Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of thematerial length scale parameter and the axial compressive force on the frequency ratios ofmicrobeams; and effect of thematerial length scale parameter on the buckling load ratios of microbeams are investigated in this paper.


Introduction
With the development of science and technology, micro/ nanostructures have become increasingly important in our life and technology.Micro/nanostructures are widely used in engineering, such as microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), sensors, actuators, and microelectronics.Although the size is very small, the response of these structures while they are in operating should be interesting.In fact, many researchers have been interested in the study of MEMS/NEMS structures [1][2][3][4].
Normally, MEMS/NEMS are often modeled into micro/ nanobeams or micro/nanoplates.The classical elastic theory is no longer effective when studying these structures.Thus, several higher-order continuum theories have been developed such as the nonlocal elasticity theory [5], surface elasticity [6], micropolar theory [7], strain gradient theory [8], and couple stress theory [9,10].Besides the two classical elastic constants (Lamé constants), the couple stress theory also contains two material constants (the length scale parameters).Although only two material constants are required, application of this theory was still difficult; therefore, Yang et al. [11] modified the couple stress theory by introducing an applicable theory in which only one additional material constant besides Lame' constants is considered.The theory of Yang et al. was then widely used by many authors to study response of elastic structures.A nonclassical Mindlin plate model was developed by Ma et al. [12] using the modified couple stress theory.Buckling analysis of functionally graded microbeams based on the modified couple stress theory was investigated by Nateghi et al. [13]; in the work, it can be observed that buckling loads deviate significantly from classical elastic theory, especially for thin beams.Bending and vibration of functionally graded (FG) microbeams were investigated by S ¸ims ¸ek and Reddy using a new higher-order beam theory and the modified couple stress theory [14]; the influences of the material length scale parameter on the bending and free vibration behavior of FG microbeams were presented.Microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material were developed by Reddy based on modified couple stress theory.The modified couple stress theory was developed for third-order shear deformation functionally graded microbeam in the work done by Salamattalab et al. [15].Roque et al. [16] used the modified couple stress theory and the meshless method to study the bending of simply supported laminated composite beams subjected to transverse loads.Lou and He [17] used the modified couple stress theory and the Kirchhoff/Mindlin plate theory together with the von Karman's geometric nonlinearity to study the nonlinear bending and free vibration responses of a simply supported FG microplate resting on an elastic foundation.Using the modified couple stress theory and Hamilton's principle, a new model contains a material length scale parameter which can capture the size effect, unlike the classical Timoshenko beam theory investigated by Ma et al. [18].Khaniki and Hashemi School [19] analysed free vibration of nonuniform microbeams based on the modified couple stress theory.A nonclassical beam theory was developed for static and nonlinear vibration analysis of microbeams resting on a three-layered nonlinear elastic foundation based on the modified couple stress theory and Euler-Bernoulli beam theory together with the von-Kármán's geometric nonlinearity; the work was done by S ¸ims ¸ek [20].Analytical solutions of stability problem for axially loaded nanobeams based on strain gradient elasticity and modified couple stress theories were presented by Akgöz and Civalek [21].And, nonlinear free vibration analysis of microbeams resting on the viscoelastic foundation was investigated by Jam et al. [22] using the modified couple stress theory.
Static, buckling, postbuckling and nonlinear vibration analyzing of beams are studied by many authors.Nonlinear dynamic response problem of Spined-Spined (S-S) and Clamped-Clamped (C-C) beams at large vibration amplitudes was studied by Azrar et al. [24].Free nonlinear vibration and postbuckling analysis of functionally graded beams resting on nonlinear elastic foundation were investigated by Fallah and Aghdam [23].S ¸ims ¸ek and Kocatürk [25] analyzed free vibration and dynamic behavior of a functionally graded simply-supported beam under a concentrated moving harmonic load.Postbuckling and free vibrations of composite beams were studied by Emam and Nayfeh [26].Nonlinear static and free vibration behaviors of a sizedependent nonlinear Timoshenko microbeam were studied by Asghari et al. [27] using the strain gradient theory and the variational approach.Response of a microbeam resting on a non-linear elastic foundation was investigated by Sari and Pakdemirli [28].The nonlocal Euler-Bernoulli beam theory was employed in the nonlinear free and forced vibration analysis of a nanobeam resting on a Pasternak foundation in the work of Togun and Bagdatlı [29].An accurate analytical solution for nonlinear vibration of microbeams based on strain gradient elasticity theory was obtained by Rajabi and Ramezani using homotopy analysis method [30].The static and dynamic response of geometrically imperfect composite beams was presented in work's Emam [31].Moeenfard et al. [32] applied the homotopy perturbation method to analyze free nonlinear vibration of clamped-clamped and clampedfree microbeams considering the effects of rotary inertia and shear deformation.The nonlinear static bending deformation, the postbucking problem, and the nonlinear free vibration of nonlinear microbeams based on the strain gradient theory were analyzed by Zhao et al. [33].Bending, buckling, and vibration of a functionally graded porous beam were presented by Fouda et al. using finite elements method [34].
The free vibration of FG beams was investigated by Alshorbagy et al. using numerical finite element method [35].And, Duy et al. [36] studied the free vibration of laminated FG-CNT reinforced composite beams by using finite element method.
The problems of nonlinear vibration described by nonlinear differential equations are very complex and therefore they are very difficult to solve.In most cases, the exact solutions of the nonlinear differential equations is not possible, and thus approximate solution techniques are very necessary.Some approximate analytical methods had been introduced such as Homotopy Perturbation Method (HPM) [37], Parameterized Perturbation Method (PPM) [38], Variational Iteration Method (VIM) [39], Variational Approach (VA) [40], Hamiltonian Approach (HA) [41], and Energy Balance Method (EBM) [42] which were introduced by He; Equivalent Linearization Method (ELM) was proposed by Caughey [43].
Until today, according to author's knowledge, there has no published article about postbuckling and nonlinear vibration of microbeams resting on three nonlinear elastic layers subjected to axial forces.Thus, in this paper, author focuses on analyzing of the postbuckling and nonlinear vibration of microbeams resting on three nonlinear elastic layers subjected to axial forces.Based on the modified couple stress theory and Hamilton's principle, the motion of microbeams is described by partial differential equation.Using Galerkin's method, this partial differential equation is reduced to the nonlinear ordinary differential equation.The Equivalent Linearization method with a weighted averaging [44,45] is applied to find the approximate solution of this problem.Comparison of the present results and the published ones shows accuracy of the present solutions.Effects of the material length scale parameter and the axial compressive load on nonlinear responds of Pinned -Pinned (P-P) and Clamped-Clamped (C-C) microbeams are investigated in this work.

