Timoshenko beam model for buckling of piezoelectric nanowires with surface effects
© Samaei et al; licensee Springer. 2012
Received: 13 October 2011
Accepted: 27 March 2012
Published: 27 March 2012
This paper investigates the buckling behavior of piezoelectric nanowires under distributed transverse loading, within the framework of the Timoshenko beam theory, and in the presence of surface effects. Analytical relations are given for the critical force of axial buckling of nanowires by accounting for the effects of surface elasticity, residual surface tension, and transverse shear deformation. Through an example, it is shown that the critical electric potential of buckling depends on both the surface stresses and piezoelectricity. This study may be helpful in the characterization of the mechanical properties of nanowires and in the calibration of the nanowire-based force sensors.
Keywordssurface elasticity buckling piezoelectric nanowire
Nanowires have attracted considerable attention in the literature for future applications as sensors, actuators, transistors, and resonators in nanoelectromechanical systems and in biotechnology . Because of these varied applications, it is very important to accurately characterize the mechanical properties of nanowires and their response to external loading. In atomistic scales, owing to the increasing ratio of surface area to volume, the stress and strain effects on surface physics become very important . In this regard, theoretical and experimental investigations have provided a better understanding of the effects of stress on surface physics [3, 4]. For example, by conducting bending tests using atomic force microscopy, Cuenot et al.  have demonstrated that the stiffness of nanowires is size-dependent, and this phenomenon has been theoretically explained by considering the surface effects [5–8]. He and Lilley  investigated the influences of surface tension on the static bending of nanowires. Wang and Feng  studied the surface effects on the buckling and vibration behaviors of nanowires, based on the Laplace-Young equation. The theoretical investigations related to the surface effects and mechanical behavior show a good agreement with the experiments and atomistic simulations [3, 6, 9].
Recently, piezoelectric nanostructures, such as nanowires, have been drawing a lot of attention due to their potential applications as nanoresonators , diodes , and nanogenerators . Piezoelectric nanomaterials exhibit size-dependent properties at nanoscale, and also, it has been demonstrated that they have larger piezoelectric constants than their bulk counterparts [13, 14]. Experimental measurements and atomistic simulations demonstrate that the elastic and fracture properties of ZnO piezoelectric nanowires vary with their cross-sectional dimensions [5, 15, 16]. Zhao et al.  found out that the effective piezoelectric coefficient of the ZnO nanowire is frequency-dependent and that it is much larger than that of the bulk material. Using the perturbation theory  and finite element method , the electrostatic potential in a bending piezoelectric nanowire was calculated. For the first time, Wang and Feng  used the Euler-Bernoulli beam model to investigate the buckling and vibration behaviors of piezoelectric nanowires by taking into account the effects of surface stresses and piezoelectricity. Also, surface effects and surface piezoelectricity are considered to study the electromechanical coupling behavior of piezoelectric nanowires with the Euler beam theory by Yan and Jiang .
The objective of the present paper is to investigate the combined surface and piezoelectric effects on the buckling of piezoelectric nanowires using the modified Timoshenko beam model. In this study, the two modified Euler beam and Timoshenko beam models have been compared, but no quantitative experimental measurement has been reported on the buckling condition of piezoelectric nanowires. A numerical example is presented in the article to demonstrate both the surface and piezoelectric effects, and then, some discussions are provided based on the obtained results.
Formulation of the problem
In the current study, a crystalline ZnO nanowire with the C6v symmetry about the poling direction along the z-axis  is considered, which has a surface layer with surface elasticity modulus (E5) [20–22], which can be determined by atomistic simulations or experiments [3, 23], surface layer thickness (t) , and constant residual surface tension (τ0) [18, 24]. The effect of the residual surface stress acting as a transverse load on the nanowire is calculated by the Laplace-Young equations .
where ξ is the slenderness ratio (slenderness ratio corresponds to length-to-the least radius of gyration of the cross section ratio) of the nanowire, and for a hinged-hinged beam, ξ = 1 . Equation 8 presents a relation between the residual surface stress of a piezoelectric nanowire and the critical electric potential in the buckling analysis. Therefore, it can be inferred that the elasticity modulus and the residual surface stress could be obtained by measuring the critical electric potentials of two nanowires with different sizes .
Example and discussion
It is found from Equation 8 that the shear deformation lowers the critical compression force of buckling in comparison with the classical Euler solution.
In this study, the thickness of the surface layer has been disregarded because it is very small relative to the sizes of the nanowire's geometrical parameters. Also, in the present investigation, the assumption is that the deformation of the nanowire is small and that the resultant principle (the principle of superposition) can be used to sum up the tensions arising from the surface and piezoelectric effects; therefore, the effect of surface tension has been modeled as a curvature-dependent transverse loading, and the piezoelectric effect has been modeled as an induced axial force in the nanowire.
In the present study, by applying the modified Timoshenko beam theory and considering the effects of surface, piezoelectricity, and shear deformation, the buckling behavior of piezoelectric nanowires was investigated. The obtained information can be used in the design and characterization of piezoelectric nanowire-based devices and instruments. The critical electric potential for the buckling of piezoelectric nanowires with the hinged-hinged boundary condition was derived analytically. The results show that, in addition to the surface effects, the shear deformation and piezoelectricity can effectively influence the buckling behavior of nanowires as well. Also, it was observed that, contrary to the surface effects, the shear deformation tends to reduce the critical electric potential and that it has a greater influence on stubby nanowires with smaller aspect ratios. Therefore, for smaller nanowire heights in the nanometer range, the effects of surface elasticity, piezoelectricity, and shear deformation should be taken into consideration so that more accurate results are obtained.
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