1 Introduction

Recently,miniature beam elements become the main constructive elements in developing nano-electromechanical systems (NEMS) such as accelerometer,nano-tweezers,and nanoswitches[ 1-3]. Hence,investigating the mechanical behavior of these components and developing new continuum beam models for small scale applications have always been attractive for researchers. It is worth to note that the presence of nano-scale phenomena is necessary to be incorporated in order to achieve reliable precise nano-scale beam models.

One important nano-scale parameter that should be considered in theoretical modeling is the surface energy. Gurtin and Murdoch^{[4, 5]} proposed a continuum theory for incorporating the influence of surface atoms on the mechanical properties of thin solids. Previous investigators have utilized this theory for modeling the buckling^[6],static bending^[7],and free vibration^[8] of nanostructures. Recently,the effects of surface characteristics on the instability threshold of elec-tromechanical structures such as double-clamped nano-bridge^[9],cantilever nano-switches^{[10, 11, 12]}, graphite NEMS^[13],and micro-plates^[14] have been explored.

Besides the surface energy,the scale dependency of material properties at the small scale is another important subject that might be crucial in modeling the ultra-small systems. For example, previous experimental findings implied that the scale dependency was an inherent nature of metallic materials^{[15, 16, 17]} and some kinds of polymers^[18]. If the characteristic size of metallic components is of the order of the internal material length scale,a hardening trend in the mechanical characteristics of the components appears. Lam et al.^[16] experimentally showed that,by decreasing the thickness of epoxy polymeric micro-beams from 115 mm to 20 mm, their bending rigidity becomes 2.4 times greater. The stiffness of polypropylene micro-beams predicted by the classical theory was four times smaller than experimental results^[17]. Some researchers experimentally evaluated the material length scale parameter for single crystal copper to be 12 μm and for polycrystalline to be 5.84 μm^{[19, 20]}. By increasing the thickness of gold film from 500 nm to 2 μm,the plastic length scale parameter of Au increased from 470 nm to 1.05 μm^[21]. Al-Rub and Voyiadjis^[22] determined the plastic length scale parameter of Cu-Ag brass in the range of 0.2 μm to 20 μm. The plastic length scale parameters of copper and nickel were evaluated to be 4 μm and and 5 μm,respectively^[23]. Material length scale parameters can also be evaluated using Molecular dynamic simulation^[24]. Material scale dependency can be modeled using non-classic continuum theories such as the non-local elasticity^{[25, 26]},the couple stress theory^{[27, 28]},and the modified couple stress theory (MCST)^[29]. One of the pioneering works in modeling the size-dependent behavior of micro-structures was conducted by Cosserat and Cosserat^[30],in the beginning of 20th century. Afterwards,more general continuum theories have been developed for linear elastic materials,in which gradients of normal strains were included,and additional material length scale parameters were therefore added as well as Lame constants^{[31, 32, 33]}. The first strain gradient theory was introduced by Mindlin and Eshel^[34],in which the potential energy-density was assumed to be depended on the stress gradient as well as the strain. The most comprehensive work was performed by Mindlin^[33] considering five additional material parameters and encompassed other non-local theories as special cases. Yang et al.^[29] proposed the MCST with only one material length scale parameter. This theory has been employed for modeling of the micro-beams by many researchers^{[35, 36, 37]}. Recently,the MCST has been used to investigate the free vibration of nano-beams^[38] and functionally graded beams^[39] and instability of NEMS^{[40, 41, 42, 43, 44]}. However,none of these works has considered the coupling between the scale dependency and surface energy in the structural models.

