A high order theory for functionally graded axisymmetric cylindrical shells
Highlights
► We develop a higher order theory for functionally graded (FG) axisymmetric cylindrical shells. ► We use for that Fourier series in terms of Legendre's polynomials expansion. ► For numerical solution the FEM implemented in the software COMSOL Multiphysics and MATLAB. ► Influence of the material graduation parameters on the stress–strain state of the cylindrical shell has been studied.
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.
Section snippets
2-D formulation
Let us consider a linear elastic axisymmetric cylindrical shell in a three-dimensional (3-D) Euclidian space domain V=Ω×[−h,h] with a smooth boundary ∂V as shown in Fig. 1. The boundary of the shell can be represented in the form ∂V=S∪Ω+∪Ω−. Here 2h is the shell's thickness, Ω is the middle surface of the shell, ∂Ω is its boundary, Ω+ and Ω− are the outer sides and S=∂Ω×[−h,h] is a lateral side.
For convenience and following in the shell theory generally accepted tradition we introduce here the
1-D formulation
We expand the physical parameters, which describe the stress–strain state of the cylindrical shell into the Legendre polynomials series along the coordinate x3. Such expansion can be done because of any function f(p), which is defined in domain −1≤p≤1 and satisfies Dirichlet's conditions (continuous, monotonous, and having finite set of discontinuity points), can be expanded into Legendre's series according formulas:
Any function of more than one
First order approximation
In the first order approximation theory only the first two series terms of Legendre's polynomials have to be taken into account. In the homogeneous case it is usually referred to as Vekua's theory of shells. In this case the stress and strain state of the shell, can be expressed in the form:
The equations of equilibrium in Eq. (2.3) now have the form:
Second order approximation
In the second order approximation only the first three series terms of Legendre's polynomials have to be taken into account. In this case the stress and strain state of the shell, can be represented in the form:
The equations of equilibrium in Eq. (2.3) now take the
Material properties of FGMs
FGMs are advanced materials and their mechanical properties can be changed continuously in the way most suitable to satisfy specific requirements. Therefore they are very useful for applications in engineering science. In the simplest FGMs, two different material ingredients change gradually from one to the other. Discontinuous changes such as a stepwise gradation of the material ingredients can also be considered as an FGM. The most familiar FGM is compositionally graded from a refractory
Conclusions
In this paper a high order theory for FG axisymmetric cylindrical shells has been developed. The proposed approach is based on the expansion of the axisymmetric equations of elasticity for FGMs into Fourier series in terms of Legendre's polynomials. Starting from the axisymmetric equations of elasticity, the stress and strain tensors, the displacement, traction and body force vectors and the material parameters of FGMs have been expanded into Fourier series in terms of Legendre's polynomials in
Acknowledgments
The work presented in this paper was supported by the German Research Foundation (DFG Project ZH 15/21-1), which is gratefully acknowledged.
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