Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations

Abstract

Vibrations of a cylindrical shell composed of three layers of different materials resting on elastic foundations are studied out. This configuration is formed by three layers of material in thickness direction where the inner and outer layers are of isotropic materials and the middle is of functionally graded material. Love shell dynamical equations are considered to describe the vibration problem. The expressions for moduli of the Winkler and Pasternak foundations are combined with the shell dynamical equations. The wave propagation approach is used to solve the present shell problem. A number of comparisons of numerical results are performed to check the validity and accuracy of the present approach.

1. Introduction

The circular cylindrical shells have been found in various engineering applications ranging from large civil and mechanical structures to small electrical components for many years. Vibrations of cylindrical shells are the most wide studied area of research because of their simple geometrical shapes. First of all, Love [1] presented the linear thin shell theory established on the Kirchhoff’s hypothesis for plates. All other shell theories have been built on this theory by modifying some physical terms involving therein. Numerical solutions of shell vibration problem started to come out in the thirties of the twentieth century and were presented by Flügge [2]. Sharma [3, 4] gave an analysis of thin circular cylinders. Sharma approximated the axial model dependence by beam functions and they used Rayleigh-Ritz technique to solve the shell problem. Paliwal et al. [5] investigated the vibrations of a thin circular cylindrical shell attached with elastic foundations. Loy et al. [6] investigated the vibrations of functionally graded material cylindrical shells, made up of FG material composed of stainless steel and nickel. The purpose of work was to examine natural frequencies, influence of the constituent volume fractions, and effects of configurations of constituent materials on their frequencies. Pardhan et al. [7] studied the vibrations of functionally graded material (FGM) cylindrical shell structured from stainless steel and zirconia. Zhang et al. [8] studied the vibration frequencies of cylindrical shells with fluid-filled. They used wave propagation approach and they compared the uncoupled frequencies with available results in the literature. Najafizadeh and Isvandzibaei [9] studied the vibrations of thin-walled cylindrical shells with ring supports composed of functionally graded material comprised of stainless steel and nickel. Arshad et al. [10] derived the frequency equation in the form of eigenvalue problem by employing Rayleigh-Ritz method. Love’s thin shell theory was used for strain-displacement and curvature-displacement relation. They studied the natural frequency for simply supported boundary conditions and compared the results with those mentioned in the literature to check the validity of the approach. Iqbal et al. [11] applied wave propagation approach to analyze vibrations of functionally graded material circular cylindrical shells. They compared their results for the validity and accuracy by the results mentioned in the literature and found an excellent agreement between two sets of results. Arshad et al. [12] analyzed an associative study for natural frequencies of two-layered cylindrical shells that was presented with one layer made up of functionally graded material and the other layer of isotropic material. Strain-displacement and curvature-displacement relationships were derived by using Love’s first shell theory. Shah et al. [13] presented an analysis on the vibrations of FG cylindrical shells founded on the Winkler and Pasternak foundations. The shell equations of motion were rearranged by introducing the moduli of the Winkler and Pasternak foundations. Shah et al. [14] studied the effects of volume fraction law on the vibration frequencies of thin FG cylindrical shells. Material parameters in shell radial direction were ranked according to the exponential law. Natural frequencies of the shell were dependent on the constituent volume fractions. The consequences were compared with those available in the literature for the validity of the presented approach. Arshad et al. [15] studied an analysis on vibrations of bilayered cylindrical shell made of two layers which were functionally graded. The thickness of the shell layers was considered to be equal and constant. Naeem et al. [16] studied the vibration frequency characteristics of functionally graded cylindrical shells using the generalized differential quadrature method. Shen et al. [17] studied the postbuckling response of a functionally graded cylindrical shell of finite length surrounded by an elastic medium and subjected to internal pressure in thermal environments. Sofiyev and Avear [18] investigated the stability of cylindrical shells that were composed of ceramic, FGM, and metal layers subjected to axial load and rested on Winkler-Pasternak foundations. Sofiyev et al. [19] analyzed the stability of functionally graded cylindrical shells subjected to combined loads with different edge conditions and resting on elastic foundations.

2. Shell Problem

A cylindrical shell of the thin wall is considered here as shown in Figure 1, with the geometrical parameters: length L, thickness ℎ, and mean radius 𝑅. The orthogonal coordinate system (𝑥,𝜑,𝑧) is taken to be at the surface of the shell where 𝑥, 𝜑, and 𝑧 represent the axial, circumferential, and radial coordinates, respectively. Young’s modulus E, the Poisson ratio 𝜈, and the mass density ρ are the shell material parameters. The axial, circumferential, and radial displacement deformations are denoted by 𝑢(𝑥,𝜑,𝑡), 𝑣(𝑥,𝜑,𝑡), and 𝑤(𝑥,𝜑,𝑡), respectively, with regard to the shell middle surface.

Figure 1 

Geometry of three-layered cylindrical shell.