In this study,the surface elasticity is incorporated with the MCST for developing a nanoscale continuum beam model. The surface layer and bulk of the beam are assumed to be elastically isotropic. The Euler-Bernoulli theory is used to model the bulk deformation,and the Gurtin-Murdoch theory is employed to describe the surface layer. In comparison with previous researches in this area,the present model takes the effect of surface-induced normal stress into account,which has not been considered in recently developed couple models^{[45, 46, 47]}. Using von-Karman strain,the influence of the geometrical nonlinearity is considered in the proposed model. As a case study,the pull-in instability of an electromechanical nano-bridge is investigated. The governing nonlinear equation of nano-bridge is solved using the reduced order method (ROM),and the obtained results are verified by comparing with the numerical solution results and those available in the literature.

2 Theory 2.1 Surface elasticity

The effect of surface layer of an elastic material can be formulated via governing equations that incorporate surface elastic constants as well as the surface residual stress^{[4, 5]}. Based on this theory,the strain energy in the surface layer with the zero thickness (US) is written as^[48]

The governing equations for the surface layer of zero thickness can be explained as^[49]

where t_α are the components of the traction vector on the surface,κ_αβ are the components of the surface curvature tensor,n_i are the components of the outward unit normal to the surface (n = n_ie_i with n_β being the two in-plane components of normal vector n),and τ_αβ are the in-plane components of the surface stress tensor. It should be noted that τβα,β shows the derivative of τ_βα respect to X_β. The in-plane components are determined as where μ₀ and λ₀ are the surface elastic constants,τ₀ is the residual surface stress,ε_αβ is the in-plane components of strain tensor,δ_αβ is the Kronecker delta,ε_pp illustrates the summation of strain components,and u_α,β shows the derivative of uα respect to X_β. The out-of-plane components are determined as^[48]

The classical assumption for zero normal stress in the Z-direction (σ_ZZ) does not satisfy the surface conditions considered in the Gurtin-Murdoch elasticity. To overcome this problem, a linear distribution for σ_ZZ through the beam thickness,which satisfies the balance conditions on the upper and lower surfaces,is considered as^[50]

where 'σ_ZZ^-' and 'σ_ZZ⁺' denote the stresses on the upper and lower surfaces,respectively.

By using (2),(4),and (5),one can obtain

2.2 MCST

Based on the MCST,the strain energy is written as^[29]

where the stress tensor σ_ij,strain tensor ε_ij,deviator part of the couple stress tensor m_ij,and symmetric curvature tensor χ_ij are defined as follows: where λ,μ,l,r,and θ are the Lame constant,the shear modulus,the material length scale parameter,the displacement vector,and the rotation vector,respectively^[29]. 3 Governing equation

In this section,the surface elasticity and the MCST are incorporated in conjunction with the von-Karman nonlinear strain to derive the governing equations of the Euler-Bernoulli beams. Figure 1 shows the schematic representation of an Euler-Bernoulli beam element.

Based on the Euler-Bernoulli beam formulation,the displacement field is given by^[51]

where U is the axial displacement of the centroid of sections,W denotes the lateral deflection of the beam,and u_X,u_Y,and u_Z are the displacements in the X-,Y-,and Z-directions, respectively. 3.1 Surface energy

By substituting (9) into (3) and (4) and considering the von-Karman nonlinear strain,one can obtain

where E₀ = λ₀ + 2μ₀ is the surface elastic modulus. By substituting (10) into (1),the surface energy can be concluded as 3.2 Size dependent strain energy of bulk

Using (8) and (9),we have

where υ and E are the Poisson's ratio and Young's modulus of elasticity,respectively.

Substituting (12) into (7),the bending strain energy U_B is derived as follows:

3.3 Work of external forces

The work by the external forces,Wext,is determined as

The external force per unit length F_ext(X,t) is the sum of the Coulomb and the dispersion forces.

The strain energy U_A,due to axial forces,can be explained as

where F_A(X,t) is the axial force. 3.4 Kinetic energy and damping loss

The kinetic energy of beam is determined as

where ρ is the density,and I is the moment of inertia of cross-section.

