On vibration analysis of functionally graded carbon nanotube reinforced magneto-electro-elastic plates with different electro-magnetic conditions using higher order finite element methods

M.Vinys .D.Hrursmpth .S.C.Kttimni

a Nonlinear Multifunctional Composites Analysis and Design (NMCAD) Laboratory.Department of Aerospace Engineering.Indian Institute of Science,Bangalore.560012.India

b Department of Mechanical Engineering.National Institute of Technology Karnataka.Surathkal.575025.India

Keywords: Carbon nanotube Magneto-electro-elastic Higher order shear deformation theory Coupled frequency Electro-magnetic conditions

ABSTRACT This article deals with evaluating the frequency response of functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plates subjected to open and closed electro-magnetic circuit conditions.In this regard finite element formulation has been derived.The plate kinematics adjudged via higher order shear deformation theory (HSDT) is considered for evaluation.The equations of motion are obtained with the help of Hamilton’s principle and solved using condensation technique.It is found that the convergence and accuracy of the present FE formulation is very good to address the vibration problem of FG-CNTMEE plate.For the first time.frequency response analysis of FG-CNTMEE plates considering the effect of various circuit conditions associated with parameters such as CNT distributions.volume fraction.skew angle.aspect ratio.length-to-thickness ratio and coupling fields has been carried out.The results of this article can serve as benchmark for future development and analysis of smart structures.

1.Introduction

In the recent years,many researchers have revealed that carbon nanotubes (CNTs) exhibit pronounced mechanical properties as opposed to continuous carbon fibers.Also.CNTs when embedded in polymer matrix provides exceptional elastic modulus and strength which has made them to evolve as a potential candidate for various engineering applications.On the other hand.the concept of functionally graded materials(FGMs)has revolutionised the engineering domain with its vast versatility.More importantly,it effectively encounters the problems witnessed in sandwich structures such as stress concentration,delamination,matrix cracks etc and enhances the structural performance.The tremendous impact made by CNTs and FGMs individually has motivated many pioneers to come up with functionally graded carbon nanotubereinforced composites (FG-CNTRC) and evaluate its mechanical and structural performance.

The nonlinear vibration response of FG-CNTRC plates was assessed by Wang et al.[1,2]and Kaci et al.[3].Liew et al.[4]evaluated both the linear and nonlinear behaviour of FG-CNTRC structures under influence of thermo-mechanical environment.Similarly,the influence of thermal loads on the vibration behaviour of FG-CNTRC plates was investigated by George et al.[5]with the aid of FE methods.Selim et al.[6]and Lei et al.[7,8]explored the element-free kp-Ritz method to understand the free vibration behaviour of FG-CNTRC plates.Through layer-wise approach.the structural response of FG-CNTRC was probed by Kumar et al.[9].The frequency response of FG-CNTRC plates was examined by Phung-Van et al.[10]under the framework of isogeometric analysis.Zhu et al.[11]examined the frequencies of FG-CNTRC plates employing first order shear deformation theory (FSDT) in conjunction with finite element (FE) methods.Duc et al.[12]investigated the influence of elastic foundations on the free vibrations of FG-CNTRC plates using FSDT.Analogously.using FSDT as kinematic model,Malekzadeh et al.[13]and Kiani et al.[14]carried out free vibration analysis through differential quadrature method(DQM) and Ritz method.respectively.Using similar approach of Ritz method,the influence of skew angles[14]and cut-outs[15,16]on vibrations of FG-CNTRC plates was demonstrated by Kiani and co researchers.The dynamic characteristics of CNTRC beams was assessed by Lin et al.[17]and the results were compared between FSDT and higher order shear deformation theory (HSDT).The vibration analysis of FG-CNTRC plates with different shapes was performed by Zhang et al.[18,19]through element free IMLS Ritz method and by Fantuzzi et al.[20]through non-uniform rational B-splines (NURBS) curves using generalized differential quadrature(GDQ) method.It is known that in order to properly define the displacement fields for thick plates.HSDT is more opt than FSDT.Additionally.HSDT enhances the computational efficiency and eliminates the requirement of shear correction factor [21-28].Consequently,the governing equations based on the HSDT emerged to analyze FG-CNTRC structures.Natarajan et al.[29]investigated the natural frequencies of sandwich plates embedded with FGCNTRC facesheets.Mehar et al.[30]developed a FE model to assess the vibration behaviour of FG-CNTRC plates on the basis of HSDT.