For a cylindrical shell, plane stress condition can be assumed. From two-dimensional Hook’s law, the constitutive relations for a thin cylindrical shell are stated mathematically as {𝜎}=[𝑄]{𝑒}, (1) where {𝜎} stands for the stress vector, {𝑒} corresponds to the strain vector, and [𝑄] denotes the reduced stiffness matrix. These stress and strain column vectors are defined as {𝜎}^𝑇=𝜎_𝑥,𝜎_𝜑,𝜎_𝑥𝜑,{𝑒}^𝑇=𝑒_𝑥,𝑒_𝜑,𝑒_𝑥𝜑, (2) where 𝜎_𝑥 and 𝜎_𝜑 are linear stresses in the 𝑥 and 𝜑-directions, and 𝜎_𝑥𝜑 is the shear stress in the 𝑥𝜑-plane. 𝑒_𝑥 and 𝑒_𝜑 are strain in the directions 𝑥 and 𝜑, and 𝑒_𝑥𝜑 the shear strain in the 𝑥𝜑-plane. The reduced stiffness matrix is written as for the plane stress condition as [𝑄]=𝑄₁₁𝑄₁₂0𝑄₁₂𝑄₂₂000𝑄₆₆. (3) So the relation (1) can be expressed as 𝜎_𝑥𝜎_𝜑𝜎_𝑥𝜑=𝑄₁₁𝑄₁₂0𝑄₁₂𝑄₂₂000𝑄₆₆𝑒_𝑥𝑒_𝜑𝑒_𝑥𝜑. (4)

The reduced stiffness 𝑄_𝑖𝑗(𝑖,𝑗=1,2,and6) for isotropic materials is defined as𝑄₁₁=𝑄₂₂=𝐸1−𝑣²,𝑄₁₂=𝑣𝐸1−𝑣²,𝑄₆₆=𝐸2().1+𝑣 (5a)(5b)(5c)

The strain vector {𝑒} in its component form is defined as𝑒_𝑥=𝑒₁+𝑧𝜅₁,𝑒_𝜑=𝑒₂+𝑧𝜅₂,𝑒_𝑥𝜑=𝛾+2𝑧𝜏, (6a)(6b)(6c)where 𝑒₁, 𝑒₂, and 𝛾 denote the reference surface strains. 𝜅₁, 𝜅₂, and 𝜏 represent the surface curvatures.

From the relations (4), (6a), (6b), and (6c), the stress vector {𝜎} in its component form is defined as𝜎_𝑥=𝑒₁+𝑧𝜅₁𝑄₁₁+𝑒₂+𝑧𝜅₂𝑄₁₂,𝜎_𝜑=𝑒₁+𝑧𝜅₁𝑄₁₂+𝑒₂+𝑧𝜅₂𝑄₂₂,𝜎_𝑥𝜑=𝛾+2𝑧𝜏𝑄₆₆. (7a)(7b)(7c)

The force resultants and moment resultants for a cylindrical shell are written in the integral forms, respectively, as 𝑁_𝑥,𝑁_𝜑,𝑁_𝑥𝜑=^ℎ/2_−ℎ/2𝜎_𝑥,𝜎_𝜑,𝜎_𝑥𝜑𝑀𝑑𝑧,_𝑥,𝑀_𝜑,𝑀_𝑥𝜑=^ℎ/2_−ℎ/2𝜎_𝑥,𝜎_𝜑,𝜎_𝑥𝜑𝑧𝑑𝑧, (8)(9) where 𝑁_𝑥, 𝑁_𝜑, and 𝑁_𝑥𝜑 are force components in axial, circumferential, and shear directions. 𝑀_𝑥, 𝑀_𝜑, and 𝑀_𝑥𝜑 are moment components in axial, circumferential, and shear directions.

The relations (6a), (6b), (6c), (8), and (9) imply that {𝑁}=[𝐾]{𝜖}, (10) where {𝑁} and {𝜖} are defined as {𝑁}^𝑇=𝑁_𝑥,𝑁_𝜑,𝑁_𝑥𝜑,𝑀_𝑥,𝑀_𝜑,𝑀_𝑥𝜑,{𝜖}^𝑇=𝑒₁,𝑒₂,𝛾,𝜅₁,𝜅₂,2𝜏, (11)(12) and [𝐾] is defined as [𝐾]=𝐴𝐵𝐵𝐷, (13) where 𝐴,  𝐵, and 𝐷 are submatrices of extensional, coupling, and boundary stiffness and are given as [𝐴]=𝐴₁₁𝐴₁₂0𝐴₁₂𝐴₂₂000𝐴₆₆,[𝐵]=𝐵₁₁𝐵₁₂0𝐵₁₂𝐵₂₂000𝐵₆₆,[𝐷]=𝐷₁₁𝐷₁₂0𝐷₁₂𝐷₂₂000𝐷₆₆. (14) From [𝐴], [𝐵], and [𝐶], a matrix [𝐾] can be written as [𝐾]=𝐴₁₁𝐴₁₂0𝐵₁₁𝐵₁₂0𝐴₁₂𝐴₂₂0𝐵₁₂𝐵₂₂000𝐴₆₆00𝐵₆₆𝐵₁₁𝐵₁₂0𝐷₁₁𝐷₁₂0𝐵₁₂𝐵₂₂0𝐷₁₂𝐷₂₂000𝐵₆₆00𝐷₆₆, (15) where 𝐴_𝑖𝑗, 𝐵_𝑖𝑗, and 𝐷_𝑖𝑗(𝑖,𝑗=1,2,and6) stand for extensional, coupling, and bending stiffness, respectively, and are defined as 𝐴_𝑖𝑗=^ℎ/2_−ℎ/2𝑄_𝑖𝑗𝐵𝑑𝑧,_𝑖𝑗=^ℎ/2_−ℎ/2𝑄_𝑖𝑗𝐷𝑧𝑑𝑧,_𝑖𝑗=^ℎ/2_−ℎ/2𝑄_𝑖𝑗𝑧²𝑑𝑧. (16) By using the relations (11)–(15), the expression (10) takes the following form: 𝑁_𝑥𝑁_𝜑𝑁_𝑥𝜑𝑀_𝑥𝑀_𝜑𝑀_𝑥𝜑=𝐴₁₁𝐴₁₂0𝐵₁₁𝐵₁₂0𝐴₁₂𝐴₂₂0𝐵₁₂𝐵₂₂000𝐴₆₆00𝐵₆₆𝐵₁₁𝐵₁₂0𝐷₁₁𝐷₁₂0𝐵₁₂𝐵₂₂0𝐷₁₂𝐷₂₂000𝐵₆₆00𝐷₆₆𝑒₁𝑒₂𝛾𝜅₁𝜅₂2𝜏. (17)

