Flutter and Divergence Instability of Axially-Moving Nanoplates Resting on a Viscoelastic Foundation

Jingbo Duan 1, Dapeng Zhang 2,* and Wenjie Wang 3,* 1 Department of Engineering Mechanics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China; duanjingbo@stdu.edu.cn 2 College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China 3 School of Engineering, University of Warwick, Coventry CV4 7AL, UK * Correspondence: zhangdapeng201812@163.com (D.Z.); Wenjie.Wang.1@warwick.ac.uk (W.W.); Tel.: +86-1850-325-6216 (D.Z.)


Introduction
Nano structures have attracted a huge amount of attention from the scientific community due to their extraordinary mechanical, electrical, thermal and other physical/chemical properties.These novel properties can be for nanoelectromechanical systems (NEMS), nanoresonators, nanosensors and nanogenerators [1,2], which are always modeled as nanobeams [3], nanorods [4], nanoribbons [5], nanoplates [6], nanoshells [7], etc to investigate the mechanical properties and other physical properties.Currently, there are three main kinds of approaches available for the study of nanomechanics, which are experimental techniques [8], molecular dynamic (MD) simulations [9] and modified continuum mechanics theories [10,11], respectively.However, nano scale experiments are exceedingly difficult and MD simulations are costly in terms of computation.As a remedy, continuum mechanics theories are a valid selection for nanomechanics.The nonlocal elasticity theory established by Eringen [12,13] is one of the typical continuum mechanics theories.It is assumed that the stress at a reference point is affected not only by the strain at that point but also by the strains at every point in the rest of the domain.Therefore, based on the nonlocal elasticity theory, a couple of researchers have applied their works to the bending, vibration and stability of nanostructures [14][15][16].and boundary conditions on the dynamic characteristics of the beam were discussed.Arani and Jalaei [32] presented transient behavior of simply-supported orthotropic single-layered graphene sheet (SLGS) resting on orthotropic visco-Pasternak foundation subjected to dynamic loads.Researches show that the structural and foundation damping coefficients are effective parameters on the transient response, particularly for large damping coefficients.Gharib et al. [33] used the micromechanics method to consider the small-scale effect in the structural equations of a piezoelectric polymeric nanocomposite panel embedded in the Pasternak-type elastic substrate.Biloueiet et al. [34] studied the nonlinear buckling of straight concrete columns armed with single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation.The column was modeled with Euler-Bernoulli beam theory.The characteristics of the equivalent composite were determined using the Mori-Tanaka model.The foundation around the column was simulated with spring and shear layer.Jamalpoor et al. [35] dealt with the free vibration and biaxial buckling of double-magneto-electro-elastic nanoplate systems (DMEENPS) subjected to initial external electric and magnetic potentials with the assumption that the two nanoplates are bonded with each other using a visco-Pasternak medium and are also limited to the external elastic substrate.Zenkour and Arefi [36] investigated the transient thermos-electro-mechanical vibration and bending analysis of a functionally graded piezoelectric nanosheet resting on visco-Pasternak foundation.The effects of some important parameters were studied on the fundamental frequencies of the system and the maximum deflection of the sheet under various thermal and electrical loadings.They also investigated the free vibration responses of a sandwich nanoplate resting on visco-Pasternak foundation based on the nonlocal Kirchhoff theory and Hamilton's principle [37].Kolahchi et al. [38] examined the dynamic buckling for a sandwich nanoplate with the surrounding medium simulated by a visco-orthotropic Pasternak foundation model based on visco-nonlocal-refined Zigzag theories.Zhang et al. [39] studied the thermo-electro-mechanical vibration responses for a rectangular piezoelectric nanoplate resting on viscoelastic foundations by using the Galerkin strip distributed transfer function method, which enables one to obtain the semi-analytical solutions of natural frequencies for piezoelectric nanoplates with arbitrary boundary conditions.Liu et al. [40] investigated the size-dependent vibrational behaviors of functionally graded (FG) magneto-electro-viscoelastic nanobeams in the presence of porosities based on the nonlocal Timoshenko beam theory in conjunction with a visco-Pasternak foundation model.Liu and Lv [41] established a theoretical framework based on interval analysis method to study the wave-dispersion behavior of CNTs embedded in a Pasternak-type elastic medium with uncertain material properties.In their work, the material properties of CNTs were considered as uncertain-but-bounded variables, which is different from probabilistic analysis method.
As reviewed above, a lot of studies on the dynamic behaviors for nanosystems with axially-moving velocity or those resting on a foundation have been separately reported in recent years.However, the literature review shows that few studies are concerned with the dynamic instability of axially-moving nanoplates resting on a viscoelastic foundation for nanomechanical systems.As well known, in engineering practices, nanoresonators, nanosensors and nanogenerators are often embedded in a moving medium, which often exhibit viscoelastic behaviors.The objective of this study is to explore dynamical instability of an axially-moving nanoplate resting on a viscoelastic foundation.With the framework of the thin plate theory, we get the governing partial differential equations according to Hamilton's principle.After examining the developed mechanics model, the effects of foundation properties, axial moving velocity, nonlocal parameter and biaxial force on the divergent instability and coupled-mode flutter of nanoplates are shown by graphs and tables in detail, which could benefit the designation of moving graphene nanosheets or other plate-like nanostructures resting on a viscoelastic medium.

