Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect

Maryam Lori Dehsaraji.Mohammad Arefi.Abbas Loghman

Department of Solid Mechanics.Faculty of Mechanical Engineering.University of Kashan.Kashan.Iran

Keywords: Thickness stretching effect Shear and normal deformation theory Vibration analysis Length scale parameter Modified couple stress theory

ABSTRACT Higher-order shear and normal deformation theory is used in this paper to account thickness stretching effect for free vibration analysis of the cylindrical micro/nano shell subjected to an applied voltage and uniform temperature rising.Size dependency is included in governing equations based on the modified couple stress theory.Hamilton’s principle is used to derive governing equations of the cylindrical micro/nano shell.Solution procedure is developed using Navier technique for simply-supported boundary conditions.The numerical results are presented to investigate the effect of significant parameters such as some dimensionless geometric parameters,material properties,applied voltages and temperature rising on the free vibration responses.

1.Introduction

The responses of materials and structures subjected to various types of loadings have been defined by constitutive relations.These relations are presented in dimensionless forms.The behaviors of structures in small scales are defined using some nonclassical theories that account size dependency [1,2].Nano structures are extensively used in various situations as nano generator.nano resonator or chemical nano sensor [3-6].There are some important works on the experimental analysis and atomistic simulations of nano-structures [7,8]to measure size effects of small scale structures.To overcome the incompleteness of the classical continuum theory in precise prediction of small scale structures,some size-dependent theories have been developed to predict the size effects.These size-dependent theories are included the strain gradient theory (SGT).modified couple stress theory (MCST),micro-polar theory.and nonlocal elasticity theory.There are some small scale parameters in the aformentioned theories for more accurate analysis of size-dependent analysis [9-11].

Yang et al.[12]used the modified couple stress theory (MCST)including only one small scale parameter.Based on the modified couple stress theory.the strain energy density function is depending on the symmetric part of the curvature tensor that leads to more easier formulation than the classical couple stress theory.There are some important works on the application of modified couple stress theory for analysis of micro or nano-sized structures such as micro actuators,carbon nanotubes,micro cantilevers,micro beams and microfilms[13-17].

Lim and He[18]studied the size-dependent nonlinear response of thin elastic films with nano-scale thickness based on a continuum approach and by applying Kirchhoff’s hypotheses.Park and Gao [19]investigated bending of a Euler-Bernoulli beam by using modified couple stress theory.Lu et al.[20]used a general thin plate-like theory with considering surface effects for analysis of thin film structures.Kong et al.[21]analyzed dynamic behavior of Euler-Bernoulli beams with simply supported boundary conditions based on the modified couple stress theory.Ma et al.[22]studied vibration and bending analysis of beams according to Timoshenko beam model.

Lam et al.[23]used the modified strain gradient elasticity theory for analysis of small scale structures as reported inReferences[24-30].Xia et al.[31]studied nonlinear static bending.free vibration and postbuckling analysis of micro-beams based on theMCST by using the Euler-Bernoulli beam theory.Xia and Wang[32]studied vibration and stability analysis of micro pipes transmitter fluid using Timoshenko beam model.Free vibration of singlelayered graphene sheet resting on Pasternak foundation with simply supported boundary condition based on the modified couple stress theory was studied by Akg¨oz and Civalek [33].Jomehzadeh et al.[34]investigated vibration analysis of microplates based on modified couple stress theory.The free vibration responses were presented in terms of different length scale parameters.different dimensionless aspect ratios and various boundary conditions.

