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

Authors: Dang Van Hieu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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 the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.


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## Body

## 1. 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 [1–4].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 Şimşek 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 Salamat-talab 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 Şimşek [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]. Şimşek 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 size-dependent 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 Bağdatlı [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 clamped-free 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.


## 2. Governing Equation

### 2.1. Modified Couple Stress Theory

According to the modified couple stress theory [11], the strain energy for an isotropic linearly elastic material in domain V can be written as(1)Us=∫V12σ:ε+m:χdV=∫V12λtrε2+με:ε+l2χ:χdV,where σ is the symmetric part of the Cauchy stress tensor, ε is the strain tensor, m is the deviatoric part of the couple stress tensor, χ is the symmetric curvature tensor, λ and μ are Lamé constants, and l is the material length scale parameter.The kinematic relations are as follows:(2)ε=12∇u+∇uT,(3)χ=12∇ω+∇ωT,where ∇ is the Hamiltonian differential operator,u is the displacement vector, and ω is the rotation vector described as:(4)ω=12curlu.Constitutive equations are described as(5)σ=λtrεI+2με,(6)m=2l2μχ,where I is the unit matrix and Lamé’s constantsλ and μ can be expressed:(7)λ=Eν1+ν1-2ν,μ=E21+ν.In (7),E and ν are the elasticity modulus and Poisson’s ratio, respectively, and μ is known as the shear modulus.