Basic Equations
A rectangular nanoplate resting on a viscoelastic foundation is shown in Figure 1.The length, width and uniform thickness of the nanoplate are denoted as l a , l b and h, respectively.The viscoelastic foundation is represented by a visco-Pasternak foundation model, whose Winkler's modulus parameter is k w , Pasternak's modulus parameter is k G and the damping parameter is c t .The axially-moving velocity of the nanoplate is υ.The nanoplate is subjected to a biaxial force N Px and N Py along with x and y directions according to the rectangular coordinate system oxyz shown in Figure 1.Based on the nonlocal elasticity theory [8], the basic constitutive equations for the nanoplate can be written as: where i u , ij σ and ij ε are the components of nonlocal displacement, stress and strain vector, respectively.ijkl c and ρ are the elastic constants and mass density of the nanoplate, respectively.
Moreover, 0 e a is the nonlocal parameter and 2 ∇ is the Laplacian operator, respectively.
For simplicity, the displacement fields of the nanoplate can be obtained by neglecting the deformation of the middle surface, given by: Based on the nonlocal elasticity theory [8], the basic constitutive equations for the nanoplate can be written as: 1 where u i , σ ij and ε ij are the components of nonlocal displacement, stress and strain vector, respectively.c ijkl and ρ are the elastic constants and mass density of the nanoplate, respectively.Moreover, e 0 a is the nonlocal parameter and ∇ 2 is the Laplacian operator, respectively.For simplicity, the displacement fields of the nanoplate can be obtained by neglecting the deformation of the middle surface, given by: where u, v and w are, respectively, the displacement components in the x, y and z directions.Based on the strain-displacement equations of thin plate theory, the nonzero strain components are given as: Based on Equations ( 1) and (3) and the classical plane stress relation, the nonlocal stress is thus derived as the following form: where E and µ denote elastic modulus and Poisson's ratio, respectively.Further, the relation of the bending moment and stress in classical elasticity theory is still valid in the nonlocal stress.Then, the nonlocal bending moment can be presented by corresponding classical counterparts as: where D = Eh 3 12(1−µ 2 ) .Subsequently, the governing equations and boundary conditions for transverse vibration of the axially-moving nanoplate resting on a viscoelastic foundation can be derived utilizing the variational approach.The strain energy Π U of the nanoplate can be stated as: where A is the area of the mid-plane of the nanoplate.The kinetic energy Π K of the moving nanoplate can be obtained as: Moreover, the work Π W done by the external forces is obtained as: where N Px and N Py are respectively the normal forces in the x and y directions.N Q is the reaction of the viscoelastic foundation, which can be obtained from: Substituting Equations ( 6)-( 9) into the Hamilton's equation Then, integrating by parts and setting the coefficient δw to zero, the governing equations of the axially-moving nanoplate resting on a viscoelastic foundation can be derived as: Appl.Sci.2019, 9, 1097 6 of 18 The corresponding boundary conditions can also be obtained as follows: where (n x , n y ) are denoted as the direction cosines of the outward unit normal to the boundaries of the mid-plane.
Using the Galerkin strip distributed transfer function method (GSDTFM), the equivalent integral of the governing Equation (11) along with y direction can be stated as: where H is an admissible function.Further, integrating by parts and considering the nonlocal bending moment equations shown in Equation ( 5), the weak form of the governing equations for the axially-moving nanoplate resting on a viscoelastic foundation can be derived as: For the sake of convenience and generality, the following dimensionless quantities are defined as: Using these dimensionless terms, the governing Equation ( 14) can be rewritten as: According to Equation ( 12), boundary conditions for the axially-moving nanoplate resting on a viscoelastic foundation can be obtained.For simplicity, in the following analysis, we only consider the simply supported boundary constraints for the nanoplate, which are given as follows:

