Abstract

This short work copes with theoretical investigations of some surface wave characteristics for transversely isotropic piezoelectromagnetic composites of class 6 mm. In the composite materials, the surface Bleustein-Gulyaev-Melkumyan wave and some new shear-horizontal surface acoustic waves (SH-SAWs) recently discovered by the author can propagate. The phase velocities 𝑉ph of the SH-SAWs can have complicated dependencies on the coefficient of the magnetoelectromechanical coupling 𝐾2em (CMEMC) which depends on the electromagnetic constant 𝛼 of the composites. Therefore, the analytical finding of the first and second partial derivatives of the 𝑉ph(𝛼) represents the main purpose of this study. It is thought that the results of this short letter can help for theoreticians and experimentalists working in the research arena of opto-acoustoelectronics to completely understand some problems of surface wave propagation in piezoelectromagnetics.

1. Introduction

Two-phase composite materials, which possess both the piezoelectric and piezomagnetic phases, are very promising composites with the magnetoelectric effect. They are very interesting for various applications in space and aircraft technologies. Several books concerning composite materials are cited in [1–3]. The geometry of a two-phase composite material can be denoted by the following connectivities: 0-0, 0-1, 0-2, 0-3, 1-1, 1-2, 1-3, 2-2, 2-3, and 3-3, where 0, 1, 2, and 3 are the dimensions of piezoelectric-piezomagnetic phases. Some latterly published papers concerning the magnetoelectric effect in different composite materials can be found in [4–9]. For example, the composite structures [10] called (2-2) composites represent laminated composite materials in which alternate layers of two different materials are bonded together to form a stratified continuum. Also, [10–13] cope with some laminate composites in which the popular Terfenol-D material is used as the piezomagnetic phase. For the study of the magnetoelectric effect, much work was described in the review paper [14] by Fiebig.

It is thought that some of the main characteristics of piezoelectric, piezomagnetic, and composite materials are the speeds of the shear-horizontal surface acoustic waves (SH-SAWs). In 1998, Gulyaev [15] has written a review of SH-SAWs in solids. However, in the beginning of this millennium, Melkumyan [16] has discovered twelve new SH-SAWs in piezoelectromagnetic composite materials. In 2010, the author of this paper has additionally discovered seven new SH-SAWs in the piezoelectromagnetic composites of class 6 mm, see the book [17]. One of the new surface Melkumyan waves [16] written in the following section was called the surface Bleustein-Gulyaev-Melkumyan wave [17]. Note that the classical SH-SAWs in purely piezoelectric materials and purely piezomagnetic materials are called the surface Bleustein-Gulyaev waves simultaneously discovered by Bleustein [18] and Gulyaev [19] to the end of the last millennium. The new SH-SAWs discovered in [17] by the author of this theoretical work depend on the speed of light in a vacuum and can represent an interest for acoustooptics and photonics researchers. Also, some peculiarities of the new SH-SAW propagation will be briefly discussed in the following section. This peculiarity allows one to assume a restriction for the electromagnetic constant 𝛼 of the complex piezoelectromagnetic composite materials.

It is also noted that SH-SAWs can easily be produced by electromagnetic acoustic transducers (EMATs), a nontrivial task for common piezoelectric transducers [20]. The EMATs can offer advantages over traditional piezoelectric transducers. Comprehensive monographs [21, 22] on the EMATs collect the research activities on this topic. Therefore, it is thought that this short theoretical work can be also useful as a small step towards new applications of the EMATs technologies. Indeed, it is believed that some characteristics of the SH-SAWs in piezoelectromagnetic composite materials can be revealed by the utilization of the electromagnetic acoustic transducers. Therefore, the following section describes the analytical finding of the first and second partial derivatives of the phase velocity 𝑉ph with respect to the electromagnetic constant 𝛼.

