Vibration analysis of fluid-conveying multi-scale hybrid nanocomposite shells with respect to agglomeration of nanofillers

Farzad Erahimi .Ali Daagh

a Department of Mechanical Engineering.Faculty of Engineering.Imam Khomeini International University.Qazvin.Iran

b School of Mechanical Engineering.College of Engineering.University of Tehran.Tehran.Iran

Keywords: Vibration Agglomeration effect Multi-scale hybrid nanocomposites Galerkin’s solution Viscous fluid flow

ABSTRACT The vibration problem of a fluid conveying cylindrical shell consisted of newly developed multi-scale hybrid nanocomposites is solved in the present manuscript within the framework of an analytical solution.The consistent material is considered to be made from an initial matrix strengthened via both macro-and nano-scale reinforcements.The influence of nanofillers’agglomeration,generated due to the high surface to volume ratio in nanostructures.is included by implementing Eshelby-Mori-Tanaka homogenization scheme.Afterwards,the equivalent material properties of the carbon nanotube reinforced(CNTR) nanocomposite are coupled with those of CFs within the framework of a modified rule of mixture.On the other hand.the influences of viscous flow are covered by extending the Navier-Stokes equation for cylinders.A cylindrical coordinate system is chosen and mixed with the infinitesimal strains of first-order shear deformation theory of shells to obtain the motion equations on the basis of the dynamic form of principle of virtual work.Next.the achieved governing equations will be solved by Galerkin’s method to reach the natural frequency of the structure for both simply supported and clamped boundary conditions.Presenting a set of illustrations.effects of each parameter on the dimensionless frequency of nanocomposite shells will be shown graphically.

1.Introduction

Once elements with at least one dimension in nano scale are selected as reinforcements in the design and fabrication of the composites.the obtained material is named a nanocomposite.Indeed.the outstanding mechanical properties of nanoparticles were appealing enough in the engineers’opinion to be employed as reinforcement in composites.One of the most famous nano size reinforcing elements is carbon nanotube (CNT).In the 2010s.with increasing application of nano structures in mechanical analyses of continuums.many researchers devoted their field of interest to investigate the static and dynamic behaviors of CNT reinforced(CNTR) nanocomposites.For example.Ke et al.[1]utilized Timoshenko beam model incorporated with von K′arm′an relations to survey nonlinear vibrational behaviors of CNTR nanocomposite beams.In another attempt.Zhu et al.(2012) used Mindlin plate model coupled with FEM to study the static and dynamic responses of CNTR nanocomposite plates.Shen and Xiang[2]investigated the thermo-elastic postbuckling problem of a cylindrical CNTR nanocomposite panel once an axial compression is applied.Another effort is made by Heshmati et al.[3]to probe the dynamic characteristics of a CNTR beam with respect to the influences of CNTs’agglomeration and waviness.Lei et al.[4]utilized a kp-Ritz method in order to survey vibration analysis of nanocomposite plates reinforced with single-walled CNTs(SWCNTs)in the framework of Mindlin plate theory.Besides.Wattanasakulpong and Chaikittiratana[5]could present an efficient model for bending,buckling and vibration behaviors of CNTR nanocomposite plates based on a higher-order plate model.Zhang and Liew [6]taken into consideration the geometrical nonlinearity in their research dealing with nonlinear large deflection behaviors of nanocomposite skew plates rested on a two-parameter elastic substrate.Also,Jam and Kiani[7]could present a solution for low-velocity impact problem of a nanocomposite beam while the structure is supposed to be affected by thermal loading.An Isogeometric analysis(IGA)in conducted by Phung-Van et al.[8]for both stability and vibration problems ofCNTR nanocomposite plates on the basis of Reddy’s plate model.Thereafter.Song et al.[9]employed a higher-order plate theory to consider impact responses of CNTR nanocomposite structures with respect to different distributions of CNTs in the initial matrix.The agglomeration effects are regarded by Tornabene et al.[10]analyzing the vibrational characteristics of nanocomposite shells containing CNTs as reinforcement parts.Lei et al.[11]developed a parametric study for dynamic behaviors of rotating cylindrical panels reinforced with CNTs on the basis of an element free kp-Ritz method.In addition.the postbuckling analysis of laminated nanocomposite plates reinforced with CNTs subjected to a bi-directional compression is performed by Zhang et al.[12].Moreover,the lowvelocity impact analysis of nanocomposite plates in thermal environments is carried out by Ebrahimi and Habibi [13].An Eshelby-Mori-Tanaka based homogenization model for CNTR nanocomposite panels is developed by García-Macías et al.[14]for the goal of investigation of the postbuckling characteristics of such structures under axial compression with respect to waviness and agglomeration effects.Another remarkable paper in this field of interest is arranged by Ansari et al.[15]dealing with the vibration and stability responses of circular sector nanocomposite plates reinforced with CNTs.Zarei et al.[16]could numerically solve the low-velocity impact problem of a nanocomposite plate with arbitrary boundary conditions (BCs) by considering both shear deformation and thermal effects.Fantuzzi et al.[17]presented a nonuniform rational B-splines curve based model for vibration problem of arbitrary shaped CNTR plates by considering agglomeration influences.Ebrahimi and Farazmandnia [18]examined the thermally affected mechanical responses of sandwich beams made of CNTR nanocomposites.The influence of the nature of the reinforcing nanosize reinforcement on the material properties of the obtained nanocomposite material was considered by some researchers while implementing an stochastic multi-scale modeling for the purpose of approximating the mechanical behaviors of the under observation nanocomposite [19-21].For more information about the mechanical behaviors of CNTR nanocomposite continuous systems.readers are advised to read complementary references [22,23].