### 2.2. The Governing Equation for Microbeam

An isotropic microbeam of a lengthL with cross-sectional dimensionsh andb is considered and shown in Figure 1. The microbeam rests on a nonlinear elastic layer with the spring constantskL,kP, andkNL 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(8)uxx,z,t=ux,t-z∂wx,t∂x,(9)uyx,z,t=0,(10)uzx,z,t=wx,t,whereu andw are the axial and the transverse displacement of any point on the neutral axis.Figure 1
An isotropic microbeam based on a three-layered nonlinear elastic foundation.The Von-Kármán’s nonlinear strain–displacement relationship with assumptions of large transverse displacements where it is assumed that the nonlinear term corresponding to∂ux/∂x is small enough to be neglected, moderate rotations and small strains for a straight beam are given by:(11)εxx=∂ux,t∂x+12∂wx,t∂x2-z∂2wx,t∂x2;εyy=εzz=εxy=εxz=εyz=0.The components of the rotation vector are as follows:(12)ωx=0,ωy=-∂wx,t∂x,ωz=0.And the expression for the components of the symmetric curvature tensor:(13)χxy=-12∂2wx,t∂x2,χxx=χyy=χzz=χxz=χyz=0.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(14)δ∫0tKe+Wext-Us-Ufdt=0,where Ke is the kinetic energy, Us is the strain energy given in (1), Uf is the strain energy induced by the nonlinear elastic layer, and Wext is the work done by the external forces.The kinetic energy can be obtained as(15)Ke=∫0LρA12∂u∂t2+12∂w∂t2dx,whereA=bhis the area of the cross-section and ρ is the mass density of the microbeam.From (1), the strain energy can be expressed by(16)Us=∫Vσijεij+mijχijdV=∫0LNx∂u∂x+12∂w∂x2-Mx+Mxy∂2w∂x2dx,Here,Nx, Mx, andMxy are the normal force, the bending moment, and a new stress resultant due to the couple stress, respectively, which are given by(17)Nx=∫AσxxdA=EA∂u∂x+12∂w∂x2,(18)Mx=∫AzσxxdA=-EI∂2w∂x2,(19)Mxy=∫AmxydA=-μAl2∂2w∂x2,whereI=bh3/12is the inertia moment of the cross-section of the microbeam.The strain energy induced by the nonlinear elastic layer can be written as(20)Uf=∫0L12kLw2+12kP∂w∂x2+14kNLw4dx.And, the work done by the external forces(21)Wext=12∫0Lp∂w∂x2dx,wherep is the initial axial compressive force as showed in Figure 2.Figure 2
Microbeams subjected to the axial compressive load: (a) P-P Microbeam and (b) C-C Microbeam.
(a) (b)Substituting (15), (16), (20), and (21) into (14) and after some mathematical transformations and collecting the coefficients of δu and δw, the following equations of motion are obtained (Note (17)–(19)):(22)δu:EA∂∂x∂u∂x+12∂w∂x2-ρA∂2u∂t2=0.(23)δw:∂2∂x2-EI∂2w∂x2-μAl2∂2w∂x2+∂∂xNx∂w∂x-p∂2w∂x2-kLw+kP∂2w∂x2-kNLw3-ρA∂2w∂t2=0.We can see that (22) and (23) are the equations of motion of the microbeam respect to the displacementsu andw. 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]:(24)Nx=EA2L∫0L∂w∂x2dx.For a uniform microbeam, the equation of motion is derived in terms ofw by substituting (24) into (23) as follows:(25)EI+μAl2∂4w∂x4+p-EA2L∫0L∂w∂x2dx∂2w∂x2+kLw-kP∂2w∂x2+kNLw3+ρA∂2w∂t2=0.We introduce the dimensionless parameters:(26)x-=xL,w-=wr,r=IA,t=t-ρAL4EI,η=lh,S=1+6η21+ν,P=pL2EI,KL=kLL4EI,KP=kPL2EI,KNL=kNLr2L4EI.Considering (26), (25) can be written in the nondimensional form as follows:(27)S∂4w-∂x-4+P-12∫01∂w-∂x-2dx-∂2w-∂x-2+KLw--KP∂2w-∂x-2+KNLw-3+∂2w-∂t-2=0.Assume that displacement functionw-(x-,t-) can be expressed as follows:(28)w-x-,t-=Qt-ϕx-.where Q(t-) is the unknown time-dependent function and ϕ(x-) 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(29)ϕx-=sin⁡πx-.+ for Clamped – Clamped microbeam(30)ϕx-=121-cos⁡2πx-.where ϕ(x-) is normalized so that ϕ(0.5)=1. Using Galerkin method, the partial differential equation (27) is transformed into the following differential equation:(31)Q¨t-+B1S+B2P-B2KP+KLQt-+B3+B4KNLQ3t-=0.Here the coefficientsB1, B2, B3,andB4 in (31) are defined as(32)B1=∫01ϕ4ϕdx-∫01ϕ2dx-,B2=∫01ϕ′′ϕdx-∫01ϕ2dx-,B3=-1/2∫01ϕ′2dx-∫01ϕ′′ϕdx-∫01ϕ2dx-,B4=∫01ϕ4dx-∫01ϕ2dx-,where ϕ(4), ϕ′′, and ϕ′ are the fourth, second, and first derivative of ϕ with respect to the axial coordinate x-, respectively.The microbeam is assumed to be subject to the following initial conditions:(33)Q0=α,Q˙0=0,where α is the dimensionless maximum vibration amplitude of oscillator.


## 2.1. Modified Couple Stress Theory

According to the modified couple stress theory [11], the strain energy for an isotropic linearly elastic material in domain V can be written as(1)Us=∫V12σ:ε+m:χdV=∫V12λtrε2+με:ε+l2χ:χdV,where σ is the symmetric part of the Cauchy stress tensor, ε is the strain tensor, m is the deviatoric part of the couple stress tensor, χ is the symmetric curvature tensor, λ and μ are Lamé constants, and l is the material length scale parameter.The kinematic relations are as follows:(2)ε=12∇u+∇uT,(3)χ=12∇ω+∇ωT,where ∇ is the Hamiltonian differential operator,u is the displacement vector, and ω is the rotation vector described as:(4)ω=12curlu.Constitutive equations are described as(5)σ=λtrεI+2με,(6)m=2l2μχ,where I is the unit matrix and Lamé’s constantsλ and μ can be expressed:(7)λ=Eν1+ν1-2ν,μ=E21+ν.In (7),E and ν are the elasticity modulus and Poisson’s ratio, respectively, and μ is known as the shear modulus.


