本文基於Reissner混合變分原理（Reissner’s mixed variational theorem，RMVT），發展有限層板法（finite layer methods，FLMs）歸一理論，並將其應用於簡支承功能性壓電材料（functionally graded piezoelectric material，FGPM）板在開路和閉路的表面條件以及機電耦合載荷下的三維（three-dimensional，3D）電彈耦合分析。文中假設板的材料性質沿著厚度方向呈指數變化，且板被細分為一定數量的有限矩形層，用三角函數和Lagrange多項式分別對每一獨立層主變量的平面內和平面外變數做插值，如彈性位移、橫向剪應力和正應力、電位勢和法線方向電位移。該有限層板法中用於描述這些主變數沿著板厚方向冪級數變化之相關階數可以自由選擇為線性、二次或三次。在板的頂層和底層表面施加四種不同的機/電荷載條件，並對此FGPM板進行相關電彈耦合分析。文中亦綜合比較本RMVT有限層板方法的解和文獻中的3D精確解，藉以評估本方法的精確度和收斂率。

Coupled electro-elastic analysis of functionally graded piezoelectric material plates using RMVT-based finite layer methods

Author: Shuang Ding
Advisor: Prof. Chih-Ping Wu
Department of Civil Engineering National Cheng Kung University

SUMMARY

A unified formulation of finite layer methods (FLMs), based on the Reissner mixed variational theorem (RMVT), is developed for the coupled electro-elastic analysis of simply-supported, functionally graded piezoelectric material (FGPM) plates. The material properties of the plate are assumed to obey an exponent-law varying exponentially through the thickness coordinate. The accuracy and convergence rate of the RMVT-based FLMs are assessed by comparing their solutions with the exact 3D piezoelectricity ones available in the literature.

Keywords: Coupled electro-elastic analysis; static; finite layer methods; functionally graded materials; piezoelectric plates.

INTRODUCTION

In recent decades, piezoelectric materials have been widely used in smart structures. However, many reports have examined conventional laminated piezoelectric structures, the material properties of which mismatch at the interfaces between adjacent layers, with reports indicating that in practical applications a number of weakness occur at these loactions (Kashtalyan and Menshykova, 2009; Woodward and Kashtalyan, 2010).

A new class of smart structures, called functionally graded piezoelectric material (FGPM) structures, the material properties of which continuously and gradually vary through the thickness coordinate, has thus been developed to overcome these drawbacks. The coupled analysis of FGPM structures has since attracted considerable attention with the aims of both improving their working performances and enhancing their lifetime.

Based on the three-dimensional (3D) piezoelectricity theory, some exact solutions for the bending, vibration and buckling analyses of simply-supported, FGPM plates have been presented to assess the accuracy and convergence rates of various related two-dimensional (2D) and approximate 3D theories (Pan and Han, 2005; Lu et al., 2005, 2006; Wu and Tsai, 2007,2009). In order to extend the scope of the coupled electro-elastic analyses of FGPM structures, a number of numerical methods combining 2D and 3D theories have been presented, such as the finite element, finite strip and meshless approaches (Carrera, 2003; Carrera et al., 2008, 2010; Wu and Li, 2010 a, b; Wu and Chang, 2012; Wu et al., 2014).

This article developed an RMVT-based FLM for the static analysis of simply-supported, FGPM FSPs with closed- and open-circuit surface conditions and subjected to four different loading conditions. The accuracy and convergence rate of the RMVT-based FLMs are assessed and the numerical examples show that FGPM FSPs overcome some of the drawbacks of conventional homogeneous FSPs.

RMVT-BASED FLMS

In this article, we consider a simply-supported, FGPM plate with the open- and closed-circuit surface conditions, and subjected to electro-mechanical loads on the top and bottom surfaces, as shown in Fig. 2a, in which the plate is artificially divided into a number of rectangular layers. A Cartesian global coordinate system is located on the middle plane of the plate, and a set of Cartesian local thickness coordinates is located at the mid-plane of each divided layer, as shown in Fig. 1b.

The elastic displacement, electric potential components, the transverse shear and normal stress components, and the normal electric displacement one, are regarded as the primary variables in these RMVT-based FLMs. The Reissner mixed variational theorem, which includes the generalized kinematic and kinetic assumptions, is used to derive the Euler-Lagrange equations of the plate for RMVT-based FLMs, and we may express the first-order variation of the Reissner energy functional as Eqs. (2.26) in the article.

