Legendre polynomial modeling for vibrations of guided Lamb waves modes in [001]c, [011]c and [111]c polarized (1-x)Pb(Mg_1/3Nb_2/3)O₃-xPbTiO₃ (x = 0.29 and 0.33) piezoelectric plates: Physical phenomenon of multiple intertwining of A_n and S_n modes

Abstract.

Guided wave devices have recently become one of the most important applications in the industry because such waves are directly related to applications in sensor technology, chemical sensing, agricultural science, fields of bio-sensing and surface acoustic wave (SAW) devices that are used in electronic filters and signal processing. On that account, this numerical investigation aims to study the propagation behavior of guided Lamb waves in a ($1-x$)Pb(Mg_1/3Nb_2/3)O₃-x PbTiO₃ [PMN-x PT] ($x=0.29$ or 0.33) piezoelectric single crystal plate. In fact, the PMN-xPT ($x=0.29$ or 0.33) piezoelectric crystals are being polarized along $[001]c$, $[011]c$ and $[111]c$ of the cubic reference directions so that the macroscopic symmetries are tetragonal 4mm, orthogonal mm2 and rhombohedral 3m, respectively. Both open- and short-circuit conditions are considered. Here, the Legendre polynomial method is proposed to solve the guided Lamb waves equations. The validity of the proposed method is illustrated by comparison with the ordinary differential equation (ODE). The convergence of this method is discussed. Consequently, the converged results are obtained with very low truncation order M . This constitutes a major advantage of the present method when compared with the other matrix methods. There is cross-crossings among multiple modes for both symmetric ($S_{n}$) and the anti-symmetric ($A_{n}$) guided Lamb waves propagation. A displacement field has been illustrated to judge whether $S_{n}$ and $A_{n}$ modes cross with each other. Moreover, electric displacement, stress field and electric potential for the open-circuit case were presented for both $S_{0}$ and $A_{0}$ Lamb modes.