A Multi-Parameter Perturbation Solution for Functionally Graded Piezoelectric Cantilever Beams under Combined Loads.

Yongsheng Lian,Xue Li,Junyi Sun,Sijie Shi,Zhixin Yang,Xiaoting He

doi:10.3390/ma11071222

Abstract

In this study, we use a multi-parameter perturbation method to solve the problem of a functionally graded piezoelectric cantilever beam under combined loads, in which three piezoelectric coefficients are selected as the perturbation parameters. First, we derive the two basic equations concerning the Airy stress function and electric potential function. By expanding the unknown Airy stress function and electric potential function with respect to three perturbation parameters, the two basic equations were decoupled, thus obtaining the corresponding multi-parameter perturbation solution under boundary conditions. From the solution obtained, we can see clearly how the piezoelectric effects influence the behavior of the functionally graded piezoelectric cantilever beam. Based on a numerical example, the variations of the elastic stresses and displacements as well as the electric displacements of the cantilever beam under different gradient exponents were shown. The results indicate that if the pure functionally graded cantilever beam without a piezoelectric effect is regarded as an unperturbed system, the functionally graded piezoelectric cantilever beam can be looked upon as a perturbed system, thus opening the possibilities for perturbation solving. Besides, the proposed multi-parameter perturbation method provides a new idea for solving similar nonlinear differential equations.

Highlights

Graded piezoelectric materials (FGPMs) have been increasingly used in piezoelectric sensors and actuators [1,2]
The pioneer work in nonlinear beam problems was done by Chien [33], in which a biparametric perturbation method (B-PPM) was initially applied to solve the classical Euler-Bernoulli equation of beams with a height difference between the two ends from a practical engineering problem
We extended the traditional single-parameter perturbation method (S-PPM) and B-PPM to multi-parameter perturbation method (M-PPM) which contains three perturbation parameters and solved the governing equations of the Functionally graded piezoelectric materials (FGPMs) cantilever beam under combined loads

Summary

Introduction

Graded piezoelectric materials (FGPMs) have been increasingly used in piezoelectric sensors and actuators [1,2]. Su et al [28] dealt with the electro-mechanical vibration characteristics of FGPM rectangular plates with different boundary conditions based on first-order shear deformation theory He et al [29] presented an electroelastic solution for FGPM beams with different moduli in tension and compression. The pioneer work in nonlinear beam problems was done by Chien [33], in which a biparametric perturbation method (B-PPM) was initially applied to solve the classical Euler-Bernoulli equation of beams with a height difference between the two ends from a practical engineering problem. Later, He et al successfully used the so-called B-PPM to solve large deflection beam problems which.

Mechanical Model and Basic Equations

Perturbation Solution

Results and Discussions

Concluding Remarks