Nonlinear Vibration Analysis of Fractional Viscoelastic Nanobeam

Considering the size-dependent influence ignored by classical continuum mechanics, a new non-classical Euler–Bernoulli beam model is proposed in this paper. The new fractional viscoelastic nanobeam model is set up using the fractional Kelvin–Voigt viscoelastic model and Hamilton’s principle. And the new model studies the total effects of nonlocal elasticity, modified couple stress, and surface energy. The model represented the fractional integral-partial differential governing equation is solved by Galerkin’s and predictor–corrector methods. First, the effects of nonlocal elasticity, modified couple stress, surface energy, and their coupling impact on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed. Then, in the frame of nonlocal couple-stress elasticity and surface energy theory, the effects of different parameters on the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are discussed. The numerical results show that the fractional order must be considered in the modeling of viscoelastic nanobeam, and the system’s damping enhances by the increase of fractional order. Moreover, owing to the correlation between fractional order and excitation frequency, the nonlinear time responses of fractional order to free and forced vibration are different.


Introduction
Small structural units, such as beams, plates and shells, are often used as components of micro-and nano-electromechanical systems (MEMS and NEMS), sensors, and actuators.Due to the high sensitivity and application potential of high-frequency devices, those have been widely concerned by the research community [1][2][3][4][5].Experiments [6,7] and atomic simulations [8][9][10] show that the mechanical and material properties of nanoscale structures are size-dependent.Although classical continuum mechanics is widely used, it cannot describe the size effect of mechanical properties and material properties of nanoscale structures [11][12][13][14].Because the classical continuum elasticity theory cannot capture the atomic properties of nanomaterials.Therefore, the high-order continuum mechanics method is widely used in nanoscale structure modeling [15][16][17][18].The development of higher-order continuum theory can be traced back to Piola's [19,20] earliest works in the 19th century and Cosserat's work in 1909.However, it was not until the 1960s that the ideas of Cosserat brothers were widely concerned by researchers, and many high-order continuum theories were developed.Generally speaking, these theories can be divided into three different theories: strain gradient theory [21,22], couple stress theory [23,24] and nonlocal elasticity theory [11,25].The surface energy effect is also significant in studying nanoscale effect.Gurtin and Murdoch [26,27] first proposed a mathematical framework for studying the interface with surface energy effect based on thin film theory.The interface with surface stress is simulated by thin film, which can only bear in-plane stress without bending stiffness.In the past ten years, this theory has received significant attention, and many studies have been done on nano solids or structures [28,29].
Since the stress-strain relationship under viscoelastic medium load is time-dependent, that is, the stress is not only related to the current strain, but also to the history strain.Therefore, viscoelastic microstructures have been studied by many researchers.With the development of fractional calculus, it is widely used in physics, mechanics, bioengineering, and other fields [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].Fractional calculus has "memory," which effectively deals with viscoelastic problems [46][47][48][49][50][51].As a new subject, fractional viscoelastic mechanics is an important content of fractional solid mechanics, an essential part of fractional continuum mechanics, and has become an international frontier mechanical subject.At the same time, the application and research of fractional calculus in the engineering field are booming.The fractional viscoelastic theory provides a new idea for the study of mechanical properties of structures.In recent years, researchers have combined fractional order theory, viscoelastic theory, and size-dependent theory to study the mechanical properties of fractional viscoelastic structures.Most of the existing literatures study the nonlinear vibration of fractional viscoelastic nanobeam with a single effect.Ansari and Oskouie et al. [52,53] analyzed the linear and nonlinear vibration of fractional viscoelastic nanobeam based on the surface stress theory and fractional viscoelastic model.Taking the Euler-Bernoulli beam and Timoshenko beam as examples, they studied the effects of different parameters, such as initial displacement, fractional order, viscoelasticity coefficient, and surface parameters, on the nonlinear vibration time response of fractional viscoelastic nanobeam.They also found that the surface effect can make nanobeam stiffer or softer, depending on the surface parameter's value.And based on the nonlocal elastic theory, the linear and nonlinear vibration problems of fractional viscoelastic nanobeam are studied [54,55].Furthermore, the effects of nonlocal parameter, fractional order, and