Solution Method
In this section, the GSDTFM was adopted to investigate the dynamic instability of the axially-moving nanoplate resting on a viscoelastic foundation.Firstly, the nanoplate was divided into a couple of strip elements along the y direction and there are NE strip elements and (NE + 1) nodal lines for the nanoplate in all, as shown in Figure 2. The length of the jth strip element is l a and its width is l e .The transverse displacement w(x, y, t) of the jth strip element can be interpolated by: where N(y) is the shape function and δ e (x, t) is the unknown displacement of the e th strip element.The expression of N(y) and δ e (x, t) are given in literature [39], which are not herein presented for the sake of brevity.are given in literature [39], which are not herein presented for the sake of brevity.According to the dimensionless quantities defined in equation (15), equation ( 18) can be rewritten as: where Substituting equation ( 19) into equation ( 16) with T = H N results in the following: , According to the dimensionless quantities defined in Equation (15), Equation ( 18) can be rewritten as: where Substituting Equation (19) into Equation ( 16) with H = N T results in the following: where To obtain the global dynamic equilibrium equations of the nanoplate, a global nodal displacement vector is defined as: Like the finite element method, the contributions of each strip element are added together and the global dynamic equilibrium equations are obtained as: where K (4) , K (2) , K (0) , K 2 , K 1 are global stiffness matrices, and M is a global mass matrix.Further, dealing with the displacement boundary conditions imposed on nodal lines, the global dynamic equilibrium equations are rewritten as: where where T 1 are row transformation matrices containing 0 and 1, which is referred to literature [39].
Next, the Fourier transform of Equation ( 25) with respect to time are given by: where δ w is the Fourier transformation form of δ w , iω is the Fourier transform parameter, ω is the angular frequency and i = √ −1.If denoting a status vector as: Equation ( 27) can be rewritten in a compact form in state space: where 1 , , ( Introducing the vector η(X, iω) and performing the Fourier transform to the boundary conditions in Equation ( 17), we get: where M b and N b are selection matrice of boundary condition at the left and right edges of nanoplate, respectively, as shown below.
Substituting Equation (33) into Equation (31) results in: For the dynamic analysis of the nanoplate, the existence of a nontrivial solution of the corresponding Equation (34) requires that the determinant of the coefficient matrix must vanish The dimensionless natural frequency ω of the axially-moving nanoplate resting on a viscoelastic foundation can be obtained by solving the above transcendental characteristic equation.

Comparison and Validation
To make a comparison with the previous literature [24], we adopted the following parameters: l a = l b = 10 nm, h = 0.34 nm, E = 1.06 TPa, µ = 0.25, ρ = 2250 kg/m and K 1 = K 2 = 80 for the axially-moving nanoplate without a foundation.The material properties of the foundation are the Winkler's modulus k w = 0, the Pasternak's modulus k G = 0, and the damping c t = 0.According to the literature [39], strip element number NE = 6 was selected for GSDTFM in all of the following numerical calculations.The numerical results of the present work were compared with the corresponding results in literature [24], as shown in Table 1.It can be observed from Table 1 that the dimensionless fundamental frequencies for the axially-moving nanoplate in this paper were in excellent agreement with those in [24], demonstrating the accuracy and efficiency of the proposed method for vibration analysis.For future comparisons with other researchers, the first dimensionless natural frequencies of the axially-moving nanoplate resting on a viscoelastic foundation before the dimensionless velocity C = 10 with small-parameter α = 0.01 and 0.10 are presented in Table 2.The material properties of the viscoelastic foundation were k w0 = 0.1 GPa/nm, k G0 = 0.25 GPa•nm and c t0 = 10 −4 GPa•ns/nm, as referred to in the literature [39].It can be seen from the tables that the dimensionless natural frequencies of the axially-moving nanoplate were complex numbers when it rested on a viscoelastic foundation.