The virtual work Wd due to the structural damping is determined as

where c_d is the damping coefficient. 3.5 Hamilton principle

The Hamilton principle can be used to derive the following governing equation of the system:

Substituting (11) and (13)-(17) into (18) and some mathematical elaborations,one can obtain

where S₀ is the cross-sectional periphery. The nonlinear governing equation of the system can be derived from the above equation as with the following boundary conditions: 4 Application

Nano-bridges are highly potential for developing new resonators^[52],switches^[53],memories^[54], and sensors^[55]. In this section,the static,dynamic,and pull-in behaviors of a nano-bridge are studied. A typical beam type nano-bridge consists of a conductive nano-beam suspended above a stationary rigid electrode. Imposing of a direct current (DC) voltage difference between the electrodes results in the nano-bridge deflection. At a critical voltage,i.e.,the pull-in voltage, the Coulomb attraction overcomes the elastic resistance,the beam spontaneously collapses towards the fixed plate,and the pull-in instability occurs. Figure 2 shows a nano-bridge with the length L,having a uniform rectangular cross-section with uniform cross-section of the thickness h and the width b,and an initial distance from a fixed substrate g subjected to the electrostatic force.

The electrostatic force per unit length of the nano-beam can be written as^[56]

where ε₀ = 8.854×10^-12 c² N^-1m^-2 is the vacuum permittivity,and ε_r is the relative permittivity of dielectric.

For nano-scale separations,two interaction regimes can be defined for determining dispersion forces. While for the large separation (above several tens of nanometers),the Casimir force is dominant for the small separation regime (below several tens of nanometers),the van der Waals (vdW) force should be considered^{[57, 58, 59]}. The Casimir force per unit length of beam can be obtained as^[12]

where ħ = 1.055×10^-34 J·s is Planck’s constant divided by 2π,and c = 2.998 × 10⁸ m/s is the light speed.

The vdW force per unit length of can be explained as^[11]

where A is the Hamaker constant. 4.1 Non-dimensional governing equation

It is assumed that the longitudinal inertia is negligible^[60]. By double integrating of (20a) and some straight forward mathematical operations,the following relation is obtained for a nano-bridge:

For the rectangular section,we have

For nano-sections,the effects of rotating inertia can be neglected due to the small values of I in nano-beams with respect to the area A. Substituting (22)-(26) into (20b),the governing equation and boundary condition of a nano-bridge are obtained as

With the definitions of x = X/L and w = W/g,the non-dimensional equation can be explained as

where the dimensionless parameters are identified as follows: 4.2 Solution methods 4.2.1 ROM

In this section,the reduced-order model is employed to solve the nonlinear governing equation of system. The displacement is expressed as a combination of a complete set of linearly independent basis functions φ_i(x). For static analysis,the displacement is assumed in the form of

Similarly,for the dynamic behavior,the displacement is considered as

The linear mode shapes of clamped-clamped beam are selected as basic functions in the Galerkin procedure

where ω_i is the ith root of the characteristic equation of a clamped-clamped beam.

Substituting (30) into (28),multiplying the result by φ_i(x),and integrating the outcome from x=0 to 1 for static problems,we have

Similarly,for the dynamic behavior,one can obtain where N is the number of considered terms of the ROM,and A_k is the Taylor expansion coefficient of right hand side of (28). The MAPLE commercial software is employed for the numerical solution of (32). 4.2.2 Numerical solution

In order to validate the ROM results,the nonlinear governing differential equations in both static and dynamic cases are solved using the MAPLE commercial software. The step size of the parameter variation is chosen based on the sensitivity of the parameter to the midpoint deflection. The pull-in parameters are determined via the slope of the w-α graphs.

4.3 Results 4.3.1 Nano-bridge deflection

Figure 3 reveals the influence of nano scale effects (i.e.,size effect,surface layer,and dispersion forces) on the static deflection of the nano-bridge,where γ,η,and α are equal to 0.25,0.25 and 5,respectively. This figure demonstrates that,for a given external voltage,considering the size effect causes the decrease of nano-bridge deflection,or it has a hardening effect on the nano-bridge. It is also observed that dispersion forces cause the increase of nano-bridge deflection due to the increasing of the external force. Considering the surface effect can reduce or enhance the beam deflection depending on the values of the surface modulus and surface stress,as shown in Fig. 3.

