Analysis of vibration characteristics of auxetic sandwich cylindrical shells resting on elastic foundation

Abstract

The main objective of this research work is focused on the vibration analysis of auxetic sandwich cylindrical shell structures resting on an elastic foundation. In the analysis, the sandwich shell structure is composed of three layers in which the middle layer consists of auxetic material with a negative Poisson’s ratio, and the two skin layers are isotropic homogeneous materials. The motion equation is extracted according to the first-order shear deformation theory (FSDT) and the Hamilton principle. The governing equations of coupled partial differential equations are solved by the generalized differential quadrature (GDQ) method, and the natural frequencies are determined. By comparing the experimental results with the numerical results calculated by commercial finite element software, the validity of the proposed theoretical model is verified. Finally, the influences of geometrical parameters and elastic foundation on the vibration behavior of sandwich shell structures have been investigated.

Keywords

Sandwich cylindrical shells Auxetic structures Frequency analysis first-order shear deformation theory Theoretical prediction Elastic foundation

Introduction

Sandwich structures have attracted much attention due to their outstanding properties of high specific stiffness with low weight, high structural efficiency, good wear resistance, and high energy absorption capability under impact load.^1–3 Based on this background, there have been intense research activities in design and analysis of the cores of sandwich structures in recent years. The core type can be any material or structure, but the three most commonly used types are stiffened core, honeycomb core, and corrugated core. Yu and Cleghorn⁴ investigated the free vibration characteristics of a simple supported rectangular symmetric honeycomb panel. Zhao et al.⁵ adopted the lateral compression tests and finite element method to solve the critical buckling performance of aluminum honeycomb panel structural system for long-span hollow-core roofs. In addition, Sofiyev et al.^6–10 analyzed the buckling, vibration, and dynamic instability of the functionally graded sandwich cylindrical shells. Li et al.¹¹ investigated the free flexural vibration characteristics of thin-walled honeycomb sandwich cylindrical shells through experiments and theoretical simulations.

It is worth noting that the studies above are excellent but mainly focus on the traditional sandwich core structures characterized by a positive Poisson’s ratio. Auxetic cell structures are amodern material structures with unique and superior mechanical properties. In general, the auxetic structure shapes can be classified into the following main groups: re-entrant/concave hexagonal shape, chiral and anti-chiral structure, quadrilateral, star-shaped, inverted tetrakaidecahedron configurations, rotating rigid body structure. Zhong et al.¹² proposed a new construction strategy of 3D dislocation chiral metamaterials, which can realize negative Poisson’s ratio effect in three directions. Meena and Singamneni¹³ designed a new auxetic structure composed of star re-entrant unit cells and S-shaped unit cells. Yang et al.¹⁴ established an analytical model of a 3D re-entrant honeycomb auxetic structure based on a large deflection beam model and a Timoshenko beam model. Recently, a brief review of works regarding design, manufacturing and applications of auxetic tubular structures can be found in.¹⁵

In recent years, cylindrical shells surrounded by the elastic medium began to be used in various engineering structures. Hence, different types of foundation models were adopted by the scientist for the engineering application. Several investigations on the vibration and buckling of the plate and shell resting on the Winkler foundation were discussed in the literature.^16–19 Another foundation is Winkler-Pasternak foundation which is derived by extension of the Winkler’s model. This foundation model is also called the two-parameter elastic foundation. The influences of two foundation parameters on the vibration and buckling of the cylindrical shell were discussed in the literature,^20–24 which can be used as a reliable source for the derivation of the basic equations, as a comparison of the results.

According to a comprehensive survey of the above literature, it is found that various researches have been carried out on the vibrational analysis of the isotropic sandwich cylindrical shell. While, very few papers are available on the sandwich cylindrical shell with auxetic structural layer resting on an elastic foundation. Motivated by this fact, it is imperative to bridge the gap in the literature. In this paper, the vibration characteristics of auxetic sandwich cylindrical shell structures resting on an elastic foundation are investigated. Through numerical results, we try to further comprehend the underlying mechanism of vibration characteristics of sandwich cylindrical shell resting on elastic foundation with auxetic structural layer.

