Higher-order electroelastic modelling of piezoelectric cylindrical nanoshell on elastic matrix

Xiao-ping Huang , Peng-fei Hu , Mohammad Arefi c,*

aCollege of Mechanical and Electrical Engineering, Nanning University, Nanning, 530000, Guangxi, China

bSchool of Materials Science and Engineering, Luoyang Institute of Technology, Luoyang, 471023, Henan, China

cDepartment of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

Keywords:Higher-order shear deformation theory Electro-elastic bending Functionally graded materials Size-dependent analysis Nonlocal parameter Cylindrical nano shell

ABSTRACT This paper develops electro-elastic relations of functionally graded cylindrical nanoshell integrated with intelligent layers subjected to multi-physics loads resting on elastic foundation.The piezoelectric layers are actuated with external applied voltage.The nanocore is assumed in-homogeneous in which the material properties are changed continuously and gradually along radial direction.Third-order shear deformation theory is used for the description of kinematic relations and electric potential distribution is assumed as combination of a linear function along thickness direction to show applied voltage and a longitudinal distribution.Electro-elastic size-dependent constitutive relations are developed based on nonlocal elasticity theory and generalized Hooke’s law.The principle of virtual work is used to derive governing equations in terms of four functions along the axial and the radial directions and longitudinal electric potential function.The numerical results including radial and longitudinal displacements are presented in terms of basic input parameters of the integrated cylindrical nanoshell such as initial electric potential, small scale parameter, length to radius ratio and two parameters of foundation.It is concluded that both displacements are increased with an increase in small-scale parameter and a decrease in applied electric potential.

1.Introduction

Small scale structures are frequently used in various situations to cover new aspects of science and engineering.These structures because of their increasingly applications have been investigated comprehensively by various researchers.Researchers and scientists have found that behavior of materials and structures in small scales differs from them in macro scale.To cover effect of sizedependency in constitutive relations and governing equations,some size-dependent theories have been developed for more accurate predicting the behaviours of small scale structures in various scales such as nano or micro.In addition,piezoelectric materials are used in various systems to measure deformations or stresses as a sensor or perform a defined work as an actuator.The combination of the piezoelectric structures in small-scales leads to significant improvement of electro-mechanical systems and structures in micro and nano scales.This paper is organized to study the effect of small scale and applied electric potential as well as geometric parameters on the electro-elastic bending analysis of functionally graded cylindrical nanoshell integrated with piezoelectric layers based on higher-order shear deformation theory.Furthermore,cylindrical shells may be used as an important element of military weapons military tanks and war cannon.These weapons are subjected to various types of loading such as mechanical,thermal and impulsive loads.Analysis of these structures is necessary for designer because of vast application of these structures in various situations.Furthermore,the piezoelectric cylindrical nano or micro shells may be used as an element of nano or micro electromechanical systems as a sensor or actuator.These instruments are used in various situations as sonar applications [1].[2] developed application of piezoelectric transducers for ballistic high-pressure measurement.Derepa et al.[3] developed application of a circular cylindrical piezoceramic transducer near a flat acoustic screen for receiving plane sound waves.The literature review is presented to show importance of the present work.

Ke et al.[4] studied thermo-electro-mechanical free vibration responses of the circular cylindrical small-scale shell using the size dependent theory and Love’s thin shell theory based on Hamilton’s principle and differential quadrature method.The mechanical,electrical and thermal pre-mechanical loads have been incorporated in external works.Tadi Beni et al.[5] employed a shear deformable model as well as a size-dependent theory in micro scales for dynamic analysis of FG cylindrical shell based on Navier’s technique, in which effect of significant parameters such as gradation of material properties, nanoscale parameter and the geometric characteristics of shell were investigated.Sahmani et al.[6] employed surface elasticity theory based on Gurtin-Murdoch elasticity theory for small-scale buckling and post-buckling analyses of the cylindrical nanoshells subjected to the thermal loads.They used boundary layer theory and perturbation technique for solution of the problem.It was concluded that surface free energy has a significant influence on the post-buckling strength of nanoshell.Effect of size-dependency based on strain gradient theory was studied on the nonlinear vibration responses of nanoporous materials by Sahmani and Aghdam [7].Bayones and Abd-Alla [8]studied effect of multi-physics loads on the dynamic responses of in-homogeneous spherical shell.