Even though the aforementioned nanocomposites possess lots of merits to be selected as the material for mechanical elements,a novel type of nanocomposites has been recently found which is able to exhibit a combined behavior of both macro- and nanocomposites.These nanocomposites are made from three phases:a primary matrix.macro scale reinforcement and nano scale reinforcement.Due to this mixture.these nanocomposites are named multi-scale hybrid nanocomposites.As a matter of fact.utilization of multi-scale hybrid nanocomposites empowers the structure to support higher critical stability limit.natural frequency and also lower deflection.Thus.it is of high importance to gain adequate knowledge about the mechanical behavior of structures consisted of this type of nanocomposites.In the recent years.some of the authors made their effort to study the mechanical responses of multi-scale hybrid nanocomposites.Rafiee et al.[24]surveyed nonlinear dynamic characteristics of piezoelectric laminated plates made from multi-scale hybrid nanocomposites.He et al.[25]explored the large amplitude nonlinear free and forced vibrational responses of multi-scale hybrid nanocomposite beams.Later,Rafiee et al.[26]investigated static and dynamic responses of thin-walled rotating multi-scale hybrid nanocomposite beams.Also.Ghorbanpour Arani et al.[27]studied the vibrational responses of doublelayered sandwich beams made from a smart core and facesheets made from multi-scale hybrid CNT/glass fiber reinforced nanocomposites.Ebrahimi and Habibi [28]tried to determine the behaviors of multi-scale hybrid nanocomposite plates in a hygrothermal environment once the structure is subjected to a lowvelocity impactor.They considered for kinematical nonlinearities on the basis of von-Karman theory.Lately,Gholami and Ansari[29]surveyed the nonlinear deflection problem of a multi-scale hybrid nanocomposite plate.Literature review reveals that the issue of investigating the free vibration problem of multi-scale hybrid nanocomposite cylinders has never been studied yet.Thus.the authors are aimed to present an analytical study concerned with the vibrational characteristics of multi-scale hybrid nanocomposites by considering the effects of viscous flow conveying in the structure and aggregation of CNTs in the media.

In this article.a combination of Eshelby-Mori-Tanaka micromechanical model and rule of mixture is implemented to homogenize the constituent material.Afterwards.first-order shell theory is extended and the motion equations are derived based on the Hamilton’s principle for a vibration problem.Next.the effect of viscous flow inside the shell are assumed to be considered by implementing the well-known Navier-Stokes equation.Next,Galerkin’s method is used to solve the governing equations.Once the participant variables are normalized.the influence of each parameter is highlighted in the framework of a group of diagrams.

