Characteristic orthogonal polynomials-Ritz method for vibration behavior of functionally graded piezoelectric plates using FSDT

Abstract

This work focuses on the free vibration characteristics of the functionally graded piezoelectric (FGPE) plate with classical and elastic constraints. First, the analytical model is established for the FGPE plate on the basis of the first-order shear deformation theory (FSDT). Besides, different distribution profiles of the material constituent are considered for the FGPE plate model, and various external initial voltages are also considered for modeling the piezoelectric effect. The displacement functions for the FGPE plate are then given in the form of third-kind orthogonal polynomials. Ultimately, the solution of the FGPE plate model is resolved by means of the Ritz method. Convergence and accuracy of the presented model are verified, and a series of parametric investigations are given.

Introduction

Based on existing investigations, the laminated piezoelectric structures are able to generate the stress concentration due to its structure characteristics. In order to reduce the stress concentration, the functionally graded piezoelectric plate is proposed and manufactured, and has been widely employed in engineering fields [1]. With the advancement of the functionally graded materials (FGMs) [2], the functionally graded piezoelectric (FGPE) plate has been extensively investigated due to its coupled properties in both mechanical and electrical fields [3], [4], [5], [6], [7], [8]. Similar to the conventional FGMs, the FGPE plate is also a kind of multilayer structure in the form of shell, plate and beam, etc., but with additional piezoelectric layer, and thus constitutes a hybrid structure with smart (or intelligent) properties [7]. The applications of the FGPE are mostly found in smart and intelligent systems, sensing and actuating systems as well as the electro-mechanical systems [9], [10], [11], [12].

Benefiting from the properties of the additional piezoelectric layer of the FGPE plates, some emerging problems may have suffered from the application of the FGPE plate [13]. For one reason, the property of the piezoelectric material layer differs from the property of the FGMs layer [4]. Therefore, the interaction, interference and coupling effect, especially at the contiguous interfaces of the piezoelectric layer and the FGMs layer, are hard to be figured out. On the other hand, it is well known that the FGMs are heterogeneous materials, the characteristics of which alter continuously in certain directions through the thickness coordinate [7]. This also increases the complexity of the FGPE plate, and hence brings about more difficulties in the acquisitions of the optimum responses (i.e., vibration, dynamic response, etc.) to externally applied electrical and mechanical loading for the FGPE plate. To solve the above-concerned issues, numerous investigations have been undertaken for better understanding of the properties of the FGPE plate, and the modeling of this kind of structure was given the first priority.

Regarding the modeling of the PGPE plate, Ke et al. [14] and Ebrahimi et al. [15] constructed a piezoelectric beam by considering the theory of nonlocal elasticity. Furthermore, the nonlinear vibration of this model is analyzed. Then, a functionally graded materials-based double-wall piezoelectric plate [16], and a piezoelectric plate reinforced by functionally graded graphene [17] are constructed for research purpose. Subsequently, Alibeigloo et al. [18] proposed an exact solution for a rectangular plate consisting of FGM and piezoelectric layers with consideration of the elastic foundation, transverse loading; Li et al. developed a size-dependent functionally graded piezo electric microplate model [19]; Fakhari et al. [20] employed higher-order shear deformation plate theory for modeling of functionally graded plate with surface-bonded piezoelectric layers; A two piezoelectric layers' FGP plate was developed by Abad [21]; Mao et al. [22] constructed a piezoelectric plate with FGM on the basis of higher-order shear plate theory and elastic piezoelectric theory. It can be found that various methods have been tried in the modeling of the FGPE plate. A detailed modeling method definitely contains necessary information for accurate description and further analysis of the FPGE plate, whilst, exceeding theories and conditions in the modeling process inevitably bring out computational burden and possible parameter interferences. Therefore, one of the research targets of this work is to figure out a unified modeling method for the FGPE plate.

Intense interests have been found in the behaviors of the FGPE plate in addition to its solely modeling [22], [23], [24], [25], of which the vibration analysis [26], [27], [28], [29] has become the most attractive one as it significantly concerns to the electrical and mechanical behavior of the FGPE plate [30]. To be more specific, active vibration [22], [28], free vibration [31], [32], [33], nonlinear free vibration and steady-state forced vibration [20] are conducted for vibration analysis of the FPGE plate as distinct investigations.

However, failures of the piezoelectric function from the interface where the piezoelectric elements exist cannot be avoided due to the material compositional gradient of the FGMs and the fragility of the piezoelectric elements [28]. Moreover, the coupling effect at the contiguous interfaces of the piezoelectric layer and the FGMs layer determines that the mechanical deformations/loads and the electric field interact with each other [30]. Therefore, an electro-mechanical vibration analysis on the vibration analysis is essential for the FGPE plate. As a result, one of the objectives of this work is to conduct the electro-mechanical vibration analysis with the constructed FGPE plate model.

With the above discussion, it is concluded that almost all of the existing investigations on the FGPE plate is constructed based on different boundary conditions including the simple supported general boundary condition and the clamped boundary condition [34]. However, the boundary conditions of the functional graded-related structure are not always ideal classical in real practice, but under diversified boundary cases. Furthermore, the customized numerical method for given illustrations of the boundary conditions also determines that it is not applicable for different boundary conditions, leading to a high sacrifice to the computational cost in adjusting the solution process. In light of the aforesaid concerns, a general functional graded model with elastic constraints has been reported in the many investigations [35], [36], [37], [38], [39]. To this end, a unified FGPE plate model with elastic constraints is desired for vibration analysis.

This work copes with free vibration characteristics of functionally graded piezoelectric (FGPE) plates considering the electro-mechanical coupling with general elastic constraints. Firstly, the modeling process are given in Section 2, which includes the theoretical formulation and solution method of the FGPE plate. Then, the convergence study is presented in Section 3, following with the validation on free vibration, and a series of parametric investigations considering coupling relations between the model parameters. Ultimately, a conclusion is summarized in the final section.