Guided wave in multilayered piezoelectric–piezomagnetic bars with rectangular cross-sections
Introduction
Over the past ten years, piezoelectric–piezomagnetic composites (PPC) have received considerable research effort with their increasing usage in various applications including sensors, actuators and storage devices [1], [2], [3], [4]. For the purpose of design and optimization of PPC transducers, wave propagation in various PPC attracted many researchers.
Chen and Shen [5] obtained effective wave velocity and attenuation factor when axial shear magneto-electro-elastic waves propagate in piezoelectric–piezomagnetic composites. By using the state space approach, Zhou et al. [6] investigated the bulk wave propagation in laminated piezomagnetic piezoelectric plates with initial stresses and imperfect interface. Using the propagator matrix and state-vector (or state space) approaches, an analytical treatment is presented for the propagation of harmonic waves in magneto-electro-elastic multilayered plates by Chen et al. [7]. By using Legendre orthogonal polynomial series expansion approach, Yu et al. investigated the guided waves in imhomogeneous magneto-electro-elastic hollow cylinders [8] and spherical curved plates [9].
In order to analyze the band gaps, wave propagation in piezoelectric–piezomagnetic periodically layered structures received attentions [10], [11], [12]. Li et al. [13] discussed the penetration depth of the Bleustein–Gulyaev waves in a functionally graded transversely isotropic electro-magneto-elastic half-space. SH waves propagating in piezoelectric–piezomagnetic layered structures with imperfect interfaces were investigated by Sun et al. [14] and Nie et al. [15]. Pang and Liu [16] discussed the reflection and transmission of plane waves at an imperfectly bonded interface between piezoelectric–piezomagnetic media. By using the Jacobi elliptic function expansion method, Xue et al. [17] investigated the solitary waves in a magneto-electro-elastic circular bar. Matar et al. [18] used Legendre and Laguerre polynomial approach for modeling of wave propagation in layered magneto-electro-elastic media. Xue and Pan [19] discussed the influences of the gradient factor on the longitudinal wave along a functionally graded magneto-electro-elastic bar. Zhang et al. [20] studied the influences of initial stresses on Rayleigh wave propagation in a magneto-electro-elastic half-space.
As can be seen from the above simple review, wave motions in many magneto-electro-elastic structures have been considered. But these structures are almost only for semi-infinite structures and one-dimensional structures, i.e. structures having a finite dimension in only one direction, such as horizontally infinite flat plates and axially infinite hollow cylinders. In practical applications, many sensitive elements have limited finite dimension in two directions. One-dimensional models are thus not suitable for these structures. On the other hand, wave propagation in purely elastic 2-D structures has been addressed by researchers. Kastrzhitskaya and Meleshko [21] proposed an exact analytical method for an effective solution of the problem of rectangular waveguide. Taweel et al. [22] used a semi-analytical finite element method to study the layered rectangular bars. Also using a semi-analytical finite element method, Hayashi et al. [23] analyzed square bars and rail waveguides. Gunawan and Hirose [24] investigated the rectangular bars by boundary element method. By means of standard commercial finite element codes, Sorohan et al. [25] investigated a layered composite plate and a square tube.
In this paper, a double orthogonal polynomial series approach is proposed to solve wave propagation in a 2-D PPC structure, namely a multilayered PPC bar with a rectangular cross-section. Traction-free and open-circuit boundary conditions are assumed in this analysis. Two cases are considered: the material stacking direction and the polarizing direction are identical and orthogonal to each other, respectively. The dispersion curves and the mechanical displacement profiles of various layered PPC rectangular bars are presented and discussed.
Section snippets
Mathematics and formulation of the problem
We consider a multilayered piezoelectric rectangular bar which is infinite in the wave propagation direction. Its width is d, the total height is , and the stacking direction is in the z-direction, as shown in Fig. 1. Its polarization direction is in the z direction. The origin of the Cartesian coordinate system is located at a corner of the rectangular cross-section and the bar lies in the positive -region, where the cross-section is defined by the region and .
For the wave
Numerical results and discussions
Based on the solution procedure as described in the previous section, a computer program in terms of the double polynomial approach has been written using Mathematica to calculate the wave dispersion curves, the mechanical displacement and electric potential distributions for the layered PPC rectangular bars.
Conclusions
In this paper, a double orthogonal polynomial series approach is proposed to solve wave propagation of 2-D layered PPC rectangular bars. The dispersion curves and mechanical displacement distributions of various layered PPC rectangular bars are presented and discussed. According to the numerical results, we can draw the following conclusions:
- (a)
Numerical comparison of the dispersion curves with reference solutions shows that the double orthogonal polynomial method can accurately and efficiently
Acknowledgments
The work was supported by the National Natural Science Foundation of China (No. 11272115) and the Outstanding Youth Science Foundation of Henan Province (No. 144100510016), China. Jiangong Yu also gratefully acknowledges the support by the Alexander von Humboldt-Foundation (AvH) to conduct his research works at the Chair of Structural Mechanics, Faculty of Science and Technology, University of Siegen, Germany.
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