2.Theory and formulation

2.1.Micromechanical homogenization scheme

In this section.the homogenization process is explained presenting a Eshelby-Mori-Tanaka model in order to capture the effect of CNTs’ agglomeration while reaching the effective mechanical properties of multi-scale hybrid nanocomposites [30].Furthermore.the rule of mixture is employed in order to account for the dispersion of CFs in the nanocomposite.The flowchart of the following homogenization method can be observed in Fig.1.First,the effective properties of CF reinforced(CFR)composites are going to be discussed as follows:

Fig.1.The flowchart of the homogenization procedure for the purpose of reaching the equivalent material properties of the hybrid nanocomposite material.

where E.G.ν and ρ stand for Young’s modulus.shear modulus,Poisson’s ratio and mass density,respectively.Also,the superscripts F and NCM denote fiber and nanocomposite matrix.respectively.Evidently.VFand VNCMare volume fractions of fiber and nanocomposite matrix.respectively.Obviously.the aforementioned volume fractions can be related to each other by:

Next.it is turn to investigate the effect of adding nanoparticles to the media.CNTs.which are employed as the nano scale reinforcements in this article.possess a remarkable stiffness incorporated with a high slenderness ratio.Due to these features,sometimes CNTs do not follow the initial uniform distribution inside the matrix.In other words,in some regions inside the continua some spherical inclusions can be found which are filled with a set of CNTs.Thus.CNTs’ concentration can be different from a region to another one.This effect is of high significance whenever the mechanical behavior of a nanocomposite is supposed to be analyzed.in this situation.the total volume of CNTs can be divided in two parts.one of them is related to the CNTs inside the inclusions and another one corresponds with CNTs which are insertedin the matrix.The volume of CNTs inside the inclusions(clusters)and the volume of CNTs inside the matrix can be related to each other as:

Now,it is turn to relate the volume of CNTs to the entire volume of the structure as follows:

where WMis the volume of the matrix which CNTs are dispersed in it.In this problem,a polymeric matrix is employed.Also,Wris the volume of CNTs.Dividing these volumes to the total volume (W),the volume fraction of each part can be written as:

As same as the volume of CNTs(W),the volume fraction of CNTs in the matrix can be divided in two parts of inside the clusters and outside of clusters.To this reason.two new parameters are introduced to formulate this issue in the following form:

where μ indicates on the volume fraction of clusters and η stands for the volume fraction of CNTs inside the clusters.It should be regarded that μ ≤ η is a limitation for this methodology.

One should be aware of the particular cases which can be generated by changing agglomeration parameters.For instance,once μ = 1.the entire matrix can be considered as a big cluster which contains all of the nanoparticles,henceforward,aggregation of nanofillers cannot be observed.However.full accumulation can occur in the situation that η = 1 (fully agglomerated CNTs).In another condition (μ≤η.η≠1).some of the nanofillers are placed inside the clusters and the others are scattered in the matrix free from any membrane (partially agglomerated CNTs).

Incorporating Eqs.(8) and (9) yields in:

Also.the variation of the Vrwith respect to the thickness direction produces mechanical properties as a function of z.The volume fraction of nanofillers in the matrix can be expresses as:

in which ρrand ρMare mass densities of CNT and matrix.respectively.In addition,wris the mass fraction of nanofillers and can be calculated by:

where Mrand MMare related to the mass of CNTs and matrix,respectively.It is worth mentioning that two versions of Vrcan be defined in the problems of which agglomeration phenomenon is studied.The main difference between these two types is about the position of agglomerated nanoparticles and the matrix.In this case,the matrix is seemed to be in the bottom and the agglomerated CNTs are assumed to be at the top of the structure.To gain more information about this issue,researchers are highly advised to read Shi et al.[30].

Now,the effective material properties can be reached following the relations of Eshelby-Mori-Tanaka micromechanical model[30].According to this model.the bulk moduli of inclusions can be written as:

where KMis the bulk moduli of matrix.Moreover,the shear moduli of inclusions can be introduced as:

where GMis the shear moduli of matrix.Next.the bulk and shear moduli of the remnant parts can be formulated as:

In Eqs.(14)-(17).the mechanical terms αr.βr.δrand ηrcan be calculated as:

where kr.lr.mr.nrand prare the elastic Hill’s coefficients of CNTs which can be different for each type of CNTs with respect to the chirality of the CNT.In this manuscript.the Hill’s constants are employed for SWCNTs with chirality of (10,10).These coefficients can be observed in the framework of Table 1.

