1 Introduction

Beams and plates of nano-scale have been effectively applied in micro-electro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS). Such micro-/nano-structural components are constructed from nano-structured materials due to possessing the small size. It is known that nano-structured materials such as nano-crystalline materials (NcMs) and nano-particle composites (NpCs) have an inhomogeneous structure, and their properties are significantly influenced by the essence of their material structure^[1-2]. In fact, NcMs are multi-phase composites consisting of grain phase, porosities, and interface phase. In NcMs, several atoms are separated from the grains and create a new phase which is called the interface. The interface phase shows a softening impact on the structure by reducing the elastic moduli^[3-4]. Also, properties of NcMs depend on the grain and porosity size which can change from 0.5 nm to 100 nm^[5].

Recently, some physical phenomena have been reported in micro-/nano-scale structures such as the translation and rotation of grains or crystals within the material structure. The translational motion of grains is the observable degree of freedom in macro-size structures. However, the rotational motion of grains inside the material shows an important effect on the behavior of a micro-/nano-structure. Several higher-order continuum theories are suggested accounting for the size influences by considering additional degrees of freedom. The modified couple stress theory^[6] has been implemented to examine the influence of the rigid rotations of grains on the behavior of nano-beams^[7]. The analysis of mechanical response of nano-crystalline nano-beams is very limited in the literature. Shaat et al.^[8] examined vibration behavior of a cracked nano-crystalline nano-beam based on the modified couple stress theory. Also, Shaat and Abdelkefi^[9] researched pull-in instability of multi-phase nano-crystalline nano-beams exposed to an electrostatic force.

Nano-structures are also significantly affected by their surface tension and surface energy^[10-12]. The free surfaces (external surfaces creating the outer boundary of materials) and interfaces (interfacial surfaces between the non-homogeneities) have a major role on the behavior of nano-structures made of nano-structured materials^[9]. Gurtin and Murdoch^[10] suggested a surface elasticity theory for modeling the continuum surface as a two-dimensional membrane with zero thickness lying on the material bulk. In the literature, there is no relevant paper addressing the effects of surface elasticity on vibration and buckling behavior of multi-phase nano-crystalline nano-beams. However, there are many papers on vibration and buckling of homogenous nano-beams incorporating surface effects. Gheshlaghi and Hasheminejad^[13] investigated effects of surface stress on nonlinear vibrational behavior of homogenous nano-beams. Also, Ansari et al.^[14] presented the vibration analysis of Timoshenko nano-beams based on the surface stress elasticity theory. Ansari et al.^[15] performed the post-buckling analysis of homogenous nano-scale beams considering effects of surface stress. Also, Sahmani et al.^[16] explored effects of surface energy on vibrational behavior of post-buckled higher order nano-beams. Ebrahimi and Boreiry^[17] examined various surface effects on vibrational behavior of nano-beams based on the classical beam theory.

