Influences of shear stresses and rotary inertia on the vibration of functionally graded coated sandwich cylindrical shells resting on the Pasternak elastic foundation

Abstract

In this study, the behavior of vibration of sandwich cylindrical shells covered by functionally graded coatings and resting on the Pasternak elastic foundation considering combined influences of shear stresses and rotary inertia are examined. It is assumed that the effective material properties of functionally graded coatings changes exponentially in thickness direction. The modified Donnell type equations of motion of functionally graded and homogeneous sandwich cylindrical shells on the Pasternak elastic foundation are deduced using the first-order shear deformation theory. Basic equations are reduced to an algebraic equation of the sixth order and numerically solving this algebraic equation gives the dimensionless fundamental frequency. The expressions for the dimensionless fundamental frequencies of functionally graded and ceramic coated sandwich cylindrical shells with and without taking into account the effects of Pasternak elastic foundation and shear stresses obtained in a special case. Calculations, the influences of an elastic foundation, compositional profiles of coatings, shear stresses, rotary inertia, and sandwich shell geometry parameters on the nondimensional fundamental frequency are described. The results are verified by comparing the obtained values with those in the existing literature.

Keywords

Functionally graded coatings first-order shear deformation theory sandwich cylindrical shells elastic foundation vibration fundamental frequency dental implant

Introduction

Functionally graded materials (FGMs) refer to the composite materials where the compositions or the microstructures are locally varied so that a certain variation of the local material properties is achieved. A study on design, processing, applications, and techniques of FGMs can be found in Kieback et al. [1]. In addition, FGMs were fabricated also for biomedical application, especially for dental implant application, and the effect of gradient structure was evaluated. The titanium/zirconium oxide and other FGM implants were fabricated to optimize both mechanical properties and biocompatibilities or change bioreactivity in each region [2,3]. FGM shell structures are generally used as structural components in missiles engine, resistant coatings in space plans, atomic reactors, spacecraft thermal shields, intelligent electrical components, submarines, turbine components, sensors, and others. A detailed review on the performance of FGMs and applications can be seen in the book of Shen [4]. The literature on the free vibration analysis of functionally graded (FG) cylindrical shells with account taken of combined effects of rotary inertia and shear deformation is relatively scarce. In addition, in the most of the existing researches in this regards, single-layer FG cylindrical shells have been analyzed [5 –18]. Sandwich constructions are widely used as structural elements in many engineering fields such as automobiles industry, aerospace industry, marine, and building constructions because of their excellent dynamic characteristics, low specific weight, outstanding banding rigidity, and maximum fatigue properties [19 –21].

The main drawback of the traditional sandwich structures is the presence of a strong discontinuity stresses, in leaf-base interface which may enter the serious problem of delamination. The significant advance in the development of an effective protective coating was associated with the development of FGMs. Using FGM, as an alternative to joining directly together two dissimilar materials such as ceramics and metal, carries several advantages including: much lower thermal stress distribution across the thickness, minimization of stress concentrations at interfaces, and an increase in bonding strength. There are many fabricating methods for producing FG materials. Thin surface FG coatings are produced by physical vapor deposition (PVD) or chemical vapor deposition, plasma spraying, self-propagating high-temperature synthesis, etc. [1,22].

Among a great number of studies on the FGM structures, an interesting issue is the static and dynamic analysis of FG-coated sandwich structures. One of the first studies on the FGM sandwich structures belongs to Zenkour [23], which are constructed the FGM sandwich plates and then discussed vibration and buckling problems. After this study, Shodja et al. [24] presented a thermo-elasticity analysis of a two-dimensional thick composite consisting of homogeneous and FG layers. Shen and Li [25] presented compressive postbuckling under thermal environments and thermal postbuckling due to heat conduction for a sandwich plate with FGM face sheets using a two-step perturbation technique. Li et al. [26] studied free vibration of sandwich rectangular plates including FGM layers based on the three-dimensional linear theory of elasticity. Zenkour and Sobhy [27] investigated the thermal buckling of FG-coated sandwich plates using the shear deformation theories (SDTs) and classical shell theory (CST). Tornabene et al. [28] studied the dynamic behavior of FG laminated doubly curved shells and panels of revolution with a free-form meridian, using the first-order shear deformation theory (FOSDT) and differential quadrature method (DQM). Neves et al. [29] investigated the free vibration and buckling analysis of isotropic and sandwich FG plates using HOSDT and meshless technique. Zaki et al. [30] examined the stresses and strains analyses of FG-coated plates, using two-dimensional elasticity theory and superposition method. Dozio [31] formulated of advanced two-dimensional Ritz-based models for accurate prediction of natural frequencies of thin and thick sandwich plates including FGM core, using an entire family of higher order layerwise and equivalent single-layer theories. Bessaim et al. [32] developed a new higher order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with FG isotropic face sheets. Zenkour [33] presented bending analysis of FG sandwich plates using a simple four-unknown shear and normal deformations theory. Fazzolari and Carrera [34] studied free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core use the Ritz minimum energy method, based on the principle of virtual displacements (PVD). Malekzadeh and Ghaedsharaf [35] presented free vibration of laminated cylindrical panels with finite length and FG core and coating employing layerwise-differential quadrature method using the three-dimensional elasticity theory. Hamidi et al. [36] developed a new four-variable refined plate theory for bending behavior of FGM sandwich plates under thermomechanical loading. Sofiyev [37] studied vibration and buckling of shear deformable sandwich cylindrical shells covered by different coatings subjected to the hydrostatic pressure. Zenkour and Alghamdi [38] discussed thermomechanical bending response of FG nonsymmetrical sandwich plates of uniform thickness (constant depth).

In some practical applications, thin cylindrical shells made of different materials are in contact with an elastic foundation. An overview of the elastic foundation models is discussed in the studies of Hui and Hansen [39], Gorbunov-Possadov et al. [40], and Hui [41]. There are very few studies on the static and dynamic behaviors of shear deformable FG sandwich structural elements resting on elastic foundations. Kiani and Eslami [42] and Kiani et al. [43] investigated thermal and mechanical buckling and postbuckling behaviors of perfect and imperfect sandwich plates with FGM face sheets under uniform temperature rise loading, employing the single mode approach combined with Galerkin technique. Sobhy [44] investigated the buckling and free vibration of plates with exponentially graded (EG) coatings surrounded by elastic medium under various boundary conditions, using the sinusoidal shear deformation theory. Kamarian et al. [45] presented free vibration analysis of three-parameter FG material sandwich plates resting on Pasternak foundations based on three-dimensional elasticity theory. Sofiyev and Kuruoglu [14] studied torsional buckling and free vibration of the cylindrical shell with FG coatings surrounded by an elastic medium using the CST. Najafov et al. [46] studied the vibration and stability of axially compressed truncated conical shells with FG middle layer surrounded by elastic medium. Najafov et al. [47] studied the stability of EG cylindrical shells with shear stresses on a Pasternak foundation. Kamarian et al. [48] investigated free vibration response of sandwich cylindrical shells with FG material face sheets resting on Pasternak foundation, based on three-dimensional elasticity theory. Iurlaro et al. [49] examined bending and free vibration analysis of FG sandwich plates using the refined ZZ theory. Tornabene et al. [50] analyzed free vibrations of free-form doubly curved shells made of FG materials using higher order ESL theories. Tornabene et al. [51] studied Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly curved and degenerate shells and panels. Tornabene et al. [52] studied stress and strain recovery for FG free-form and doubly curved sandwich shells using higher order ESL theory.