Based on the implemented homogenization scheme.the equivalent bulk moduli of the nanocomposite can be computed using the following formula:

where νoutis the Poisson’s ratio of the matrix and can be defined as:

Also.the equivalent shear moduli can be computed as:

Finally,the equivalent Young moduli and Poisson’s ratio of CNTR nanocomposites can be written in the following form:

Moreover.the equivalent density of the CNTR nanocomposite can be formulated utilizing the fundamentals of mixture’s rule as:

It is worth mentioning that in the above homogenization procedure,the bonding between the CNTs and the matrix is considered to be ideal.However.this assumption is a simplifying one; hence,volunteers are advised to read complementary references dealing with this issue [32-34].

2.2.First-order shear deformable shell theory

The kinematic relations of the nanocomposite shell are going to be derived in this section.The geometry and coordinate system of the structure are shown in Fig.2.Now,the displacement fields of a shell can be expressed as follows based on the first-order shear deformable shell theory [35-37]:

in which u.v and w are axial.circumferential and lateraldisplacements,respectively.Furthermore,θxand θφare the rotation components about axial and circumferential directions.respectively.Henceforward.the nonzero strains of a shell type element can be written in the following form:

Fig.2.Geometry and coordinate systems of a fluid conveying cylindrical shell.

Table 1 The material properties of the matrix.carbon fiber and carbon nanotube.

2.3.Derivation of motion equations

Herein.the dynamic form of principle of virtual work.or Hamilton’s principle,will be extended for cylindrical shells in order to reach the Euler-Lagrange equations of a nanocomposite shell.Hamilton’s principle can be defined in the following form:

where U,K and V are strain energy,kinetic energy and work done by external loading.respectively.The variation of strain energy for a linear elastic solid can be expressed as:

Moreover.variation of kinetic energy can be written as:

Now.the variation of external loadings must be formulated.Herein.the viscous fluid flow in assumed to be an axially symmetric.Newtonian and laminal flow.Henceforward.the Navier-Stokes equation can be employed.The momentum equation of the flow can be written in the following form [38,39]:

in which P and ρbare fluid pressure and density of the fluid,respectively.Due to the reciprocal identity between the speed and acceleration of fluid and structure in the contact points.the following relations can be developed [38,39]:

where the differentiation with respect to time is Eulerian time derivative that can be defined as:

in which vxcorresponds with the average velocity of the flow.Furthermore,the shear stresses in Eq.(30)can be expressed in the following form [38,39]:

where μ is the viscosity of the conveying fluid.Now,the variation of work done by external loadings can be formulated in the following form:

where Nr.Nxand Nφ are radial.axial and circumferential loadings,respectively.Once Eqs.(33)-(36) are inserted in Eq.(37).the final form of the work done by external forces can be achieved.Now,once Eqs.(31).(32) and (37) are inserted in Eq.(30).the motion equations of cylindrical shells can be written as:

where

and

where σij.εkland Cijklare components of Cauchy stress.strain and elasticity tensors.respectively.Integrating from above equation over the shell’s thickness.the following relation can be achieved:

where

in which κsis shear correction factor.

2.4.Constitutive equations

The stress-strain relationship of a multi-scale hybrid nanocomposite can be expressed in the following form:

2.5.Governing equations

The coupled partial differential governing equations of a multiscale hybrid nanocomposite shell can be formulated in the following form:

3.Solution procedure

In this part,the Galerkin’s method is utilized in order to achieve the natural frequency of nanocomposite shells.According to this analytical method.the displacement field can be expressed in the following form:

in which Umn.Vmn.Wmn,Θxmnand Θφmn are unknown coefficients.Moreover.n is circumferential wave number and Xmis a function which is arranged to satisfy the axial boundary conditions (BCs).The preliminary assumptions for simply support and clamped BCs are:

?Simply supported - simply supported (S-S):

?Clamped- clamped (C-C):

Readers are advised to read more references about the influences of various types of BCs on the mechanical response of the continuous systems[40].Now,the circular natural frequency of the shell can be obtained once Eq.(53)is substituted in Eqs.(48)-(52).Indeed.the following eigenvalue problem should be solved:

where Δ is a column vector including unknown coefficients.Also,K,M and C denote stiffness,mass and damping matrices,respectively.

The corresponding arrays of such matrices can be found looking for the Appendix at the end.Here,Xmfunctions corresponding with S-S and C-C edge conditions can be assumed to be as:

Table 2 Comparison of natural frequencies of cylindrical shells for both S-S and C-C boundary conditions.

Fig.3.Variation of dimensionless natural frequency of multi-scale hybrid nanocomposite shells versus circumferential wave number for various flow velocities for(a)S-S and(b)CC edge supports.