Governing Equation
. .Modified Couple Stress eory.According to the modified couple stress theory [11], the strain energy for an isotropic linearly elastic material in domain  can be written as where  is the symmetric part of the Cauchy stress tensor,  is the strain tensor, m is the deviatoric part of the couple stress tensor,  is the symmetric curvature tensor,  and  are Lamé constants, and  is the material length scale parameter.The kinematic relations are as follows: where ∇ is the Hamiltonian differential operator, u is the displacement vector, and  is the rotation vector described as: Constitutive equations are described as where I is the unit matrix and Lamé's constants  and  can be expressed: In (7), E and ] are the elasticity modulus and Poisson's ratio, respectively, and  is known as the shear modulus.
. .e Governing Equation for Microbeam.An isotropic microbeam of a length L with cross-sectional dimensions h and b is considered and shown in Figure 1.The microbeam rests on a nonlinear elastic layer with the spring constants k  , k  , and k  of the Winkler elastic layer, Pasternak elastic layer, and nonlinear elastic layer, respectively.The displacement components of an initially straight beam on the basis of Euler-Bernoulli beam theory can be written as (, , ) =  (, ) , where u and w are the axial and the transverse displacement of any point on the neutral axis.The Von-Kármán's nonlinear strain-displacement relationship with assumptions of large transverse displacements where it is assumed that the nonlinear term corresponding to   / is small enough to be neglected, moderate rotations and small strains for a straight beam are given by: The components of the rotation vector are as follows: And the expression for the components of the symmetric curvature tensor: Hamilton's principle can be applied to derive the equations of motion of a microbeam.For a microbeam resting the nonlinear elastics foundation, the principle can be expressed as where   is the kinetic energy,   is the strain energy given in (1),   is the strain energy induced by the nonlinear elastic layer, and   is the work done by the external forces.
The kinetic energy can be obtained as where A=bh is the area of the cross-section and  is the mass density of the microbeam.
From (1), the strain energy can be expressed by  Here, N  , M  , and M  are the normal force, the bending moment, and a new stress resultant due to the couple stress, respectively, which are given by where I=bh 3 / is the inertia moment of the cross-section of the microbeam.The strain energy induced by the nonlinear elastic layer can be written as And, the work done by the external forces where p is the initial axial compressive force as showed in Figure 2. Substituting ( 15), ( 16), (20), and ( 21) into ( 14) and after some mathematical transformations and collecting the coefficients of  and , the following equations of motion are obtained (Note ( 17)-( 19)): We can see that ( 22) and ( 23) are the equations of motion of the microbeam respect to the displacements u and w.To reduce the governing equations into a single equation for the transverse deformation, the inplane inertia is assumed to be negligible, the axial normal force can be obtained as follows [18,31]: For a uniform microbeam, the equation of motion is derived in terms of w by substituting ( 24) into (23) as follows: We introduce the dimensionless parameters: Considering ( 26), ( 25) can be written in the nondimensional form as follows: Assume that displacement function (, ) can be expressed as follows: where () is the unknown time-dependent function and () is the basis function which satisfies the kinematic boundary conditions.For P-P and C-C microbeams, the basis functions are selected as + for Pinned-Pinned microbeam + for Clamped -Clamped microbeam where () is normalized so that (0.5)= 1.Using Galerkin method, the partial differential equation ( 27) is transformed into the following differential equation: Here the coefficients B 1 , B 2 , B 3 , and B 4 in (31) are defined as where  (4) ,   , and   are the fourth, second, and first derivative of  with respect to the axial coordinate , respectively.The microbeam is assumed to be subject to the following initial conditions: where  is the dimensionless maximum vibration amplitude of oscillator.

