Influence of flexoelectric, small-scale, surface and residual stress on the nonlinear vibration of sigmoid, exponential and power-law FG Timoshenko nano-beams

Abstract

This research deals with the nonlinear vibration of the functionally graded nano-beams based on the nonlocal elasticity theory considering surface and flexoelectric effects. The flexoelectric functionally graded nano-beam is resting on nonlinear Pasternak foundation. Cubic nonlinearity is assumed for foundation. It is assumed that the material properties of the nano-beam change continuously along the thickness direction according to different patterns of material distribution. In order to include coupling of strain gradients and electrical polarizations in equation of motion, the nonlocal, nonclassical nano-beam model containing flexoelectric effect is employed. In addition, the effects of surface elasticity, di-electricity, and piezoelectricity as well as bulk flexoelectricity are accounted in constitutive relations. The governing equations of motion are derived using Hamilton principle based on first shear deformation beam theory and the nonlocal strain gradient elasticity theory considering residual surface stresses. The differential quadrature method is used to calculate nonlinear natural frequency of flexoelectric functionally graded nano-beam as well as nonlinear vibrational mode shape. After validation of the present numerical results with those results available in literature, full numerical results are presented to investigate the influence of important parameters such as flexoelectric coefficients of the surface and bulk, residual surface stresses, nonlocal parameter, length scale effects (strain gradient parameter), cubic nonlinear Winkler and shear coefficients, power gradient index of functionally graded material, and geometric dimensions on the nonlinear vibration behaviors of flexoelectric functionally graded nano-beam. The numerical results indicate that, considering the flexoelectricity leads to the decrease of the bending stiffness of the flexoelectric functionally graded nano-beams.

On the Dynamics of a Visco–Piezo–Flexoelectric Nanobeam

“…Regardless of time dependency and on the basis of Equation (45), the natural frequencies of the beam were extracted nondimensionally to be better plotted in an illustration as Ω = ω L 2 h ρ C 11 and X = x L . In terms of predicting the static and dynamic responses of piezoelectric-flexoelectric nanostructures, the previous published research studied many conditions, e.g., the effect of nonlocal parameter [29,34,38], different edge conditions [31], thermal environment [36], surface effect [36,37], elastic substrate [39], etc. ; therefore, the present study focuses on the influence of flexoelectricity on a nanobeam with internal viscoelasticity.…”

“…Generally, the natural frequencies of the problem were computed by the use of the Galerkin method. Arefi et al [37] investigated the effects of residual stress, surface, small scale, and flexoelectricity on a Timoshenko piezoelectric nanobeam while considering functionality in different cases, i.e., simple, sigmoid, and exponential power indices. A cubically nonlinear foundation was modeled and placed.…”

A gradient electromechanical theory for thin dielectric curved beams considering direct and converse flexoelectric effects

“…Experimental and computational analysis of flexoelectric structures incorporating converse flexoelectricity has been studied in recent literature [33][34][35][36]. Furthermore, electric field based formulations have been used to analyze flexoelectric beams [37][38][39][40] for different trasnducer applications. However, the existing electric field-strain gradient flexoelectric theories do not consider the higher-order effects due to electrical field gradient in the constitutive relations and also ignore the electrical size effects that contribute to an increase in electric permittivity of the material at lower scales [19].…”

Surface effect on vibration characteristics of bi-directional functionally graded nanobeam using Eringen’s nonlocal theory

“…Ebrahimi et al [31] explore the free vibrations of FG Euler-Bernoulli nanobeam with surface layers resting on a viscoelastic foundation in the thermal field based on NSGT and G-M theory. Arefi et al [32] studied the vibration behavior of FG Timoshenko nanobeam resting on the nonlinear foundation for various distributions of material properties using NSGT and surface elasticity theory by considering the effect of di-electricity, piezoelectricity, and flexoelectricity. Ebrahimi and Heidari [33] explore the nonlinear free vibrations of rectangular FG Reddy nanoplates embedded in an elastic medium based on ENT and G-M surface theory.…”