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

Ebrahimi, Farzad; Daman, Mohsen

doi:https://doi.org/10.1155/2016/9848343

Journal of Engineering

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Research Article | Open Access

Volume 2016 | Article ID 9848343 | https://doi.org/10.1155/2016/9848343

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

Farzad Ebrahimi¹and Mohsen Daman¹

Academic Editor: Parviz Malekzadeh

Received26 May 2016

Accepted28 Jul 2016

Published30 Aug 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.

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 Figure 1, with the radius curvature and thickness as and , respectively. In addition surface properties are assumed for surface elasticity, surface tension, and surface density.

Based on the Timoshenko beam model, the displacement fields of an arbitrary point of the circular curved nanobeam can be defined as follows:where and give the middle displacements in and directions, respectively. is the rotation of transverse normal.

The relation between length of circular curved beam and the angle of curvature of beam can be written as [32]The normal and shear strains at any point of the beam space can be denoted as follows:where and are the normal and shear strains in the main surface. is the curvature change. They are explained in terms of the middle surface displacements and rotation as [36]By exerting the integration of the stresses with respect to the cross-section, the normal force and moment resultants can be determined aswhere deputes the transverse shear force resultant.

2.2. Hamilton Principle

The strain energy of the circular curved nanobeams for free vibration analysis can be expressed asMoreover, the kinetic energy () of the circular curved nanobeam can be defined as [36]where the inertia components are given aswhere 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]:

3. Mathematical Process

The Lagrangian functional () of the beams for free vibration analysis is demonstrated in terms of the energy expressions asBy substituting (6)–(9) into (10) and employing the Hamilton’s principle as (11), governing equation can be determined: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:where , , and are the cross-sectional area, mass density, and surface density of the circular curved nanobeam, respectively. In (12) is the width of circular curved nanobeam.

Omitting the normal stress resultant 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 and then insert the obtained relation for into the other. Also by neglecting products of small quantities [47], derivation of the obtained result with respect to , and streamline, this relation determinesThen again, the constitutive equation of elastic surface materials for the general case is defined as [10]where is surface residual stress and and are the lame constants for the surface material. Plus and minus sign refer to and 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]Considering inextensible deformation of the curved element at , 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 and , the stress-strain relation for the surface material can be given asIn 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: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: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]:where stands for gradient operator and is the nonlocality. The size effect on the response of nanostructure is taken into account by the scale length . However, the relation between nonlocality and scale length can be written as follows:In one-dimensional problems, the nonlocal constitutive relations of an elastic material can be written as follows:where and depute the nonlocal stress and strain at any point, respectively. Also, 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 aswhere is defined as follows:By inserting (22) in (13), the governing equation is obtained as follows:where is related to Winkler and Pasternak elastic foundations which is defined as

(i) Vibration Equation. By employing (25) into (24), the governing equation is determined as follows:

(ii) Critical Buckling Load Equationwhere is defined in (24).

4. Solution Method

The nanoring or nanoarch with total central angle α is considered. The analytical solution for free vibration of nanoarches can be written asin which 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:where is the corresponding deformation shape of the circular curved nanobeam and nanoring, and is the conventional imaginary number . 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 are , , and , and the surface elastic properties are , , , and [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 Figures 2 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.

5.2. The Effect of Temperature Change

To illustrate effects of temperature on vibration and buckling of circular curved nanobeams, buckling and vibration nondimensional thermal ratio parameters are defined as follows:where and represent the dimensionless critical buckling load when the influence of temperature change is assumed or eliminated, respectively. Also and represent the dimensionless natural frequency when the effect of temperature change is considered or neglected, respectively.

As it is clear in Figures 4 and 5, for all types of opening angles considered, the vibration and critical buckling load thermal ratios and 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.

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 Figures 6 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.

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, Tables 3 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.

According to Tables 3 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 Figures 8, 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.

The nondimensional natural frequency and critical buckling load as a function of thickness, for three different Pasternak elastic foundation values, are illustrated in Figures 9 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 thickness h, 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.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

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Copyright © 2016 Farzad Ebrahimi and Mohsen Daman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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