Nevertheless, the vibration behavior of sandwich cylindrical shells covered by FG coatings, resting on an elastic foundation and considering combined effects of shear stresses and rotation has not been studied sufficiently. The purpose of this study is the solution of such problem. The effective material properties of FG coatings are assumed to be graded in the thickness direction according to an exponential law distribution. The nondimensional fundamental frequencies of FG-coated sandwich cylindrical shells with and without taking into account influences of an elastic foundation, shear stresses, and rotary inertia are obtained. Calculations, the influences of Pasternak elastic foundation, compositional profile of coatings, shear stresses, rotary inertia, and sandwich shell geometry parameters on the nondimensional fundamental frequencies are described. The results are verified by comparing the obtained values with those in the existing literature.

Formulation of the problem

Figure 1 shows the nomenclature of a sandwich circular cylindrical shell resting on the Pasternak elastic foundation. The most simple representation of a foundation response is Winkler foundation which consists of an infinite set of uncoupled springs (Figure 1) with a spring constant $K w (N / m 3)$ , i.e. modulus of subgrade reaction. In the case of the Pasternak elastic foundation model, the foundation is capable of transferring loads in horizontal direction through a shear layer on the top of the Winkler foundation (Figure 1). It should be marked that the shear layer is not an elastic layer of finite thickness, it is a mathematical representation of the behavior of the reaction of the subgrade reaction. Most earthen soils can be appropriately represented by a mathematical model from Pasternak, whereas sandy soils and liquids can be represented by Winkler’s model. The load–displacement relationship of the Pasternak foundation is assumed to be $p 0 = K w w - K p (w, xx + w, yy)$ , where p₀ is the force per unit area, $K p (N / m)$ is shearing layer stiffness of the foundation, w is the displacement, and a comma denotes partial differentiation with respect to the corresponding coordinates [42]. This model simply reduces to the Winkler’s type when $K p = 0$ . Let $Φ (x, y)$ be the Airy stress function for the stress resultants defined by $N x = h Φ, yy, N xy = - h Φ, xy$ , and $N y = h Φ, xx$ . The coordinate system is placed on the reference surface of the FGM sandwich cylindrical shell, with the origin O at its end and the coordinate axes x, y, and ζ in the axial, circumferential, and the inward normal directions, as shown in Figure 1. The length, radius, and total thickness of the sandwich cylindrical shell are $L, R$ , and h, respectively. The inner and outer surfaces of the sandwich cylindrical shell are covered by FG coatings consisting of different materials. The cross-section of the FG sandwich cylindrical shells is shown in Figure 2. The thickness of each coating is $h FG$ , while the thickness of the metal core is h_m, respectively. Three types of sandwich cylindrical shells, namely (a) the sandwich cylindrical shell with FG coatings and metal-rich core (FG–M–FG), (b) the sandwich cylindrical shell with ceramic-rich coatings and metal-rich core (C–M–C), and (c) the metal-rich cylindrical shell with metal-rich coatings (M–M–M) are considered (see Figure 2).

The material properties of FG coatings are varying smoothly in the thickness direction only. We assume that the composition is varied from the interfaces to the outer and inner surfaces, i.e. the outer ( $ζ = h 0 = - h / 2$ ) and inner ( $ζ = h 3 = + h / 2$ ) surfaces of the cylindrical shell are core, whereas the interface $(h 1, h 2)$ is metal rich. The volume fractions of the FG coatings varied according to a simple power law function of ζ while that of the core equals unity, and they are given as

V (1) = \frac{2 \bar{ζ} + 1}{2 ζ 1 + 1}, ζ 0 \leq \bar{ζ} < ζ 1, (ζ 0 = - 0.5) V (2) = 1, ζ 1 \leq \bar{ζ} \leq ζ 2 V (3) = \frac{2 \bar{ζ} - 1}{2 ζ 2 - 1}, ζ 2 < \bar{ζ} \leq ζ 3 (ζ 3 = 0.5)

(1)

where

\bar{ζ} = ζ / h; ζ i - 1 = h i - 1 / h (i = 1, 2, \dots, 4)

V (1) = V (3) = 1

represent a metal-rich cylindrical shell and

V (1) = V (3) = 0

represent a ceramic-metal-ceramic (C–M–C) sandwich cylindrical shell.

The above assumption reflects a simple rule of mixtures used to obtain the effective material properties of FG coatings. The effective material properties, like Young’s modulus, Poisson’s ratio, and mass density of outer FG coating by the rule of mixture may be expressed as [44]

E_{FG}^{(1)} (\bar{ζ}) = E c e V (1) ln (E m / E c), ν_{FG}^{(1)} (\bar{ζ}) = ν c e V (1) ln (ν m / ν c), ρ_{FG}^{(1)} (\bar{ζ}) = ρ m e V (1) ln (ρ m / ρ c)

(2)

and the inner FG coating can be expressed as

E_{FG}^{(3)} (\bar{ζ}) = E c e V (3) ln (E m / E c), ν_{FG}^{(3)} (\bar{ζ}) = ν c e V (3) ln (ν m / ν c), ρ_{FG}^{(3)} (\bar{ζ}) = ρ m e V (3) ln (ρ m / ρ c)

(3)

where

E m, ν m, ρ m

and

E c, ν c, ρ c

are the Young’s modulus, Poisson’s ratio, and density of the metal and ceramic surfaces of FG coatings.

The variation of Young’s modulus, Poisson’s ratio, and density of a sandwich cylindrical shell covered by FG coatings are given as [37]

[E (\bar{ζ}), ν (\bar{ζ}), ρ (\bar{ζ})] = {E_{FG}^{(1)}, ν_{FG}^{(1)}, ρ_{FG}^{(1)} - 0.5 \leq \bar{ζ} < ζ 1 E_{m}^{(2)}, ν_{m}^{(2)}, ρ_{m}^{(2)} ζ 1 \leq \bar{ζ} \leq ζ 2 E_{FG}^{(3)}, ν_{FG}^{(3)}, ρ_{FG}^{(3)} ζ 2 < \bar{ζ} \leq 0.5

(4)

Fundamental relations and basic equations

For the kth layer of FG sandwich cylindrical shells, the constitutive equations are [25,37,53]

[σ_{x}^{(k)} σ_{y}^{(k)} σ_{yz}^{(k)} σ_{xz}^{(k)} σ_{xy}^{(k)}] = [Q_{11}^{(k)} Q_{12}^{(k)} 000 Q_{12}^{(k)} Q_{11}^{(k)} 00000 Q_{44}^{(k)} 00000 Q_{55}^{(k)} 00000 Q_{66}^{(k)}] [ɛ x ɛ y γ yz γ xz γ xy]

(5)

where

σ_{x}^{(k)}, σ_{y}^{(k)}, σ_{yz}^{(k)}, σ_{xz}^{(k)}, σ_{xy}^{(k)}

(k = 1, 2, 3) are the stresses in the layers, k is the layer number,

ɛ x, ɛ y, γ yz, γ xz, γ xy

are the strains of the sandwich cylindrical shell, and the quantities

Q_{ij}^{(k)} (i, j = 1, 2, 6)

in the layers are

Q_{11}^{(1)} = \frac{E_{FG}^{(1)}}{1 - (ν_{FG}^{(1)}) 2}, Q_{12}^{(1)} = ν_{FG}^{(1)} Q_{11}^{(1)}, Q_{44}^{(1)} = Q_{55}^{(1)} = Q_{66}^{(1)} = \frac{E_{FG}^{(1)}}{2 [1 + ν_{FG}^{(1)}]}, Q_{11}^{(2)} = \frac{E_{m}^{(2)}}{1 - (ν_{i}^{(2)}) 2}, Q_{12}^{(2)} = ν_{m}^{(2)} Q_{11}^{(2)}, Q_{44}^{(2)} = Q_{55}^{(2)} = Q_{66}^{(2)} = \frac{E_{m}^{(2)}}{2 [1 + ν_{m}^{(2)}]}, Q_{11}^{(3)} = \frac{E_{FG}^{(3)}}{1 - (ν_{FG}^{(3)}) 2}, Q_{12}^{(3)} = ν_{FG}^{(3)} Q_{11}^{(3)}, Q_{44}^{(3)} = Q_{55}^{(3)} = Q_{66}^{(3)} = \frac{E_{FG}^{(3)}}{2 [1 + ν_{FG}^{(3)}]}