viscoelasticity coefficient on the linear and nonlinear time responses of fractional viscoelastic nanobeams are discussed in detail, and the time responses of linear and nonlinear models are compared.In addition, they also studied the linear and nonlinear vibrations of fractional viscoelastic Timoshenko micro-/ nano-beam using the strain gradient theory [56].Furthermore, the effects of fractional order, viscoelasticity coefficient, and length scale parameter on the time response of small-scale beam are studied.Hosseini-Hashemi et al. [57] studied the vibration analysis of two-phase local/nonlocal viscoelastic nanobeam with surface effect.The results show that the both parts of the complex frequencies are affected by the surface elasticity, which mainly depend on the length and cross section size of the nanobeam.Based on the modified couple stress theory, Loghman et al. [58,59] studied the nonlinear free and forced vibrations of fractional viscoelastic nanobeam.The results show that increasing the fractional order can increase the system's damping, and fractional order modeling can change the structure's performance.Therefore, the influence of fractional derivative must be considered in modeling viscoelastic behavior.In addition, because there is an essential correlation between the fractional derivative and the excitation frequency, the influence of the fractional derivative on the forced vibration and free vibration response is different.Nešić et al. [60] studied the nonlinear vibration of a nonlocal functionally graded beam on the fractional visco-Pasternak foundation.The numerical results show that the response amplitude is greatly affected by the parameters of the power law exponent and visco-Pasternak foundation, and the length scale and the nonlocal parameters have little impact on it.Meanwhile, the fractional order's effect on the system's damping is discussed.
With the advent of nanotechnology, many factors have led to the breakdown of classical continuum mechanics.The main influencing factors are surface energy, long-range cohesion interaction, and microstructure degree of freedom.The above literature survey shows that the current research is based on the influence of different parameters on the nonlinear vibration time response of fractional viscoelastic nanobeam under a single factor.However, there is no research on the comprehensive influence of multiple factors.And small-scale and surface effects are two important characteristics of submicron structures, which must be considered when using continuum mechanics methods to study the mechanical behavior of submicron structures.Therefore, we need to conduct the research.This paper will study the comprehensive effects of these three factors to develop a new non-classical beam model, that is, the nonlocal couple-stress elastic continuum model, considering the influence of surface energy.The advantage of this model is that it can study the small size and surface effect comprehensively.To illustrate the problem, taking the fractional viscoelastic Euler-Bernoulli nanobeam as an example, the fractional Kelvin-Voigt viscoelastic model is used to model and analyze the nanobeam.The governing equation considering the simultaneous effects of nonlocal couple-stress elasticity and surface energy theory is derived using Hamilton's principle.In the process solving, the Galerkin method and the predictor-corrector method are used to solve the fractional integral-partial differential equation.First, the effects of nonlocal elasticity, modified couple stress, surface energy theory, and their coupling effect on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed.Then, in the context of nonlocal couple-stress elasticity and surface energy theory, the effects of different parameters on the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are discussed.The model proposed in this paper can be used as a basic model to study the influence of the small-scale and surface effects on the vibration behavior of fractional viscoelastic nanostructures.

Theoretical Formulations
In this paper, a fractional viscoelastic nanobeam with length L, width b, and thickness h excited by transverse harmonic is considered, as shown in Fig. 1.Based on the Euler-Bernoulli beam theory, the effects of the rotary inertia and shear stresses are neglected, and the motion equation of fractional viscoelastic nanobeam subjected to a transverse harmonic excitation is derived.
According to the Euler-Bernoulli beam theory, the component of the displacement field at any point in the beam can be expressed as [52] in which u x , u y , u z are the displacements of any point on the beam along the x, y, and z axes, respectively.u(x, t), w(x, t) are the axial displacement of the section centroid and the transverse deflection of the beam, respectively.
To establish a new model considering both small-scale and surface effects, the influence of different parameters on the nonlinear vibration time response of fractional viscoelastic nanobeam is further studied.Consider the influence of a group of size-dependent factors neglected in classical continuum mechanics, including microstructure degree of freedom, surface energy, and long-range cohesive interaction.Their corresponding theories are given below.