The developments in the field of smart materials also have made it feasible to unite the benefits of multifunctionality with structural robustness and adaptability.One such category is magneto-electroelastic(MEE)smart structures where energy interactions between magnetic,electric and elastic phases can be witnessed.This has led to the replacement of many conventional materials in applications such as nano-electromechanical systems (NEMS).nano-probes,atomic force microscope (AFM).nano-actuators and nano sensors.Therefore.it is very crucial to assess the associated electromagnetism phenomenon either experimentally or with the aid of sophisticated computational techniques [31-35].Many literatures have focused on unveiling the various coupled characteristics of functionally graded magneto-electro-elastic (FGMEE) structures.Few are summarised here for the benefit of the readers.Pan and Han[36]examined the free vibrations of exponentially graded MEE plates.Ramirez et al.[37]assessed the frequency response of FGMEE plates.Considering plane stress state.the structural response of FGMEE beams was analytically studied by Huang et al.[38].Through finite element(FE)procedure,Bhangale and Ganesan[39]investigated the free vibration behaviour of FGMEE plates.Milazzo [40,41]developed two-dimensional refined equivalent single layer models and studied the dynamic response of FGMEE plate in detail.The effect of internal crack on the dynamic performance of FGMEE structures was probed by Feng and Su[42].Vinyas et al.[43-46]derived a FE formulation and demonstrated the vibration control abilities of various kinds of MEE skew plates treated with active constrained layer damping.Also,through the sensitivity analysis [47]the performance optimization of MEE structures can also be achieved effectively.Wu et al.[48]demonstrated the wave propagation characteristics in FGMEE plates.Wu and Tsai [49]proposed modified Pagano method to assess the dynamic behaviour of FGMEE plate subjected to closed-circuit condition.Hou and Leung [50]addressed the transient response problem of MEE hollow cylinders through orthogonal expansion technique.Bhangale and Ganesan [51]analysed the frequency response of FGMEE cylindrical shells.Among the static analysis,Wu and Tsai[52]in their work adopted an asymptotic approach and evaluated the influence of various forms of loading on the static response of doubly curved FGMEE shells.Li et al.[53]investigated the effect of uniform load on FGMEE circular plate.Sladek et al.[54]evaluated the bending response of FGMEE circular plate by employing a meshless method.Vinyas and co-researchers designed stepped functionally graded magneto-electro-elastic (SFG-MEE) structures and their static behaviour was analysed under different working environments[55-61].Very recently.Mohammadimehr et al.[62]reported on free vibration of CNTMEE cylindrical shell using FSDT.Also,with the recent development in technology.the machine learning techniques can also be adopted for accurate and effective analysis of MEE structures in the near future[63-65].

The extensive literature review suggests that the structural assessment of CNTMEE structures is still an over-looked area which requires more attention.Alongside.the opportunities to collectively exploit the benefits of FG-CNTs.MEE material.FE methods and kinematics accuracy of HSDT have motivated the authors to take up this current work.The coupled response of the MEE smart structure is significantly influenced by the applied magnetic and electric constraints.Hence this works makes the first attempt to study the influence of open and closed circuit electromagnetic boundary conditions associated with different parameters.on the frequency response of FG-CNTMEE plates under the framework of FE methods.The C1continuity requirement of HSDT is fulfilled by considering second order derivative terms as higher order rotational degrees of freedom.Therefore,its computational efficiency is found to be better than the numerical software in terms of computational time and costs.The equations of motion are developed by using Hamilton’s principle and HSDT which are solved by incorporating condensation technique.A detailed evaluation of various parameters such as CNT distribution,CNT volume fraction,piezoelectric matrix material.skew angle.coupling fields.aspect ratio (a/h).the length-to-width ratio(a/b) has been carried out.

