Abstract
This paper analyses the nonstationary free vibration and nonlinear dynamic behavior of the viscoelastic nanoplates. For this purpose, a sizedependent theory is developed in the framework of the consistent couple stress theory for viscoelastic materials. The previously presented modified couple stress theory was based on some consideration making it partially doubtful to apply. This paper uses the recent findings for the mentioned problem and develops it to analyze the nonlinear dynamic behavior of nanoplates with nonlinear viscoelasticity. The material is supposed to follow the Leaderman integral nonlinear constitutive relation. In order to capture the geometrical nonlinearity, the vonKarman strain displacement relation is used. The viscous parts of the sizeindependent and sizedependent stress tensors are calculated in the framework of the Leaderman integral and the resultant virtual work terms are obtained. The governing equations of motion are derived using the Hamilton principle in the form of the nonlinear second order integropartial differential equation with coupled terms. These coupled sizedependent viscoelastic equations are solved using the forthorder Rungekutta and Harmonic balance method after simplifying by the expansion theory. The shorttime Fourier transform is performed to examine the system free vibration. In addition, frequency and forceresponses of the nanosystem subjected to distribute harmonic load are presented. The obtained results show that the viscoelastic modelbased vibration is nonstationary unlike the elastic model. Moreover, the damping mechanism of the viscoelasticity is amplitude dependent and the contribution of the viscoelastic damping terms at higher forcing conditions becomes noticeable.
Keywords
nonlinear viscoelastic; consistent couple stress; Harmonic balance method; nonlinear dynamic; nanoplate
1 INTRODUCTION
Micronano elements are extensively used in tiny biostructural and mechanical applications (Pandey et al. 2009Pandey, J., Chu, W., Kim, C., Lee, C. and Ahn, S. (2009). Bionano reinforcement of environmentally degradable polymer matrix by cellulose whiskers from grass. Composites Part B: Engineering 40: 676680.,Feng et al. 2012Feng, L., Wu, L., Wang, J., Ren, J., Miyoshi, D., Sugimoto, N. and Qu, X. (2012). Detection of a Prognostic Indicator in Early‐Stage Cancer Using Functionalized Graphene‐Based Peptide Sensors. Advanced Materials 24: 125131.,Murmu and Adhikari 2012Murmu, T. and Adhikari, S. (2012). Nonlocal frequency analysis of nanoscale biosensors. Sensors and Actuators A: Physical 173: 4148.,Xiao et al. 2017Xiao, Y., Huang, W., Tsui, C.P., Wang, G., Tang, C.Y. and Zhong, L. (2017). Ultrasonic atomization based fabrication of bioinspired micronanobinary particles for superhydrophobic composite coatings with lotus/petal effect. Composites Part B: Engineering 121: 9298.). Micronanomechanical resonators that can reach to very high frequencies up to GHz (Husain et al. 2003Husain, A., Hone, J., Postma, H.W.C., Huang, X., Drake, T., Barbic, M., Scherer, A. and Roukes, M. (2003). Nanowirebased veryhighfrequency electromechanical resonator. Applied Physics Letters 83: 12401242.,Huang et al. 2005Huang, X., Feng, X., Zorman, C., Mehregany, M. and Roukes, M. (2005). VHF, UHF and microwave frequency nanomechanical resonators. New Journal of Physics 7: 247.,Baghelani 2016Baghelani, M. (2016). Design of a multifrequency resonator for UHF multiband communication applications. Microsystem Technologies 22: 25432548.) are very important types of these elements highlighted due to their exemplary characteristics. In order to tune the resonators for better sensing performance, one should recognize their dynamic characteristics, carefully (Ekinci et al. 2004Ekinci, K., Huang, X. and Roukes, M. (2004). Ultrasensitive nanoelectromechanical mass detection. Applied Physics Letters 84: 44694471.,Braun et al. 2005Braun, T., Barwich, V., Ghatkesar, M.K., Bredekamp, A.H., Gerber, C., Hegner, M. and Lang, H.P. (2005). Micromechanical mass sensors for biomolecular detection in a physiological environment. Physical Review E 72: 031907.,Tajaddodianfar et al. 2017Tajaddodianfar, F., Yazdi, M.R.H. and Pishkenari, H.N. (2017). Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method. Microsystem Technologies 23: 19131926.). Hence, analyzing the dynamic characteristics of these elements is a critical issue to improve their performance. The mentioned resonators are often made from micronano plate and shell elements. The experimental results for instance, in polymers and metals showed that the mechanical behaviors of the micronano structures have considerable amount of size effects (Poole et al. 1996Poole, W., Ashby, M. and Fleck, N. (1996). Microhardness of annealed and workhardened copper polycrystals. Scripta Materialia 34: 559564.,Lam et al. 2003Lam, D.C., Yang, F., Chong, A., Wang, J. and Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids 51: 14771508.). It is worth noting that in classical continuum theories, CT, the size effects are not considered. Hence, the material lengthscale parameter is not considered in these theories. In order to compensate this shortage, some nonclassical elasticity theories were developed (Mindlin and Tiersten 1962Mindlin, R. and Tiersten, H. (1962). Effects of couplestresses in linear elasticity. Archive for Rational Mechanics and analysis 11: 415448., Koiter 1964Koiter, W. (1964). Couplestress in the theory of elasticity, North Holland Pub, Eringen and Edelen 1972Eringen, A.C. and Edelen, D. (1972). On nonlocal elasticity. International Journal of Engineering Science 10: 233248., Yang et al. 2002Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures 39: 27312743., Lam et al. 2003). In this regard, Mindlin and Tiersten (1962) and Koiter (1964), developed a couple stress theory for elastic models based on the rotation vector as a curvature tensor. However, this theory suffers from some problems: first, the spherical part of the couplestress tensor is indeterminate and second, the body couple appears in the constitutive relation of the forcestress tensor. Therefore, its original form has not been used widely. In order to compensate the mentioned problems, the modified couple stress theory (MCST) was developed by Yang et al. (2002). According to the MCST, when the resultant vectors of couples, moments of couples and applied forces become zero, the system is in equilibrium. In this theory, true continuum representations of deformation has been used. However, the obtained equations were not consistent with proper boundary conditions and the virtual work principle energy conjugacy Hadjesfandiari and Dargush (2011Hadjesfandiari, A.R. and Dargush, G.F. (2011). Couple stress theory for solids. International Journal of Solids and Structures 48: 24962510.). In addition, the resulting couplestress and stress tensors are symmetric and this contrasts with the original form of the couple stress theory, specifically for the couple stress tensor. Recently, Hadjesfandiari and Dargush developed a new theory based on the original theory by using true continuum kinematical displacement and rotation and found a reasonable solution for the associated problems (Hadjesfandiari and Dargush 2011Hadjesfandiari, A.R. and Dargush, G.F. (2011). Couple stress theory for solids. International Journal of Solids and Structures 48: 24962510., ^{2013}Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265., Hadjesfandiari et al., 2013Hadjesfandiari, A.R., Dargush, G.F. and Hajesfandiari, A. (2013). Consistent skewsymmetric couple stress theory for sizedependent creeping flow. Journal of NonNewtonian Fluid Mechanics 196: 8394.). Their new sizedependent theory is called the consistent couple stress theory (CCST). In their theory, the couplestress tensor becomes skewsymmetric while the skewsymmetric components of the rotation vector gradient are considered as a consistent curvature tensor. The two skewsymmetric tensors are conjugate with each other in calculation of the virtual work Hadjesfandiari and Dargush (2011)Hadjesfandiari, A.R. and Dargush, G.F. (2011). Couple stress theory for solids. International Journal of Solids and Structures 48: 24962510.. In the current paper, for the first time, the recent theory that (CCST) is developed for the viscoelastic model to study nonstationary free vibration and nonlinear dynamic behavior of a viscoelastic nanoplate subjected to harmonic load.