2. Theoretical Investigations

According to the recent work by Melkumyan [16] concerning wave propagation in piezoelectromagnetic materials of class 6 mm, the velocity 𝑉BGM for the shear-horizontal surface Bleustein-Gulyaev-Melkumyan wave can be written in the explicit form [17] as follows: 𝑉BGM=𝑉tem𝐾1−2em1+𝐾2em21/2.(2.1) In (2.1), the velocity 𝑉tem of the piezo-magnetoelectromechanical shear-horizontal bulk acoustic wave (SH-BAW) and the coefficient of the magnetoelectromechanical coupling 𝐾2em (CMEMC) are defined as follows: 𝑉tem=𝑉𝑡41+𝐾2em1/2,(2.2)𝐾2em=𝜇𝑒2+𝜀ℎ2−2𝛼𝑒ℎ𝐶𝜀𝜇−𝛼2.(2.3) In (2.2), the velocity 𝑉𝑡4 of the purely mechanical SH-BAW is determined as follows: 𝑉𝑡4=𝐶𝜌,(2.4) where 𝜌 is the mass density. In (2.3) and (2.4), there are the following material constants: the elastic stiffness constant 𝐶, piezoelectric constant 𝑒, piezomagnetic coefficient ℎ, dielectric permittivity coefficient 𝜀, magnetic permeability coefficient 𝜇, and electromagnetic constant 𝛼. The material constants are described in the well-known handbook [23] on electromagnetic materials.

Formula (2.1) for the surface Bleustein-Gulyaev-Melkumyan (BGM) wave corresponds to the first case of the electrical and magnetic boundary conditions at the interface between the composite surface and a vacuum. This case is for the electrically closed surface (electrical potential 𝜑=0) and the magnetically open surface (magnetic potential 𝜓=0) using the mechanical boundary condition of the mechanically free interface. The realization of different boundary conditions is described in an excellent theoretical work [24]. In addition to the first case, it is also possible to treat the second case of electrical and magnetic boundary conditions for the mechanical boundary condition. This second case represents the continuity of both the normal components of 𝐷3 and 𝐵3 at the interface, where 𝐷3 and 𝐵3 are the components of the electrical displacement and the magnetic flux, respectively. This leads to the following velocities for the SH-SAWs discovered by the author in the recent theoretical work [17]: 𝑉new1=𝑉tem⎡⎢⎢⎣𝐾1−2em−𝐾2𝑒+𝛼2𝐶2𝐿𝜀0𝐾/𝜀2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜇/𝜇02⎤⎥⎥⎦1/2,(2.5)𝑉new2=𝑉tem⎡⎢⎢⎣𝐾1−2em−𝐾2𝑚+𝛼2𝐶2𝐿𝜇0𝐾/𝜇2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜀/𝜀02⎤⎥⎥⎦1/2.(2.6)

In expressions (2.5) and (2.6) there is already dependence on the vacuum characteristics such as the dielectric permittivity constant 𝜀0=10−7/(4𝜋𝐶2𝐿)=8.854187817×10−12 [F/m] and the magnetic permeability constant 𝜇0=4𝜋×10−7 [H/m] = 12.5663706144×10−7 [H/m], where 𝐶𝐿=2.99782458×108 [m/s] is the speed of light in a vacuum: 𝐶2𝐿=1𝜀0𝜇0.(2.7) Also, expression (2.5) depends on the well-known coefficient of the electromechanical coupling 𝐾2𝑒 (CEMC) for a purely piezoelectric material (see below), and expression (2.6) depends on the well-known coefficient of the magnetomechanical coupling 𝐾2𝑚 (CMMC) for a purely piezomagnetic material: 𝐾2𝑒=𝑒2,𝐾𝜀𝐶2𝑚=ℎ2.𝜇𝐶(2.8)

Therefore, it is possible to obtain the first and second derivatives of the velocities 𝑉BGM, 𝑉new1, and 𝑉new2 with respect to the electromagnetic constant 𝛼 as the results of the theoretical investigations for this short report. Note that these investigations were not carried out in the recent book [17] due to some mathematical complexities. Therefore, this report continues the theoretical investigations of the book [17]. These investigations are useful because it is possible that the functions 𝑉BGM(𝛼>0), 𝑉new1(𝛼>0), and 𝑉new2(𝛼>0) can have some peculiarities, namely, the SH-SAWs cannot exist for some large values of 𝛼2→𝜀𝜇 when 𝐾2em→∞; see formula (2.3). Note that papers [25, 26] studied some composites with the electromagnetic constant 𝛼<0, for which these peculiarities do not exist. Therefore, it allows one to suppose that the right sign for the electromagnetic constant 𝛼 is negative.