The (coupling stiffness) 𝐵_𝑖𝑗’s are vanished for isotropic cylindrical shells and are nonzero in case of FGM cylindrical shells.

For a thin-walled cylindrical shell that performs vibration, the formula for its strain energy denoted by U is written as 1𝑈=2^𝐿₀₀^2𝜋{𝜖}^𝑇[𝐾]{𝜖}𝑅𝑑𝜑𝑑𝑥. (18)

By substitution of {𝜖}^𝑇,[𝐾],and{𝜖} from (12) and (15) in the above expression, the new expression of the shell strain energy is written as 1𝑈=2^𝐿₀₀^2𝜋𝐴₁₁𝑒²₁+𝐴₂₂𝑒²₂+2𝐴₁₂𝑒₁𝑒₂+𝐴₆₆𝛾²+2𝐵₁₁𝑒₁𝜅₁+2𝐵₁₂𝑒₁𝜅₂+2𝐵₁₂𝑒₂𝜅₁+2𝐵₂₂𝑒₂𝜅₂+4𝐵₆₆𝛾𝜏+𝐷₁₁𝜅²₁+𝐷₂₂𝜅²₂+2𝐷₁₂𝜅₁𝜅₂+4𝐷₆₆𝜏²𝑑𝜑𝑑𝑥. (19) Also the kinetic energy of the cylindrical shell, represented by 𝑇, is expressed as 𝑇=^𝐿₀₀^2𝜋𝜌_𝑡𝜕𝑢𝜕𝑡²+𝜕𝑣𝜕𝑡²+𝜕𝑤𝜕𝑡²𝑅𝑑𝜑𝑑𝑥, (20) where 𝑡 symbolizes the time variable and 𝜌_𝑡 denotes the mass density per unit length and is defined as 𝜌_𝑡=^ℎ/2_−ℎ/2𝜌𝑑𝑧. (21)

The present problem is performed by the Love’s [1] first order thin shell theory and the analysis of expressions strain-displacement and curvature-displacement relationships are taken from this theory and are written as 𝑒₁,𝑒₂,𝛾=𝜕𝑢,1𝜕𝑥𝑅𝜕𝑣,𝜕𝜑+𝑤𝜕𝑣+1𝜕𝑥𝑅𝜕𝑢𝜅𝜕𝜑₁,𝜅₂=−𝜕,𝜏₂𝑤𝜕𝑥²1,−𝑅²𝜕²𝑤𝜕𝜑²−𝜕𝑣1𝜕𝜑,−𝑅𝜕²𝑤−𝜕𝑥𝜕𝜑𝜕𝑣.𝜕𝑥 (22) In a cylindrical shell executing vibration. The expressions for surface strains 𝑒₁,𝑒₂, and 𝛾 and the curvatures 𝜅₁,𝜅₂, and 𝜏 from the relations (22), respectively, are substituted into (19), and the expression for strain energy is obtained as 1𝑈=2^𝐿₀₀^2𝜋𝐴₁₁𝜕𝑢𝜕𝑥²+𝐴₂₂𝑅²𝜕𝑣𝜕𝜑+𝑤²𝐴+2¹²𝑅𝜕𝑢×𝜕𝑥𝜕𝑣𝜕𝜑+𝑤+𝐴₆₆𝜕𝑣+1𝜕𝑥𝑅𝜕𝑢𝜕𝜑²−2𝐵₁₁×𝜕𝑢𝜕𝜕𝑥²𝑤𝜕𝑥²𝐵−2¹²𝑅²𝜕𝑢𝜕𝜕𝑥²𝑤−𝜕𝑥𝜕𝜑𝜕𝑣𝐵𝜕𝑥−2¹²𝑅𝜕𝑣𝜕𝜕𝜑+𝑤²𝑤𝜕𝑥²𝐵−2²²𝑅³×𝜕𝑣𝜕𝜕𝜑+𝑤²𝑤𝜕𝜑²−𝜕𝑣𝐵𝜕𝜑−8⁶⁶𝑅×𝜕𝑣+1𝜕𝑥𝑅𝜕𝑢𝜕𝜕𝜑²𝑤−𝜕𝑥𝜕𝜑𝜕𝑣𝜕𝑥+𝐷₁₁𝜕²𝑤𝜕𝑥²²+𝐷₂₂𝑅⁴𝜕²𝑤𝜕𝜑²−𝜕𝑣𝜕𝜑²𝐷+2¹²𝑅²𝜕²𝑤𝜕𝑥²𝜕²𝑤𝜕𝜑²−𝜕𝑣𝐷𝜕𝜑+4⁶⁶𝑅²×𝜕²𝑤−𝜕𝑥𝜕𝜑𝜕𝑣𝜕𝑥²𝑅𝑑𝜑𝑑𝑥. (23)

