NONLINEAR VIBRATIONS OF FUNCTIONALLY GRADED CYLINDRICAL SHELLS: EFFECT OF THE GEOMETRY

In this paper, the effect of the geometry on the nonlinear vibrations of functionally graded (FGM) cylindrical shells is analyzed. The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. In the linear analysis, after spatial discretization, mass and stiff matrices are computed, natural frequencies and mode shapes of the shell are obtained. In the nonlinear analysis, the three displacement fields are re-expanded by using approximate eigenfunctions obtained by the linear analysis; specific modes are selected. The Lagrange equations reduce nonlinear partial differential equations to a set of ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response of the shell. A convergence analysis is carried out to determine the correct number of the modes to be used. The analysis is focused on determining the nonlinear character of the response as the geometry of the shell varies. INTRODUCTION FGMs are composite materials obtained by combining different constituent materials, which are distributed along the thickness in accordance with a volume fraction law. The idea of FGMs was first introduced in 1984/87 by a group of Japanese material scientists [1]. Loy et al. [2] analyzed the vibrations of FGM cylindrical shells considering simply supported boundary conditions. Leissa [3] studied the linear dynamics of shells with different topologies and materials. Yamaki [4] studied buckling and post-buckling of the shells in linear and nonlinear fields, reporting solution methods, numerical and experimental results. A modern treatise on the shells dynamics and stability can be found in Ref. [5], where also FGMs are analyzed. Pellicano et al. [6] considered the effect of the geometry on the nonlinear vibrations of homogeneous isotropic shells, leading to similar conclusions of the present work. The method of solution used in the present work was developed in Ref. [7]. In this paper, the effect of the geometry on the nonlinear vibrations of FGM cylindrical shells is analyzed; the Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration. The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields. Simply supported boundary conditions are considered. The FGM is made of stainless steel and nickel, the material properties are graded along the thickness according to a volume fraction law. The solution method consists of two steps: 1) linear analysis and eigenfunctions evaluation; 2) nonlinear analysis, using an eigenfunction-based expansion. In the linear analysis, the displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable. A Ritz based method allows to obtain approximate natural frequencies and mode shapes. In the nonlinear analysis, the three displacement fields are reexpanded by using the approximate eigenfunctions; an energy approach based on Lagrange equations is considered to reduce the nonlinear partial differential equations to a set of nonlinear ordinary differential equations. Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load; a convergence analysis is carried out in order to obtain the correct number of the axisymmetric and asymmetric modes. The effect of the geometry on the nonlinear vibrations of the shells is analyzed, and a comparison of nonlinear amplitude-frequency curves of the FGM shells with different geometries is carried out.


INTRODUCTION
FGMs are composite materials obtained by combining different constituent materials, which are distributed along the thickness in accordance with a volume fraction law.The idea of FGMs was first introduced in 1984/87 by a group of Japanese material scientists [1].Loy et al. [2] analyzed the vibrations of FGM cylindrical shells considering simply supported boundary conditions.Leissa [3] studied the linear dynamics of shells with different topologies and materials.Yamaki [4] studied buckling and post-buckling of the shells in linear and nonlinear fields, reporting solution methods, numerical and experimental results.A modern treatise on the shells dynamics and stability can be found in Ref. [5], where also FGMs are analyzed.Pellicano et al. [6] considered the effect of the geometry on the nonlinear vibrations of homogeneous isotropic shells, leading to similar conclusions of the present work.The method of solution used in the present work was developed in Ref. [7].In this paper, the effect of the geometry on the nonlinear vibrations of FGM cylindrical shells is analyzed; the Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration.The shell deformation is described in terms of longitudinal, circumferential and radial displacement fields.Simply supported boundary conditions are considered.The FGM is made of stainless steel and nickel, the material properties are graded along the thickness according to a volume fraction law.The solution method consists of two steps: 1) linear analysis and eigenfunctions evaluation; 2) nonlinear analysis, using an eigenfunction-based expansion.In the linear analysis, the displacement fields are expanded by means of a double mixed series based on harmonic functions for the circumferential variable and Chebyshev polynomials for the longitudinal variable.A Ritz based method allows to obtain approximate natural frequencies and mode shapes.In the nonlinear analysis, the three displacement fields are reexpanded by using the approximate eigenfunctions; an energy approach based on Lagrange equations is considered to reduce the nonlinear partial differential equations to a set of nonlinear ordinary differential equations.Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to an harmonic external load; a convergence analysis is carried out in order to obtain the correct number of the axisymmetric and asymmetric modes.The effect of the geometry on the nonlinear vibrations of the shells is analyzed, and a comparison of nonlinear amplitude-frequency curves of the FGM shells with different geometries is carried out.
where  ̃ and   are the material property and the volume fraction of the constituent material .The material property  ̃ of a constituent material can be described as a function of the environmental temperature (K) by Touloukian's relation [2] (the index  is dropped for the sake of simplicity) where  ,  ,  ,  and  are the coefficients of temperature of the constituent material.In the case of a FGM thin circular cylindrical shell with a uniform thickness ℎ and a reference surface at its middle surface, the volume fraction   of a constituent material can be written as [2] where the power-law exponent  is a positive real number, (0 ≤  ≤ ∞), and  describes the radial distance measured from the middle surface of the shell, (−ℎ/2 ≤  ≤ ℎ/2), see Fig. 1.For a FGM thin cylindrical shell made of two different constituent materials, the volume fractions   and   can be written in the following form [2]   ( Young's modulus , Poisson's ratio  and mass density  are expressed as [2]   (, ) = ( () −  ()