Fig.4.Variation of dimensionless natural frequency of multi-scale hybrid nanocomposite shells versus volume fraction of CNTs inside the clusters for various flow velocities for(a)S-S and (b) C-C edge supports.

Fig.5.Variation of dimensionless natural frequency of multi-scale hybrid nanocomposite shells versus mass fraction of CNTs for various flow velocities once both S-S and C-C edge supports are included as well as different types of agglomerated nanocomposites.

4.Numerical results and discussion

In this section,a series of illustrations are presented in order to clarify the effect of various parameters on the vibration responses of multi-scale hybrid nanocomposite structures by considering the effect of a viscous fluid flow.Basically.the cylindrical shell is supposed to be made of epoxy and it is reinforced with CF and CNT.Material properties of CFs are achieved from Ref.[28].The mechanical properties of SWCNT (10,10) which are used here can be found in reference [31].Moreover.the material properties of the matrix are as same as those implemented in reference [17].The presented results are validated by comparing the results of this model with those of former researches presented by Loy et al.[41]and Ke et al.[42].The results of this comparison are tabulated in Table 2 for first four circumferential wave numbers.According to this table,the presented methodology is an efficient method for the goal of estimating the vibrational responses of cylindrical shells.

In this research.thickness is supposed to be h = 5 cm.Also,length and radius of the shell are seemed to be 20 times greater than the shell’s thickness in the future illustrations.In this section,effects of both S-S and C-C edge conditions are covered.Here.the dimensionless form of natural frequency’s real part can be presented as:

Fig.3 is presented in order to clarify how the flow velocity canaffect the dimensionless frequency for both S-S and C-C multi-scale hybrid nanocomposite shells.To this purpose.the variation of dimensionless frequency versus circumferential wave number is plotted once the flow velocity is varied.First of all.it can be well observed that C-C shells are able to tolerate higher dynamic loadings in comparison with S-S ones.Furthermore.the influence of viscous fluid which is considered to be going through the shell is a decreasing effect.In other words.the dimensionless frequency becomes smaller as flow velocity is assumed to be a high value.This decreasing change can be better seen in the mid-range circumferential wave numbers.

Fig.6.Variation of dimensionless natural frequency of S-S multi-scale hybrid nanocomposite shells versus volume fraction of CNTs inside the clusters for various flow velocities and mass fractions of nanofillers.

Besides.the variation of natural frequency against volume fraction of CNTs inside the clusters is drawn in Fig.4 for both S-S and C-C hybrid nanocomposite shells for various amounts of flow velocity.Once again.it is observed that C-C shells provide greater frequencies compared with S-S structures.Clearly.it can be seen that the flow velocity plays a damping role for the dimensionless natural frequency in every desirable volume fraction of CNTs inside the clusters.On the other hands,it can be found that the frequency becomes smaller as agglomeration coefficient (volume fraction of CNTs inside the clusters,η)rises.As a matter of fact,the more is η,the higher is the number of aggregated nanofillers.Thus.the uniform dispersion of nanotubes cannot be satisfied and the mechanical response becomes smaller.

In another graphical study.the influence of nanofillers’ mass fraction is covered as well as the effects of BC and flow velocity in the framework of Fig.5.Herein,it can be seen that the influences of BCs are as same as those presented in the former illustrations.Moreover.the effect of adding the velocity of the fluid flow is a decreasing effect in all amounts of nanofillers’ mass fraction.However.the effect of increasing the mass fraction of nanotubes directly depends on the agglomeration regime.Actually.in the cases of partially agglomerated nanofillers for the nanocomposite(η ＜ 1).the dimensionless frequency can be intensified by making an increase in the amount of nanotubes’ mass fraction.Reversely,natural frequency becomes smaller once fully agglomerated nanofillers are implemented in the structure.The physical reason of this issue is that in the fully agglomerated situation all of the CNTs are inside the clusters and increasing their mass fraction cannot be resulted in an increase in the structure’s stiffness; whereas.in the partially agglomerated condition.some of the CNTs are out of inclusions and they are better candidates for the goal of increasing the natural frequency if their mass fraction is added.

