Vibrational Study of Fluid-Filled Functionally Graded Cylindrical Shells Resting on Elastic Foundations

Vibrational characteristics of functionally graded cylindrical shells filled with fluid and placed on Winkler and Pasternak elastic foundations are investigated. Love’s thin-shell theory is utilized for strain-displacement and curvature-displacement relationships. Shell dynamical equations are solved by using wave propagation approach. Natural frequencies for both empty and fluid-filled functionally graded cylindrical shells based on elastic foundations are determined for simply-supported boundary condition and compared to validate the present technique. Results obtained are in good agreement with the previous studies. It is seen that the frequencies of the cylindrical shells are affected much when the shells are filled with fluid, placed on elastic foundations, and structured with functionally graded materials. The influence of Pasternak foundation is more pronounced than that of Winkler modulus.


Introduction
Cylindrical shells have vast applications in many engineering fields such as aerospace, civil, chemical, mechanical, naval, and nuclear.First thin-shell theory was developed by Love [1].He based his theory on Kirchhoff 's approximations for plate and beam problems.Arnold and Warburton [2] did some pioneering work on the vibration characteristics of cylindrical shells.Best reviews on the shell vibration characteristics have been given by Leissa [3], Blevins [4], and Markus [5].Different types of materials like isotropic, laminated and composite like functionally graded materials are used in fabricating shells.
Loy et al. [6] did vibration frequency analysis of functionally graded cylindrical shells.They have employed the Raleigh-Ritz numerical technique to evaluate natural frequencies for simply-supported cylindrical shells.The axial modal dependence was approximated by trigonometric functions.They studied their characteristics with regard to circumferential wave numbers and geometrical parameters of length-to-radius and thickness-to-radius ratios.Pradhan et al. [7] did work on vibration characteristics of functionally graded cylindrical shells made of stainless steel and zirconia for a number of boundary conditions.They also used the Rayleigh-Ritz method to solve the shell governing equations.Characteristic beam functions were utilized for the axial modal displacement deformations.In 2002, Naeem [8] worked on vibration analysis of nonrotating and rotating FGM circular cylindrical shells using Rayleigh-Ritz method and Galerkin technique, respectively.Ritz polynomial functions for nonrotating shells and characteristic beam functions for rotating ones approximate axial modal dependences.Najafizadeh and Isvandzibaei [9] have studied the vibration characteristics of functionally graded material cylindrical shells with ring support.They based their analysis on higher-order plate theory.Arshad et al. [10] studied the natural frequencies of FGM cylindrical shells by exponential volume fraction law with base e (= 2.718 . ..).Shah et al.  [11] further modified and extended this volume fraction law to a general base (b > 0).Arshad et al. [12] also extended their work of reference [10] to study the natural frequencies of FGM cylindrical shells under the influence of various boundary conditions with exponential volume fraction law.Arshad et al. [13,14] also evaluated natural frequencies of bi-layered cylindrical shells by inserting different materials in the layers of the shells.Rayleigh-Ritz numerical technique has been employed to solve the governing equations of motions in the references [9][10][11][12][13][14]. Fluid-filled cylindrical shells have paramount importance in engineering and industries to design pressure vessels, fluid tanks and underground pipe fitting.Junger [15,16] maidenly studied the structural response of thin-shell in an acoustic fluid media by studying the free and forced vibrations of cylinders submerged in an acoustic medium.Chung et al. [17] gave an analytical and experimental study of the vibrational characteristics of thin circular cylindrical shells filled with fluid.Goncalves and Batista [18] gave a theoretical study for the free vibration of vertical shells with simply-supported end condition, partially filled with or submerged in a fluid.They used Sanders' shell theory for acquiring shell frequency equation coupling with shell dynamical equation and the acoustic medium of fluid equations.Amabili [19] has studied the natural frequencies and mode shapes of a simply-supported circular cylindrical shell placed horizontally partially filled with liquid.He [20] further extended his study to investigate the free vibrations of circular cylindrical shells and tubes entirely filled with a dense and partially immersed in various fluids under different end conditions.Sharma et al. [21] have determined the natural frequencies of vertical cantilever fluid-filled composite shells by utilizing the Fourier series of trigonometric functions to approximate the axial modal dependence.Zhang et al. [22] solved the shell frequency equations with wave propagation approach to study the vibrational response of cylindrical shells containing fluid.Natsuki et al. [23] used this technique for the vibrational analysis of carbon nanotubes having fluid inside.Iqbal et al. [24] investigated the vibration characteristics of FGM circular cylindrical shells using wave propagation approach.Shah et al. [25] studied the vibration characteristics of functionally graded cylindrical shells resting on Winkler and Pasternak elastic foundations by using wave propagation approach.This approach seem to be a simple and easily applicable to study vibration characteristics of shells involving fluids.In the wave propagation method, the axial modal dependence is approximated by simple wave number characteristic formulas.Paliwal et al. [26] and Paliwal and Pandey [27] have studied the free vibrations of circular cylindrical shells resting on elastic foundations.