Analysing of Postbuckling and Free Nonlinear Vibration
. .Analysing of Postbuckling.From (31), the postbuckling load-deflection relation of the microbeam can be obtained as If neglecting the contribution of Q in (34), the linear buckling load can be determined as In the next sub-section, we will find approximate solution () of (31).
. .Analysing of Free Nonlinear Vibration.In this subsection, we use the equivalent linearization method with a weighted averaging [44,45] to find approximate solution of (31).First proposed by Caughey [43], the equivalent linearization method is an effective tool for analyzing random oscillators.This method can easily be applied to deterministic oscillations.In [44,45], Anh et al. had developed this method by using the weighted averaging instead of the convenial (classical) averaging [43,46].The equivalent linearization method with a weighted averaging was then used effectively to analyze strong nonlinear oscillations [45,47,48].
First, to simplify (31), we introduce new coefficients: Considering (36), (31) becomes The linearized equation of (37) has the form where the coefficient of linear term in (38)  2 can be determined by using the mean square criterion which minimizes the error between two equations ( 37) and ( 38): Thus, from we get in (41) is the approximate frequency of the microbeam.In (41), the symbol ⟨•⟩ denotes the time-averaging operator in classical meaning [46]:

Mathematical Problems in Engineering
For a -frequency function, the averaging process is taken during one period T, namely, The averaging values in (42) and ( 43) are called the classical/conventional averaging values.In this paper, the weighted averaging value proposed by Anh [44,45] is used to calculate averaging values in (41) instead of the conventional averaging values in (42) or (43).The idea of the proposed method is described as follows: replacing the constant coefficient 1/ in ( 42) and ( 43) by a weighted coefficient function ℎ().Thus, we get so-called a weighted average value: with the weighted coefficient function ℎ() satisfying the following condition: The weighted coefficient function ℎ() was in detail discussed in [44,45].Here, we use a specific form of the weighted coefficient function [42] as below: where s is a positive constant called the adjustment parameter.Equation (44) will take the form of ( 42) and ( 43) when s=0.
The solution of the linearized (38) is given by With the periodic solution of linearized (38) given in (47), the averaging values in (41) can be calculated by using (44) with the weighted coefficient function given in (46) and Laplace transform as follows: Substituting the averaging values in ( 48) and ( 49) into (41), we get the approximate frequency of oscillation: The approximate frequency obtained in (50) is not only a function of the initial amplitude  but also a function of the parameter s (in the expression of the weighted function ℎ() =  2  − ).With  = 2, we get the amplitude-frequency relationship: Thus, the approximate solution of oscillator is Substituting ( 29) and ( 30) into (32) and performing the integrations in (32), we can obtain the following expressions for the nonlinear frequency (Note (36)