2.Materials and methods

The present work considers MEE composite plate made of carbon nanotubes (CNTs) reinforced in piezoelectric matrix.Here,CNTs act as piezomagnetic fibers.Apart from uniformly distributed(UD) CNTs.this work also considers various functionally graded CNT distribution patterns such as FG-X.FG-O and FG-V whose graphical and mathematical equations are illustrated in Table 1.Through the rule of mixture,the effective properties of FG-CNTMEE material are estimated as follows [15]:

Table 1 CNT distribution with corresponding mathematical expression.

Where.E.G.ρ and υ denote the elastic modulus.shear modulus,density and Poisson’s ratio respectively.The superscripts CNT and m represents corresponding values for CNT fiber and piezoelectric matrix.respectively.

VCNTand Vmrepresent CNT and matrix volume fractions,respectively.Meanwhile.η1，η2and η3are CNT/matrix efficiency parameters which are determined by molecular dynamics simulation.

The coupled constitutive equation of FG-CNTMEE material can be represented as follows [23]:

where.[C].[e].[q].[m].[η]and [μ]are the elastic stiffness.piezoelectric,magnetostrictive,electromagnetic,dielectric and magnetic permeability coefficient matrices.respectively.Also.｛σ｝.｛D｝ and｛B｝.represent the stress tensor.electric displacement and the magnetic flux.respectively; ｛ε｝.｛E｝ and ｛H｝ are the linear strain tensor.electric field and magnetic field.respectively.

The different FG-CNTMEE material properties can be estimated using the mixture rule as follows [23]:

The various matrices appearing in Eq.(2) can be subdivided based on bending and shear material constants for the sake of simplicity [25].

3.Higher order shear deformation theory

The kinematic model of the FG-CNTMEE plate is assumed to follow higher-order shear deformation theory(HSDT)according to which the displacement fields can be represented as:

where,u0,v0,and w0are the mid-plane displacements along the x,y.and z-axes.respectively at z = 0.The slopes of xz plane and yz plane are denoted by θxand θy.respectively.The bending strains｛εb｝ and shear strains ｛εs｝ can be shown as:

4.Finite element formulation

The FE model is developed through eight noded isoparametric element.The degrees of freedom(DOFs)associated with each node are bifurcated into displacement quantities translationalrotationalhigher order rotationalelectric potential φ and magnetic potential ψ.The DOFs corresponding to the ith node can be expressed through shape function matrices as follows:

in which

Through the FE parameters mentioned in Eq.(7) - Eq.(9).the strains represented in Eq.(6) can be expressed as

5.Equation of motion

The Hamilton’s principle as adopted for FG-CNTMEE plate can be expressed as follows [22]:

where.Tp.Tkare the potential and kinetic energies.respectively.Further,Ω denotes the volume of the plate.On making use of Eq.(2)- Eq.(11).Eq.(12) can be re-written as follows:

The present study assumes that the free charge and current densities are absent and electromagnetic fields exhibit quasi-static behaviour.Further.using the Maxwell’sequations.the electric intensity(E)and the magnetic intensity(H)are related to the electric potential (φ) and magnetic potential(ψ).respectively as [22]:

Table 2 The material properties of CNT fiber and piezoelectric matrix used for CNTMEE plate [23,62].

Fig.1.Schematic representation of FG-CNTMEE plate.

The transformation of Eq.(14) to accommodate skew angle variations is illustrated in Appendix.