The Modified Couple Stress Theory (MCST)
According to MCST, the strain energy density of materials is not only related to the strain tensor ij , but also to the symmetric rotation gradient tensor ij .Therefore, the strain energy of nanobeam can be expressed as [58,59,61] in which ij , ij are the stress tensor and strain tensor, m ij , ij are higher-order couple stress tensor and symmetric rotation (1) gradient tensor.The symmetric rotation gradient tensor ij is related to the rotation vector , and the formula is as follows [59,61] in which u is the displacement vector.
Using the fractional Kelvin-Voigt viscoelastic model [62], the higher-order couple stress tensor is defined as in which is the order of the fractional derivative, g is vis- coelasticity coefficient, l is the length scale parameter, is the shear modulus.
The fractional derivative in the Kelvin-Voigt viscoelastic constitutive model is the Riemann-Liouville type, which is defined by the following formula [52,63] The rotation vector component and symmetric rotation gradient tensor can be obtained by calculation Thus, the component of the couple stress tensor is According to the assumption of von-Kármán geometric nonlinearity, the relationship between strain and displacement can be expressed as [52] According to the linear fractional Kelvin-Voigt viscoelastic model, the normal stress is as follows: [52,53]

The Surface Elasticity Theory
The surface effect is one of the unique properties of nanomaterials, which greatly affects the mechanical properties of nano-materials.It is also an essential aspect of the study of nanostructures.According to the Gurtin-Murdoch surface elasticity theory, the surface constitutive equations are as follows: [52] in which , = x, y , s and s denote the surface Lamé con- stants, is the Kronecker delta, and s is the residual surface stress under unstrained conditions.
According to Eqs. ( 15) and ( 16), the surface stress components can be calculated as follows: According to the surface equilibrium condition in the Gurtin-Murdoch model, zz cannot be ignored.To solve this problem, we assume that zz varies linearly through the beam thickness and satisfies the surface balance conditions.Thus, zz can be expressed as [64] (11) where the superscripts s+ and s− indicate the upper layer and lower layer of the surface, respectively.Substituting Eq. ( 18) into Eq.( 19), zz can be expressed as Therefore, by including the zz based on the Eq. ( 20) in the component of stress for the bulk of the nanobeam [65], xx can be modified as

The Nonlocal Differential Constitutive Relation
In the nonlocal elasticity theory, stress at a point in the continuum is related not only of the strains at that point, but also of the strains at all other points of the continuum.According to nonlocal elasticity theory [12][13][14], the nonlocal constitutive formula is as follows: in which * ij is the nonlocal stress tensors, ij is the classical stress tensors, ∇ 2 = 2 ∕ x 2 is the Laplacian operator.e 0 is the material constant, a is the internal characteristic length, and c is the external characteristic length.
By applying Eq. ( 22) into Eqs.( 11), (17), and ( 21), the following are the relations between the nonlocal and local stress: 1 − (e 0 a) in which e 0 a is the nonlocality parameter.

Derivation of Governing Equations
The first variation in strain energy of nonlocal couple-stress fractional viscoelastic nanobeam considering surface stress effect can be calculated as in which Here, N * xx and M * xx denote the nonlocal axial stress resultant and bending moment resultant of the bulk material, N s * xx (25) 1 − (e 0 a)2 2 and M s * xx denote the nonlocal surface axial stress resultant and bending moment resultant of the surface material, and Y * cxy denotes the nonlocal couple-stress moment of the bulk material.A is the cross-sectional area of the nanobeam and A is the boundary of A.
The kinetic energy of fractional viscoelastic nanobeam can be written as The variation of the kinetic energy is The external distributed force on the nanobeam is F(x) cos( t) , as shown in Fig. 1.The variation of virtual work by external force and non-conservative viscous damping force are as follows: in which c 0 is viscous damping with coefficient.
