Table of Contents
ISRN Applied Mathematics
VolumeΒ 2011, Article IDΒ 408529, 8 pages
http://dx.doi.org/10.5402/2011/408529
Research Article

Analytical Investigation of Surface Wave Characteristics of Piezoelectromagnetics of Class 6 mm

International Institute of Zakharenko Waves (IIZWs), Krasnoyarsk-37, Krasnoyarsk 17701, Russia

Received 13 March 2011; Accepted 22 May 2011

Academic Editors: Y.-D.Β Kwon and W.Β Yeih

Copyright Β© 2011 Aleksey A. Zakharenko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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=𝑉tem𝐾1βˆ’2em1+𝐾2emξ‚Ά2ξƒ­1/2.(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=𝑉𝑑4ξ€·1+𝐾2emξ€Έ1/2,(2.2)𝐾2em=πœ‡π‘’2+πœ€β„Ž2βˆ’2π›Όπ‘’β„ŽπΆξ€·πœ€πœ‡βˆ’π›Ό2ξ€Έ.(2.3) In (2.2), the velocity 𝑉𝑑4 of the purely mechanical SH-BAW is determined as follows: 𝑉𝑑4=ξ‚™πΆπœŒ,(2.4) where 𝜌 is the mass density. In (2.3) and (2.4), there are the following material constants: the elastic stiffness constant 𝐢, piezoelectric constant 𝑒, piezomagnetic coefficient β„Ž, dielectric permittivity coefficient πœ€, magnetic permeability coefficient πœ‡, 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 πœ‘=0) and the magnetically open surface (magnetic potential πœ“=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 𝐡3 at the interface, where 𝐷3 and 𝐡3 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=𝑉temβŽ‘βŽ’βŽ’βŽ£ξƒ©πΎ1βˆ’2emβˆ’πΎ2𝑒+𝛼2𝐢2πΏξ€·πœ€0𝐾/πœ€ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ‡/πœ‡0ξ€Έξƒͺ2⎀βŽ₯βŽ₯⎦1/2,(2.5)𝑉new2=𝑉temβŽ‘βŽ’βŽ’βŽ£ξƒ©πΎ1βˆ’2emβˆ’πΎ2π‘š+𝛼2𝐢2πΏξ€·πœ‡0𝐾/πœ‡ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ€/πœ€0ξ€Έξƒͺ2⎀βŽ₯βŽ₯⎦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 πœ€0=10βˆ’7/(4πœ‹πΆ2𝐿)=8.854187817Γ—10βˆ’12 [F/m] and the magnetic permeability constant πœ‡0=4πœ‹Γ—10βˆ’7 [H/m] = 12.5663706144Γ—10βˆ’7 [H/m], where 𝐢𝐿=2.99782458Γ—108 [m/s] is the speed of light in a vacuum: 𝐢2𝐿=1πœ€0πœ‡0.(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 𝐾2π‘š (CMMC) for a purely piezomagnetic material: 𝐾2𝑒=𝑒2,πΎπœ€πΆ2π‘š=β„Ž2.πœ‡πΆ(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 𝛼2β†’πœ€πœ‡ when 𝐾2emβ†’βˆž; 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: πœ•π‘‰BGM=π‘‰πœ•π›ΌBGM𝑉temπœ•π‘‰temβˆ’π‘‰πœ•π›Ό2tem𝑉BGM𝐾2emξ€·1+𝐾2emξ€Έ3πœ•πΎ2em,πœ•π›Ό(2.9)πœ•π‘‰new1=π‘‰πœ•π›Όnew1𝑉temπœ•π‘‰temβˆ’π‘πœ•π›Ό1𝑉2tem𝑉new1πœ•π‘1πœ•π›Ό,(2.10)πœ•π‘‰new2=π‘‰πœ•π›Όnew2𝑉temπœ•π‘‰temβˆ’π‘πœ•π›Ό2𝑉2tem𝑉new2πœ•π‘2πœ•π›Ό,(2.11) where πœ•π‘‰tem=π‘‰πœ•π›Ό2𝑑42𝑉temπœ•πΎ2emπœ•π›Ό.(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 πœ•πΎ2em=2ξ€·πœ•π›Όπ›ΌπΎ2emξ€Έβˆ’π‘’β„Ž/πΆπœ€πœ‡βˆ’π›Ό2.(2.13)