In fact, the literature survey indicates that all of the previous papers on nano-crystalline nano-beams have not considered nonlocal effects in their analysis. It is reported that the mechanical behavior of nano-beams is significantly influenced by the presence of nonlocality. Therefore, there is a strong scientific need to investigate the behavior of nano-crystalline nano-beams incorporating both surface elasticity and nonlocal effects. Up to now, several investigations have been performed to incorporate the nonlocal effects in the vibration and buckling analysis of nano-beams based on Eringen's nonlocal elasticity theory^[18-23]. Reddy^[24] presented nonlocal modeling of nano-beams for the static, buckling, and vibration analysis of small scale beams based on the Euler-Bernoulli, Timoshenko and third-order theories. Eltaher et al.^[25] explored both nonlocal and surface energy effects on vibrational responses of nano-beams. Şimşek^[26] examined nonlocal effects on the nonlinear free vibration of nano-beams under different boundary conditions. Eltaher et al.^[27] performed the vibration analysis of nonlocal nano-beams with the finite element approach. Berrabah et al.^[28] proposed different higher order nonlocal beam models for the static, vibration, and buckling analysis of nano-scale beams. Also, Tounsi et al.^[29] investigated thermal buckling responses of nano-beams by developing a nonlocal higher order beam model. Investigation of vibration behavior of preloaded nonlocal coupled nano-beams is performed by Murmu and Adhikari^[30]. Zenkour et al.^[31] studied vibration of nano-beams using a nonlocal thermo-elasticity model without energy dissipation implementing a state space method. Ebrahimi et al.^[32] examined the application of the differential transformation approach in the vibration analysis of nonlocal inhomogeneous nano-beams. Also, Ansari et al.^[33] carried out the nonlocal nonlinear free vibration analysis of fractional viscoelastic nano-beams. Ebrahimi and Barati^[34-37] proposed a nonlocal third-order shear deformable beam model for the vibration and buckling analysis of nano-beams under various physical fields. Most recently, Ebrahimi and Barati^[38-40] explored thermal and hygro-thermal effects on nonlocal vibration behavior of small scale of temperature-dependent nonhomogeneous nano-scale beams. All of these papers based on Eringen's nonlocal elasticity theory only introduced a stiffness-softening effect. In the recent era of nano-science and nano-technology, some factors are responsible for the deficiency of the nonlocal elasticity theory. Among them are the effects of micro-structure degrees of freedoms and the surface energy. In only one paper, Attia and Mahmoud^[41] investigated combined effects of nonlocal stress field, couple stress and surface energy on the mechanical behavior of nano-beams.

Also, it is evident that all the aforementioned papers related to vibration and buckling of nonlocal nano-beams have not used size-dependent material properties, and they are independent of grain and void sizes. Therefore, it is crucial to consider size-dependent material properties in the analysis of nonlocal nano-beams by using a micro-mechanical model in which the effects of nano-grains, nano-voids, and interface size are considered. Based on the above studies, the analysis of vibration behavior of nano-crystalline nano-beams, considering combined effects of nonlocal stress field, couple stress, and surface energy, is very important for the accurate analysis of nano-scale beams by taking into account both size-dependency of structure and material properties.

In this paper, a nonlocal couple stress beam model is developed for the vibration analysis of preloaded nano-crystalline nano-beams incorporating effects of surface energy. The model contains the modified couple-stress theory to explore the influence of rotational degree of freedom of particles. Moreover, a nonlocal elasticity theory is used to study the nonlocal and long-range interactions between the particles. A micro-mechanical model is used to describe a multi-phase nano-beam with grain and void sizes dependent material properties. The governing differential equations of motion are derived by using Hamilton's principle, and the differential transform method (DTM) is used to solve the equations for various boundary conditions. The results of present study are compared with those of previously published papers. The effects of couple stress, nonlocality, surface energy, nano-grain/nano-void size, nano-void percentage, interface thickness, axial preload, and different boundary conditions on natural frequencies of nano-crystalline nano-beam are investigated.

2 Theory and formulation 2.1 Effective elastic constants of nano-crystalline nano-beam

Consider a nano-crystalline silicon nano-beam which is a three-phase composite with nano-grains and nano-voids randomly distributed in the interface region, as shown in Fig. 1. In this figure, a representative volume element (RVE) is proposed in which distinct surface phases of inhomogeneities are indicated. A size-dependent micro-mechanical model^[8] is used to describe the effective material constants. In this model, effects of the size of grains and voids and their surface energies are included in the Mori-Tanaka micro-mechanical model. Elastic properties of interface or grain boundary, nano-grains and nano-voids are presented in Table 1. According to the suggested model, the elastic properties of a two-phase RVE considering nano-grains can be expressed as

(1)

(2)

in which A^(k), B^(k), A^(µ), and B^(µ) are four scalars determined in the context of the used micro-mechanical model. Based on the Mori-Tanaka model for two-phase composites with spherical inclusion, the effective bulk modulus k_eff and shear modulus µ_eff can be defined as^[2]

(3)

(4)

where

(5)

(6)

(7)

In the above equations,

(8)

where and are the surface bulk modulus and shear modulus, respectively, , and R_g is the average radius of nano-grains.

