Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory
Introduction
Since the publication of ZnO piezoelectric nanostructures by Pan and his colleagues [1] in Science magazine, several piezoelectric materials (e. g. ZnO, ZnS, PZT, GaN, BaTiO3) and their nanostructures (e. g. nanowires, nanorings, nanohelices, nanosprings, nanobelts) have attracted the attention of numerous research workers [2], [3], [4], [5], [6]. Due to the novel mechanical, electrical and other physical and chemical properties of piezoelectric materials over their bulk counterparts [2] as well as their inherent electro-thermo-mechanical coupling effect that can generate mechanical deformations and electrical fields under the action of electrical and thermal loads (i.e. the direct effect) and mechanical loads (i.e. the converse effect), respectively [7], [8], they have found a wide range of applications in many nanodevices such as nanoresonators [9], field effect transistors [10] light-emitting diodes [11] chemical sensors [12] and nanogenerators [13]. These materials are capable of simultaneously sensing/actuating mechanical, electrical and even thermal effects. They also have the capability of simultaneously generating control forces to remove the unfavorable effects or to improve the favorable ones resulting in the enhancement of lifetime and working performance of devices [14], [15]. In such nanodevices, the role of size effect becomes prominent when the dimensions of nanostructures vary from several hundred nanometers to just a few nanometers [16], [17]. In many experimental studies and atomistic simulations, the deformation behavior of piezoelectric nanostructures is shown to be size-dependent. For instance, Chen et al. [18] demonstrated that, for nanowires with diameters lower than about 120 nm, Young modulus increases dramatically as the diameter decreases. Stan et al. [19] found that decreasing the diameter of ZnO nanowires causes the lateral shear modulus and radial indentation modulus to increase significantly. Zhao et al. [20] using the piezoresponse force microscopy reported that the effective piezoelectric coefficient of ZnO nanowires is much higher than that of bulk ZnO. Agrawal et al. [21] observed that Young modulus of ZnO nanowires approximately increases from 140 to 160 GPa as the diameter decreases from 80 to 20 nm.
Since the classical continuum theory fails to predict the size-dependent response of nanostructures, different types of higher-order theories such as couple stress theory (CST) [22], [23], [24], strain gradient theory (SGT) [25], micropolar theory [26], nonlocal elasticity theory [27], [28], [29] and surface elasticity theory [30] have been developed to characterize the size effect in nanostructures through introducing an intrinsic length scale parameter. Amongst these higher-order theories, the nonlocal elasticity theory proposed by Eringen [27], [31] has been broadly adopted in the analysis of size effect of nanostructures. In this theory, the internal length scale is introduced into the constitutive equations as a material parameter and the stress at a reference point is assumed to be a function of strain field at every point in the body. In recent years, many studies employed nonlocal elasticity theory to model the size-dependent mechanical properties of different nanostructures such as single-walled and double-walled carbon nanotubes, single-layered and multi-layered graphene sheets, mass sensors, nanowires and phononic crystals [32], [33], [34], [35], [36], [37], [38], [39], [40]. Moreover, by developing the nonlocal nanobeam model [41], [42], nonlocal nanoplate model [43], [44], [45] and nanoshell model [46], the bending, buckling, linear and nonlinear vibration, postbuckling and wave-propagation behaviors were fully discussed.
More recently, the nonlocal elasticity theory has been adopted to study the size-dependent mechanical behaviors of piezoelectric nanostructures. Based on the nonlocal theory and Timoshenko beam theory, Ke and Wang [47] and Ke et al. [48] investigated the linear and nonlinear vibration of piezoelectric nanobeams and explored the influences of nonlocal parameter and electro-thermo-mechanical loads on the vibration behavior. Arani et al. [49] on the basis of nonlocal Timoshenko beam theory analyzed the nonlinear vibration characteristics of single-walled Boron Nitride nanotubes. Li et al. [50], using nonlocal Timoshenko beam theory and von Kármán geometric nonlinearity, studied static buckling and postbuckling of size-dependent piezoelectric Timoshenko nanobeams subjected to thermo-electro-mechanical loadings. In this study, the effects of nonlocal parameter, temperature rise and external electric voltage on the size-dependent buckling and postbuckling responses of piezoelectric nanobeams were investigated. Asemi et al. [51] developed a nonlinear continuum model to analyze the large amplitude vibration of nanoelectromechanical resonators using piezoelectric nanofilms under external electric voltage and indicated that their nonlocal model with reasonable small scale parameters gives more precise estimation of natural frequencies than the classical theory. Some researchers also studied the surface effect on the piezoelectric nanostructures using surface elasticity theory. In this respect, Huang and Yu [52] discussed the effect of surface piezoelectricity on the electromechanical behavior of a piezoelectric ring and indicated that surface piezoelectricity plays a significant role in the electromechanical behavior of piezoelectric nanostructures. Zhang and Wang [53] developed a sandwich plate model to study the vibration behavior of piezoelectric nanofilms with surface effect. Yan and Jiang [54] investigated the vibration and buckling characteristics of piezoelectric nanobeams with the consideration of surface effect. In another study, these authors analyzed the surface effect on the electromechanical coupling and bending behaviors of piezoelectric nanowires [55]. Li et al. [56] studied the surface effects including surface elasticity, surface piezoelectricity and residual surface stress on the postbuckling response of piezoelectric as a result of an electric field.