Flutter and Divergent Instability
The flutter instability refers to a dynamic process of structures transiting from stability to instability with a nonvanishing frequency, whereas the divergence instability refers to a structure passing from stability to instability at a zero frequency.In the following text, the mode-couple flutter and divergence instability of an axially-moving nanoplate resting on a visoelastic foundation with simply supported boundary condition are investigated.The small parameter of the nanoplate was α = 0.001 and other parameters were the same as Section 4.1.
For comparison, the first four dimensionless natural frequencies ω 11 , ω 12 , ω 21 and ω 22 of the axially-moving nanoplate without a foundation with respect to moving speeds are plotted in Figure 3.As shown in the figure, the dimensionless frequencies were real numbers at the dimensionless speed C = 0. Along with the increase of axially moving speed, the first four dimensionless natural frequencies changed but remained real numbers before C = 12.8.This showed that the nanoplate was stable before C = 12.8.At the speed of C = 12.8, the real part of ω 11 decreased to zero and its imaginary part had two branches.This indicated the nanoplate began divergence instability on its first order mode.C = 12.8 was a threshold critical value of ω 11 , denoted as the divergence speed C div1 .Moreover, ω 11 of the nanoplate was unstable in the interval [12.8, 14.1], but ω 12 , ω 21 and ω 22 still kept stable.When the axially moving speed further increased to C = 16.2, ω 11 and ω 21 of the nanoplate coupled with each other into a pair of complex conjugate frequency.This means that the nanoplate exhibited a coupled-mode flutter instability.C = 16.2 was denoted as the flutter speed C flu1 .When C = 17.9, denoted as the divergence speed C div2 , the real part of ω 12 decreased to zero with its imaginary part becoming two branches, representing the nanoplate becoming divergent instable again on its second order mode.The nanoplate stayed unstable again during the interval [17.9, 25.5].At the speed of C = 26.2,denoted as the flutter speed C flu2 , the nanoplate exhibited coupled-mode flutter instability again with ω 12 and ω 22 merging together.
In contrast with Figure 3, Figure 4 presents the first four order dimensionless natural frequencies ω 11 , ω 12 , ω 21 and ω 22 of the axially-moving nanoplate resting on a viscoelastic foundation with respect to moving speeds.As can be seen, the dimensionless frequencies only had a real component at the speed of C = 0, but the values were slightly larger than those without a foundation.This implies that the rigidity of the nanoplate was enhanced due to the viscoelastic foundation.When the moving speed C > 0, the first four dimensionless frequencies ω 11 , ω 12 , ω 21 and ω 22 became complex numbers, which were significantly different from vibrations of the nanoplate without a foundation.Also, the divergence instability type of the axially moving nanoplate resting on a viscoelastic foundation was similar to that without a foundation.However, the divergence speeds for ω 11 and ω 21 (C div1 = 13.3 and C div2 = 18.3) were larger than those without a foundation (C div1 = 12.8 and C div2 = 17.9).This means the divergence speed was more obviously affected by the viscoelastic foundation.Particularly, it is worth pointing out that the phenomenon of mode-couple flutter instabilities of ω 11 coupling ω 21 and ω 12 coupling ω 22 disappeared when the moving nanoplate rested on a viscoelastic foundation.
frequencies changed but remained real numbers before C = 12.8.This showed that the nanoplate was stable before C = 12.8.At the speed of C = 12.8, the real part of ω11 decreased to zero and its imaginary part had two branches.This indicated the nanoplate began divergence instability on its first order mode.C = 12.8 was a threshold critical value of ω11, denoted as the divergence speed Cdiv1.Moreover, ω11 of the nanoplate was unstable in the interval [12.8, 14.1], but ω12, ω21 and ω22 still kept stable.When the axially moving speed further increased to C = 16.2, ω11 and ω21 of the nanoplate coupled with each other into a pair of complex conjugate frequency.This means that the nanoplate exhibited a coupled-mode flutter instability.C = 16.2 was denoted as the flutter speed Cflu1.When C = 17.9, denoted as the divergence speed Cdiv2, the real part of ω12 decreased to zero with its imaginary part becoming two branches, representing the nanoplate becoming divergent instable again on its second order mode.The nanoplate stayed unstable again during the interval [17.9, 25.5].At the speed of C = 26.2,denoted as the flutter speed Cflu2, the nanoplate exhibited coupled-mode flutter instability again with ω12 and ω22 merging together.In contrast with Figure 3, Figure 4 presents the first four order dimensionless natural frequencies ω11, ω12, ω21 and ω22 of the axially-moving nanoplate resting on a viscoelastic foundation with respect to moving speeds.As can be seen, the dimensionless frequencies only had a real component at the speed of C = 0, but the values were slightly larger than those without a foundation.This implies that the rigidity of the nanoplate was enhanced due to the viscoelastic foundation.When the moving speed C > 0, the first four dimensionless frequencies ω11, ω12, ω21 and ω22 became complex numbers, which were significantly different from vibrations of the nanoplate without a foundation.Also, the divergence instability type of the axially moving nanoplate resting on a viscoelastic foundation was similar to that without a foundation.However, the divergence speeds for ω11 and ω21 (Cdiv1 = 13.3 and Cdiv2 = 18.3) were larger than those without a foundation (Cdiv1 = 12.8 and Cdiv2 = 17.9).This means the divergence speed was more obviously affected by the viscoelastic foundation.Particularly, it is worth pointing out that the phenomenon of mode-couple flutter instabilities of ω11 coupling ω21 and ω12 coupling ω22 disappeared when the moving nanoplate rested on a viscoelastic foundation.From the Figures 3 and 4, it was found that the viscoelastic foundation had a significant impact on the dynamic instability of the nanoplate.So, the effects of viscoelastic foundation model parameters including the Winkler's modulus kw, the Pasternak's modulus kG, and the damping ct on the dimensionless complex frequencies of the axially-moving nanoplate are respectively investigated in the following text.