ISRN Mechanical Engineering
In the present study, the vibration characteristics of fluidfilled cylindrical shells fabricated from functionally graded materials resting on the elastic foundations are investigated.The acoustic wave equation of fluid is coupled with the shell dynamical equations.This gives the mathematical relationship between solid and fluid dynamics.The influence of FGM configurations on shell vibrations is analyzed for various physical and geometrical parameters.For straindisplacement and curvature displacement relationships, Love's thin-shell theory is used and wave propagation approach is employed to solve the shell dynamical equations involving the Winkler and Pasternak elastic foundation and fluid loading terms.A good agreement is found between the present and the literature's results on comparison.

Theoretical Formulation
A circular cylindrical shell is considered for its vibration characteristics with length L, thickness h, and mean radius R as shown in Figure 1.The material properties are Young's modulus E, the Poisson ratio ν, and the mass density ρ.An orthogonal coordinate system (x, θ, z) is assumed to be established at the shell mid surface where x, θ, and z represent the axial, circumferential, and radial coordinates, respectively.u, v and w are axial, circumferential and radial deformation displacements, respectively, of the shell mid surface.
The equations of motion for a cylindrical shell derived from the Love's thin-shell theory can be written as where N x , N θ , and N xθ are force resultants and M x , M θ , M xθ moment resultants along the normal and shear directions, respectively.They are defined as where e 1 , e 2 , and γ are the reference surface strains and k 1 , k 2 and τ are the surface curvatures.A i j , B i j , and D i j (i, j = 1, 2 and 6) are the membrane, coupling, and flexural stiffnesses defined as The reduced stiffnesses Q i j (i, j = 1, 2 and 6) for isotropic materials are defined as The cylindrical shell considered in this study is assumed to be of thin with thickness to radius ratio less than 1/20.For strain displacements and the curvature displacements Love's  thin-shell theory is utilized and these quantities are defined as ( The above-mentioned expressions for the surface strains and curvatures are substituted into (2) to get the forces and moments of the shell and then putting them into shell dynamical ( 1)-( 3) and introducing the fluid acoustic pressure inside the cylinder and the Winkler and Pasternak foundations terms (Kw − G∇ 2 w) in the z-direction outside the cylinder.The equations of motion for a modified cylindrical shell can be written in the differential operator form as: where L i j (i, j = 1, 2, 3) are the differential operators with respect to x and θ given in Appendix A. G represents the shear modulus of the material used for the elastic foundation and K for the Winkler foundation modulus, and the expression for the differential operator ∇ 2 is The Winkler model is a special case of the Pasternak model when G = 0.The equation of motion of the fluid can be written in the cylindrical coordinate system (x, θ, r) as: where t is the time, P is the acoustic pressure, and c is the speed of sound in the fluid.
Wave propagation approach has been employed by many researchers (Zhang et al. [22,29,30], Natsuki et al. [23], Li et al. [31], Liu et al. [32], and Li [28]) to evaluate the approximate solution of the shell dynamic equations.This numerical technique is used in this study as it is very simple and easily applicable to determine the shell frequencies.For separating the spatial and temporal variables, the following shapes of modal displacement functions are assumed as The coefficients A m , B m , C m denote the wave amplitudes in the x, θ, and z directions, respectively.n denotes the circumferential waves number, ω is the natural angular frequency for the cylindrical shell, and k m is the axial wave number that has been specified in the reference [18] for a number of boundary conditions.The acoustic pressure field in the cylindrical shell containing fluid, which satisfies the acoustic wave (8), can be expressed as where J n is the Bessel function of first kind with order n, and k r is radial wave number.Coupling condition of the fluid with the shell wall is given by at r = R and where ρ f stands for the density of the contained fluid and the prime on the J n / represents the differentiation of the Bessel function with respect to the argument k r R. By substituting the expressions for u, v, and w from ( 9) into ( 6) and simplifying the algebraic expressions and rearranging the terms, the shell frequency equation leads to the following eigenvalue form: where S i j (i, j = 1, 2, 3) are some matrix coefficients depending on shell parameters and the type of boundary conditions specified at the ends of an FG cylindrical shell and are given in Appendix B. Equation ( 13) is solved for shell frequencies and mode shapes by using MATLAB software.The three frequencies are obtained corresponding to the axial, circumferential, and radial displacements.

