IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, vol. 60, no. 10, OCTOBER 2013 2189 0885–3010/$25.00 © 2013 IEEE Propagation of SH Waves in Functionally Gradient Electromagnetoelastic Half-Space Brij Mohan Singh and Jon Rokne Abstract—The propagation of shear-horizontal (SH) waves is studied for a functionally gradient magnetoelectric (ME) material. The material properties of the ME half-space are normal to the free surface. The ME open conditions are ap- plied to the free surface. Dispersion relations are obtained in explicit form for different forms of the nonhomogeneities. I. Introduction C omposite materials consisting of a piezoelectric (PE) and piezomagnetic (PM) phase exhibit a magneto- electric (ME) efect that is absent in pure PM media and pure PE media. This phenomenon is called the ME efect. In such cases, the coupling efect of PE/PM composites can even be obtained that is a hundred times larger than that of a single-phase ME material. Consequently, PE/ PM composites are extensively used as electric packag- ing, sensors and actuators, e.g., magnetic feld probes, acoustic/ultrasonic devices, hydrophones, and transduc- ers, where the responsibility of the material is electro- magneto-mechanical energy conversions. Recently, the theoretical work for investigation on functionally graded PE/PM material has received considerable attention in the literature. In the early 1970s, magnetoelectroelastic (MEE) materials such as BaTiO 3 –CoFe 2 O 4 composite were produced to provide the PE/PM property [1]–[3]. Electric and magnetic behavior in ME materials should satisfy the Maxwell equations. Following this approach, new types of surface waves in PE and ME materials have been investigated in [4]–[6]. An overall and detailed review of this research topic can be found in the recent papers of Fiebig [7] and Nan et al. [8]. Within the framework of linear MEE, a horizontal shear surface wave on the free surface of the ME half-space adjoining vacuum supports has been studied by Wang et al. [9] and Liu et al. [10]. Soh and Liu [11] have developed the existence conditions for interfacial shear-horizontal (SH) waves in PE/PM bi- materials. The dispersion characteristics of SH waves in PE/PM layered half-space were studied by Du et al. [12] with ini- tial stress and Liu et al. [13]. The refection and refraction of plane waves at the PE and PM interface were inves- tigated by Pang et al. [14]. Wang et al. [15] revealed the band structures of elastic wave propagation in two-dimen- sional PE/PM phononic crystals. Chen et al. [16] investi- gated the propagation of harmonic waves in multilayered MEE plates. Pang et al. [17] analyzed the propagation of Rayleigh-type surfaces waves in PM half-space carrying a PE layer and discussed in detail the efects of electromag- netic and PE properties on the phase velocity. Scattering problems of anti-plane shear waves in PE/ PM materials were studied by Du et al. [18] and Liang [19]. Wave propagation in PE/PM layered periodic com- posites was discussed by Pang et al. [20]. Over the past decade, wave propagation in functionally graded PE materials and functionally graded ME materi- als have been widely studied. Because of their superior- ity over many composite materials, functionally graded materials have been used in spacefight, aerospace, and electronic felds. To improve the efciency and natural life of the SAW devices, functionally graded magnetoelastic materials (FGMEMs) are considered for their application in SAW devices. Hence, the study of wave propagation behaviors and characteristics in FGMEM has become a topic of practical importance. Liu and Tani [21]–[24] have investigated surface waves in functionally graded piezo- electric materials (FGPMs). Han and Liu [25] investigated frequency and group velocity dispersion behaviors and characteristic surfaces of waves in FGPM cylinders. Chen et al. [26] considered the free vibration of nonho- mogeneous transversely isotropic MEE plates. Scattering of harmonic anti-plane shear stress waves by a crack in functionally graded PE/PM materials was studied by Li- ang [19]. Dynamic stress for a circular hole in functionally graded PE/PM material subject to shear waves was stud- ied by Wang et al. [27]. Functionally graded PE/PM ma- terials are an emerging class of materials. These compos- ites are referred to as smart or intelligent materials which provide feedback on the internal state of the structure. The goal of this paper is to fnd the solution of the equilibrium equation when the variation of all material parameters are general functions diferent than those of other papers in the literature for gradient solids. Making use of this solution in the present paper, we report on the discovery of a type of MEE surface waves in functionally gradient half-space materials. The material properties are normal to the free surface. Making use of the ME open conditions to the free surface, the exact dispersion rela- tions are obtained. The fnal solution is given in a special case. The geometry of the problem is shown in Fig. 1. Manuscript received February 28, 2013; accepted July 8, 2013. The authors are with the Department of Computer Science, The Uni- versity of Calgary, Calgary, AB, Canada (e-mail: rokne@ucalgary.ca). DOI http://dx.doi.org/10.1109/TUFFC.2013.2809