# Investigating Surface Effects on Thermomechanical Behavior of Embedded Circular Curved Nanosize Beams

Authors: Farzad Ebrahimi; Mohsen Daman
Journal: Journal of Engineering
 (2016)

---

## Abstract

To investigate the surface effects on thermomechanical vibration and buckling of embedded circular curved nanosize beams, nonlocal elasticity model is used in combination with surface properties including surface elasticity, surface tension, and surface density for modeling the nanoscale effect. The governing equations are determined via the energy method. Analytically Navier method is utilized to solve the governing equations for simply supported nanobeam at both ends. Solving these equations enables us to estimate the natural frequency and critical buckling load for circular curved nanobeam including Winkler and Pasternak elastic foundations and under the effect of a uniform temperature change. The results determined are verified by comparing the results with available ones in literature. The effects of various parameters such as nonlocal parameter, surface properties, Winkler and Pasternak elastic foundations, temperature, and opening angle of circular curved nanobeam on the natural frequency and critical buckling load are successfully studied. The results reveal that the natural frequency and critical buckling load of circular curved nanobeam are significantly influenced by these effects.


---

## Body

## 1. Introduction

Nanomaterials are attracting many researchers over the recent years due to their improvement of the quality properties. Atomistic modeling and experimental researches show that the size effect gains important when the dimensions of structures become very small. Due to this fact, the size effect plays an important role on the mechanical behavior of micro- and nanostructures [1]. Among various nanostructures, nanobeams have more important applications [2, 3]. Reddy [4] has investigated nonlocal theories for bending, buckling, and vibration of beams. Moreover, Demir and Civalek [5] have surveyed torsional and longitudinal vibration and wave response of microtubules based on the nonlocal continuum and nonlocal discrete theories. Meanwhile a nonlocal beam theory is presented by Thai [6], in this research; bending, buckling, and vibration of nanobeams have been investigated. However, the nonlinear vibration of the piezoelectric nanobeams based on the Timoshenko beam theory and nonlocal modeling has been investigated by Ke et al. [7]. In addition Murmu and Adhikari [8] have investigated the nonlocal transverse vibration of double-nanobeam-system. In this research, an analytical method has been developed for determining the natural frequencies of the nonlocal double-nanobeam-system. Also Eltaher et al. [9] have presented free vibration analysis of functionally graded (FG) size-dependent nanobeams using finite element method.Because the nanobeams has the high proportion of the surface to volume, the surface stress effects have important role in their mechanics behavior of these structures. The effect of the residual surface stress on the beam gives the distributed loading on the two surfaces. Hence Gurtin and Murdoch [10] have considered surface stress effects. In this theory the surface is considered as a part of (nonphysical) the two-dimensional body with zero thickness (mathematically) which has covered the total volume. This theory has been used in many researches about nanostructures [11–14]. Nonlinear free vibration of functionally graded nanobeams with surface effects has been investigated by Sharabiani and Yazdi [15]. Furthermore, the nonlinear free vibration of nanobeams with considering surface properties has been studied by Nazemnezhad et al. [16]. However, Hosseini-Hashemi and Nazemnezhad [17] have presented nonlinear free vibration of functionally graded nanoscale beams with surface properties. Also, Ansari et al. [18] have investigated nonlinear forced vibration characteristics of nanobeams including surface stress effect. In this study, a new formulation of the Timoshenko beam theory has been developed through the Gurtin-Murdoch elasticity theory in which the effect of surface stress has been incorporated. Moreover, the surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported nonuniform cross-section nanobeams have been investigated by Malekzadeh and Shojaee [19] simultaneously. In addition, Ebrahimi et al. [20] have presented surface and nonlocal effects on buckling and vibrational analysis of nanotubes with various boundary conditions.In the field of elastic foundation there are linear and nonlinear ones which are called Winkler and Pasternak, respectively. Zhao et al. [21] have investigated the axial buckling of a nanowire resting on Winkler-Pasternak substrate medium with the Timoshenko beam theory. In addition, simple analytical expressions have been presented by Fallah and Aghdam [22] for large amplitude free vibration and postbuckling analysis of functionally graded beams rest on nonlinear elastic foundation. Furthermore, Jang et al. [23] have presented a new method of analyzing the nonlinear deflection behavior of an infinite beam on a nonlinear elastic foundation. Also, Niknam and Aghdam [24] have obtained a closed form solution for both natural frequency and buckling load of nonlocal functionally graded beams resting on nonlinear elastic foundation. Moreover, the static instability of a nanobeam with geometrical imperfections with elastic foundation has been investigated by Mohammadi et al. [25]. In this paper, size-dependent effect is included in the nonlinear model. Nevertheless, differential transformation method has been used to predict the buckling behavior of single walled carbon nanotube on Winkler foundation under various boundary conditions by Pradhan and Reddy [26].In recent years vibrations of curved nanobeams and nanorings have been done in many empirical experiments and dynamic molecular simulations [27]. Hence some researchers are interested in studying of vibration curved nanobeams and nanorings. Malekzadeh et al. [28] have presented out-of-plane free vibration of functionally graded circular curved beams in thermal environment. However, Yan and Jiang [29] have investigated the electromechanical response of a curved piezoelectric nanobeam with the consideration of surface effects. In addition, a new numerical technique, the differential quadrature method, has been developed for dynamic analysis of the nanobeams in the polar coordinate system by Kananipour et al. [30]. Moreover, Khater et al. [31] have investigated the effect of surface energy and thermal loading on the static stability of nanowires. In this research, nanowires have been considered as curved fixed-fixed Euler-Bernoulli beams and have used Gurtin-Murdoch theory to represent surface effects. The model has taken into account both von Kármán strain and axial strain. Also, Wang and Duan [27] have surveyed the free vibration problem of nanorings/arches. In this research the problem was formulated on the framework of nonlocal elasticity theory. Furthermore, DQ thermal buckling analysis of embedded curved carbon nanotubes via nonlocal elasticity model has been presented by Setoodeh et al. [32]. Nevertheless, explicit solution has been shown for size- and geometry-dependent free vibration of curved nanobeams including surface effects by Assadi and Farshi [33].Recently, nanostructures have generated a great deal of interest from research. These distinguished properties make them apt for potential applications in nanoelectromechanical systems (NEMS) such as nanosensors [34, 35] and nanoactuators [36].Ebrahimi and Salari [37] presented a semianalytical method for vibrational and buckling analysis of FG nanobeams considering the physical neutral axis position. Recently Ebrahimi and Barati [38–41] presented static and dynamic modeling of a thermopiezoelectrically actuated nanosize beam subjected to a magnetic field. Most recently small scale effects on hygrothermomechanical vibration of temperature-dependent nonhomogeneous nanoscale beams are investigated by Ebrahimi and Barati [42]. They [43] also proposed a nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams. The influences of various thermal environments on buckling and vibration of nonlocal temperature-dependent FG beams is analyzed by Ebrahimi and Salari [44] using Navier analytical solution. In another work, Ebrahimi and Salari [45] investigated thermomechanical vibration of FG nanobeams with arbitrary boundary conditions applying differential transform method. Also, Ebrahimi et al. [46] explored the effects of linear and nonlinear temperature distributions on vibration of FG nanobeams.To the best of the author’s knowledge, there is no study regarding the surface effects on thermomechanical vibration and buckling of embedded circular curved nanosize beams. Therefore, there is a strong scientific need to understand the thermomechanical vibration and buckling behavior of circular curved nanobeams with surface effects in considering the effect of elastic foundations. Curvature rather exists in all of the real beams and nanobeams. Moreover, in previous researches in order to streamline mathematical equations, straight beam models have been used, whilst curved beam models are more practicable than straight ones. Analyzing of the vibration treatment of nanostructures is an important topic in the design process of the nanodevices. The aim of this research is to survey the effects of temperature changes and Winkler and Pasternak elastic foundation on natural frequencies and critical buckling loads of curved nanobeams with and without surface properties. So the paper has investigated the effects of surface density, surface elasticity, and surface residual stress.


