions must be the same because this is the transversely isotropic case. Note that the high symmetry propagation directions are well-known and can be also found in [56] [57] . Also, it is worth noting that in the high symmetry propagation direction, the following independent nonzero material parameters exist: the stiffness constant C, piezomagnetic coefficient h, piezoelectric constant e, dielectric permittivity coefficient ε, magnetic permeability coefficient μ, and electromagnetic constant α, where C = C₄₄ = C₆₆, e = e₁₆ = e₃₄, h = h₁₆ = h₃₄, ε = ε₁₁ = ε₃₃, μ = μ₁₁ = μ₃₃, and α = α₁₁ = α₃₃ [53] -[55] , see also the famous books cited in [58] [59] .

The boundary conditions in the case when the treated material simultaneously possesses the piezoelectric, piezomagnetic, and magnetoelectric effects are perfectly described in [60] . To obtain the dispersion relations for the case of the mechanically free, electrically closed, and magnetically closed surfaces of the piezoelectromagnetic plate, the following points must be passed through:

• consider thermodynamic variables and functions;

• write constitutive relations;

• thermodynamically define material constants;

• compose equilibrium equations;

• exploit the electrostatics and magnetostatics in the quasi-static approximation;

• constitute coupled equations of motion in the differential forms;

• represent the tensor form of the equations of motion;

• treat the suitable high symmetry propagation directions for SH-waves;

• find the eigenvalues and the corresponding eigenvectors;

• employ the mechanical, electrical, and magnetic boundary conditions at the upper and lower surfaces of the piezoelectromagnetic plate.

As a result, the corresponding dispersion relations can be obtained for this case of the mechanical, electrical, and magnetic boundary conditions. Following book [53] , the following two dispersion relations for the determination of the velocities V_new10 and V_new11 of the tenth and eleventh new SH-waves propagating in the piezoelectromagnetic plate can be written:

, (1)

. (2)

In Equations (1) and (2), the velocities V_new10 and V_new11 are numbered similar to those used in [53] to avoid any confusion. It is also central to state that dispersion relations (1) and (2) are valid for calculation of the velocities of the fundamental modes when the velocities V_new10 and V_new11 are smaller than the speed V_tem of the shear-horizontal bulk acoustic wave (SH-BAW) coupled with both the electrical and magnetic potentials. The value of the SH-BAW velocity V_tem can be evaluated with the following expression:

, (3)

where r is the mass density of the piezoelectromagnetic plate.

In Expressions (1), (2), and (3), the following material parameter is also present:

. (4)

The material parameter defined by Expression (4) is called the coefficient of the magnetoelectromechanical coupling (CMEMC). This dimensionless coefficient represents the material characteristic of the two-phase materials. However, in the dispersion relations written above the second dimensionless coefficient denoted by can be found. This coefficient of the magnetomechanical coupling (CMMC) represents a material characteristic of a purely piezomagnetic material and is defined by the following expression:

. (5)

It is transparent in dispersion relations (1) and (2) that for kd → ¥, both the velocities V_new10 and V_new11 of the new dispersive SH-waves in the piezoelectromagnetic plate will approach some nondispersive SH-SAW velocity recently discovered by Melkumyan [61] . This SH-SAW velocity is called the piezoelectric exchange surface Melkumyan (PEESM) wave [6] and can be defined by the following expression:

. (6)

Also, it is possible to discuss in this report that for the case of a very small value of the electromagnetic constant α, this constant can be neglected, namely α = 0. This discussion is missed in [53] . Therefore, Expression (4) for the CMEMC can be reduced to the following form:

. (7)

where represents the coefficient of the electromechanical coupling (CEMC). It is a material characteristic of a pure piezoelectrics.

Using Expression (7) instead of (4), it is possible to write the following definitions instead of the SH-BAW and SH-SAW velocities defined by Expressions (3) and (6), respectively:

, (8)

. (9)

So, it is now possible to report the obtained results concerning the calculation of the dispersion curves for concrete transversely isotropic composite materials, namely to investigate the dependencies of the velocities V_new10 and V_new11 on the normalized half-thickness kd of the piezoelectromagnetic plate, where k and d respectively stand for the wavenumber in the propagation direction and the plate half-thickness. This is the main purpose of this paper. It is possible to briefly discuss the piezoelectromagnetic composites given in Table 1 for comparison. The table lists the material parameters for two famous composites such as BaTiO₃-CoFe₂O₄ and PZT-5H-Terfenol-D. The material parameters of the composites were borrowed from papers [4] -[6] . In the table, one can find that the value of εμ for the BaTiO₃-CoFe₂O₄ composite is an order larger than that for the other composite. Also, both composites have the dominant piezoelectric phase because the CMMC is significantly smaller than the CEMC. It is well-known that the PZT-5H-Terfenol-D composite can possess a large value of the electromagnetic constant α that is the characteristic of the magnetoelectric effect. Therefore, to study this composite is preferable in this short report.