6.Problem statement

In this study,the frequency characteristics of FG-CNTMEE plates are evaluated through FE formulation.The material properties of FG-CNTMEE depicted in Table 2 are considered in this work.According to Fig.1(a).a， b and h represents the length.width and thickness of the FG-CNTMEE plate.Further.λ is the skew angle of the plate and x’-,y’-and z’-axes are the transformation axes.In this study,a=b=0.1 m;h=0.03 m are considered for evaluation.The various CNT distribution patterns and its mathematical expressions are shown in Table 1.Further,the open and closed electro-magnetic boundary conditions are enforced on the FG-CNTMEE plates,which can be represented as follows:

Table 3 Convergence of the natural frequency of MEE plate with mesh size.

Analogously,the mechanical boundary constraints employed in this work can be explicitly represented as

Table 4 Convergence and comparison of the fundamental frequency of MEE plates(a = b = 1 m; h = 0.3 m;λ = 15°).

Table 5 Validation of the FE formulation for skew MEE plates(BFB stacking sequence.SSSS boundary condition).

7.Results and discussion

In this section the coupled structural response of FG-CNTMEE plates subjected to various electromagnetic (EM) boundary conditions are studied via FE formulation depicted in earlier section.Initially.to verify the credibility of the proposed formulation to incorporate coupling fields.geometrical skewness and FG-CNTs,the results of present study are compared with previously published literature [15,66,67].The convergence study depicted in Table 3 and Table 4 suggest that for a mesh size of 10 × 10 the results converge with that of Moita et al.[66].In addition.the verification results illustrated in Table 5 -and 6 affirm that there exist a good correlation between the results and hence the proposed FE formulation is efficient enough to accommodate geometrical skewness and FG-CNTs.Therefore.it is justified that the proposed formulation can be extended to examine the influence of EM boundary conditions on the frequencies of FG-CNTMEE plate.Also.parametric study has been performed to assess theeffect of CNT distributions.volume fraction.skew angle.aspect ratio.length-to-thickness ratio and coupling fields on frequency response of FG-CNTMEE plates with different EM conditions.

Table 6 Validation for skew FG-CNT plates(λ=30ο;V*CNT=0.12;CCCC boundary condition).

7.1.Effect of open and closed circuit conditions

The variation in the first three natural frequencies of FGCNTMEE plate with UD distribution and V*CNT=0.12 subjected to different EM conditions is shown in Table 7.Two variants of FGCNTMEE plates with all sides clamped (CCCC) and all sides simply supported (SSSS) are considered for evaluation.It was found that for both the cases of constraints a better frequency response was observed for open circuit EM condition in contrast to the closed circuit EM condition.This may be attributed to the fact that open circuit EM condition exhibit an enhanced energy release capability which adds up to the stiffness of the plate.Also,it is worth noticing that the EM conditions have a pronounced effect on the higher modes.

7.2.Effect of piezoelectric matrix material

Table 8 encapsulates the effect of piezoelectric matrix materials viz.PVDF and BaTiO3on the fundamental natural frequency of FGCNTMEE plate.As seen from the table.a predominant influence of PVDF over BaTiO3is noticed due to higher piezoelectric coefficients.In addition,the significant influence of EM condition prevails on the FG-CNTMEE plate with PVDF as the matrix as opposed to BaTiO3

Table 7 Effect of electromagnetic boundary conditions on first three natural frequencies ( × 103 rad·s-1) of CNTMEE plate.

Table 8 Effect of electromagnetic boundary conditions and piezoelectric matrix material on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate(SSSS).

7.3.Effect of CNT distribution and volume fraction

The influence of different CNT distribution patterns and volume fraction on the frequency response of FG-CNTMEE plates with PVDF and BaTiO3as matrix is illustrated in Fig.2(a) - Fig.2(d).It can be inferred from the figure that among the various CNT distributions selected.FG-X pattern yields a higher frequency.This may be attributed to the higher flexural stiffness exhibited by FG-X distribution.In addition.a predominant influence of EM conditions can be noticed in the order of FG-X ＞ UD ＞ FG-V ＞ FG-O.Thus.byadjusting the distribution of CNTs along the thickness direction,the desired stiffness of the plates can be achieved and noted that reinforcement distributed close to the top and bottom of plates are more efficient than those distributed nearby the mid-plane for increasing the stiffness of plates.