Analysis of the nano and micro shells has attracted various researchers due to various applications in technical applications[35-38].Tadi Beni et al.[39]derived equations of motion of functionally graded cylindrical shell based on modified couple stress theory and first order shear deformation shell theory.Khalili et al.[40]studied dynamic behavior of functionally graded (FG) beams under moving loads by using Euler-Bernoulli beam theory to investigate the effect of power index parameter and moving load on the dynamic behavior.Li et al.[41]studied bending of simplysupported micro plate under external load based on the strain gradient elasticity theory.Ashoori et al.[42]obtained governing equations of the Kirchhoff micro plate model by using the modified strain gradient elasticity theory.The accuracy and trueness of the present theory was justified using comparison with previous works.Dynamic behavior of nanotubes based on nonlocal theory and using the three dimensional theory of elasticity was studied by Alibeigloo et al.[43]to study the effect of small scale parameter on the natural frequency.Zeighampour and tadibeni [44]used modified couple stress theory for dynamic analysis of transporter fluid carbon nanotube.The obtained results have been presented in terms of small scale parameter and fluid velocity based on the classical theory and the modified couple stress theory.Dynamic stability of FGM micro cylindrical shells based on MCST theory was studied by Sahmani et al.[45].They showed that by decreasing the size parameter.the domain of the instability district of system increases.

The piezoelectric cylindrical shells because their direct and reverse piezoelectric effects.are extensively used as useful structures in the design of spatial transporters[46-51]in various modes such as vibration,buckling and bending behaviors subjected to the electric and thermal loads.Mohammadimehr et al.[52]presented the thermo-electro-mechanical buckling and vibration analysis of a double-bonded nano composite piezoelectric plate reinforced by a boron nitride nanotube by using the Mori-Tanaka method and modified couple stress theory.Mosallaie Barzoki et al.[53]analyzed the thermo-electro-mechanical buckling analysis of a piezoelectric polymeric cylindrical shell reinforced by double-walled boron nitride nanotubes.Liu et al.[54]studied nonlocal thermo-electromechanical vibration analysis of the piezoelectric nanoplate.The vibration responses have been presented in terms of the nonlocal parameter.axial force.temperature change and external electric voltage.Hadi et al.[55]employed couple-stress theory for vibration analysis of Euler-Bernoulli nano-beams made of fgms with various boundary conditions using generalized differential quadrature method.Zeighampour and Tadi Beni [56]used the couple stress theory for vibration analysis of a shear deformable cylindrical shell model.Also free vibration analysis of FGM cylindrical nanoshells was studied by Tadi Beni et al.[57].Zenkour[58]used a new theory including four unknown shear and normal deformations for bending of FG plate.Linear and nonlinear vibration of carbon nanotubes were investigated using perturbation technique and representative volume element method [59,60].Modal analysis of carbon nanotubes reinforced by polymer based on finite element method was presented by Fereidoon et al.[61].Some works presented a computational design methodology for topology optimization of piezoelectric and flexoelectric materials [62,63].Also different methods are developed for analyzing cracks and fractures in structures [64-66].Rabczuk et al.[67]treated fluid and fracturing structure by using a meshfree method under impulsive loads.Nanthakumar et al.[68]solved inverse problem of detecting inclusion interfaces in a piezoelectric structure.Anitescu et al.[69]developed a model for solving the partioal differential equations using rtificial neural networks and an adaptive collocation strategy.Guo et al.[70]proposed a deep collocation method for bending analysis of a thin plate based on Kirchhoff theory.Rabczuk et al.[71]proposed a new novel nonlocal parameter for solution of partial differential equations based on variational principle.

Literature review was completed using summarizing some important works on the analysis of cylindrical micro/nano shells,higher order shear deformation theory and size dependent analysis.Based on the best author’s knowledge.there is no published works on the free vibration analysis of functionally graded cylindrical piezoelectric nano/micro shell with accounting thickness stretching effect.The thickness stretching effect is accounted based on shear and normal deformation theory.Size dependency is included in this analysis based on modified couple stress theory.The governing equations of motion are derived based on Hamilton’s principle.The accuracy and trueness of present numerical results are justified using comparison with existing numerical results of literature.The effect of significant parameters such as small scale parameters,length to radius ratio,radius to thickness ratio,applied voltage and temperature change on the natural frequencies of micro/nano shell.

