Preface

Functionally graded materials (FGMs) are inhomogeneous materials, consisting of two (or more) different materials, engineered to have a continuously varying spatial composition profile. This is achieved by gradually varying the volume fraction of the constituent materials. The concept is to produce a composite material by varying the microstructure from one material to another material with a specific gradient. The materials can be designed for specific functions and applications. FGMs offer great promise in applications where the operating conditions are extreme. For example, wear-resistant linings for handling large, heavy, abrasive ore particles; rocket heat shields; heat exchanger tubes; thermoelectric generators; heat-engine components; plasma facings for fusion reactors; and electrically insulating metal/ ceramic joints. They are also ideal for minimizing thermomechanical mismatch in metal-ceramic bonding.

This book is a result of contributions of experts from the international scientific community working in different aspects of functionally graded materials and structures and reports on the latest research and development findings on this topic through original and innovative research studies. Through its six chapters, the reader will have access to works related to processing, characteristics, modeling, and applications of functionally graded materials and structures.

The book contains up-to-date publications from leading experts and the edition is intended to provide valuable recent information to the professionals involved in functionally graded materials and structure analysis and applications. The text is addressed not only to researchers, but also to professional engineers, students, and other experts in a variety of disciplines, both academic and industrial seeking to gain a better understanding of what has been done in the field recently, and what open problems are in this area.

I hope that the readers will find this book useful and inspiring by examining the recent developments in functionally graded materials and structures.

Lastly, I would like to thank all the authors for their excellent contributions in different areas covered by this book, and the IntechOpen team, especially the process manager Ms. Ivana Barac, for their support and patience throughout the publishing process.

> **Dr. Farzad Ebrahimi** Department of Mechanical Engineering, Imam Khomeini International University, Qazvin, Iran

Chapter 1

Abstract

vibration of AFG beam.

1. Introduction

(length) direction.

1

Free Vibration of Axially

Functionally Graded Beam

Dongxing Cao, Bin Wang, Wenhua Hu and Yanhui Gao

Axially functionally graded (AFG) beam is a special kind of nonhomogeneous functionally gradient material structure, whose material properties vary continuously along the axial direction of the beam by a given distribution form. There are several numerical methods that have been used to analyze the vibration characteristics of AFG beams, but it is difficult to obtain precise solutions for AFG beams because of the variable coefficients of the governing equation. In this topic, the free vibration of AFG beam using analytical method based on the perturbation theory and Meijer G-Function are studied, respectively. First, a detailed review of the existing literatures is summarized. Then, based on the governing equation of the AFG Euler-Bernoulli beam, the detailed analytic equations are derived on basis of the perturbation theory and Meijer G-function, where the nature frequencies are demonstrated. Subsequently, the numerical results are calculated and compared, meanwhile, the analytical results are also confirmed by finite element method and the published references. The results show that the proposed two analytical methods are simple and efficient and can be used to conveniently analyze free

Keywords: axially functionally graded beams, free vibration, natural frequency, asymptotic perturbation method, Meijer G-function, finite segment model

Functionally gradient materials (FGMs) make a composite material by varying the microstructure from one material to another material with a specific gradient. It can be designed for specific function and applications. If it is for thermal or corrosive resistance or malleability and toughness, both strengths of the material may be used to avoid corrosion, fatigue, fracture, and stress corrosion cracking. FGMs are usually made into several structures, such as beams [1–4], plates [5–8], and shells [9–12]. In this area, the variation of material properties in functionally graded beams may be oriented in transverse (thickness) direction or/and longitudinal/axial

For functionally graded beams with thickness-wise gradient variation, there have been many studies devoted to this topic. Lee et al. [13] establish an accurate transfer matrix method to analyze the free vibration characteristics of FGM beams whose Young's modulus and density vary continuously with the height of the beam section through power law distribution. Su et al. [14] developed the dynamic stiffness method to investigate the free vibration behavior of FGM beams. Jing et al. [15]
