Elsevier

European Journal of Mechanics - A/Solids

Volume 44, March–April 2014, Pages 222-233
European Journal of Mechanics - A/Solids

Three-dimensional elasticity solution for vibration analysis of functionally graded hollow and solid bodies of revolution. Part I: Theory

https://doi.org/10.1016/j.euromechsol.2013.11.004Get rights and content

Highlights

  • A semi-analytical method is presented for vibration analysis of functionally graded bodies of revolution.

  • Theoretical model is formulated using a modified variational principle based on 3-D theory of elasticity.

  • Displacement variations of each segment are represented by Fourier series and orthogonal polynomials.

  • The method is capable of handling free, steady-state and transient vibration problems.

Abstract

This is the first of two companion papers which collectively present a novel semi-analytical method and its associated applications for linear vibration analyses of functionally graded bodies (either hollow or solid) of revolution with arbitrary boundary conditions. A modified variational principle combined with a multi-segment partitioning procedure is employed to formulate the theoretical model in the context of three-dimensional theory of elasticity. Displacement variations of each body segment are represented by Fourier series for the circumferential variable and orthogonal polynomials for the meridional and normal variables. The effective material properties of functionally graded bodies are assumed to vary continuously in the normal direction according to general four-parameter power-law distributions in terms of volume fractions of the constituents, and are estimated by Voigt's rule of mixture and Mori–Tanaka's homogenization scheme. The proposed method is capable of handling various combinations of boundary constraints in a unified fashion, including free, simply-supported, clamped and elastic-supported boundary conditions, and allows the use of different polynomials as displacement functions for meridional and normal variables, such as Chebyshev and Legendre orthogonal polynomials as well as hybrid polynomials. Moreover, it permits to deal with the lower- and high-order vibration problems of functionally graded bodies of revolution subjected to dynamic loads of arbitrary type. In Part I, attention is principally focused on the theoretical development and solution methodology of the method. Comprehensive studies on the convergence, accuracy, stability and efficiency of the method are addressed in Part II, where parametric studies concerning the influences of the geometrical parameters, material distributions as well as boundary conditions on free, steady-state and transient vibrations of functionally graded cylinders, cones and spheres are also investigated in detail.

Introduction

Functionally graded materials (FGMs) are classified as novel composite materials that consist of two or more distinct material phases with a continuously variable composition. Such materials possess a number of advantages that make them preferable in potential applications, including improved residual stress distribution, enhanced thermal properties, higher fracture toughness, and reduced stress intensity factors (Birman and Byrd, 2007). In recent years, hollow and solid bodies of revolution (e.g., cylinders, cones, spheres, ellipsoids and hyperboloids) made of FGMs have been utilized in a variety of engineering applications due to their mechanical merits and structural efficiencies. Rocket nozzles, missiles, gas pipelines, and spacecraft heat shields are practical examples of such structures. During their operation, these structural components are commonly subjected to varying levels of external dynamic loads, which may cause fatigue damage and result in severe reduction in the strength and stability of the structures. Consequently, accurate prediction of the dynamic behaviors of FGM bodies of revolution is of paramount importance for engineering design and manufacture. This paper and its companion are concerned with the theoretical development and associated applications of a novel formulation for the free, steady-state and transient vibration analyses of hollow and solid FGM bodies of revolution based on the three-dimensional (3-D) theory of elasticity.