## 2.2. The Governing Equation for Microbeam

An isotropic microbeam of a lengthL with cross-sectional dimensionsh andb is considered and shown in Figure 1. The microbeam rests on a nonlinear elastic layer with the spring constantskL,kP, andkNL 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(8)uxx,z,t=ux,t-z∂wx,t∂x,(9)uyx,z,t=0,(10)uzx,z,t=wx,t,whereu andw are the axial and the transverse displacement of any point on the neutral axis.Figure 1
An isotropic microbeam based on a three-layered nonlinear elastic foundation.The Von-Kármán’s nonlinear strain–displacement relationship with assumptions of large transverse displacements where it is assumed that the nonlinear term corresponding to∂ux/∂x is small enough to be neglected, moderate rotations and small strains for a straight beam are given by:(11)εxx=∂ux,t∂x+12∂wx,t∂x2-z∂2wx,t∂x2;εyy=εzz=εxy=εxz=εyz=0.The components of the rotation vector are as follows:(12)ωx=0,ωy=-∂wx,t∂x,ωz=0.And the expression for the components of the symmetric curvature tensor:(13)χxy=-12∂2wx,t∂x2,χxx=χyy=χzz=χxz=χyz=0.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(14)δ∫0tKe+Wext-Us-Ufdt=0,where Ke is the kinetic energy, Us is the strain energy given in (1), Uf is the strain energy induced by the nonlinear elastic layer, and Wext is the work done by the external forces.The kinetic energy can be obtained as(15)Ke=∫0LρA12∂u∂t2+12∂w∂t2dx,whereA=bhis the area of the cross-section and ρ is the mass density of the microbeam.From (1), the strain energy can be expressed by(16)Us=∫Vσijεij+mijχijdV=∫0LNx∂u∂x+12∂w∂x2-Mx+Mxy∂2w∂x2dx,Here,Nx, Mx, andMxy are the normal force, the bending moment, and a new stress resultant due to the couple stress, respectively, which are given by(17)Nx=∫AσxxdA=EA∂u∂x+12∂w∂x2,(18)Mx=∫AzσxxdA=-EI∂2w∂x2,(19)Mxy=∫AmxydA=-μAl2∂2w∂x2,whereI=bh3/12is the inertia moment of the cross-section of the microbeam.The strain energy induced by the nonlinear elastic layer can be written as(20)Uf=∫0L12kLw2+12kP∂w∂x2+14kNLw4dx.And, the work done by the external forces(21)Wext=12∫0Lp∂w∂x2dx,wherep is the initial axial compressive force as showed in Figure 2.Figure 2
Microbeams subjected to the axial compressive load: (a) P-P Microbeam and (b) C-C Microbeam.
(a) (b)Substituting (15), (16), (20), and (21) into (14) and after some mathematical transformations and collecting the coefficients of δu and δw, the following equations of motion are obtained (Note (17)–(19)):(22)δu:EA∂∂x∂u∂x+12∂w∂x2-ρA∂2u∂t2=0.(23)δw:∂2∂x2-EI∂2w∂x2-μAl2∂2w∂x2+∂∂xNx∂w∂x-p∂2w∂x2-kLw+kP∂2w∂x2-kNLw3-ρA∂2w∂t2=0.We can see that (22) and (23) are the equations of motion of the microbeam respect to the displacementsu andw. 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]:(24)Nx=EA2L∫0L∂w∂x2dx.For a uniform microbeam, the equation of motion is derived in terms ofw by substituting (24) into (23) as follows:(25)EI+μAl2∂4w∂x4+p-EA2L∫0L∂w∂x2dx∂2w∂x2+kLw-kP∂2w∂x2+kNLw3+ρA∂2w∂t2=0.We introduce the dimensionless parameters:(26)x-=xL,w-=wr,r=IA,t=t-ρAL4EI,η=lh,S=1+6η21+ν,P=pL2EI,KL=kLL4EI,KP=kPL2EI,KNL=kNLr2L4EI.Considering (26), (25) can be written in the nondimensional form as follows:(27)S∂4w-∂x-4+P-12∫01∂w-∂x-2dx-∂2w-∂x-2+KLw--KP∂2w-∂x-2+KNLw-3+∂2w-∂t-2=0.Assume that displacement functionw-(x-,t-) can be expressed as follows:(28)w-x-,t-=Qt-ϕx-.where Q(t-) is the unknown time-dependent function and ϕ(x-) 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(29)ϕx-=sin⁡πx-.+ for Clamped – Clamped microbeam(30)ϕx-=121-cos⁡2πx-.where ϕ(x-) is normalized so that ϕ(0.5)=1. Using Galerkin method, the partial differential equation (27) is transformed into the following differential equation:(31)Q¨t-+B1S+B2P-B2KP+KLQt-+B3+B4KNLQ3t-=0.Here the coefficientsB1, B2, B3,andB4 in (31) are defined as(32)B1=∫01ϕ4ϕdx-∫01ϕ2dx-,B2=∫01ϕ′′ϕdx-∫01ϕ2dx-,B3=-1/2∫01ϕ′2dx-∫01ϕ′′ϕdx-∫01ϕ2dx-,B4=∫01ϕ4dx-∫01ϕ2dx-,where ϕ(4), ϕ′′, and ϕ′ are the fourth, second, and first derivative of ϕ with respect to the axial coordinate x-, respectively.The microbeam is assumed to be subject to the following initial conditions:(33)Q0=α,Q˙0=0,where α is the dimensionless maximum vibration amplitude of oscillator.


