Vibration analysis of size dependent micro FML cylindrical shell reinforced by CNTs based on modified couple stress theory

1 Introduction

Compared to metal, the application of composites has been grown in the previous decades due to of required to light weight and high strength structures and design ability to construct the structures with high stress tolerance in an arbitrary direction in modern engineering. More information about laminated composite structures and functionally graded materials (FGMs) can be observed with a deep consideration to the complementary references (Ghasemi and Mohandes, 2016; Mohammadimehr and Mohandes, 2015; Zhou et al., 2022; Chen et al., 2021; Chen et al., 2021; Mou and Bai, 2018; Baghlani et al., 2020; Khayat et al., 2020; Khayat et al., 2019; Khayat et al., 2017). Although composite materials have excellent strength, their fragile behavior under impact loading is an essential problem. This deficiency could be led to some critical problems namely crack, delamination and debonding, while ductile materials tolerate impact loading. So, using the combination of these two materials, which composite layers are bounded by aluminum metal face sheets to protect the composite materials, Fiber Metal Laminates (FMLs) were produced for more availability of the advantages of composites as well as metals in industries. FMLs have promoted their individual negative properties such as poor impact resistance of composites and poor fatigue strength of metals. Due to promoted mechanical properties of FMLs against composites and metals separately, their applications have been gradually increased in various industries such as fan blades in aeroengine, panels in military and civil aircrafts, solar panels in satellites. In addition, FMLs are applicable for complex environment under intense impact accidents namely hailstone, tool collisions, birds, and runway debris or under severe dynamic loading such as continuous drastic spinning imbalance loads. With increasing the usage of sandwich and FMLs in various industries, numerous investigations have been leaded to vibration of these structures (Mohandes et al., 2018; Ghasemi and Mohandes, 2019; Iriondo et al., 2015; Ghasemi and Mohandes, 2019; Khalili et al., 2010; Zhang et al., 2021; Liu et al., 2021) and the other structures (Wang et al., 2022; Ma et al., 2022; Tao et al., 2017). Tao et al. (Fu et al., 2014) obtained frequencies, mode shapes and deflection of FML Euler-Bernoulli beams with a single-open sided crack subjected to different boundary conditions. Some researchers (Fu and Shao, 2014) studied nonlinear geometrical dynamic of FML beam under thermal and mechanical loadings utilizing differential quadrature method. Nonlinear dynamic response of FML rectangular plate under thermal loading with interfacial damage was conducted by Fu and Shao (Ghasemi and Mohandes, 2020). Ghasemi and Mohandes (Mahmood et al., 2021) employed developed couple stress theory (MCST) to consider dynamic of micro FML cylindrical shell.