The Lagrangian energy functional represented by Π for a circular cylindrical shell is defined by differentiating its strain and kinetic energies as Π=𝑇−𝑈. (24)

Hamilton’s principle is employed to the Lagrangian energy functional Π, and introducing the Winkler and Pasternak foundations (𝐾𝑤−𝐺∇²𝑤) term in the 𝑧-direction is inducted and appended with the equations of motion to derive the shell motion equation written in a differential operator form as 𝐿₁₁𝑢+𝐿₁₂𝑣+𝐿₁₃𝑤=𝜌_𝑡𝜕²𝑢𝜕𝑡²,𝐿₂₁𝑢+𝐿₂₂𝑣+𝐿₂₃𝑤=𝜌_𝑡𝜕²𝑣𝜕𝑡²,𝐿₃₁𝑢+𝐿₃₂𝑣+𝐿₃₃𝑤=𝜌_𝑡𝜕²𝑢𝜕𝑡²+𝐾𝑤−𝐺∇²𝑤, (25) where 𝐿_𝑖𝑗(𝑖,𝑗=1,2,3) state the differential operators with regard to 𝑥 and 𝜑 and are given in Appendix A, where 𝐺 stands for Pasternak elastic foundation and 𝐾 for the Winkler foundation modulus. The expression for the differential operator ∇² is ∇²=𝜕²𝜕𝑥²+1𝑅²𝜕²𝜕𝜑². (26) The Pasternak model reduces to the Winkler model when 𝐺 vanishes.

Using the expressions for the differential operators 𝐿_𝑖𝑗(𝑖,𝑗=1,2,3), the shells dynamical equations are written as 𝑒₁₁𝜕²𝑢𝜕𝑥²+𝑒₁₂𝜕²𝑢𝜕𝜑²+𝑒₁₃𝜕²𝑣𝜕𝑥𝜕𝜑+𝑒₁₄𝜕𝑤𝜕𝑥+𝑒₁₅𝜕³𝑤𝜕𝑥³+𝑒₁₆𝜕³𝑤𝜕𝑥𝜕𝜑²=𝜌_𝑡𝜕²𝑢𝜕𝑡²,𝑒₂₁𝜕²𝑢𝜕𝑥𝜕𝜑+𝑒₂₂𝜕²𝑣𝜕𝑥²+𝑒₂₃𝜕²𝑣𝜕𝜑²+𝑒₂₄𝜕³𝑤𝜕𝑥²𝜕𝜑+𝑒₂₅𝜕𝑤𝜕𝜑+𝑒₂₆𝜕³𝑤𝜕𝜑³=𝜌_𝑡𝜕²𝑣𝜕𝑡²,𝑒₃₁𝜕³𝑢𝜕𝑥³+𝑒₃₂𝜕𝑢𝜕𝑥+𝑒₃₃𝜕³𝑢𝜕𝑥𝜕𝜑²+𝑒₃₄𝜕³𝑣𝜕𝑥²𝜕𝜑+𝑒₃₅𝜕³𝑣𝜕𝜑³+𝑒₃₆𝜕𝑣𝜕𝜑+𝑒₃₇𝜕⁴𝑤𝜕𝑥⁴+𝑒₃₈𝜕⁴𝑤𝜕𝑥²𝜕𝜑²+𝑒₃₉𝜕⁴𝑤𝜕𝜑⁴+𝑒₃₁₀𝜕²𝑤𝜕𝑥²+𝑒₃₁₁𝜕²𝑤𝜕𝜑²+𝑒₃₁₂𝑤=𝜌_𝑡𝜕²𝑤𝜕𝑡²𝜕+𝐾𝑤−𝐺²𝑤𝜕𝑥²+𝑒₃₁₃𝜕²𝑤𝜕𝜑², (27a)(27b)(27c) where 𝑒^′_𝑖𝑗𝑠 are defined in Appendix B.

3. Methodology

The following modal displacement shape functions are adopted to separate the time and space variables: 𝑢𝑥,𝜑,𝑡=𝐴cos𝑛𝜑𝑒^{𝑖(𝜔𝑡−𝑘𝑚𝑥)},𝑣𝑥,𝜑,𝑡=𝐵sin𝑛𝜑𝑒^{𝑖(𝜔𝑡−𝑘𝑚𝑥)},𝑤𝑥,𝜑,𝑡=𝐶cos𝑛𝜑𝑒^{𝑖(𝜔𝑡−𝑘𝑚𝑥)}, (28) in the longitudinal, circumferential, and transverse directions, respectively. The constants 𝐴, 𝐵, and 𝐶 are the amplitudes of vibrations in the 𝑥, 𝜑, and 𝑧 directions, respectively, n is the number of circumferential waves, and 𝑘_𝑚 stands for axial wave number that is associated with a boundary condition is axial wave number given for four types of boundary conditions in Table 1.