## 3. Analysing of Postbuckling and Free Nonlinear Vibration

### 3.1. Analysing of Postbuckling

From (31), the postbuckling load-deflection relation of the microbeam can be obtained as(34)PNL=-B1S-B2KP+KL+B3+B4KNLQ2B2.If neglecting the contribution ofQ in (34), the linear buckling load can be determined as(35)PL=-B1S-B2KP+KLB2.In the next sub-section, we will find approximate solution Q(t-) of (31).


### 3.2. Analysing of Free Nonlinear Vibration

In this sub-section, 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:(36)γ1=B1S+B2P-B2KP+KL,γ2=B3+B4KNL.Considering (36), (31) becomes(37)Q¨t-+γ1Qt-+γ2Q3t-=0.The linearized equation of (37) has the form(38)Q¨t-+ω2Qt-=0,where the coefficient of linear term in (38) ω2 can be determined by using the mean square criterion which minimizes the error between two equations (37) and (38):(39)e2Q=γ1Q+γ2Q3-ω2Q2→Minω2Thus, from(40)∂e2Q∂ω2=0,we get(41)ω2=γ1Q2+γ2Q4Q2.ω in (41) is the approximate frequency of the microbeam. In (41), the symbol ∙ denotes the time-averaging operator in classical meaning [46]:(42)Qt-=limT→+∞⁡1T∫0TQt-dt-.For a ω-frequency function, the averaging process is taken during one periodT, namely,(43)Qωt-=1T∫0TQωt-dt=12π∫02πQτdτ,τ=ωt-.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/T in (42) and (43) by a weighted coefficient function h(t-). Thus, we get so-called a weighted average value:(44)Qt-w=∫0+∞ht-Qt-dt-,with the weighted coefficient function h(t-) satisfying the following condition:(45)∫0+∞ht-dt-=1.The weighted coefficient function h(t-) was in detail discussed in [44, 45]. Here, we use a specific form of the weighted coefficient function [42] as below:(46)ht-=s2ωt-e-sωt-,wheres is a positive constant called the adjustment parameter. Equation (44) will take the form of (42) and (43) whens=0. The solution of the linearized (38) is given by(47)Qt-=αcos⁡ωt-,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:(48)Q2w=α2cos2⁡ωtw=∫0+∞α2s2ωt-e-sωt-cos2⁡ωt-dt-=∫0+∞α2s2τe-sτcos2⁡τdτ=α2s4+2s2+8s2+42.(49)Q4w=α4cos4⁡ωtw=∫0+∞α4s2ωt-e-sωt-cos4⁡ωt-dt-=∫0+∞α2s2τe-sτcos4τdτ=α4s8+28s6+248s4+416s2+1536s2+42s2+162.Substituting the averaging values in (48) and (49) into (41), we get the approximate frequency of oscillation:(50)ωNL=γ1+γ2α2s8+28s6+248s4+416s2+1536s4+2s2+8s2+162.The approximate frequency obtained in (50) is not only a function of the initial amplitude α but also a function of the parameters (in the expression of the weighted function h(t-)=s2ωt-e-sωt-). With s=2, we get the amplitude-frequency relationship:(51)ωNL=γ1+0.72γ2α2.Thus, the approximate solution of oscillator is(52)Qt-=αcos⁡γ1+0.72γ2α2t-.Substituting (29) and (30) into (32) and performing the integrations in (32), we can obtain the following expressions for the nonlinear frequency (Note (36) and (51)):+ For P-P microbeam(53)ωNL=π4S+π2KP+KL-π2P+0.72π44+34KNLα2.+ For C-C microbeam(54)ωNL=163π4S+43π2KP+KL-43π2P+0.72π43+3548KNLα2.