Recently, some experiments have discovered that the viscoelasticity widely present in materials of NEMS and MEMS such as silicon Elwenspoek and Jansen (2004Elwenspoek, M. and Jansen, H.V. (2004). Silicon micromachining, Cambridge University Press), polysilicon Teh and Lin (1999Teh, K.S. and Lin, L. (1999). Timedependent buckling phenomena of polysilicon micro beams. Microelectronics journal 30: 11691172.) and gold films Yan et al. (2009Yan, X., Brown, W., Li, Y., Papapolymerou, J., Palego, C., Hwang, J. and Vinci, R. (2009). Anelastic stress relaxation in gold films and its impact on restoring forces in MEMS devices. Journal of Microelectromechanical Systems 18: 570576.). Furthermore, the experimental investigations conducted by Su et al. (2012Su, Y., Wei, H., Gao, R., Yang, Z., Zhang, J., Zhong, Z. and Zhang, Y. (2012). Exceptional negative thermal expansion and viscoelastic properties of graphene oxide paper. Carbon 50: 28042809.) revealed that the viscoelastic phenomena exist in graphene oxide sheets. The tensile tests on this specimen displayed clear hysteresis loops, indicating the viscoelasticity of the graphene oxide. Furthermore, the viscoelastic properties can exhibit at some nanostructures (Karlièiæ et al. 2014Karlièiæ, D., Koziæ, P. and Pavloviæ, R. (2014). Free transverse vibration of nonlocal viscoelastic orthotropic multinanoplate system (MNPS) embedded in a viscoelastic medium. Composite Structures 115: 8999.,Mohammadimehr et al. 2015Mohammadimehr, M., Navi, B.R. and Arani, A.G. (2015). Free vibration of viscoelastic doublebonded polymeric nanocomposite plates reinforced by FGSWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures 131: 654671.,Khaniki and HosseiniHashemi 2017Khaniki, H.B. and HosseiniHashemi, S. (2017). Dynamic response of biaxially loaded doublelayer viscoelastic orthotropic nanoplate system under a moving nanoparticle. International Journal of Engineering Science 115: 5172.). Zhang et al. (2016Zhang, Y., Pang, M. and Fan, L. (2016). Analyses of transverse vibrations of axially pretensioned viscoelastic nanobeams with small size and surface effects. Physics letters A 380: 22942299.) studying the vibration of nanobeams. Wang et al. (2015Wang, Y., Li, F.M. and Wang, Y.Z. (2015). Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E: Lowdimensional Systems and Nanostructures 67: 6576.) and Ebrahimi and Hosseini (2016Ebrahimi, F. and Hosseini, S. (2016). Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. Journal of Thermal Stresses 39: 606625.) examined the nonlinear vibration of viscoelastic nanoplates. In addition, Hashemi et al. (2015) investigated the free vibration of viscoelastic nano graphene sheets.
Generally, in a structure made from viscoelastic material, a quota of energy of deformation is recoverable and the other quota is irrecoverable. Mainly, in the viscoelastic materials, the stressstrain relations cause the equations of motion to become in integrodiﬀerential type. As a result, the dynamic behavior of the viscoelastic nanoplates are more complicated than their elastic counterparts.
Viscoelastic models have been used for dynamic/statics analyses of macroscale plates. For instance, nonlinear vibration of viscoelastic thin plates subjected to transverses harmonic force was discussed at the first resonances Amabili (2016Amabili, M. (2016). Nonlinear vibrations of viscoelastic rectangular plates. Journal of Sound and Vibration 362: 142156.). Dynamic stability of the viscoelastic plates with longitudinally varying tensions subjected to axial acceleration was studied by Tang et al. (2016Tang, Y., Zhang, D., Rui, M., Wang, X. and Zhu, D. (2016). Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Applied Mathematics and Mechanics 37: 16471668.). In addition, An and Chen (2016An, F. and Chen, F. (2016). Bifurcations and chaos of the nonlinear viscoelastic plates subjected to subsonic flow and external loads. Chaos, Solitons & Fractals 91: 7885.) studied the nonlinear dynamic behavior of the viscoelastic plates subjected to external force and subsonic flow. They found the critical conditions that chaos could take place. Furthermore, Chen and Cheng (2005)Chen, L.Q. and Cheng, C.J. (2005). Instability of nonlinear viscoelastic plates. Applied Mathematics and computation 162: 14531463. discussed the instability problem of an isotropic, homogeneous, rectangular plate made from viscoelastic material subjected to a prescribed periodic inplane load. Cveticanin et al. (2012Cveticanin, L., KalamiYazdi, M., Askari, H. and Saadatnia, Z. (2012). Vibration of a twomass system with noninteger order nonlinear connection. Mechanics Research Communications 43: 2228.) studied the vibration of the twomass system with viscoelastic connection. However, the large amount of previous studies in the literature on the static/dynamic deformation of the micronanostructures considered only the elastic models of the systems (Jomehzadeh et al. 2011Jomehzadeh, E., Noori, H. and Saidi, A. (2011). The sizedependent vibration analysis of microplates based on a modified couple stress theory. Physica E: Lowdimensional Systems and Nanostructures 43: 877883.,Farajpour et al. 2012Farajpour, A., Shahidi, A., Mohammadi, M. and Mahzoon, M. (2012). Buckling of orthotropic micro/nanoscale plates under linearly varying inplane load via nonlocal continuum mechanics. Composite Structures 94: 16051615.,Ke et al. 2012Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012). Free vibration of sizedependent Mindlin microplates based on the modified couple stress theory. Journal of Sound and Vibration 331: 94106.). For example, Ke et al. (2013)Ke, L., Yang, J., Kitipornchai, S., Bradford, M. and Wang, Y. (2013). Axisymmetric nonlinear free vibration of sizedependent functionally graded annular microplates. Composites Part B: Engineering 53: 207217. employed the MCST to investigate free vibration of functionally graded (FG) axisymmetric Mindlin microplates with vonKarman nonlinearity. Asghari (2012Asghari, M. (2012). Geometrically nonlinear microplate formulation based on the modified couple stress theory. International Journal of Engineering Science 51: 292309.) derived the sizedependent motion equations for geometrically nonlinear microplates with arbitrary shapes by means of the MCST. Moreover, Lou and He (2015Lou, J. and He, L. (2015). Closedform solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory. Composite Structures 131: 810820.) examined the free vibration and nonlinear bending responses of a FGM Kirchhoff and Mindlin microplate resting on an elastic substrate by means of the MCST.