Figure 5 shows the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Winkler's modulus kw = 0, kw0, 2kw0 (the Pasternak's modulus kG = 0 and the damping ct = 0), respectively.Note that the instability type of the axially-moving nanoplate along with the increase of the Winkler's modulus kw included divergence instability and mode-couple flutter instability.For divergence instability, as the increase of the Winkler's modulus kw, divergence instability zeros for ω11 and ω12 were enlarged and the corresponding divergence speeds Cdiv1 and Cdiv2 were increased.For mode-couple flutter instability, flutter speeds Cflu1 and Cflu2 were also increased by raising the kw.This indicates that the divergence instability and flutter instability is delayed when the kw increases.Besides, the dimensionless natural From the Figures 3 and 4, it was found that the viscoelastic foundation had a significant impact on the dynamic instability of the nanoplate.So, the effects of viscoelastic foundation model parameters including the Winkler's modulus k w , the Pasternak's modulus k G , and the damping c t on the dimensionless complex frequencies of the axially-moving nanoplate are respectively investigated in the following text.
Figure 5 shows the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Winkler's modulus k w = 0, k w0 , 2k w0 (the Pasternak's modulus k G = 0 and the damping c t = 0), respectively.Note that the instability type of the axially-moving nanoplate along with the increase of the Winkler's modulus k w included divergence instability and mode-couple flutter instability.For divergence instability, as the increase of the Winkler's modulus k w , divergence instability zeros for ω 11 and ω 12 were enlarged and the corresponding divergence speeds C div1 and C div2 were increased.For mode-couple flutter instability, flutter speeds C flu1 and C flu2 were also increased by raising the k w .This indicates that the divergence instability and flutter instability is delayed when the k w increases.Besides, the dimensionless natural frequencies ω 11 and ω 12 increased before their divergence instability along with the increase of the k w , whereas ω 21 and ω 22 increased before their flutter instability.Figure 6 gives the vibration of the divergence speeds C div1 and C div2 and the flutter speeds C flu1 and C flu2 along with the increase of k w .The changes of flutter speeds were dominant over divergence speeds, which indicates that the flutter instability of the nanoplate is more sensitive to the Winkler's modulus k w .
parameters including the Winkler's modulus kw, the Pasternak's modulus kG, and the damping ct on the dimensionless complex frequencies of the axially-moving nanoplate are respectively investigated in the following text.
Figure 5 shows the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Winkler's modulus kw = 0, kw0, 2kw0 (the Pasternak's modulus kG = 0 and the damping ct = 0), respectively.Note that the instability type of the axially-moving nanoplate along with the increase of the Winkler's modulus kw included divergence instability and mode-couple flutter instability.For divergence instability, as the increase of the Winkler's modulus kw, divergence instability zeros for ω11 and ω12 were enlarged and the corresponding divergence speeds Cdiv1 and Cdiv2 were increased.For mode-couple flutter instability, flutter speeds Cflu1 and Cflu2 were also increased by raising the kw.This indicates that the divergence instability and flutter instability is delayed when the kw increases.Besides, the dimensionless natural frequencies ω11 and ω12 increased before their divergence instability along with the increase of the kw, whereas ω21 and ω22 increased before their flutter instability.Figure 6 gives the vibration of the divergence speeds Cdiv1 and Cdiv2 and the flutter speeds Cflu1 and Cflu2 along with the increase of kw.The changes of flutter speeds were dominant over divergence speeds, which indicates that the flutter instability of the nanoplate is more sensitive to the Winkler's modulus kw. Figure 7 presents the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Pasternak's modulus kG = 0, kG0, 2kG0 (the Winkler's modulus kw = 0 and the damping ct = 0), respectively.Similarly to the Pasternak's modulus parameter kw, the instability type of the axially-moving nanoplate with kG = 0, kG0, 2kG0 still included divergence instability and flutter instability.However, for divergence instability, the instability zeros for ω11 and ω12 had a slightly change with the increase of the Pasternak's modulus kG, although the corresponding divergence speeds Cdiv1 and Cdiv2 were still increased.For mode-couple flutter instability, flutter speeds Cflu1 and Cflu2 were also increased with the increase of the Pasternak's modulus kG.Also, the dimensionless natural frequencies ω11 and ω12 were increased before their divergence instability and ω21 and ω22 were also increased before their flutter instability with the increase of the Pasternak's modulus kG.The vibrations of divergence speeds Cdiv1 and Cdiv2 and flutter Figure 7 presents the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Pasternak's modulus k G = 0, k G0 , 2k G0 (the Winkler's modulus k w = 0 and the damping c t = 0), respectively.Similarly to the Pasternak's modulus parameter k w , the instability type of the axially-moving nanoplate with k G = 0, k G0 , 2k G0 still included divergence instability and flutter instability.However, for divergence instability, the instability zeros for ω 11 and ω 12 had a slightly change with the increase of the Pasternak's modulus k G , although the corresponding divergence speeds C div1 and C div2 were still increased.For mode-couple flutter instability, flutter speeds C flu1 and C flu2 were also increased with the increase of the Pasternak's modulus k G .Also, the dimensionless natural frequencies ω 11 and ω 12 were increased before their divergence instability and ω 21 and ω 22 were also increased before their flutter instability with the increase of the Pasternak's modulus k G .The vibrations of divergence speeds C div1 and C div2 and flutter speeds C flu1 and C flu2 along with the increase of the Pasternak's modulus k G are plotted in Figure 8, illustrating that changes of the flutter speeds are no more dominant than those of the divergence speeds.