## 2. Theories and Relations

### 2.1. Displacement Fields

Consider a free vibration of curved nanobeam as is shown in Figure1, with the radius curvature and thickness as R and h, respectively. In addition surface properties are assumed for surface elasticity, surface tension, and surface density.Figure 1
Geometry of an element of a circular curved nanobeam with surface layers.Based on the Timoshenko beam model, the displacement fields of an arbitrary point of the circular curved nanobeam can be defined as follows:(1)Uθ,z,t=uθ,t+zφθθ,t,Wθ,z,t=wθ,t,where u and w give the middle displacements in θ and z directions, respectively. φθ is the rotation of transverse normal.The relation between length of circular curved beamα and the angle of curvature of beam θ can be written as [32](2)α=Rθ.The normal and shear strains at any point of the beam space can be denoted as follows:(3)εθ=11+z/Rεθ0+zχθ,λθz=11+z/Rγθz0,where εθ0 and γθz0 are the normal and shear strains in the main surface. χθ is the curvature change. They are explained in terms of the middle surface displacements and rotation as [36](4)εθ0=∂uR∂θ+wR;χθ=∂φθR∂θ;λθz0=∂wR∂θ-uR+φθ.By exerting the integration of the stresses with respect to the cross-section, the normal force and moment resultants can be determined as(5)NQ=b∫-h/2h/2σθτθzdz;M=b∫-h/2h/2σθzdz,where Q deputes the transverse shear force resultant.


### 2.2. Hamilton Principle

The strain energyUs of the circular curved nanobeams for free vibration analysis can be expressed as(6)Us=12∫θNεθ0+Mχθ+Qγθz0Rdθ.Moreover, the kinetic energy (T) of the circular curved nanobeam can be defined as [36](7)T=12∫θI-0∂u∂t2+I-0∂w∂t2Rdθ,where the inertia components are given as(8)I-0=I0+I1R,I0,I1=b∫-h/2h/2ρ1,zdz,where ρ is the density of the circular curved nanobeam. The work done corresponding to temperature changes and elastic foundations can be written in the following form [32]:(9)Wext=12∫θNTR2∂w∂θ2+fRdθ.