In the table, the value of the CMEMC for the PZT-5H-Terfenol-D composite is ~ 0.8. Therefore, the value of the CMEMC for a hypothetic (composite) material graphically studied in Figure 1 is chosen as = 0.8. This figure shows the dependence on the second parameter such as that can be found in dispersion relations (1) and (2). To understand the influence of this parameter on the fundamental mode dispersion relations, the values of were chosen as follows: = 0.1, 0.4, and 0.7. In Figure 1, dispersion relations (1) and (2) are shown by the grey and black lines, respectively. It is natural that when the value of is significantly smaller that the value of (case of = 0.1 in the figure) both the normalized velocities V_new10/V_tem and V_new11/V_tem can be situated well below the SH-BAW velocity V_tem coupled with both the electrical and magnetic potentials. In contrast, the large value of = 0.7, which is slightly below the value of = 0.8, leads to the case when the velocities V_new10/V_tem and V_new11/V_tem for the corresponding fundamental modes of the new

dispersive SH-waves are positioned just below V_tem. This already looks like the dispersion relations shown in Figure 2 for the BaTiO₃-CoFe₂O₄ composite possessing the significantly smaller value of the CMEMC ~ 0.16 listed in the table.

Figure 2 shows dispersion relations (1) and (2) for the fundamental modes of the new dispersive SH-waves propagating in the piezoelectromagnetic plates. Two composite materials listed in the table such as BaTiO₃-CoFe₂O₄ and PZT-5H-Terfenol-D are compared. It is clearly seen in Figure 2 that the BaTiO₃-CoFe₂O₄ composite with smaller value of the CMEMC illuminates the weakly dispersive behaviors of the velocities V_new10/V_tem and V_new11/V_tem. However, the values of the velocity V_new10 at the plate half-thickness kd → 0 can be situated significantly below the value of the SH-BAW velocity V_tem. This is true because the BaTiO₃- CoFe₂O₄ composite has the value of V_tem twice as much compared with the PZT-5H-Terfenol-D composite. This peculiarity manifests that to use plates instead of the corresponding bulk samples can be preferable for investigation of the SH-wave propagation. The plates are also used to further miniaturize various technical devices based on smart materials. It is well-known that the sensitivity of some technical devices (for instant, biological and chemical sensors, delay lines) based on different dispersive and non-dispersive SH-waves can be more significant. It is also flagrant that the piezoelectromagnetic SH-waves can be produced by the electromagnetic

acoustic transducers (EMATs) [62] . This non-contact method (EMAT) can offer a series of advantages in comparison with the traditional piezoelectric transducers [63] [64] . The results of this short report can be also useful for constitution of technical devices with a higher level of integration such as lab-on-a-chip, etc.

3. Conclusion

The propagation of new dispersive acoustic SH-waves in the transversely isotropic PEM plates was considered. The two-phase PEM composites such as BaTiO₃-CoFe₂O₄ and PZT-5H-Terfenol-D were studied. For the plates, the case of the mechanically free, electrically closed, and magnetically closed surfaces was treated. The fundamental modes of two new dispersive SH-waves were numerically calculated. It was found that for large values of kd, the velocities of both the new dispersive SH-waves can approach the nondispersive SH-SAW velocity of the PEESM wave. For small values of kd, the value of the corresponding new SH-wave in the plate can be situated significantly below the value of the SH-BAW velocity V_tem. This can be convenient for experimental studies of the new dispersive SH-waves propagating even in such PEM materials as BaTiO₃-CoFe₂O₄ in comparison with the nondispersive PEESM wave. It is thought that such new dispersive SH-waves can be also exploited in the nondestructive testing and evaluation. It is natural that various technical devices based on dispersive SH-waves and two-phase smart materials can be constituted, for instance, dispersive wave delay lines.

References