Functionally Graded Materials
Functionally graded materials are advanced materials and are used in engineering and technology applications due to their mechanical properties.Their best use is found in the thermal environment for their superb properties.They are fabricated from two or more materials.The material properties of their constituents are functions of the temperature and the volume fractions.If P i represents a material property of the ith constituent material of an FGM consisting of k constituent materials, then its effective material P is written as where V i is the volume fraction of the ith constituent material.Also the sum of volume fractions of the constituent materials is equal to one, that is, The volume fraction depends on the thickness variable and is defined as for a cylindrical shell.R i and R 0 denote inner and outer radii of the shell and z is the thickness variable in the radial direction.N is known as power law exponent and is a nonnegative real number and lies between zero and infinity.When a cylindrical shell is considered to be consisting of two constituent functionally graded materials M 1 and M 2 , the volume fraction V 1 of the outer shell surface is obtained from (16) as  where h is the shell uniform thickness.The effective Young's modulus E, Poisson's ratio ν, and the mass density ρ are given by where E 1 , E 2 are Young's moduli, ν 1 , ν 2 Poisson's ratios, and ρ 1 , ρ 2 the mass densities of the constituent materials M 1 and M 2 , respectively.From the relations (18), the following things are noted, that is, at the inner surface the FGM properties are those of the constituent material M 2 and at the outer surface, they are those of material M 1 .Thus the FGM properties change continuously from the material M 2 at the inner surface to the material M 1 at the outer surface.

Results and Discussions
In Table 1, a comparison of frequency parameter Ω = ωR (1 − ν 2 )ρ/E is made against circumferential wave number n with Loy et al. [6] for a simply-supported isotropic cylindrical shell with shell parameters m = 1, L/R = 20, h/R = 0.01, and ν = 0.3.Table 2 shows the comparison of the variation of frequency parameter of isotropic cylindrical shells with circumferential wave number n and h/R ratio to that of Loy et al. [33] for simply-supported end condition having shell parameters (m = 1, L/R = 20).In  values of power law exponent N = 0.5, 2, and 15 with shell geometrical and material parameters (m = 1, L/R = 20, h/R = 0.002, and ν = 0.3).It is observed that results agree very well with one another.This shows accuracy and validity of the present approach.In Tables 4, 5, and 6, variations of natural frequencies (Hz) against circumferential wave number n are given for types I & II FG cylindrical shells on elastic foundation.Shells parameters are given in the tables.In Table 4, Pasternak foundation G is zero and Winkler foundation K is taken equal to 1.5 × 10 7 .Values of exponential volume fraction law are N = 0.5, 0.7, 1, 2, 5, 15, and 30 in all cases.It is observed that frequencies of both shells increase vertically with increasing values of circumferential wave number n, whereas it slowly decreases for shell type I but increases for shell type II with increasing value of exponential volume fraction law.Frequency of the pure stainless steel is more than frequency of pure nickel.Frequency of FG cylindrical shell remains between the frequency of stainless steel and nickel when FG cylindrical shell is on elastic foundation in Tables 5 and 6; all parameters of shell are same as given in Table 4 but value of K = 0 and G = 1.5 × 10 7 .In this case, frequency increases rapidly with increasing values of circumferential wave number n as compared with frequency when G = 0, K = 1.5 × 10 7 .It means that influence of G is more than that of K on the shell frequencies.Here again frequencies decrease and increase with increasing values of power law exponent, N for shell types I & II, respectively, but remain between the frequencies of pure stainless steel and pure nickel.In Table 6, variation of natural frequencies (Hz) against circumferential wave number n is given when both K and G are nonzero.Natural frequencies increase rapidly with increasing value of circumferential wave number n but it decreases and increases with increasing values of N and it always remains between the frequencies of pure stainless steel and nickel.In Tables 7, 8, and 9, variations of natural frequencies (Hz) against circumferential wave number n of type I & II functionally graded fluid-filled cylindrical shells on elastic foundation are given.Parameters are given in the tables.In Table 7, G = 0 and K = 1.5 × 10 7 .Value of K = 0 and G = 1.5 × 10 7 in Table 8, and K = 1.5 × 10 7 , G = 1.5 × 10 7 in Table 9.In Table 7, it is observed that frequencies increase with circumferential wave number n.It decreases and increases very slowly with increasing value of N .It is observed that their frequencies remain within the frequencies of pure stainless steel and pure nickel fluidfilled cylindrical shells on elastic foundation, and it is also observed that frequency of fluid-filled FGM cylindrical shell is much less than that of empty FG cylindrical shell on the elastic foundation.Tables 8 and 9 also show that the influence of G is more than that of K when fluid-filled FG cylindrical shell is on elastic foundation.Frequency increases with increasing values of K and G.These tables show that influence of Pasternak foundation is more than that of Winkler foundation.Also it is clear that frequency of fluidfilled FG cylindrical shell is much less than that of empty FG cylindrical shell on elastic foundation.