The governing differential equations of fractional viscoelastic nanobeam are derived using Hamilton's principle: Substituting Eqs. ( 26), ( 29)-(31) into Eq.( 32), the governing equations of motion of the nonlinear nanobeam are obtained as ( 28) Also, the associated boundary conditions at x = 0 and x = L are Herein, the simply supported boundary conditions (SS) and clamped boundary conditions (C) at x = 0 and x = L are, respectively To obtain the nonlocal governing equations of viscoelastic nanobeam considering surface stress effect and modified couple stress, according to the constitutive Eqs. ( 23)-( 25) and the stress resultants in Eq. ( 27), the nonlocal xx and Y * cxy can be defined as ( 34) (42) Furthermore, as the axial inertia is neglected, Eq. ( 33) is simplified as 45) into Eq.( 42) and integrate with respect to x, it can be concluded Using Eq. ( 34) and Eqs. ( 43)-( 45), the nonlocal governing equation of the fractional viscoelastic nanobeam considering the surface stress effect and the modified coupling stress can be expressed as Substituting Eqs. ( 39), (40) into Eq.( 47) and using Eq. ( 46), the governing equation can be derived as ( 43) in which By introducing the following non-dimensional parameters, (48) in which I = ∫ A z 2 dA = bh 3 ∕12 denotes the moment of iner- tia of the rectangular cross-section.Eq. ( 48) can be rewritten as The dimensionless equation of simply supported boundary conditions (SS) and clamped boundary conditions (C) after simplification at x = 0 and x = 1 are, respectively in which the superscript * in Eqs. ( 51), (52), and (53) are dropped for simplicity.

Solution Procedure
In practical applications, it is difficult to obtain analytical solutions for many equations.Numerical schemes have been widely used due to the features of no analytical solutions, easy programming, strong adaptability, and the ability to solve very complex problems.There are many methods for vibration analysis, while solving this kind of problem with the numerical scheme is the most direct method for vibration analysis, which is one of the effective means to study dynamic response, and can analyze the dynamic response of the system under arbitrary excitation.Furthermore, Galerkin's and predictor-corrector methods are one of the appropriate methods to solve the nonlinear vibration problem of fractional viscoelastic nanobeam.In this subsection, Galerkin's and predictor-corrector ( 51) x 2 cos(Ωt) methods are used to study the geometrically nonlinear free and forced vibrations of fractional viscoelastic nanobeam.First, the fractional integral-partial differential governing equation is converted into a fractional ordinary differential equation by the Galerkin method.Then, the obtained equation is expressed in a more effective state-space form.Eventually, the predictor-corrector method is used to solve the nonlinear fractional timedependent equations and obtain the time response of fractional viscoelastic nanobeam.

Galerkin Method
To solve Eq. ( 51), the Galerkin method is used.Therefore, the solution of Eq. ( 51) can be approximated as in which p n (x) = sin(n x) is the mode function correspond- ing to the simply supported beam and n (t) denotes the time- dependent function.It can be proved that w(x, t) satisfies boundary condition Eq. ( 52).Substituting Eq. ( 54) into Eq.( 51) and applying the Galerkin method, Eq. ( 51) can be written as where the external force is distributed according to the shape of the first mode i.e., F(x) = fp 1 (x) , f is the amplitude.To solve Eq. ( 55), rewrite it to the following state space form in which X j1 0 represents the initial value of j .The fractional derivative in Eqs. ( 56) and ( 57) is of Caputo type, which is defined by the following formula The relationship between Riemann-Liouville fractional derivative and Caputo fractional derivative is as follows: (56)

Predictor-Corrector Method
Equation ( 57) is solved by the predictor-corrector method [66].In the process of solving, the initial velocity is taken to be zero and the explanation of this method is as follows: in which  > 0 , and approximate the solution of Eq. (60)  with Volterra integral equation The integral form of Eq. ( 61) is discretized in the time domain, and the time domain grid points are set to in which the total number of grid points is S. Further, Eq. ( 61) can be written in the following discrete form (60) The above process is the Adams-Bashforth-Moulton predictor-corrector method [67][68][69], commonly used to study fractional order system dynamics.
With the influence of a single size-dependent factor, the influence of different parameters on the nonlinear vibration time response of fractional viscoelastic nanobeam has been studied previously based on Galerkin's and predictor-corrector methods [52][53][54][55][56].However, the study of the effects of different parameters on the nonlinear vibration time response of the fractional viscoelastic nanobeam under the combined influence of multiple factors is absent.Therefore, the comprehensive impact of multiple factors will be studied in the following subsection.

