Frzd Erhimi .Pendr Hfezi .Ali Dgh

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: Buckling Graphene oxide powder (GOP) Nanocomposite First-order shell theory

ABSTRACT In this study,the buckling analysis of a Graphene oxide powder reinforced(GOPR)nanocomposite shell is investigated.The effective material properties of the nanocomposite are estimated through Halpin-Tsai micromechanical scheme.Three distribution types of GOPs are considered.namely uniform.X and O.Also.a first-order shear deformation shell theory is incorporated with the principle of virtual work to derive the governing differential equations of the problem.The governing equations are solved via Galerkin’s method,which is a powerful analytical method for static and dynamic problems.Comparison study is performed to verify the present formulation with those of previous data.New results for the buckling load of GOPR nanocomposite shells are presented regarding for different values of circumferential wave number.Besides.the influences of weight fraction of nanofillers.length and radius to thickness ratios and elastic foundation on the critical buckling loads of GOP-reinforced nanocomposite shells are explored.

1.Introduction

The use of laminated composite structures in aerospace.shipbuilding,civil and mechanical structural applications has witnessed a tremendous increase in recent decades.primarily due to their innumerable attractive mechanical properties [1].Many studies have been conducted to investigate mechanical behavior of laminated composite structures.There are many researches in the area of effect of delamination on mechanical behavior of laminated composite structures [2,3].Piezoelectric composites are gaining growing attention in recent years due to their coupled mechanical and electrical properties through different engineering applications.Wei and Shen [4]developed a new refined layerwise finite element model representing more refined description of the kinematics of laminated structures so that the interlaminar stresses and electric potential can be obtained more accurately.Muc et al.[5]examined buckling behavior of biaxially compressed laminated plates and shallow cylindrical panels having two symmetric piezoelectric patches on the top and the bottom of laminates.In another study.Lezgy-Nazargah et al.[6]investigated the application of FGPMs for shunted passive vibration damping of laminated composite beams.Also several researchers have shown interest to study the static and dynamic behavior of plates and shells made of laminated composites [7-12].

In addition,over the last few decades,advanced fiber-reinforced composite laminated structures are increasingly in the fields of aviation.aerospace.marine.and other industries due to excellent mechanical performances.The main aim of the researches in this field is the development of new class of materials which can be employed to design stiffer structures with limited weight,increase the safety requirements.improve the dynamic behavior.and reduce delamination phenomena [13-15].Xie et al.[16]and Xu et al.[17]investigated the shock and spallation behavior of a carbon fiber reinforced polymer composite and static and dynamic bending behaviors of carbon fiber reinforced composite cantilever cylinders respectively.In the field of natural fiber reinforced composites Yu et al.[18]developed a novel process for production of bamboo fiber reinforced composite.

Carbon nanotube reinforced composites are the novel materials which have outstanding properties and can be used in high-tech industries.There are many studies about different mechanical behavior of CNTRCs.Ebrahimi and Farazmandnia [19]developed ahigher-order shear deformation beam theory for free vibration analysis of FG-CNTRC sandwich beams in thermal environment.Formica et al.[20]investigated CNTRTs by employing an equivalent continuum model based on Eshelby-Mori-Tanaka approach.Sobhy and Zenkour [21]analyzed magnetic field effect on thermomechanical buckling and vibration of viscoelastic sandwich nanobeams with FG CNT reinforced face sheets in humid environment.In another research.Ansari et al.[22]studied nonlinear forced vibration analysis of FG CNT reinforced composite Timoshenko beams [23,24].analyzed wave propagation in CNT reinforced nanobeams using extended Hamilton principle and considering elastic foundation and thermal effects.In order to include shear deformation and thickness stretching effects,the quasi-3D theories,which are based on a higher-order shear deformation and thickness of the in-plane and transverse displacements.are used in works done by many researchers[25-27].

In addition to CNT reinforced materials.in some scientific attempts,mechanical elements are reinforced by other carbon based gadget.Graphene platelets (GPLs).Graphene platelets due to its fantastic properties is an appropriate candidate as a reinforcement in composite materials [28-30].Zhao et al.[31]investigated the bending and vibration behaviors of a novel class of functionally graded trapezoidal reinforced with GPLs by using finite element analysis.Song et al.[32]surveyed both free and forced vibrations of GPL reinforced FG plates.Later.Song et al.[33]performed another research on bending and buckling analyses of FG composite plates reinforced with GPLs.In another research.Yang et al.[34]investigated buckling and postbuckling behaviors of GPL reinforced FG multilayer beams by using first-order shear deformation method.Also.analysis of free vibration and elastic buckling of FG porous beams reinforced by graphene platelets have been done by Kitipornchai et al.[35]by using Timoshenko beam theory and Ritz method.