Recently, in some research studies, the viscoelasticity effects on the static/dynamic behavior of the micronanostructures were studied (Hashemi et al. 2015Hashemi, S.H., Mehrabani, H. and AhmadiSavadkoohi, A. (2015). Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium. Composites Part B: Engineering 78: 377383.,Wang et al. 2015Wang, Y., Li, F.M. and Wang, Y.Z. (2015). Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E: Lowdimensional Systems and Nanostructures 67: 6576.,Tang et al. 2016Tang, Y., Zhang, D., Rui, M., Wang, X. and Zhu, D. (2016). Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Applied Mathematics and Mechanics 37: 16471668.,Khaniki and HosseiniHashemi 2017Khaniki, H.B. and HosseiniHashemi, S. (2017). Dynamic response of biaxially loaded doublelayer viscoelastic orthotropic nanoplate system under a moving nanoparticle. International Journal of Engineering Science 115: 5172.). For instance, Liu et al. (2017Liu, J., Zhang, Y. and Fan, L. (2017). Nonlocal vibration and biaxial buckling of doubleviscoelasticFGMnanoplate system with viscoelastic Pasternak medium in between. Physics letters A 381: 12281235.) discussed the vibration of a doubleviscoelastic FGM nanoplate, and Pouresmaeeli et al. (2013Pouresmaeeli, S., Ghavanloo, E. and Fazelzadeh, S. (2013). Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Composite Structures 96: 405410.) found a closedform solution for the vibration of viscoelastic orthotropic nanoplates resting on viscoelastic substrate in the framework of the nonlocal plate theory. In their study, the KelvinVoigt model was used to model the medium. In another research, Jamalpoor et al. (2017Jamalpoor, A., Bahreman, M. and Hosseini, M. (2017). Free transverse vibration analysis of orthotropic multiviscoelastic microplate system embedded in viscoPasternak medium via modified strain gradient theory. Journal of Sandwich Structures & Materials 1099636216689384.) used the Hamilton's principle and found an analytical solution for outofplane vibration of multiviscoelastic Kirchhoff microplates with freechain and clampedchain boundary conditions in the framework of the modified strain gradient theory. In addition, Farokhi and Ghayesh (2017Farokhi, H. and Ghayesh, M.H. (2017). Viscoelasticity effects on resonant response of a shear deformable extensible microbeam. Nonlinear Dynamics 87: 391406.) studied the effects of the viscoelasticity on dynamic response of a shear deformable microbeam in the framework of the MCST. In another study, the sizedependent instability and forced vibration of a viscoelastic sinusoidal shear deformation microplate with axially moving condition is examined based on the MCST Ghorbanpour Arani and Haghparast (2017Ghorbanpour Arani, A. and Haghparast, E. (2017). Sizedependent vibration of axially moving viscoelastic microplates based on sinusoidal shear deformation theory. International Journal of Applied Mechanics 9: 1750026.). In all above investigations, the KelvinVigot model was used enabling modeling the linear viscoelastic materials. However, large amount of the viscoelastic materials are not linear and in order to have accurate solutions for the behavior of viscoelastic structures, the nonlinearity of these materials should be modeled. A comparative research study Smart and Williams (1972Smart, J. and Williams, J. (1972). A comparison of singleintegral nonlinear viscoelasticity theories. Journal of the Mechanics and Physics of Solids 20: 313324.) discovered that when prediction and simplicity become important the Leaderman integral model Leaderman (1962Leaderman, H. (1962). Large longitudinal retarded elastic deformation of rubberlike network polymers. Transactions of the Society of Rheology 6: 361382.) is one of the useful representations of the nonlinear viscoelastic properties. Thus, this model is used in this paper. In addition, in all the mentioned studies, the governing motion equations were achieved but the solutions of the nonlinear dynamic characteristics were not provided.
According to the descriptions in the previous paragraphs, all of the studies performed analysis on the elastic or linear viscoelastic nanomico structures. This paper, for the ﬁrst time, studies the nonstationary free vibration and nonlinear dynamic behavior of a nanoplate with nonlinear viscoelasticity by means of the CCST. This paper used the CCST and developed it to analyze the nonlinear dynamic characteristics of a nanoplate with nonlinear viscoelasticity. The material is supposed to follow the Leaderman integral nonlinear relation. Additionally, in many applications, such as resonators (Ghayesh et al. 2013aGhayesh, M.H., Farokhi, H. and Amabili, M. (2013a). Nonlinear behaviour of electrically actuated MEMS resonators. International Journal of Engineering Science 71: 137155., bGhayesh, M.H., Farokhi, H. and Amabili, M. (2013b). Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Composites Part B: Engineering 50: 318324.), the nanoplate undergoes largeamplitude deformations. Therefore, it is necessary to employ a nonlinear straindisplacement relation. Therefore, in this study, the vonKarman relation is used to capture the geometrical nonlinearity. The viscous parts of the sizeindependent and sizedependent stress tensors are derived by means of the Leaderman integral and their virtual work terms are obtained. The governing equations of motion are derived using the Hamilton's principle in the form of the nonlinear secondorder integropartial differential equation with coupled terms. The nonlinear vibration equations are solved previously with Hamiltonian approach, He's variational approach, global error minimization and Jacobi collocation method (Askari et al. 2013Askari, H., Nia, Z.S., Yildirim, A., Yazdi, M.K. and Khan, Y. (2013). Application of higher order Hamiltonian approach to nonlinear vibrating systems. Journal of Theoretical and Applied Mechanics 51: 287296.,Yazdi and Tehrani 2015Yazdi, M. and Tehrani, P. (2015). The energy balance to nonlinear oscillations via Jacobi collocation method. Alexandria Engineering Journal 54: 99103.,Yazdi 2016Yazdi, M. (2016). Approximate solutions to nonlinear oscillations via an improved He's variational approach. Karbala International Journal of Modern Science 2: 289297.,Yazdi and Tehrani 2016Yazdi, M.K. and Tehrani, P.H. (2016). Frequency analysis of nonlinear oscillations via the global error minimization. Nonlinear Engineering 5: 8792.). In this paper, the sizedependent viscoelastically coupled equations are solved with incorporating the expansion theory and Harmonic balance method, HBM,. Frequency and force responses of the nanosystem subjected to distributive harmonic load are obtained and validated with the forthorder Rungekutta method. The effects of the initial excitation values and lengthscale amounts as well as the viscoelastic parameter on the system vibration are also examined. In the following, the viscoelastic model and elastic model with linear damping resonance frequency are compared with each other. Furthermore, the shorttime Fourier transform is performed to investigate the system free vibration.