## 3.1. Analysing of Postbuckling

From (31), the postbuckling load-deflection relation of the microbeam can be obtained as(34)PNL=-B1S-B2KP+KL+B3+B4KNLQ2B2.If neglecting the contribution ofQ in (34), the linear buckling load can be determined as(35)PL=-B1S-B2KP+KLB2.In the next sub-section, we will find approximate solution Q(t-) of (31).


## 3.2. Analysing of Free Nonlinear Vibration

In this sub-section, 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:(36)γ1=B1S+B2P-B2KP+KL,γ2=B3+B4KNL.Considering (36), (31) becomes(37)Q¨t-+γ1Qt-+γ2Q3t-=0.The linearized equation of (37) has the form(38)Q¨t-+ω2Qt-=0,where the coefficient of linear term in (38) ω2 can be determined by using the mean square criterion which minimizes the error between two equations (37) and (38):(39)e2Q=γ1Q+γ2Q3-ω2Q2→Minω2Thus, from(40)∂e2Q∂ω2=0,we get(41)ω2=γ1Q2+γ2Q4Q2.ω in (41) is the approximate frequency of the microbeam. In (41), the symbol ∙ denotes the time-averaging operator in classical meaning [46]:(42)Qt-=limT→+∞⁡1T∫0TQt-dt-.For a ω-frequency function, the averaging process is taken during one periodT, namely,(43)Qωt-=1T∫0TQωt-dt=12π∫02πQτdτ,τ=ωt-.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/T in (42) and (43) by a weighted coefficient function h(t-). Thus, we get so-called a weighted average value:(44)Qt-w=∫0+∞ht-Qt-dt-,with the weighted coefficient function h(t-) satisfying the following condition:(45)∫0+∞ht-dt-=1.The weighted coefficient function h(t-) was in detail discussed in [44, 45]. Here, we use a specific form of the weighted coefficient function [42] as below:(46)ht-=s2ωt-e-sωt-,wheres is a positive constant called the adjustment parameter. Equation (44) will take the form of (42) and (43) whens=0. The solution of the linearized (38) is given by(47)Qt-=αcos⁡ωt-,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:(48)Q2w=α2cos2⁡ωtw=∫0+∞α2s2ωt-e-sωt-cos2⁡ωt-dt-=∫0+∞α2s2τe-sτcos2⁡τdτ=α2s4+2s2+8s2+42.(49)Q4w=α4cos4⁡ωtw=∫0+∞α4s2ωt-e-sωt-cos4⁡ωt-dt-=∫0+∞α2s2τe-sτcos4τdτ=α4s8+28s6+248s4+416s2+1536s2+42s2+162.Substituting the averaging values in (48) and (49) into (41), we get the approximate frequency of oscillation:(50)ωNL=γ1+γ2α2s8+28s6+248s4+416s2+1536s4+2s2+8s2+162.The approximate frequency obtained in (50) is not only a function of the initial amplitude α but also a function of the parameters (in the expression of the weighted function h(t-)=s2ωt-e-sωt-). With s=2, we get the amplitude-frequency relationship:(51)ωNL=γ1+0.72γ2α2.Thus, the approximate solution of oscillator is(52)Qt-=αcos⁡γ1+0.72γ2α2t-.Substituting (29) and (30) into (32) and performing the integrations in (32), we can obtain the following expressions for the nonlinear frequency (Note (36) and (51)):+ For P-P microbeam(53)ωNL=π4S+π2KP+KL-π2P+0.72π44+34KNLα2.+ For C-C microbeam(54)ωNL=163π4S+43π2KP+KL-43π2P+0.72π43+3548KNLα2.