Karami et al.[9] employed a variational method for wave propagation analysis of the isotropic doubly-curved nanoshells based on a new size-dependent higher-order shear deformation theory.To account size dependency,nonlocal strain gradient theory including two small-scale parameters was used.Zeighampour et al.[10] studies wave propagation characteristics of a single walled carbon nanotube based on thin shell theory and nonlocal strain gradient theory.Viscoelasticity was assumed for material properties based on the Kelvin-Voigt model.Wave propagation equations were derived based on the Hamilton’s principle.They studied the influence of various wavenumbers, damping parameter,nanoscale parameter and different geometric characteristics of the model on the phase velocity.Buckling analysis of piezoelectric cylindrical nanoshell subjected to compressive loads and applied external voltages was performed by Jiabin et al.[11].For more accurate modelling the nanoshell structure, Reddy’s third-order shear deformation theory was employed.In addition, because of importance of surface effect in small scales,surface elasticity theory was included in the governing equations.The buckling loads were examined in terms of significant parameters such as nonlocal effects, boundary conditions, shell length and radius, material properties and applied voltages.Effect of a higher order shear deformation theory was studied on the electro-elastic results of functionally graded plates and shells by Mohammadimehr et al.[12] and ˙Zur et al.[13]; respectively.Arefi et al.[14] investigated exact solution of a functionally graded piezoelectric cylindrical shell subjected to thermal,mechanical and electrical loads.Effect of non-uniform pressure and short length was studied on the elastic analysis of functionally graded cylindrical shell by Khoshgoftar et al.[15].

Dynamic analysis of a cylindrical nanoshell was investigated by Rouhi et al.[16] using nonlinear shell model with accounting surface elasticity theory in which geometric nonlinearity was included based on von Karman relations.Size dependency was included in governing equations based on nonlocal elasticity theory.Modified couple stress theory was developed by Mehralian et al.[17]for sizedependent buckling analysis of shear deformable functionally graded cylindrical nanoshell based on FSDT.Functionality was assumed based on power-law distribution along the thickness direction.The critical pressure was calculated in terms of main parameters of the problem such as dimensionless geometrical parameters, material length scale parameter, length, thickness,applied voltage and in-homogeneous index.Magneto-electrothermo-elastic vibration analysis of cylindrical nanoshell resting on Winkler’s foundation was studied by Ghadiri and Safarpour[18]based on first-order shear deformation theory and modified couple stress theory as a size-dependent theory.The effect of thermal,mechanical, electrical and magnetic loads were examined on the vibration responses of cylindrical nanoshell.In addition, the effect of dimensionless geometric parameters and material length scale parameter as well as wave numbers have been studied on the responses.Razavi et al.[19] studied electro-mechanical vibration analysis of functionally graded piezoelectric cylindrical nanoshell.Size effects were included in governing equations using consistent CST.The Hamilton’s principle was used to derive both governing equations and boundary conditions.Solution was developed using Navier and Galerkin methods to examine effect of various boundary conditions on the responses.Free vibration characteristics were calculated in terms of important parameters such as some dimensionless geometric parameters, and length scale parameter.Vibration and stability analysis of a functionally graded laminated composite cylindrical shell reinforced with carbon nanotubes was studied by Chakraborty et al.[20] based on a semi-analytical approach.Zhang and Zhang [21] studied buckling and dynamic analysis of FG nanoporous metal foam nanoshells based on FSDT and SGT.They assumed two symmetric and unsymmetric nanoporosity distributions for nanoporous materials.

Ebrahimi et al.[22]studied the effect of Graphene oxide powder on the buckling behavior of a shell in which the effective material properties were estimated using the Halpin-Tsai micromechanical model.Dastjerdi et al.[23] studied the effect of viscoelasticity on the time-dependent bending analysis of the rotating spherical nanostructures using the Hamilton’s principle, the Eringen’s nonlocal elasticity, first-order shear deformation theory and semianalytical polynomial method.Demir and Civalek [24] employed the Eringen’s nonlocal elasticity theory for static bending analysis of micro beams with various boundary conditions.Civalek et al.[25] studied the effect of carbon nanotube reinforcements on the free vibration responses of microbeams based on the Hamilton’s principle and the Navier’s solution method.The effect of micro length scale parameter was studied on the responses.Jalaei and Civalek[26]studied the effect of porosity and small scales in micro size on the axial instability analysis of a microbeam subjected to magnetic field.They studied effect of some main parameters such as power-law index, structural damping, porosity and small-scale parameter on the dynamic instability results.