2 Viscoelastically sizedependent coupled nonlinear models
In the CCST, the equilibrium equations are formulated as Hadjesfandiari and Dargush (2013Hadjesfandiari, A.R., Dargush, G.F. and Hajesfandiari, A. (2013). Consistent skewsymmetric couple stress theory for sizedependent creeping flow. Journal of NonNewtonian Fluid Mechanics 196: 8394.):
In the above equations, σ_{ji} and µ_{ji} are the force and couplestress tensors while f _{i} and 𝜀 _{ijk} are the body force per volume unit and the permutation tensor. The µ_{ji} equals to zero for the classical continuum mechanics. According to Hadjesfandiari and Dargush, in materials with couple stresses, body couples can be converted to a surface traction and equivalent body force Hadjesfandiari and Dargush (2013)Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.. Particularly, as mentioned before, they showed that the couplestress tensor is skewsymmetric. Therefore, the forcestress tensor can be divided into the skewsymmetric and symmetric parts:
In the above equation,
In this paper, the continuum mechanic kinematics is defined based on the inﬁnitesimal deformation. Thus, the gradient of displacement vector is divided into skewsymmetric and symmetric components
where
where ε_{ij} and ω_{ij} are symmetric strain and skewsymmetric rotation tensors. The rotation vector is deﬁned as
Decomposing the rotation vector gradient into two components results in:
where the 𝜒_{ij} and 𝜅_{ij} are symmetric and skewsymmetric tensors obtained from applying the strain and rotation operators to the rotation vector
According to the couple stress theory, the symmetric tensor 𝜒_{ij}, as curvature tensor, is an important part in calculating strain energy. However, this tensor denotes the pure twists along the principal axes. Consequently, this symmetric tensor cannot participate as a fundamental element in measuring deformation in continuum mechanics Hadjesfandiari and Dargush (2013Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.). However, in the CCST the skewsymmetric tensor
After defining the kinematic parameters, the force and couplestresses can be formulated. For the elastic isotropic materials, Hadjesfandiari and Dargush (2013Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.) demonstrated that the couple stress can be defined as:
Furthermore, the skewsymmetric and symmetric components of the forcestress tensor are formulated as Hadjesfandiari and Dargush (2013Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.).
where l is the material lengthscale parameter and the
Therefore, according to the CCST, the strain energy for an isotropic material with volume V can be formulated as:
As mentioned before, the material of the nanoplate is assumed to have nonlinear viscoelasticity and follows from the nonlinear Leaderman's integral relation. According to this relation, Christensen and Freund (1971Christensen, R.M. and Freund, L. (1971). Theory of viscoelasticity. Journal of Applied Mechanics 38: 720.), for a viscoelastic structure the components of the couplestress and forcestress tensors can be written as
where
Based on the definition in Eq. (17), the Eqs. (1416) can be rewritten as
where λ_{0} and μ_{0} are the Lame's constants in time zero and
Considering the Cartesian system (x, y, z) where xyplane is coincident with the geometrical midplane of the undeformed nanoplate, the displacement field according to the Kirchhoff’s plate theory JE. (1989JE., L. (1989). Boundary stabilization of thin plates, SIAM (Philadelphia).) can be formulated as:
The variables u, v and w are the timedependent displacements of the midsurface in the x, y and z directions, respectively. Considering of the vonKarman nonlinearity, the strain components of any point in the nanoplate are expressed as
where 𝜀_{ij} (i,j= x, y, z) are strain components.
By substituting Eq. (20) in Eq. (5), the component of the rotation tensors are obtained as
Similarly, from Eqs. (8) and (20), one can write
Replacing Eq. (21) into Eq. (18), the symmetrical stress components can be written as
Similarly, after substituting Eq. (23) into Eq. (19), the skewsymmetric couple stress can be written as
In order to derive the governing motion equations of the viscoelastic nanoplate, the wellknown Hamilton's principle is used
where, δ is the variational operator, U and W are the kinetic energy, elastic strain energy and nonconservative forces such as viscous dissipation or virtual work of external forces, respectively. Therefore, the virtual work of the nonconservative forces can be decomposed into the virtual work of external forces δW_{ext} and the virtual work of viscous dissipative forces δW_{vis}. Hence, we get
Replacing Eq. (27) into Eq. (26), the generalized Hamilton's principle can be rewritten as
According to the CCST, the first variational of elastic strain energy is obtained from Eq. (29)
Integration on volume for homogenous rectangular nanoplate can be written as
Substituting Eqs. (30), (21) and (23) into Eq. (29), integrating by parts and after some algebraic processes, the Eq. (29) can be written as Eq. (31)
where
By integrating stress and couple along the thickness of the nanoplate, the resultants are obtained as
Similarly, the first variational of virtual work of the viscous dissipative forces can be defined as
Substituting Eqs. (30), (21) and (23) into Eq. (34), and integrating by parts results in the Eq. (35)
where
Similarly, the viscous stress and couple of resultants can be deﬁned as
The first variational of kinetic energy is given as
In the Eq. (38) and throughout this paper, the overhead “·” and“··” denote, respectively, the ﬁrst and second time derivatives. In addition, ρ is the mass density of the nanoplate. Derivation of Eq. (20) with respect to the time and replacing in Eq. (27) gives Eq.(39)
Where
From the general expression of the external forces work in the CCST Hadjesfandiari and Dargush (2013Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.), the first variational of virtual work performed by the applied forces on the viscoelastic nanoplate in the time interval [0, T] can be calculated as Ma et al. (2011Ma, H., Gao, X.L. and Reddy, J. (2011). A nonclassical Mindlin plate model based on a modified couple stress theory. Acta mechanica 220: 217235.)
where Γ is the the middle surface boundary of the nanoplate Reddy and Kim (2012Reddy, J. and Kim, J. (2012). A nonlinear modified couple stressbased thirdorder theory of functionally graded plates. Composite Structures 94: 11281143.). In addition, (fx ,f _{y} , f _{z}) and the (c _{x} , c _{y} , c _{z}) are, respectively, the body forces and the body couples, and (q _{x} , q _{y} , q _{z}), (t _{x} , t _{y} , t _{z}) and (s _{x} , s _{y} , s _{z}) are, respectively, the tractions applied on Γ, the surface couple and Cauchy traction applied on A. In this paper, it is assumed that the nanoplate is only subjected to transverse force.
Replacing the expressions for δU, δW _{vis}, δK and δW _{ext} from the Eqs. (31), (35), (39) and (41) into Eq. (3) and integrating by parts, the sizedependent viscoelastically coupled governing equations of the motion based on the CCST can be obtained as Eq. (42)
where
The Equation (42) is the system of nonlinear integraldiﬀerential partial equations for a viscoelastic nanoplate based on the CCST. Proposed model includes an added lengthscale parameter. As expected, the added material constant affects both of the current and the past history conditions, simultaneously. Additionally, with ignoring the past history term, the Eq. (42) is abridged into the sizedepended CCST for elastic nanoplate, and with ignoring the size effect, namely l=0, it is abridged into the macro scale viscoelastic plate model. These facts clearly reveal the accuracy of our calculations during the derivation of governing equations.
The motion equations can be presented in the displacement terms (u,v,w) as
Where
For homogenous rectangular nanoplate, I _{1} becomes zero.