Figure 7 presents the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the Pasternak's modulus kG = 0, kG0, 2kG0 (the Winkler's modulus kw = 0 and the damping ct = 0), respectively.Similarly to the Pasternak's modulus parameter kw, the instability type of the axially-moving nanoplate with kG = 0, kG0, 2kG0 still included divergence instability and flutter instability.However, for divergence instability, the instability zeros for ω11 and ω12 had a slightly change with the increase of the Pasternak's modulus kG, although the corresponding divergence speeds Cdiv1 and Cdiv2 were still increased.For mode-couple flutter instability, flutter speeds Cflu1 and Cflu2 were also increased with the increase of the Pasternak's modulus kG.Also, the dimensionless natural frequencies ω11 and ω12 were increased before their divergence instability and ω21 and ω22 were also increased before their flutter instability with the increase of the Pasternak's modulus kG.The vibrations of divergence speeds Cdiv1 and Cdiv2 and flutter speeds Cflu1 and Cflu2 along with the increase of the Pasternak's modulus kG are plotted in Figure 8, illustrating that changes of the flutter speeds are no more dominant than those of the divergence speeds.
(a) (b)    Figure 9 illustrates the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the damping ct = 0, ct0, 2ct0(the Winkler's modulus kw = 0 and the Pasternak's modulus kG = 0), respectively.Obviously, the instability type of the axially-moving nanoplate was altered with the appearance of the damping.When the damping ct = 0, there was divergence instability in ω11, ω12 and flutter instability in ω11 coupling ω21 and ω12 coupling ω22.However, when the damping ct > 0, the flutter instability disappeared and only divergence instability in ω11 and ω12 was left.Furthermore, the modes ω11 /ω21 and ω12/ω22, which coupled when the damping ct = 0, separated with the appearance of the damping ct.It was found that the damping ct had almost no influences on the divergence speeds Cdiv1/Cdiv2 and the real parts of the first two dimensionless complex frequencies Re(ω11)/Re(ω12) before their divergence instability, but intensively affected the imaginary part of dimensionless complex frequencies.Figure 9 illustrates the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the damping c t = 0, c t0 , 2c t0 (the Winkler's modulus k w = 0 and the Pasternak's modulus k G = 0), respectively.Obviously, the instability type of the axially-moving nanoplate was altered with the appearance of the damping.When the damping c t = 0, there was divergence instability in ω 11 , ω 12 and flutter instability in ω 11 coupling ω 21 and ω 12 coupling ω 22 .However, when the damping c t > 0, the flutter instability disappeared and only divergence instability in ω 11 and ω 12 was left.Furthermore, the modes ω 11 /ω 21 and ω 12 /ω 22 , which coupled when the damping c t = 0, separated with the appearance of the damping c t .It was found that the damping c t had almost no influences on the divergence speeds C div1 /C div2 and the real parts of the first two dimensionless complex frequencies Re(ω 11 )/Re(ω 12 ) before their divergence instability, but intensively affected the imaginary part of dimensionless complex frequencies.
Figure 9 illustrates the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a foundation with the damping ct = 0, ct0, 2ct0(the Winkler's modulus kw = 0 and the Pasternak's modulus kG = 0), respectively.Obviously, the instability type of the axially-moving nanoplate was altered with the appearance of the damping.When the damping ct = 0, there was divergence instability in ω11, ω12 and flutter instability in ω11 coupling ω21 and ω12 coupling ω22.However, when the damping ct > 0, the flutter instability disappeared and only divergence instability in ω11 and ω12 was left.Furthermore, the modes ω11 /ω21 and ω12/ω22, which coupled when the damping ct = 0, separated with the appearance of the damping ct.It was found that the damping ct had almost no influences on the divergence speeds Cdiv1/Cdiv2 and the real parts of the first two dimensionless complex frequencies Re(ω11)/Re(ω12) before their divergence instability, but intensively affected the imaginary part of dimensionless complex frequencies.Figure 10 plots the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a viscoelastic foundation with the small-parameter α = 0.001, 0.050 and 0.100, respectively.As the small-parameter α increased, divergence instability zeros for ω11 and ω12 were enlarged but the corresponding divergence speeds Cdiv1 and Cdiv2 were decreased.The changes of divergence speeds Cdiv1 and Cdiv2 along with the increase of the small-parameter α are presented in Figure 10 plots the change of dimensionless complex frequencies for the axially-moving nanoplate resting on a viscoelastic foundation with the small-parameter α = 0.001, 0.050 and 0.100, respectively.As the small-parameter α increased, divergence instability zeros for ω 11 and ω 12 were enlarged but the corresponding divergence speeds C div1 and C div2 were decreased.The changes of divergence speeds C div1 and C div2 along with the increase of the small-parameter α are presented in Figure 11.It was seen that the divergence speeds C div1 and C div2 decreased significantly when α > 0.02.
In Figure 12, the changes of dimensionless complex frequencies for the axially-moving nanoplate resting on a viscoelastic foundation with the biaxial force K = 40, 60 and 80 are presented.The biaxial force K clearly influenced the divergence instability zero and the divergence speed.Raising the biaxial force K, divergence instability zeros for ω 11 and ω 12 were slightly enlarged and the corresponding divergence speeds C div1 and C div2 were increased as shown in Figure 13.In Figure 12, the changes of dimensionless complex frequencies for the axially-moving nanoplate resting on a viscoelastic foundation with the biaxial force K = 40, 60 and 80 are presented.The biaxial force K clearly influenced the divergence instability zero and the divergence speed.Raising the biaxial force K, divergence instability zeros for ω11 and ω12 were slightly enlarged and the

