Effect of surface layer thickness on buckling and vibration of nonlocal nanowires
Introduction
Nanowires (NWs) show great promise for a wide diversity of device applications, such as actuators, resonators, sensors, probes and transistors in nano-electromechanical systems (NEMS) [1], [2]. It is of great importance to exactly characterize the mechanical properties of NWs for these applications.
In nanoscale, owing to the drastic increasing ratio of surface area to volume, the physical and mechanical properties of NWs exhibit size-dependent [3], [4]. The size effect of Youngʼs modulus of nanoscaled structures is studied by the core–shell model. Chen et al. [5] experimentally reported the size effect of Youngʼs modulus in 〈0001〉 oriented ZnO NWs as a function of the diameters from 17 nm to 550 nm for the first time. Agrawal et al. [6] presented an ingeniously experimental and computational approach to quantify the size-dependent elastic properties of ZnO NWs and interpreted their results with core–shell structures. Yu et al. [7] attributed the size dependence of the apparent Youngʼs modulus of the silver NWs to the surface effect, which included the effect of the surface stress, the oxidation layer and the surface roughness. Furthermore, Zheng et al. [8] systematically investigated the elastic modulus of the NWs with an arbitrary regular polygonal cross-sectional shape and explained various phenomena associated with the size effect of the Youngʼs modulus of NWs. These reports [5], [6], [7], [8], [9] indicated that the Youngʼs modulus of nanostructures increased dramatically with decreasing diameters.
During the past several years, the core–shell model [5] was used by many researchers to describe the size-dependent properties of NWs. Wang et al. [9], [10], [11], [12], [13], [14], [15] analyzed the surface effect on the axial buckling and transverse vibration of NWs using the Euler beam (EB) and Timoshenko beam (TB) model and their studies had implications on measuring the mechanical properties of NWs. He and Lilley [16], [17], [18] also extensively estimated the surface effect on the static bending behavior and resonance frequencies of NWs incorporated into the EB theory via the Young–Laplace equation for three different boundary conditions. Furthermore, Chiu and Chen [19], [20] presented a high-order surface stresses model, considering the surface moment along the surface layer, to study of the effective size-dependent Youngʼs modulus, maximum deflection, buckling and natural frequency of NWs. Malekzadeh et al. [21], Sharabiani and Yazdi [22], Gheshlaghi and Hasheminejad [23], Fu et al. [24], [25] investigated the surface effect on the nonlinear free vibration behavior of nanobeams where the effect of the geometry nonlinearity is considered. Koochi et al. [26], [27], [28], [29], [30] studied the influence of surface effect and size effect on the pull-in performance of nano-actuators in the presence of dispersion forces and their research is very useful for NEMS engineering design. Lei et al. [31] discovered that the influences of surface effect on the vibrational frequency of nanotubes were more pronounced under the condition of higher vibration modes. The results demonstrate that surface elasticity plays an important role in the behavior of NWs.
Gurtin et al.ʼs theory [32] has been widely employed to analyze the performance of nanoscaled structures and the thickness of surface layer is assumed to be zero in their theory. However, the zero thickness assumption of surface layer may lead to an unrealistic physical image and is invalid when the cross-sectional size of the nanomaterial is considerable to the thickness of surface layer. Therefore, it is necessary to take surface layer thickness as a factor for the core–shell model when the effect of surface elasticity is introduced. Recently, based on the assumption of exponential decay function of surface elasticity, Yao and Yun [33] analyzed the effect of surface layer thickness on the EB buckling behavior of ZnO NWs.
In this Letter, the buckling and vibration behavior of nonlocal NWs considering the effect of surface elasticity is investigated. The modified core–shell model is developed and validated by the experimental data [5]. The buckling and vibration behavior of NWs is studied by using this modified model and compared with the results reported in [9], [10], [11], [12], [13], [14].
Section snippets
Modified core–shell model
The size-dependent phenomena of nanomaterials have been reported by many experiments [5], [6], [7], [8]. In order to explain these phenomena, some theories have been proposed [12] such as the strain gradient theory [34], [35], [36], the modified couple stress theory [37], the microcontinuum theory [38], the nonlocal elasticity theory [39], [40], [41], [42].
Recently, the theory of surface elasticity as another efficient continuum beam theory was presented by Gurtin et al. [43] to explain the
Results and discussion
To verify the modified core–shell model, comparisons between the effective Youngʼs modulus of this model, the reported experimental data [5] and the theoretical results by He and Lilley [16] are presented in Fig. 2. The material parameters of NWs [5], [54] presented here are , , Poissonʼs ratio , surface stress of , . Fig. 2 shows that the result of modified model fits well with the experimental result of Chen et al. [5] when . Therefore, the
Conclusions
The buckling and vibration behavior of nonlocal NWs incorporated the effect of surface elasticity have been investigated. The modified core–shell model, which accounts for surface layer thickness, is developed to depict the size effect of Youngʼs modulus of NWs and verified with the reported experimental data [5]. The modified model is more suitable for depicting the size effect of Youngʼs modulus when surface layer thickness becomes larger. Significant size effect on the critical buckling
Acknowledgements
This work was supported by the National Science Fund for Excellent Young Scholars (Grant No. 11322215) and National Natural Science Foundation of China (Grant No. 11342001 and 11072147), and sponsored by Shanghai Rising-Star Program under Grant No. 11QA1403400.
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