The large amount of experimental results (Lee et al. 2005Lee, H.J., Zhang, P. and Bravman, J.C. (2005). Stress relaxation in freestanding aluminum beams. Thin Solid Films 476: 118124.,Yan et al. 2009Yan, X., Brown, W., Li, Y., Papapolymerou, J., Palego, C., Hwang, J. and Vinci, R. (2009). Anelastic stress relaxation in gold films and its impact on restoring forces in MEMS devices. Journal of Microelectromechanical Systems 18: 570576.) proposed standard anelastic solid model for viscoelastic material. In this paper, this model is used to define the elastic modulus relaxation function. Therefore, it is defined as
where γ is the relaxation coefficient. In addition, the E(t) value at t=0 indicates the initial elastic modulus E _{0}. Introducing the following nondimensional quantities
Eq. (45) can then be expressed in dimensionless form as
In order to nondimensionalize the governing equation, the following dimensionless parameters are introduced
where c is the damping coefficient.
The solutions of the simply supported immovable rectangular nanoplates can be assumed as Niyogi (1973Niyogi, A. (1973). Nonlinear bending of rectangular orthotropic plates. International Journal of Solids and Structures 9: 11331139.)
where α=mπ and β=nπ. The simply supported immovable boundary conditions of the rectangular nonlinear nanoplate are satisfied with this consideration in Eq. (49). In addition, the u and v consideration satisfy the Eq. (43a) and Eq.(43b), simultaneously. As mentioned above, in this paper, it is assumed that the nanoplate is subjected only to the harmonic distributed force, f cosΩt per unit of the surface, in the z direction. The nondimensional transverse harmonic load amplitude
where
Inserting Eq. (49) into Eq. (43c), the residue
Using BubnovGalerkin approach and setting the integral to zero, the expression of Ë is obtained as Eq. (54)
The above equation can be solved with the fourthorder RungeKutta method after some algebraic processes Fu and Zhang (2009Fu, Y. and Zhang, J. (2009). Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam. Acta Mechanica Sinica 25: 211218.). Furthermore, in this paper, the HBM method is applied to solve the above equation Mickens (2010Mickens, R.E. (2010). Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods, World Scientific). Considering the periodic solution of the firstorder approximation as the following form and substituting in Eq.(54), one gets Eq.(56)
Where
By ignoring higher frequency, 3Ω, and applying the steadystate condition the following polynomial equation is obtained
where
Solving the Eq.(58) gives the amplitude of the nanosystem response subjected to harmonic force. The real and positive roots of Eq.(58) are acceptable.
3 Free vibration
Nonlinear system frequencies highly depend on the vibration amplitude. The presence of the past history terms in Eq. (54) obtained from viscoelastic model changes the vibration amplitude over time. Therefore, the system nonlinearity effect changes over time and causes to change the system vibration frequency with time. In order to evaluate the initial excitation value and the viscoelastic relaxation time effects on the system natural frequencies, the shorttime Fourier transform (STFT) is performed on the nanosystem model. The nanosystem under consideration is assumed to be made of epoxy with the following mechanical and geometric properties:
STFT spectrum at l_{0} =0.1 and 𝛾=1: (ad) Dimensionless frequency respect dimensionless time for V_{0}=75, 50, 10 and 1, respectively.
Fig. 1 displays the STFT spectrum for the nanosystem with l _{o}=0.1 and 𝛾=1 at four different initial excitation values equaling to 75, 50, 10 and 1 (1/s), respectively. The variation of natural frequency of the transverse motion over time spectrum is shown in this figure. Fig.1 (ac) shows that for the initial excitation values equaling to 75, 50 and 10, the natural frequency decreases as the time increases. Therefore, the viscoelastic nanosystem vibration is nonstationary at these conditions. Moreover, the presence of the nonlinear terms in the vibration equation causes to higher natural frequencies at bigger initial values. However, the Fig (1d) shows that at initial value equaling 1 (1/s), the viscoelastic model frequency does not change vs. time. This occurs because the nanosystem nonlinearity is weaker at smaller vibration amplitude.
In order to understand the vibration response of the viscoelastic nanoplate, the deflectiontime responses of the central point are presented in Fig.2 for two different initial conditions. The dashed and solid lines denote the deflection responses predicted by the Runge Kutta method for lengthscale ratios 0.1 and 0.25, respectively. It can be seen that the vibration frequency displayed by the dashed line is smaller than the solid line.
Vibration response of the center deﬂection vs. time for the viscoelastic nanoplate at diferente lengthscale ratios; (a) X_{0}=1, V_{0}=0 and (b) X_{0}=0, V_{0}=1
4 Dynamic response analysis
The nonlinear dynamic response of the nanosystem is studied in this section with illustrating the frequency and forceresponses obtained based on HBM method. Furthermore the outcomes are validated by the RungeKutta method and previous results.
Fig. 3 demonstrates the frequency responses of the viscoelastic nanoplate for the outofplane and inplane motions. The dimensionless relaxation coefficient 𝛾 is set to 5 and amplitude of the dimensionless distributed transverse force f is set to 30. The horizontal axis values, i.e., distributed load frequencies, are normalized with natural frequencies of ù_{1,1} = 16, obtained via an eigenvalue analysis. The solid and dashed lines are predicted by the HBM method and the dotted symbols are obtained by Rungekutta method.
Frequencyresponse curves of viscoelastic nanosystem. a the outofplane motion maximum amplitude at midplane ; b, c the inplane motions maximum amplitude at x=y = 3/4; l_{0}=0.1, f=30 and 𝛾=5
This figure shows that all of the motions have hardening type nonlinearities. Moreover, two saddle node bifurcations are seen in the figure corresponding to Ω =1.50 ù_{1},_{1} and Ω =1.19 ù_{11,1}. The ﬁrst one corresponds to instability beginning and the second one corresponds to stability recapturing. The transverse and longitudinal responses of the nanosystem at x=y=0.5 and x=y=3/4, respectively for Ω = 1.50 ù_{1,1} are plotted in Fig. 4. It is observed that the nanosystem has a periodic motion. The results show that the inplane motions, the frequency is much larger than the outofplane motion. Moreover, the transverse motion amplitude equals to 0.1 that can be found at lower branch around the first saddle node bifurcations in Fig. (3a).
Dynamic response of the viscoelastic nanosystem at Ω =1.5 ù_{1,1}; a and b,c time history of the outofplane and inplane motions, respectively
The frequency response of the viscoelastic and elastic models are compared with each other at different applied force amplitudes in Fig.5. The outofplane motion response at x=y=0.5 and inplane motion response at x=y=3/4 are depicted in this figure. The dimensionless relaxation coefficient is set to 𝛾 =5 for viscoelastic model and in elastic model ã is set to zero in Eq.(54) and linear damping with damping “c” is added to the model. The value of the dimensionless damping coefficient is tuned in order to have similar prediction at smaller forcing amplitude c =0.93. The results show that these models predict different response amplitudes at smaller frequency ratios and especially at resonance frequency. As seen in the figure, the maximum amplitude of the elastic model with linear damping is bigger than the viscoelastic model counterpart. More especially, the maximum amplitudes at f =30 for outofplane motion are 1.5 and 1.63 for the viscoelastic and elastic models, respectively. This figure shows the importance of using the viscoelastic model at higher applied forces with respect to the elastic model with linear damping.