SANDERS-KOITER NONLINEAR THEORY OF SHELLS
In Figure 1, a FGM circular cylindrical shell having radius , length  and thickness ℎ is shown; a cylindrical coordinate system (; , , ) is considered in order to take advantage from the axial symmetry of the structure, the origin  of the reference system is located at the center of one end of the shell.Three displacement fields are represented in Fig. 1: longitudinal (, , ), circumferential (, , ) and radial (, , ).The Sanders-Koiter nonlinear theory of shells is an eight-order theory based on the Love's "first approximation" [3].The strain components (  ,   ,   ) at an arbitrary point of the shell are related to the middle surface strains ( , ,  , ,  , ) and to the changes in the curvature and torsion (  ,   ,   ) of the middle surface of the shell by the following relationships [4]   =  , +     =  , +     =  , +   (8) where  is the distance of the arbitrary point of the cylindrical shell from the middle surface and (, ) are the longitudinal and angular coordinates of the shell, see Fig. 1.The middle surface strains and changes in curvature and torsion are given by [4] where ( = /) represents the nondimensional longitudinal coordinate.
In the case of FGMs, the stresses are related to the strains as follows [5] where () is the Young's modulus and () is the Poisson's ratio (  = 0, plane stress hypotheses).
The elastic strain energy  of a cylindrical shell is given by [5] The kinetic energy  of a cylindrical shell (rotary inertia effect is neglected) is given by [5] where () is the mass density of the shell.
The virtual work  done by the external forces is written as [5]  =  ∫ ∫ (   +    +   ) ( 13) where (  ,   ,   ) are the distributed forces per unit area acting in longitudinal, circumferential and radial direction.
The nonconservative damping forces are assumed to be of viscous type and are taken into account by using Rayleigh's dissipation function (viscous damping coefficient )

VIBRATION ANALYSIS
In order to carry out the dynamic analysis of the shell, a two-steps procedure is considered [7]: i) the Rayleigh-Ritz method is applied to the linearized formulation of the problem, in order to obtain an approximation of the eigenfunctions; ii) the displacement fields are re-expanded using the approximate eigenfunctions, the Lagrange equations are considered in conjunction with the fully nonlinear expression of the potential energy, in order to obtain a set of nonlinear ordinary differential equations in modal coordinates.

NUMERICAL RESULTS
In this section, the nonlinear vibrations of FGM shells with different modal shape expansions and geometries are analyzed.Analyses are carried out on a FGM made of stainless steel and nickel, its properties are graded along the thickness according to a volume fraction distribution, where  is the power-law exponent.The material properties are reported in Tab.1-2 [2].