## 2.1. Displacement Fields

Consider a free vibration of curved nanobeam as is shown in Figure1, with the radius curvature and thickness as R and h, respectively. In addition surface properties are assumed for surface elasticity, surface tension, and surface density.Figure 1
Geometry of an element of a circular curved nanobeam with surface layers.Based on the Timoshenko beam model, the displacement fields of an arbitrary point of the circular curved nanobeam can be defined as follows:(1)Uθ,z,t=uθ,t+zφθθ,t,Wθ,z,t=wθ,t,where u and w give the middle displacements in θ and z directions, respectively. φθ is the rotation of transverse normal.The relation between length of circular curved beamα and the angle of curvature of beam θ can be written as [32](2)α=Rθ.The normal and shear strains at any point of the beam space can be denoted as follows:(3)εθ=11+z/Rεθ0+zχθ,λθz=11+z/Rγθz0,where εθ0 and γθz0 are the normal and shear strains in the main surface. χθ is the curvature change. They are explained in terms of the middle surface displacements and rotation as [36](4)εθ0=∂uR∂θ+wR;χθ=∂φθR∂θ;λθz0=∂wR∂θ-uR+φθ.By exerting the integration of the stresses with respect to the cross-section, the normal force and moment resultants can be determined as(5)NQ=b∫-h/2h/2σθτθzdz;M=b∫-h/2h/2σθzdz,where Q deputes the transverse shear force resultant.


## 2.2. Hamilton Principle

The strain energyUs of the circular curved nanobeams for free vibration analysis can be expressed as(6)Us=12∫θNεθ0+Mχθ+Qγθz0Rdθ.Moreover, the kinetic energy (T) of the circular curved nanobeam can be defined as [36](7)T=12∫θI-0∂u∂t2+I-0∂w∂t2Rdθ,where the inertia components are given as(8)I-0=I0+I1R,I0,I1=b∫-h/2h/2ρ1,zdz,where ρ is the density of the circular curved nanobeam. The work done corresponding to temperature changes and elastic foundations can be written in the following form [32]:(9)Wext=12∫θNTR2∂w∂θ2+fRdθ.


## 3. Mathematical Process

The Lagrangian functional (L) of the beams for free vibration analysis is demonstrated in terms of the energy expressions as(10)L=T-Us-Wext.By substituting (6)–(9) into (10) and employing the Hamilton’s principle as (11), governing equation can be determined:(11)∫0tδT-Us-Wextdt=0.Equation (11) can be satisfied only if the coefficients of the virtual displacements are zero. Hence, motion equations of circular curved nanobeams with considering surface properties and elastic foundations are obtained as follows:(12)∂N∂θ+Q=ρAR+2bRρs∂2u∂t2,-N+∂Q∂θ-fR+NTR∂2w∂θ2=ρAR+2bRρs∂2w∂t2,∂M∂θ-QR=0,where A, ρ, and ρs are the cross-sectional area, mass density, and surface density of the circular curved nanobeam, respectively. In (12) b is the width of circular curved nanobeam.Omitting the normal stress resultantN from (12) obtains the relation between radial displacement and bending moment component. For this aim it is essential to differentiate the first of the two equations in (12) with respect to θ and then insert the obtained relation for ∂N/∂θ into the other. Also by neglecting products of small quantities ∂u/∂θ=-w [47], derivation of the obtained result with respect to θ, and streamline, this relation determines(13)1R∂4M∂θ4+∂2M∂θ2+NTR∂4w∂θ4-∂2f∂θ2R=ρAR+2Rbρs∂4w∂θ2∂t2-∂2w∂t2.Then again, the constitutive equation of elastic surface materials for the general case is defined as [10](14)ταβ±=τδαβ+μs-τuαβ±+uβα±+λs+τuγγ±δαβ+τuαβ±,where τ is surface residual stress and λs and μs are the lame constants for the surface material. Plus and minus sign refer to S+ and S- surfaces, respectively. Other essential relation for derivation of the modified differential equation is the tangential strain in terms of planar displacement component, which can be given as [47](15)εθθ=1R-w+∂u∂θ-xR∂∂θu+∂w∂θ.Considering inextensible deformation of the curved element at x=0, it can be concluded that employing the tangential strain statement by assuming the corresponding out-of-plane contractions for a curved beam at (14) and assuming the same surface properties for S+ and S-, the stress-strain relation for the surface material can be given as(16)τθθ±=τ±h2μs+λs1-ν-ντ2R2w+∂2w∂θ2.In addition, by integrating the strain components on the cross-section the resultant bending moment exerting on the cross-section of the circular curved beam can be determined as follows:(17)M=-b∫-h/2h/2Eεθθxdx+bh2τθθ+-τθθ-.By employing (15) and (16) in (17) the bending moment of the cross-section can be obtained with respect to the radial displacement as follows:(18)M=EIR2+bh22μs+λs1-ν-ντ2R2w+∂2w∂θ2.Inserting (18) into (13) yields the modified governing equation of motion for the circular curved beam as a pure function of the radial displacement components, which must be solved to obtain the natural frequencies.