Fig. 6 demonstrates the viscoelasticity effect on the resonance frequency of the viscoelastic nanoplate. In order to depict this figure, the resonance frequency, excitation frequency of the maximum amplitude, corresponds to the frequency response curve of each applied force amplitude that is obtained for elastic model with c=0.93 and viscoelastic models with 𝛾 =5. Then, these resonance frequencies are plotted versus forcing amplitudes at Fig.6. It can be seen that, at the forcing amplitude equaling to 1, both models predict the same results but with increasing the applied force amplitude, the difference between the two models becomes more obvious. More especially, at the same forcing frequency, the viscoelastic model predicts smaller resonance frequency than the elastic model with linear damping. For instance, at f=100, the viscoelastic model resonance occurs at Ω = 2.7 ù_{1,1} while the elastic one predicts at Ω = 4.4 ù_{1,1} i.e. with 63% difference. It can be concluded that damping mechanism of the viscoelasticity is amplitudedependent. In addition, the viscoelasticity reduces the hardening behavior of the nanosystem. Therefore, it is expected that viscoelastic model predicts more reliable dynamic behavior than the elastic model with linear damping.
The effect of the dimensionless relaxation coefficient (𝛾) on the resonance frequency and maximum amplitude of the outofplane oscillation, at x=y = 0.5, of the nanosystem is highlighted in Fig. 7(ab). The forcing amplitude is selected f=30. In Fig. 7 the frequency response is depicted for various dimensionless relaxation coefficients (from 0 to 5); then, the resonance frequency and its corresponding amplitude are plotted in Fig.7. It can be seen that, as the dimensionless relaxation coefficient is increased, the resonance frequency and its corresponding amplitude are decreased to smaller values due to the dissipation of the nanosystem energy. Hence, the nanosystem hardening behavior reduces with increasing the relaxation coefficient.
Frequency response of the viscoelastic, 𝛾=5, and elastic with c=0.93, damping nanoplate for different applied force amplitudes: a the outofplane motion maximum amplitude at midplane ; b, c the inplane motions maximum amplitude at x=y = 3/4, l_{0}=0.1(solid line elastic model with linear damping and dashed line viscoelastic model)
Applied Force amplitude versus the resonance frequency for elastic model with linear damping, c=0.93,and viscoelastic model of the nanosystem
Dimensionless relaxation coefficient effect on resonance frequency and maximum amplitude of the oscillation (at f=30 and l_{0}=0.1)
The frequency response of the viscoelastic nanoplate predicted by means of the CCST and the classical continuum theory are demonstrated in Fig. 8 for the case of f=30, l _{o}=0.25 and 𝛾 =5.This figure highlights more the importance of employing the CCST in comparison with the classical continuum mechanics theory. As seen in the ﬁgure, both theories predict the hardening type nonlinear behavior. The peakamplitude values of the outofplane and inplane motions are larger for the case of the classical continuum theory. Furthermore, it can be seen that the resonance frequency are Ω=23.90 and Ω=24.24 for the case of the classical continuum and CCS theories, respectively. The natural frequency predicted for this nanosystem via CCST theory is larger than classical continuum mechanics one and this is the reason why the resonant region shifts to the larger excitation frequencies. Furthermore, dynamic response of the viscoelastic nanosystem at Ω= 24.24 for the transverse and longitudinal motions are depicted in the framework of the CCST and classical continuum theories in Fig.9. It can be seen that both theories predict periodic motions.
Frequency response of the viscoelastic nanoplate, predicted by means of the classical continuum theory and CCST. a the outofplane motion maximum amplitude at midplane; b, c the inplane motions maximum amplitude at x =y= 3/4, respectively; f = 30, 𝛾= 5 and l_{0}=0.25
The forceresponse of the viscoelastic nanoplate obtained by means of the classical continuum and CCS theories are depicted in Fig.10. The dimensionless relaxation coefficient, excitation frequency and lengthscale parameter are set to 5, 18 and 0.25, respectively. The solid and dashed lines are predicted by the HBM and the dotted symbols are obtained by the RungeKutta method. This ﬁgure reveals that the CCST and classical continuum theory predict different response paths. Particularly, response amplitude of the CCST increases slowly with the applied force amplitude and no jumps or bifurcations are seen in its response. In addition, the overall amplitude predicted with this theory is much smaller than the classical continuum theory one at bigger applied force. However for the classical continuum theory, as force amplitude is increased, the amplitude of the response increases and then jumps to a higher value. Decreasing the applied force amplitude, because of the nonlinearity that exists in the nanosystem, causes the response amplitude to decrease and then second jump happens to the smaller amplitude response. These characteristics are for all the outofplane and inplane motions of the viscoelastic nanoplate.
Dynamic response of the viscoelastic nanoplate at Ω= 24.24; ac time history of the outofplane and inplane motions. (Solid and dashed lines correspond to classical continuum theory and CCST results, respectively)
Force response of the viscoelastic nanoplate, 𝛾= 5, predicted by means of classical continuum theory and CCST at Ω=18. a the outofplane motion maximum amplitude at midplane; b, c the inplane motions maximum amplitude at x =y= 3/4, respectively (dotted symbols predicted by RungKutta method)
Force response of viscoelastic nanoplate at different frequency ratio; a the outofplane motion maximum amplitude at midplane ; b, c the inplane motions maximum amplitude at x=y = 3/4; l_{0}=0.1, and 𝛾=5
The force response of the mentioned, outofplane and inplane, motions at different normalized frequency and 𝛾 =5 are plotted in Fig. 11. It can be seen that as the applied force amplitude increases, the response amplitude also increases gradually for the Ω/ ù_{1,1}≤1 while no bifurcations and jumps are seen in the response path. However, at the normalized frequencies equaling to 1.1 and 1.05, more than unity, by increasing the applied force amplitude, the response amplitude becomes larger and then shifts to a larger value. This phenomenon, saddle node bifurcation, relates to jumping between stable branches. In addition, as the force amplitude is increased, the response amplitudes for all of these motions increase while the magnitude and the increasing rate of the responses are different.
The accuracy of the HBM performed in the present work are also validated by plotting the transverse motion frequency response for an elastic macroplate with linear damping, simplified model with l _{0}=0, and comparing it with the given frequency response in Amabili (2004Amabili, M. (2004). Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Computers & structures 82: 25872605.) in Fig.12. As the figure shows, there is a good agreement between results. Therefore, the validity of the current simulation and the accuracy of the numerical calculations are partially proved.
The marcoplate transverse motion frequency response: dotted symbol and solid line obtained by present model and Amabili (2004Amabili, M. (2004). Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Computers & structures 82: 25872605.), respectively.
5 Conclusion
This paper analyzed the nonstationary free vibration and nonlinear dynamic behavior of the viscoelastic microplates. For this purpose, a sizedependent model was developed for viscoelastic material based on the CCST. As described, the first version of the couple stress theory suffers from some problems. Hence, its first version was not used widely. In addition, its modification called MCST used some doubtful assumptions. This paper used the recent theory, CCST, which solved the associated problems. The material was supposed to follow the Leaderman integral nonlinear relation. Additionally, in order to capture the geometrical nonlinearity, the vonKarman straindisplacement relation was used. The viscous parts of the sizeindependent and sizedependent stress tensors are derived by means of the Leaderman integral and their virtual work terms are obtained. The governing equations of motion were derived using the Hamilton's principle in the form of the nonlinear secondorder integropartial differential equations with coupled terms. These sizedependent viscoelastically coupled equations are solved with incorporating the expansion theory and HBM. The shorttime Fourier transform were performed to investigate the system free vibration. The effects of the initial excitation values and length scales as well as the viscoelastic parameter on the system vibration are also examined. In addition, frequency and force response curves of the nanosystem subjected to distributive harmonic load were obtained based on the HBM and forthorder RungeKutta method.