In the recent years, modern industries need materials which are applicable in different environmental circumstances such as variety of thermal, mechanical, and chemical properties as well as reasonable price. Carbon nanotubes (CNTs) are expected to have proper industrial potential due to their excellent thermal, mechanical and electrical properties such as great aspect ratio, high thermal, low density, high tensile strength, high elastic modulus, and electrical conductivities (Ghasemi and Mohandes, 2016; Ninh et al., 2021). They have attracted much attention and redundant applications in nano-engineering applications namely actuators, oil probes, nano reactors, chemical sensors, and support catalysts. Based on the mentioned benefits of CNTs in the field of structural mechanics, numerous studies have been conducted by researchers which are including both static and dynamic behaviors of nanocomposite structures (Ebrahimi and Seyfi, 2021; Miao et al., 2021; Moradi-Dastjerdi and Behdinan, 2021; Sheybani et al., 2021; Mohandes and Ghasemi, 2019; Ghasemi and Mohandes, 2019; Mohandes and Ghasemi, 2019; Mohammadimehr et al., 2016; Huang et al., 2021; Zerrouki et al., 2021; Arshid et al., 2021; Heidari et al., 2021; Bendenia et al., 2020; Pan et al., 2021; Behdinan et al., 2020; Qin et al., 2020; Qin et al., 2019; Liu et al., 2022; Sobhani and Masoodi, 2021; Sobhani et al., 2022; Sobhani et al., 2021; Rezaiee-Pajand et al., 2020; Nasution et al., 2022; Khayat et al., 2021; Khayat et al., 2021; Khayat et al., 2021; Khayat et al., 2021; Khayat et al., 2022; Mohammadimehr et al., 2018). Mohammadimehr et al. (Chakraborty et al., 2019) reached natural frequencies of cylindrical composite panel undergoing magneto-electro fields using first order shear deformation theory (FSDT). The composite panel was reinforced by different distributions of CNTs which were arranged in Poly-vinylidene fluoride matrix. Chakraborty et al. (SafarPour et al., 2019) analyzed nonlinear vibration and stability of pre-buckled and post-buckled laminated composite cylindrical panels reinforced by CNTs. They found out that the volume fraction of CNTs had a remarkable influence on vibration and post-buckling responses of this panel. SafarPour et al. (Zhu et al., 2012) considered the impact of critical voltage on vibration and buckling of rotating reinforced CNTs cylindrical shell. Free vibration and bending of reinforced composite plate based on FSDT were studied by Zhu et al. (Guo and Zhang, 2016). The outputs indicated that the natural frequencies and mode shapes were affected by different volume fraction of CNTs and width-to-thickness ratio. Guo and Zhang (Bousahla et al., 2020) considered nonlinear vibration of composite plates reinforced by CNTs undergoing combined in-plane and transverse excitations for simply supported boundary condition. Bousahla et al. (Shen and Xiang, 2012) focused on vibration and buckling of composite beam reinforced by CNTs.

Nonlinear vibration of composite shells with or without CNTs is one of the main matters of engineering and it has developed recently. In nonlinear vibration analysis of composite shells, it is significant to survey their frequencies and mode shapes, due to the structures of composite shell which often operate to dynamic loads in different situation. Shen and Xiang (Shen et al., 2017) proposed the vibration of reinforced composite cylinders under thermal loading using von-Karman assumption based on HSDT. Shen et a. (Li et al., 2020) considered the effects of graphene reinforced composites on the nonlinear vibration of laminated cylindrical shells undergoing thermal loading based on Reddy’s third order shear deformation theory. Some researchers (Liu et al., 2021) promoted a new analytical model for prediction of strain parameters of composite shell reinforced by CNTs with partial constrained layer damping. They used Jones-Nelson nonlinear theory to analyze material properties as nonlinearity. Some researchers (Li et al., 2021) presented a novel method to optimize the nonlinear forced vibrations of FG shells fabricated applying piezoelectric materials in multi-physics fields undergoing electro-thermo-mechanical loadings. Both material and geometrical nonlinearities for vibration of fiber reinforced polymer composite cylindrical shells within thermal environment has been studied (Li et al., 2021). Some researchers (Li et al., 2021) considered nonlinear forced vibration of FGM sandwich cylindrical shells with porosities on an elastic substrate. The researchers (Ebrahimi and Beni, 2016) have been concentrated on the experimental and theoretical analyses of nonlinear vibration of fiber-reinforced composite cylindrical shells with bolted joint boundary limitations.

