On the dynamic response of bi-directional functionally graded nanobeams under moving harmonic load accounting for surface effect

Abstract

This paper presents an investigation of the dynamic behavior of bi-directionally functionally graded (BDFG) micro/nanobeams excited by a moving harmonic load. The formulation is established in the context of the surface elasticity theory and the modified couple stress theory to incorporate the effects of surface energy and microstructure, respectively. Based on the generalized elasticity theory and the parabolic shear deformation beam theory, the nonclassical governing equations of the problem are obtained using Lagrange’s equation accounting for the physical neutral plane concept. The material properties of the beam smoothly change along both the axial and thickness directions according to power-law distribution, accounting for the gradation of the material length scale parameter and the surface parameters, i.e., residual surface stress, two surface elastic constants, and surface mass density. Using trigonometric Ritz method (TRM), the trial functions denoting transverse, axial deflections, and rotation of the cross sections of the beam are expressed in sinusoidal form. Then, with the aid of Lagrange’s equation, the system of equations of motion are derived. Finally, Newmark method is employed to find the dynamic responses of BDFG subjected to a moving harmonic load. To validate the present formulation and solution method, some comparisons of the obtained fundamental frequency and dynamic response with those available in the literature are performed. A parametric study is performed to extensively explore the impact of the key parameters such as the gradient indices in both directions, moving speed, and excitation frequency of the acting load on the dynamic response of BDFG nanobeams. The obtained results can serve as a guideline for assessing the multi-functional and optimal design of micro/nanobeams acted upon by a moving load.

Reformulated Strain Gradient (Rsg) Elasticity Theory for Free Vibration of Thermal Bi-Directional Fg Microbeam

“…The gradient index exhibits a slightly lesser influence on the dynamic behavior in the thickness direction compared to its effect in the length direction, as observed by Ref. [19].…”

Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution

“…Because of nonsymmetric material properties' gradation of the BDFG beam about its midplane, the geometrical neutral plane (GNP) does not coincide with the physical neutral plane (PNP) [57,58], as shown in Figure 1. The difference between the locations of GNP and PNP is evaluated by [29,59]:…”

“…Since there is no previous work considering bidirectional sigmoidal graded beams, the SBDFG is validated with power law results for the k z = 1 case and also for the homogeneous case k z = k x = 0. Table 1 compares the peak deflections w p and the corresponding absolute velocities v p of an SS transverse power functionally graded (TPFG) SUS304/Al2O3 beam with the results reported by [29,35,69,70] at Ω f = 0 (ignoring temperature effect). In Table 1, the beam is composed of a mixture of metal SUS304 stainless steel (E m = 210 GPa and mass density of ρ B m = 7800 kg/m 3 ) and ceramic (Al 2 O 3 ) (E c = 390 GPa, and ρ B c = 3960 kg/m 3 ).…”

“…Ahmadi [6] exploited the meshless method to examine the dynamic response of 2D-FG nonlocal nanobeams by using FOSDT. Attia and Shanab [29] investigated the size-dependent bending, buckling, and vibration responses of 2D-FG micro/nanoscale beams including the microstructure surface energy-based theory. Zhao et al [30] presented a comprehensive review on the mechanical analyses of FG graphene platelet-reinforced composite structure.…”

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

“…Recently, Attia et al [93] developed a closed-form solution using Laplace transform to study the dynamic response of sigmoid 2DFGM microbeams under moving harmonic load and thermal environmental conditions for a simply-supported boundary condition. Adopting the MCST and GM-SET, the dynamical performance of a moving load of perfect 2DFGM nanobeams was studied by Attia and Shanab [94].…”