The STFT analysis showed that the vibration of the nanosystem with viscoelastic model is nonstationary at higher initial excitation values unlike the elastic model. However, the system frequencies do not change with time at smaller initial values. Moreover, the presence of the nonlinear terms in vibration equation causes higher natural frequencies at larger initial values.
The nonlinear analysis showed that outofplane and inplane motions displayed hardening type nonlinearities. Moreover, two saddle node bifurcations are seen in the frequency response of this nanosystem. In addition, the resonance frequency and its corresponding amplitude for all of these motions are increased with increasing the amplitude of the applied force. These reveal that the nanosystem displayed stronger hardening type nonlinearities at larger forcing amplitudes. In addition, it was shown that damping mechanism of the viscoelasticity is amplitudedependent. Moreover, the viscoelasticity reduces the hardening behavior of the nanosystem. More speciﬁcally, the difference between the elastic model with linear damping and viscoelastic model was more obvious at larger amplitude of the applied force. The obtained results showed that, as the dimensionless relaxation coefficient is increased, the resonance frequency and its corresponding amplitude decreased to smaller values due to the dissipation of the nanosystem energy. Hence, the nonlinearities of the nanosystem become weaker. It was observed that, the CCST predicted the resonant frequency at bigger excitation frequencies than the classical continuum theory. Furthermore, the CCST predicted no saddle node bifurcation where the classical continuum theory predicts two nodes. Furthermore, the overall amplitudes predicted by this theory for outofplane and inplane motions were much larger than the classical continuum ones at higher applied forces.
6 Nomenclature
 a,b nanoplate length and width
 E relaxation function
 c damping coefficient
 f Body force
 G modulus of rigidity
 h nanoplate thickness
 K kinetic energy
 t time
 U elastic strain energy
 A,V nanoplate Area and Volume
 W nonconseravtive forces virtual work
 X_{0}, V_{0} Initial displacement and velocity conditions
 ó Force stress tensors
 𝜀 permutation tensor
 𝜅 curvature tensor
 µ Couplestress tensors
 ω Rotation tensor
 l lengthscale parameter
 µ Lame constants
 𝜒 curvature tensor
 λ Lame constants
 õ Poisson ratio
 𝛾 relaxation coefficient
 l _{0} lengthscale ratio
 𝜌 Density
 Ω External load frequency
 e Elastic
 v Plastic
 vis Viscous forces
 ext External forces

Greek symbols

Subscript
References
 Amabili, M. (2004). Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Computers & structures 82: 25872605.
 Amabili, M. (2016). Nonlinear vibrations of viscoelastic rectangular plates. Journal of Sound and Vibration 362: 142156.
 An, F. and Chen, F. (2016). Bifurcations and chaos of the nonlinear viscoelastic plates subjected to subsonic flow and external loads. Chaos, Solitons & Fractals 91: 7885.
 Asghari, M. (2012). Geometrically nonlinear microplate formulation based on the modified couple stress theory. International Journal of Engineering Science 51: 292309.
 Askari, H., Nia, Z.S., Yildirim, A., Yazdi, M.K. and Khan, Y. (2013). Application of higher order Hamiltonian approach to nonlinear vibrating systems. Journal of Theoretical and Applied Mechanics 51: 287296.
 Baghelani, M. (2016). Design of a multifrequency resonator for UHF multiband communication applications. Microsystem Technologies 22: 25432548.
 Braun, T., Barwich, V., Ghatkesar, M.K., Bredekamp, A.H., Gerber, C., Hegner, M. and Lang, H.P. (2005). Micromechanical mass sensors for biomolecular detection in a physiological environment. Physical Review E 72: 031907.
 Chen, L.Q. and Cheng, C.J. (2005). Instability of nonlinear viscoelastic plates. Applied Mathematics and computation 162: 14531463.
 Christensen, R.M. and Freund, L. (1971). Theory of viscoelasticity. Journal of Applied Mechanics 38: 720.
 Cveticanin, L., KalamiYazdi, M., Askari, H. and Saadatnia, Z. (2012). Vibration of a twomass system with noninteger order nonlinear connection. Mechanics Research Communications 43: 2228.
 Ebrahimi, F. and Hosseini, S. (2016). Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates. Journal of Thermal Stresses 39: 606625.
 Ekinci, K., Huang, X. and Roukes, M. (2004). Ultrasensitive nanoelectromechanical mass detection. Applied Physics Letters 84: 44694471.
 Elwenspoek, M. and Jansen, H.V. (2004). Silicon micromachining, Cambridge University Press
 Eringen, A.C. and Edelen, D. (1972). On nonlocal elasticity. International Journal of Engineering Science 10: 233248.
 Farajpour, A., Shahidi, A., Mohammadi, M. and Mahzoon, M. (2012). Buckling of orthotropic micro/nanoscale plates under linearly varying inplane load via nonlocal continuum mechanics. Composite Structures 94: 16051615.
 Farokhi, H. and Ghayesh, M.H. (2017). Viscoelasticity effects on resonant response of a shear deformable extensible microbeam. Nonlinear Dynamics 87: 391406.
 Feng, L., Wu, L., Wang, J., Ren, J., Miyoshi, D., Sugimoto, N. and Qu, X. (2012). Detection of a Prognostic Indicator in Early‐Stage Cancer Using Functionalized Graphene‐Based Peptide Sensors. Advanced Materials 24: 125131.
 Fu, Y. and Zhang, J. (2009). Nonlinear static and dynamic responses of an electrically actuated viscoelastic microbeam. Acta Mechanica Sinica 25: 211218.
 Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013a). Nonlinear behaviour of electrically actuated MEMS resonators. International Journal of Engineering Science 71: 137155.
 Ghayesh, M.H., Farokhi, H. and Amabili, M. (2013b). Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Composites Part B: Engineering 50: 318324.
 Ghorbanpour Arani, A. and Haghparast, E. (2017). Sizedependent vibration of axially moving viscoelastic microplates based on sinusoidal shear deformation theory. International Journal of Applied Mechanics 9: 1750026.
 Hadjesfandiari, A.R. and Dargush, G.F. (2011). Couple stress theory for solids. International Journal of Solids and Structures 48: 24962510.
 Hadjesfandiari, A.R. and Dargush, G.F. (2013). Fundamental solutions for isotropic sizedependent couple stress elasticity. International Journal of Solids and Structures 50: 12531265.
 Hadjesfandiari, A.R., Dargush, G.F. and Hajesfandiari, A. (2013). Consistent skewsymmetric couple stress theory for sizedependent creeping flow. Journal of NonNewtonian Fluid Mechanics 196: 8394.