Numerical Results and Discussions
To verify accuracy of the present solution, comparison of the present solution with the exact solution is investigated.
Note that the exact frequency of ( 37) is given by (55).Tables 1 and 2 list values of the exact frequencies and the present frequencies with some different values of the material length scale parameter (l/h), the coefficients of elastic foundation (K  , K  and K  ), the axial compressive load (P), and the initial amplitude  for P-P and C-C microbeams, respectively.We can see a good agreement between the current frequency and the exact frequency from these tables.Equation ( 37) is the cubic Duffing oscillator, the exact frequency can be given by [49] The relative error of the present solution is shown in Figure 3 with l/h=0.5,K  =50, K  =100, K  =50, and P=20.We can see from this figure and Tables 1 and 2 that the relative error of the present solution reaches to 0.15% when the initial amplitude  increases.

Comparison of frequency ratios of macrobeams for various values of the dimensionless initial amplitude (𝛼= , ,
, ) is presented in Table 3.Note that frequency ratios of macrobeams can be obtained by letting l/h= .Again accuracy of the present solution can be observed.The variation of the frequency ratios of microbeams (  /  ) with the material length scale parameter is shown in Figures 4 and 5.In these figures, we can see that the frequency ratios decrease when the material length scale parameter increases.Figure 4 is plotted with values of parameter: Poisson's ratio ]=0.3, the coefficients of the nonlinear elastic layer as K  =50, K  =50, and K  =50, the axial compressive load P=20, and some values of the initial amplitude  (=0.1,0.5, 1.0, 1.5 and 2.0).And, Figure 5 is plotted with values of parameter: Poisson's ratio ]=0.3, the coefficients of the nonlinear elastic layer as K  =100, K  =100 and K  =100, the initial amplitude =0.5, and some values of the axial compressive load P (P=0, 5, 10, 15, 20, and 25).We can see that with increasing in the length scale parameter (l/h) leads to decreasing in the frequency ratio of microbeam.With selected values of parameters of system and the initial amplitude =1.5, from Figure 4, we see that the frequency ratios decrease 1.18% and 4.42% when the length scale parameter increases 40% for P-P microbeam and C-C microbeam, respectively.Effects of the material length scale parameter (l/h) on the frequency ratio of microbeam (  /  ) based on parameters of system are presented in Figures 6-10.It can be observed in Figures 6, 9, and 10 that the frequency ratios of microbeams increase as the initial amplitude (), nonlinear foundation parameter (K  ) and axial load (P) increase.However, the frequency ratios of microbeams decrease as the Winkler parameter (K  ) and Pasternak parameter (K  ) increase (in Figures 7 and 8).In these figures, the Poisson's ratio is chosen equal to 0.3 (]=0.3).
Next, we will examine effect of the material length scale parameter (l/h) on the buckling load ratios of microbeam based on the initial amplitude and the coefficients of elastic layer.The results are presented in Figures 11-14.We can see from Figures 11 and 14 that with increasing in values       of the initial amplitude () and the nonlinear foundation parameter (K  ) leads to increasing in values of the buckling load ratios of microbeams.The opposite can be observed in Figures 12 and 13 and values of the buckling load ratios of microbeams descrease when the Winkler parameter (K  ) and the Pasternak parameter (K  ) increase.Noted that the postbuckling load P  is determined by maximizing (34) in one period of vibration of microbeams.Finally, in Figures 15-18, effect of the axial compressive load (P) on the frequency ratios of microbeams is investigated.It can be concluded from Figures 15-18 that when increasing of value of the axial load (P), value of frequency ratios increases.However, for each specific value of the axial load (P), the frequency ratios of microbeams increase as the initial amplitue () and the nonlinear foundation parameter (K  ) increases while the frequency ratios of microbeams decrease as the Winkler parameter (K  ) and the Pasternak parameter (K  ) increase.
Compressed by the axial force (P), microbeams will be buckled as value of P is very large, namelly the phenomenon of instability will occur.When value of P is very large, nonlinear frequencies of microbeams trend to zero and vibration amplitudes of microbeams will increase infinitely.This phenomenon can be observed in Figures 19 and 20 20, instability occurs with only one value of the material length scale parameter l/h=0.0.