2.Governing equations of motion

The FGP cylindrical micro/nano shell with length L,radius R and thickness h under the external electric voltage V0and uniform temperature change ΔT.in thickness direction is shown in Fig.1.FGPM is usually made of the combination of two materials with electric properties in which the material properties varies continuously and uniformly from PZT_5H properties at the internal surface of the cylindrical micro/nano shell to the properties of the PZT_4 at the external surface.Volume fractions of PZT_4 and PZT_5H according to power law distribution are expressed as:

In which N is power index.

The displacement field based on higher order shear and normal deformation theory to account thickness stretching is expressed as:

In which.u0(x，θ) and v0(x，θ) are displacements of middle surface.w(x，θ) and φ(x，θ) are the bending and shear components of the radial displacement,and χ(x，θ)is an extra function of x and θ to shows variation of radial displacement along the radial direction.The shape functions appeared in Eq.(2) are suggested as [58]:

Fig.1.The schematic figure of a cylindrical micro/nano shell subjected to applied voltage.

Based on the modified couple stress theory,the strain energy is expressed as [52]:

where σij， εij， Diand mijare the components of the Cauchy stress tensor.strain tensor.electric displacement vector and the higher order stress tensor.In addition.Eiand χijare the components of electric field vector and the symmetrical part of rotation gradient tensor.The constitutive relations are defined as:

In which Cijkl， emij， Di，λij，Piand βijare the stiffness coefficients,piezoelectric coefficients.electric displacement vector.dielectric constants,pyroelectric constants and thermal moduli respectively.Also φ， Em， θ and ΔT are electric potential.electric field vector,rotation vector and temperature change,respectively.Based on the displacement field relation (2).the components of strain are expressed as:

Moreover.the non-classical strain components are obtained as:

The constitutive relationships for piezoelectric cylindrical shell in general state are given by Ref.[53]:

According to Ref.[53],the electric potential as a combination of a cosine and linear variation,that satisfied the Maxwall equation,is assumed as follows [72-74]:

In which.β =h/π ， Q(x，θ) and V0represent variation of the electric potential in the mid-plane and the external electric voltage,respectively.The electric field components are defined as[72-74]:

By substitution of strain components.we obtain the stress and electric displacement components as:

Substitution of strain components into strain energy gives:

In which the resultant components are defined in Appendix A.

The variation of kinetic energy is defined as:

Substitution of components of displacement field from relation (2) into leads to:

The unknown coefficients in Eq.(17)are defined in Appendix B.The work done by external loadings such as thermal and electrical loads are expressed as:

In which(NTx，NTθ)and(NEx，Npθ)are thermal and electrical loads in x and θ directions respectively.Hamilton principle is used to derive governing equations of motion as:

By substituting Eqs.(15).(17) and (18) into Eq.(19).the six governing equations of motion are derived as:

The coefficients a1…a16， b1…b18， C1…C27， d1…d26， E1…E26and f1…f14are expressed in Appendix B.

3.Solution procedure

The thermal and electrical resultant components are defined as[53]:

In this section,the solution of Eq.(20)for the simply supported FGP cylindrical nano/micro shell subjected to temperature changeΔT and external electric voltage V0.is investigated.The unknown functions based on the Navier solution are expressed as:

Table 1 Comparison of dimensionless natural frequency for different thickness to radius ratios and various circumferential wave numbers (m = 1).

Table 2 Comparison of dimensionless natural frequency for different thickness to radius ratios and various circumferential wave numbers.

In which U1mn，V1mn，W1mn，χ1mn，Uφ1mn and Q1mnare amplitudes of unknown functions,and m,n are axial and circumferential wave numbers respectively.Thus.by substituting approximate solutions from Eq.(22) into Eq.(20).the governing equations are obtained as known format [K]6×6｛d｝6×1+ [M]6×6｛d…｝6×1= 0.in which [K]and [M]is stiffness and mass matrix respectively.The harmonic unknown functions are assumed as:｛d｝ =[U1mnV1mnχ1mn W1mnφ1mn Q1mn]Teiωt in which ω is natural frequency.To investigate the shell frequency,the non-trivial solutions are obtained by solving the determinant of the coefficients matrix.