## 4. 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 (KL,KP andKNL), the axial compressive load (P), and the initial amplitude α 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.Table 1
Comparison of the present frequency with the exact frequency of P-P microbeam (ν=0.3).
α l/h K L K P K NL P ω exact ω present R. error (%) 0.1 0.25 10 10 10 5 13.6049 13.6046 0.0026 0.5 10 10 10 5 16.4131 16.4128 0.0018 0.75 10 10 10 5 20.2456 20.2454 0.0012 1.0 10 10 10 5 24.6288 24.6286 0.0008  1 0.25 50 50 50 20 22.7555 22.7185 0.1626 0.5 50 50 50 20 24.5384 24.5636 0.1418 0.75 50 50 50 20 27.2517 27.2198 0.1169 1.0 50 50 50 20 30.6492 30.6204 0.0937Table 2
Comparison of the present frequency with the exact frequency of C-C microbeam (ν=0.3).
α l/h K L K P K NL P ω exact ω present R. error (%) 0.1 0.25 10 10 10 5 13.6049 13.6046 0.0026 0.5 10 10 10 5 16.4131 16.4128 0.0018 0.75 10 10 10 5 20.2456 20.2454 0.0012 1.0 10 10 10 5 24.6288 24.6286 0.0008  1 0.25 50 50 50 20 22.7555 22.7185 0.1626 0.5 50 50 50 20 24.5384 24.5636 0.1418 0.75 50 50 50 20 27.2517 27.2198 0.1169 1.0 50 50 50 20 30.6492 30.6204 0.0937Equation (37) is the cubic Duffing oscillator, the exact frequency can be given by [49](55)ωexact=2π42∫0π/2dt/γ2α2cos2⁡t+γ2α2+2γ1.The relative error of the present solution is shown in Figure3 withl/h=0.5,KL=50,KP=100,KNL=50, andP=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 α increases.Figure 3
Relative error of the present frequencies of P-P and C-C microbeams versus the initial amplitude.Comparison of frequency ratios of macrobeams for various values of the dimensionless initial amplitude (α=1, 2, 3, 4) is presented in Table 3. Note that frequency ratios of macrobeams can be obtained by lettingl/h=0. Again accuracy of the present solution can be observed.Table 3
Comparison of frequency ratios of macrobeams.
α P-P macrobeam C-C macrobeam Present Fallah et al. [23] Azrar et al. [24] Present Fallah et al.[23] Azrar et al. [24] 1 1.0863 1.0897 1.0891 1.0223 1.0552 1.0221 2 1.3115 1.3229 1.3177 1.0863 1.2056 1.0856 3 1.6186 1.6393 1.6256 1.1853 1.4214 1.1831 4 1.9698 1.9999 - 1.3115 1.6776 1.3064The variation of the frequency ratios of microbeams (ωNL/ωL) 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 ν=0.3, the coefficients of the nonlinear elastic layer asKL=50,KP=50, andKNL=50, the axial compressive loadP=20, and some values of the initial amplitude α (α=0.1, 0.5, 1.0, 1.5 and 2.0). And, Figure 5 is plotted with values of parameter: Poisson’s ratio ν=0.3, the coefficients of the nonlinear elastic layer asKL=100,KP=100 andKNL=100, the initial amplitude α=0.5, and some values of the axial compressive loadP (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 α=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.Figure 4
Variation of the frequency ratios with the length scale parameter for some values of the initial amplitude: (a) P-P microbeam and (b) C-C microbeam.
(a) (b)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.
(a) (b)Effects of the material length scale parameter (l/h) on the frequency ratio of microbeam (ωNL/ωL) 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 (α), nonlinear foundation parameter (KNL) and axial load (P) increase. However, the frequency ratios of microbeams decrease as the Winkler parameter (KL) and Pasternak parameter (KP) increase (in Figures 7 and 8). In these figures, the Poisson’s ratio is chosen equal to 0.3 (ν=0.3).Figure 6
Effect of the material length scale parameter on frequency ratios of microbeams based on the initial amplitude (α) withKL=50,KP=10,KNL=10, andP=20; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 7
Effect of the material length scale parameter on frequency ratios of microbeams based on the Winkler parameter (KL) with α=1,KP=10,KNL=10, andP=10; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 8
Effect of the material length scale parameter on frequency ratios of microbeams based on the Pasternak parameter (KP) with α=1,KL=50,KNL=10, andP=10; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 9
Effect of the material length scale parameter on frequency ratios of microbeams based on the nonlinear foundation parameter (KNL) with α=1,KL=50,KP=50, andP=10; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 10