Nonlinear Response Convergence Analysis
The convergence analysis is carried out on a simply supported shell excited with an harmonic force; the excitation frequency is close to mode (, ).The convergence is checked by adding suitable modes to the resonant one, i.e., asymmetric modes ( × ,  × )  = 1,3  = 1,2,3 due to the presence of the quadratic and the cubic nonlinearities; axisymmetric modes (, 0)  = 1,3,5,7 due to the quadratic nonlinearities.The convergence analysis is then developed by introducing a different number of asymmetric and axisymmetric modes in the expansions of the displacement fields , , , see Tab. 3. The FGM cylindrical shell is excited by means of an external modally distributed radial force   =  ,6 sin  cos 6 cos Ω; the amplitude of excitation is  ,6 = 0.0012ℎ  ,6 and the frequency of excitation  is close to the mode (1,6), Ω ≅  ,6 .The external forcing  ,6 is normalized with respect to mass, acceleration and thickness; the damping ratio is equal to  ,6 = 0.0005.The nonlinear amplitudes  , ,  , ,  , of the expansions (24) refer to the displacement fields , ,  of the mode (1,6), respectively.In Figure 2, a comparison of nonlinear amplitude-frequency curves of the cylindrical shell (ℎ/ = 0.002, / = 20,  = 1) with different nonlinear expansions is shown; the shell is very thin and long.The nonlinear 6 dof model describes a wrong softening nonlinear behaviour, while the higher-order nonlinear expansions converge to a hardening nonlinear behaviour.In Figure 3, a comparison of nonlinear amplitude-frequency curves of the cylindrical shell (ℎ/ = 0.025, / = 20,  = 1) with different nonlinear expansions is shown; the shell is quite thick and long.The nonlinear 6 dof model describes a wrong hardening nonlinear behaviour, and the higher-order nonlinear expansions converge to a softening nonlinear behaviour.
In Figure 4, a comparison of nonlinear amplitude-frequency curves of the cylindrical shell (ℎ/ = 0.050, / = 20,  = 1) with different nonlinear expansions is shown; the shell is thick and long.The nonlinear 6 dof model describes a wrong softening nonlinear behaviour, while the higher-order nonlinear expansions converge to a hardening nonlinear behaviour.The fundamental role of the axisymmetric and higher-order asymmetric modes is clarified in order to obtain the actual character of the shell nonlinearity.
From the convergence analysis, it can be observed that the 9 dof model gives satisfactory results with the minimal computational effort; therefore, in the following analyses the 9 dof model will be used.In particular, the following 9 dof model will be considered for studying a generic resonant mode (, ):  modes (, ), (1,0), (3,0) for the field   modes (, ), (, 2), (3, 2) for the field   modes (, ), (1,0), (3,0) for the field  After selecting such modes, each expansion present in the Eqns.(24) is reduced to a three-terms modal expansion; the resulting nonlinear system has 9 dof.The expression of the resulting discretized nonlinear equations of motion and the method used to compute the nonlinear response amplitude curves are described in Ref. [7].

Effect of the Geometry
In this part, the role of the geometric parameters ℎ, ,  and in particular their ratios ℎ/ and / on the nonlinear response of the FGM cylindrical shells is clarified.In Figure 5 The nonlinear response of the thicker circular cylindrical shell (ℎ/ = 0.050, / = 20,  = 1) is more hardening than the thinner one (ℎ/ = 0.002, / = 20,  = 1), a wide interval of thickness gives rise to softening type behaviour.In order to determine the influence of the geometry on the nonlinear vibration, a parametric analysis is carried out by varying the fundamental ratios (ℎ/) and (/).
In Figure 6, the effect of the geometry on the nonlinearity type is analyzed considering few numerical test-cases: square marks describe a softening behaviour, while circle marks describe a hardening behaviour.In Figure 6, the dashed lines, which are referred to homogeneous shells, are reproduced from Ref. [6]: such lines represent the boundaries of the hardening/softening regions.The present analysis shows that the FGM shells behave similarly to homogeneous ones: very short shells (/ < 0.5) and thick shells (ℎ/ > 0.045) present a hardening nonlinear behaviour; conversely, a softening nonlinearity is found in a wide range of shell geometries.However, for sufficiently long (/ > 5) and thin (ℎ/ < 0.005) shells, the system can be hardening again.This confirms for the FGM shells the results available in literature concerning homogeneous isotropic shells.

CONCLUSIONS
In this paper, the effect of the geometry on the nonlinear vibrations of FGM cylindrical shells is analyzed.The Sanders-Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration.
The functionally graded material is made of stainless steel and nickel, the material properties are graded along the thickness according to a volume fraction law.Numerical analyses are carried out in order to characterize the nonlinear response when the shell is subjected to a harmonic external load.A convergence analysis is carried out by introducing in the longitudinal, circumferential and radial displacement fields a different number of asymmetric and axisymmetric modes; the role of the axisymmetric and higher-order asymmetric modes is clarified in order to obtain the actual character of the shell nonlinearity.

Figure 1 .
Figure 1.GEOMETRY OF THE FGM CYLINDRICAL SHELL.(a) COMPLETE SHELL; (b) CROSS-SECTION OF THE SHELL SURFACE.

Figure 6 .
Figure 6.EFFECT OF THE GEOMETRY ON THE NONLINEAR RESPONSE OF THE FGM SHELL.CIRCLE BLUE MARKS: HARDENING; SQUARE RED MARKS: SOFTENING; DASHED LINES: BOUNDARIES BETWEEN HARDENING AND SOFTENING REGIONS (FROM REF.[6], HOMOGENEOUS MATERIALS).

Table 3 .
MODAL EXPANSION FOR THE NONLINEAR ANALYSIS.