### 3.1. The Nonlocal Circular Curved Nanobeam Model

Eringen’s nonlocal model states that the stress at any physical point of a body is assumed to be a function of the strain field for the whole body. The differential form of this theory is as follows [48]:(19)1-μ∇2σkl=tkl,where ∇2 stands for gradient operator and μ is the nonlocality. The size effect on the response of nanostructure is taken into account by the scale length e0a. However, the relation between nonlocality μ and scale length e0a can be written as follows:(20)μ=e0a2.In one-dimensional problems, the nonlocal constitutive relations of an elastic material can be written as follows:(21)σxx-μ∂2σxx∂x2=Eεxx,where σ and ε depute the nonlocal stress and strain at any point, respectively. Also, E represents Young’s modulus. By integrating (19) and (20) over cross-sectional area of the beam, force-strain and moment-strain of nonlocal circular beam model will be given as(22)M-μR2∂2M∂θ2=-DR2w+∂2w∂θ2,where D is defined as follows:(23)D=EI+bh22μs+λs1-ν-ντ2.By inserting (22) in (13), the governing equation is obtained as follows:(24)DR3∂6w∂θ6+2∂4w∂θ4+∂2w∂θ2+NTR∂4w∂θ4-μR3NT∂6w∂θ6-∂2f∂θ2R+μR∂4f∂θ4=-ρAR+2bRρs∂4w∂θ2∂t2-∂2w∂t2+μR2∂6w∂θ4∂t2-∂4w∂θ2∂t2,where f is related to Winkler and Pasternak elastic foundations which is defined as (25)f=-Kww+KpR2∂2w∂θ2.(i) Vibration Equation. By employing (25) into (24), the governing equation is determined as follows:(26)DR3∂6w∂θ6+2∂4w∂θ4+∂2w∂θ2+NTR∂4w∂θ4-μR3NT∂6w∂θ6+KwR∂2w∂θ2-KpR∂4w∂θ4-μKwR∂4w∂θ4+μKpR3∂6w∂θ6=-ρAR+2bRρs∂4w∂θ2∂t2-∂2w∂t2+μR2∂6w∂θ4∂t2-∂4w∂θ2∂t2.(ii) Critical Buckling Load Equation (27) - D R 3 ∂ 5 w ∂ θ 5 + ∂ 3 w ∂ θ 3 + ∂ w ∂ θ - ∂ f ∂ θ + μ R 2 ∂ 3 f ∂ θ 3 + N T R ∂ 3 w ∂ θ 3 - μ R 3 N T ∂ 5 w ∂ θ 5 + N 0 R ∂ 3 w ∂ θ 3 - μ R 3 N 0 ∂ 5 w ∂ θ 5 = 0 ,where f is defined in (24).


## 3.1. The Nonlocal Circular Curved Nanobeam Model

Eringen’s nonlocal model states that the stress at any physical point of a body is assumed to be a function of the strain field for the whole body. The differential form of this theory is as follows [48]:(19)1-μ∇2σkl=tkl,where ∇2 stands for gradient operator and μ is the nonlocality. The size effect on the response of nanostructure is taken into account by the scale length e0a. However, the relation between nonlocality μ and scale length e0a can be written as follows:(20)μ=e0a2.In one-dimensional problems, the nonlocal constitutive relations of an elastic material can be written as follows:(21)σxx-μ∂2σxx∂x2=Eεxx,where σ and ε depute the nonlocal stress and strain at any point, respectively. Also, E represents Young’s modulus. By integrating (19) and (20) over cross-sectional area of the beam, force-strain and moment-strain of nonlocal circular beam model will be given as(22)M-μR2∂2M∂θ2=-DR2w+∂2w∂θ2,where D is defined as follows:(23)D=EI+bh22μs+λs1-ν-ντ2.By inserting (22) in (13), the governing equation is obtained as follows:(24)DR3∂6w∂θ6+2∂4w∂θ4+∂2w∂θ2+NTR∂4w∂θ4-μR3NT∂6w∂θ6-∂2f∂θ2R+μR∂4f∂θ4=-ρAR+2bRρs∂4w∂θ2∂t2-∂2w∂t2+μR2∂6w∂θ4∂t2-∂4w∂θ2∂t2,where f is related to Winkler and Pasternak elastic foundations which is defined as (25)f=-Kww+KpR2∂2w∂θ2.(i) Vibration Equation. By employing (25) into (24), the governing equation is determined as follows:(26)DR3∂6w∂θ6+2∂4w∂θ4+∂2w∂θ2+NTR∂4w∂θ4-μR3NT∂6w∂θ6+KwR∂2w∂θ2-KpR∂4w∂θ4-μKwR∂4w∂θ4+μKpR3∂6w∂θ6=-ρAR+2bRρs∂4w∂θ2∂t2-∂2w∂t2+μR2∂6w∂θ4∂t2-∂4w∂θ2∂t2.(ii) Critical Buckling Load Equation (27) - D R 3 ∂ 5 w ∂ θ 5 + ∂ 3 w ∂ θ 3 + ∂ w ∂ θ - ∂ f ∂ θ + μ R 2 ∂ 3 f ∂ θ 3 + N T R ∂ 3 w ∂ θ 3 - μ R 3 N T ∂ 5 w ∂ θ 5 + N 0 R ∂ 3 w ∂ θ 3 - μ R 3 N 0 ∂ 5 w ∂ θ 5 = 0 ,where f is defined in (24).