The first partial derivatives of the velocities 𝑉BGM, 𝑉new1, and 𝑉new2 with respect to the constant 𝛼 can be written in the following forms: 𝜕𝑉BGM=𝑉𝜕𝛼BGM𝑉tem𝜕𝑉tem−𝑉𝜕𝛼2tem𝑉BGM𝐾2em1+𝐾2em3𝜕𝐾2em,𝜕𝛼(2.9)𝜕𝑉new1=𝑉𝜕𝛼new1𝑉tem𝜕𝑉tem−𝑏𝜕𝛼1𝑉2tem𝑉new1𝜕𝑏1𝜕𝛼,(2.10)𝜕𝑉new2=𝑉𝜕𝛼new2𝑉tem𝜕𝑉tem−𝑏𝜕𝛼2𝑉2tem𝑉new2𝜕𝑏2𝜕𝛼,(2.11) where 𝜕𝑉tem=𝑉𝜕𝛼2𝑡42𝑉tem𝜕𝐾2em𝜕𝛼.(2.12) In (2.9)–(2.12), the first partial derivative of the CMEMC 𝐾2em with respect to the electromagnetic constant 𝛼 is defined by 𝜕𝐾2em=2𝜕𝛼𝛼𝐾2em−𝑒ℎ/𝐶𝜀𝜇−𝛼2.(2.13)

Using (2.5) and (2.6), the functions 𝑏1 and 𝑏2 in (2.10) and (2.11) are determined as follows: 𝑏1=𝐾2em−𝐾2𝑒+𝛼2𝐶2𝐿𝜀0𝐾/𝜀2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜇/𝜇0,𝑏2=𝐾2em−𝐾2𝑚+𝛼2𝐶2𝐿𝜇0𝐾/𝜇2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜀/𝜀0.(2.14) Therefore, the first partial derivatives of the 𝑏1 and 𝑏2 with respect to the constant 𝛼 can be expressed in the following forms: 𝜕𝑏1=𝜕𝛼1−𝑏11+𝜇/𝜇0𝜕𝐾2em/𝜕𝛼+2𝛼𝐶2𝐿𝜀0𝐾/𝜀2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜇/𝜇0+𝛼2𝐶2𝐿𝜀0/𝜀𝜕𝐾2em/𝜕𝛼+𝑒ℎ/𝛼2𝐶1+𝐾2em1+𝜇/𝜇0,𝜕𝑏2=𝜕𝛼1−𝑏21+𝜀/𝜀0𝜕𝐾2em/𝜕𝛼+2𝛼𝐶2𝐿𝜇0𝐾/𝜇2em−𝑒ℎ/𝛼𝐶1+𝐾2em1+𝜀/𝜀0+𝛼2𝐶2𝐿𝜇0/𝜇𝜕𝐾2em+/𝜕𝛼𝑒ℎ/𝛼2𝐶1+𝐾2em1+𝜀/𝜀0.(2.15)