(6)

The shear stresses in the layers of sandwich cylindrical shells covered by FG coatings on the basis of FOSDT vary depending on the thickness coordinate as follows [53 –56]

σ_{xz}^{(k)} = f_{1, ζ}^{(k)} (ζ) φ (x, y), σ_{yz}^{(k)} = f_{2, ζ}^{(k)} (ζ) ψ (x, y), (k = 1, 2, 3)

(7)

where a comma denotes derivation with respect to ζ,

φ (x, y)

and

ψ (x, y)

are the angles of rotation of a normal to the reference surface, and

f_{1}^{(k)} (ζ)

and

f_{2}^{(k)} (ζ)

represent the posteriori-specified shape functions in the layers which, through their derivatives, determine the through-the-thickness distribution of the transverse shear stresses

σ_{xz}^{(k)}

and

σ_{yz}^{(k)}

in the layers.

Substituting equation (7) into the third and fourth equations of system (5), we get

γ xz = f_{1, ζ}^{(k)} a_{55}^{(k)} (\bar{ζ}) φ (x, y), γ yz = f_{1, ζ}^{(k)} (ζ) a_{44}^{(k)} (\bar{ζ}) ψ (x, y)

(8)

where the following definitions apply

a_{55}^{(k)} (\bar{ζ}) = 1 / Q_{55}^{(k)} (\bar{ζ}), a_{44}^{(k)} (\bar{ζ}) = 1 / Q_{44}^{(k)} (\bar{ζ})

(9)

Due to assumptions of the FOSDT, we obtain [53 –55]

u x, z = - w, x + γ xz, u y, z = - w, y + γ yz

(10)

where u_x and u_y are displacements in the x and y directions of the any point of sandwich shell, respectively.

Integration of equation (10) with respect to ζ from zero to ζ with the condition that for $ζ = 0$ , $u x = u (x, y)$ , and $u y = v (x, y)$ , the following expressions for the in-plane displacements of any point in the layers of a sandwich cylindrical shell covered by FG coatings are obtained

u z = w, u x = u - ζ w, x + I_{01}^{(k)} φ, u y = v - ζ w, y + I_{02}^{(k)} ψ

(11)

where

u x = u (x, y)

and

u y = v (x, y)

are displacements along coordinates x and y, respectively, and the following definitions apply

I_{01}^{(k)} = \int_{0}^{ζ} a_{55}^{(k)} (\bar{ζ}) f_{1, ζ}^{(k)} (ζ) d ζ; I_{02}^{(k)} = \int_{0}^{ζ} a_{44}^{(k)} (\bar{ζ}) f_{2, ζ}^{(k)} (ζ) d ζ, (k = 1, 2, 3)

(12)

The strains $ɛ x, ɛ y, γ yz, γ xz, γ xy$ are related to the displacements $u x, u y, u z$ by the following equations

ɛ x = u x, x, ɛ y = u y, y + w / R, γ yz = u y, z + u z, y, γ xz = u z, x + u x, z, γ xy = u x, y + u y, x

(13)

Substituting u_x and u_y from equation (11) into equation (13) we obtain expressions for the strains at ζ distance from the reference surface of FG-coated sandwich cylindrical shell based on the FOSDT

(ɛ x ɛ y γ xy) = (e x - ζ w, xx + I_{01}^{(k)} φ, x e y - ζ w, yy + I_{02}^{(k)} ψ, y γ 0 xy - 2 ζ w, xy + I_{01}^{(k)} φ, y + I_{02}^{(k)} ψ, x)

(14)

where a comma denotes partial differentiation with respect to the corresponding coordinates and

e x, e y, γ 0 xy

are strains on the reference surface.

The force and moment resultants of sandwich cylindrical shells covered by FG coatings are given by [53 –55]

(N x, N y, N xy, Q x, Q y) = \sum_{k = 1}^{3} \int_{h k - 1}^{h k} [σ_{x}^{(k)}, σ_{y}^{(k)}, σ_{xy}^{(k)}, σ_{xz}^{(k)}, σ_{yz}^{(k)}] d ζ,

(15)

(M x, M y, M xy) = \sum_{k = 1}^{3} \int_{h k - 1}^{h k} ζ [σ_{x}^{(k)}, σ_{y}^{(k)}, σ_{xy}^{(k)}] d ζ

(16)

where N_x, N_y, and

N xy

are the in-plane force resultants; Q_x and Q_y are the transverse shear force resultants; and M_x, M_y, and

M xy

are the moment resultants.

The governing equations of motion for a FG-coated sandwich cylindrical shell including the shear stresses and rotary inertia and resting on a Pasternak elastic foundation are given as [53]

\frac{\partial M x}{\partial x} + \frac{\partial M y}{\partial y} - Q x = - ρ 1 \frac{\partial 3 w}{\partial x \partial t 2} + ρ 2 \frac{\partial 2 φ}{\partial t 2} \frac{\partial M xy}{\partial x} + \frac{\partial M y}{\partial y} - Q y = - ρ 1 \frac{\partial 3 w}{\partial y \partial t 2} + ρ 3 \frac{\partial 2 ψ}{\partial t 2} \frac{\partial 2 e x}{\partial y 2} + \frac{\partial 2 e y}{\partial x 2} - \frac{\partial 2 γ 0 xy}{\partial x \partial y} = - \frac{1}{R} \frac{\partial 2 w}{\partial x 2} \frac{\partial Q x}{\partial x} + \frac{\partial Q y}{\partial y} + \frac{N y}{R} - K w w + K p (\frac{\partial 2 w}{\partial x 2} + \frac{\partial 2 w}{\partial y 2}) = ρ t \frac{\partial 2 w}{\partial t 2}

(17)

where t is a time variable,

ρ i (i = 1, 2, 3)

and

ρ t

are inertia terms and the following definitions apply

ρ t = \int_{- h / 2}^{- h 1} ρ_{FG}^{(1)} d ζ + ρ_{m}^{(2)} (h 2 + h 1) + \int_{h 2}^{h / 2} ρ_{FG}^{(3)} d ζ, ρ 1 = \int_{- h / 2}^{- h 1} ζ 2 ρ_{FG}^{(1)} d ζ + ρ_{m}^{(2)} \int_{- h 1}^{h 2} ζ 2 d ζ + \int_{h 2}^{h / 2} ζ 2 ρ_{FG}^{(3)} d ζ, ρ 2 = \int_{- h / 2}^{- h 1} ζ ρ_{FG}^{(1)} I_{01}^{(1)} d ζ + ρ_{m}^{(2)} \int_{- h 1}^{h 2} ζ I_{01}^{(2)} d ζ + \int_{h 2}^{h / 2} ζ ρ_{FG}^{(3)} I_{01}^{(3)} d ζ, ρ 3 = \int_{- h / 2}^{- h 1} ζ ρ_{FG}^{(1)} I_{02}^{(1)} d ζ + ρ_{m}^{(2)} \int_{- h 1}^{h 2} ζ I_{02}^{(2)} d ζ + \int_{h 2}^{h / 2} ζ ρ_{FG}^{(3)} I_{02}^{(3)} d ζ

(18)

By substituting equations (5), (14), (16), (17) and expression for the Airy stress function into the set of equations (17), the reduced equations of motion in terms of $Φ, w, φ, ψ$ can be written as

[L 11 L 12 L 13 L 14 L 21 L 22 L 23 L 24 L 31 L 32 L 33 L 34 L 41 L 42 L 43 L 44] [Φ w φ ψ] = 0

(19)

where

L ij (i, j = 1, 2, 6)

are differential operators and are given in Appendix 1.