The present form of Eqs.(3) and (4) cannot be used for multi-phase composites, since it is unable to capture the effect of nano-voids or other inclusions. To overcome this problem, the decoupling method introduced by Huang et al.^[42] is implemented to decompose a multi-phase composite into a set of two-phase composites. Based on the decoupling method, a two-phase composite for every kind of inclusion is considered with a matrix material as the matrix of multi-phase composite. Hence, the effective bulk modulus K_MP and shear modulus µ_MP of a multi-phase composite can be represented as

(9)

(10)

in which k_Hn and µ_Hn are effective elastic constants of each two-phase composite. Also, N is the number of phases in the multi-phase composite. It should be mentioned that a multi-phase composite is decoupled into N-1 two-phase composites.

For an NcM, the atoms inside the crystallites vibrate at their equilibrium position r₀ with an elastic modulus of E(r₀) which is identical to that of a perfect crystal, E_g. Also, the average atomic spacing r in the grain boundary regions is larger than r₀, with an elastic modulus of E_in = E(r), which has the following relationship with an elastic modulus of the perfect crystal:

(11)

in which the interatomic spacing r is related to the mass density of interface ρ(r),

(12)

where ρ(r₀) is the mass density of perfect crystal. Also, for a nano-crystalline Si, the values of m and n are equal to 8 and 12, respectively.

By defining η =E_in /E_g, the shear and bulk moduli of interface should be related to those of the grain,

(13)

Hence, implementing the decoupling method leads to the following relations for the elastic properties of an NcM:

(14)

(15)

in which using Eqs.(3) and (4) gives

(16)

(17)

(18)

(19)

Hence, the effect of nano-voids is included in the present size-dependent micro-mechanical model. Also, the grain volume fraction f_g as a function of porosity percent f_v can be determined as follows:

(20)

in which R_g and T_in are the average radius of grain and the interface thickness, respectively. Finally, Young's modulus and Poisson's ratio of an NcM can be obtained as

(21)

(22)

Also, the effective mass density of nano-crystalline is determined by

(23)

Therefore, by using Eqs.(21)-(23), the size-dependent material properties of multi-phase composites could be obtained incorporating the surface energy effects of inclusions.

2.2 Kinematic relations

The classical beam theory introduces the following displacement fields:

(24)

(25)

where u and w are displacements of the mid-surface in axial and lateral directions, respectively. Hence, the nonzero normal strains are

(26)

where ε_xx⁰ and k⁰ denote the extensional and bending strains, respectively.

2.2.1 Modified couple stress theory

According to the modified couple stress model, the strain energy U of an elastic material occupying the region Ω is related to the strain and curvature tensors,

(27)

where σ, ε, m, and χ are the Cauchy stress tensor, the classical strain tensor, the deviatoric parts of the couple stress tensor, and the symmetric curvature tensor, respectively. The strain and curvature tensors can be defined as

(28)

(29)

where u and θ are the components of the displacement and rotation vectors, respectively,

(30)

in which e_ijk is the permutation symbol. The constitutive relationships can be expressed as

(31)

(32)

where δ_ij is the Kroenke delta, and l is the material length scale parameter which reflects the effect of couple stress. Also, the Lame's constants can be defined as

(33)

(34)

2.2.2 Surface elasticity theory

For a nano-crystalline nano-beam by ignoring any residual stresses in the bulk due to the surface stress, the relevant stress-strain relationship can be expressed as

(35)

Herein, the Gurtin-Murdoch elasticity theory is used in which the surface of nano-beam can be regarded as a two-dimensional membrane linking to the underlying bulk material. Therefore, the non-zero surface stresses can be expressed as

(36)

(37)

where τ_s is the surface residual stress, and λ_s and γ_s are the surface Lame's constants. For a nano-beam, the non-zero components of surface stress can be expressed as