The second partial derivatives of the velocities 𝑉BGM, 𝑉new1, and 𝑉new2 with respect to the electromagnetic constant 𝛼 read 𝜕2𝑉BGM𝜕𝛼2=𝑉BGM𝑉tem𝜕2𝑉tem𝜕𝛼2+1𝑉tem𝜕𝑉BGM𝜕𝛼𝜕𝑉tem−𝑉𝜕𝛼BGM𝑉2tem𝜕𝑉tem𝜕𝛼2−𝑉2tem𝑉BGM𝐾2em1+𝐾2em3𝜕2𝐾2em𝜕𝛼2−2𝑉tem𝑉BGM𝜕𝑉tem−𝑉𝜕𝛼tem𝑉BGM2𝜕𝑉BGM𝐾𝜕𝛼2em1+𝐾2em3𝜕𝐾2em−𝜕𝛼1−3𝐾2em1+𝐾2em𝑉2tem𝑉BGM11+𝐾2em3𝜕𝐾2em𝜕𝛼2,𝜕(2.16)2𝑉new1𝜕𝛼2=𝑉new1𝑉tem𝜕2𝑉tem𝜕𝛼2−𝑉2tem𝑉new1𝜕𝑏1𝜕𝛼2−𝑏1𝑉2tem𝑉new1𝜕2𝑏1𝜕𝛼2−3𝑏1𝑉tem𝑉new1𝜕𝑉tem𝜕𝛼𝜕𝑏1𝜕𝛼+𝑏1𝑉tem𝑉new12𝜕𝑉new1𝜕𝛼𝜕𝑏1,𝜕𝜕𝛼(2.17)2𝑉new2𝜕𝛼2=𝑉new2𝑉tem𝜕2𝑉tem𝜕𝛼2−𝑉2tem𝑉new2𝜕𝑏2𝜕𝛼2−𝑏2𝑉2tem𝑉new2𝜕2𝑏2𝜕𝛼2−3𝑏2𝑉tem𝑉new2𝜕𝑉tem𝜕𝛼𝜕𝑏2𝜕𝛼+𝑏2𝑉tem𝑉new22𝜕𝑉new2𝜕𝛼𝜕𝑏2,𝜕𝛼(2.18) where 𝜕2𝑉tem𝜕𝛼2=𝑉2𝑡42𝑉tem𝜕2𝐾2em𝜕𝛼2−𝑉2𝑡42𝑉2tem𝜕𝑉tem𝜕𝛼𝜕𝐾2em.𝜕𝛼(2.19) In (2.16) and (2.19), the second partial derivative of the 𝐾2em with respect to the constant 𝛼 is defined as follows: 𝜕2𝐾2em𝜕𝛼2=2𝐾2em+4𝛼𝜕𝐾2em/𝜕𝛼𝜀𝜇−𝛼2.(2.20) In (2.17) and (2.18), the second partial derivatives of the 𝑏1 and 𝑏2 with respect to the 𝛼 are 𝜕2𝑏1𝜕𝛼2=𝐵11+𝐾2em1+𝜇/𝜇0,𝜕2𝑏2𝜕𝛼2=𝐵21+𝐾2em1+𝜀/𝜀0,(2.21) where 𝐵1=1−𝑏1𝜇1+𝜇0𝜕2𝐾2em𝜕𝛼2𝜇−21+𝜇0𝜕𝑏1𝜕𝛼𝜕𝐾2em𝜕𝛼+2𝐶2𝐿𝜀0𝜀𝐾2em−𝑒ℎ𝛼𝐶+4𝛼𝐶2𝐿𝜀0𝜀𝜕𝐾2em+𝜕𝛼𝑒ℎ𝛼2𝐶+𝛼2𝐶2𝐿𝜀0𝜀𝜕2𝐾2em𝜕𝛼2−2𝑒ℎ𝛼3𝐶,𝐵2=1−𝑏2𝜀1+𝜀0𝜕2𝐾2em𝜕𝛼2𝜀−21+𝜀0𝜕𝑏2𝜕𝛼𝜕𝐾2em𝜕𝛼+2𝐶2𝐿𝜇0𝜇𝐾2em−𝑒ℎ𝛼𝐶+4𝛼𝐶2𝐿𝜇0𝜇𝜕𝐾2em+𝜕𝛼𝑒ℎ𝛼2𝐶+𝛼2𝐶2𝐿𝜇0𝜇𝜕2𝐾2em𝜕𝛼2−2𝑒ℎ𝛼3𝐶.(2.22)

It is obvious that the first partial derivatives of the velocities 𝑉BGM, 𝑉new1, and 𝑉new2 with respect to the electromagnetic constant 𝛼 have dimension of (m/s)2 and can represent some squares in the corresponding two-dimensional (2D) spaces of velocities. Analogically, the second partial derivatives of the velocities with respect to the constant 𝛼 can represent some volumes with dimensions of (m/s)3 in the corresponding 3D spaces of velocities. Indeed, it is also possible to graphically investigate the complicated first and second partial derivatives of the velocities obtained in formulae (2.9)–(2.11) and from (2.16) to (2.18). However, this does not represent the purpose of this short report.

3. Conclusion

This short theoretical report further developed the study of the recently published book [17]. In this work, on some wave properties of composite materials, the propagation peculiarities of new shear-horizontal surface acoustic waves (SH-SAWs) recently discovered in book [17] were theoretically studied and briefly discussed. Therefore, the analytical finding of the first and second partial derivatives of the phase velocity with respect to the electromagnetic constant 𝛼 represented the main purpose of this study. This theoretical work can be useful for theoreticians and experimentalists working in the arena of acoustooptics, photonics, and opto-acoustoelectronics. Also, the theoretical study of this short paper can be useful for investigations of cubic piezoelectromagnetics like the researches carried out for cubic piezoelectrics [27] and cubic piezomagnetics [28].

Acknowledgment

The author would like to thank the referees for useful notes.