The material’s stiffness and strength can be increased with decreasing the size scale that is named size effects. The classical continuum mechanics theory is unable to count the size effects in micro scale structures, while some higher order continuum theories have been used to specify the size effect. The strain gradient approach and couple stress theory (CST) are used for micro structures with a bit difference. The variable for describing curvature is introduced using strain and rotation for strain gradient and couple stress theories, respectively. As a conclusion, it can be understood that the CST is a specific form of the strain gradient theory. Ebrahimi and Beni (Akgoz and Civalek, 2011) obtained natural frequencies of short piezoelectric cylindrical nanotube using shear deformable cylindrical theory according to consistent CST. Akgoz and Civalek (Akgoz and Civalek, 2016; Habibi et al., 2019) used strain gradient elasticity and MCST for buckling and bending analysis of micro beams. Habibi et al. (Zeighampour and Beni, 2014) applied MCST for nonlinear free vibration of magneto-electro-elastic Euler-Bernoulli nano beams under thermal loading. Zeighampour and Beni (Zeighampour and Beni, 2014) utilized strain gradient theory for developing cylindrical thin-shell model to analyze free vibration of CNTs subjected to simply-supported boundary condition. In the same study (Pashmforoush, 2020) they studied free vibration of thin conical shells utilizing MCST. Pashmforoush (Mehralian and Beni, 2018) considered low velocity impact response of composites reinforced by CNTs based on finite element model using Hashin’s criterion and cohesive zone modeling. Mehralian and Beni (Soleimani and Beni, 2018) applied strain gradient theory for dynamic replies of bimorph FG piezoelectric cylindrical shell via FSDT. The natural frequencies of axisymmetric shell element using MCST were obtained by Soliemani and Beni (Zeighampour and Beni, 2017). Zeighampour and Beni (Sobhani et al., 2022) studied wave propagation of FG micro cylindrical shell reinforced by CNTs using strain gradient theory based on shear deformable shell theory.

Although CNTs have some efficiencies that promote electro-thermo-mechanical properties of structures and these benefits are existed when they uniformly dispersed in the matrix, they have a deficiency which is agglomeration. They tend to be agglomerated in a cluster because of their maximum aspect ratio (usually>1000) and low bending stiffness because of small diameter and/or elastic modulus in the radial direction (Allahkarami and Nikkhah-Bahrami, 2017). Allahkarami and Nikkhah-bahrami (Ebrahimi et al., 2019) studied free vibration of CNTs reinforced micro curved Timoshenko beam with considering agglomeration factors using generalized differential quadrature method. Another article in this interest field was arranged by Ebrahimi et al. (Ebrahimi et al., 2019) dealing with the efficiency of agglomeration on the vibration of multi-scale hybrid plates reinforced by CNTs. Ebrahimi et al. (García-Macías and Castro-Triguero, 2018) could analytically solve the equations of motion for wave frequency and phase velocity of multi-scale hybrid CNTs reinforced beams with agglomeration effects. García-Macías and Castro-Triguero (Daghigh and Daghigh, 2019) presented the effects of waviness and agglomeration on vibration of CNTs reinforced skew plates. Daghigh and Daghigh (Afshari and Amirabadi, 2021) successfully probed free vibration of agglomerated composite plates reinforced by CNTs mounted on elastic foundation under thermal loading. Afshari and Amirabadi (Ghasemi et al., 2019) surveyed the effect of rotational speed on agglomerated CNTs reinforced truncated conical shells based on FSDT using a semi-analytical solution. The issue of vibration of FML CNTs reinforced cylindrical shell has been studied (Syah et al., 2021) with the consideration of agglomeration effects. Some researchers (Yang et al., 2002) highlighted the influences of rotation on the frequencies of FML rotating CNTs reinforced circular cylindrical shells with agglomerated nanoparticles.

Then, no literatures have been reported on the vibration response of agglomerated micro cylindrical shells specially CNTs/fiber/polymer/metal laminated (CNTFPML) micro cylindrical shell. It means the effect of agglomeration in micro phase of CNTFPML for vibration analysis has not been considered, yet. The vibration analysis of this material can be applied in aerospace industries because of proper properties of metal namely impact, damage tolerance and ductility plus the appropriate characteristics of the composites such as excellent fatigue resistance, stiffness to weight ratios, high strength and acceptable corrosion. In this investigation, the agglomeration effects on this micro cylindrical shell are studied on the basis of Love’s first approximation theory. The structure is including four phases: CNTs, fiber, polymer matrix and metal layers which are attached together. The equations of motion for this CNTFPML micro cylindrical shell are obtained utilizing Hamilton’s principle so that the size effect is considered in these equations using MCST. In addition, the frequencies of structure are calculated utilizing the beam modal function model that is semi-analytical model for considering different boundary conditions.