Table 1 

These axial wave numbers 𝑘_𝑚 are selected to satisfy boundary conditions at both edges of a cylindrical shell. 𝜔 denotes the natural angular frequency for the cylindrical shell.

4. Derivation of Frequency Equation

Making substitution for the displacement functions u, 𝑣, and 𝑤 from the expressions (28) in system of (27a), (27b), and (27c), the shell algebraic equations are written as:𝑐₁₁𝐴+𝑐₁₂𝐵+𝑐₁₃𝐶=𝜔²𝜌_𝑡𝑐𝐴,₂₁𝐴+𝑐₂₂𝐵+𝑐₂₃𝐶=𝜔²𝜌_𝑡𝑐𝐵,₃₁𝐴+𝑐₃₂𝐵+𝑐₃₃𝐶+𝐾𝑤+𝐺𝑘²_𝑚+𝑒₃₁₃𝑛²=𝜔²𝜌_𝑡𝐶, (29a)(29b)(29c)where 𝑐^′_𝑖𝑗𝑠(𝑖,𝑗=1,2,3) are given in Appendix C.

After the arrangement of terms, the algebraic simultaneous equations (29a)–(29c) are written in matrix notation as 𝑐₁₁𝑐₁₂𝑐₁₃−𝑐₂₁𝑐₂₂𝑐₂₃−𝑐₁₃𝑐₂₃𝑐₃₃𝐴𝐵𝐶=𝜔²𝜌_𝑡𝐴𝐵𝐶100010001. (30)

For forming the shell frequency equation, the determinant of the matrix coefficients is vanished for nontrivial eigenvalue of the shell frequency. Either the eigenvalue problem obtained above is solved using computer program or the determinant is expanded and the frequencies equation is achieved in the form of polynomial equation involving 𝜔².

5. Results and Discussion

Numerical results for simply supported-simply supported cylindrical shell are calculated and compared with those found in the open literature in order to examine validity, efficiency, and accuracy of wave propagation approach. In Table 2, the nondimensional frequency parameters Ω=𝜔𝑅(1−𝑣²)𝜌/𝐸 for an isotropic cylindrical shell are compared with those ones obtained by Naeem and Sharma [20]. The simply supported boundary conditions are described on shell ends. In this comparison, the shell parameters: length 𝐿=6𝑚, thickness ℎ=0.002𝑚, and 𝑅=1𝑚, are taken, respectively. The axial mode is assumed to be 𝑚=1, and the circumferential wave numbers 𝑛 are chosen from 1–10. Material properties of the shell are mass density (𝜌), Poisson’s ratio (𝜈), and Young’s modulus (𝐸) and their values are given as 𝜌=7850kg/m³, 𝜈=0.3, and 𝐸=2.1×10¹¹ N/m².

Table 2 

Comparison of frequency parameter Ω=𝜔𝑅(1−𝜈²)𝜌/𝐸 for SS-SS cylindrical shell (𝑚=1, 𝐿=6𝑚, ℎ=0.002𝑚, 𝑅=1𝑚, 𝜈=0.3).

In Table 3, the frequency parameters Ω for a cylindrical shell with simply supported boundary conditions are compared with those results determined by Zhang et al. [8]. The geometrical parameters are assumed to be as 𝐿=20𝑚, ℎ=0.002𝑚, and 𝑅=1𝑚. For the parameters 𝐿/𝑅, a value of m=1 is used and 𝑛 are selected 0–4 in the comparison. The material properties of the shell are given as 𝜌 = 7850 kg/m³, 𝜈=0.3, and 𝐸=2.1×10¹¹ N/m².

Table 3 

Comparison of frequency parameter Ω=𝜔𝑅(1−𝜈²)𝜌/𝐸 for SS-SS cylindrical shell (𝑚=1, 𝐿/𝑅=20, 𝜈=0.3).

6. Three-Layered Cylindrical Shells

A number of results for the proposed three-layered functionally graded cylindrical shells are determined for various sets of material and geometrical parameters. The inner and outer layers of the shell are comprised of isotropic material, whereas the middle layer is assumed to be functionally graded as shown in Figure 1. In general, vibration characteristics are most influenced by Young’s modulus. In this study, the Poisson’s ratio is presumed to be constant for functionally graded materials, whereas the Young’s modulus depends on intrinsic thickness variable (z) as well as the Young’s modulus of constituent materials forming functionally graded layers. Here two configurations of a cylindrical shell are considered to suggest with regard to the shell layer thickness. In the first configuration, the thickness of each layer is supposed to be of ℎ/3 while in the second configuration, the thickness of each of the inner and outer layers is of ℎ/4 and that of the middle layer is of ℎ/2.