Equation (19) is governing equations for the motion of FG sandwich cylindrical shell including shear stresses and rotary inertia, and resting on the Pasternak elastic foundation.

Solution of basic equations

Assuming that the sandwich cylindrical shell covered by FG coatings is simply supported at both ends. For this case, the corresponding boundary conditions may be specified as [4,53]

w = 0, M x = 0, ψ = 0, Φ, yy = 0 when x = 0, x = L

(20)

The Airy stress function Φ, the displacement components w, and the shear rotations φ and ψ are assumed in the form of [51,52]

Φ = C ϕ \sin (β 1 x) \sin (β 2 y) \cos ω t, w = C f \sin (β 1 x) \sin (β 2 y) \cos ω t, φ = C φ \cos (β 1 x) \sin (β 2 y) \cos ω t, ψ = C ψ \sin (β 1 x) \cos (β 2 y) \cos ω t

(21)

where ω is the angular frequency of FG-coated sandwich cylindrical shell on the Pasternak elastic foundation;

C ϕ, C f, C φ, C ψ

are unknown functions to be determined;

β 1 = \frac{m π}{L}

and

β 2 = \frac{n}{R}

, in which m is the half wave number in axial direction and n is the circumferential wave number.

Making use of the orthogonality condition, we multiply set of equation (19) by $\sin (β 1 x) \sin (β 2 y)$ and substituting equation (21) into a set of resulting equations, and then applying the Galerkin method that gives the following set of equations

[U 11 - U 12 + U 120 ω 2 U 13 - U 130 ω 2 U 14 U 21 - U 22 + U 220 ω 2 U 23 U 24 - U 240 ω 2 U 31 - U 32 U 33 U 34 U 41 - U 42 + U 420 ω 2 U 43 U 44] [C ϕ C f C φ C ψ] = [0000]

(22)

where the following definitions apply

U 11 = [(C 1 - C 5) β_{1}^{2} β_{2}^{2} + C 2 β_{1}^{4}] h, U 12 = (C 4 + C 6) β_{1}^{2} β_{2}^{2} + C 3 β_{1}^{4}, U 120 = ρ 1 β_{1}^{2}, U 13 = C 7 β_{1}^{3} + C 11 β 1 β_{2}^{2} + I 5 β 1, U 130 = ρ 2 β 1, U 14 = (C 8 + C 12) β 2 β_{1}^{2}, U 21 = [C 2 β_{2}^{4} + (C 1 - C 5) β_{1}^{2} β_{2}^{2}] h, U 22 = (C 6 + C 4) β_{1}^{2} β_{2}^{2} + C 3 β_{2}^{4}, U 220 = ρ 1 β_{2}^{2}, U 23 = (C 9 + C 11) β 1 β_{2}^{2}, U 24 = C 10 β_{2}^{3} + C 12 β_{1}^{2} β 2 + I 6 β 2, U 240 = ρ 3 β 2, U 31 = [B 1 β_{1}^{4} + (2 B 2 + B 5) β_{1}^{2} β_{2}^{2} + B 1 β_{2}^{4}] h, U 32 = B 4 β_{1}^{4} + (2 B 3 - B 6) β_{1}^{2} β_{2}^{2} + B 4 β_{2}^{4} + β_{1}^{2} / R, U 33 = B 9 β_{1}^{3} + (B 7 + B 11) β 1 β_{2}^{2}, U 34 = (B 10 + B 12) β_{1}^{2} β 2 + B 8 β_{2}^{3}, U 41 = β_{1}^{2} h / R, U 42 = K w + K p (β_{1}^{2} + β_{2}^{2}), U 420 = ρ t, U 43 = I 5 β 1, U 44 = I 6 β 2

(23)

in which

C i, B i (i = 1, 2, \dots, 12)

and

I j (j = 5, 6)

are parameters depending on the material properties, sandwich shell characteristics, and are given in detail in Appendix 1.

For the nontrivial solution, the determinant of system of equation (22) must be zero, i.e.

ω 6 + F 1 ω 4 + F 2 ω 2 + F 3 = 0

(24)

where

F i (i = 1, 2, 3)

are coefficients depending on the FG sandwich shell characteristics and a Pasternak elastic foundation coefficients and are given in detail in Appendix 2.

The solution for equation (24) consists of six roots, and three positive roots for each wave numbers (m,n) are the natural frequencies. The smallest positive root is applied in the present study of the fundamental frequency, $ω_{SDT}^{rotwp}$ , of FG-coated sandwich cylindrical shells including the shear stresses and rotary inertia, and resting on the Pasternak elastic foundation. Equation (24) is solved with computer programming using MAPLE 14 package for numerical investigation. The nondimensional fundamental frequency of FG-coated sandwich cylindrical shell including the shear stresses and rotary inertia and resting on the Pasternak elastic foundation is expressed as

ω_{1 SDT}^{rotwp} = ω_{SDT}^{rotwp} \frac{L 2}{h} \sqrt{\frac{ρ_{m}^{(2)}}{E_{m}^{(2)}}}

(25)

As the rotary inertia is not taken into account, i.e. $ρ i = 0 (i = 1, 2, 3)$ , from equation (22), we obtain an expression for the nondimensional frequency of FG-coated sandwich cylindrical shell with the shear stresses and resting on the Pasternak elastic foundation based on the SDT

ω_{1 SDT}^{wp} = \frac{L 2}{h} \sqrt{\frac{ρ_{m}^{(2)}}{E_{m}^{(2)}}} \sqrt{\frac{U 41 T 41 + U 43 T 43 + U 44 T 44 + U 42 T 42}{T 42 ρ t}}

(26)

where

T 41 = | U 12 U 13 U 14 U 22 U 23 U 24 U 32 U 33 U 34 |, T 42 = | U 11 U 13 U 14 U 21 U 23 U 24 U 31 U 33 U 34 |, T 43 = | U 11 U 12 U 14 U 21 U 22 U 24 U 31 U 32 U 34 |, T 44 = - | U 11 U 12 U 13 U 21 U 22 U 23 U 31 U 32 U 33 |

(27)

The equations of motion for FG-coated sandwich cylindrical shells resting on the Pasternak elastic foundation based on CST are obtained as

[\tilde{L} 11 \tilde{L} 12 \tilde{L} 21 \tilde{L} 22] [Φ w] = [00]

(28)

where

\tilde{L} 11 = C 2 h (•), xxxx + 2 h (C 1 - C 5) (•), xxyy + C 2 h (•), yyyy + h (•), xx / R \tilde{L} 12 = - C 3 (•), xxxx - (C 4 + 2 C 6 + C 3) (•), xxyy - C 4 (•), yyyy - K w + K p [(•), xx + (•), yy] - ρ t (•), tt \tilde{L} 21 = B 1 h (•), xxxx + h (2 B 2 + B 5) (•), xxyy + hB 1 (•), yyyy \tilde{L} 22 = - (2 B 3 - B 6) (•), xxyy - B 4 (•), yyyy - B 4 (•), xxxx + (•), xx / R

(29)

Substituting first and second expressions of equation (21) into system of equation (28), after some mathematical operations, for the nondimensional frequencies of FG sandwich cylindrical shells resting on a Pasternak elastic foundation, on the basis of CST, the following expression is obtained