ISRN Mechanical Engineering
Figure 2 shows a similar and opposite behaviour of the natural frequencies (Hz) of both the configurations of fluidfilled FGM type I & type II cylindrical shells for the same values of Winkler and Pasternak elastic foundations and shell parameters shown in this figure.It is clear from this figure that natural frequencies of shell type I decrease while that of shell type II increase with the increasing values of power law exponent N .Rate of decrement of shell type I and increment of shell type II is the same for the ascending values of N and frequencies of both types of the shells seem to intersect at N = 1.
In Figures 3(a)-3(d), variation of natural frequencies (Hz) of fluid-filled isotropic as well as FGM type I cylindrical shells resting on elastic foundations is sketched against L/R ratio with shell parameters m = 1, n = 4, and h/R = 0.001, by keeping Winkler elastic modulus (K = 1.5×10 7 ) constant and Pasternak elastic modulus G varied from 1.5 × 10 7 to 4.5 × 10 7 with 1 × 10 7 class interval, respectively.Nickel and stainless steel are used in isotropic cylindrical shells, whereas these isotropic materials are also used in the fabrication of FGM cylindrical shells as constituents.Influence of G is studied in both isotropic and three FGM cylindrical shells corresponding to power law exponents N = 0.5, 1, 5, respectively.
It is observed from these figures that natural frequencies (Hz) of these cylindrical shells first decrease rapidly from L/R = 5 to L/R = 10, for L/R > 10 these frequencies decline very slowly by increasing values of L/R ratio.Natural frequencies of FG cylindrical shells always lie between the natural frequencies of isotropic cylindrical shells to which FG    It is observed from these figures that natural frequencies (Hz) of these cylindrical shells decrease simultaneously from h/R = 0.002 to h/R = 0.01, afterward it decreases sharply from h/R = 0.01 to 0.02, 0.025, 0.03, and 0.035 in figures 5(a)-5(d) respectively and afterwards these frequencies increase rapidly with h/R ratio.It is further noted that natural frequencies of FG cylindrical shells always lie between the natural frequencies of isotropic cylindrical shells to which FG cylindrical shells are fabricated.For lower values of G, natural frequencies of cylindrical shells with small h/R ratio are low as compared to high h/R ratio but for higher values of G, this order of frequencies is reversed with h/R ratio.For lower values of h/R ratio, natural frequencies of the cylindrical shells seem to be converged, but for higher values of h/R ratio, these frequencies show diverging behaviour.Natural frequencies (Hz) of isotropic cylindrical shell made   Influence of K is studied on the natural frequencies of these cylindrical shells.
The same behaviour is observed as in Figures 5(a)-5(d) except that Winkler modulus K also has no effect on the order of natural frequencies of isotropic as well as FGM cylindrical shells.Natural frequencies of cylindrical shells increased with K but very slowly.Rate of increment for lower values of h/R ratio is the same for higher values of h/R ratio with increasing values of K. Rate of increment of natural frequencies of the cylindrical shells with Pasternak modulus G is very high as compared to Winkler modulus.It meansthat, Pasternak modulus G directly influences the natural frequencies of the cylindrical shells under considered.Similar and opposite behaviour of shell frequencies can be observed for FGM shell type II on elastic foundations.

