In this paper, nonlinear vibration analysis of Functionally graded piezoelectric (FGP) beam with porous materials is investigated based on the Timoshenko beam theory. Material properties of FG porous beam are described according to the rule of mixture which is modified to approximate material properties with porosity phases. Ritz method is used to obtain the governing equation which is then solved by a direct iterative method to determine the nonlinear vibration frequencies of FGP porous beam subjected to different boundary conditions. The effects of external electric voltage, material distribution profile, porosity volume fraction, slenderness ratios and boundary conditions on the nonlinear vibration characteristics of the FGP porous beam are discussed in detail. The results indicate that piezoelectric layers have a significant effect on the nonlinear frequencies. Also, it is found that porosity has a considerable influence on the nonlinear frequency, especially when the electric voltage is applied.

A novel two-dimensional elasticity solution is presented in this paper, specifically designed for studying the vibration of Functionally graded porous (FGP) beams. The kinetics of the beam are defined by two-dimensional elasticity theory, and Lagrange’s equations are used to derive the governing equations of motion. The Ritz method devises the expansion of displacement variables in polynomial and trigonometric series in the thickness and axial directions. Furthermore, microvoids can emerge as a result of technical issues during the manufacture of Functionally graded materials (FGMs), leading to the development of porosities. The porosity distribution functions, one for three porosity distributions: uniform porosity (UP), non-uniform porosity-I (NUP-I), and non-uniform porosity-II (NUP-II), are considered in the problem. This study investigates the impact of the gradation exponents (p) in the z-direction, the slenderness ratio (L/h), the distribution of porosity, the porosity coefficient (e), and various boundary conditions on the natural frequencies. A comparison with the findings from higher-order shear deformation theory (HSDT) validated the accuracy and effectiveness of the proposed methodology.

Functionally graded material (FGM) is an in-homogeneous composite, constructed from various phases of material elements, often ceramic and metal. This work aims to examine the behavior of free vibration in porous Functionally graded beams (FGBs) in 2 directions (2D) by using nth-order shear deformation theory. With the help of Hamilton's principle and Reddy's beam theory, equilibrium equations for free vibration were derived. Boundary conditions such as Simply Supported – Simply Supported (SS), Clamped – Clamped (CC) and Clamped-Free (CF) were employed. A unique shear shape function was derived and nth-order theory was adapted to take into account the effect of transverse shear deformation to get zero shear stress conditions at the top and bottom surfaces of the beam. Based on power law, FGB properties were changed in length and thickness directions. The displacement functions in axial directions were articulated in algebraic polynomials, including admissible functions which were used to fulfill different boundary conditions. Convergence and verification were performed on computed results with findings of previous studies. It was found that the results obtained using the nth-order theory were in agreement and allows for better vibration analysis in a porous material.

In this article, thermal stability of beams made of Functionally graded material (FGM) is considered. The derivations of equations are based on the one-dimensional theory of elasticity. The material properties vary continuously through the thickness direction. Tanigawa's model for the variation of Poisson's ratio, the modulus of shear stress, and the coefficient of thermal expansion is considered. The equilibrium and stability equations for the Functionally graded beam under thermal loading are derived using the variational and force summation methods. A beam containing six different types of boundary conditions is considered and closed form solutions for the critical normalized thermal buckling loads related to the uniform temperature rise and axial temperature difference are obtained. The results are reduced to the buckling formula of beams made of pure isotropic materials….

In this paper, the static bending, free vibration, and dynamic response of Functionally graded piezoelectric beams have been carried out by finite element method under different sets of mechanical, thermal, and electrical loadings. The beam with Functionally graded piezoelectric material (FGPM) is assumed to be graded across the thickness with a simple power law distribution in terms of the volume fractions of the constituents. The electric potential is assumed linear across the FGPM beam thickness. The temperature field is assumed to be of uniform distribution over the beam surface and through the beam thickness. The governing equations are obtained using potential energy and Hamilton's principle based on the Euler-Bernoulli beam theory that includes thermo-piezoelectric effects. The finite element model is derived based on the constitutive equation of piezoelectric material accounting for coupling between the elasticity and the electric effect by two nodes Hermitian beam element. The present finite element is modelled with displacement components and electric potential as nodal degrees of freedom. The temperature field is calculated by post-computation through the constitutive equation.Results are presented for two-constituent FGPM beam under different mechanical boundary conditions. Numerical results include the influence of the different power law indexes, the effect of mechanical, thermal, and electrical loadings and the type of in-plane boundary conditions on the deflection, stress, natural frequencies, and dynamic response. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.

Pores affect Functionally graded materials. Further characteristics may be added if pores expand from the surface to the interior. Functionally graded porous beam (FGPB) bending response is analyzed using a specific shear shape function that accounts for both uniform and uneven porosity distributions. Power law changes the material characteristics of FGPBs with uniform and uneven porosity distributions along length and thickness. In order to determine the maximum transverse deflections, axial stresses, transverse shear stresses, and normal stresses in simply-supported and clamped-clamped beams, numerical calculations are performed with various gradation exponents, aspect ratios (L/h), and porosity levels (both even and uneven). The obtained results are compared with earlier investigations and justified.

The present study investigates buckling characteristics of both nonlinear symmetric power and sigmoidFunctionally graded (FG) beams. The volume fractions of metal and ceramic are assumed to be distributed througha beam thickness by the sigmoid-law distribution (S-FGM), and the symmetric power function (SP-FGM). Thesefunctions have smooth variation of properties across the boundary rather than the classical power law distributionwhich permits gradually variation of stresses at the surface boundary and eliminates delamination. The Voigt modelis proposed to homogenize micromechanical properties and to derive the effective material properties. The Euler-Bernoulli beam theory is selected to describe Kinematic relations. A finite element model is exploited to formstiffness and buckling matrices and solve the problem of eignivalue numerically. Numerical results present theeffect of material graduations and elasticity ratios on the buckling behavior of FG beams. The proposed model ishelpful in stability of mechanical systems manufactured from FGMs.