Theoretical formulation

Geometrical modeling of sandwich cylindrical shell and auxetic core layer

In this study, we consider that the sandwich cylindrical shell has three layers, consisting of two skin layers and one core layer, wherein the core layer is made of auxetic material with a negative Poisson’s ratio, as shown in Figure 1. As illustrated in Figure 2, the sandwich cylindrical shell structure has the total thickness $h$ and length $L$ , the upper and lower skin layer thicknesses are assumed to have an equal thickness $h_{f}$ , the height of the core is $h_{c}$ , in particular, the auxetic unit cell with a re-entrant shape is considered as the most popular structure exhibiting negative effective Poisson’s ratio, the equivalent elasticity material properties of auxetic core layer are determined by using the following formulas^25–26

\begin{array}{l} E_{1}^{c} = E \frac{η_{3}^{3} (η_{1} - \sin φ)}{\cos ϕ [1 + (\tan^{2} ϕ + η_{1} \sec^{2} ϕ) η_{3}^{2}]}, E_{2}^{c} = E \frac{η_{3}^{3}}{\cos φ (η_{1} - \sin ϕ) (\tan^{2} ϕ + η_{3}^{2})} \\ G_{12}^{c} = G \frac{η_{3}^{2}}{η_{1} (1 + 2 η_{1}) \cos ϕ}, G_{23}^{c} = G \frac{η_{3} \cos ϕ}{η_{1} - \sin ϕ} \\ G_{13}^{c} = G \frac{η_{3}}{2 \cos ϕ} [\frac{η_{1} - \sin ϕ}{1 + 2 η_{1}} + \frac{η_{1} + 2 \sin^{2} ϕ}{2 (η_{1} - \sin ϕ)}], μ_{12}^{c} = - \frac{\sin ϕ (1 - η_{3}^{2}) (η_{1} - \sin ϕ)}{\cos^{2} ϕ [1 + (\tan^{2} ϕ + \sec^{2} ϕ η_{1}) η_{3}^{2}]} \\ μ_{21}^{c} = - \frac{\sin ϕ (1 - η_{3}^{2})}{(\tan^{2} ϕ + η_{3}^{2}) (η_{1} - \sin ϕ)}, ρ^{c} = ρ \frac{η_{3} (η_{1} + 2)}{2 \cos ϕ (η_{1} - \sin ϕ)} \end{array}

(1)

where superscript “

c

” represents auxetic core layer,

E_{1}^{c}

E_{2}^{c}

G_{12}^{c}

G_{23}^{c}

G_{13}^{c}

μ_{12}^{c}

and

μ_{21}^{c}

denote the auxetic core layer elastic modulus, shear modulus and Poisson’s ratio, respectively, the subscripts “1”, “2” and “3” represent the

x

θ

and

z

directions,

t

is the cell wall thickness,

l

is the sidewall length,

d

is the length of the vertical cell rib,

ϕ

is the inclined angle, wherein

η_{1} = d / l

η_{3} = t / l

, as shown in Figure 2.

The governing equations of sandwich cylindrical shell

According to the first-order shear deformation theory (FSDT), the strain fields of sandwich cylindrical shells are written as follows²⁷

{\begin{matrix} ε_{x} \\ ε_{θ} \\ γ_{x θ} \end{matrix}} = {\begin{matrix} \frac{\partial u}{\partial x} + z \frac{\partial ϕ_{x}}{\partial x} \\ \frac{w}{R} + \frac{\partial v}{R \partial θ} + z \frac{\partial ϕ_{θ}}{R \partial θ} \\ \frac{\partial u}{R \partial θ} + \frac{\partial v}{\partial x} + z \frac{\partial φ_{x}}{R \partial θ} + z \frac{\partial ϕ_{θ}}{\partial x} \end{matrix}}, {\begin{matrix} γ_{θ z} \\ γ_{x z} \end{matrix}} = {\begin{matrix} ϕ_{θ} - \frac{v}{R} + \frac{\partial w}{R \partial θ} \\ ϕ_{x} + \frac{\partial w}{\partial x} \end{matrix}}

(2)

The constitutive equations of the layer $k$ of the sandwich cylindrical shell can be written according to the Hooke’s law as follows

{\begin{matrix} σ_{x}^{k} \\ σ_{θ}^{k} \\ τ_{x θ}^{k} \end{matrix}} = [\begin{matrix} Q_{11}^{k} & Q_{12}^{k} & 0 \\ Q_{12}^{k} & Q_{22}^{k} & 0 \\ 0 & 0 & Q_{66}^{k} \end{matrix}] {\begin{matrix} ε_{x} \\ ε_{θ} \\ γ_{x θ} \end{matrix}}, {\begin{matrix} τ_{θ z}^{k} \\ τ_{x z}^{k} \end{matrix}} = [\begin{matrix} Q_{44}^{k} & 0 \\ 0 & Q_{55}^{k} \end{matrix}] {\begin{matrix} γ_{θ z} \\ γ_{x z} \end{matrix}}

(3)

where “

T

“, “

c

” and “

B

” represent the top, auxetic core, and bottom layers, and

\begin{array}{l} Q_{11}^{c} = \frac{E_{1}^{c}}{1 - μ_{12}^{c} μ_{21}^{c}}, Q_{12}^{c} = \frac{μ_{12}^{c} E_{2}^{c}}{1 - μ_{12}^{c} μ_{21}^{c}}, Q_{22}^{c} = \frac{E_{2}^{c}}{1 - μ_{12}^{c} μ_{21}^{c}}, Q_{66}^{c} = G_{12}^{c}, Q_{44}^{c} = G_{23}^{c}, Q_{55}^{c} = G_{13}^{c} \\ Q_{11}^{T} = Q_{22}^{T} = \frac{E}{1 - μ^{2}}, Q_{12}^{T} = \frac{μ E}{1 - μ^{2}}, Q_{66}^{T} = Q_{44}^{T} = Q_{55}^{T} = \frac{E}{2 (1 + μ)} \\ Q_{11}^{B} = Q_{22}^{B} = \frac{E}{1 - μ^{2}}, Q_{12}^{B} = \frac{μ E}{1 - μ^{2}}, Q_{66}^{B} = Q_{44}^{B} = Q_{55}^{B} = \frac{E}{2 (1 + μ)} \end{array}

(4)

where

E

and

μ

are Young’s modulus and Poisson’s ratios of the upper and lower skin layers. By integrating the constitutive equations over the thickness direction, the force resultants (

N_{x}, N_{θ}, N_{x θ}

), shear force resultants (

Q_{θ z}, Q_{x z}

), and moment resultants (

M_{x}, M_{θ}, M_{x θ}

) are obtained as

{\begin{matrix} N_{x} \\ N_{θ} \\ N_{x θ} \\ M_{x} \\ M_{θ} \\ M_{x θ} \\ Q_{θ z} \\ Q_{x z} \end{matrix}} = [\begin{matrix} A_{11} & A_{12} & 0 & B_{11} & B_{12} & 0 & 0 & 0 \\ A_{12} & A_{22} & 0 & B_{12} & B_{22} & 0 & 0 & 0 \\ 0 & 0 & A_{66} & 0 & 0 & B_{66} & 0 & 0 \\ B_{11} & B_{12} & 0 & D_{11} & D_{12} & 0 & 0 & 0 \\ B_{12} & B_{22} & 0 & D_{12} & D_{22} & 0 & 0 & 0 \\ 0 & 0 & B_{66} & 0 & 0 & D_{66} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & κ A_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & κ A_{55} \end{matrix}] {\begin{matrix} ε_{x} \\ ε_{θ} \\ γ_{x θ} \\ κ_{x} \\ κ_{θ} \\ κ_{x θ} \\ γ_{θ z} \\ γ_{x z} \end{matrix}}

(5)

where

κ

is the shear correction factor and it was selected as

5 / 6

κ_{x}

κ_{θ}

and

κ_{x θ}

are the curvature and twist changes, see²⁷ for more details. The coefficients

A_{i j}

B_{i j}

and

D_{i j}

are expressed as

(A_{i j}, B_{i j}, D_{i j}) = \int_{- h / 2}^{- h_{c} / 2} Q_{i j}^{B} (1, z, z^{2}) d z + \int_{- h_{c} / 2}^{h_{c} / 2} Q_{i j}^{c} (1, z, z^{2}) d z + \int_{h_{c} / 2}^{h / 2} Q_{i j}^{T} (1, z, z^{2}) d z

(6)

In addition, the strain energy $U_{s}$ of sandwich cylindrical shell is obtained as

U_{s} = \frac{1}{2} \int_{- h / 2}^{h / 2} \int_{0}^{L} \int_{0}^{2 π} (N_{x} ε_{x} + N_{θ} ε_{θ} + N_{x θ} ε_{x θ} + M_{x} κ_{x} + M_{θ} κ_{θ} + M_{x θ} κ_{x θ} + Q_{x z} γ_{x z} + Q_{θ z} γ_{θ z}) R d θ d x d z

(7)

The kinetic energy $T_{s}$ of the sandwich cylindrical shell is

T_{s} = \frac{1}{2} \int_{- h / 2}^{h / 2} \int_{0}^{L} \int_{0}^{2 π} [{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}] ρ (z) R d θ d x d z

(8)

Also, the cylindrical shell is considered to be supported by the Winkler-Pasternak elastic foundations, wherein the Winkler foundation model is called the one parameter foundation, which is designed to describe the mechanical behavior of elastic supports. Interaction between lateral springs is ignored in this model. The Winkler-Pasternak foundation model added the additional shear layer above the spring and thus, the model is characterized by two independent elastic constants, which are derived by extension of the Winkler’s model. The strain energy $U_{F}$ of the Winkler-Pasternak elastic foundation can be expressed as

U_{F} = \frac{1}{2} K_{w} \int_{0}^{L} \int_{0}^{2 π} w^{2} R d θ d x + \frac{1}{2} K_{s} \int_{0}^{L} \int_{0}^{2 π} [{(\frac{\partial w}{\partial x})}^{2} + {(\frac{\partial w}{R \partial θ})}^{2}] R d θ d x

(9)

where

K_{w}

denotes the radial stiffness of the spring layer and the

K_{s}

denotes the transverse stiffness of the shear layer showing the effect of the shear interactions of the vertical elements (Figure 1). Considering that the cylindrical shell uses cylindrical coordinates, the value range of

θ

is set from

0

2 π

Finally, based on the Hamilton’s principle, one gets

\int_{0}^{t} (δ U_{s} + δ U_{F} - δ T_{s} - δ T_{q}) d t = 0

(10)

where

T_{q}

is the virtual work of the external loads which is absent for the free vibration problem. According to equations (7–10), the equations of motion for the sandwich cylindrical shell are obtained as follows

\begin{array}{l} δ u : N_{x, x} + R^{- 1} N_{θ x, θ} = I_{1} \ddot{u} + I_{2} {\ddot{ϕ}}_{x} \\ δ v : N_{x θ, x} + R^{- 1} (N_{θ, θ} + Q_{θ z}) = I_{1} \ddot{v} + I_{2} {\ddot{ϕ}}_{θ} \\ δ w : Q_{x z, x} + R^{- 1} (Q_{θ z, θ} - N_{θ}) + R^{- 1} {(R N_{x} w_{, x} + N_{x θ} w_{, θ})}_{, x} \\ + R^{- 1} {(R^{- 1} N_{θ} w_{, θ} + N_{x θ} w_{, x})}_{, θ} - K_{w} w + K_{s} \nabla^{2} w = I_{1} \ddot{w} \\ δ ϕ_{x} : M_{x, x} + R^{- 1} M_{θ x, θ} - Q_{x} = I_{2} \ddot{u} + I_{3} {\ddot{ϕ}}_{x} \\ δ ϕ_{θ} : M_{x θ, x} + R^{- 1} M_{θ, θ} - Q_{θ} = I_{2} \ddot{v} + I_{3} {\ddot{ϕ}}_{θ} \end{array}

(11)

in which two dots on the top of the variable represent the second derivative with respect to time,

(I_{1}, I_{2}, I_{3})

denote the mass moment of inertia, and

(I_{1}, I_{2}, I_{3}) = \int_{- h / 2}^{- h_{c} / 2} ρ^{f} (1, z, z^{2}) d z + \int_{- h_{c} / 2}^{h_{c} / 2} ρ^{c} (1, z, z^{2}) d z + \int_{h_{c} / 2}^{h / 2} ρ^{f} (1, z, z^{2}) d z

(12)

in which

ρ_{f}

stands for the density of the upper and lower skin layers and

ρ_{c}

represents the density of the auxetic core layer. The boundary conditions of the sandwich cylindrical shell are as follows

(1) S i m p l y \sup p o r t e d e d g e (S) : N_{x} = v = w = M_{x} = ϕ_{θ} = 0 a t x = 0 o r L

(13)

(2) S l i d i n g \sup p o r t e d e d g e (R) : u = N_{x θ} = Q_{x} = ϕ_{x} = M_{x θ} = 0 a t x = 0 o r L

(14)

(3) C l a m p e d e d g e (C) : u = v = w = ϕ_{x} = ϕ_{θ} = 0 a t x = 0 o r L

(15)

(4) F r e e e d g e (F) : N_{x} = N_{x θ} = Q_{x} = M_{x} = M_{x θ} = 0 a t x = 0 o r L

(16)

Solving the equation of motion

At this stage, the solution procedure is presented based on the GDQ method. The rth-order derivative of the function $f (x)$ at any sample grid point i can be approximated by the following formulation²⁸

f_{i}^{(r)} = \sum_{j = 1}^{N} E_{i j}^{(r)} f_{j}

(17)

where

N

is the total number of grid points in the

x

-directions,

E_{i j}^{(r)}

is the weighting coefficient of the rth-order derivative,

f_{i}^{(r)}

indicates the rth-order derivative of the function

f (x)

at a sample grid point

i

. Also, the weighting coefficients related to the rth-order derivative can be obtained by the following equation²⁸

E_{i k}^{(r)} = {\begin{matrix} r [W_{i i}^{(r - 1)} W_{i k}^{(1)} - \frac{W_{i k}^{(r - 1)}}{(x_{i} - x_{k})}] i \neq k, i, k = 1,2, \dots N \\ - \sum_{j = 1, j \neq i}^{N} W_{i j}^{(r)} i = k, i = 1,2, \dots N \end{matrix}

(18)

The displacement of the cylindrical shell can be defined as harmonic forms by the following equations

{\begin{matrix} u (x, θ, t) \\ v (x, θ, t) \\ φ_{x} (x, θ, t) \\ φ_{θ} (x, θ, t) \\ w (x, θ, t) \end{matrix}} = [\begin{matrix} \sin (n θ) & 0 & 0 & 0 & 0 \\ 0 & \cos (n θ) & 0 & 0 & 0 \\ 0 & 0 & \sin (n θ) & 0 & 0 \\ 0 & 0 & 0 & \cos (n θ) & 0 \\ 0 & 0 & 0 & 0 & \sin (n θ) \end{matrix}] {\begin{matrix} U (x) \cos (ω t) \\ V (x) \cos (ω t) \\ ϕ_{x} (x) \cos (ω t) \\ ϕ_{θ} (x) \cos (ω t) \\ W (x) \cos (ω t) \end{matrix}}

(19)

where in the above equation, the

n

indicates the wave number through the circumferential direction.

Implementing the given boundary conditions and substituting for the displacement components from equation (19), and applying the differential quadrature procedure, the equations of motion can be expressed in the following discrete form

[\begin{matrix} K_{bb} & K_{bd} \\ K_{db} & K_{dd} \end{matrix}] [\begin{matrix} δ_{b} \\ δ_{d} \end{matrix}] = ω^{2} [\begin{matrix} 0 & 0 \\ 0 & M_{dd} \end{matrix}] [\begin{matrix} δ_{b} \\ δ_{d} \end{matrix}]

(20)

where the subscripts

b

and

d

indicate the grid points used for writing the boundary conditions and the governing differential equation,

K_{bb}

and

K_{dd}

indicate the stiffness matrices of boundary and interior points, while

δ_{b}

and

δ_{d}

are the displacement vectors of boundary and interior points,

M_{dd}

are the mass matrices of interior points.

The natural frequencies of the shell structure can be determined by utilizing eigenvalue of the linear equation system as follows

| K- ω^{2} M | =0

(21)

where

K = K_{dd} - K_{db} K_{bb}^{-1} K_{bd}

M = M_{dd}

. By solving the eigenvalue problem given in equation (21), the natural frequencies of the sandwich cylindrical shell can be obtained.

Numerical results for free vibration of sandwich cylindrical shell

For numerical investigation, the sandwich cylindrical shell with thickness $h = 0.007 m$ and length $L = 0.104 m$ is considered, and the material parameters of the sandwich cylindrical shell are $E = 210 \times 10^{9} P a$ , $μ = 0.3$ and $ρ = 7800 k g / m^{3}$ , the upper and lower skin layer thicknesses are assumed to have an equal thickness $h_{f} = 0.002 m$ . The auxetic core unit geometrical dimensions are $h_{c} = 0.003 m$ , $ϕ = 60^{\circ}$ , $t = 0.001 m$ , $d = 0.0105 m$ and $l = 0.0046 m$ , i.e., $η_{1} = d / l = 2.28$ , $η_{3} = t / l = 0.21$ . For convenience, the dimensionless frequency parameter is used in presenting the numerical results, which is equal to

Ω = ω R \sqrt{ρ (1 - μ^{2}) / E}

(22)

where,

ω

is the frequency parameter. Unless specified otherwise, the material and geometry parameters are used as mentioned above.

Verification study

To evaluate the reliability of the present theory formulation and the method, the commercial software ABAQUS is firstly used to obtain the finite element results for verification. In addition, we conducted a modal test prepared sandwich cylindrical shell so as to prove the validity of the present theory formulation and the method. The experiment equipment components used for the modal test are illustrated in Figure 3 and Figure 4 shows the physical figure. In this text, the sandwich cylindrical shell structure is hung by an elastic string to simulate the free-free boundary condition, the method of force-hammer excitation was chosen for this text, in which the hammer was used to excite the sandwich cylindrical shell at the point on the shell, while the micro-accelerometer was placed on the shell structure and has been used to measure the vibration response of sandwich cylindrical shell, and the data analysis system (LMS Test. Lab) was used to process the test vibration data, from which the natural frequencies of the shell were extracted.

It is necessary to point out that in order to demonstrate the convergence of the GDQ method further, a study of the number of grid points is also done. The results, illustrated in Table 1, compare the first six natural frequencies obtained from the finite element model, experiment, and theoretical prediction. The error 1 between the FEM results and theoretical results, and the error 2 between the experimental results and theoretical results, are defined as

e r r o r 1 (%) = \frac{| Ω_{T h e o r y} - Ω_{F E M} |}{Ω_{F E M}} \times 100 %, e r r o r 2 (%) = \frac{| Ω_{T h e o r y} - Ω_{E x p e r i m e n t} |}{Ω_{E x p e r i m e n t}} \times 100 %

where,

Ω_{T h e o r y}

Ω_{F E M}

and

Ω_{E x p e r i m e n t}

are the natural frequencies of theoretical, FEM and experimental results.

Table 1.

The first six dimensionless frequencies ( $Ω$ ) obtained from the FEM, experiment and theoretical results under the free-free boundary condition.

Mode sequence	FEM results	Experimental results	Theory results			Error 1, %	Error 2, %
Mode sequence	FEM results	Experimental results	N = 5	N = 10	N = 15	Error 1, %	Error 2, %
1	0.1283	0.1239	0.1388	0.1259	0.1259	1.876	2.62
2	0.1557	0.1507	0.1819	0.1632	0.1632	4.817	8.29
3	0.3527	0.3442	0.3819	0.3567	0.3567	1.389	3.89
4	0.3889	0.3809	0.4316	0.4024	0.4024	3.471	5.64
5	0.5991	0.5764	0.6247	0.6116	0.6116	2.086	6.11
6	0.6336	0.6129	0.6962	0.6422	0.6422	1.35	4.78

As can be seen from Table 1, when the number of grid points equals to 15, the calculation results based on this method are in good agreement with finite element results and experimental modal test results, and the fast convergence of this method is also shown in the table. Up to the first six natural frequencies, the errors in the theoretical results with respect to the FEM results are less than 4.817%, whereas the errors in the theoretical results with respect to the experimental results are less than 8.29%. In addition, Table 2 compares the FEM and experimental results for the natural mode shapes of sandwich cylindrical shell, wherein the FEM and the experiments have similar mode shape patterns. According to these results, there is a good agreement between the numerical analysis, FEM, and experiments in terms of the natural frequency and mode shapes indicating that the present theory formulation and the method are highly reliable. Note that, for the following analysis, we will choose 15 grid points to perform the parameter study.

Table 2.

Comparison of mode shapes of sandwich cylindrical shell under the free-free boundary condition.

Mode type	Experimental results	FEM results
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6

Structural parametric analysis of sandwich cylindrical shell

In this section, the vibration behaviors of sandwich cylindrical shells with different geometrical parameters are discussed, wherein the dimensionless first natural frequency parameter $Ω_{1} = ω_{1} R \sqrt{ρ (1 - μ^{2}) / E}$ is selected. In Figure 5, the effect of core-thickness ratio $h_{c} / h$ on the dimensionless first natural frequency parameters $Ω_{1}$ of sandwich cylindrical shell with different length-radius ratio $L / R$ is shown under different boundary conditions, where four different length-radius ratio ( $L / R = 2$ , 3, 4 and 5) are chosen in this example. As can be seen from Figure 5, the first natural frequency of the sandwich cylindrical shell decreases as the core-thickness ratio $h_{c} / h$ increases form 0.1 to o.7 under F-F, R-R, S-S and C-C four boundary conditions. We also found that for a specific value of the core-thickness ratio $h_{c} / h$ , the first natural frequency decreases with the rise of the length-radius ratio $L / R$ . Such behavior is due to the fact that when the sandwich shell thickness is kept constant, the increase of core thickness leads to the increase of the spacing between the upper and lower skin layer shell, which weakens the overall bending stiffness of shell structure, similarly, when the sandwich shell radius remains unchanged, the increase of shell length also decreases the overall bending stiffness of shell structure. The reduction of the overall bending stiffness of sandwich shell structure will lead to large deformation and low natural frequency.

As the core layer is one of the most critical parameters in the whole sandwich shell structure, it is essential to evaluate its effect on the vibration characteristics of sandwich shell structures. Thus the influences of two geometric parameters $η_{1}$ and $η_{3}$ (defined in Eq. (1)) of the core layer as well as the cell inclined angle on the first natural frequency are discussed in detail here, where four different geometric parameters $η_{1} = d / l$ ( $η_{1} = 3$ , 5, 7 and 9) and $η_{3} = t / l$ ( $η_{3} = 0.3$ , 0.5, 0.7 and 0.9) are chosen in this example.

As can be seen from Figures 6 and 7, for F-F boundary condition, the first natural frequency of sandwich cylindrical shells increases first and then decreases when the inclined cell angle $φ$ is varied form $- 80^{\circ}$ to $80^{\circ}$ under various values of the geometric parameter $η_{1}$ and $η_{3}$ , besides, we also noticed that the highest natural frequencies of sandwich cylindrical shells could be obtained when the cell inclined angle $φ$ is set between $- 30^{\circ}$ and $- 10^{\circ}$ . In contrast to the F-F boundary condition, the first natural frequency of shell structure decreases first and then increases under the C-C boundary condition and for a specific value of inclined cell angle, the first natural frequency increases with the rise of $η_{1}$ and $η_{3}$ . For Figures 6(b) and 7(b), another exciting point attracting one’s attention is that for various values of the geometric parameter $η_{1}$ and $η_{3}$ , the first natural frequency curve varying with the inclined cell angle has approximate axisymmetric characteristics about cell inclined angle $φ = 0^{\circ}$ .In addition, the comparison of the first natural frequency amplitude fluctuation for the four different ratios of geometric parameters $η_{1}$ and $η_{3}$ shows that the vibration characteristics of larger $η_{1}$ and $η_{3}$ ratio seem to be better.