This paper studies the free vibration behavior of piezoelectric nanobeams in the pre- and postbuckling domains using the nonlocal elasticity theory, the Timoshenko beam theory and von Kármán geometric nonlinearity. The current work can be regarded as the extension of the work by Li et al. [50] to a dynamical case. The piezoelectric nanobeam is assumed to be subjected to the combined thermo-electro-mechanical loads. The nonlinear governing differential equations and the associated boundary conditions are established through Hamilton principle. After discretizing the governing differential equations using generalized differential quadrature (GDQ) technique, Newton–Raphson scheme is utilized to solve the eigenvalue problem. Thereafter, based on a numerical solution strategy, a dynamic analysis is conducted to predict the natural frequencies of piezoelectric nanobeams corresponding to both prebuckling and postbuckling domains. The effects of nonlocal parameter, temperature rise and external electric voltage on the thermo-electro-mechanical vibration characteristics of piezoelectric nanobeams with different types of boundary conditions are discussed in detail.
Section snippets
Nonlocal theory for piezoelectric materials
According to the Eringen's nonlocal elasticity theory [27], [31], the stress at a point x in a body depends not only on the strain at that point but also on those at all points of the body. In the nonlocal elasticity theory, some phenomena corresponding to atomic and molecular scales such as high frequency vibration and wave dispersion can be explained satisfactorily. Lately, Zhou and his colleagues [57], [58], [59] have extended the Eringen's nonlocal elasticity theory to the piezoelectric
Formulation of motion and corresponding boundary conditions
Based on the nonlocal theory of piezoelectric materials presented in the previous section, here, the vibration of the postbuckled piezoelectric nanobeam under thermo-electro-mechanical loading is analyzed. Consider a piezoelectric nanobeam with length L and thickness h subjected to an applied voltage ϕ(x,z,t) and a uniform temperature change ΔT as depicted in Fig. 1. The poling direction of piezoelectric medium is taken to be parallel to the positive z- axis in which (x,z) is the coordinate
Postbuckling analysis
In this section, first of all, the governing equations and associated boundary conditions are discretized using GDQ scheme [63]. The one-dimensional functions (ζ), w(ζ) and ψ(ζ) are considered in the domain 0 < ζ < 1 in which the grid points in ζi direction are generated using the shifted Chebyshev–Gauss–Lobatto grid points as
The column vectors U,W and Ψ with N entries equal to the number of grid points are introduced as follows
Numerical results and discussion
In this section, based on the nonlocal Timoshenko beam theory, the natural frequencies of piezoelectric nanobeams under thermo-electro-mechanical loadings corresponding to both prebuckling and postbuckling domains are predicted. It should be noted that the critical axial load is a threshold through which the system passes from the prebuckling domain to the postbuckling one. The nanobeam is composed of PZT-4 with material properties outlined in Table 1 [66], [67]. Selected numerical results are
Conclusion
The nonlocal elasticity theory was employed to study the size-dependent free vibration behavior of postbuckled piezoelectric Timoshenko nanobeams subjected to thermo-electro-mechanical loads. Based on the Hamilton principle, the governing differential equations of motion incorporating von Kármán geometric nonlinearity and the associated boundary conditions were derived and then discretized by means of GDQ technique. After solving the eigenvalue problem using Newton–Raphson method, the natural
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