Using (2.5) and (2.6), the functions 𝑏1 and 𝑏2 in (2.10) and (2.11) are determined as follows: 𝑏1=𝐾2emβˆ’πΎ2𝑒+𝛼2𝐢2πΏξ€·πœ€0𝐾/πœ€ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ‡/πœ‡0ξ€Έ,𝑏2=𝐾2emβˆ’πΎ2π‘š+𝛼2𝐢2πΏξ€·πœ‡0𝐾/πœ‡ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ€/πœ€0ξ€Έ.(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: πœ•π‘1=ξ€Ίπœ•π›Ό1βˆ’π‘1ξ€·1+πœ‡/πœ‡0ξ€Έξ€»ξ€·πœ•πΎ2emξ€Έ/πœ•π›Ό+2𝛼𝐢2πΏξ€·πœ€0𝐾/πœ€ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ‡/πœ‡0ξ€Έ+𝛼2𝐢2πΏξ€·πœ€0/πœ€ξ€Έξ€·πœ•πΎ2em/πœ•π›Ό+π‘’β„Ž/𝛼2𝐢1+𝐾2emξ€Έξ€·1+πœ‡/πœ‡0ξ€Έ,πœ•π‘2=ξ€Ίπœ•π›Ό1βˆ’π‘2ξ€·1+πœ€/πœ€0ξ€Έξ€»ξ€·πœ•πΎ2emξ€Έ/πœ•π›Ό+2𝛼𝐢2πΏξ€·πœ‡0𝐾/πœ‡ξ€Έξ€·2emξ€Έβˆ’π‘’β„Ž/𝛼𝐢1+𝐾2emξ€Έξ€·1+πœ€/πœ€0ξ€Έ+𝛼2𝐢2πΏξ€·πœ‡0/πœ‡ξ€Έξ€·ξ€·πœ•πΎ2emξ€Έ+ξ€·/πœ•π›Όπ‘’β„Ž/𝛼2𝐢1+𝐾2emξ€Έξ€·1+πœ€/πœ€0ξ€Έ.(2.15)