The stiffness moduli 𝐴_𝑖𝑗, 𝐵_𝑖𝑗, and 𝐷_𝑖𝑗 are modified in according to the thickness of material layers when inner and outer layers are isotropic and middle, is functionally graded as𝐴_𝑖𝑗=𝐴^{in(isotropic)}_𝑖𝑗+𝐴^𝑚(FG)_𝑖𝑗+𝐴^{out(isotropic)}_𝑖𝑗,𝐵_𝑖𝑗=𝐵^{in(Isotropic)}_𝑖𝑗+𝐵^𝑚(FG)_𝑖𝑗+𝐵^{out(isotropic)}_𝑖𝑗,𝐷_𝑖𝑗=𝐷^{in(isotropic)}_𝑖𝑗+𝐷^𝑚(FG)_𝑖𝑗+𝐷^{out(isotropic)}_𝑖𝑗, (31a)(31b)(31c)where 𝑖,𝑗=1,2,6 and in(isotropic), 𝑚(FG), and out(isotropic) are associated with inner isotropic, middle functionally graded and outer isotropic layers of cylindrical shell, respectively. Here, by considering the constituent materials of stainless steel and aluminum for isotropic layers and also the FG layers which are structured from two kinds of materials, nickel and zirconia four types of shells are obtained and are listed Table 4.

Table 4 

Description of cylindrical shells.

Material properties of isotropic materials: steel and aluminum are given in Table 5, whereas the material properties of the constituent materials forming functionally graded layers are listed in Table 6.

Table 5 

Material properties of isotropic materials.

Table 6 

Material properties of Nickel and Zirconia.

7. Functionally Graded Materials

The FGMs are much advanced materials and are used in engineering science and technology. Material properties of an FGM are the functions of the temperature and the volume fractions. These properties of a constituent material are managed by a volume fraction. If 𝑃_𝑖 represents a material property of the 𝑖th constituent material of an FGM consisting of 𝑘 constituent materials, then the effective property 𝑃 of the FGM is written as 𝑃=^𝑘_𝑖=1𝑃_𝑖𝑉_𝑖, (32) 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 1, that is, ^𝑘_𝑖=1𝑉_𝑖=1. (33) The volume fraction depends on the thickness variable and is defined as 𝑉_𝑖=𝑍−𝑅_𝑖𝑅_𝑜−𝑅_𝑖^𝑁, (34) for a cylindrical shell. 𝑅_𝑖 and 𝑅_𝑜 denote inner and outer radii of the shell, respectively, and 𝑧 is the thickness variable in the radial direction. 𝑁 is known as the power law exponent. It is a nonnegative real number and lies between zero and infinity. For a cylindrical shell, the volume fraction is assumed as 𝑉_𝑖=𝑍+0.5ℎℎ^𝑁, (35) where ℎ is the shell uniform thickness. When the shell is considered to consist of two materials, the effective Young’s modulus 𝐸, the poisson ratio 𝜐, and the mass density 𝜌 are given by 𝐸=𝐸₁−𝐸₂𝑍+0.5ℎℎ^𝑁+𝐸₂,𝜐=𝜐₁−𝜐₂𝑍+0.5ℎℎ^𝑁+𝜐₂,𝜌=𝜌₁−𝜌₂𝑍+0.5ℎℎ^𝑁+𝜌₂. (36a)(36b)(36c)

8. Frequency Analysis of Cylindrical Shells

8.1. Variations of Natural Frequencies (Hz) against Circumferential Wave Number (n)

In Table 7, natural frequencies (Hz) of four types of cylindrical shells are given against the circumferential wave numbers (n). The boundary conditions applied on the shell ends are simply supported. The shell parameter data is 𝑚=1,𝐿=20,ℎ=0.002,and𝑅=1. For functionally graded layer, the power law exponent (p) is 5. The numerical results exhibit the well-known characteristic of shell vibration frequencies, that is, the value of the frequency first decreases achieves its lowest value and then it begins to increase with the circumferential wave modes (n).

Table 7 

Variation of natural frequencies (Hz) of a three-layered simply supported-simply supported cylindrical shell (𝑚=1, 𝑝=5, 𝐿=20𝑚, ℎ=0.002𝑚, 𝑅=1𝑚).

For Type 1 and 2, the outer and inner layers are of steel. This shows that the values of shell frequencies increase with interchanging the order of constituent materials forming the functionally graded middle layer. For Type 1, the inner and outer layer is of steel whereas in Type 3, the inner is of steel and the outer is of aluminum. The frequency increases with the interchange of inner and outer layers material whereas the configuration of functionally graded layer is the same. Also in Type 4, the inner and outer layers possess the same materials as in Type 3, where the materials of middle layer have been replaced by changing the order of zirconia and nickel. This shows that the frequency has been increased by changing the materials of functionally graded layer.

In Tables 8 and 9, variations of natural frequencies for Type 1 for two configurations with respect to thickness of layers are shown against circumferential wave numbers (n). It is seen from Table 8 that by decreasing the thickness of the middle functionally graded layer, the frequencies diminish.

Table 8 

Variation of natural frequencies (Hz) of a three-layered simply supported-simply supported cylindrical shell for two configurations with respect to thickness of shell layers (𝑚=1, 𝑝=5, 𝐿=20𝑚, ℎ=0.002𝑚, 𝑅=1𝑚).

Table 9 

Variation of natural frequencies (Hz) of a three-layered clamped-clamped cylindrical shell for two configurations with respect to thickness of shell layers (𝑚=1, 𝑝=5, 𝐿=20𝑚, ℎ=0.002𝑚, 𝑅=1𝑚).