2 Formulation

In this section, the necessary equations related to dynamic analysis of CNTFPML micro cylindrical shell (Figure 1) with considering agglomeration relations are derived. As depicted in Fig. 1, the composite section of cylindrical shell has been considered cross-ply [Al/0°/90°/0°] so that the metal thickness is same as each layer of composites.

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Fig. 1

CNTFPML micro cylindrical shell (Mahmood et al., 2021)

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Also, in the following flowcharts 1, the steps for constructing the whole structure as well as the procedure of finding natural frequencies of the micro agglomerated CNTFPML structure are shown.

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Flowchart 1

Steps for Obtaining natural frequencies of Micro agglomerated CNTFPML cylindrical shell.

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3 Governing equations of motion

Hamilton’s principle is applied to derive the governing equations of motion with boundary limitations for micro cylindrical shell as follows (Ma et al., 2008; Reddy, 2017; Beni et al., 2016):

L represents Lagrangian function. Moreover, $\mathrm{\delta }\mathrm{T}$ , $\mathrm{\delta }\mathrm{U}$ and $\mathrm{\delta }\mathrm{W}$ express virtual kinetic energy, virtual strain energy and virtual work.

In consideration of Eq. (1), virtual form of strain energy is calculated as (Mahmood et al., 2021):

The stress resultants $\mathrm{N}$ and $\mathrm{M}$ which are forces and moments (Reddy, 2004):

$\mathrm{Y}$ and $\mathrm{P}$ , which are couple moment and high order resultant of normal stress, are illustrated (Reddy, 2004):

The kinetic energy and work done using external loading (in which q is zero in this study) are obtained as (Shi et al., 2004):

The stiffness coefficients for micro CNTFPML cylindrical shells can be described as (Mahmood et al., 2021):

${\mathrm{A}}_{\mathrm{i}\mathrm{j}}$ , ${\mathrm{B}}_{\mathrm{i}\mathrm{j}}$ and ${\mathrm{D}}_{\mathrm{i}\mathrm{j}}$ denote extensional, coupling and bending stiffnesses. In addition, the distances of the middle surface of the shell to outer and inner surfaces of the composite $\mathrm{k}\mathrm{t}\mathrm{h}$ layer are shown as ${\mathrm{h}}_{\mathrm{k}}$ and ${\mathrm{h}}_{\mathrm{k}-1}$ . Here, ${\mathrm{Q}}_{\mathrm{i}\mathrm{j}}^{\mathrm{k}}$ stand for the coefficients of transformed reduced stiffness for the $\mathrm{k}\mathrm{t}\mathrm{h}$ layer of nanocomposites. Then, ${\mathrm{h}}_{\mathrm{m}}$ and ${\mathrm{Q}}_{\mathrm{i}\mathrm{j}}^{\mathrm{m}}$ are the thickness and decreased stiffness of the metal layer, respectively.

The equations of motion can be obtained using substitution Eqs. (12), (15) and (16) into Hamilton’s principle. It is worth noticing that the equations of motion are obtained based on stress resultants, while they shall be according to displacements for better application. So, the stress resultants shall be rewritten based on changes in the curvature of the middle surface and middle surface strains at first. Since then, the equations of motion for micro CNTFPML cylindrical shell shall be calculated based on displacements by introducing the resulted stress resultants and Eq. (14) into Eq. (20) as given:

4 Agglomeration

The agglomeration of CNTs would influence on the elastic attributes of CNTRCs that this effect can be focused using a micromechanical model, which has been presented here. The agglomeration causes some CNTs are appeared in the cluster (concentrated region) and the others are distributed within the matrix (Fig. 2a) (Dabbagh et al., 2020). So, there are some limited clusters including CNTs which are settled in the whole matrix as well as distributed CNTs outside the clusters. So, CNTs’ concentration which creates the degradation of elastic properties would be different from the other regions.