Conclusions
The dynamic instability of axially-moving nanoplates resting on a viscoelastic foundation was investigated based on the nonlocal elasticity theory.The viscoelastic foundation, nonlocal effect and biaxial loadings of the nanoplate are considered in the established model to capture the size-dependent mechanical property.The main conclusions are drawn as follow: (i) The viscoelastic foundation has a dominant impact on vibrations of the axially-moving nanoplate.
The dimensionless natural frequencies of the axially-moving nanoplate resting on a viscoelastic foundation are complex numbers when the dimensionless moving speed C > 0. (ii) The Winkler's modulus k w of the viscoelastic foundation significantly affects both the instability zone and the divergence speed.The Pasternak's modulus k G of the viscoelastic foundation affects mainly the divergence speed.The damping c t of the viscoelastic foundation has almost no influence on the instability zone or divergence speed.(iii) Mode-couple flutter disappears when the axially-moving nanoplate rests on a viscoelastic foundation.This change of instability types is mainly influenced by the damping c t of the viscoelastic foundation.(iv) Small-parameter α affects both the divergence instability zero and the divergence speed of the axially-moving nanoplates resting on a viscoelastic foundation.Such influences become more substantial when α > 0.02.(v) The divergence instability is also quite sensitive to the biaxial force, which influences the divergence instability zero and the corresponding divergence speed.
Appl.Sci.2019, 9, x FOR PEER REVIEW 4 of 19 viscoelastic foundation is represented by a visco-Pasternak foundation model, whose Winkler's modulus parameter is kw, Pasternak's modulus parameter is kG and the damping parameter is ct.The axially-moving velocity of the nanoplate isυ .The nanoplate is subjected to a biaxial force NPx and NPy along with x and y directions according to the rectangular coordinate system oxyz shown in Figure 1.
displacement of the e th strip element.The expression of ( )