Numerical Results and Discussion
In this subsection, the nonlinear time response of vibration of fractional viscoelastic nanobeam based on nonlocal couple-stress elasticity and surface energy theory is studied.The following parameters are used in the numerical simulation [52]: In the following analyses, CL, NL, SE, and CS refer, respectively, to the classical (l = = 0) , and couple stress ( s = s = s = s = = 0) .NL-SE denotes a nonlocal nanobeam with surface effect without considering the modified couple stress theory; NL-CS denotes a nonlocal couple-stress elasticity nanobeam without surface effect; CS-SE denotes a nanobeam considering the modified couple stress theory and surface effect without nonlocal elasticity; NL-CS-SE denotes a nanobeam considering nonlocal couple-stress elasticity and surface effect.Since we mainly studied the influence of different parameters on the nonlinear vibration time response of fractional viscoelastic nanobeam under the comprehensive influence of multiple factors, one term of Galerkin can be used to explain the problem, and most of the existing studies use one term of Galerkin to calculate this kind of problem.Therefore, this paper selects one term of Galerkin for calculation, and the time step is 0.01.Fig. 2 shows a comparison of the results of nonlinear time response of free vibration of a fractional viscoelastic simply supported nanobeam considering only the surface stress effect with those of Oskouie et al. [52].It can be seen from the figure that there will be a deviation between the two curves over time, as Oskouie et al. [52] did not consider the thickness h when calculating the axial stress resultant ( N s xx ) caused by the surface stress effect in Eq. ( 41).Moreover, s+ 2 w∕ t 2 , s− 2 w∕ t 2 in Eq. ( 19) and bh 3 ∕12 + s bh 2 ∕2 + h 3 ∕6 2 w∕ x t 2 in Eq. ( 28) have the little effect ignored by Oskouie et al. [52].It can also be found from the figure that the nonlinear vibration frequency in this paper is greater than Oskouie et al. [52].It is mainly because considering thickness h in the calculation of axial stress resultant ( N s xx ) caused by the surface stress effect will make the stiffness of the nanobeam higher.
Figure 3 shows the effects of NL, SE and CS on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam, respectively.As can be seen from Fig. 3, compared with the classical beam theory, CS will increase the amplitude peak and frequency of nonlinear time response, and the maximum dimensionless amplitude attenuation speed is greater, while NL will reduce the amplitude peak and frequency of nonlinear response, and the maximum dimensionless amplitude attenuation speed is smaller.This is because the modified couple stress theory improves the stiffness of the nanobeam, and nonlocal parameters have to soften the effect on the beam.Moreover, compared with the classical beam theory, the influence of SE on the nonlinear response is related to the value of surface parameters.Under the parameter conditions in this paper, SE will increase the amplitude peak and frequency of the nonlinear time response.This is because the surface effect increases the stiffness of the nanobeam.It is further found that CS has more obvious stiffness hardening effect than SE under the parameter conditions in this paper.
From the above analysis, it can be seen that NL, SE and CS, a group of size-dependent factors, impact the nonlinear vibration-time response of fractional viscoelastic nanobeam.Therefore, their coupling effects on the nonlinear vibration time response of fractional viscoelastic nanobeam will be discussed below.
Figure 4 shows the effects of NL coupled with CS, SE and CS-SE on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that compared with NL, NL-CS, NL-SE and NL-CS-SE will increase the amplitude peak and frequency of nonlinear vibration time response of nanobeam.This is because both CS and SE will increase the stiffness of nanobeam.By comparing NL, NL-CS and NL-SE, it can be seen that the addition of CS has a more significant impact on the frequency of the nonlinear vibration-time response of the nanobeam, while the addition of SE has a more significant impact on the amplitude peak.Comparing NL-CS-SE with NL-SE and NL-CS, respectively, it can be seen that the addition of CS will increase the amplitude peak and frequency of nonlinear vibration response, and the addition of SE will increase the amplitude peak, but has little effect on the frequency.
Figure 5 shows the effects of NL-SE and NL-CS on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.By comparing NL-SE with SE and NL-CS with CS, it can be seen that the addition of NL reduces the amplitude peak and frequency of nonlinear vibration-time response of nanobeam.This is because nonlocal parameter have to soften the effect on nanobeam.Moreover, by comparing SE with CS and NL-SE with NL-CS, it can be seen that CS can make the nonlinear vibration response frequency of the nanobeam higher, but the amplitude peak smaller.Figure 6 shows the effects of CS-SE and NL-CS-SE on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.By comparing CS-SE with CS and SE, respectively, it can be seen that the addition of SE will increase the amplitude peak of nonlinear vibration of nanobeam, and the addition of CS will increase the amplitude peak and frequency of nonlinear vibration of nanobeam.By comparing CS-SE with NL-CS-SE, it can be seen that the addition of NL will reduce the frequency and amplitude peak of nonlinear vibration of nanobeam.This is due to the softening effect of nonlocal parameters on the beam.