Fig.2.Effect of volume fraction and CNT distributions on FG-CNTMEE plates.

Fig.3.Effect of skew angle on FG-CNTMEE plates with different CNT volume fractions.

Meanwhile.it is also witnessed that for the CNTs distribution pattern considered.the volume fraction of CNTs has a greater influence on the frequency response.A higher volume fraction of CNTs enhances the frequency of the plate as the stiffness is improved.A minute variation in the volume of CNTs results in drastic increase in the natural frequency.As expected.the results suggest that the discrepancies between the open circuit and closed circuit natural frequencies improve with higher volume fraction as noticed from Table 8.

7.4.Effect of skew angle (λ)

Fig.3(a) - Fig.3(f).illustrate the effect of EM conditions associated with the different skew angles of FG-CNTMEE plates.Based on the results of these figures.it is worth to noticing that with an improvement in the skew angle.the fundamental natural frequency improves.Further.it can also be witnessed that at lowerskew angle,the influence of EM conditions are negligible.However,as the skew angles improve.its effect also becomes proportionally significant.This may be due to the fact that at higher skew angles the stiffness and the energy release capabilities of the plate enhances due to minimum area.In addition,a predominant influence of FG-X distribution it observed here as well.

Fig.4.Effect of a/h ratio associated with EM conditions on the fundamental natural frequency of FG-CNTMEE plates with different CNT volume fractions (SSSS condition).

7.5.Effect of aspect (a/h) ratio

In order to evaluate the effect of a/h ratios in conjunction with EM conditions,numerical analysis has been performed considering FG-CNTMEE plate with various CNT volume fractions and results are highlighted in Fig.4 and Fig.5.respectively for SSSS and CCCC conditions.It can be seen from these figures that irrespective of the CNT distribution patterns.lower aspect ratio has a pronounced effect on the frequency.This is due to higher degree of coupling displayed by thick FG-CNTMEE plates.In addition,the influence of EM conditions on the frequency diminishes as the plate moves from thick to thin category.In other words.predominant effect of EM conditions is witnessed for FG-CNTMEE plates with less a/h ratios rather than higher a/h ratios.

7.6.Effect of a/b ratio

The results related to frequency of FG-CNTMEE plate with different a/b ratio is summarised in Fig.6(a) - Fig.6(d).It can be clearly seen from these figures that for all the CNT distribution patterns considered.a higher value of a/b ratio results in dramatically reduced frequency values.This may be due to the nearing of beam behaviour with higher a/b ratio.Additionally,the influence of EM conditions are negligible at higher a/b ratio which gradually improves and magnifies at the lower a/b ratio.A similar conclusion drawn with reference to a/h ratio can be witnessed here as well.

7.7.Effect of boundary conditions

The influence of EM circuit conditions with different essential mechanical boundary conditions is evaluated in Table 9.As expected,the significant effect of open circuit conditions over closed circuit improves with more number of clamped edges owing to higher energy release from open circuit condition and higher rigidity of the plate with more number of clamped edges.

7.8.Effect of coupling fields

The coupled natural frequency of FG-CNTMEE plates is the summation of contribution from magnetic.electric and elastic fields.It becomes very much crucial to evaluate the effects of different fields.Tables 10-13 show the influence of coupling fields associated with the EM circuit conditions on the natural frequencies of FG-CNTMEE plate with various skew angles.It can be witnessed from this analysis that as opposed to closed circuit,opencircuit condition has a predominant effect on the percentage difference between the natural frequencies obtained from complete coupling and elastic field alone.Further.as depicted from Fig.7(a)and Fig.7(b).the percentage difference between completely coupled and elastic frequencies magnifies with higher skew angle.Also.FG-CNTMEE plate with FG-X distribution subjected to open circuit condition exhibit a significant effect.

Fig.5.Effect of a/h ratio associated with EM conditions on the fundamental natural frequency of FG-CNTMEE plates with different CNT volume fractions (CCCC condition).