1 e, 1 , k 3 e, 1 , k 2 e, 2
denote stiffness matrices of the strip element and m e denotes mass matrice of the strip element.

Figure 3 .
Figure 3. Dimensionless complex frequencies vs axially moving speed for the nanoplate without a foundation.

22 Figure 3 .
Figure 3. Dimensionless complex frequencies vs axially moving speed for the nanoplate without a foundation.Appl.Sci.2019, 9, x FOR PEER REVIEW 13 of 19

Figure 4 .
Figure 4. Dimensionless complex frequencies vs axially moving speed for the nanoplate resting on a viscoelastic foundation.

22 Figure 4 .
Figure 4. Dimensionless complex frequencies vs axially moving speed for the nanoplate resting on a viscoelastic foundation.

Figure 6 .
Figure 6.Effect of the Winkler's modulus k w on the flutter and divergence speed of the axially-moving nanoplate.

Figure 7 .
Figure 7. Effect of the Pasternak's modulus kG on dimensionless complex frequencies of the axially-moving nanoplate.

Figure 7 .
Figure 7. Effect of the Pasternak's modulus k G on dimensionless complex frequencies of the axially-moving nanoplate.Appl.Sci.2019, 9, x FOR PEER REVIEW 15 of 19

Figure 8 .
Figure 8.Effect of the Pasternak's modulus kG on the flutter and divergence speed of the axially-moving nanoplate.

Figure 8 .
Figure 8.Effect of the Pasternak's modulus k G on the flutter and divergence speed of the axially-moving nanoplate.

Figure 9 .
Figure 9.Effect of the Pasternak's modulus kG on dimensionless complex frequencies of the axially-moving nanoplate.

Figure 9 .
Figure 9.Effect of the Pasternak's modulus k G on dimensionless complex frequencies of the axially-moving nanoplate.

Figure 10 .
Figure10.Effect of the small-parameter α on dimensionless complex frequencies of the axially-moving nanoplate resting on a viscoelastic foundation.

Figure 11 .Figure 12 .
Figure 11.Effect of the small-parameter α on the divergence speed of the axially-moving nanoplate.

Figure 11 .
Figure 11.Effect of the small-parameter α on the divergence speed of the axially-moving nanoplate.

Figure 10 .
Figure10.Effect of the small-parameter α on dimensionless complex frequencies of the axially-moving nanoplate resting on a viscoelastic foundation.

Figure 11 .Figure 12 .
Figure 11.Effect of the small-parameter α on the divergence speed of the axially-moving nanoplate.

Figure 12 . 19 Figure 13 .
Figure 12.Effect of the small-parameter α on dimensionless complex frequencies of the axially-moving nanoplate resting on a viscoelastic foundation.Appl.Sci.2019, 9, x FOR PEER REVIEW 17 of 19

Figure 13 .
Figure13.Effect of the axis force K on the divergence speed of the axially-moving nanoplate resting on a viscoelastic foundation.
or ∂M xx ∂x +

Table 1 .
Dimensionless natural frequencies for the axially-moving nanoplate without a foundation.