4.Numerical results and discussion

In this section.the free vibration of an FG piezoelectric micro/nano shell with simply supported boundary conditions is investigated based on the modified couple stress theory.The obtained results are validated using comparison with previous valid works.After validation.the numerical results are presented in terms of significant parameters such as dimensionless size parameter l/h,power index N,length-to-radius ratio L/R,temperature change and external electric voltage.

4.1.Validation of results

The present section investigates the verification of present numerical results using comparison with some valid references.The verification is performed with the relevant references (Alibeigloo and Shaban [43].Tadi Beni et al.[57].Zeighampour and Tadi Beni[56]).

The material properties for the present comparison are assumed as follows.

Tables 1 and 2 list comparison of dimensionless natural frequency for different thickness to radius ratios h/R and various circumferential wave numbers n for m = 1.The dimensionless natural frequency is assumed asOne can conclude that the present results with considering thickness stretching effect leads to significant improvements of numerical results respect to previous results without thickness stretching effect.

Table 3 Material properties of the PZT-4 and the PZT-5H.[53].

4.2.Effect of length scale parameter on dimensionless natural frequency

The complete numerical results are presented in this section in terms of significant parameters of the cylindrical micro/nano shell.The material properties of cylindrical micro/nano shell are expressed in Table 3.The dimensionless natural frequency is assumed as:The effect of the dimensionless length scale parameter and the various power index N on natural frequencies according to the classical continuum theory(l=0)and size dependent modified couple stress theory (l = 3 h) are presented in Fig.2.Fig.2 shows that increase in the length scale parameter.leads to increase in dimensionless natural frequency,also the highest natural frequency is related to PZT4 and the lowest is for PZT5H.When the material length scale parameter changes from l/h=0 to l/h=5,the dimensionless natural frequency changes from 0.2539 to 0.3751 for N=0 in shells made from PZT4,and from 0.2360 to 0.34812 for N = ∞in shells made from PZT-5H.

Fig.2.Variation of dimensionless natural frequencies in terms of dimensionless small scale parameter l/h for various gradient index N.

Fig.3.Variation of dimensionless natural frequencies in terms of dimensionless length to radius ratio L/R based on classical and modified couple stress theory for various gradient index N.

4.3.Effect of dimensionless length to radius ratio L/R on dimensionless natural frequencies

Fig.3 presents variation of dimensionless natural frequencies of micro/nano shell in terms of various length to radius ratio L/R based on the size-dependent theory (l = 3h) and classical continuum theory for different materials.The numerical results indicate that the dimensionless natural frequencies are decreased with increase of length to radius ratio L/R.In addition,the natural frequencies for modified couple stress theory is significantly more than the results of classical continuum theory.It is concluded that with length to ratio more than 10,L/R ＞10,the natural frequencies are closed to an assymptotic value.

Fig.4.Variation of dimensionless natural frequencies in terms of axial and circumferential wave numbers based on classical and modified couple stress theory.

Fig.5.Variation of dimensionless natural frequencies in terms of circumferential wave numbers for various length to radius ratio L/R based on classical and modified couple stress theory.

Fig.6.Variation of dimensionless natural frequency in terms of circumferential and axial wave numbers based on modified couple stress theory (l = 3h).

Fig.7.Variation of dimensionless natural frequencies in terms of circumferential wave numbers for various thickness to radius ratio h/R based on classical and modified couple stress theory.

4.4.Effects of axial and circumferential wave numbers on dimensionless natural frequencies

Figs.4-6 show the variation of dimensionless natural frequencies of cylindrical micro/nano shell based on calssical and modified couple stress theories for different circumferential and axial wave numbers n,m.It is observed that increasing in axial and circumferential wave numbers increases the dimensionless natural frequency for both theories.This increasing for modified couple stress theory is more significant than classic theory.