A body of revolution, either hollow or solid, is obtained by the rotation of a plane figure 3600 about an axis in its plane. In particular, if it is hollow and its thickness is much smaller compared to its radii of curvature, it may also be called a shell of revolution. In the case of thin and moderately thick shells of revolution, the full 3-D vibration problems of these structures may be characterized with reduced 2-D shell models (i.e., the classical, first- and higher-order shell models) by approximating the field quantities of interest, such as displacements and stresses, in terms of certain averages over the thickness direction (Carrera, 2003, Leissa, 1973, Liew et al., 2011, Qatu et al., 2010). The classical or thin-shell theories (e.g., Donnell's, Love's, Sanders' and Flügge's shell theories) based on the Kirchhoff–Love's hypothesis have been widely acclaimed for their simplicity in providing approximate solutions for the vibrations of thin shells in the lower-frequency range. However, as the thickness or the transverse shear flexibility of the shell increases, the classical theories lose their validity because of the significant effects of transverse shear deformation and rotary inertia. Shear deformation theories provide a better prediction of the dynamic behaviors of shells than those classical ones by accounting for transverse shear deformation of shells. In particular, the first-order shear deformation theory (FSDT) assumes constant shear strains across the shell thickness and violates stress-free boundary conditions on the top and bottom surfaces of the shell. Higher-order shear deformation theories (HSDTs) are based on the assumption of nonlinear strain variation through the shell thickness, which are better than FSDTs for the vibration analyses of slightly thick shells; they, however, introduce sophisticated formulations and boundary terms that are not easily applicable or yet understood, and are still inadequate for the analysis of very thick shells. It should be born in mind that all 2-D shell theories are approximate in nature because they are developed on the basis of certain kinematic assumptions, and the application range of these theories is defined by the 3-D theory of elasticity. In the context of the elasticity theory, no hypotheses about the distribution field of deformations and stresses are adopted, and contributions of all stresses and strains are considered by accounting for all the elastic constants. Actually, a 3-D analysis not only provides a full spectrum of vibration results for arbitrarily thick FGM bodies of revolution but also allows further physical insights, which cannot otherwise be predicted by 2-D shell theories. Consequently, the analysis based on elasticity theory that takes into account all 3-D variations of stresses and strains is probably the only way to provide accurate responses for hollow and solid FGM bodies of revolution.

The determination of the linear vibration behaviors of FGM bodies of revolution based on various structural theories has been a problem of great technical interest over the past few years. Restricting our attention to the 3-D elasticity solutions, the important works related to the 3-D vibration analyses of FGM bodies of revolution, including FGM cylinders, cones and spheres, are briefly reviewed here. Hollow FGM cylinders have attracted more attention than any other shapes, not only due to their technical importance, but also because the mathematical theory is relatively simple. Regarding the free vibration problems, Vel (2010) presented a power-series method in conjunction with the 3-D elasticity theory for the analysis of FGM cylindrical shells having simply-supported boundary conditions. Santos et al. (2009) developed an axisymmetric finite element model to study the free vibrations of FGM cylindrical shells using the 3-D theory of elasticity. They reduced the 3-D equations of motion to 2-D representations by expanding the displacement field in terms of Fourier series for the circumferential variable. Yas and Sobhani Aragh (2011) carried out a 3-D free vibration analysis for a four-parameter functionally graded fiber-oriented cylindrical panel with simply-supported boundary conditions. The differential quadrature method was employed to obtain the natural frequencies. Pilafkan et al. (2013) considered the free vibrations of thick bidirectional FGM cylindrical shells using a radial point interpolation method. Chen et al. (2004) investigated the free vibration behaviors of a hollow FGM cylinder filled with compressible fluid by using the state-space method. On the basis of the differential quadrature method and the state-space approach, Alibeigloo et al. (2012) analyzed the free vibrations of a functionally graded cylindrical shell embedded in piezoelectric layers with different boundary conditions. Wu and Yang (2011) performed a free vibration analysis for simply-supported hollow FGM cylinders using the element-free Galerkin method in conjunction with a differential reproducing kernel interpolation method. Mollarazi et al. (2012) investigated the free vibrations of FGM cylinders by using a meshfree method. Wu and Li (2012) developed a modified Pagano method for predicting the vibration behaviors of simply-supported, multilayered FGM cylinders with a constant rotational speed with respect to the meridional direction of the cylinders. Very few theoretical studies have been reported on the forced vibration problems (including the steady-state and transient vibrations) of hollow and solid FGM cylinders. Santos et al. (2008) carried out a thermoelastic analysis for FGM cylindrical shells subjected to a transient thermal shock loading, in which an axisymmetric finite element model using the 3-D elasticity theory was employed. Hasheminejad and Ahamdi-Savadkoohi (2010) investigated the steady-state sound radiation of an arbitrarily thick hollow FGM cylinder of infinite length subjected to concentrated mechanical forces. The available 3-D elasticity solutions for the vibration problems of FGM cones are rather limited. The only work on such problems in the open literature is that of Malekzadeh et al. (2012), who investigated the vibration behaviors of truncated FGM conical shells subjected to thermal environment. They used the differential quadrature method to solve the thermo-mechanical governing equations. There is also a considerable lack of information pertaining to the vibrations of hollow and solid FGM spheres based on the 3-D theory of elasticity. Using the Frobenius power series method, Chen et al. (1999) analyzed the free vibration of a spherically isotropic hollow sphere made of a functionally graded material and filled with a compressible fluid medium. Later, Chen et al. (2002) investigated the free vibration behaviors of functionally graded piezoceramic hollow spheres by using a laminated approximation method.