8.2. Frequency Analysis of Cylindrical Shells on Elastic Foundations

Figure 3 represents variations of natural frequencies (Hz) for three-layered cylindrical shells for SS-SS edge conditions with elastic foundations, that is, 𝐺=1×10⁹, 𝐾=1×10⁹ N·m. The middle layer is functionally graded material. For both Types 1 and 2 of the cylindrical shells, the frequencies increase considerably by adding the elastic foundations, that is, Pasternak and Winkler models. Also, the values of natural frequencies increase gradually against circumferential wave modes (𝑛). The minimum frequency is associated with 𝑛=1. This shows that the shell vibration is similar to that of beam type for these boundary conditions.

In Table 10, natural frequencies (Hz) for SS-SS three-layered cylindrical shells are listed for Type 1, 2, 3, and 4 cylindrical shells. The shells are based on the Pasternak and Winkler foundations as shown in Figure 2. The values of the shell frequencies show considerable increments by inducting the elastic moduli. Type 1 has the highest frequency, followed by Type 2, Type 4, and Type 3. This also exhibits the effects of material thickness of the layers forming cylindrical shell.

Table 10 

Variation of natural frequencies (Hz) of a simply supported-simply supported three-layered cylindrical shell on elastic foundations (𝑚=1, 𝑝=1, 𝐿=0.41, 𝑅=0.3015, ℎ=0.001, 𝜈=0.3, 𝐺=1.5×10⁷ N·m, 𝐾=2.5×10⁷ N·m).

Figure 2 

Cylindrical shell with elastic foundations.

Figure 3 

Variations of natural frequencies (Hz) for three-layered SS-SS cylindrical shells (𝐺=1×109 N·m, 𝐾=1×109).

Table 11 displays the variations of natural frequencies (Hz) for four cylindrical shells: Type 1, Type 2, Type 3, and Type 4 for simply supported boundary conditions. The half axial wave mode number (m) is assumed to be one. Here, 𝐾=2.5×10⁷ N·m and 𝐺 is varied. As 𝐺 is increased, the frequency increases gradually but slowly for the present four types of shells. Also for circumferential wave numbers (n), the frequency increases as n is increased. The minimum frequency corresponds to the circumferential wave number 𝑛=1. The vibration becomes beam-type because the fundamental circumferential mode is 1.

Table 11 

Variations of natural frequencies (Hz) of SS-SS cylindrical shell for different values of G, where 𝐾=2.5×10⁷ N·m, ℎ=0.001, 𝑚=1, 𝑅=1, 𝐿=0.41.

In Table 12, the natural frequency variations (Hz) for four cylindrical shells Type 1, Type 2, Type 3, and Type 4 are given against circumferential wave numbers (n). The value of Pasternak modulus (G) is maintained and fixed, that is, 𝐺=2.5×10⁷ N·m for the half axial wave mode 𝑚=1. The values of shell frequencies do not increase much with increasing value of 𝐾. The smallest frequency is associated with 𝑛=1, that is, the vibration is similar as that of a beam. Thus the effect of 𝐾 is negligible for keeping 𝐺 constant.

Table 12 

Variations of natural frequencies (Hz) of SS-SS cylindrical shell for different values of K, where 𝐺=2.5×10⁷ N·m, ℎ=0.001, 𝑚=1, 𝑅=1, 𝐿=0.41.

9. Conclusion

In this study, vibrations of three-layered cylindrical shells are investigated for various shell parameters. Three shell layers are designed by isotropic and functionally graded materials. In the configuration, outer and inner layers are of isotropic layers, whereas the middle layer is assumed to be of functionally graded layers. This forms a combination of isotropic and functionally graded layers. In a shell thickness direction, material composition of a functionally graded material is controlled by volume fraction law. Love shell dynamical equations are considered to describe the vibration problem. Pasternak and Winkler foundations are appended in the transverse direction. The boundary conditions applied on the shell ends are simply supported. Wave propagation approach is utilized to frame the shell frequency equation for a cylindrical shell in the eigenvalue form. The influences of the configurations of the cylindrical shells are analyzed with interchange of materials of shell layers. From this analysis, it is observed that the change of material nature of the shell functionally graded layer influences the shell frequencies. As the thickness of the middle layer is decreased, the vibration frequencies diminish prominently. For cylindrical shells of Types 1 and 2, the vibration frequencies increase considerably by adding the elastic foundations, that is, Pasternak and Winkler models. This analysis can be applied to extract vibration frequencies for the present cylindrical shell configurations for various boundary conditions.

Appendices

A.

