Wave propagation in layered piezoelectric rectangular bar: An extended orthogonal polynomial approach
Introduction
In the past four decades, wave propagation in piezoelectric structures has received considerable attention from engineering and scientific communities because of their applications in ultrasonic nondestructive evaluation and transducer design and optimization. For these piezoelectric devices, layered model consisting of piezoelectric and non-piezoelectric layers stacked in a certain sequence is common. Many solution methods have been used to investigate the wave propagation in multilayered piezoelectric structures, including analytical method [1], [2] and various numerical method. The mostly used method is the transfer matrix method (TMM) [3], [4], [5] and the finite element method (FEM) [6].
Because the TMM and FEM suffer from numerical instability in some particular cases, some improvements have been developed, such as the recursive asymptotic stiffness matrix method [7], [8], the surface impedance matrix method [9], [10], the scattering-matrix method [11] and the reverberation-ray matrix method [12].
In 1972, the orthogonal polynomial approach has been developed to analyze linear acoustic waves in homogeneous semi-infinite wedges [13]. After that, this approach has been used to solve various wave propagation and vibration problems, including surface acoustic waves in layered semi-infinite piezoelectric structures [14], [15], Lamb-like waves in multilayered piezoelectric plates, [16] multilayered piezoelectric curved structures [17], [18] and multilayered magneto-electro-elastic plates [19].
So far, investigations on wave propagation in multilayered piezoelectric structures are 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. But in practical applications, many piezoelectric elements have finite dimensions in two directions. One-dimensional models are thus not suitable for these structures. To the extent of the authors’ knowledge, wave propagation in multilayered 2-D piezoelectric structure has not been reported. This paper proposes an extension of the orthogonal polynomial series approach to solve wave propagation problems in a 2-D piezoelectric structure, namely, multilayered piezoelectric bar with a rectangular cross-section. In particular, two cases are considered: the material stacking direction and the polarization direction are parallel and perpendicular, respectively. Traction-free and open-circuit boundary conditions are assumed in this analysis. The wave dispersion curves and the electric potential profiles of various multilayered piezoelectric rectangular bars are presented and discussed.
Section snippets
Problem formulation and solution method
We consider a multilayered piezoelectric rectangular bar which is infinite in the wave propagation direction. Its width is d, the total height is h = hN, 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 y–z-region, where the cross-section is defined by the region 0 ⩽ z ⩽ h and 0 ⩽ y ⩽ d.
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 extended polynomial approach has been written using Mathematica to calculate the wave dispersion curves, the displacement and the electric potential distributions for the layered piezoelectric rectangular bars.
Conclusions
This paper extends the orthogonal polynomial approach to solve wave propagation problems in 2-D multilayered piezoelectric rectangular bars. The dispersion curves, the displacement and the electric potential distributions of various layered rectangular bars are obtained. According to the present numerical results, we can draw the following conclusions:
- (a)
Numerical comparison of the wave dispersion curves show that the extended orthogonal polynomial method can accurately and efficiently solve the
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No. 11272115), China and the Outstanding Youth Science Foundation of Henan Polytechnic University (No. J2013-08), China. Jiangong Yu also gratefully acknowledges the support by the Alexander von Humboldt Foundation (AvH), Germany to conduct his research works at the Chair of Structural Mechanics, Faculty of Science and Technology, University of Siegen, Germany.
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