ω_{1 CST}^{wp} = \frac{L 2}{h} \sqrt{} \frac{ρ_{m}^{(2)}}{E_{m}^{(2)} ρ t} {C 3 β_{1}^{4} + 2 (C 4 + C 6) β_{1}^{2} β_{2}^{2} + C 3 β_{2}^{4} + K w + K p (β_{1}^{2} + β_{2}^{2}) + [β_{1}^{2} / R - C 2 β_{1}^{4} - 2 (C 1 - C 5) β_{1}^{2} β_{2}^{2} - C 2 β_{2}^{4}] \times \frac{β_{1}^{2} / R + B 4 β_{1}^{4} + (2 B 3 - B 6) β_{1}^{2} β_{2}^{2} + B 4 β_{2}^{4}}{B 1 β_{1}^{4} + (2 B 2 + B 5) β_{1}^{2} β_{2}^{2} + B 1 β_{2}^{4}}}

(30)

The expressions for nondimensional frequencies of the monolayer (M–M–M) cylindrical shells based on CST and FOSDT can be obtained by letting $V (1) = V (3) = 1$ in equations (24), (26), and (30).

The expressions for nondimensional frequencies of the sandwich homogeneous (C–M–C) cylindrical shells based on CST and FOSDT can be obtained by letting $V (1) = V (3) = 0$ in equations (24), (26), and (30).

The nondimensional fundamental frequencies of sandwich cylindrical shells on the basis of CST and FOSDT are obtained by minimizing equations (24), (26), and (30), respectively, with respect to circumferential wave number (n), because of $m = 1$ .

Numerical analysis

Convergence study

As a check on the numerical accuracy of the theory and formulation, the values of the fundamental frequencies are compared with those in the literature.

Example 1: The nondimensional fundamental frequency

ω_{1 SDT}^{rot} = ω_{SDT}^{rot} (L 2 / 10 h) \sqrt{ρ_{0}^{(k)} / E_{02}^{(k)}}

, for (0/90/0) sandwich homogeneous orthotropic cylindrical shells including shear stresses and rotary inertia is compared with the results of Reddy and Liu [57] and Ferreira et al. [58] and presented in Table 1. The following material properties in the layers are used [18]:

E_{01}^{(k)} / E_{02}^{(k)} = 25;

G_{012}^{(k)} / E_{02}^{(k)} = G_{013}^{(k)} / E_{02}^{(k)} = 0.5;

G_{023}^{(k)} / E_{02}^{(k)} = 0.2; ν_{12}^{(k)} = 0.25

ρ_{0}^{(k)} = 1

(k = 1,2,3). The sandwich cylindrical shell characteristics are taken to be

R / L = 5; 10

and

L / h = 100

Table 1.

Comparison of the nondimensional fundamental frequency for (0/90/0) sandwich cylindrical shell including shear stresses and rotary inertia.

$ω_{1 SDT}^{rot} = ω_{SDT}^{rot} (L 2 / 10 h) \sqrt{ρ_{0}^{(k)} / E_{02}^{(k)}}$ , (n)
R/L	5	10
Reddy and Liu [57]	20.332	16.6325
Ferreira et al. [58]	20.4856	16.8441
Present study	19.556 (20)	16.638(31)

The thickness of the layers is the same. It is observed that our results for various $R / L$ ratios are very convenient with the results of Reddy and Liu [57] and Ferreira et al. [58].

Example 2: The nondimensional fundamental frequency,

ω_{1 SDT}^{rot} = 104 \times ω_{SDT}^{rot} h \sqrt{ρ c / E c}

for pure FGM (Al/Al₂O₃) cylindrical shell by taking into account the effects of shear deformations and rotary inertia is compared with the results of Matsunaga [8] and presented in Table 2. The Young’s modulus and mass density are assumed to be in terms of a simple-power law distribution and Poisson’s ratio is assumed to be constant in Matsunaga [8]. The cylindrical shell characteristics and FG material properties are given as

R / h = 1000

L / R = 2; 5

and (Al):

E m = 7 \times 1010 Pa,

ν m = 0.3

ρ m = 2702 kg / m 3

and (Al₂O₃):

E c = 3.8 \times 1011 Pa,

ν c = 0.3

ρ c = 3800 kg / m 3

. The volume fraction index N presented in Table 2 and the stresses shape functions are

f 1 (ζ) = f 2 (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)

. It is seen that the results obtained in this study are in harmony with the results of Matsunaga [8].

Table 2.

Comparison of nondimensional fundamental frequency for the pure FGM cylindrical shell with those of Matsunaga [8].

$ω_{1 SDT}^{rot} = 104 \times ω_{SDT}^{rot} h \sqrt{ρ c / E c}$ ,(n)
N	0.5	1.0	4.0	10
L/R	Matsunaga [8]
2	0.3368(9)	0.3068(9)	0.2571(9)	0.2351(9)
5	0.1314(6)	0.1201(6)	0.1004(6)	0.09247(6)
	Present study
2	0.3401(10)	0.3101(10)	0.2599(9)	0.2378(9)
5	0.1348(6)	0.1233(6)	0.1032(6)	0.09514(6)

Vibration analysis of FG sandwich cylindrical shells on a Pasternak elastic foundation

In all the following numerical calculations, the shear stresses shape functions distributed parabolic manner across the layers of cylindrical shells, $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$ , see Timarci and Soldatos [55]. In addition, in Tables 5 and 6, three different shear stresses shape functions, such as parabolic, hyperbolic cosine and uniform functions, i.e. $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$ , $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = h \sinh (ζ / h) - ζ \cosh (1 / 2)$ , and $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ$ , respectively, are used.

In numerical calculations, two sets of material mixture for FG coatings are considered. The FG₁ coating is considered to be silicon nitride and stainless steel, referred to as Si₃N₄/SUS304 and FG₂ coating is considered to be zirconium oxide and titanium alloy, referred to as ZrO₂/Ti6Al4V. The following types of sandwich shells are discussed:

FG 1 - M 1 - FG 1 or FG 1 - SUS 304 - FG 1 C 1 - M 1 - C 1 or Si 3 N 4 - SUS 304 - Si 3 N 4 FG 2 - M 2 - FG 2 or FG 2 - Ti 6 Al 4 V - FG 2 C 2 - M 2 - C 2 or ZrO 2 - Ti 6 Al 4 V - ZrO 2

Typical values for effective modulus of elasticity E_f (in Pa) and Poisson’s ratio

ν f

of these materials adopted as Shen [4] and are listed in Table 3. The densities of silicon nitride, stainless steel, zirconium oxide, and titanium alloy are constant and are taken to be 2370, 8166, 5680, and 4420 kg/m³, respectively. In Table 3,

E 0 m, ν 0 m, ρ 0 m

and

E 0 c, ν 0 c, ρ 0 c

are Young’s modulus and Poisson’s ratio of pure metal and pure ceramic materials, respectively.

Table 3.

Mechanical properties of constituent materials for FG coatings [4].