## 4. Solution Method

The nanoring or nanoarch with total central angleα is considered. The analytical solution for free vibration of nanoarches can be written as(28)wθ,t=Wθeiωntin which ωn is the natural frequency of the nanoring. Navier solution is employed for simply-simply supported circular curved nanobeam. Therefore (27) can be rewritten as follows:(29)wθ,t=sin⁡nπαθeiωnt,where sin⁡nπ/αθ is the corresponding deformation shape of the circular curved nanobeam and nanoring, and i is the conventional imaginary number -1. Substituting (29) into (26) and (27), natural frequencies and critical buckling load of the circular curved nanobeam and nanoring with surface properties and elastic foundation can be obtained.


## 5. Numerical Results

In this section, numerical results are presented for the vibration and buckling of circular curved nanobeam embedded in an elastic medium with surface properties. In the first step, the accuracy study of the present analytically method is illustrated. The bulk elastic properties areE=177.3Gpa, ρ=7000Kg/m3, and ν=0.27, and the surface elastic properties are λs=-8N/m, μs=2.5N/m, τ=1.7N/m, and ρs=7×10-6Kg/m2 [8].


### 5.1. Validation of the Results

To validate results, elastic foundations, thermal effect, and nonlocal coefficient are eliminated and simply-simply supported boundary conditions are considered. Therefore, the plot of frequency ratio versus thickness of curved nanobeam is illustrated in Figures2 and 3. The results by comparison with [33] have been reached to reasonable results. In this survey the validity of our research can be represented as is shown in Figures 2 and 3. However, to validate results with nonlocal coefficient, the accuracy of the nonlocal natural frequencies of straight nanotube is investigated [4, 6]. The nondimensional fundamental frequencies and critical buckling loads of the nonlocal nanotube without consideration of the surface properties are compared to the results presented by Reddy [4], Thai, [6], and Ebrahimi et al. [20] and are listed in Tables 1 and 2, respectively. It is observed that the present results agree very well with those given by [4, 6, 20] and that increasing the nonlocality parameter tends to decrease the natural frequency and critical buckling load. The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanostructures and hence decrease the values of natural frequencies and critical buckling loads.Table 1
Comparison of the dimensionless fundamental natural frequencies of simply supported nanobeam.
L / h μ Thai [6] Reddy [4] Ebrahimi et al. [20] Present study 10 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569  20 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569  100 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569Table 2
Comparison of the dimensionless critical buckling load of simply supported nanobeam.
L / h μ Thai [6] Reddy [4] Ebrahimi et al. [20] Present study 10 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761  20 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761  100 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761Figure 2
Comparison between the effect of various surface properties on natural frequency with respect to thicknessh (nm).Figure 3
Comparison between the effect of surface tension and surface elasticity on natural frequency with respect to thicknessh (nm).


### 5.2. The Effect of Temperature Change

To illustrate effects of temperature on vibration and buckling of circular curved nanobeams, bucklingηThermal and vibration ξThermal nondimensional thermal ratio parameters are defined as follows:(30)ηThermal=NThermalNNon-thermal,ξThermal=ΩThermalΩNon-thermal,where NThermal and NNon-thermal represent the dimensionless critical buckling load when the influence of temperature change is assumed or eliminated, respectively. Also ΩThermal and ΩNon-thermal represent the dimensionless natural frequency when the effect of temperature change is considered or neglected, respectively.As it is clear in Figures4 and 5, for all types of opening angles considered, the vibration and critical buckling load thermal ratios ηThermal and ξThermal reduce with increasing the temperature change. In other words, with increasing of the temperature, the nondimensional natural frequency and critical buckling load reduce. Furthermore, the slope of vibration and buckling thermal ratio versus temperature change increases as the opening angles become greater.Figure 4
Buckling thermal ratio parameter of circular curved nanobeam with respect to temperature changeμ=1,Kw=1010,Kp=10-6,R=10nm.Figure 5
Vibration thermal ratio parameter of circular curved nanobeam with respect to temperature changeμ=1,Kw=1010,Kp=10-6,R=10nm.


### 5.3. Influence of the Nonlocal Parameter

Next, In order to illustrate the effect of the nonlocal parameter on the circular curved nanobeam in thermal environment, the nondimensional fundamental natural frequency and critical buckling load for various values of nonlocal parameters are illustrated in Figures6 and 7. It is obviously seen from Figures 6 and 7 that the critical buckling load and frequency parameter decrease with the increase of values of the nonlocal parameter.Figure 6
Dimensionless natural frequency with respect to thicknessh for various nonlocalities Kw=1010,Kp=10-6,R=30nm,ΔT=100.Figure 7
Dimensionless critical buckling load with respect to thicknessh for various nonlocalities Kw=1010,Kp=10-6,R=30nm,ΔT=100.The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanostructures and furthermore decrease the values of dimensionless natural frequencies and critical buckling loads.


