Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading

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Abstract

In this paper, a formulation for the dynamic stability analysis of functionally graded shells under harmonic axial loading is presented. A profile for the volume fraction is assumed and a normal-mode expansion of the equations of motion yields a system of Mathieu–Hill equations the stability of which is analyzed by the Bolotin’s method. The present study examines the effects of the volume fraction of the material constituents and their distribution on the parametric response, in particular the positions and sizes of the instability regions.

Introduction

The use of functionally graded materials has gained much popularity in recent years especially in extreme high temperature environments. Functionally graded materials are composite materials, which are microscopically inhomogeneous, and the mechanical properties vary smoothly or continuously from one surface to the other. It is this continuous change that results in gradient properties in functionally graded materials. Typically, these materials are made from a mixture of ceramic and metal, or a combination of different metals. Unlike fiber-matrix composites which have a mismatch of mechanical properties across an interface of two discrete materials bonded together and may result in debonding at high temperatures, functionally graded materials have the advantage of being able to withstand high temperature gradient environments while maintaining their structural integrity. The ceramic material provides high temperature resistance due to its low thermal conductivity while the ductile metal component prevents fracture due to thermal stresses.

Functionally graded materials are now being strongly considered as a potential structural material for future high-speed spacecraft. They are also developed now for the general use as structural components in high temperature environments. Many studies have examined functionally graded materials as thermal barriers. With the increased usage of these materials, it is also important to understand the dynamics of functionally graded material structures. A few studies have addressed this. The elastic problem of thick-walled tubes of a functionally graded material under internal pressure in the case of plane strain has been studied (Fukui and Yamanaka, 1992). Rooney and Ferrari (1994) presented the solution to the problem of torsion of an inhomogeneous functionally graded shaft with rectangular cross-section where a methodology was developed to reduce the problem to the solution of a simple linear ordinary differential equation. A formulation of the stability problem for functionally graded hybrid composite plates was presented by Birman (1995) where a micromechanical model was employed to solve the buckling problem for a rectangular plate subjected to uniaxial compression. The correlation between hardness and residual stress in layered functionally graded materials was examined by Omori et al. (1995). It was found that the functionally graded materials considered exhibited two kinds of residual stresses, a local stress which is stored in each layer, and a layer stress is dispersed throughout each layer of the whole material. Durodola and Adlington (1996) presented the use of numerical methods to assess the effect of various forms of gradation of material properties to control deformation and stresses in rotating axisymmetric components such as disks and rotors.

Studies of buckling of thin-walled isotropic cylinders under axial compression, torsional loadings, bending, hydrostatic pressure and lateral pressure have been extensively covered in the literature. However, structural components under periodic loads can undergo parametric resonance which may occur over a range of forcing frequencies and if the load is compressive to the structure, resonance or instability can and usually occurs even if the magnitude of the load is below the critical buckling load of the structure. It is thus, of prime importance to investigate the dynamic stability of dynamic systems under periodic loads. The parametric resonance of the cylindrical shells under axial loads has become a popular subject of study. It was first examined by Bolotin, 1964, Yao, 1965 and Vijayaraghavan and Evan-Iwanowski (1967). For thin cylindrical shells under periodic axial loads, the method of solution is almost always to first reduce the equations of motion to a system of Mathieu–Hill equations. The dynamic stability for such a system of equations can then be analyzed by a number of methods.

In this paper, the parametric resonance or dynamic stability of functionally graded cylindrical shells under periodic axial loading is studied using Bolotin’s first approximation. This work is a natural extension of a previous piece of work by the present authors (Lam and Ng, 1997) on the dynamic stability of isotropic cylindrical shells. This piece of work was motivated by the increased general use of functionally graded materials and also a need to understand their dynamic responses.

Section snippets

Theory and formulation

The functionally graded cylindrical shell as shown in Fig. 1 is assumed to be thin and of length L, thickness h and radius R. The x-axis is taken along a generator, the circumferential arc length subtends an angle θ, and the z-axis is directed radially inwards. The periodic extensional axial load per unit length is given byNa=N0+NscosPt,where P is the frequency of excitation in radians per unit time. The equations of motion according to Donnell's theory are thus given byNxx+1RNθt2ut2,

Stability analysis

Eq. (35) is in the form of a second order differential equation with periodic coefficients of the Mathieu–Hill type. The regions of unstable solutions are separated by periodic solutions having period T and 2T with T=2π/P. The solutions with period 2T are of greater practical importance as the widths of these unstable regions are usually larger than those associated with solutions having period T. Using Bolotin’s (1964) first approximation, the periodic solutions with period 2T can be sought in

Numerical results and discussion

The ceramic material used in this study is silicon nitride and the metal material used is nickel. The densities and Poisson's ratios of the materials are in this case independent of the temperature. The density of silicon nitride is taken to be 2370 kg/m3 and that of nickel is 8900 kg/m3. The Poisson's ratio is 0.24 for silicon nitride and 0.31 for nickel. The elastic moduli are however, temperature dependent and are obtained from Touloukian (1967) asEsn=348.43×1091−3.070×10−4T+2.160×10−7T2

Conclusions

The dynamic stability of simply-supported cylindrical shells of functionally graded material under combined static and periodic axial forces was investigated. Results were found to vary significantly when material distribution was varied by changing the values of the power law exponent which controls the volume fraction of the different materials in the FGM shell. It was also found that reasonable control can be achieved on the natural frequencies and dynamic instability regions by correctly

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