Effect of the graphene nanoplatelets was studied on the vibration, static and dynamic analyses of composite reinforced structures by Li et al.[27] and Shi et al.[28].Al-Furjan et al.[29,30].[31,32]presented nonlinear analysis of a small scale nanocomposite annular plate reinforced with carbon nanotubes subjected to various loadings based on Hamilton’s principle and differential quadrature method.Al-Furjan et al.[29,30].[31,32] investigated application of modified couple stress theory to size dependent vibration analysis of a microplate resting on a viscoelastic foundation obeying Kelvin-Voight model.After derivation of the governing equations, the vibration results were obtained using generalized differential quadrature method.Effect of residual stress and pressure was studied on the vibration analysis of polynomial viscoelastic composite shell, imperfect annular system and circular plates reinforced by graphene nanoplatelets with honeycomb core by Dai et al.[33];Bai et al.[34];and Habibi et al.[35].The numerical methods such as generalized differential quadrature and the Runge-Kutta methods were used for solution of the governing equations.Effect of movability was studied on the stability and vibration analyses of viscoelastic macro and nano beams made of axially functionally graded materials by Refs.[36,37].Effect of small scale effects in micro and nano scales based on modified strain gradient theory was studied on the free and forced vibration, and stability analyses of cylindrical shells reinforced by graphene nanoplatelets was studied by Esmailpoor Hajilak et al.[38-40].Application of higher-order shear deformation theory in the framework of three-dimensional theory of elasticity was developed to stress, strain and vibration analyses of graphene nanoplatelets reinforced composite disks with viscoelastic properties based on fourth-order Runge-Kutta numerical and Generalized differential quadrature methods.Effect of a nonlinear foundation was studied on the large amplitude vibration analysis of a composite disk reinforced by carbon nanotubes in macro and nano sizes by Shariati et al.[41].[42] studied the critical voltages and temperatures of a foundation.

2.Electro-elastic constitutive relations

This paper studies electro-elastic bending analysis of threelayered functionally graded cylindrical nanoshell integrated with piezoelectric hollow cylinders at top and bottom.Two-dimensional analysis is presented here for electro elastic analysis of cylindrical nanoshell.The proposed model may be used as a sensor or an actuator in an electromechanical system.This model can be applied as a sensor to detect deformations or as an actuator to perform a defined work.

The constitutive relations are developed for functionally graded core and piezoelectric cylinders as follows respectively:

Functionally graded core: constitutive relations [37]

micro disk reinforced by the graphene nanoplatelets integrated with the piezoelectric actuators.Ghabussi et al.[43] studied frequency analysis of the graphene nanoplatelets composite circular microplate in the framework of a numerical-based generalized

differential quadrature method.

A literature survey was performed with a focus on the important related works of cylindrical small-scale shells, various lower and higher-order shear deformation theories and small scale dependent theories.Based on the literature review and best author’s knowledge, it is confirmed that there is no published work on the application of various advanced theories such as nonlocal elasticity theory,third-order shear deformation theory and piezoelasticity to electro-elastic bending analysis of three-layered nano cylindrical shell.For higher-order modelling and accounting thickness stretching, the radial displacement is modelled as a third-order function along the thickness direction.This paper uses the thirdorder shear deformation theory for axial and radial displacements along the thickness direction.The principle of virtual work is used for derivation of the governing equations.The piezoelectric layer is subjected to applied electric potential as well as external pressure.The cylindrical nanoshell is rested on Pasternak’s foundation.Electro-elastic results are presented in terms of main important parameters such as small scale parameter, applied electric potential, length to radius ratio, and two parameters of Pasternak’s

In which σ，ε，，are stress,strain,electric displacement and electric field components, respectively.Furthermore,，，εare stiffness, piezoelectric and dielectric coefficients, respectively.

The constitutive relations are obtained using the principle of virtual work δ-δ=0,in which δis variation of strain energy and δis variation of external work.The principle of strain energy is developed as follows [28,37]:

Fig.1.The schematic figure of a three-layered cylindrical nanoshell integrated with piezoelectric layers.

To complete necessary relation, the electric potential function should be assumed as:

The strain components are derived using displacement components.The displacement components are derived using thirdorder shear deformation theory along the axial and radial directions as follows [44]:

Fig.2.The changes of in terms of the small scale parameter ξ（nm） for Ψ0 =0，0.001，0.002，0.003 V.

Fig.3.The changes of in terms of the small scale parameter ξ（nm） for Ψ0 =0，0.001，0.002，0.003 V.