 Hashemi, S.H., Mehrabani, H. and AhmadiSavadkoohi, A. (2015). Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium. Composites Part B: Engineering 78: 377383.
 Huang, X., Feng, X., Zorman, C., Mehregany, M. and Roukes, M. (2005). VHF, UHF and microwave frequency nanomechanical resonators. New Journal of Physics 7: 247.
 Husain, A., Hone, J., Postma, H.W.C., Huang, X., Drake, T., Barbic, M., Scherer, A. and Roukes, M. (2003). Nanowirebased veryhighfrequency electromechanical resonator. Applied Physics Letters 83: 12401242.
 Jamalpoor, A., Bahreman, M. and Hosseini, M. (2017). Free transverse vibration analysis of orthotropic multiviscoelastic microplate system embedded in viscoPasternak medium via modified strain gradient theory. Journal of Sandwich Structures & Materials 1099636216689384.
 JE., L. (1989). Boundary stabilization of thin plates, SIAM (Philadelphia).
 Jomehzadeh, E., Noori, H. and Saidi, A. (2011). The sizedependent vibration analysis of microplates based on a modified couple stress theory. Physica E: Lowdimensional Systems and Nanostructures 43: 877883.
 Karlièiæ, D., Koziæ, P. and Pavloviæ, R. (2014). Free transverse vibration of nonlocal viscoelastic orthotropic multinanoplate system (MNPS) embedded in a viscoelastic medium. Composite Structures 115: 8999.
 Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2012). Free vibration of sizedependent Mindlin microplates based on the modified couple stress theory. Journal of Sound and Vibration 331: 94106.
 Ke, L., Yang, J., Kitipornchai, S., Bradford, M. and Wang, Y. (2013). Axisymmetric nonlinear free vibration of sizedependent functionally graded annular microplates. Composites Part B: Engineering 53: 207217.
 Khaniki, H.B. and HosseiniHashemi, S. (2017). Dynamic response of biaxially loaded doublelayer viscoelastic orthotropic nanoplate system under a moving nanoparticle. International Journal of Engineering Science 115: 5172.
 Koiter, W. (1964). Couplestress in the theory of elasticity, North Holland Pub
 Lam, D.C., Yang, F., Chong, A., Wang, J. and Tong, P. (2003). Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids 51: 14771508.
 Leaderman, H. (1962). Large longitudinal retarded elastic deformation of rubberlike network polymers. Transactions of the Society of Rheology 6: 361382.
 Lee, H.J., Zhang, P. and Bravman, J.C. (2005). Stress relaxation in freestanding aluminum beams. Thin Solid Films 476: 118124.
 Liu, J., Zhang, Y. and Fan, L. (2017). Nonlocal vibration and biaxial buckling of doubleviscoelasticFGMnanoplate system with viscoelastic Pasternak medium in between. Physics letters A 381: 12281235.
 Lou, J. and He, L. (2015). Closedform solutions for nonlinear bending and free vibration of functionally graded microplates based on the modified couple stress theory. Composite Structures 131: 810820.
 Ma, H., Gao, X.L. and Reddy, J. (2011). A nonclassical Mindlin plate model based on a modified couple stress theory. Acta mechanica 220: 217235.
 Mickens, R.E. (2010). Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods, World Scientific
 Mindlin, R. and Tiersten, H. (1962). Effects of couplestresses in linear elasticity. Archive for Rational Mechanics and analysis 11: 415448.
 Mohammadimehr, M., Navi, B.R. and Arani, A.G. (2015). Free vibration of viscoelastic doublebonded polymeric nanocomposite plates reinforced by FGSWCNTs using MSGT, sinusoidal shear deformation theory and meshless method. Composite Structures 131: 654671.
 Murmu, T. and Adhikari, S. (2012). Nonlocal frequency analysis of nanoscale biosensors. Sensors and Actuators A: Physical 173: 4148.
 Niyogi, A. (1973). Nonlinear bending of rectangular orthotropic plates. International Journal of Solids and Structures 9: 11331139.
 Pandey, J., Chu, W., Kim, C., Lee, C. and Ahn, S. (2009). Bionano reinforcement of environmentally degradable polymer matrix by cellulose whiskers from grass. Composites Part B: Engineering 40: 676680.
 Poole, W., Ashby, M. and Fleck, N. (1996). Microhardness of annealed and workhardened copper polycrystals. Scripta Materialia 34: 559564.
 Pouresmaeeli, S., Ghavanloo, E. and Fazelzadeh, S. (2013). Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Composite Structures 96: 405410.
 Reddy, J. and Kim, J. (2012). A nonlinear modified couple stressbased thirdorder theory of functionally graded plates. Composite Structures 94: 11281143.
 Smart, J. and Williams, J. (1972). A comparison of singleintegral nonlinear viscoelasticity theories. Journal of the Mechanics and Physics of Solids 20: 313324.
 Su, Y., Wei, H., Gao, R., Yang, Z., Zhang, J., Zhong, Z. and Zhang, Y. (2012). Exceptional negative thermal expansion and viscoelastic properties of graphene oxide paper. Carbon 50: 28042809.
 Tajaddodianfar, F., Yazdi, M.R.H. and Pishkenari, H.N. (2017). Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method. Microsystem Technologies 23: 19131926.
 Tang, Y., Zhang, D., Rui, M., Wang, X. and Zhu, D. (2016). Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions. Applied Mathematics and Mechanics 37: 16471668.
 Teh, K.S. and Lin, L. (1999). Timedependent buckling phenomena of polysilicon micro beams. Microelectronics journal 30: 11691172.
 Wang, Y., Li, F.M. and Wang, Y.Z. (2015). Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory. Physica E: Lowdimensional Systems and Nanostructures 67: 6576.
 Xiao, Y., Huang, W., Tsui, C.P., Wang, G., Tang, C.Y. and Zhong, L. (2017). Ultrasonic atomization based fabrication of bioinspired micronanobinary particles for superhydrophobic composite coatings with lotus/petal effect. Composites Part B: Engineering 121: 9298.
 Yan, X., Brown, W., Li, Y., Papapolymerou, J., Palego, C., Hwang, J. and Vinci, R. (2009). Anelastic stress relaxation in gold films and its impact on restoring forces in MEMS devices. Journal of Microelectromechanical Systems 18: 570576.
 Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002). Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures 39: 27312743.
 Yazdi, M. (2016). Approximate solutions to nonlinear oscillations via an improved He's variational approach. Karbala International Journal of Modern Science 2: 289297.
 Yazdi, M. and Tehrani, P. (2015). The energy balance to nonlinear oscillations via Jacobi collocation method. Alexandria Engineering Journal 54: 99103.
 Yazdi, M.K. and Tehrani, P.H. (2016). Frequency analysis of nonlinear oscillations via the global error minimization. Nonlinear Engineering 5: 8792.
 Zhang, Y., Pang, M. and Fan, L. (2016). Analyses of transverse vibrations of axially pretensioned viscoelastic nanobeams with small size and surface effects. Physics letters A 380: 22942299.

Available Online: August 06, 2018
Publication Dates

Publication in this collection
2018
History

Received
17 Feb 2018 
Reviewed
12 July 2018 
Accepted
30 July 2018