### 5.4. The Effect of Surface Properties

To illustrate surface effects in thermal environment for vibration and buckling of circular curved nanobeams, Tables3 and 4 are presented. According to the tables, the dimensionless frequency and critical buckling loads are investigated for different values of temperature changes and nonlocality with various opening angles, with and without surface properties.Table 3
Comparison of the dimensionless fundamental natural frequencies of circular curved nanobeam with and without surface effects with various temperature changes and nonlocalityR = 20 nm, Kw=1010, and Kp=10-6.
Δ T μ θ = π / 3 θ = π / 2 θ = 2 π / 3 With surface effects Without surface effects With surface effects Without surface effects With surface effects Without surface effects 0 0 9.6217 11.4112 12.3122 14.5812 16.5025 19.5307 1 9.3899 11.1352 12.1984 14.4455 16.4623 19.4828 2 9.1919 10.8994 12.1026 14.3312 16.4287 19.4426 3 9.0207 10.6954 12.0208 14.2337 16.4002 19.4086 4 8.8710 10.5171 11.9501 14.1493 16.3756 19.3793  250 0 9.5838 11.3666 12.2531 14.5113 16.4347 19.4504 1 9.3512 11.0865 12.1388 14.3750 16.3943 19.4023 2 9.1523 10.8527 12.0425 14.2601 16.3606 19.3620 3 8.9803 10.6478 11.9602 14.1621 16.3319 19.3278 4 8.8300 10.4687 11.8892 14.0773 16.3073 19.2984  500 0 9.5459 11.3218 12.1937 14.4411 16.3665 19.3699 1 9.3123 11.0436 12.0788 14.3041 16.3261 19.3216 2 9.1126 10.8057 11.9820 14.1887 16.2922 19.2811 3 8.9398 10.5999 11.8994 14.0901 16.2634 19.2468 4 8.7888 10.4200 11.8280 14.0049 16.2386 19.2172Table 4
Comparison of the dimensionless critical buckling loads of circular curved nanobeam with and without surface effects with various temperature changes and nonlocalityR = 20 nm, Kw=1010, and Kp=10-6.
Δ T μ θ = π / 3 θ = π / 2 θ = 2 π / 3 With surface effects Without surface effects With surface effects Without surface effects With surface effects Without surface effects 0 0 14.5910 14.6596 26.8786 26.9274 55.7991 55.8259 1 13.8967 13.9591 26.3843 26.4287 55.5279 55.5523 2 13.3168 13.3740 25.9715 26.0122 55.3014 55.3237 3 12.8252 12.8781 25.6215 25.6592 55.1094 55.1300 4 12.4032 12.4523 25.3211 25.3561 54.9445 54.9637  250 0 14.4766 14.5451 26.6212 26.6700 55.3414 55.3682 1 13.7822 13.8446 26.1268 26.1713 55.0702 55.0946 2 13.2023 13.2596 25.7140 25.7548 54.8437 54.8660 3 12.7108 12.7637 25.3641 25.4017 54.6516 54.6723 4 12.2888 12.3379 25.0636 25.0986 54.4868 54.5060  500 0 14.3622 14.4307 26.3637 26.4125 54.8837 54.9105 1 13.6678 13.7302 25.8694 25.9138 54.6125 54.6368 2 13.0879 13.1452 25.4565 25.4973 54.3859 54.4083 3 12.5963 12.6492 25.1066 25.1442 54.1939 54.2146 4 12.1743 12.2235 24.8062 24.8412 54.0291 54.0483According to Tables3 and 4, it is obviously seen that the dimensionless frequency and critical buckling load decrease with considering surface properties and also decrease when opening angle increases. It is interesting to say that natural frequencies and critical buckling load also decrease with increase in temperature changes. However, to compare Table 3 with Table 4 it can be seen that surface properties are more sensitive in vibration analysis than buckling analysis. The results in Tables 3 and 4 can be used for design of circular curved nanobeams and nanorings in future.