The second partial derivatives of the velocities 𝑉BGM, 𝑉new1, and 𝑉new2 with respect to the electromagnetic constant 𝛼 read πœ•2𝑉BGMπœ•π›Ό2=𝑉BGM𝑉temπœ•2𝑉temπœ•π›Ό2+1𝑉temπœ•π‘‰BGMπœ•π›Όπœ•π‘‰temβˆ’π‘‰πœ•π›ΌBGM𝑉2temξ‚΅πœ•π‘‰temξ‚Άπœ•π›Ό2βˆ’π‘‰2tem𝑉BGM𝐾2emξ€·1+𝐾2emξ€Έ3πœ•2𝐾2emπœ•π›Ό2βˆ’ξƒ¬2𝑉tem𝑉BGMπœ•π‘‰temβˆ’ξ‚΅π‘‰πœ•π›Όtem𝑉BGMξ‚Ά2πœ•π‘‰BGMξƒ­πΎπœ•π›Ό2emξ€·1+𝐾2emξ€Έ3πœ•πΎ2emβˆ’ξ‚΅πœ•π›Ό1βˆ’3𝐾2em1+𝐾2em𝑉2tem𝑉BGM1ξ€·1+𝐾2emξ€Έ3ξ‚΅πœ•πΎ2emξ‚Άπœ•π›Ό2,πœ•(2.16)2𝑉new1πœ•π›Ό2=𝑉new1𝑉temπœ•2𝑉temπœ•π›Ό2βˆ’π‘‰2tem𝑉new1ξ‚΅πœ•π‘1ξ‚Άπœ•π›Ό2βˆ’π‘1𝑉2tem𝑉new1πœ•2𝑏1πœ•π›Ό2βˆ’3𝑏1𝑉tem𝑉new1πœ•π‘‰temπœ•π›Όπœ•π‘1πœ•π›Ό+𝑏1𝑉tem𝑉new1ξ‚Ά2πœ•π‘‰new1πœ•π›Όπœ•π‘1,πœ•πœ•π›Ό(2.17)2𝑉new2πœ•π›Ό2=𝑉new2𝑉temπœ•2𝑉temπœ•π›Ό2βˆ’π‘‰2tem𝑉new2ξ‚΅πœ•π‘2ξ‚Άπœ•π›Ό2βˆ’π‘2𝑉2tem𝑉new2πœ•2𝑏2πœ•π›Ό2βˆ’3𝑏2𝑉tem𝑉new2πœ•π‘‰temπœ•π›Όπœ•π‘2πœ•π›Ό+𝑏2𝑉tem𝑉new2ξ‚Ά2πœ•π‘‰new2πœ•π›Όπœ•π‘2,πœ•π›Ό(2.18) where πœ•2𝑉temπœ•π›Ό2=𝑉2𝑑42𝑉temπœ•2𝐾2emπœ•π›Ό2βˆ’π‘‰2𝑑42𝑉2temπœ•π‘‰temπœ•π›Όπœ•πΎ2em.πœ•π›Ό(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: πœ•2𝐾2emπœ•π›Ό2=2𝐾2emξ€·+4π›Όπœ•πΎ2emξ€Έ/πœ•π›Όπœ€πœ‡βˆ’π›Ό2.(2.20) In (2.17) and (2.18), the second partial derivatives of the 𝑏1 and 𝑏2 with respect to the 𝛼 are πœ•2𝑏1πœ•π›Ό2=𝐡1ξ€·1+𝐾2emξ€Έξ€·1+πœ‡/πœ‡0ξ€Έ,πœ•2𝑏2πœ•π›Ό2=𝐡2ξ€·1+𝐾2emξ€Έξ€·1+πœ€/πœ€0ξ€Έ,(2.21) where 𝐡1=ξ‚Έ1βˆ’π‘1ξ‚΅πœ‡1+πœ‡0πœ•ξ‚Άξ‚Ή2𝐾2emπœ•π›Ό2ξ‚΅πœ‡βˆ’21+πœ‡0ξ‚Άπœ•π‘1πœ•π›Όπœ•πΎ2emπœ•π›Ό+2𝐢2πΏπœ€0πœ€ξ‚€πΎ2emβˆ’π‘’β„Žξ‚π›ΌπΆ+4𝛼𝐢2πΏπœ€0πœ€ξ‚΅πœ•πΎ2em+πœ•π›Όπ‘’β„Žπ›Ό2𝐢+𝛼2𝐢2πΏπœ€0πœ€ξƒ©πœ•2𝐾2emπœ•π›Ό2βˆ’2π‘’β„Žπ›Ό3𝐢ξƒͺ,𝐡2=ξ‚Έ1βˆ’π‘2ξ‚΅πœ€1+πœ€0πœ•ξ‚Άξ‚Ή2𝐾2emπœ•π›Ό2ξ‚΅πœ€βˆ’21+πœ€0ξ‚Άπœ•π‘2πœ•π›Όπœ•πΎ2emπœ•π›Ό+2𝐢2πΏπœ‡0πœ‡ξ‚€πΎ2emβˆ’π‘’β„Žξ‚π›ΌπΆ+4𝛼𝐢2πΏπœ‡0πœ‡ξ‚΅πœ•πΎ2em+πœ•π›Όπ‘’β„Žπ›Ό2𝐢+𝛼2𝐢2πΏπœ‡0πœ‡ξƒ©πœ•2𝐾2emπœ•π›Ό2βˆ’2π‘’β„Žπ›Ό3𝐢ξƒͺ.(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.

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