8.Conclusions

In this article.the coupled frequency response of FG-CNTMEE plates has been evaluated considering the effect of open and closed circuit electro-magnetic boundary conditions.To this end,higher order finite element formulation has been developed and equations of motion are arrived via Hamilton’s principle.The C1continuity requirement of HSDT is fulfilled by considering second order derivative terms as higher order rotational degrees of freedom.Therefore.its computational efficiency is found to be better than the numerical software in terms of computational time and costs.

The following conclusions can be drawn out of the numerical evaluation.

· In contrast to closed circuit condition a beneficial effect of open circuit condition on the natural frequencies is witnessed.The maximum percentage difference open and closed circuit natural frequencies of CNTMEE plate is about 2.43%and 1.89%,for CCCC and SSSS conditions.respectively

· Alongside.FG-X CNT distribution exhibit more coupling frequency when subjected to open circuit condition,due to higher flexural rigidity offered.For V*CNT= 0.28 and V*CNT= 0.12.the percentage difference between the open and closed circuit fundamental natural frequencies is found to be maximum and minimum with 3.67% and 1.93%.respectively.

· At higher skew angle,CNT volume fraction and length-to-width ratio the discrepancies between open and closed circuit frequencies magnifies.Meanwhile.a reverse trend is noticed for aspect ratio.

· The coupling fields exhibit a higher frequency than elastic field alone and this difference improves with volume fraction of CNT and skew angle.For a higher skew angle of 60°.Open circuit condition yields an improvement in coupled frequency by 3.65%,where as the closed circuit condition results in 3.51%enhancement.

It is believed that the proposed formulation and the results obtained may serve as benchmark solutions for future design and analysis of smart structures.However.the limitation of this formulation is that it does not consider the influence of external environment or loadings on the frequency analysis,which may be a novel research gap for the future evaluation.

Fig.6.Effect of a/b ratio associated with EM conditions on the fundamental natural frequency of FG-CNTMEE plates with aspect ratio and CNT distributions.

Table 9 Effect of mechanical boundary conditions on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate with different electromagnetic boundary conditions.

Table 10 Effect of coupling fields on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate with different electromagnetic boundary conditions and skew angles (FG-X distribution; BaTiO3 material).

Table 11 Effect of coupling fields on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate with different electromagnetic boundary conditions and skew angles (UD distribution; BaTiO3 material).

Table 12 Effect of coupling fields on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate with different electromagnetic boundary conditions and skew angles (FG-V distribution; BaTiO3 material).

Table 13 Effect of coupling fields on fundamental natural frequency ( × 103 rad·s-1) of CNTMEE plate with different electromagnetic boundary conditions and skew angles (FG-O distribution; BaTiO3 material).

Fig.7.Effect of skew angles,EM conditions and FG-CNT distributions on%difference in fundamental natural frequency of FG-CNTMEE plate with different coupling fields(PVDF piezoelectric matrix).

Acknowledgements

The first author acknowledges the support of Indian Institute of Science.Bangalore.through C.V.Raman Post-doctoral fellowship,under Institution of Eminence scheme.

Appendix

The derivative of shape function matrices appearing in Eq.(10)can be represented by

The explicit representation of the stiffness matrices appearing in Eq.(14) can be shown as follows:

where,

The various rigidity matrices contributing to Eq.(A-3) can be denoted as follows:

Skew transformation

To incorporate skewed edges in the FE formulation and modify the stiffness matrices of the elements lying along the skewed edges transformation of the Cartesian co-ordinate axes along x’-.y’- and z’-directions is performed.The transformed DOFs pertaining to displacements can be shown as follows:

The transformed stiffness and mass matrices can be represented as follows:

where [T1]and [T2]are banded transformation matrices with respect to skew angle λ and are given as:

Finally,Eq.(14)can be condensed to a more generalized form as follows:

where.[Keq]is the equivalent stiffness matrix.

Declaration of interest

None.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.