(38)

(39)

The stresses at surface layers must verify the following equilibrium relations:

(40)

(41)

in which (τ_βi)⁺ and (τ_βi)^- are surface stresses, and (σ_iz)⁺ and (σ_iz)^- are bulk stresses. Inserting Eqs.(38) and (39) into Eqs.(40) and (41) gives

(42)

(43)

In this study, σ_zz is considered to be in the following form:

(44)

Inserting Eqs.(42) and (43) into Eq.(44) leads to

(45)

2.2.3 Nonlocal constitutive relations

According to Eringen's nonlocal elasticity model, the stress state at a point inside a body is regarded to be a function of strains of all points in the neighbor regions. The equivalent differential form of nonlocal constitutive equation can be expressed as

(46)

where ∇² is the Laplacian operator. Thus, the scale length e₀a considers the influence of the small scale on the response of nano-structures. Finally, the constitutive relations of nonlocal nano-beams can be expressed as^[41]

(47)

(48)

(49)

3 Governing equations

To derive the governing equation, the extended Hamilton's principle is applied as follows:

(50)

Here, U, T, and V are the strain energy, kinetic energy, and external forces work, respectively. The strain energy can be written as

(51)

Inserting Eq.(26) into Eq.(51) gives

(52)

in which N and M denote the axial force and bending moment, respectively, and Y₁ is the couple stress moment. The variation of kinetic energy can be expressed as

(53)

in which the mass moment of inertia can be expressed as

(54)

Also, the variation of work done by external loads can be written as

(55)

where N⁰ is the axial load, and τ₀ is the residual surface stress. The following equations are obtained by inserting Eqs.(52)-(55) into Eq.(50) when the coefficients of δu and δw equal zero:

(56)

(57)

Integrating Eqs.(47)-(49) over the cross-section area provides the following nonlocal relations for a nano-crystalline beam model:

(58)

(59)

(60)

(61)

(62)

in which the cross-sectional rigidities are defined as follows:

(63)

(64)

(65)

By employing Eqs.(58)-(62), the equations of motion of the nano-crystalline nano-beam in terms of the displacements are calculated as

(66)

4 DTM

In this section, by implementing the DTM, the governing equation of a nano-crystalline nano-beam is transformed into an algebraic equation, and a closed-form series solution can be obtained easily. Effective material properties of nano-crystalline nano-beam for various grain sizes are presented in Table 2. The DTM contains several transformation rules for the governing equations of motion and boundary conditions in order to transform them into a set of algebraic equations, as presented in Tables 3 and 4. In this approach, the differential inverse transformation of Y(k) and the differential transformation of kth derivative function y(x) are respectively expressed as^[32]

(67)

(68)

in which Y(k) is the transformed function of an original function y(x). Therefore, the following relation can be obtained from the above equations:

(69)

In real applications, the function y(x) in Eq.(69) can be written in a finite form as

(70)

in which N is determined by the convergence of the eigenvalues. Based on the transformation rules, the transformed form of the governing equation around x₀=0 can be expressed as

(71)

where U(k) and W(k) are the transformed functions of u and w, respectively. Also, the transformed form of boundary conditions based on the DTM can be expressed as follows^[32].

For simply supported-simply supported (S-S):

(72a)

For clamped-clamped supported (C-C):

(72b)

For clamped-simply supported (C-S):

(72c)

By using the transformed governing equation and boundary conditions, one can obtain the following eigenvalue problem:

(73)

where C corresponds to the missing boundary conditions at x=0. By setting the determinant of Eq.(33) to zero as a non-trivial solution, one can obtain the natural frequencies. Here, the Newton-Raphson method is used to solve the governing equation. Solving Eq.(73), the ith estimated eigenvalue for the nth iteration (ω = ω_i⁽ⁿ⁾) may be obtained, and the total number of iterations is related to the accuracy of calculations which can be determined by the following equation:

(74)

