M.A. Eltaher

Free vibration analysis of functionally graded size-dependent nanobeams

Abstract This paper presents free vibration analysis of functionally graded (FG) size-dependent nanobeams using finite element method. The size-dependent FG nanobeam is investigated on the basis of the nonlocal continuum model. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. The material properties of FG nanobeams are assumed to vary through the thickness according to a power law. The nanobeam is modeled according to Euler–Bernoulli beam theory and its equations of motion are derived using Hamilton’s principle. The finite element method is used to discretize the model and obtain a numerical approximation of the equation of motion. The model is validated by comparing the obtained results with benchmark results. Numerical results are presented to show the significance of the material distribution profile, nonlocal effect, and boundary conditions on the dynamic characteristics of nanobeams.

Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams

This paper is proposed to study the coupled effects of surface properties and nonlocal elasticity on vibration characteristics of nanobeams by using a finite element method. Nonlocal differential elasticity of Eringen is exploited to reveal the long-range interactions of a nanoscale beam. To incorporate surface effects, Gurtin-Murdoch model is proposed to satisfy the surface balance equations of the continuum surface elasticity. Euler-Bernoulli hypothesis is used to model the bulk deformation kinematics. The surface layer and bulk of the beam are assumed elastically isotropic. Galerkin finite element technique is employed for the discretization of the nonlocal mathematical model with surface properties. An efficiently finite element model is developed to descretize the beam domain and solves the equation of motion numerically. The output results are compared favorably with those published works. The effects of nonlocal parameter and surface elastic constants on the vibration characteristics are presented. Also, the effectiveness of finite element method to handle a complex geometry is illustrated. The present model can be used for free vibration analysis of single-walled carbon nanotubes with essential, natural and nonlinear boundary conditions.

Static and buckling analysis of functionally graded Timoshenko nanobeams

Investigation of static and buckling behaviors of nonlocal functionally graded (FG) Timoshenko nanobeam is the main objective of this paper. Eringen nonlocal differential constitutive equation is exploited to describe the size dependency of nanostructure beam. The material properties of FG nanobeam are assumed to vary through the thickness direction by power-law. The kinematic assumption of beam is assumed by Timoshenko theory, which accommodates for thin and moderated thick beam, and hence, considers the shear effect. The equilibrium equations are derived using the principle of the minimum total potential energy. A finite element method is proposed to obtain a numerical solution of equilibrium equations. Model validation is presented and compared with peer works. The results show and address the significance of the material distribution profile, size-dependence, and boundary conditions on the bending and buckling behavior of nanobeams. Also, the significant effects of neutral axis position on static and buckling behaviors are figured out.

Mechanical analysis of higher order gradient nanobeams

This paper figures out effective of a new proposed beam element to handle mechanical behaviors of size-dependent nanobeams on the basis of the higher order gradient model. A higher order strain gradient model is derived from the nonlocal Eringen differential equation. The kinematic assumption of nanobeam is proposed by Euler-Bernoulli theory. Galerkin finite element technique is employed for discretizing a beam domain and solves the equation of motion numerically. The output results are compared and verified with those previously published works. Static, buckling and dynamic behaviors of nonlocal and higher order gradient beams are investigated and discussed. Also, the effectiveness of a higher order gradient model to incorporate the size-dependent of nanobeams with different boundary conditions is presented. The applicability of proposed model to analyze a carbon nanotubes is illustrated.

Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity

ABSTRACT This manuscript illustrates coupled effects of nonlocal elasticity and surface properties on static and vibration characteristics of piezoelectric nanobeams using thin beam theory. The mechanical and piezoelectric surface nanoscale properties are governed by Gurtin–Murdoch model. The length scale effect is imposed to the problem by including nonlocal elasticity theory to describe the long–range atoms interactions. The governing equations are derived by Hamilton’s principle and solved numerically using the finite-element method. The proposed model is verified and validated with previous published works. Numerical results illustrate the effects of nonlocal parameter, surface elasticity, and boundary conditions on the bending and dynamic characteristics of the nanobeam. It is found that, the nonlocal effect and the surface piezoelectricity effect play a significant role on the static deflection and the natural frequencies. The obtained results are in good agreement with previous published works. This study should have useful insights on the design, fabrication and applications of piezoelectric-beam-based nanodevices.