A high order theory for functionally graded axisymmetric cylindrical shells

Abstract

In this paper a high order theory for functionally graded (FG) axisymmetric cylindrical shells based on the expansion of the axisymmetric equations of elasticity for functionally graded materials (FGMs) into Fourier series in terms of Legendre's polynomials is presented. Starting from the axisymmetric equations of elasticity, the stress and strain tensors, the displacement, traction and body force vectors are expanded into Fourier series in terms of Legendre's polynomials in the thickness coordinate. In the same way the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke's law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients is obtained. In particular the first and second order approximations of the exact shell theory are considered in more details. The obtained boundary-value problems are solved by the finite element method (FEM) with COMSOL Multiphysics and MATLAB software. Numerical results are presented and discussed.

Introduction

In recent years, a new type of composite materials, the so-called FGMs have been introduced and applied in a wide range of engineering sciences, as reflected in numerous papers on this subject [1], [2]. FGMs are advantageous over the homogeneous materials with only one material constituent, because they consist of two or more material constituents and combine the desirable properties of each constituent. FGMs received a great deal of attention because of their distinctive material properties, which may vary continuously in one (or more) direction(s), in the case of plates and shells usually in the thickness direction. The gradual change of material properties makes FGMs superior to conventional laminated composite materials, which are prone to debonding due to the mismatch of the mechanical properties of the constituents. The concept of FGMs has been applied to electronics, optics, chemistry, biomedical and many other fields. Typical FGMs are made of a mixture of ceramic and metal or a combination of different metals, and can be easily manufactured by varying the volume fractions of their constituents. A representative example for FGMs is the metal/ceramic FGMs, which are compositionally graded from a ceramic phase to a metal phase. Metal/ceramic FGMs can incorporate advantageous properties of both ceramics and metals such as the excellent heat, wear, and corrosion resistances of ceramics and the high strength, high toughness, good machinability and bonding capability of metals.

The FG thin-walled structures, such as plates and shells, have many engineering applications, especially in reactor vessels, turbines and many other fields such as in civil, mechanical and aerospace engineering [3]. Laminated composites are commonly used in many engineering structures. In conventional laminated composite structures, homogeneous elastic laminas are bonded together to obtain enhanced mechanical properties [4]. However, the abrupt change in the material properties across the interface between different materials may result in large interlaminar stresses leading to delamination or interface cracking. One efficient way to remove this disadvantage is to use FGMs in which the material properties vary continuously in the space. This can be realized by gradually changing the volume fraction of the constituent materials, usually in the thickness direction of the plates and shells. In contrast to classical laminates, the stress distributions in FGMS are rather smooth.

Many different theories of FG plates and shells have been developed so far. These models are mostly based on the Kirchhoff–Love, Timoshenko–Mindlin hypothesis or used more complicated high order theories. More details related to this topic can be found in numerous publications (see for example [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]). The material properties of FG plates and shells can be described by various functional relations, for instance the power-law function [12], [13], exponential function [18], [19], or sigmoid function [8], [17] to describe the spatial change of the volume fractions of the material constituents. Vibration analysis of the conical, cylindrical and annular shell structures has been done in [20], [21], [22], [23], [24], [25]

In this paper we present a new theory for FG axisymmetric cylindrical shells based on the expansion of the axisymmetric equations of elasticity for FGMs into Fourier series in terms of Legendre's polynomials. Such an approach has been used widely for the development of various theories for isotropic [26], [27] and anisotropic [28], [29], [30] plates and shells. This method has been also utilized in previous publications for the development of thermoelasticity theory of plates and shells with considering mechanical and thermal contact [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. In this study we expand the physical quantities and the material parameters of the FGMs into Fourier series in terms of Legendre's polynomials and derive Hooke's law relating the Fourier series expansion for stresses and strains. Then we obtain a system of differential equations and the corresponding boundary conditions for the Fourier series expansion coefficients. Special attention of the study is devoted to the first and second order approximations of the exact shell theory. The arising boundary-value problems are solved numerically by using the FEM implemented in COMSOL Multiphysics software. Numerical examples are presented and discussed to demonstrate the accuracy of the developed high order shell theory.