## 5.1. Validation of the Results

To validate results, elastic foundations, thermal effect, and nonlocal coefficient are eliminated and simply-simply supported boundary conditions are considered. Therefore, the plot of frequency ratio versus thickness of curved nanobeam is illustrated in Figures2 and 3. The results by comparison with [33] have been reached to reasonable results. In this survey the validity of our research can be represented as is shown in Figures 2 and 3. However, to validate results with nonlocal coefficient, the accuracy of the nonlocal natural frequencies of straight nanotube is investigated [4, 6]. The nondimensional fundamental frequencies and critical buckling loads of the nonlocal nanotube without consideration of the surface properties are compared to the results presented by Reddy [4], Thai, [6], and Ebrahimi et al. [20] and are listed in Tables 1 and 2, respectively. It is observed that the present results agree very well with those given by [4, 6, 20] and that increasing the nonlocality parameter tends to decrease the natural frequency and critical buckling load. The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanostructures and hence decrease the values of natural frequencies and critical buckling loads.Table 1
Comparison of the dimensionless fundamental natural frequencies of simply supported nanobeam.
L / h μ Thai [6] Reddy [4] Ebrahimi et al. [20] Present study 10 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569  20 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569  100 0 9.8293 9.8696 9.8696 9.8696 1 9.3774 9.4159 9.4159 9.4159 2 8.9826 9.0195 9.0195 9.0195 3 8.6338 8.6693 8.6693 8.6693 4 8.3228 8.3569 8.3569 8.3569Table 2
Comparison of the dimensionless critical buckling load of simply supported nanobeam.
L / h μ Thai [6] Reddy [4] Ebrahimi et al. [20] Present study 10 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761  20 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761  100 0 9.8696 9.8696 9.8696 9.8696 1 8.9830 8.9830 8.9830 8.9830 2 8.2426 8.2426 8.2426 8.2426 3 7.6149 7.6149 7.6149 7.6149 4 7.0761 7.0761 7.0761 7.0761Figure 2
Comparison between the effect of various surface properties on natural frequency with respect to thicknessh (nm).Figure 3
Comparison between the effect of surface tension and surface elasticity on natural frequency with respect to thicknessh (nm).


## 5.2. The Effect of Temperature Change

To illustrate effects of temperature on vibration and buckling of circular curved nanobeams, bucklingηThermal and vibration ξThermal nondimensional thermal ratio parameters are defined as follows:(30)ηThermal=NThermalNNon-thermal,ξThermal=ΩThermalΩNon-thermal,where NThermal and NNon-thermal represent the dimensionless critical buckling load when the influence of temperature change is assumed or eliminated, respectively. Also ΩThermal and ΩNon-thermal represent the dimensionless natural frequency when the effect of temperature change is considered or neglected, respectively.As it is clear in Figures4 and 5, for all types of opening angles considered, the vibration and critical buckling load thermal ratios ηThermal and ξThermal reduce with increasing the temperature change. In other words, with increasing of the temperature, the nondimensional natural frequency and critical buckling load reduce. Furthermore, the slope of vibration and buckling thermal ratio versus temperature change increases as the opening angles become greater.Figure 4
Buckling thermal ratio parameter of circular curved nanobeam with respect to temperature changeμ=1,Kw=1010,Kp=10-6,R=10nm.Figure 5
Vibration thermal ratio parameter of circular curved nanobeam with respect to temperature changeμ=1,Kw=1010,Kp=10-6,R=10nm.


## 5.3. Influence of the Nonlocal Parameter

Next, In order to illustrate the effect of the nonlocal parameter on the circular curved nanobeam in thermal environment, the nondimensional fundamental natural frequency and critical buckling load for various values of nonlocal parameters are illustrated in Figures6 and 7. It is obviously seen from Figures 6 and 7 that the critical buckling load and frequency parameter decrease with the increase of values of the nonlocal parameter.Figure 6
Dimensionless natural frequency with respect to thicknessh for various nonlocalities Kw=1010,Kp=10-6,R=30nm,ΔT=100.Figure 7
Dimensionless critical buckling load with respect to thicknessh for various nonlocalities Kw=1010,Kp=10-6,R=30nm,ΔT=100.The reason is that the presence of the nonlocal effect tends to decrease the stiffness of the nanostructures and furthermore decrease the values of dimensionless natural frequencies and critical buckling loads.


