Abstract

In this study, the stochastic finite element solution is given to obtain the variability in the natural frequency of functionally graded material (FGM) beam. The elastic modulus is assumed to vary in the thickness direction and the width of the beam to vary as well in the longitudinal direction following the exponential law. The random material properties of elastic modulus and mass density of the FGM beam are assumed to be one-dimensional homogeneous stochastic processes. The stochastic finite element analysis of FGM beam is performed in conjunction with Monte Carlo simulation (MCS) employing the spectral representation method for 16, the description of random processes of the random material properties under consideration. The response variability of the natural frequency due to random elastic modulus is evaluated for various states of randomness. Furthermore, the investigation on the effect of the correlation between random elastic modulus and random mass density on the response variability is addressed in detail as well.

Keywords: functionally graded materials, finite element method, FGM beam, Monte Carlo simulation

## 1. Introduction

Functionally graded materials (FGM) have received considerable attention in many engineering applications, since the theory of FGM was firstly introduced in 1984. In general, FGM is made from the volume fractions of two or more material components that have continuous variation of material properties from one surface to another [1]. Nowadays, FGM suits the specific demand in different engineering applications, especially for high temperature environment applications of heat exchanger tubes, thermal barrier coating for turbine blades, thermoelectric generators, furnace linings, electrically insulated metal ceramic joints, space/aerospace industries, automotive applications, biomedical area, etc.

The manufacturing of FGM with fully specified profile of material gradation, however, is very difficult causing significant variability in their mechanical and structural properties. Therefore, proper handling of the randomness in the material properties is required for accurate prediction of structural response for safe and reliable design. The stochastic analysis is a useful analytical tool to predict the

response of structures with random material properties. In this direction, there is a reasonable body of recent research on the effect of uncertainties in material properties on the mechanical behavior of FGM. Investigators used stochastic simulation to study the effect of microstructural randomness on stress in FGM [2]. Ferrante et al. studied the effect of non-Gaussian porosity randomness on the response of functionally graded plate [3]. Yang et al. dealt with the stochastic bending response of moderately thick FGM plates [4, 5]. The effect of random material properties on post buckling response of FGM plate are presented in Lal et al. [6].

However, the above mentioned literatures are for the static analysis. To the best of author's knowledge, few limited works have been done on the eigen analysis of FGM structures involving randomness in system parameters. Certain efforts have been made in the past to predict the dynamic behavior of structures with randomness. In most of the studies conducted, investigators dealt with the free vibration of functionally graded laminates with random material properties using first-order perturbation technique (FOPT) incorporating mixed type and semi-analytical approach to derive the standard eigenvalue problem [7]. Some of these papers presented the stochastic finite element method (SFEM) to investigate the natural frequency of functionally graded plates based on the higher-order shear deformation theory (HSDT) utilizing first-order reliability method and second-order reliability method [8]. In most cases, Jagtap et al. [9] examined the stochastic nonlinear free vibration response of FGM plate using HSDT with von-Karman kinematic nonlinearity via direct iterative stochastic finite element method. Shegokar et al. investigated the stochastic finite element nonlinear free vibration analysis of FGM beam with random material properties due to thermo-piezoelectric loadings [10]. The above mentioned literatures investigated the free vibration and nonlinear behavior of FGM beam and plate. The material properties, such as Young's modulus, shear modulus, and Poisson's ratio of FGM, are modeled as independent random variables.

In this chapter, the stochastic finite element solution is suggested to obtain the variability in the natural frequency of functionally graded material (FGM) beam. The elastic modulus and width of the FGM beam are assumed to vary in thickness and longitudinal directions following the exponential law. The uncertain material properties, such as modulus of elasticity and mass density of the FGM beam, are considered to be a one-dimensional homogeneous stochastic process. The stochastic finite element analysis of FGM beam is performed using the spectral representation method for the description of randomness in conjunction with Monte Carlo simulation (MCS). The response variability of natural frequency due to random elastic modulus and mass density in FGM beam is given. Furthermore, the effect of correlation between the two random parameters is observed as well.

E zð Þ¼ ð Þ Ec � Em g zð Þþ Em

The Stochastic Finite Element in the Natural Frequency of Functionally Graded Material Beams

where E denotes the effective Young's modulus of elasticity, and Em and Ec represent the Young's modulus of metal and ceramic, respectively. The parameters g(z) and p represent the volume fraction of the metal and ceramic exponent,

In exponential law, for the material parameter of Young's modulus E, mass density ρ of the beam and the width of the beam b, with absolute values for z coordinate, which endows the symmetric characteristic to the beam with respect to

<sup>β</sup>j j <sup>z</sup> ; <sup>ρ</sup>ð Þ¼ <sup>z</sup> <sup>ρ</sup>0<sup>e</sup>

In Eq. (3), E0, ρ<sup>0</sup> are the values of the Young's modulus and mass density at the mid-plane (z = 0) of the beam. The parameter β in the exponent characterizes the material property variation along the thickness direction. The width of the beam b varies according to the non-uniformity parameter ψ along the axis of the beam. In order to give the insight into the varying characteristics of the FGM beam under

, � 0:5h≤ z≤ 0:5h, 0≤ p ≤ ∞

<sup>β</sup>j j <sup>z</sup> ;b xð Þ¼ <sup>b</sup>0<sup>e</sup>

ψx

with

Figure 2.

Figure 1.

Model of FGM beams.

DOI: http://dx.doi.org/10.5772/intechopen.86013

respectively.

mid-plane.

63

g zð Þ¼ <sup>1</sup>

2 þ z h <sup>p</sup>

The variation of elasticity modulus along the corresponding directions.

E zð Þ¼ E0e