Effect of the material length scale parameter on frequency ratios of microbeams based on the axial load (P) with α=1,KL=50,KP=50, andKNL=20; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)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 (α) and the nonlinear foundation parameter (KNL) 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 (KL) and the Pasternak parameter (KP) increase. Noted that the postbuckling loadPNL is determined by maximizing (34) in one period of vibration of microbeams.Figure 11
Effect of the material length scale parameter on buckling load ratios of microbeams based on the initial amplitude (α) withKL=50,KP=10, andKNL=10; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 12
Effect of the material length scale parameter on buckling load ratios of microbeams based on the Winkler parameter (KL) withKP=5,KNL=10, and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 13
Effect of the material length scale parameter on buckling load ratios of microbeams based on the Pasternak Parameter (KP) withKL=100,KNL=100, and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 14
Effect of the material length scale parameter on buckling load ratios of microbeams based on the nonlinear foundation parameter (KNL) withKL=50,KP=10, and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Finally, in Figures15–18, 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 (α) and the nonlinear foundation parameter (KNL) increases while the frequency ratios of microbeams decrease as the Winkler parameter (KL) and the Pasternak parameter (KP) increase.Figure 15
Effect of the axial load on frequency ratios of microbeams based on the initial amplitude (α) withKL=50,KP=10, andKNL=50; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 16
Effect of the axial load on frequency ratios of microbeams based on the Winkler parameter (KL) withKP=100,KNL=100 and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 17
Effect of the axial load on frequency ratios of microbeams based on the Pasternak parameter (KP) withKL=200,KNL=100 and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Figure 18
Effect of the axial load on frequency ratios of microbeams based on the nonlinear foundation parameter (KNL) withKL=100,KP=100 and α=0.5; (a) P-P microbeam and (b) C-C microbeam.
(a) (b)Compressed by the axial force (P), microbeams will be buckled as value ofP is very large, namelly the phenomenon of instability will occur. When value ofP 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 for P-P microbeam and C-C microbeam, respectively. In these figures, a specific case with the foundation parametersKL=100,KP=50,KNL=20, the Poisson's ratio ν=0.3 and the initial amplitude α=0.3 is considered, the critical values of axial load for the P-P microbeam asPcr=70.3606, 73.2076, 81.7486, 95.9836 and 115.9126, and for C-C microbeam asPcr= 97.3092, 108.6972, 142.8612, 199.8013 and 279.5173 corresponding to values of the material length scale parameterl/h=0.0, 0.25, 0.50, 0.75 and 1.0, respectively. Assusing that microbeams are subjected to an axial compressive loadP=100, the results are presented in Figures 19 and 20 for P-P microbeam and C-C microbeam, respectively. Figure 19 is drawn for P-P microbeam, the instability occurs with four values of the material length scale parameterl/h (l/h=0, 0.25, 0.5 and 0.75) because corresponding critical values of axial compressive load are less than value of the initial axial loadP (P=100). For C-C microbeam in Figure 20, instability occurs with only one value of the material length scale parameterl/h=0.0.Figure 19
Nonlinear frequency and time history of deflection of P-P microbeam.
(a) (b)Figure 20
Nonlinear frequency and time history of deflection of C-C microbeam.
(a) (b)


## 5. 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 (ωNL/ωL) 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 (ωNL/ωL) and effect of the material length scale parameter (l/h) on the buckling load ratios (PNL/PL) 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 ratio decrease with increasing of the Winkler parameter and the Pasternak parameter.



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