Conclusion
In this study, vibration frequency analysis of empty as well as fluid-filled FGM cylindrical shells based on elastic foundations is presented for simply-supported boundary conditions.Wave propagation approach is used to derive the shell frequency equation in the form of the eigenvalue problem including the elastic foundation and fluid loading terms.The influence of elastic moduli is more pronounced on the shell frequencies.The frequency of the fluid-filled FG cylindrical shell is much less than that of empty FG cylindrical shell based on elastic foundations.

Figure 1 :
Figure 1: View of circumferential and longitudinal cross section of the FGM cylindrical shell resting on elastic foundations.
)-4(d), variation of natural frequencies (Hz) of fluid-filled isotropic as well as FGM type I cylindrical shells resting on elastic foundations is drawn with L/R ratio having shell parameters m = 1, n = 4, and h/R = 0.001, by keeping Pasternak elastic modulus (G = 1.5 × 10 7 ) while Winkler elastic modulus K is varied from 1.5×10 7 to 4.5×10 7 with 1 × 10 7 class interval, respectively.Influence of K is studied on these cylindrical shells.These figures have all the properties of Figures3(a)-3(d) except that Winkler modulus does not affect the order of natural frequencies of isotropic as well as FGM cylindrical shells.In Figures5(a)-5(d), variation of natural frequencies (Hz) of fluid-filled isotropic as well as FGM type I cylindrical shells placed on elastic foundations is sketched against h/R ratio with shell parameters m = 1, n = 4, and L/R = 20, by taking Winkler elastic modulus (K = 1.5 × 10 7 ) constant and Pasternak elastic modulus G varied from 1.5 × 10 7 to 4.5 × 10 7 with class interval 1 × 10 7 .Influence of G is studied on the natural frequencies of these cylindrical shells.
)-6(d), variation of natural frequencies (Hz) of fluid-filled isotropic as well as FGM type I cylindrical shells placed on elastic foundations is drawn with increasing values of h/R ratio having shell parameters m = 1, n = 4, L/R = 20¸considering Pasternak elastic modulus (G = 1.5 × 10 7 ) constant and Winkler elastic modulus K varied from 1.5 × 10 7 to 4.5 × 10 7 with class interval 1 × 10 7 .

Table 3
[6]omparison of natural frequencies (Hz) for simply-supported type I and type II FGM cylindrical shell is made with Loy et al.[6]against circumferential wave number n for various

Table 4 :
Variation of natural frequencies (Hz) against circumferential wave number n of an empty FG cylindrical shell on elastic foundation.

Table 5 :
Variation of natural frequencies (Hz) against circumferential wave number n of an empty FG cylindrical shell on elastic foundation.

Table 6 :
Variation of natural frequencies (Hz) against circumferential wave number n of an empty FG cylindrical shells on elastic foundation.

Table 7 :
Variation of natural frequencies (Hz) against circumferential wave number n of fluid-filled FG cylindrical shell on elastic foundation.

Table 8 :
Variation of natural frequencies (Hz) against circumferential wave number n of fluid-filled FG cylindrical shell on elastic foundation.

Table 9 :
Variation of natural frequencies (Hz) against circumferential wave number n of fluid-filled FG cylindrical shell on elastic foundation.(G=1.5 × 10 7 , K = 1.5 × 10 7 ) Types I & II (m = 1, h/R = 0.002, L/R = 20).× 10 7 , this order of natural frequencies remains the same.Similarly, natural frequencies of FG cylindrical shells with low value of power law exponent say N < 1 are higher in type I and lower in type II FGM cylindrical shells, whereas for higher value of power law exponent N say N > 1, the natural frequencies are lower for type I and higher in type II FGM cylindrical shells.It is clear from these figures that variation of G does not affect the order of natural frequencies of these cylindrical shells.It is further noted that for increasing values of G, natural frequencies of the cylindrical shells also increased; this rate of increment is higher for low values of h/R ratio than those of higher values of h/R ratio.It is further observed that for smaller values of G say 1.5 × 10 7 , natural frequencies of type I & II FGM cylindrical shells with lower values of h/R ratio are smaller than those with higher values of h/R, but for higher values of G say 4.5 × 10 7 , the order of frequencies of the FGM shells is reversed.It means that Pasternak modulus G has prominent effect on the natural frequencies of the cylindrical shells.In Figures6(a