Materials (in Pa)		E ₀	$E - 1$	E ₁	E ₂	E ₃	E_f
C₁	Silicon nitride (Si₃N₄)	348.43 × 10⁹	0	−3.070 × 10^–4	2.160 × 10^–7	−8.946 × 10^–11	3.22271 × 10¹¹
M₁	Stainless steel (SUS304)	201.04 × 10⁹	0	3.079 × 10^–4	−6.534 × 10^–7	0	2.07788 × 10¹¹
C₂	Zirconium-oxide (ZrO₂)	244.27 × 10⁹	0	−1.371 × 10^–3	1.214 × 10^–6	−3.681 × 10^–10	1.68063 × 10¹¹
M₂	Titanium Alloy (Ti6Al4V)	122.56 × 10⁹	0	−4.586 × 10^–4	0	0	1.05698 × 10¹¹
		$ν 0$	$ν - 1$	$ν 1$	$ν 2$	$ν 3$	$ν f$
C₁	Silicon nitride (Si₃Ni₄)	0.2400	0	0	0	0	0.2400
M₁	Stainless steel (SUS304)	0.3262	0	−2.002 × 10^–4	3.797 × 10^–7	0	0.317756
C₂	Zirconium- oxide (ZrO₂)	0.2882	0	1.133 × 10^–4	0	0	0.297996
M₂	Titanium Alloy (Ti6Al4V)	0.2884	0	1.121 × 10^–4	0	0	0.298099

For these examples, the cylindrical shell characteristics are taken to be $R / h = 25$ –45 and $L / R = 0.2$ –0.6. Numerical results for the vibration of monolayer metal-rich (M₁ and M₂) shells, homogenous sandwich (C₁–M₁–C₁ and C₂–M₂–C₂) shells, and FG sandwich (FG₁–M₁–FG₁ and FG₂–M₂–FG₂) shells with and without shear stresses, rotary inertia, and Pasternak elastic foundation are presented in Tables 4 to 6, and Figures 3 and 4. The circumferential wave number in brackets corresponds to the nondimensional fundamental frequency. The present results are compared with those of metal-rich and ceramic-coated cylindrical shells with and without an elastic foundation based on FOSDT as well as CPT by estimating the percentage differences of nondimensional fundamental frequencies, respectively, as

\frac{ω_{1}^{wp} - ω 1}{ω 1} \times 100 %, \frac{ω_{1}^{wp} - ω_{1 met}^{wp}}{ω_{1 met}^{wp}} \times 100 %, \frac{ω_{1 SDT}^{wp} - ω_{1 CST}^{wp}}{ω_{1 CST}^{wp}} \times 100 %,

Table 4.

Variation of the nondimensional fundamental frequencies for sandwich and monolayer cylindrical shells with and without an elastic foundation, based on SDT and CST versus the ratio, h_m/h_FG.

$h m / h FG$	$ω_{1 SDT}^{rot}$ (n)	$ω 1 SDT$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{rot}$ (n)	$ω 1 SDT$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{rot}$ (n)	$ω 1 SDT$ (n)	$ω 1 CST$ (n)
Unconstrained sandwich and monolayer shells
	C₁–M₁–C₁			FG₁–M₁–FG₁			M₁
2	1.923(1)	1.938(1)	2.029(1)	1.587(1)	1.601(1)	1.717(1)	1.291(1)	1.308(1)	1.374(1)
4	1.712(1)	1.726(1)	1.809(1)	1.462(1)	1.475(1)	1.604(1)
6	1.610(1)	1.625(1)	1.704(1)	1.389(1)	1.402(1)	1.548(1)
8	1.549(1)	1.564(1)	1.640(1)	1.333(1)	1.344(1)	1.514(1)
	C₂–M₂–C₂			FG₂–M₂–FG₂			M₂
2	3.048(1)	3.093(1)	3.247(1)	2.858(1)	2.895(1)	3.105(1)	2.671(1)	2.706(1)	2.839(1)
4	3.007(1)	3.050(1)	3.203(1)	2.774(1)	2.813(1)	3.049(1)
6	2.961(1)	3.004(1)	3.154(1)	2.710(1)	2.747(1)	3.011(1)
8	2.924(1)	2.966(1)	3.113(1)	2.651(1)	2.686(1)	2.984(1)
Sandwich and monolayer shells on the Pasternak foundation
	C₁–M₁–C₁			FG₁–M₁–FG₁			M₁
2	2.171(1)	2.187(1)	2.268(1)	1.829(1)	1.845(1)	1.947(1)	1.521(1)	1.541(1)	1.598(1)
4	1.945(1)	1.962(1)	2.035(1)	1.702(1)	1.718(1)	1.829(1)
6	1.839(1)	1.856(1)	1.926(1)	1.631(1)	1.646(1)	1.772(1)
8	1.776(1)	1.794(1)	1.861(1)	1.577(1)	1.591(1)	1.737(1)
	C₂–M₂–C₂			FG₂–M₂–FG₂			M₂
2	3.752(1)	3.807(1)	3.933(1)	3.648(1)	3.695(1)	3.861(1)	3.552(1)	3.599(1)	3.700(1)
4	3.746(1)	3.801(1)	3.924(1)	3.596(1)	3.647(1)	3.832(1)
6	3.725(1)	3.779(1)	3.899(1)	3.556(1)	3.604(1)	3.809(1)
8	3.705(1)	3.758(1)	3.875(1)	3.517(1)	3.563(1)	3.792(1)

Table 5.

Variation of the nondimensional fundamental frequencies for sandwich and monolayer cylindrical shells with and without a Pasternak elastic foundation, based on SDT and CST versus the ratio, L/R.

	C₁–M₁–C₁		FG₁–M₁–FG₁		M₁
Unconstrained sandwich and monolayer shells
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$ .
$L / R$	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)
0.2	1.712(1)	1.809(1)	1.462(1)	1.604(1)	1.291(1)	1.374(1)
0.4	2.696(3)	2.725(3)	2.393(3)	2.436(3)	2.095(3)	2.121(3)
0.6	4.076(4)	4.100(4)	3.635(4)	3.670(4)	3.179(5)	3.206(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = h \sinh (ζ / h) - ζ \cosh (1 / 2)$
0.2	1.712(1)	1.809(1)	1.462(1)	1.604(1)	1.291(1)	1.374(1)
0.4	2.696(3)	2.725(3)	2.393(3)	2.436(3)	2.095(3)	2.121(3)
0.6	4.076(4)	4.100(4)	3.635(4)	3.670(4)	3.179(5)	3.206(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ$
0.2	1.724(1)	1.809(1)	1.485(1)	1.604(1)	1.301(1)	1.374(1)
0.4	2.700(3)	2.725(3)	2.400(3)	2.436(3)	2.098(3)	2.121(3)
0.6	4.078(4)	4.100(4)	3.640(4)	3.670(4)	3.183(5)	3.206(4)
Sandwich and monolayer shells on the Pasternak foundation
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$
0.2	1.818(1)	1.911(1)	1.572(1)	1.707(1)	1.397(1)	1.476(1)
0.4	3.201(2)	3.224(2)	2.898(2)	2.930(2)	2.584(3)	2.607(2)
0.6	5.429(4)	5.452(4)	4.965(4)	4.996(4)	4.466(4)	4.488(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = h \sinh (ζ / h) - ζ \cosh (1 / 2)$
0.2	1.818(1)	1.911(1)	1.570(1)	1.707(1)	1.397(1)	1.476(1)
0.4	3.201(2)	3.224(2)	2.898(2)	2.930(2)	2.584(3)	2.607(2)
0.6	5.429(4)	5.452(4)	4.965(4)	4.996(4)	4.466(4)	4.488(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ$
0.2	1.830(1)	1.911(1)	1.593(1)	1.707(1)	1.405(1)	1.476(1)
0.4	3.203(2)	3.224(2)	2.903(2)	2.930(2)	2.586(3)	2.609(2)
0.6	5.431(4)	5.452(4)	4.969(4)	4.996(4)	4.467(4)	4.488(4)

Table 6.

Variation of the nondimensional fundamental frequencies for sandwich and monolayer cylindrical shells with and without a Pasternak elastic foundation, based on SDT and CST versus the ratio, L/R.