4.5.Effect of thickness-to-radius ratio on dimensionless natural frequency

Fig.7 shows the effect of thickness to radius ratio h/R of micro/nano shell as well as circumferential wave numbers on the dimensionless natural frequency.It is observed that the natural frequencies are increased with increase of thickness to radius ratio h/R for both theories of classic and modified couple stress.In addition,increase in circumferential wave number leads to increase in dimensionless natural frequency.

Fig.8.Variation of dimensionless natural frequency in terms temperature rising and external applied voltage based on modified couple stress theory (l = 3h).

4.6.Effect of uniform temperature changes on dimensionless natural frequency

Fig.8 shows the effect of the uniform temperature changes ΔT on the dimensionless frequency of micro/nano shells based on the modified couple stress theory for different applied voltages.The numerical results indicate that with increasing the temperature rise.stiffness of micro/nano shell decreases that causes the fundamental dimensionless frequency decreases.It is also observed that increasing the applied voltage reduces the natural frequency.

4.7.Effects of external electric voltage on dimensionless natural frequency

The effect of external electric voltage V0on the dimensionless natural frequency of micro/nano shells based on the modified coupling stress theory is illustrated in Fig.9.The numerical results indicate that increasing in electric voltage applied leads to decrease in fundamental dimensionless frequency.It is concluded that increase of external electric voltage in a piezoelectric cylindrical shell leads to decrease of stiffness.

4.8.Effect of thickness stretching

As said in Abstract.the main novelty of the present paper is conclusion on the effect of thickness stretching.To conclude the effect of thickness stretching effect in improvement of numerical results respect to the previous cases.the natural frequencies are depicted with and without thickness stretching effect.Shown in Fig.10 is comparison between natural frequencies of cylindrical micro/nano shell with and without thickness stretching effect in terms of thickness to radius ratio h/R.It is concluded that accounting thickness stretching leads to increase of the natural frequencies rather than the cases that ignore thickness stretching.This improvement becomes very important for modified couple stress theory.

Table 4 lists variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of dimensionless small scale parameter l/h for various power index N.It is observed that the dimensionless natural frequencies are increased with increase ofdimensionless small scale parameter l/h while increase of power index N leads to decrease of natural frequencies.Listed in Table 5 is variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of various temperature rising and external electric voltage.It is concluded that the natural frequencies are decreased with increase of temperature rising and external electric voltage.

Fig.9.Variation of dimensionless natural frequency in terms temperature rising and external applied voltage based on modified couple stress theory (l = 3h).

Fig.10.Comparison between natural frequencies of cylindrical micro/nano shell with and without thickness stretching effect in terms of thickness to radius ratio h/R.

5.Conclusion

Size dependent free vibration analysis of a functionally graded cylindrical micro/nano shell was studied in this paper based on a new higher-order shear and normal deformation theory and modified couple stresstheory.The cylindricalmicro/nano shell wassubjected to thermal and electrical loads.Hamilton’s principle was used to derive governing equations of motion.The analytical solution was proposed for the cylindrical shell with simply-supported boundary conditions.The main numerical results of the present paper are summarized as:

Comparison between numerical results with and without thickness stretching effect show that accounting thicknessstretching effect leads to significant improvement of natural frequencies specially for modified couple stress theory.

Table 4 Variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of dimensionless small scale parameter l/h for various power index N.

Table 5 Variation of dimensionless natural frequencies of cylindrical micro/nano shell in terms of various temperature rising and external electric voltage.

Size dependency was included in constitutive relations based on the modified couple stress theory.The numerical results show that considering size-dependency based on the modified couple stress theory and increase of small scale parameter leads to increase of stiffness of material and increase of natural frequencies.

Investigation on the effect of applied electric voltage and uniform temperature rising indicates that the natural frequencies are decreased with increase of these parameters.

The numerical results have been presented in terms of dimensionless geometric parameters.The numerical results indicate that the natural frequencies are decreased with increase of length to radius ratio L/R and decrease of thickness to radius ratio h/R.

Acknowledgment

The authors would like to thank the Iranian Nanotechnology Development Committee for their financial support.

Appendix A

Appendix B

1.Unknown constants in governing equations