Mechanics of Functionally Graded Material Structures

The following sections are included:

• Dedication
• Preface
• Table of Contents

The following sections is included:

• Introduction

In structural engineering, functionally graded materials (FGMs) are mostly employed as constituent materials of rectangular plates. Thus, this chapter is devoted to a brief overview about the most common elastic plate theories, starting from the classical (Leissa, 1973), and the shear deformation theories (Mindlin, 1951; Reddy, 1984). It is well known that these structural models are based on assumptions concerning the kinematics of deformation or the stress through the thickness of the plate…

The superior properties of advanced composite materials, such as specific high strength and high stiffness, have led to their widespread use in high performance aircrafts, spacecrafts, automobile parts and space structures. In conventional laminated composite structures, homogeneous elastic laminae are bonded together to obtain enhanced mechanical and material properties…

This chapter deals with the dynamic analysis of rectangular plates made of functionally graded materials (FGMs). There are several papers devoted to vibrations of FG structures in the literature. Among these, Loy et al. (1999a) and Pradhan et al. (2000) examined the free vibration of FG cylindrical shells by using the Rayleigh–Ritz method.The first-order deformation theory is employed by Praveen and Reddy (1998) to analyze the nonlinear static and dynamic response of heated FG ceramic-metal plates subjected to dynamic lateral loads by the finite element method. Reddy (2000) developed both theoretical and finite element formulations for thick FG plates according to the higher order deformation theory, and studied the nonlinear dynamic response of FG plates subjected to sudden applied uniform pressure. Yang and Shen (2002) dealt with free vibration and transient response due to impulsive lateral patch load for initially stressed FG thin plates by means of the classical plate theory together with a differential quadrature method. After, the same authors developed their studies on free and forced vibration of FG plates inYang and Shen (2002) employing Reddy's higherorder theory. Higher order theory together with a meshless Petrov–Galerkin procedure was employed for the static and dynamic analysis of FG plates in Qian et al. (2004)…

In this chapter, a three-dimensional solution for the problem of transversely loaded, all round clamped rectangular plates of arbitrary thickness is presented within the linear, small deformation theory of elasticity. The energy functional defined in Chapter 3 is simplified in order to derive the governing equation of the functionally graded (FG) plate.Young's modulus is assumed to vary in the thickness direction…

In this part, we deal with the vibrational behavior of inhomogeneous beams and circular plates, utilizing the semi-inverse method developed by the first author and extensively discussed in his monograph (Elishakoff, 2005). The main thread of his methodology is that the knowledge of the mode shape encountered in vibration or buckling problem is postulated. The candidate mode shapes can be adopted from relevant static, dynamic or buckling problems. In this study, the exact mode shapes are sought as polynomial functions, in the context of vibration tailoring, i.e. designing the structure that possesses the pre-specified value. Apparently, for the first time in the literature, several closed-form solutions for vibration tailoring have been derived for vibrating inhomogeneous beams and circular plates. Twelve new closed-form solutions for vibration tailoring have been derived for an inhomogeneous polar orthotropic plate that is either clamped or simply supported around its circumference. Also, the vibration tailoring of a polar orthotropic circular plate with translational spring is analyzed. There is considerable potential of utilizing the developed method for the design of functionally graded materials.

Analysis and design of beams engages a significant place in the field of Mechanical, Civil, Aerospace and Ocean Engineering. Abeam is a structural element that carries load mainly in bending. The loads carried by a beam are transferred to supports, which in turn transfer the force to adjacent structural members…

In this chapter, we consider free vibration of inhomogeneous Bernoulli–Euler beam which is elastically restrained at one end and clamped at the other, as shown in Fig. 6.1. The closed-form solution is obtained for the beam of constant material density and constant cross-section but of modulus of elasticity, which varies in a polynomial manner. The semi-inverse method is utilized; namely the fundamental mode of vibration is postulated as a polynomial. It turns out that such a formulation leads to infinite number of solutions, one can obtain a unique solution by introducing an additional requirement. It is shown that if in addition to the fundamental mode shape the natural frequency is specified also, the unique solution is derived.

Some intriguing results are reported in conjunction with closed-form solutions obtained for a clamped-free vibrating inhomogeneous column under an axial concentrated load using the semi-inverse method. Fourthorder polynomial is postulated for both the vibration mode shape and buckling displacement. Solution is provided for the flexural rigidity and the natural frequency. It is shown that for each level of axial loading, there may exist up to five flexural rigidities satisfying the governing differential equation and boundary conditions.

The titled problem is solved by the aid of the semi-inverse formulation. At the first glance, a surprising demand is posed: Design of a radially heterogeneous polar orthotropic circular plate whose fundamental mode shape coalesces with the static deflection of a homogeneous circular plate of the same radius and thickness. Compatible polynomial variations of the radial and circumferential flexural rigidities are introduced. It turns out that there is an infinite number of polar–orthotropic circular plates that solve the posed semi-inverse problem, depending on the free parameter represented by one of the coefficients of flexural rigidities. This allows us to provide a unique solution to the titled problem: There is a single plate that possesses the specified fundamental natural frequency.

Semi-inverse problem of free vibration of simply-supported inhomogeneous polar orthotropic plates is studied. The mode shape is postulated in the form of the classic formula by Lekhnitskii, namely, the static deflection of the associated uniform polar-orthotropic circular plate under uniform loading. The ratios of the circumferential rigidity to the radial rigidity are identified as integer numbers, so that the candidate mode shapes constitute the polynomial functions. The flexural rigidities themselves are also represented by polynomials. The semi-inverse problem of identifying the coefficients of the flexural rigidities is solved analytically. It appears remarkable, that for numerous cases, the simple method developed in this study provides novel closed-form solutions for the design of the polar-orthotropic circular plates with pre-specified mode shapes.

In this case, the same method of analysis is used for the semi-inverse problem of free vibration of clamped–clamped inhomogeneous polar orthotropic plates, as in Chapter 9 for the semi–inverse problem of free vibration of simply-supported inhomogeneous polar orthotropic plates. Several closedform solutions are found by using an exact mode shape derived by Lekhnitskii (1968) for the clamped edge of uniform polar orthotropic circular plate under uniform loading, as shown in Fig. 10.1.

In this chapter the results of the paper by Elishakoff and Meyer (2005f) will be generalized for the polar orthotropic circular plate with translational spring. By using the semi-inverse method and postulating the mode shape as a polynomial, we derive closed-form solutions.

In this book we pursued two objectives: To deal with both traditional, thickness-wise grading, in Part I, along with non-traditional, axial or radial gradings in Part II. These two different approaches open the possibility of multi-directional grading in order to achieve optimal solutions for tailoring the material properties with the desired performance of the structure…

The following sections are included:

• Appendix A: A Novel Formulation Leading to Closed-Form Solutions for Buckling of Circular Plates
• Appendix B: Inverse Vibration Problem for Inhomogeneous Circular Plate with Translational Spring
• Appendix C: Apparently First Closed-Form Solutions for Non-Symmetric Vibrations of Inhomogeneous Circular Plates
• Appendix D: Closed-Form Solution for Axisymmetric Vibration of Inhomogeneous Simply-Supported Circular Plates

The following sections are included:

• References
• Author Index
• Subject Index