The mechanical loads , the electric potentials and the electric normal displacements are expressed as the double Fourier series. By means of the separation of variables, the primary field variables of each individual layer are expanded as the forms of a double Fourier series so that the boundary conditions of the simply supported edges are exactly satisfied.

Imposing the stationary principle of the Reissner energy functional, we obtain the Euler-Lagrange equations of the plate as Eqs. (2.35). Using this unified formulation of RMVT-based FLMs, we may analyze the 3D coupled electro-elastic behaviors of FGPM plates and laminated homogeneous piezoelectric ones with closed- and open-circuit surface conditions and under electro-mechanical loads.

RESULTS AND DISCUSSION

Single-layered FGPM plates

In this section, the static behavior of a simply-supported, single-layered FGPM plate is investigated. PZT-4 is used as the reference material. The material properties of the plate are assumed to vary exponentially through the thickness coordinate.

Tables 2 and 3 and Fig. 4 show the accuracy and convergence rate of the FLMs with different orders compared with the results of Lu et al. (2006) and Brischetto and Carrera (2009). It can be seen that the accuracy and convergence rate for various FLMs are > > , in which the symbol “>” means more accurate and more rapid. It is also shown that the convergence solutions are in excellent agreement with the exact 3D solutions and CUF ones, in which the relative errors of various FLM solutions of all electric and elastic variables will be lower than 1% as compared with the 3D solutions.

Multi-layered piezoelectric laminated plates

In this section, the static behavior of a simply-supported, single-layered FGPM plate is investigated. The plate is consist of two layers of composite material and two additional piezoelectric layers on the top and bottom surfaces.

Tables 4 and 5 and Fig. 6 show the accuracy and convergence rate of the FLMs with different orders compared with the results of Heyliger (1994). We can get a same conclusion as that of the previous example. The RMVT-based FLMs with cubic orders are thus used in the later work in this article.

Two-layered FGPM film/substrate plates

In this section, we consider a simply-supported, two-layered FGPM film-substrate plates (FSP) under electro-mechanical loads. The substrate is a homogenous PZT-4 layer, and the film is an FGPM one bounded on the top surface of the substrate.

Figs 8-11 show the through-thickness distributions of electric and elastic variables induced in the FGPM film-substrate plate. It can be seen in Figs. 8 and 10 that in the cases of applied mechanical loads, the in-plane elastic displacement and stress, transverse shear stress, and transverse normal stress variables appear to be the linear, parabolic and higher-order polynomial variations through the thickness for the homogeneous piezoelectric plates, while those for FGPM FSPs change more dramatically than those for homogeneous piezoelectric ones. The effects of different surface conditions on the through-thickness distributions of elastic variables are very minor in the FGPM FSPs, but significant for those of electric variables. The results in Figs. 9 and 11 show the effects of different surface conditions on the through-thickness distributions of both electric and elastic variables are significant when the electric loads are applied.

Figs 12 and 13 show the through-thickness distributions of elastic and electric variables induced in the FGPM FSPs and homogeneous ones for the loading Case 1. It can be seen in Figs. 12 (c) and 13 (c) that the transverse shear stresses in the homogeneous FSPs change dramatically through the thickness when the deviations of the material properties between the film and substrate layer become greater, while this situation will be reduced when we use an FGPM film. Figures. 12(b) and 13(b) show the in-plane stresses induced in the homogenous FSPs change abruptly at the film-substrate interface due to the mismatched material properties occurring at this location, while those induced in the FGPM FSPs vary smoothly through the thickness coordinate and are continuous at the film-substrate interface.

CONCLUSION

In this article, we developed an RMVT-based FLM for the static analysis of simply-supported, FGPM FSPs with closed- and open-circuit surface conditions and subjected to four different loading conditions.

In the implementation of various FLMs, the results show that > > , in which the symbol “>” means more accurate results and a more rapid convergence rate.

In the numerical example, it is shown that FGPM film reduce the situation that the transverse shear stresses change dramatically through the thickness coordinate in the homogeneous FSPs, especially when the deviations of the material properties between the film and substrate layers is great. The in-plane stresses induced at the film-substrate interface for the homogeneous FSPs change abruptly due to the mismatched material properties occurring at that location, while those for the FGPM FSPs vary smoothly through the thickness coordinate of the plate and are continuous across the film-substrate interface.

Moreover, the through-thickness distributions of electric and elastic variables induced in the FGPM FSPs and homogeneous ones appear to be layer-wise higher-order polynomial variations.

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