From the above analysis, it can be seen that NL, SE, CS and their coupling effects all affect the nonlinear vibration response of fractional viscoelastic nanobeam.The current research is based on the influence of different parameters on the nonlinear vibration time response of fractional viscoelastic nanobeam under a single factor, but there is little research on the comprehensive influence of multiple factors.However, small-scale and surface effects are two critical characteristics of submicron structures, which must be considered when using continuum mechanics methods to study the mechanical behavior of submicron structures.Therefore, based on the nonlocal couple-stress elasticity and surface energy theory, the effects of initial displacement, fractional order, viscoelasticity coefficient, surface parameters, dimensionless length scale parameter, dimensionless nonlocal parameter, length-to-thickness ratio, damping coefficient, force amplitude, and excitation frequency on the time responses of nonlinear free and forced vibration of nanobeam are studied in the following.
The comparison of linear and nonlinear time responses of free vibration of fractional viscoelastic simply supported nanobeam with different initial displacements is shown in Fig. 7.It can be seen from the figure that due to the viscous effect and damping, the maximum dimensionless amplitude attenuates with time.The larger the initial displacement, the larger the nonlinear frequency, the more pronounced the attenuation of the maximum dimensionless amplitude, the greater the attenuation speed, and the greater the influence of geometric nonlinearity.This is because the nonlinear vibration frequency increases with the increase of the amplitude peak.Moreover, with the decrease of the initial displacement, the nonlinear time response of the nanobeam converges to the linear time response, and the nonlinear frequency is close to the linear frequency.Compared with linear nanobeam, nonlinear fractional viscoelastic nanobeam have higher stiffness.
Figure 8 shows the effect of fractional order on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.It can be seen that the fractional order has a strong influence on the nonlinear vibration-time response of nanobeam.As seen from the figure, with the increase of fractional order, the amplitude peak and vibration frequency decrease.Moreover, it can be seen from Fig. 8 that the system's damping increases with the increase of fractional order and the free vibration response is damped faster by increasing the fractional order.For instance, as can be observed from the figure, for = 0.9 after time t = 2 , the response is damped, but for other examples of , this damping happens slower.It can be found that the greater the fractional derivative, the more obvious the influence on the peak of the response amplitude.For instance, increasing from = 0.3 to = 0.6 can decrease peaks of the response amplitude about 20 percent, but increasing from = 0.6 to = 0.9 affects peaks of the response amplitude about 50 percent.Figure 9 analyzes the influence of the viscoelasticity coefficient on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.When the viscoelasticity coefficient is zero (g=0), it is the elastic model.It can be seen from the figure that the system damping increases with the increase of the viscoelasticity coefficient.Moreover, increasing the viscoelasticity coefficient will increase the damping speed, reduce the amplitude peak, and suppress the vibration behavior of the system faster.Therefore, the viscoelastic model has both damping and viscous effects compared with the elastic model.
Figure 10 shows the effect of surface elastic modulus s + 2 s on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.When the surface elastic modulus is positive, the stiffness of the structure is enhanced, while when the surface elastic modulus is negative, the stiffness of the structure is reduced.Therefore, compared with the zero value of the surface elastic modulus, its positive and negative values increase and decrease the nonlinear vibration frequency of the fractional viscoelastic nanobeam, respectively.The higher the surface elastic modulus, the higher the nonlinear frequency.Moreover, the amplitude peak when the surface elastic modulus is zero is between the amplitude when it is positive and negative.It is further found that the nonlinear vibration frequency is more affected by the surface elastic modulus than the amplitude peak.
Figure 11 shows the effect of residual surface stress s on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that the nonlinear vibration frequency and amplitude peak when the residual surface stress is zero are between the nonlinear vibration frequency and amplitude peak when it is positive and negative.Because the increase of residual surface stress will exert pre-tensile stress on the structure, thus increasing the stiffness of the fractional viscoelastic nanobeam.Therefore, compared with the surface residual stress taking zero, its positive and negative values increase and decrease the nonlinear vibration frequency, respectively.The nonlinear vibration frequency and amplitude peak increase with the increase of residual surface stress.The damping of the system decreases with the increase of residual surface stress.