## 5.4. The Effect of Surface Properties

To illustrate surface effects in thermal environment for vibration and buckling of circular curved nanobeams, Tables3 and 4 are presented. According to the tables, the dimensionless frequency and critical buckling loads are investigated for different values of temperature changes and nonlocality with various opening angles, with and without surface properties.Table 3
Comparison of the dimensionless fundamental natural frequencies of circular curved nanobeam with and without surface effects with various temperature changes and nonlocalityR = 20 nm, Kw=1010, and Kp=10-6.
Δ T μ θ = π / 3 θ = π / 2 θ = 2 π / 3 With surface effects Without surface effects With surface effects Without surface effects With surface effects Without surface effects 0 0 9.6217 11.4112 12.3122 14.5812 16.5025 19.5307 1 9.3899 11.1352 12.1984 14.4455 16.4623 19.4828 2 9.1919 10.8994 12.1026 14.3312 16.4287 19.4426 3 9.0207 10.6954 12.0208 14.2337 16.4002 19.4086 4 8.8710 10.5171 11.9501 14.1493 16.3756 19.3793  250 0 9.5838 11.3666 12.2531 14.5113 16.4347 19.4504 1 9.3512 11.0865 12.1388 14.3750 16.3943 19.4023 2 9.1523 10.8527 12.0425 14.2601 16.3606 19.3620 3 8.9803 10.6478 11.9602 14.1621 16.3319 19.3278 4 8.8300 10.4687 11.8892 14.0773 16.3073 19.2984  500 0 9.5459 11.3218 12.1937 14.4411 16.3665 19.3699 1 9.3123 11.0436 12.0788 14.3041 16.3261 19.3216 2 9.1126 10.8057 11.9820 14.1887 16.2922 19.2811 3 8.9398 10.5999 11.8994 14.0901 16.2634 19.2468 4 8.7888 10.4200 11.8280 14.0049 16.2386 19.2172Table 4
Comparison of the dimensionless critical buckling loads of circular curved nanobeam with and without surface effects with various temperature changes and nonlocalityR = 20 nm, Kw=1010, and Kp=10-6.
Δ T μ θ = π / 3 θ = π / 2 θ = 2 π / 3 With surface effects Without surface effects With surface effects Without surface effects With surface effects Without surface effects 0 0 14.5910 14.6596 26.8786 26.9274 55.7991 55.8259 1 13.8967 13.9591 26.3843 26.4287 55.5279 55.5523 2 13.3168 13.3740 25.9715 26.0122 55.3014 55.3237 3 12.8252 12.8781 25.6215 25.6592 55.1094 55.1300 4 12.4032 12.4523 25.3211 25.3561 54.9445 54.9637  250 0 14.4766 14.5451 26.6212 26.6700 55.3414 55.3682 1 13.7822 13.8446 26.1268 26.1713 55.0702 55.0946 2 13.2023 13.2596 25.7140 25.7548 54.8437 54.8660 3 12.7108 12.7637 25.3641 25.4017 54.6516 54.6723 4 12.2888 12.3379 25.0636 25.0986 54.4868 54.5060  500 0 14.3622 14.4307 26.3637 26.4125 54.8837 54.9105 1 13.6678 13.7302 25.8694 25.9138 54.6125 54.6368 2 13.0879 13.1452 25.4565 25.4973 54.3859 54.4083 3 12.5963 12.6492 25.1066 25.1442 54.1939 54.2146 4 12.1743 12.2235 24.8062 24.8412 54.0291 54.0483According to Tables3 and 4, it is obviously seen that the dimensionless frequency and critical buckling load decrease with considering surface properties and also decrease when opening angle increases. It is interesting to say that natural frequencies and critical buckling load also decrease with increase in temperature changes. However, to compare Table 3 with Table 4 it can be seen that surface properties are more sensitive in vibration analysis than buckling analysis. The results in Tables 3 and 4 can be used for design of circular curved nanobeams and nanorings in future.


## 6. Influence of the Winkler and Pasternak Elastic Foundations

In order to investigate the influence of the elastic foundation surrounding the circular curved nanobeam in thermal environment, the nondimensional fundamental natural frequency and critical buckling load for various values of Winkler and Pasternak parameters are illustrated in Figures8, 9, 10, and 11. It can be observed that the nondimensional natural frequency and critical buckling load increase when the Winkler elastic foundation becomes stiffer. It is also seen that the influence of thickness on the nondimensional natural frequency and critical buckling load is higher than the Winkle parameter.Figure 8
Dimensionless natural frequency with respect to thicknessh for various Winkler elastic foundations Kp=10-6,R=30nm,ΔT=100,μ=1,b=50nm,θ=π/2.Figure 9
Dimensionless critical buckling load with respect to thicknessh for various Winkler elastic foundations Kp=10-6,R=30nm,ΔT=100,μ=1,b=50nm,θ=π/2.Figure 10
Dimensionless natural frequency with respect to thicknessh for various Pasternak elastic foundations Kw=1010,R=30nm,ΔT=100,μ=1,b=50nm,θ=π/2.Figure 11
Dimensionless critical buckling load with respect to thicknessh for various Pasternak elastic foundations Kw=1010,R=30nm,ΔT=100,μ=1,b=50nm,θ=π/2.The nondimensional natural frequency and critical buckling load as a function of thickness, for three different Pasternak elastic foundation values, are illustrated in Figures9 and 10, respectively. It can be observed that the nondimensional natural frequency and critical buckling load increase when the Pasternak elastic foundation becomes greater. It is also seen that the effect of thickness on the nondimensional natural frequency and critical buckling load is higher than the Pasternak parameter. Moreover, by comparing Figures 7 and 8 with Figures 9 and 10, it should be noted that the influence of the Pasternak elastic foundation on the nondimensional natural frequency and critical buckling load is higher than the effect of the Winkler elastic foundation.


## 7. Conclusion

In this research, the surface properties on dimensionless natural frequency and critical buckling load of circular curved nanobeam embedded in an elastic medium were studied with various opening angles in the thermal environment, using the nonlocal elasticity model. Hamilton’s principle was employed to derive the governing equations. Next, the analytically exact solution was employed to solve the governing equations for simply supported curved nanobeam. The effects of the surface properties, thickness of circular curved nanobeam, Winkler and Pasternak elastic foundations, opening angle, temperature changes, and nonlocal parameter were investigated on the frequency and critical buckling load parameters of the circular curved nanobeams. It is clearly observed that, by increasing thicknessh, the surface effects tend to vanish. Furthermore it is shown that natural frequencies and critical buckling load also decrease with increase in temperature changes. In addition, results revealed that the elastic foundations and surface properties play an important role in vibration and buckling behavior of circular curved nanobeams. However it is interesting to say that the surface properties are more sensitive in vibration analysis than buckling analysis.



---

*Source: 9848343-2016-08-30.xml*