In this study, ε =0.000 1 is considered in the procedure of finding eigenvalues which results in 4 digit precision in the estimated eigenvalues. In this study, 30 iterations are considered for convergence of obtained results^[32]. The following relation is accomplished in order to compute the non-dimensional natural frequencies:

(75)

5 Numerical results and discussion

Based on a size-dependent micro-mechanical model, the effects of nonlocality parameter, surface elasticity, grain size, grain rotation, porosities and interface on the natural frequencies of the preloaded nano-crystalline nano-beam will be explored. The nano-crystalline silicon is a three-phase composite with two inhomogeneity types. Phase 2 is nano-crystals (grains) of an average radius R_g and Young's modulus E_g. Phase 3 is nano-void (pores) of an average radius R_v=R_g and Young's modulus E_v=0 GPa. Here, the surface of voids could be supposed to have the same surface parameters of the grains' surface. In fact, the surface parameters of solids rely on the intermolecular bonds at the surface, the nature of the surrounding medium, and the bulk material parameters. Also, the nano-beam thickness in this study is taken as h=100 nm.

The results are first validated with those of nonlocal nano-beams without the couple stress effect. Table 5 presents the comparison of the natural frequency of a nonlocal beam of S-S boundary conditions with those presented by Reddy^[24] using an Euler-Bernoulli beam model. According to this table, the results are presented for different nonlocal parameters, and good agreement is observed. Also, Table 6 presents the verification of natural frequency of couple stress based beams with those of Ref.[43] for various thickness-to-length scale parameters (h/l).

In Fig. 2, the variation of the dimensionless frequency of nano-crystalline nano-beam versus the nonlocal parameter (µ) for various gain/void sizes is depicted for the material length scale parameter l=0.1h and porosity percentage f_v=10%. It is evident from this figure that the magnitudes of natural frequencies become smaller as the nonlocal parameter increases for all values of grains and porosities radius. In fact, inclusion of nonlocal parameter has a stiffness-softening impact on the nano-beam structure. It is also observed that the vibration frequency of nano-beam is significantly influenced by the size of grains and porosities inside the material. In fact, a multi-phase nano-crystalline nano-beam has lower frequencies than a single phase Si nano-beam for every value of nonlocal parameters. A reduction in the magnitude of natural frequency is obtained by reducing the grain and porosity sizes from 100 nm to 20 nm. However, an increment in the magnitude of natural frequency is seen by reducing the average radius from 20 nm to 0.5 nm. Such different observations are related to the ratio of the bulk to the surface material of inhomogeneities. In fact, the surface energy of an inhomogeneity within the material structure becomes more important by reducing the inhomogeneity size to a few nano-meters. Hence, when the radius is 0.5 nm, the surface energy of inhomogeneities is very prominent to indicate a stiffness-hardening impact.

The influence of porosity percentage on vibration frequencies of nano-beams versus the nonlocal parameter for different boundary conditions (S-S, C-S, and C-C) is shown in Fig. 3 when l=0.1h and R_g=20 nm. In this figure, a softening mechanism is illustrated due to the porosities or nano-voids inside the material structure. Therefore, an increase in the porosity percentage leads to reduction in the nano-beam rigidity and magnitude of natural frequency. Therefore, the natural frequencies of a nano-crystalline nano-beams are overestimated by neglecting the porosity effect. These observations are valid for every kind of boundary condition. However, the maximum and minimum frequencies are respectively obtained for C-C and S-S nano-crystalline nano-beams.

Effects of the nano-beam thickness on the dimensionless frequency for different values of the average radius are presented in Fig. 4 when µ =0.1, l=0.2h and f_v=10%. The vibration frequencies may increase or decrease with the reduction in the inhomogeneities sizes. For all sizes of grains and porosities, the vibration frequency of nano-crystalline nano-beam decreases with an increase in the thickness value. It can be observed that effects of the size of inhomogeneities on vibration frequencies depend on the value of nano-beam thickness. In fact, the thicker nano-beams are more affected by the size of inhomogeneities.