Fig.4.The changes of in terms of Ψ0 for various n = 1，2，3.

Fig.5.The changes of in terms of Ψ0 for various n = 1，2，3.

Fig.6.Two-dimensional variation ofin terms of ξ and Ψ0.

Fig.7.Two-dimensional variation of in terms of ξ and Ψ0.

Fig.8.Two-dimensional change of in terms ξ along the thickness direction.

Fig.9.Two-dimensional change ofin terms of Ψ0 and (L/R).

Fig.10.Two-dimensional change of in terms of Ψ0 and (L/R).

Fig.11.The changes of in terms of ξ for various K1.

Fig.12.The changes of in terms of ξ for various K2.

Fig.13.The changes of in terms of ξ for various K1.

Fig.14.The changes of in terms of ξ for various K2.

where,are the axial and radial displacement.The strain components are obtained as:

Fig.15.The changes of in terms of ξ and K2.

Fig.16.The changes of in terms of ξ and K2.

Substitution of displacement components into straindisplacement relations leads to strain components as:

Based on definition of the electric field and the strain field components, the constitutive electro-elastic relations are developed as:

FG-Core:

Piezoelectric layers:

Substitution of variation of the strain components and the electric field components into variation of strain energy using definition of resultant components leads to:

where:

Variation of the strain energy is updated as:

The external work can be completed by considering the internal pressureand reaction of elastic foundation=-?.Based on this relation,，are Winkler’s and Pasternak’s parameters of foundation.Considering both forces,the external work is completed as:

Finally,the governing equations are derived using the principle of virtual work as:

Finally, the resultant components are derived as:

And

And the governing equations are derived as:

3.Analytical solution, numerical results and discussion

The governing equations of cylindrical nanoshell integrated with piezoelectric layers were derived in form of a second-order system of differential equations.Solution[[45-48]]is assumed as:

In which element of the stiffness matrix are defined as:

The material properties of core and integrated piezoelectric face-sheets are assumed as:

Two dimensional variation of radial and axial displacements in terms of nonlocal parameter and Pasternak’s coefficient of foundation are presented in Figs.15 and 16,respectively.It is concluded that the radial displacement is decreased with an increase in Pasternak’s coefficient of foundation, while the axial displacement is increased.Furthermore, increase in nonlocal parameter leads to significant increase of radial and axial displacements.

4.Conclusions

Higher-order shear deformation theory and nonlocal piezoelasticity relations were employed in this paper for electro-elastic bending analysis of functionally graded three-layered cylindrical nanoshell integrated with piezoelectric layers subjected to mechanical and electrical loads.The kinematic relation was developed based on third-order shear deformation theory.To include size effect in the governing equations,the nonlocal theory was applied to derive the governing equations.The piezoelectric layers are subjected to the applied electric potential.The governing equations were derived using the principle of virtual work and the generalized Hooke’s law with accounting transverse shear strain and stress.The numerical results including the radial and longitudinal displacements are presented in terms of main parameters of the problem such as small scale parameter, in-homogeneous index,applied electric potential and length to radius ratio.The main conclusions of this paper are expressed as follows:

? The effect of nanoscale parameter was studied on the variation of the radial and longitudinal displacements of the cylindrical nanoshell.It indicates that an increase in small scale parameter leads to enhance of displacements because of a decrease in nanoshell stiffness.

? The non-homogeneity was assumed for gradation of the material properties of the cylindrical noncore.It indicates that increase of non-homogeneous parameter leads to a decrease in displacements of the nanoshell.

? The effect of the initial voltage of the piezoelectric layers was studied on the electro elastic results of the cylindrical nanoshell.It was deduced that a decrease behavior in radial and axial displacement may be observed with enhance of initial voltage.

? Investigating on the effect of length to radius ratio (), indicates that the both displacements are increased with an increase in length to radius ratio (), because a decrease in the stiffness of the nanoshell.

? The numerical results indicate that an increase in both parameters of the elastic foundation leads to a decrease in radial displacement and an increase in axial displacement.One can conclude that an increase in two parameters of Pasternak’s foundation leads to an increase in stiffness in radial direction and a decrease in stiffness along the axial direction.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This work was supported by the Research team project of Nanning University (2018KYTD03), the Science and Technology Planning Project of Yongning Zone of Nanning(20180205A).Henan Province Doctor Startup Fund of China under Grant No.2012BZ01.