The time history of mid-point displacement of nano-bridge is shown in Fig. 4 for various nano-scale effects. This figure reveals that the maximum beam deflection is decreased by considering the size effect,while dispersion forces consideration increase that. Similar to the static behavior,Fig. 4 reveals that considering the surface energy can reduce or enhance the nanobridge deflection depending on the values of the surface residual stress and surface modulus.

4.3.2 Pull-in behavior

Figure 5 shows the impacts of surface layer and dispersion forces on the pull-in α_PI voltage of nano-bridge. As shown,an increase in the surface residual stress enhances the instability voltage of the nano-bridge. It should be noted that t₀ = 0 is subjected to neglecting surface stress condition. Figure 5 depicts that the surface stress can induce softening or stiffening effects depending on its sign. For positive values of the surface stress,the surface energies increase the instability threshold,while for negative values of the surface stress,the surface layer reduces the instability voltage. Furthermore,Fig. 5 shows that by incorporating the dispersion forces, the pull-in voltage is decreased. The dynamic pull-in voltage of nano-bridge is smaller than the static pull-in voltage as a result of inertia forces.

The effects of the dispersion forces and the size dependency on the instability voltages of nano-bridge α_PI are shown in Fig. 6. These figures reveal that the higher value of the scale parameter (l/h) results in the higher instability voltage. Figure 6 also shows that the results of the ROM are in good agreement with those obtained from the numerical method. By comparing the results in Figs. 5 and 6,it is found that the size effect always has a hardening effect,i.e., increasing the pull-in voltage and the surface effect can induce softening or stiffening effects depending on its sign.

In the above presented results,the effects of the damping on the obtained pull-in voltages are neglected. The influence of damping on the dynamic behavior of vibrating nano-bridges is illustrated in Fig. 7 for ĉ = 0.5. The obtained results show that there is a stable focus point when the damping parameter is taken into account. It can be concluded that the oscillations in the nano-bridge are converged near a focus point due to the consideration of damping. On the other hand,the second equilibrium point is unstable saddle point for any ĉ value. When the actuation voltage reaches the pull-in voltage,the trajectories which are approached to the stable focus due to the damping effect,diverge and the nano-bridge become unstable.

4.3.3 Verification

The influence of surface elasticity on the behavior of a clamped-clamped nano-bridge made of silver with E = 76 GPa,ν = 0.3,E₀ = 1.22 N/m and τ₀ = 0.89 N/m was already studied by Wang and Wang^[48]. The geometrical parameters of the nano-switch were L = 1 μm, h = 50 nm,b = 5h and g = 50 nm. They used the linear fringing field for electrical force and ignored the effect of dispersion forces and size effects. The variation of mid-point deflection of the nano-switch for various applied voltages is shown in Fig. 8. This figure reveals that,by ignoring the size effect for both classical theory and surface elasticity,the data obtained by the presented model in this investigation are in very good agreement with those presented by Wang and Wang^[48],however,by considering the size effect,the beam deflections are decreased,and pull-in voltage is increased significantly.

5 Conclusions

In this paper,using von-Karman strain,a nonlinear beam model based on the Gurtin- Murdoch’s surface elasticity and the MCST is presented considering the effects of surfaceinduced normal stress. The model is able to simultaneously take the effects of surface energy and microstructure into account. As the case study,the proposed model is employed to simulate the static and dynamic pull-in instability of electromechanical nano-bridges. The results demonstrate that considering the size dependency causes the nano-bridge to deflect as a harder structure. On the other hand,the consideration of surface energy could increase or decrease the pull-in voltage of the system depending on the sign of surface stresses.