Figure 12 shows the effect of surface mass density s on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.Because increasing the surface mass density increases the density of the structure.Therefore, the nonlinear vibration frequency decreases with the increase of the surface mass density.The amplitude peak increases with the increase of surface mass density.Figures 10,11,and 12 show the influence of surface parameters on the nonlinear vibration-time response of fractional viscoelastic nanobeam.According to the above analysis, considering the surface stress effect may make the nanobeam harder or softer, depending on the value of the surface parameters.
Figure 13 shows the effect of dimensionless nonlocal parameter on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that the nonlinear vibration frequency and amplitude peak decrease with the increase of nonlocal parameter.It is because the larger the nonlocal parameter, the more pronounced the stiffness softening effect of the nanobeam.Moreover, the attenuation speed of the maximum dimensionless amplitude decreases with the increase of the dimensionless nonlocal parameter.
Figure 14 shows the effect of dimensionless length scale parameter on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that with the increase of the ratio, the nonlinear vibration frequency and amplitude peak value increase.It is because the increase of the length scale parameter will increase the stiffness of the nanobeam.Moreover, compared with the amplitude peak, the nonlinear vibration frequency is more affected by the length scale parameter.Figures 13 and 14 show the influence of small-scale effect on the nonlinear vibration-time response of fractional viscoelastic nanobeam.
Figure 15 shows the effect of different length-to-thickness ratios on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam when the nanobeam thickness is fixed.It can be seen from the figure that the frequency increases with the increase of the ratio, and the amplitude peak decreases with the increase of the ratio.Moreover, with the increase this ratio, the free vibration response of the system can be damped faster.Therefore, increasing or decreasing the ratio can change the frequency and maximum amplitude of the response.
Figure 16 shows the effect of damping coefficient on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam.When the damping coefficient is zero ( c 0 = 0 ), it is undamped free vibration.It can be seen that the amplitude peak and vibration frequency decrease with the increase of the damping coefficient.Compared with the undamped free vibration of viscoelastic nanobeam, the influence of damped vibration on the amplitude peak is more prominent and attenuates faster.The greater the damping, the faster the equilibrium state will be reached.Damped vibration is a more practical kind of motion.Above, the effects of initial displacement, fractional order, viscoelasticity coefficient, surface parameters, dimensionless length scale parameter, dimensionless nonlocal parameter, length-to-thickness ratio, and damping coefficient on the nonlinear time response of free vibration of fractional viscoelastic simply supported nanobeam are analyzed.Next, the effects of force amplitude, fractional order, viscoelasticity coefficient, and excitation frequency on the nonlinear time response of forced vibration of fractional viscoelastic simply supported nanobeam are studied.
Figure 17 shows the effect of force amplitude on the nonlinear time response of forced vibration of fractional viscoelastic simply supported nanobeam.It can be seen that the response amplitude of the nonlinear forced vibration of the nanobeam increases with the increase of the force amplitude, and the larger the force amplitude, the greater the change of the amplitude peak.Moreover, the vibration period decreases with the increase of this parameter.l/h=0 l/h=0.25 l/h=0.5 Figure 18 shows the effect of fractional order on the nonlinear time response of forced vibration of fractional viscoelastic simply supported nanobeam.As can be seen from the figure, the change of the amplitude peak decreases as the fractional order increases.Moreover, increasing the fractional order will increase the system's damping and vibration period, reducing the response's amplitude peak.
Figure 19 shows the effect of the viscoelasticity coefficient on the nonlinear time response of forced vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that the change of the amplitude peak of vibration of the viscoelastic model is smaller than that of the elastic model.Moreover, the damping of the system increases with the increase of the viscoelasticity coefficient, resulting in the decrease of the amplitude peak of the response.
