Numerical Analysis of Piezoelectric Nanobeams with Flexoeletric Effect

Guo and Zhao [1] developed a theoretical background aimed at a study of the sizedependent bending elastic properties of nanobeams with surface effects.Lim and Wang [2] employed the exact variational nonlocal stress modelling with asymptotic higher order strain gradients for nanobeams.Wang et al.[3] analysed vibrations of initially stressed micro-and nanobeams. Challamel and Wang [4] solved a paradox associated with the small-scale effect for a nonlocal cantilever beam. Lim et al. [5] illustrated the stiffness strengthening effects of nonlocal stress and axial tension on free vibration of cantilever nanobeams.Aydogdu [6] proposed a general nonlocal beam theory to analyse bending, buckling and free vibration of nanobeams based on the classical Euler-Bernoulli, Timoshenko, Reddy and Levinson theories.Zhang et al.[7] analysed bending and vibration of hybrid nonlocal beams. Murmu and Adhikari [8] studied nonlocal transverse vibration of doublenanobeam systems within the framework of Eringen's nonlocal elasticity theory. The following results were reported: nonlocal natural frequencies are smaller than the corresponding local frequencies, small-scale effects are higher with increasing values of nonlocal parameter, whereas increase of stiffness of coupling springs reduces the nonlocal effects.Li et al.[9] derived a sixth-order PDE governing dynamics of simply supported nanobeams under initial axial force based on nonlocal elasticity theory. The effects of the nonlocal nanoscale and dimensionless axial force on the first twice mode frequencies were presented and discussed.Hosseini-Hashemi et al.[10] investigated the surface effects (elasticity, stress and density) of free vibrations of the Euler-Bernoulli and Timoshenko nanobeams employing the nonlocal elasticity theory. The governing PDEs were studied with regard to three different boundary conditions, i.e. simple-simple, clamped-simple and clamped-clamped. In particular, it was shown that rotary inertia and shear deformation had more effects on the surface than the nonlocal parameter.Simsek [11] proposed a novel size-dependent beam model for nonlinear free vibration of a functionally graded nanobeam by matching the nonlocal strain gradient theory and the Euler-Bernoulli beam theory with an account of the von Kármán's nonlinearity. Hamilton's principle yielded the governing PDEs and boundary conditions. A few case studies were supplemented pointing out the important features of the strain gradient length scale, the nonlocal parameters, vibration amplitude and various material compositions.Hashemi and Khaniki [12] studied free vibrations of a Timoshenko nanobeam with variable cross section in frame of the nonlocal elasticity theory. The smallscale effects were modelled by Eringen's nonlocal elasticity theory. They derived an analytical solution with regard to the Timoshenko nanobeams for three different

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“…Phunpeng et al [39] studied piezoelectric and flexoelectric effects of nanobeam by using finite element method.…”

Section: Electric Fieldmentioning

confidence: 99%