Also,the coupled effects of mass fraction of nanofillers and the flow velocity are again discussed in the framework of Fig.6 to put emphasize on the qualitative effect of CNTs’ mass fraction on the dimensionless frequency of multi-scale hybrid nanocomposite shells with S-S edge supports.According to the diagram,the former declarations in the previous paragraph can be seen again.It can be well observed that the mechanical response can be amplified by adding the magnitude of mass fraction of nanotubes once the term η is lower than approximately 0.95 in the presence and absence ofviscous fluid.Actually.it can be noted that in the cases of utilizing fully agglomerated nanosize reinforcements we cannot expect to possess an increasing effect by adding the mass fraction of nanotubes.Similarly,in this diagram it can be found that the capacity of the nanocomposite shell for enduring mechanical vibrations can be lessened while a viscous fluid is seemed to be conveying in the structure.

Fig.7.Variation of dimensionless natural frequency of S-S multi-scale hybrid nanocomposite shells versus flow velocity for various gradient indices for(a)η=0.5 and(b)η=0.8.

Fig.8.Variation of dimensionless natural frequency of multi-scale hybrid nanocomposite shells versus circumferential wave number for various CFs’ volume fractions and flow velocities for (a) S-S and (b) C-C edge supports.

Afterwards.it is turn to pay attention to the effect of material composition term(P)as well as the velocity of the fluid in Fig.7 for multi-scale hybrid nanocomposite shells with S-S BC with different volume fractions of CNTs inside the inclusions.One can easily found that the natural frequency becomes smaller while the gradient index is aggrandized (higher P parameters).It is of importance to point that this decreasing impact can be better observed once a lower amount is assigned to the volume fraction of CNTs inside the clusters.As well as former illustrations.here.a raise in the flow velocity results in a decrease in the magnitude of dimensionless frequency of the shell.

Furthermore.the variation of dimensionless frequency against circumferential wave number is again plotted in Fig.8 in order to investigate the effect of CFs’ volume fraction on the frequency of nanocomposite shells in the presence and absence of viscous fluid.Based on this figure.it is clear that an increase in the volume fraction of CFs reveals a raise in the amount of the structure’s natural frequency.Again.one can observe the damping effect of flow velocity on the vibrational characteristics of multi-scale hybrid shells.Also.the effect of viscous flow can be better seen in low amounts of CFs’volume fraction.Moreover,it can be seen that C-C shells are able to endure greater frequencies.particularly in midrange circumferential wave numbers.

Finally.Fig.9 is depicted to show how volume fraction of clusters can affect the dimensionless frequency in various conditions.The dimensionless frequency that can be tolerated by the shell goes through an increasing path once the volume fraction of clusters increases.The main reason of such a phenomenon can be understand by referring to the primary definition of this parameter.In other words,all of the clusters in a nanocomposite continuum can be unified in a big cluster while the term μ is added.Therefore,this big inclusion can operate as same as the structure itself in the situation which no cluster is available in the shell.Moreover,it can be seen that the dimensionless frequency becomes smaller in great volume fractions of CNTs inside the clusters.Again.C-C shells can tolerate higher vibrational loadings in comparison with S-S ones.

5.Conclusion

The issue of analyzing vibrational behaviors of fluid conveying multi-scale hybrid nanocomposite cylinders with respect to the CNTs’ agglomeration was undertaken in this research.The firstorder shear deformation shell theorem was incorporated with an energy based variational approach in order to derive the motion equations.The effective material properties were achieved by considering the effects of nanofillers’agglomeration within a twostep homogenization technique.Here.the most important concluding remarks are going to be reviewed as follows:

· Natural frequency can be enlarged by adding mass fraction of CNTs once η ＜ 0.95;whereas.this effect is completely different in η higher than the specified amount.

· Another alternative to strengthen the shell against vibration phenomenon is to add the volume fraction of clusters in order to make one big cluster that is able to cover all of the nanofillers.

· On the other hand.dimensionless frequency becomes smaller once volume fraction of CNTs inside the clusters and gradient index are raised.

· An increase in the value of CFs’ volume fraction reveals higher natural frequencies.

· The vibration frequency decreases once a viscous fluid is passing through the nanocomposite shell.

· Effect of changing the BCs from S-S to C-C can be better seen in mid-range circumferential wave numbers.

Declaration of competing interest

The authors declare no conflict of interest in preparing this article.

Funding acknowledgement

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

Appendix

The components of stiffness matrix can be written as:

The only nonzero array of the damping matrix can be stated as:

Also.the nonzero arrays of mass matrix are in the following form: