Research paperPropagation behavior of ultrasonic Love waves in functionally graded piezoelectric-piezomagnetic materials with exponential variation
Introduction
The study of the Love wave through layered piezoelectric and (or) piezomagnetic materials has received significant attention owing to its diverse engineering applications such as the surface acoustic wave (SAW) sensors, the filters, and the delay lines (Boxberger et al., 2011). A functionally graded material (FGM) is a type of composite with varying material properties in one or more directions from one pure state to another. It differs from traditional materials that are characterized by a disparity of the material distribution. FGM materials can not only decrease the thermal stress, residual stress and stress concentration but also eliminate interface effect (Kugler et al., 2016). Therefore, they are applied to various industrial cases, such as communication devices, electronics, thermoelectric generators, solar cells, medical devices as well as applications in extreme environments (Jha et al., 2013). Previously, the interface between the film and the substrate created an area of high stress that could result in quick fatigue for the structure. However, by using a functionally graded material, the sharp transition between materials was removed, increasing the durability of the system (Qiu et al., 2003). On the other hand, the magneto-electro-elastic coupled properties of the piezoelectric and piezomagnetic materials lead to their widespread use in engineering applications including sensors and actuators (Fukao et al., 1971; Bernhard et al., 1997). To achieve more effective usage of magneto-electro-elastic coupling effects in instruments, researchers have been manufactured many types of composite materials consisting of piezoelectric and piezomagnetic phases (Spaldin et al., 2005). With the current technological developments, FGM materials with piezoelectric and magnetic effects can be manufactured and used in the SAW device with more efficiency. In order to study the static as well as the dynamic characteristics of the multilayered plates or the hetero-structure materials, many methods such as the WKB method (Singhal et al., 2018; Qian et al 2009), the Peano-series method (Shuvalov et al 2008), the Green's function method (Eskandari et al., 2012; Eskandari et al., 2010) and the state vector approach (Zhou et al 2012; Wang et al., 2003), have been used in literature. Some studies have been carried out on piezoelectric–piezomagnetic materials, such as Lamb waves in a functionally graded piezoelectric–piezomagnetic plate by Cao et al. (2012). Love waves in a piezoelectric piezomagnetic layered structure are studied by Liu et al. (2008). The Bleustein–Gulyaev waves in a functionally graded transversely isotropic electro-magneto elastic half-space are studied by Li et al. (2013). The author has also studied the problem of the Rayleigh wave behavior in a functionally graded magneto-electro-elastic material and developed the related equations (Ezzin et al 2017a; Ezzin et al 2017b). To date, to the best of the author's knowledge, only a few results about the issue of the transverse surface wave in a piezomagnetic homogenous substrate covered by a functionally graded piezoelectric material with exponential law variation of these properties have been obtained. Owing to the significant importance of Love waves in many fields such as engineering, geophysics and so on, as well as the advantages of the FGM materials cited above and the magneto-electric coupling in piezoelectric-piezomagnetic material, the current research answers this need and sets forth to provide a numerical solution for Love wave in a FGPPM material. This paper is organized as follows: Section 2 includes a detailed development of the global stiffness matrix method (GSMM) for an arbitrary functionally graded multilayered system, with a description of mechanical, electrical and magnetically boundary conditions. Section 3 presents various numerical results. Firstly, the gradient coefficients are varied to illustrate their effects on phase velocities, group velocities and the magneto-electromechanical coupling factor. Then, the gradient coefficient is fixed while the properties of the piezoelectric film are changed to show their effect on phase velocity for the first Love mode. Finally, the cutoff wavenumber is numerically evaluated for different gradient coefficients. Modal shapes are also performed for some physical quantities to further investigate the behavior of the wave across the film/substrate system. The conclusions are drawn in Section 4.
Section snippets
Theoretical background
Consider a surface wave propagating in heterostructure consisting of a piezomagnetic homogenous substrate perfectly bound to a transversely isotropic functionally graded piezoelectric layer, as shown schematically in Fig. 1. For illustration, a semi-infinite CoFe2O4 substrate carrying a piezoelectric layer PZT5H on the top assumed hexagonal, has been adopted. Rectangular Cartesian coordinates system (x, y, z) is selected and the propagation of Love wave along y-direction with a polarization
Implementation of a functionally graded piezoelectric material
Similar to the model used by Loy et al. (1999) and Chen et al. (1999), we assume that the variation of all the piezoelectric material properties follows the following exponential law along the z-axis direction, where α is the gradient factor, χik represents all properties among {cijkl, eijk, Eij, gij, ρ}, which represent the elastic, piezoelectric, dielectric constants, permeability magnetic constant and the density, respectively. The superscript ‘‘0’’ denotes the property of the homogeneous
Program description: stiffness matrix method (MATLAB code)
To overcome the problem of Love wave propagation in FGPPM material MATLAB is used as software and the Stiffness matrix as a method. The numerical procedure applied to obtain one velocity of the dispersion diagram is explained in the following steps:
- (a)
Input data: elastic constant, piezoelectric constant, piezomagnetic constant, dielectric permittivity, magnetic permeability, density respectively: Cijkl, eikl,qikl,ɛik , μik,ρ.
- (b)
Choose a non-dimensional wave-number and sweep the phase velocity.
- (c)
Solve
Result and discussion
In numerical analysis, the hetero-structure is a functionally graded piezoelectric film of 1=mm thick made of PZT-5H and a piezomagnetic substrate CoFe2O4 with the material properties given below (Table 1). The numerical development presented in the previous section is applied so as to find out the response of functional graded piezoelectric–piezomagnetic materials to the mechanical traction (Table 1).
Conclusion
The response of a non-homogeneous layer with an exponential variation in magneto-electromechanical properties has been presented. The effect of gradient coefficient on phase, group velocity, coupling factor, and cutoff wave-number number are deeply discussed. An interesting phenomenon is presented in this research which presents a higher peak of the magneto-electromechanical coupling coefficient in some wave number value and a less penetration depth by compared to the homogeneous
Declaration of Competing Interest
None.
Acknowledgments
This work was supported by the National Natural Science Foundation of China [grant numbers 11502108 and 1611530686], the State Key Laboratory of Mechanics and Control of Mechanical Structures at NUAA [grant number MCMS-E-0520K02], the Key Laboratory of Impact and Safety Engineering, Ministry of Education at Ningbo University [grant number CJ201904], and the Fundamental Research Funds for the Central Universities [grant number NE2020002, NS2019007] and a project Funded by the Priority Academic
Author statement
Hamdi Ezzin: Implementation of the computer Code, Writing - original draft preparation, Bin Wang: Visualization, Writing - reviewing and editing, Zhenghua Qian: Supervision, Funding acquisition.
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