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Fig. 2

Agglomeration of CNTs in a cluster model (Syah et al., 2021)

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Thus, the CNTs volume inside ${\mathrm{V}}_{\mathrm{r}}^{\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{r}}$ and outside ${\mathrm{V}}_{\mathrm{r}}^{\mathrm{m}}$ the cluster constitute the total volume of CNTs which is indicated by ${\mathrm{V}}_{\mathrm{r}}$ (Hedayati and Sobhani Aragh, 2012):

Also, the total volume of representative volume element (RVE) $\mathrm{V}$ (Hedayati and Sobhani Aragh, 2012):

The MT approach would be applied to predict the effective elastic attributes of cluster and matrix of nanocomposite. Therefore, the effective bulk moduli ${\mathrm{K}}_{\mathrm{i}\mathrm{n}}$ and ${\mathrm{K}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and the effective shear moduli ${\mathrm{G}}_{\mathrm{i}\mathrm{n}}$ and ${\mathrm{G}}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ of inclusions and remnant parts are expressed (Syah et al., 2021):

${\mathrm{E}}_{\mathrm{L}},{\mathrm{E}}_{\mathrm{T}},{\mathrm{E}}_{\mathrm{Z}},{\mathrm{G}}_{\mathrm{T}\mathrm{Z}},{\mathrm{G}}_{\mathrm{Z}\mathrm{L}},{\mathrm{G}}_{\mathrm{L}\mathrm{T}}$ and ${\mathrm{\upsilon }}_{\mathrm{L}\mathrm{T}}$ represent the modulus and Poisson’s ratio of equivalent fiber. Via MT procedure, the equivalent bulk $\mathrm{K}$ and shear $\mathrm{G}$ modulus of nanocomposites would be computed as (Kamarian et al., 2016):

Then the effective Young's modulus $\mathrm{E}$ and Poisson's ration $\mathrm{\upsilon }$ of the composite can be defined as given (Tsai et al., 2003):

5 Mechanical properties of CNTRCs

The mechanical characteristics of straight CNT inserted in matrix can be predicted using an equivalent solid long fiber with 2.374 nm diameter bonded to surrounding resin which is depicted in Fig. 3.

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Fig. 3

Conversion strategy (Ghasemi and Mohandes, 2017)

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In this manuscript, the extended rule of mixture is utilized to focus on the influence of the CNTs on the properties of shell reinforced composites. Longitudinal modulus $E_{\mathit{LEF}}$ , transverse modulus $E_{\mathit{TEF}}$ , shear modulus $G_{\mathit{EF}}$ Poisson's ratio ${\upsilon }_{\mathit{EF}}$ , and volume fraction $V_{\mathit{EF}}$ of equivalent fiber can be obtained utilizing rule of mixture (ROM) (Shokrieh and Rafiee, 2010):

6 Beam modal function model’s solution:

A wide range of solution methods exist including numerical (Liu et al., 2016; Civalek, 2004; Talebitooti, 2013; Mohandes and Ghasemi, 2016; Ghasemi and Mohandes, 2016; Tornabene et al., 2015; Zhong and Yu, 2009; Al-Furjan et al., 2020; Al-Furjan et al., 2021; Al-Furjan et al., 2020; Al-Furjan et al., 2020; Al-Furjan et al., 2020), analytical (Bourada et al., 2020; Zhang et al., 2021; Lam and Loy, 1994; Callahan and Baruh, 1999), semi-analytical ones which are applied to analyze the vibration and dynamic of different structures. Although analytical methods are more accurate than numerical ones, various boundary conditions can be analyzed using numerical approaches. In this manuscript, beam modal function model (Mohammadimehr et al., 2016), that is a semi-analytical method with great accuracy, is utilized to consider dynamic of CNTFPML cylindrical shell with various boundary conditions. Thus, the following expression including mode shapes in the longitudinal $\mathrm{U}(\mathrm{x})$ , torsional $\mathrm{V}(\mathrm{x})$ and flexural $\mathrm{W}(\mathrm{x})$ directions is assumed (Wang and Lai, 2000):