Figure 5 illustrates the effect of the couple stress parameter (rigid rotation of grains) on natural frequencies of nano-crystalline nano-beam for various inhomogeneity sizes when µ=0.1 and f_v=10%. The plotted curves of this figure reflect the prominent influences of the micro-rotations when dimensionless frequencies are obtained by changing the couple stress parameter l. It is noticed that enlargement of the couple stress parameter leads to an increase in the frequency values which highlights the stiffness enhancement of nano-beam. Enlargement of dimensionless frequency with respect to the couple stress parameter depends on the value of inhomogeneity sizes. The lowest frequency is observed for a nano-crystalline nano-beam with R_g = R_v = 20 nm.

Figure 6 depicts the variation of the dimensionless frequency of the nano-crystalline nano-beam with respect to the axial preload for various inhomogeneity sizes and boundary conditions when µ =0.1 and f_v=10%. As mentioned, by reducing the grain and porosity sizes from 100 nm to 20 nm, the magnitude of the natural frequency will decrease. However, by reducing the average radius from 20 nm to 0.5 nm, the magnitude of the natural frequency will rise, which is a pre-buckling domain. Increasing the axial load leads to reduction in the stiffness of nano-crystalline nano-beam and natural frequencies. In fact, as the axial load increases, the vibration frequency decreases until it becomes close to zero with a critical buckling load. After this critical load, the increase in the axial load yields larger frequencies. It can be concluded that the largest and smallest critical loads are obtained when R_g=R_v=100 nm and R_g=R_v= 20 nm, respectively. This figure indicates the prominence of accurate modeling of nano-beams by incorporating the essential measures to describe the size-dependency of material structure.

Effects of elasticity theories on vibration frequencies of nano-crystalline nano-beams versus the axial preload when f_v=10% and R_g=20 nm are demonstrated in Fig 7. In this figure, the NL model represents the frequency result of nonlocal theory without surface and couple stress effects, and NL-CS and NL-CS-SE correspond to nonlocal couple stress theories without and with surface effect, respectively. The most important observation from this figure is that the NL-CS-SE theory gives larger critical loads than NL and NL-CS theories for a nano-crystalline nano-beam. In fact, inclusion of surface effect enhances the stiffness of nano-crystalline nano-beam, and the critical buckling load increases. Therefore, the frequency results are underestimated by neglecting effects of surface energy. However, inclusion of the couple stress effect (grain rotation) which is ignored in most of previous papers on nonlocal nano-beams leads to larger critical loads. Therefore, for accurate prediction of vibration behavior of nano-beams, a contribution of nonlocal, couple stress and surface elasticity theories is required. Also, a nano-crystalline nano-beam with the C-C boundary condition possesses larger critical loads than S-S nano-crystalline nano-beams. This is due to the fact that stronger supports at ends make the nano-beam more rigid and critical loads larger.

6 Conclusions

In this article, the vibrational behavior of a multi-phase nano-crystalline nano-beam under the axial preload is examined in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nano-crystalline nano-beams and the size effects are incorporated. This non-classical nano-beam model contains the couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is used to study the nonlocal and long-range interactions between the particles. The present model can be degenerated into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton's principle is used to derive the governing equations which are solved with the DTM. It is known that inclusion of nonlocal parameter leads to lower frequencies by reducing the bending stiffness of nano-crystalline nano-beam, regardless of the size of inhomogeneities. As the axial load increases, the vibration frequency decreases until it becomes close to zero with a critical load. After this critical load, the increase in the axial load yields larger frequencies. Also, inclusion of surface effects gives larger critical loads when the surface effects are ignored. It is observed that the couple stress effect leads to larger frequencies and critical loads. By ignoring the couple stress effect in the analysis of nano-crystalline nano-beam, the frequencies and critical loads are underestimated. The vibration frequencies may increase or decrease with the reduction in the inhomogeneity sizes. Also, an increase in the porosity percentage leads to reduction in the nano-beam rigidity and magnitude of natural frequency.