Figure 20 shows the effect of dimensionless excitation frequency on the nonlinear time response of forced vibration of fractional viscoelastic simply supported nanobeam.It can be seen from the figure that the nonlinear vibration frequency of nanobeam increases with the increase of excitation frequency.However, with the increase in excitation frequency, the amplitude peak increases first and then decreases.This is because resonance occurs when the excitation frequency is equal to the natural frequency of the system, in which case the amplitude peak reaches the maximum value.Figure 21 shows the nonlinear time response of forced vibration of a fractional viscoelastic simply supported nanobeam under different fractional orders and different excitation frequencies.It can be observed that there is a correlation between the effect of the excitation frequency and the fractional order.The influence of fractional order in the case of near resonance is significantly more potent than in other cases.In the case of far from resonance, increasing from 0.3 to 0.9, the influence on the amplitude peak of the steady-state response is less than that of near resonance.Therefore, choosing the proper fractional order in the resonance case is crucial for viscoelastic material modeling.Moreover, it can be seen that there is a difference between the forced and free responses of fractional viscoelastic nanobeam.

Conclusion
In this paper, the nonlinear vibration of fractional viscoelastic nanobeam is studied.Based on nonlocal couple-stress elasticity and surface energy theory, the fractional viscoelastic nanobeam are modeled using Hamilton's principle and the fractional Kelvin-Voigt viscoelastic model.The advantage of this new model is that it can consider both the small-scale effect and the surface effect.First, the effects of nonlocal elasticity, modified couple stress, surface energy, and their coupling impact on the nonlinear time response of free vibration of fractional viscoelastic nanobeam are analyzed.Then, the nonlinear time responses of free and forced vibration of fractional viscoelastic nanobeam are studied in the context of the nonlocal couple-stress elasticity and the surface energy theory.The Galerkin method and predictor-corrector method are used in the solving process to solve the fractional integral-partial differential equation.
In the free vibration analysis, the effects of initial displacement, fractional order, viscoelasticity coefficient, surface parameters, dimensionless length scale parameter, dimensionless nonlocal parameter, length-to-thickness ratio, and damping coefficient on the nonlinear vibration time response of fractional viscoelastic nanobeam are studied.The numerical results show that with the decrease of the initial displacement, the response of the nonlinear solution of the fractional viscoelastic nanobeam converges to the response of the linear solution, and the frequency of the nonlinear model is higher than that of the linear model.The system's damping increases with the increase of initial displacement, fractional order, viscoelasticity coefficient, damping coefficient, and length-to-thickness ratio.Moreover, the system's vibration frequency increases with the increase of initial displacement, surface elastic modulus, residual surface stress, dimensionless length scale parameter, length-tothickness ratio, and decreases with the increase of fractional order, surface mass density, dimensionless nonlocal parameter, and damping coefficient.The amplitude peak increases with the increase of initial displacement, residual surface stress, dimensionless length scale parameter, and decreases with the increase of fractional order, viscoelasticity coefficient, dimensionless nonlocal parameter, length-to-thickness ratio, and damping coefficient.The effects of force amplitude, fractional order, viscoelasticity coefficient, and excitation frequency on the nonlinear time response of forced vibration of fractional viscoelastic nanobeam are also studied in this paper.It can be found that the amplitude peak increases with the increase of force amplitude and decreases with the increase of fractional order and viscoelasticity coefficient.With the increase of excitation frequency, the amplitude peak increases first and then decreases, reaching the maximum at resonance.The system's damping increases with the increase of fractional order and viscoelasticity coefficient.Moreover, the nonlinear vibration frequency of fractional viscoelastic nanobeam increases with the decrease of fractional order and the increase of force amplitude and excitation frequency.The change of the amplitude peak increases with the increase of force amplitude and the decrease of fractional order, and the change of the viscoelastic model is less than that of the elastic model.It should be pointed out that the fractional order has different effects on the responses of forced and free vibration.It can be observed that there is a correlation between the fractional order and excitation frequency in nonlinear forced vibration.It can be seen from the observation that the fractional order near the resonance has a significant influence on the amplitude peak.Therefore, choosing the proper fractional order in the resonance case is very crucial for viscoelastic material modeling.
The results presented in this paper can be applied to the damping modeling of fractional viscoelastic nanostructures, as well as the design and optimization of nanoscale sensors.Meanwhile, this study accurately measures the mechanical properties of viscoelastic nanobeam and design-related measurement techniques and devices based on them.Moreover, the nonlinear dynamic characteristics of the forced vibration response and the influence of the fractional derivative on the forced vibration at resonance need to be further studied.