	C₂–M₂–C₂		FG₂–M₂–FG₂		M₂
Unconstrained sandwich and monolayer shells
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$
$L / R$	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)	$ω_{1 SDT}^{Rot}$ (n)	$ω 1 CST$ (n)
0.2	1.712(1)	1.809(1)	1.462(1)	1.604(1)	1.291(1)	1.374(1)
0.4	2.696(3)	2.725(3)	2.393(3)	2.436(3)	2.095(3)	2.121(3)
0.6	4.076(4)	4.100(4)	3.635(4)	3.670(4)	3.179(5)	3.206(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = h \sinh (ζ / h) - ζ \cosh (1 / 2)$
0.2	1.712(1)	1.809(1)	1.462(1)	1.604(1)	1.291(1)	1.374(1)
0.4	2.696(3)	2.725(3)	2.393(3)	2.436(3)	2.095(3)	2.121(3)
0.6	4.076(4)	4.100(4)	3.635(4)	3.670(4)	3.179(5)	3.206(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ$
0.2	1.724(1)	1.809(1)	1.485(1)	1.604(1)	1.301(1)	1.374(1)
0.4	2.700(3)	2.725(3)	2.400(3)	2.436(3)	2.098(3)	2.121(3)
0.6	4.078(4)	4.100(4)	3.640(4)	3.670(4)	3.183(5)	3.206(4)
Sandwich and monolayer shells on the Pasternak foundation
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$
0.2	1.818(1)	1.911(1)	1.572(1)	1.707(1)	1.397(1)	1.476(1)
0.4	3.201(2)	3.224(2)	2.898(2)	2.930(2)	2.584(3)	2.607(2)
0.6	5.429(4)	5.452(4)	4.965(4)	4.996(4)	4.466(4)	4.488(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = h \sinh (ζ / h) - ζ \cosh (1 / 2)$
0.2	1.818(1)	1.911(1)	1.570(1)	1.707(1)	1.397(1)	1.476(1)
0.4	3.201(2)	3.224(2)	2.898(2)	2.930(2)	2.584(3)	2.607(2)
0.6	5.429(4)	5.452(4)	4.965(4)	4.996(4)	4.466(4)	4.488(4)
	$f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ$
0.2	1.830(1)	1.911(1)	1.593(1)	1.707(1)	1.405(1)	1.476(1)
0.4	3.203(2)	3.224(2)	2.903(2)	2.930(2)	2.586(3)	2.609(2)
0.6	5.431(4)	5.452(4)	4.969(4)	4.996(4)	4.467(4)	4.488(4)

The variation of the nondimensional fundamental frequencies for ceramic-coated (C₁–M₁–C₁ and C₂–M₂–C₂) sandwich, FG-coated (FG₁–M₁–FG₁ and FG₂–M₂–FG₂) sandwich, and monolayer (M₁ and M₂) cylindrical shells considering effects of shear stresses and rotary inertia, and with and without a Pasternak elastic foundation versus $h m / h FG$ are presented in Table 4. The shell characteristics are $R / h = 25$ and $L / R = 0.2$ , the Pasternak foundation stiffness are taken to be $K w = 1011 N / m 3$ and $K p = 8 \times 106 N / m$ , and the shear stresses shape function is $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$ . The nondimensional fundamental frequencies for FG and ceramic-rich coated sandwich and monolayer metal-rich cylindrical shells are increasing with the existence of the elastic foundation. The nondimensional fundamental frequencies for FG₁–M₁–FG₁ and C₁–M₁–C₁ sandwich cylindrical shells with and without a Pasternak elastic foundation decrease with increasing the ratio, $h m / h FG$ . When comparing the values of $ω_{1 SDT}^{Rot}$ for FG₁–M₁–FG₁ sandwich cylindrical shells on the Pasternak elastic foundation with those of C₁–M₁–C₁ sandwich and M₁ metal-rich cylindrical shells on the Pasternak elastic foundation, respectively, the influences of compositional profile of FG₁ coatings on the values of $ω_{1 SDT}^{Rot}$ decrease from 15.75 to 11.2% and 20.25 to 3.68%, respectively, while these influences based on the CST decrease from 14.15 to 6.66% and 21.84 to 8.7%, respectively, as $h m / h FG$ increases from 2 to 8. When comparing the values of $ω_{1 SDT}^{Rot}$ for FG₂–M₂–FG₂ sandwich cylindrical shells on the Pasternak elastic foundation with those of C₂–M₂–C₂ sandwich and M₂ metal-rich cylindrical shells resting on the Pasternak elastic foundation, respectively, the influences of compositional profile of FG₂ coatings on the values of $ω_{1 SDT}^{Rot}$ increase from 2.77 to 5.07% and decrease from 2.7 to (–0.99)%, respectively, while these influences based on the CST increase from 1.83 to 2.14% and decrease from 4.35 to 2.49%, respectively, as $h m / h FG$ increases from 2 to 8. The combined influences of shear stresses and rotary inertia on the nondimensional fundamental frequencies for FG₁–M₁–FG₁ and C₁–M₁–C₁ sandwich cylindrical shells on the Pasternak elastic foundation increase from 6.06 to 9.21% and 4.28 to 4.57%, respectively, whereas the influence of shear stresses on the nondimensional fundamental frequencies for FG₁–M₁–FG₁ sandwich cylindrical shells on the Pasternak elastic foundation increases from 5.54 to 8.41% and for C₁–M₁–C₁ sandwich cylindrical shells remains constant (3.60%), respectively, as $h m / h FG$ increases from 2 to 8. The combined influences of shear stresses and rotary inertia on the nondimensional fundamental frequencies for FG₂–M₂–FG₂ and C₂–M₂–C₂ sandwich cylindrical shells on the Pasternak elastic foundation increase from 5.52 to 7.25% and decrease from 4.60 to 4.39%, respectively, whereas the influence of shear stresses on the nondimensional fundamental frequencies for FG₂–M₂–FG₂ sandwich cylindrical shells on the Pasternak elastic foundation increase from 4.30 to 6.04% and for C₂–M₂–C₂ sandwich cylindrical shells remains constant (3.10%), respectively, as $h m / h FG$ increases from 2 to 8. When comparing the values of $ω_{1 SDT}^{Rot}$ for FG₁–M₁–FG₁ and C₁–M₁–C₁ sandwich cylindrical shells on the Pasternak elastic foundation with those of unconstrained cylindrical shells, the influence of a Pasternak elastic foundation on the values of $ω_{1 SDT}^{Rot}$ increases from 15.25 to 18.30% and 12.90 to 14.65%, respectively, as $h m / h FG$ increases from 2 to 8. When comparing the values of $ω_{1 SDT}^{Rot}$ for FG₂–M₂–FG₂ and C₂–M₂–C₂ sandwich cylindrical shells on the Pasternak elastic foundation with those of unconstrained cylindrical shells, the influence of elastic foundation on the values of $ω_{1 SDT}^{Rot}$ increases from 27.64 to 32.67% and 23.10 to 26.71%, respectively, as $h m / h FG$ increases from 2 to 8. The combined effect shear stresses and rotary inertia on the nondimensional fundamental frequencies is higher than the separately effect of shear stresses.