Recently.another Carbon derivative.has attracted researchers’attention due to its unique properties.This nanofiller is named graphene oxide powder (GOP) and has marvelous mechanical properties in both longitudinal and transverse directions.In the first research procured on the GOP reinforced (GOPR) nanocomposites [36].investigated mechanical behavior of GOPR multilayer nanocomposite beams by first-order shear deformation theory and Hamilton’s principle.They considered for the influences of various distribution types of GOPs in their analysis.It must be considered that even though the authors made a giant scientific research.no analysis can be found dealing with the buckling responses of nanocomposite shells reinforced with GOPs.In order to cover this problem.the authors are interested to inform that the first-order shear deformation shell theory is incorporated with the static form of the principle of virtual work in order to find the critical buckling load of nanocomposite shells.The cylinder is supposed to be rested on a two-parameter elastic medium containing both linear and shear layers.The motion equations are solved via the well-known Galerkin’s method for the simplysupported (S-S) boundary condition (BC).

2.Theory and formulation

2.1.Homogenization of GOP reinforced nanocomposite

In this section.the equivalent material properties of the GOP-reinforced polymeric nanocomposite will be enriched using the Halpin-Tsai micromechanical method.It is worth mentioning that there exists some multiscale modeling methods which can underestimate the mechanical properties of the polymeric materials without employment of the micromechanical methods[37-41].Also.in some of the modeling studies about the nanocomposite materials.the effects of the interphase and bonding condition are included[42,43].As can be seen in Fig.1,the structure is consisted of an initial polymer matrix that is strengthened via a group of GOP fibers.The volume fraction of the nanofillers in the media can be calculated as follows:

where GOP and M subscripts are related to GOP reinforcements and the matrix,respectively.In addition,ρ stands for mass density and WGOPdenotes GOP weight fraction.Afterwards.it is necessary to earn the effective Young’s modulus and the effective Poisson’s ratio of the nanocomposite.Herein.the Halpin - Tsai homogenization technique is extended for derivation of the material properties[36].Now.the Young’s modulus can be written as:

where Eland Etaccount for longitudinal and transverse Young’s modulus of the composite.respectively.These elastic parameters can be calculated as [36]:

where

in which EGOPand EMstand for GOPs and matrix Young modulus,respectively.Also,the geometry factors(ξl， ξt)can be computed in the following form [36]:

in which dGOPand hGOPare related to diameter and thickness ofGOPs.respectively.Now.the effective Poisson’s ratio of the composite can be achieved by using the rule of mixture in the following form:

Fig.1.Geometry and coordinate system of a cylindrical shell.

where VGOPand VMcorrespond with the volume fractions of GOPs and matrix.respectively.It is worth mentioning that the effective mass density can be computed in the same form as Poisson’s ratio is achieved in Eq.(6).The volume fractions are related to each other as:

2.2.First-order shear deformable shell theory

The kinematic relations of the nanocomposite shell are going to be derived in this section.Herein,the First-order shear deformation theory of the shells is used in order to reach the motion equations of the cylinder.However.the classical theory of the shells can be employed in the cases of investigation thin-walled shell-type elements which can be resulted in reducing the computational costs[44,45].The geometry and coordinate system of the structure are shown in Fig.1.Now.the displacement fields of a shell can be expressed as follows based on the first-order shear deformable shell theory:

in which u.v and w are axial.circumferential and lateral displacements.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:

2.3.Derivation of motion equations

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

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

Finally.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.Furthermore.kwand kpare Winkler and Pasternak coefficients of the elastic substrate.respectively.In the present problem.the applied buckling load can be inserted in the formulation by replacing the axial loading with Nbthat is the buckling load.Now.once Eqs.(12) and (11) are inserted in Eq.(10).the motion equations of cylindrical shells can be written as:

where

in which κsis shear correction factor.

2.4.Constitutive equations

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

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:

2.5.Governing equations

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

2.6.Solution procedure

In this part,Galerkin’s method is utilized in order to achieve the buckling load 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 BCs.The preliminary assumptions for S-S BC is:

Now,the buckling load of the shell can be obtained once Eq.(27)is substituted in Eqs.(22)-(26).Indeed.the following eigenvalue problem should be solved:

where Δ is a column vector including unknown coefficients.Also,K stands for stiffness matrix.

The corresponding arrays of stiffness matrix can be found looking for the Appendix at the end.Here.Xmfunctions corresponding with S-S edge condition can be assumed to be as:

3.Numerical results and discussion

In this section.some numerical results are given to investigate effect of various parameters on the buckling behavior of GOPR nanocomposite shells.Fundamentally.the shell is supposed to be made of epoxy and it is reinforced with GOP fibers.The material properties of the constituent materials are adopted as same as those presented by Ref.[36].Besides.the shell’s thickness is assumed to be h=5 cm.In the following examples,the effects of a group of parameters on the stability behaviors of the continuous system will be studied; however.the sensitivity analysis (SA) will not be carried out here.Readers are advised to refer to the complementary references for the goal of being familiar with this type of analysis [38,39].Also.accuracy of this formulation validated by previous papers dealing with the buckling analysis of shells (see Table 1).The difference between the results of our model with those reported by Wang et al.[46]originates from the fact that they implemented FE analysis;whereas,in this paper,the buckling loads were extracted using an analytical method.Therefore.it is natural to see stiffer responses once the FEM is utilized.The reliable agreement between the results of this work with those reported in the open literature can guarantee the accuracy of the presented methodology.Herein.the following dimensionless parameters are defined for the sake of simplicity:

Fig.2 illustrates the effect of using various weight fractions on the variation of buckling loads of GOPR shells with respect to constant ratio of L/h and R/h at L/h=R/h=20.It is observable that a GOPR shell with lower amount of weight fractions has lower critical buckling loads compared with GOPR shells with higher weight fractions.So.having less weight fraction percentage reduces the buckling loads of a GOPR shell.Thus.it is better to increase the weight fraction of GOPs while the nanocomposite shell is going to be subjected to great buckling loads.

Influences of Winkler (Kw) and Pasternak (Kp) foundation parameters on the buckling loads of GOPR nanocomposite shells are highlighted in Fig.3 for different weigh fractions of GOPs.It is found that presence of elastic medium has a significant effect on the buckling behavior of GOPR shells.In fact,elastic medium makes the structure stiffer and due to this fact the buckling loads increase at any circumferential wave number.Besides that.buckling loads of GOPR nanocomposite shell depend on the value of circumferential wave number.It is observed that increasing the value of circumferential leads to reduction in dimensionless buckling loads of GOPR nanocomposite shell at first and then the buckling load start to increase continuously in each magnitude of Winkler and Pasternak coefficients regardless to value of weight fraction.As shown in Fig.2,again one should pay attention that higher stability limits are reachable by adding the weight fraction of GOPs.

Moreover.Figs.4 and 5 demonstrate the effect of different radius-to-thickness and length-to-thickness ratios.respectively as well as the effect of GOPs’weight fraction.It is observed that there is no remarkable difference in the buckling behaviors of nanocomposite shells in lower values of circumferential wave numbers.However,once a high value is assigned to the circumferential wavenumber.lower radius-to-thickness ratio and higher length-tothickness at constant L/h and R/h ratios.respectively.increase the stiffness of the nanocomposite and lead to higher buckling loads.Also,as same as Fig.2 it is obvious that regardless of the impact of the circumferential wave number,higher values of weight fraction lead to higher buckling loads in constant values of either Length-tothickness or radius-to-thickness ratios.

Fig.2.Variation of dimensionless buckling load of S-S nanocomposite shells against circumferential wave number for various weight fractions of GOPs (L/h = R/h = 20).

Table 1 Comparison of the critical buckling load of cylindrical shells.

Fig.3.Variation of the dimensionless buckling load of S-S nanocomposite shells versus circumferential wave number for various foundation parameters at(a)WGOP=1%and(b)WGOP = 2% (L/h = R/h = 20).

Fig.6 is depicted for the purpose of putting emphasize on the influences of elastic foundation’s coefficients on the dimensionless buckling loads of GOPR nanocomposite shells.It is clear that with the increase of Winkler and Pasternak parameters.the critical buckling load increases linearly.due to the enhancement in stiffness of the GOPR nanocomposite.Furthermore.it is obvious that Pasternak foundation possesses a more powerful effect in the enlargement of buckling loads rather than Winkler foundation.

Fig.4.Variation of the dimensionless buckling load of S-S nanocomposite shells versus circumferential wave number for different radius-to-thickness ratios and weight fractions of GOPs (L/h = 25).

Fig.5.Variation of the dimensionless buckling load of S-S nanocomposite shells versus circumferential wave number for different length-to-thickness ratios and weight fractions of GOPs (R/h = 25).

4.Conclusion

In this paper.first-order shear deformation shell theory was employed to investigate buckling behavior of a GOPR nanocomposite shell.Material homogenization was done by Halpin-Tsai method.The governing differential equations were solved by implementing Galerkin’s method and the obtained solutions are in excellent agreement with those derived from earlier works.Fromthe analytical results.it was found that critical buckling load of GOPR nanocomposite shells increases significantly with enlargement of the value of GOPs’ weight fraction regardless of other parameters.It was also observed that in larger values of circumferential wave numbers,lower radius-to-thickness ratio and higher length-to-thickness at constant L/h and R/h ratios.respectively,leads to higher critical buckling loads.Also,it was seen that both Winkler and Pasternak coefficients have enough potential to increase the critical buckling loads but Pasternak coefficient has much more impact on the buckling loads rather than Winkler’s one.

Fig.6.Variation of dimensionless buckling load of S-S nanocomposite shells versus foundation coefficients for various weight fractions of GOPs whenever(a)Winkler coefficient is varied and (b) Pasternak coefficient is varied (L/h = R/h = 20).