In addition, the influences of two geometric parameters $η_{1}$ and $η_{3}$ of the core layer as well as the inclined cell angle on the Poisson’s ratio $μ_{21}^{c}$ (defined in Eq. (1)) of the core layer are also shown in Figure 8. It can be observed that when the inclined cell angle $φ$ is set between $- 80^{\circ}$ and $0^{\circ}$ , the cell of the core layer exhibits a positive Poisson’s ratio value, while when the inclined cell angle $φ$ is set between $0^{\circ}$ and $80^{\circ}$ , the cell of the core layer exhibits a negative Poisson’s ratio value and when the inclined cell angle remains unchanged, the negative Poisson’s ratio $μ_{21}^{c}$ increases with the rise of the geometric parameters $η_{1}$ and $η_{3}$ . Besides, we also found that except for the case of $η_{1} = 1$ in Figure 8, the curves of Poisson’s ratio $μ_{21}^{c}$ are centrally symmetrical about inclined cell angle $φ = 0^{\circ}$ .

The effect of elastic foundations on the dimensionless first natural frequency of sandwich cylindrical shells is shown in Figure 9. The modulus values of the Pasternak foundation ( $K_{s}$ ) and the Winkler foundation ( $K_{w}$ ) are varied from $K_{0}$ to $7 K_{0}$ , where $K_{0} = 1 \times 10^{7} k N / m^{3}$ . It is found from the Figure 9 that the first natural frequency of the sandwich cylindrical shell decreases as the length-thickness ratio $L / h$ increases from 10 to 25 under the F-F boundary condition. Such behavior is due to the fact that when the sandwich shell thickness is kept constant, the stiffness of the sandwich shell structure will decrease with the increase of shell length, resulting in the reduction of the natural frequency of the shell structure.

From Figure 9(a)–(c), it can be observed that when the Pasternak foundation ( $K_{s}$ ) and the Winkler foundation ( $K_{w}$ ) are increased and the length-thickness ratio is fixed, the first natural frequency of shell structures increases with the increase of the elastic foundation modulus values. Moreover, comparing Figure 9(a) and 9(b), it can be found that the Pasternak foundation ( $K_{s}$ ) has larger influence on the natural frequencies of sandwich cylindrical shell structures than the Winkler foundation ( $K_{w}$ ). To illustrate this point, Figure 9(d) shows a global observation concerning the influence of different elastic foundation variations on the natural frequencies of sandwich cylindrical shells, it is clearly evident from Figure 9(d) that the influence of Pasternak foundation ( $K_{s}$ ) on increasing the first natural frequency is more than the the Winkler foundation ( $K_{w}$ ).

Conclusions

The vibration responses of sandwich cylindrical shells resting on elastic foundations with auxetic structural layers is discussed. By comparing the experimental results with the numerical results calculated by the commercial finite element software, the validity of the proposed theoretical model is verified. Based on the developed theoretical model, the influences of geometrical parameters and elastic foundation on the vibration behavior of sandwich shell structures have been investigated.

The results revealed that under different boundary conditions, the first natural frequency decreases with the rise of the length-radius ratio $L / R$ , core-thickness ratio $h_{c} / h$ and length-thickness ratio $L / h$ due to the weakness of the structural stiffness. Moreover, it was found that under the C-C boundary condition, the first natural frequency increases with the rise of $η_{1}$ and $η_{3}$ , whereas under F-F boundary condition, the first natural frequency increases first and then decreases when the inclined cell angle $φ$ varies from $- 80^{\circ}$ to $80^{\circ}$ . In addition, when the inclined cell angle $φ$ is set between $0^{\circ}$ and $80^{\circ}$ , the cell of the core layer exhibits a negative Poisson’s ratio value and the curves of Poisson’s ratio $μ_{21}^{c}$ are centrally symmetrical about the inclined cell angle $φ = 0^{\circ}$ . The first natural frequency of sandwich cylindrical shell structure increases with the increase of elastic foundation modulus values, and the Pasternak foundation ( $K_{s}$ ) has larger influence on the natural frequencies of shell structures than the Winkler foundation ( $K_{w}$ ).

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

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Yunnan Fundamental Research Projects (Project No. 202201AT070145 and 202101BE070001-005) and the Scientific Research Fund of Yunnan Provincial Department of Education (Project No. 2021J0053).

ORCID iD

Tao Fu

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