Tables 5 and 6 exhibit the combined and separate effects of shear stresses, rotary inertia, and Pasternak elastic foundation on the nondimensional fundamental frequencies and corresponding circumferential wave numbers for sandwich (FG₁–M₁–FG₁, FG₂–M₂–FG₂, C₁–M₁–C₁, and C₂–M₂–C₂) and monolayer (M₁ and M₂) cylindrical shells versus the aspect ratio, $L / R$ for $R / h = 25$ , $h m / h FG = 4$ , $K w = 2 \times 1010 N / m 3$ and $K p = 7 \times 106 N / m$ . Also, the results predicted by various shear stresses shape functions, such as parabolic, hyperbolic cosine and uniform functions, approach each other as the ratio $L / R$ increases, as shown in Tables 5 and 6. The nondimensional fundamental frequencies and corresponding wave numbers of sandwich cylindrical shells with and without a Pasternak elastic foundation increase monotonically, as $L / R$ increases. The effect of a Pasternak elastic foundation on the nondimensional fundamental frequencies for sandwich cylindrical shells increases, as $L / R$ increases. However, the effect of Pasternak elastic foundation on the nondimensional fundamental frequencies for FG₁–M₁–FG₁ and FG₂–M₂–FG₂ cylindrical shells is higher than the C₁–M₁–C₁ and C₂–M₂–C₂ cylindrical shells, respectively. In addition, the effect of the Pasternak elastic foundation on the nondimensional fundamental frequencies for FG₂–M₂–FG₂ cylindrical shells is higher than the FG₁–M₁–FG₁ cylindrical shells. For example, the effect of Pasternak elastic foundation on the nondimensional fundamental frequencies for FG₁–M₁–FG₁ and FG₂–M₂–FG₂ cylindrical shells increases from 7.52 to 36.59%, 7.39 to 36.59%, 7.27 to 36.51% and from 14.13 to 63.81%, 13.88 to 63.76%, 13.55 to 63.63%, whereas for C₁–M₁–C₁ and C₂–M₂–C₂ sandwich cylindrical shells increases from 6.19 to 33.19%, 6.19 to 33.19%, 6.15 to 33.18% and from 11.44 to 57.69%, 11.44 to 57.69%, 11.26 to 57.63% for parabolic, hyperbolic cosine, uniform functions, respectively, as $L / R$ increases from 0.2 to 0.6. The nondimensional fundamental frequencies for all above-mentioned cylindrical shells with and without an elastic foundation are approximately same for parabolic and hyperbolic cosine manners, due to the similarity of distributions of these shape functions in the thickness direction of the FG-coated sandwich shell, while these values are slightly different for uniformly manner of the shape function for short shells with and without elastic foundation.

From Tables 5 and 6, it can be also seen that the difference between the nondimensional fundamental frequencies for the sandwich cylindrical shells with and without elastic foundation obtained by using the parabolic or hyperbolic cosine functions and those obtained by the uniform shape function decreases, as $L / R$ increases. The effect of shear stresses on the nondimensional fundamental frequencies decreases with the increasing of the ratio, R/h. The combined influences of shear stresses and rotary inertia on the nondimensional fundamental frequencies for FG₁–M₁–FG₁ and FG₂–M₂–FG₂ cylindrical shells with and without an elastic foundation decrease, as $L / R$ increases from 0.2 to 0.6.

Figure 3 shows the combined and separate effects of shear stresses and rotary inertia, and Pasternak elastic foundation on the $ω_{1 SDT}^{Rot}$ for FG₁–M₁–FG₁, FG₂–M₂–FG₂, C₁–M₁–C₁, and C₂–M₂–C₂ sandwich cylindrical shells versus $L / R$ for the hyperbolic cosine shape function. The sandwich shell characteristics and Pasternak elastic foundation coefficients are $R / h = 25$ , $h m / h FG = 4$ , $K w = 2 \times 1010 N / m 3$ , and $K p = 7 \times 106 N / m$ . In the preparation of Figure 3, we used the values given in Tables 5 and 6. It is observed that the nondimensional fundamental frequencies of sandwich cylindrical shells with and without a Pasternak elastic foundation increase monotonically, as $L / R$ increases from 0.2 to 0.6. The combined effect of shear stresses and rotary inertia on the $ω_{1 SDT}^{Rot}$ for sandwich cylindrical shells with and without a Pasternak elastic foundation decreases, as $L / R$ increases. The effect of FG coatings on the $ω_{1 SDT}^{Rot}$ for sandwich cylindrical shells also decreases considering the elastic foundation.

Figure 4(a) and (b) shows the variation of nondimensional fundamental frequencies for sandwich (FG₁–M₁–FG₁, FG₂–M₂–FG₂, C₁–M₁–C₁, C₂–M₂–C₂) cylindrical shells with and without a Pasternak elastic foundation, on the basis of FOSDT and CST, against the ratio, $R / h$ . Data for the calculations are taken to be $L / R = 0.2$ , $h m / h FG = 4$ , $K w = 8 \times 1010 N / m 3$ , and $K p = 7.5 \times 106 N / m$ and the shear stresses shape function is $f_{1}^{(k)} (ζ) = f_{2}^{(k)} (ζ) = ζ (1 - 4 ζ 2 / 3 h 2)$ . The nondimensional fundamental frequencies for FG and ceramic-rich-coated sandwich cylindrical shells with and without the Pasternak elastic foundation increase monotonically, with increasing of the ratio, $R / h$ . It is clear that the nondimensional fundamental frequency of the FG₁–M₁–FG₁ sandwich cylindrical shell is lower than the FG₂–M₂–FG₂ sandwich cylindrical shell based on FOSDT and CST. When comparing the values of $ω_{1 SDT}^{Rot}$ for FG₁–M₁–FG₁ and FG₂–M₂–FG₂ sandwich cylindrical shells on the Pasternak elastic foundation with those of C₁–M₁–C₁ and C₂–M₂–C₂ sandwich cylindrical shells on the Pasternak elastic foundation, respectively, the influences of FG₁ and FG₂ coatings on the values of $ω_{1 SDT}^{Rot}$ decrease from 12.76 to 8.62% and 4.39 to 0.03%, respectively, while these influences based on the CST decrease from 10.24 to 8.20% and from 2.62 to 0.21%, respectively, as $R / h$ increases from 25 to 45. From the data presented in these figures, it can be seen that the differences between the nondimensional frequencies of sandwich cylindrical sells with and without an elastic foundation obtained by using the FOSDT and those obtained by the CST decrease, as the ratio, $R / h$ increases. The influence of a Pasternak elastic foundation on the values of the $ω_{1 SDT}^{Rot}$ for FG₁–M₁–FG₁ and C₁–M₁–C₁ cylindrical shells increases from 14.09 to 51.37% and 11.68 to 45.77%, whereas this influence on the $ω_{1 SDT}^{Rot}$ for FG₂–M₂–FG₂ and C₂–M₂–C₂ cylindrical shells increases from 25.59 to 86.87% and 21.18 to 77.19, as $R / h$ increases from 25 to 45.

Conclusion

In this study, the vibration behaviors of FG sandwich cylindrical shell resting on the Pasternak elastic foundation considering effects of shear stresses and rotary inertia are examined. The effective material properties of FG coatings are assumed to be graded in the thickness direction according to an exponential law distribution. The modified Donnell-type equations of motion of FG and homogeneous sandwich cylindrical shells on the Pasternak elastic foundation are deduced using the FOSDT. The basic equations are reduced to an algebraic equation of the sixth order and solved with computer programming using Maple 14, obtained six roots that three positive roots are the natural frequencies. The smallest positive root is applied in the present study of the fundamental frequency of FG-coated sandwich cylindrical shells including the shear stresses and rotary inertia, and resting on the Pasternak elastic foundation. In addition, the expression for the nondimensional fundamental frequency of FG-coated sandwich cylindrical shells with and without taking into account influences of an elastic foundation and shear deformation is obtained. Calculations, the influences of an elastic foundation, shear stresses, rotary inertia, and sandwich shell geometry parameters on the nondimensional fundamental frequencies are described. The results are verified by comparing the obtained values with those in the existing literature.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Appendix 1

Appendix 2

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