Conclusion
Postbuckling and free nonlinear vibration of microbeams resting on three nonlinear elastic layers subjected to axial compressive load are investigated in this work for the first time.The equation of motion of microbeam resting on three nonlinear elastics layers can be obtained by using the modifided couple stress theory and Von Karman's assumption.The equivalent linearization method with a weighted averaging is used to get the frequency-amplitude and postbuckling load-deflection relationships of microbeam.Accuracy of the present solution is verified by comparing the current solution with the exact and published ones.Effects of the length scale parameter (l/h) and the axial compressive load (P) on respond of microbeam are studied in this paper.The frequency ratio (  /  ) decreases as the length scale parameter increases, and the axial compressive load has the reverse effect on the frequency ratio.
Effects of the material length scale parameter (l/h) and the axial compressive load (P) on the frequency ratios (  /  ) and effect of the material length scale parameter (l/h) on the buckling load ratios (P  /P  ) based on different parameters such as the initial amplitude and the coefficients of elastic foundation are investigated in this paper.The frequency ratio and the buckling load ratio increase when the initial amplitude and the nonlinear foundation parameter increase.On the other hand, the frequency ratio and the buckling load

Figure 1 :
Figure 1: An isotropic microbeam based on a three-layered nonlinear elastic foundation.

Figure 3 :Figure 4 :
Figure 3: Relative error of the present frequencies of P-P and C-C microbeams versus the initial amplitude.

Figure 5 :Figure 6 :
Figure 5: Variation of the frequency ratios with the length scale parameter for some values of the axial compressive load: (a) P-P microbeam and (b) C-C microbeam.

Figure 7 :Figure 8 :
Figure 7: Effect of the material length scale parameter on frequency ratios of microbeams based on the Winkler parameter (K  ) with =1, K  =10, K  =10, and P=10; (a) P-P microbeam and (b) C-C microbeam.

Figure 9 :Figure 10 :
Figure 9: Effect of the material length scale parameter on frequency ratios of microbeams based on the nonlinear foundation parameter (K  ) with =1, K  =50, K  =50, and P=10; (a) P-P microbeam and (b) C-C microbeam.

Figure 11 :Figure 12 :
Figure 11: Effect of the material length scale parameter on buckling load ratios of microbeams based on the initial amplitude () with K  =50, K  =10, and K  =10; (a) P-P microbeam and (b) C-C microbeam.

Figure 13 :Figure 14 :
Figure 13: Effect of the material length scale parameter on buckling load ratios of microbeams based on the Pasternak Parameter (K  ) with K  =100, K  =100, and = .; (a) P-P microbeam and (b) C-C microbeam.

Figure 15 :Figure 16 :
Figure 15: Effect of the axial load on frequency ratios of microbeams based on the initial amplitude () with K  =50, K  =10, and K  =50; (a) P-P microbeam and (b) C-C microbeam.

Figure 17 :Figure 18 :
Figure 17: Effect of the axial load on frequency ratios of microbeams based on the Pasternak parameter (K  ) with K  =200, K  =100 and  = 0.5; (a) P-P microbeam and (b) C-C microbeam.

Figure 19 :
Figure 19: Nonlinear frequency and time history of deflection of P-P microbeam.

Figure 20 :
Figure 20: Nonlinear frequency and time history of deflection of C-C microbeam.

Table 1 :
Comparison of the present frequency with the exact frequency of P-P microbeam (] = 0.3).

Table 2 :
Comparison of the present frequency with the exact frequency of C-C microbeam (] = 0.3).

Table 3 :
Comparison of frequency ratios of macrobeams.