The above review clearly indicates that there exists a notable body of literature on the free vibrations of FGM bodies of revolution with particular geometrical shapes, such as cylinders, cones and spheres, but a general 3-D elasticity solution for both free and forced vibrations of axisymmetric FGM bodies with arbitrarily smooth shaped meridian seems to be nonexistent. In particular, to the best of the authors' knowledge, rigorous analytical or numerical solutions for transient response of FGM cylinders, cones and spheres under arbitrary time-dependent dynamic forces are not available. Moreover, the extensive literature review also reveals that most of the previous research efforts were restricted to vibration problems of particular types of FGM bodies of revolution with limited sets of classical boundary conditions, e.g., a hollow FGM cylinder with free, simply-supported, or clamped boundary conditions. It is well recognized that there exist some deviations from these ideal boundary conditions for FGM bodies of revolution in practical engineering applications, and the mechanical interaction of the structure and the support foundation is an important issue. Consequently, more realistic models are needed that will be capable of representing the complexities of boundary conditions encountered in engineering practice. The objective of the present work is to develop a unified semi-analytical method in the context of 3-D theory of elasticity for predicting the linear vibration behaviors of hollow and solid FGM bodies of revolution with arbitrary boundary conditions. The effective material properties, estimated using either the Voigt' rule of mixture or Mori–Tanaka's homogenization scheme, are assumed to be functionally graded in the normal direction according to two general four-parameter power-law distributions in terms of volume fractions of the constituents. A modified variational principle in conjunction with a multi-segment partitioning technique is employed to formulate the theoretical model based on the 3-D theory of elasticity. The discretized governing equations of motion for FGM bodies of revolution are obtained by expanding the displacement components in terms of Fourier series for the circumferential coordinate and orthogonal polynomials for the meridional and normal coordinates. By this way, a 3-D problem is transformed into a set of 2-D analytical problems corresponding to the circumferential waves of the Fourier expansion. As it will be shown in what follows, the formulation is particularly attractive since once can choose different polynomials as displacement functions for a practical analysis, such as Chebyshev orthogonal polynomials of first and second kind, Legendre orthogonal polynomials of first kind, and hybrid orthogonal polynomials. Moreover, the present method is capable of handling various types of boundary constraints in a unified fashion, including free, simply-supported, clamped and elastic-supported boundary conditions, and permits to deal with the linear vibration problems for functionally graded bodies of revolution subjected to dynamic loads of arbitrary type. In Part I, attention is principally focused on the theoretical development and solution methodology of the method. Applications of the method to various FGM bodies of revolution, including hollow and solid FGM cylinders, cones and spheres, are addressed in Part II of this paper.

Section snippets

Preliminaries

In order to specify the deformation state of a FGM body of revolution, it is necessary to define a set of generalized coordinates that uniquely locates every point in the body in space. We consider the general case and introduce three orthogonal curvilinear coordinates denoted by α, β and γ as shown in Fig. 1. These coordinates may be derived from a set of Cartesian coordinates (i.e., x, y and z) by using a transformation that is locally invertible (a one-to-one map) at each point. The

Conclusions

This paper describes a unified semi-analytical method for the vibration analysis of hollow and solid FGM bodies of revolution with arbitrary boundary conditions. The theoretical model is formulated by using a modified variational principle in conjunction with a multi-segment partitioning technique based on the 3-D linear theory of elasticity. The material properties of FGM bodies of revolution are assumed to vary continuously in the normal direction following two general four-parameter

References (30)

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