𝐿₁₁=𝐴₁₁𝜕²𝜕𝑥²+𝐴₆₆𝑅²𝜕²𝜕𝜑²,𝐿₁₂=𝐴₁₂+𝐴₆₆𝑅𝜕²+𝐵𝜕𝑥𝜕𝜑¹²+2𝐵₆₆𝑅²𝜕²,𝐿𝜕𝑥𝜕𝜑₁₃=𝐴₁₂𝑅𝜕𝜕𝑥−𝐵₁₁𝜕³𝜕𝑥³−𝐵₁₂+2𝐵₆₆𝑅²𝜕³𝜕𝑥𝜕𝜑²,𝐿₂₁=𝐴₁₂+𝐴₆₆𝑅𝜕²+𝐵𝜕𝑥𝜕𝜑¹²+𝐵₆₆𝑅²𝜕²,𝐿𝜕𝑥𝜕𝜑₂₂=𝐴₆₆+3𝐵₆₆𝑅+2𝐷₆₆𝑅²𝜕²𝜕𝑥²+𝐴₂₂𝑅²+2𝐵₂₂𝑅³+𝐷₂₂𝑅⁴𝜕²𝜕𝜑²,𝐿₂₃=𝐴₂₂𝑅²+𝐵₂₂𝑅³𝜕−𝐵𝜕𝜑𝑅²²³+𝐷₂₂𝑅⁴𝜕³𝜕𝜑³−𝐵₁₂+2𝐵₆₆𝑅+𝐷₁₂+2𝐷₆₆𝑅²𝜕³𝜕𝑥²,𝐿𝜕𝜑₃₁𝐴=−¹²𝑅𝜕𝜕𝑥+𝐵₁₁𝜕³𝜕𝑥³+𝐵₁₂+2𝐵₆₆𝑅²𝜕³𝜕𝑥𝜕𝜑²,𝐿₃₂𝐴=−²²𝑅²+𝐵₂₂𝑅³𝜕+𝐵𝜕𝜑𝑅²²³+𝐷₂₂𝑅⁴𝜕³𝜕𝜑³+𝐵₁₂+2𝐵₆₆𝑅+𝐷₁₂+4𝐷₆₆𝑅²𝜕³𝜕𝑥²,𝐿𝜕𝜑₃₃𝐴=−²²𝑅²+2𝐵₁₂𝑅𝜕²𝜕𝑥²+2𝐵₂₂𝑅³𝜕²𝜕𝜑²−𝐷₁₁𝜕⁴𝜕𝑥⁴𝐷−2¹²+2𝐷₆₆𝑅²𝜕⁴𝜕𝑥²𝜕𝜑²−𝐷₂₂𝑅⁴𝜕⁴𝜕𝜑⁴. (A.1)

B.

𝑒₁₁=𝐴₁₁,𝑒₁₂=𝐴₆₆𝑅²,𝑒₁₃=𝐴₁₂+𝐴₆₆𝑅+2𝐵₆₆+𝐵₁₂𝑅²,𝑒₁₄=𝐴₁₂𝑅,𝑒₁₅=−𝐵₁₁,𝑒₁₆=−𝐵₁₂+2𝐵₆₆𝑅²,𝑒₂₁=𝐴₁₂+𝐴₆₆𝑅+𝐵₆₆+𝐵₁₂𝑅²,𝑒₂₂=𝐴₆₆+3𝐵₆₆𝑅+2𝐷₆₆𝑅²,𝑒₂₃=𝐴₂₂𝑅²+2𝐵₂₂𝑅³+𝐷₂₂𝑅⁴,𝑒₂₄𝐵=−¹²+2𝐵₆₆𝑅+2𝐷₆₆+𝐷₁₂𝑅²,𝑒₂₅=𝐴₂₂𝑅²+𝐵₂₂𝑅³,𝑒₂₆𝐵=−²²𝑅³+𝐷₂₂𝑅⁴,𝑒₃₁=𝐵₁₁=−𝑒₁₅,𝑒₃₂𝐴=−¹²𝑅=−𝑒₁₄,𝑒₃₃=𝐵₁₂+2𝐵₆₆𝑅²,𝑒₃₄=𝐵₁₂+2𝐵₆₆𝑅+4𝐷₆₆+𝐷₁₂𝑅²=−𝑒₂₄+2𝐷₆₆𝑅²,𝑒₃₅=𝐵₂₂𝑅³+𝐷₂₂𝑅⁴=−𝑒₂₆,𝑒₃₆𝐴=−²²𝑅²+𝐵₂₂𝑅³=−𝑒₂₅,𝑒₃₇=−𝐷₁₁,𝑒₃₈=−2𝐷₁₂+4𝐷₆₆𝑅²,𝑒₃₉𝐷=−²²𝑅⁴,𝑒₃₁₀=2𝐵₁₂𝑅,𝑒₃₁₁=2𝐵₂₂𝑅³,𝑒₃₁₂𝐴=−²²𝑅²,𝑒₃₁₃=1𝑅². (B.1)

C.

𝑐₁₁=𝑘²_𝑚𝑒₁₁+𝑛²𝑒₁₂,𝑐₁₂=𝑖𝑛𝑘_𝑚𝑒₁₃,𝑐₁₃=𝑖𝑘_𝑚𝑒₁₄−𝑛²𝑒₁₆−𝑘²_𝑚𝑒₁₅,𝑐₂₁=−𝑖𝑛𝑘_𝑚𝑒₂₁,𝑐₂₂=𝑘²_𝑚𝑒₂₂+𝑛²𝑒₂₃,𝑐₂₃=−𝑛𝑘²_𝑚𝑒₂₄+𝑛𝑒₂₅−𝑛³𝑒₂₆,𝑐₃₁=−𝑖𝑘_𝑚𝑒₁₄−𝑛²𝑒₁₆−𝑘²_𝑚𝑒₁₅,𝑐₃₂=−𝑛𝑘²_𝑚𝑒₂₄+𝑛𝑒₂₅−𝑛³𝑒₂₆,𝑐₃₃=𝑒₃₁₂+𝑛²𝑒₃₁₁+𝑘²_𝑚𝑒₃₁₀−𝑛⁴𝑒₃₉−𝑛𝑘²_𝑚𝑒₃₈−𝑘⁴_𝑚𝑒₃₇+𝐾+𝐺𝑘²_𝑚+𝑒₃₁₃𝑛². (C.1)