It is worth noticing that the exact value of $\mathrm{\alpha }$ , which is related to boundary condition, for cylindrical shells. it should be identified using the axial modal parameter $\mathrm{m}$ for a given circumferential modal parameter $\mathrm{n}$ . The amount of $\alpha$ would be obtained using the following assumption: the flexural mode shapes of the cylindrical shell in the axial direction are in the identical form with the flexural vibration of beam within the same boundary limitations. Then, the beam modal function has been applied to the acquire the magnitude of $\mathrm{\alpha }$ refered to proper limitations (Mahmood et al., 2021). For instance, the modal wave number would be approximately obtained through beam function model as following (Mahmood et al., 2021):

Moreover, the $\mathrm{\alpha }$ for a cylindrical shell under clamped boundary conditions would be derived by:

Moreover, the $\mathrm{\alpha }$ for a cylindrical shell with free-free boundary conditions can be given by:

Like the Eq. (38), the modes $m=0$ and $m=1$ are two inextensional modes of circular cylindrical shells so that these two modes correlated to rigid body translation and rotation modes for a free-free beam. Therefore, they are trivial modes with zero frequencies for the beams and they cannot be estimated for cylindrical shells. So, the non-dimensional form of equations of motion:

7 Numerical results and discussion

The work is focused on an investigation of the agglomeration effects on free vibration of micro CNTFPML cylindrical shells in terms of semi-analytical approach. It should be noted that the composite sector of FML cylindrical shell is reinforced by agglomerated CNTs so that this section is combined to thin metal layers. Initially, the non-dimensional frequencies $\mathrm{\Omega }=\mathrm{\omega }\sqrt{(\mathrm{\rho }{\mathrm{R}}^2/\mathrm{E})}$ of classical macro CNTFPML cylindrical should has been compared to other research study (Syah et al., 2021) in Table 2. Material characteristics of CNTFPML cylindrical shell used in this case is CARALL reinforced by CNTs for $\mathrm{n}=1$ and $\mathrm{m}=1$ . The geometric characteristics of cylindrical shell are considered $\mathrm{L}=10\times \mathrm{R}$ ; $\mathrm{R}=23.437$ . It can be found out that the presented results in this investigation compare excellent with the results of Ghasemi et al. (Syah et al., 2021). Besides, the non-dimensional frequencies of the presented study would be compared with the frequency form $\mathrm{\Omega }=\mathrm{R}\mathrm{\omega }\sqrt{(1-{\mathrm{\upsilon }}^2)\mathrm{\rho }/\mathrm{E}}$ of the other research which are isotropic macro cylindrical shells with different boundary conditions. That is worth noticing the material properties of studied isotropic cylindrical shell are $\mathrm{\upsilon }=0.3$ and $\mathrm{E}=200\mathrm{G}\mathrm{P}\mathrm{a}$ . In addition, the cylindrical shell geometry to compare with wave propagation method illustrated below via Table 3, $\mathrm{h}/\mathrm{R}=0.002$ and $\mathrm{L}/\mathrm{R}=20$ and for comparison with Ritz procedure presented in Table 4 are $\mathrm{h}/\mathrm{R}=0.01$ and $\mathrm{L}/\mathrm{R}=20$ . The outputs illustrate excellent correlation among the method illustrated in this manuscript and wave propagation and Ritz procedures.

Next, the CNTFPML cylindrical shell with considering agglomeration impacts is studied with the following circumstances:

The metal layers are constructed using aluminum and the composite layers are consisting of carbon/epoxy with cross-ply lay-ups $[\mathrm{A}\mathrm{l}/0^{°}/{90}^{°}/0^{°}]$ . Also, geometric attributes of cylindrical shell are considered $\mathrm{L}=10\times \mathrm{R}$ , $\mathcal{l}=0.0937\times {10}^{-6}$ , $\mathrm{R}=23.437\times {10}^{-6}$ subjected to simply supported boundary condition. Moreover, the agglomerated CNTs are distributed within the resin symmetrically. Further, the mechanical properties of fiber, metal, CNT and matrix utilized in this manuscript are presented in Table 5.